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12232e6b52064dcc | Article 25 of 215 | Back to Science News Previous | Next
New Evidence Could Overthrow the Standard View of Quantum Mechanics
New Evidence Could Overthrow the Standard View of Quantum Mechanics
New Evidence Could Overthrow the Standard View of Quantum Mechanics
May 21, 2016
Of the many counterintuitive features of quantum mechanics, perhaps the most challenging to our notions of common sense is that particles do not have locations until they are observed. This is exactly what the standard view of quantum mechanics, often called the Copenhagen interpretation, asks us to believe. Instead of the clear-cut positions and movements of Newtonian physics, we have a cloud of probabilities described by a mathematical structure known as a wave function. The wave function, meanwhile, evolves over time, its evolution governed by precise rules codified in something called the Schrödinger equation. The mathematics are clear enough; the actual whereabouts of particles, less so. Until a particle is observed, an act that causes the wave function to "collapse," we can say nothing about its location. Albert Einstein, among others, objected to this idea. As his biographer Abraham Pais wrote: "We often discussed his notions on objective reality. I recall that during one walk Einstein suddenly stopped, turned to me and asked whether I really believed that the moon exists only when I look at it."
But there's another view-one that's been around for almost a century-in which particles really do have precise positions at all times. This alternative view, known as pilot-wave theory or Bohmian mechanics, never became as popular as the Copenhagen view, in part because Bohmian mechanics implies that the world must be strange in other ways. In particular, a 1992 study claimed to crystalize certain bizarre consequences of Bohmian mechanics and in doing so deal it a fatal conceptual blow. The authors of that paper concluded that a particle following the laws of Bohmian mechanics would end up taking a trajectory that was so unphysical-even by the warped standards of quantum theory-that they described it as "surreal."
Nearly a quarter-century later, a group of scientists has carried out an experiment in a Toronto laboratory that aims to test this idea. And if their results, first reported earlier this year, hold up to scrutiny, the Bohmian view of quantum mechanics-less fuzzy but in some ways more strange than the traditional view-may be poised for a comeback.
Saving Particle Positions
Bohmian mechanics was worked out by Louis de Broglie in 1927 and again, independently, by David Bohm in 1952, who developed it further until his death in 1992. (It's also sometimes called the de Broglie-Bohm theory.) As with the Copenhagen view, there's a wave function governed by the Schrödinger equation. In addition, every particle has an actual, definite location, even when it's not being observed. Changes in the positions of the particles are given by another equation, known as the "pilot wave" equation (or "guiding equation"). The theory is fully deterministic; if you know the initial state of a system, and you've got the wave function, you can calculate where each particle will end up.
That may sound like a throwback to classical mechanics, but there's a crucial difference. Classical mechanics is purely "local"-stuff can affect other stuff only if it is adjacent to it (or via the influence of some kind of field, like an electric field, which can send impulses no faster than the speed of light). Quantum mechanics, in contrast, is inherently nonlocal. The best-known example of a nonlocal effect-one that Einstein himself considered, back in the 1930s-is when a pair of particles are connected in such a way that a measurement of one particle appears to affect the state of another, distant particle. The idea was ridiculed by Einstein as "spooky action at a distance." But hundreds of experiments, beginning in the 1980s, have confirmed that this spooky action is a very real characteristic of our universe.
In the Bohmian view, nonlocality is even more conspicuous. The trajectory of any one particle depends on what all the other particles described by the same wave function are doing. And, critically, the wave function has no geographic limits; it might, in principle, span the entire universe. Which means that the universe is weirdly interdependent, even across vast stretches of space. The wave function "combines-or binds-distant particles into a single irreducible reality," as Sheldon Goldstein, a mathematician and physicist at Rutgers University, has written.
The differences between Bohm and Copenhagen become clear when we look at the classic "double slit" experiment, in which particles (let's say electrons) pass through a pair of narrow slits, eventually reaching a screen where each particle can be recorded. When the experiment is carried out, the electrons behave like waves, creating on the screen a particular pattern called an "interference pattern." Remarkably, this pattern gradually emerges even if the electrons are sent one at a time, suggesting that each electron passes through both slits simultaneously.
Those who embrace the Copenhagen view have come to live with this state of affairs-after all, it's meaningless to speak of a particle's position until we measure it. Some physicists are drawn instead to the Many Worlds interpretation of quantum mechanics, in which observers in some universes see the electron go through the left slit, while those in other universes see it go through the right slit-which is fine, if you're comfortable with an infinite array of unseen universes.
By comparison, the Bohmian view sounds rather tame: The electrons act like actual particles, their velocities at any moment fully determined by the pilot wave, which in turn depends on the wave function. In this view, each electron is like a surfer: It occupies a particular place at every specific moment in time, yet its motion is dictated by the motion of a spread-out wave. Although each electron takes a fully determined path through just one slit, the pilot wave passes through both slits. The end result exactly matches the pattern one sees in standard quantum mechanics.
For some theorists, the Bohmian interpretation holds an irresistible appeal. "All you have to do to make sense of quantum mechanics is to say to yourself: When we talk about particles, we really mean particles. Then all the problems go away," said Goldstein. "Things have positions. They are somewhere. If you take that idea seriously, you're led almost immediately to Bohm. It's a far simpler version of quantum mechanics than what you find in the textbooks." Howard Wiseman, a physicist at Griffith University in Brisbane, Australia, said that the Bohmian view "gives you a pretty straightforward account of how the world is…. You don't have to tie yourself into any sort of philosophical knots to say how things really are."
But not everyone feels that way, and over the years the Bohm view has struggled to gain acceptance, trailing behind Copenhagen and, these days, behind Many Worlds as well. A significant blow came with the paper known as "ESSW," an acronym built from the names of its four authors. The ESSW paper claimed that particles can't follow simple Bohmian trajectories as they traverse the double-slit experiment. Suppose that someone placed a detector next to each slit, argued ESSW, recording which particle passed through which slit. ESSW showed that a photon could pass through the left slit and yet, in the Bohmian view, still end up being recorded as having passed through the right slit. This seemed impossible; the photons were deemed to follow "surreal" trajectories, as the ESSW paper put it.
The ESSW argument "was a striking philosophical objection" to the Bohmian view, said Aephraim Steinberg, a physicist at the University of Toronto. "It damaged my love for Bohmian mechanics."
But Steinberg has found a way to rekindle that love. In a paper published in Science Advances, Steinberg and his colleagues-the team includes Wiseman, in Australia, as well as five other Canadian researchers-describe what happened when they actually performed the ESSW experiment. They found that the photon trajectories aren't surrealistic after all-or, more precisely, that the paths may seem surrealistic, but only if one fails to take into account the nonlocality inherent in Bohm's theory.
The experiment that Steinberg and his team conducted was analogous to the standard two-slit experiment. They used photons rather than electrons, and instead of sending those photons through a pair of slits, they passed through a beam splitter, a device that directs a photon along one of two paths, depending on the photon's polarization. The photons eventually reach a single-photon camera (equivalent to the screen in the traditional experiment) that records their final position. The question "Which of two slits did the particle pass through?" becomes "Which of two paths did the photon take?"
Importantly, the researchers used pairs of entangled photons rather than individual photons. As a result, they could interrogate one photon to gain information about the other. When the first photon passes through the beam splitter, the second photon "knows" which path the first one took. The team could then use information from the second photon to track the first photon's path. Each indirect measurement yields only an approximate value, but the scientists could average large numbers of measurements to reconstruct the trajectory of the first photon.
The team found that the photon paths do indeed appear to be surreal, just as ESSW predicted: A photon would sometimes strike one side of the screen, even though the polarization of the entangled partner said that the photon took the other route.
But can the information from the second photon be trusted? Crucially, Steinberg and his colleagues found that the answer to the question "Which path did the first photon take?" depends on when it is asked.
At first-in the moments immediately after the first photon passes through the beam splitter-the second photon is very strongly correlated with the first photon's path. "As one particle goes through the slit, the probe [the second photon] has a perfectly accurate memory of which slit it went through," Steinberg explained.
But the farther the first photon travels, the less reliable the second photon's report becomes. The reason is nonlocality. Because the two photons are entangled, the path that the first photon takes will affect the polarization of the second photon. By the time the first photon reaches the screen, the second photon's polarization is equally likely to be oriented one way as the other-thus giving it "no opinion," so to speak, as to whether the first photon took the first route or the second (the equivalent of knowing which of the two slits it went through).
The problem isn't that Bohm trajectories are surreal, said Steinberg. The problem is that the second photon says that Bohm trajectories are surreal-and, thanks to nonlocality, its report is not to be trusted. "There's no real contradiction in there," said Steinberg. "You just have to always bear in mind the nonlocality, or you miss something very important."
Faster Than Light
Some physicists, unperturbed by ESSW, have embraced the Bohmian view all along and aren't particularly surprised by what Steinberg and his team found. There have been many attacks on the Bohmian view over the years, and "they all fizzled out because they had misunderstood what the Bohm approach was actually claiming," said Basil Hiley, a physicist at Birkbeck, University of London (formerly Birkbeck College), who collaborated with Bohm on his last book, The Undivided Universe. Owen Maroney, a physicist at the University of Oxford who was a student of Hiley's, described ESSW as "a terrible argument" that "did not present a novel challenge to de Broglie-Bohm." Not surprisingly, Maroney is excited by Steinberg's experimental results, which seem to support the view he's held all along. "It's a very interesting experiment," he said. "It gives a motivation for taking de Broglie-Bohm seriously."
On the other side of the Bohmian divide, Berthold-Georg Englert, one of the authors of ESSW (along with Marlan Scully, George Süssman and Herbert Walther), still describes their paper as a "fatal blow" to the Bohmian view. According to Englert, now at the National University of Singapore, the Bohm trajectories exist as mathematical objects but "lack physical meaning."
On a historical note, Einstein lived just long enough to hear about Bohm's revival of de Broglie's proposal-and he wasn't impressed, dismissing it as too simplistic to be correct. In a letter to physicist Max Born, in the spring of 1952, Einstein weighed in on Bohm's work:
Have you noticed that Bohm believes (as de Broglie did, by the way, 25 years ago) that he is able to interpret the quantum theory in deterministic terms? That way seems too cheap to me. But you, of course, can judge this better than I.
But even for those who embrace the Bohmian view, with its clearly defined particles moving along precise paths, questions remain. Topping the list is an apparent tension with special relativity, which prohibits faster-than-light communication. Of course, as physicists have long noted, nonlocality of the sort associated with quantum entanglement does not allow for faster-than-light signaling (thus incurring no risk of the grandfather paradox or other violations of causality). Even so, many physicists feel that more clarification is needed, especially given the prominent role of nonlocality in the Bohmian view. The apparent dependence of what happens here on what may be happening there cries out for an explanation.
"The universe seems to like talking to itself faster than the speed of light," said Steinberg. "I could understand a universe where nothing can go faster than light, but a universe where the internal workings operate faster than light, and yet we're forbidden from ever making use of that at the macroscopic level-it's very hard to understand."
Kali Yantra |
00870177aa1f8bce | Laplace–Runge–Lenz vector
In classical mechanics, the Laplace–Runge–Lenz (LRL) vector is a vector used chiefly to describe the shape and orientation of the orbit of one astronomical body around another, such as a binary star or a planet revolving around a star. For two bodies interacting by Newtonian gravity, the LRL vector is a constant of motion, meaning that it is the same no matter where it is calculated on the orbit;[1][2] equivalently, the LRL vector is said to be conserved. More generally, the LRL vector is conserved in all problems in which two bodies interact by a central force that varies as the inverse square of the distance between them; such problems are called Kepler problems.[3][4][5][6]
The hydrogen atom is a Kepler problem, since it comprises two charged particles interacting by Coulomb's law of electrostatics, another inverse-square central force. The LRL vector was essential in the first quantum mechanical derivation of the spectrum of the hydrogen atom,[7][8] before the development of the Schrödinger equation. However, this approach is rarely used today.
In classical and quantum mechanics, conserved quantities generally correspond to a symmetry of the system.[9] The conservation of the LRL vector corresponds to an unusual symmetry; the Kepler problem is mathematically equivalent to a particle moving freely on the surface of a four-dimensional (hyper-)sphere,[10] so that the whole problem is symmetric under certain rotations of the four-dimensional space.[11] This higher symmetry results from two properties of the Kepler problem: the velocity vector always moves in a perfect circle and, for a given total energy, all such velocity circles intersect each other in the same two points.[12]
The Laplace–Runge–Lenz vector is named after Pierre-Simon de Laplace, Carl Runge and Wilhelm Lenz. It is also known as the Laplace vector,[13][14] the Runge–Lenz vector[15] and the Lenz vector.[8] Ironically, none of those scientists discovered it.[15] The LRL vector has been re-discovered and re-formulated several times;[15] for example, it is equivalent to the dimensionless eccentricity vector of celestial mechanics.[2][14][16] Various generalizations of the LRL vector have been defined, which incorporate the effects of special relativity, electromagnetic fields and even different types of central forces.[17][18][19]
A single particle moving under any conservative central force has at least four constants of motion: the total energy E and the three Cartesian components of the angular momentum vector L with respect to the center of force.[20][21] The particle's orbit is confined to the plane defined by the particle's initial momentum p (or, equivalently, its velocity v) and the vector r between the particle and the center of force[20][21] (see Figure 1). This plane of motion is perpendicular to the constant angular momentum vector L = r × p; this may be expressed mathematically by the vector dot product equation rL = 0. Given its mathematical definition below, the Laplace–Runge–Lenz vector (LRL vector) A is always perpendicular to the constant angular momentum vector L for all central forces (AL = 0). Therefore A always lies in the plane of motion. As shown below, A points from the center of force to the periapsis of the motion, the point of closest approach, and its length is proportional to the eccentricity of the orbit.[1]
The LRL vector A is constant in length and direction, but only for an inverse-square central force.[1] For other central forces, the vector A is not constant, but changes in both length and direction. If the central force is approximately an inverse-square law, the vector A is approximately constant in length, but slowly rotates its direction.[14] A generalized conserved LRL vector can be defined for all central forces, but this generalized vector is a complicated function of position, and usually not expressible in closed form.[18][19]
The LRL vector differs from other conserved quantities in the following property. Whereas for typical conserved quantities, there is a corresponding cyclic coordinate in the three-dimensional Lagrangian of the system, there does not exist such a coordinate for the LRL vector. Thus, the conservation of the LRL vector must be derived directly, e.g., by the method of Poisson brackets, as described below. Conserved quantities of this kind are called "dynamic", in contrast to the usual "geometric" conservation laws, e.g., that of the angular momentum.
History of rediscoveryEdit
The LRL vector A is a constant of motion of the Kepler problem, and is useful in describing astronomical orbits, such as the motion of planets and binary stars. Nevertheless, it has never been well-known among physicists, possibly because it is less intuitive than momentum and angular momentum. Consequently, it has been rediscovered independently several times over the last three centuries.[15]
Jakob Hermann was the first to show that A is conserved for a special case of the inverse-square central force,[22] and worked out its connection to the eccentricity of the orbital ellipse. Hermann's work was generalized to its modern form by Johann Bernoulli in 1710.[23] At the end of the century, Pierre-Simon de Laplace rediscovered the conservation of A, deriving it analytically, rather than geometrically.[24] In the middle of the nineteenth century, William Rowan Hamilton derived the equivalent eccentricity vector defined below,[16] using it to show that the momentum vector p moves on a circle for motion under an inverse-square central force (Figure 3).[12]
At the beginning of the twentieth century, Josiah Willard Gibbs derived the same vector by vector analysis.[25] Gibbs' derivation was used as an example by Carl Runge in a popular German textbook on vectors,[26] which was referenced by Wilhelm Lenz in his paper on the (old) quantum mechanical treatment of the hydrogen atom.[27] In 1926, Wolfgang Pauli used the LRL vector to derive the energy levels of the hydrogen atom using the matrix mechanics formulation of quantum mechanics,[7] after which it became known mainly as the Runge–Lenz vector.[15]
Mathematical definitionEdit
An inverse-square central force acting on a single particle is described by the equation
The corresponding potential energy is given by . The constant parameter k describes the strength of the central force; it is equal to GMm for gravitational and keQq for electrostatic forces. The force is attractive if k > 0 and repulsive if k < 0.
Figure 1: The LRL vector A (shown in red) at four points (labeled 1, 2, 3 and 4) on the elliptical orbit of a bound point particle moving under an inverse-square central force. The center of attraction is shown as a small black circle from which the position vectors (likewise black) emanate. The angular momentum vector L is perpendicular to the orbit. The coplanar vectors p × L and (mk/r)r are shown in blue and green, respectively; these variables are defined below. The vector A is constant in direction and magnitude.
The LRL vector A is defined mathematically by the formula[1]
• m is the mass of the point particle moving under the central force,
• p is its momentum vector,
• L = r × p is its angular momentum vector,
• r is the position vector of the particle (Figure 1),
• is the corresponding unit vector, i.e., , and
• r is the magnitude of r, the distance of the mass from the center of force.
The SI units of the LRL vector are joule-kilogram-meter (J⋅kg⋅m). This follows because the units of p and L are kg⋅m/s and J⋅s, respectively. This agrees with the units of m (kg) and of k (N⋅m2).
This definition of the LRL vector A pertains to a single point particle of mass m moving under the action of a fixed force. However, the same definition may be extended to two-body problems such as the Kepler problem, by taking m as the reduced mass of the two bodies and r as the vector between the two bodies.
Since the assumed force is conservative, the total energy E is a constant of motion,
The assumed force is also a central force. Hence, the angular momentum vector L is also conserved and defines the plane in which the particle travels. The LRL vector A is perpendicular to the angular momentum vector L because both p × L and r are perpendicular to L. It follows that A lies in the plane of motion.
Alternative formulations for the same constant of motion may be defined, typically by scaling the vector with constants, such as the mass m, the force parameter k or the angular momentum L.[15] The most common variant is to divide A by mk, which yields the eccentricity vector, [2][16] a dimensionless vector along the semi-major axis whose modulus equals the eccentricity of the conic:
An equivalent formulation[14] multiplies this eccentricity vector by the major semiaxis a, giving the resulting vector the units of length. Yet another formulation[28] divides A by , yielding an equivalent conserved quantity with units of inverse length, a quantity that appears in the solution of the Kepler problem
where is the angle between A and the position vector r. Further alternative formulations are given below.
Derivation of the Kepler orbitsEdit
Figure 2: Simplified version of Figure 1, defining the angle θ between A and r at one point of the orbit.
The shape and orientation of the orbits can be determined from the LRL vector as follows.[1] Taking the dot product of A with the position vector r gives the equation
where θ is the angle between r and A (Figure 2). Permuting the scalar triple product yields
Rearranging yields the solution for the Kepler equation
This corresponds to the formula for a conic section of eccentricity e
where the eccentricity and C is a constant.[1]
Taking the dot product of A with itself yields an equation involving the total energy E,[1]
which may be rewritten in terms of the eccentricity,[1]
Thus, if the energy E is negative (bound orbits), the eccentricity is less than one and the orbit is an ellipse. Conversely, if the energy is positive (unbound orbits, also called "scattered orbits"[1]), the eccentricity is greater than one and the orbit is a hyperbola.[1] Finally, if the energy is exactly zero, the eccentricity is one and the orbit is a parabola.[1] In all cases, the direction of A lies along the symmetry axis of the conic section and points from the center of force toward the periapsis, the point of closest approach.[1]
Circular momentum hodographsEdit
Figure 3: The momentum vector p (shown in blue) moves on a circle as the particle moves on an ellipse. The four labeled points correspond to those in Figure 1. The circle is centered on the y-axis at position A/L (shown in magenta), with radius mk/L (shown in green). The angle η determines the eccentricity e of the elliptical orbit (cos ηe). By the inscribed angle theorem for circles, η is also the angle between any point on the circle and the two points of intersection with the px axis, px = ±p0, which only depend on E, but not L.
The conservation of the LRL vector A and angular momentum vector L is useful in showing that the momentum vector p moves on a circle under an inverse-square central force.[12][15]
Taking the dot product of
with itself yields
Further choosing L along the z-axis, and the major semiaxis as the x-axis, yields the locus equation for p,
In other words, the momentum vector p is confined to a circle of radius mk/L = L/ centered on (0, A/L).[29] The eccentricity e corresponds to the cosine of the angle η shown in Figure 3.
In the degenerate limit of circular orbits, and thus vanishing A, the circle centers at the origin (0,0). For brevity, it is also useful to introduce the variable .
This circular hodograph is useful in illustrating the symmetry of the Kepler problem.
Constants of motion and superintegrabilityEdit
The seven scalar quantities E, A and L (being vectors, the latter two contribute three conserved quantities each) are related by two equations, AL = 0 and A2 = m2k2 + 2 mEL2, giving five independent constants of motion. (Since the magnitude of A, hence the eccentricity e of the orbit, can be determined from the total angular momentum L and the energy E, only the direction of A is conserved independently; moreover, since A must be perpendicular to L, it contributes only one additional conserved quantity.)
This is consistent with the six initial conditions (the particle's initial position and velocity vectors, each with three components) that specify the orbit of the particle, since the initial time is not determined by a constant of motion. The resulting 1-dimensional orbit in 6-dimensional phase space is thus completely specified.
A mechanical system with d degrees of freedom can have at most 2d − 1 constants of motion, since there are 2d initial conditions and the initial time cannot be determined by a constant of motion. A system with more than d constants of motion is called superintegrable and a system with 2d − 1 constants is called maximally superintegrable.[30] Since the solution of the Hamilton–Jacobi equation in one coordinate system can yield only d constants of motion, superintegrable systems must be separable in more than one coordinate system.[31] The Kepler problem is maximally superintegrable, since it has three degrees of freedom (d = 3) and five independent constant of motion; its Hamilton–Jacobi equation is separable in both spherical coordinates and parabolic coordinates,[17] as described below.
Maximally superintegrable systems follow closed, one-dimensional orbits in phase space, since the orbit is the intersection of the phase-space isosurfaces of their constants of motion. Consequently, the orbits are perpendicular to all gradients of all these independent isosurfaces, five in this specific problem, and hence are determined by the generalized cross products of all of these gradients. As a result, all superintegrable systems are automatically describable by Nambu mechanics,[32] alternatively, and equivalently, to Hamiltonian mechanics.
Maximally superintegrable systems can be quantized using commutation relations, as illustrated below.[33] Nevertheless, equivalently, they are also quantized in the Nambu framework, such as this classical Kepler problem into the quantum hydrogen atom.[34]
Evolution under perturbed potentialsEdit
Figure 5: Gradually precessing elliptical orbit, with an eccentricity e = 0.667. Such precession arises in the Kepler problem if the attractive central force deviates slightly from an inverse-square law. The rate of precession can be calculated using the formulae in the text.
The Laplace–Runge–Lenz vector A is conserved only for a perfect inverse-square central force. In most practical problems such as planetary motion, however, the interaction potential energy between two bodies is not exactly an inverse square law, but may include an additional central force, a so-called perturbation described by a potential energy h(r). In such cases, the LRL vector rotates slowly in the plane of the orbit, corresponding to a slow apsidal precession of the orbit.
By assumption, the perturbing potential h(r) is a conservative central force, which implies that the total energy E and angular momentum vector L are conserved. Thus, the motion still lies in a plane perpendicular to L and the magnitude A is conserved, from the equation A2 = m2k2 + 2mEL2. The perturbation potential h(r) may be any sort of function, but should be significantly weaker than the main inverse-square force between the two bodies.
The rate at which the LRL vector rotates provides information about the perturbing potential h(r). Using canonical perturbation theory and action-angle coordinates, it is straightforward to show[1] that A rotates at a rate of,
where T is the orbital period, and the identity L dt = m r2 was used to convert the time integral into an angular integral (Figure 5). The expression in angular brackets, h(r)⟩, represents the perturbing potential, but averaged over one full period; that is, averaged over one full passage of the body around its orbit. Mathematically, this time average corresponds to the following quantity in curly braces. This averaging helps to suppress fluctuations in the rate of rotation.
This approach was used to help verify Einstein's theory of general relativity, which adds a small effective inverse-cubic perturbation to the normal Newtonian gravitational potential,[35]
Inserting this function into the integral and using the equation
to express r in terms of θ, the precession rate of the periapsis caused by this non-Newtonian perturbation is calculated to be[35]
which closely matches the observed anomalous precession of Mercury[36] and binary pulsars.[37] This agreement with experiment is strong evidence for general relativity.[38][39]
Poisson bracketsEdit
The unscaled functionsEdit
The algebraic structure of the problem is, as explained in later sections, SO(4)/Z2 ~ SO(3) × SO(3).[11] The three components Li of the angular momentum vector L have the Poisson brackets[1]
where i=1,2,3 and εijs is the fully antisymmetric tensor, i.e., the Levi-Civita symbol; the summation index s is used here to avoid confusion with the force parameter k defined above. Then since the LRL vector A transforms like a vector, we have the following Poisson bracket relations between A and L:[40]
Finally, the Poisson bracket relations between the different components of A are as follows:[41]
where is the Hamiltonian. Note that the span of the components of A and the components of L is not closed under Poisson brackets, because of the factor of on the right-hand side of this last relation.
Finally, since both L and A are constants of motion, we have
The Poisson brackets will be extended to quantum mechanical commutation relations in the next section and to Lie brackets in a following section.
The scaled functionsEdit
As noted below, a scaled Laplace–Runge–Lenz vector D may be defined with the same units as angular momentum by dividing A by . Since D still transforms like a vector, the Poisson brackets of D with the angular momentum vector L can then be written in a similar form[11][8]
The Poisson brackets of D with itself depend on the sign of H, i.e., on whether the energy is negative (producing closed, elliptical orbits under an inverse-square central force) or positive (producing open, hyperbolic orbits under an inverse-square central force). For negative energies—i.e., for bound systems—the Poisson brackets are[42]
We may now appreciate the motivation for the chosen scaling of D: With this scaling, the Hamiltonian no longer appears on the right-hand side of the preceding relation. Thus, the span of the three components of L and the three components of D forms a six-dimensional Lie algebra under the Poisson bracket. This Lie algebra is isomorphic to so(4), the Lie algebra of the 4-dimensional rotation group SO(4).[43]
By contrast, for positive energy, the Poisson brackets have the opposite sign,
In this case, the Lie algebra is isomorphic to so(3,1).
The distinction between positive and negative energies arises because the desired scaling—the one that eliminates the Hamiltonian from the right-hand side of the Poisson bracket relations between the components of the scaled LRL vector—involves the square root of the Hamiltonian. To obtain real-valued functions, we must then take the absolute value of the Hamiltonian, which distinguishes between positive values (where ) and negative values (where ).
Casimir invariants and the energy levelsEdit
The Casimir invariants for negative energies are
and have vanishing Poisson brackets with all components of D and L,
C2 is trivially zero, since the two vectors are always perpendicular.
However, the other invariant, C1, is non-trivial and depends only on m, k and E. Upon canonical quantization, this invariant allows the energy levels of hydrogen-like atoms to be derived using only quantum mechanical canonical commutation relations, instead of the conventional solution of the Schrödinger equation.[8][43] This derivation is discussed in detail in the next section.
Quantum mechanics of the hydrogen atomEdit
Figure 6: Energy levels of the hydrogen atom as predicted from the commutation relations of angular momentum and Laplace–Runge–Lenz vector operators; these energy levels have been verified experimentally.
Poisson brackets provide a simple guide for quantizing most classical systems: the commutation relation of two quantum mechanical operators is specified by the Poisson bracket of the corresponding classical variables, multiplied by .[44]
By carrying out this quantization and calculating the eigenvalues of the C1 Casimir operator for the Kepler problem, Wolfgang Pauli was able to derive the energy levels of hydrogen-like atoms (Figure 6) and, thus, their atomic emission spectrum.[7] This elegant 1926 derivation was obtained before the development of the Schrödinger equation.[45]
A subtlety of the quantum mechanical operator for the LRL vector A is that the momentum and angular momentum operators do not commute; hence, the quantum operator cross product of p and L must be defined carefully.[8] Typically, the operators for the Cartesian components As are defined using a symmetrized (Hermitian) product,
Once this is done, one can show that the quantum LRL operators satisfy commutations relations exactly analogous to the Poisson bracket relations in the previous section—just replacing the Poisson bracket with times the commutator.[46][47]
From these operators, additional ladder operators for L can be defined,
These further connect different eigenstates of L2, so different spin multiplets, among themselves.
A normalized first Casimir invariant operator, quantum analog of the above, can likewise be defined,
where H−1 is the inverse of the Hamiltonian energy operator, and I is the identity operator.
Applying these ladder operators to the eigenstates |mn〉 of the total angular momentum, azimuthal angular momentum and energy operators, the eigenvalues of the first Casimir operator, C1, are seen to be quantized, n2 − 1. Importantly, by dint of the vanishing of C2, they are independent of the ℓ and m quantum numbers, making the energy levels degenerate.[8]
Hence, the energy levels are given by
which coincides with the Rydberg formula for hydrogen-like atoms (Figure 6). The additional symmetry operators A have connected the different ℓ multiplets among themselves, for a given energy (and C1), dictating n2 states at each level. In effect, they have enlarged the angular momentum group SO(3) to SO(4)/Z2 ~ SO(3) × SO(3).[48]
Conservation and symmetryEdit
The conservation of the LRL vector corresponds to a subtle symmetry of the system. In classical mechanics, symmetries are continuous operations that map one orbit onto another without changing the energy of the system; in quantum mechanics, symmetries are continuous operations that "mix" electronic orbitals of the same energy, i.e., degenerate energy levels. A conserved quantity is usually associated with such symmetries.[1] For example, every central force is symmetric under the rotation group SO(3), leading to the conservation of the angular momentum L. Classically, an overall rotation of the system does not affect the energy of an orbit; quantum mechanically, rotations mix the spherical harmonics of the same quantum number l without changing the energy.
Figure 7: The family of circular momentum hodographs for a given energy E. All the circles pass through the same two points on the px axis (see Figure 3). This family of hodographs corresponds to one family of Apollonian circles, and the σ isosurfaces of bipolar coordinates.
The symmetry for the inverse-square central force is higher and more subtle. The peculiar symmetry of the Kepler problem results in the conservation of both the angular momentum vector L and the LRL vector A (as defined above) and, quantum mechanically, ensures that the energy levels of hydrogen do not depend on the angular momentum quantum numbers l and m. The symmetry is more subtle, however, because the symmetry operation must take place in a higher-dimensional space; such symmetries are often called "hidden symmetries".[49]
Classically, the higher symmetry of the Kepler problem allows for continuous alterations of the orbits that preserve energy but not angular momentum; expressed another way, orbits of the same energy but different angular momentum (eccentricity) can be transformed continuously into one another. Quantum mechanically, this corresponds to mixing orbitals that differ in the l and m quantum numbers, such as the s (l = 0) and p (l = 1) atomic orbitals. Such mixing cannot be done with ordinary three-dimensional translations or rotations, but is equivalent to a rotation in a higher dimension.
For negative energies – i.e., for bound systems – the higher symmetry group is SO(4), which preserves the length of four-dimensional vectors
In 1935, Vladimir Fock showed that the quantum mechanical bound Kepler problem is equivalent to the problem of a free particle confined to a three-dimensional unit sphere in four-dimensional space.[10] Specifically, Fock showed that the Schrödinger wavefunction in the momentum space for the Kepler problem was the stereographic projection of the spherical harmonics on the sphere. Rotation of the sphere and re-projection results in a continuous mapping of the elliptical orbits without changing the energy, an SO(4) symmetry sometimes known as Fock symmetry;[50] quantum mechanically, this corresponds to a mixing of all orbitals of the same energy quantum number n. Valentine Bargmann noted subsequently that the Poisson brackets for the angular momentum vector L and the scaled LRL vector A formed the Lie algebra for SO(4).[11][42] Simply put, the six quantities A and L correspond to the six conserved angular momenta in four dimensions, associated with the six possible simple rotations in that space (there are six ways of choosing two axes from four). This conclusion does not imply that our universe is a three-dimensional sphere; it merely means that this particular physics problem (the two-body problem for inverse-square central forces) is mathematically equivalent to a free particle on a three-dimensional sphere.
For positive energies – i.e., for unbound, "scattered" systems – the higher symmetry group is SO(3,1), which preserves the Minkowski length of 4-vectors
Both the negative- and positive-energy cases were considered by Fock[10] and Bargmann[11] and have been reviewed encyclopedically by Bander and Itzykson.[51][52]
The orbits of central-force systems – and those of the Kepler problem in particular – are also symmetric under reflection. Therefore, the SO(3), SO(4) and SO(3,1) groups cited above are not the full symmetry groups of their orbits; the full groups are O(3), O(4), and O(3,1), respectively. Nevertheless, only the connected subgroups, SO(3), SO(4) and SO(3,1), are needed to demonstrate the conservation of the angular momentum and LRL vectors; the reflection symmetry is irrelevant for conservation, which may be derived from the Lie algebra of the group.
Rotational symmetry in four dimensionsEdit
Figure 8: The momentum hodographs of Figure 7 correspond to stereographic projections of great circles on the three-dimensional η unit sphere. All of the great circles intersect the ηx axis, which is perpendicular to the page; the projection is from the North pole (the w unit vector) to the ηxηy plane, as shown here for the magenta hodograph by the dashed black lines. The great circle at a latitude α corresponds to an eccentricity e = sin α. The colors of the great circles shown here correspond to their matching hodographs in Figure 7.
The connection between the Kepler problem and four-dimensional rotational symmetry SO(4) can be readily visualized.[51][53][54] Let the four-dimensional Cartesian coordinates be denoted (w, x, y, z) where (x, y, z) represent the Cartesian coordinates of the normal position vector r. The three-dimensional momentum vector p is associated with a four-dimensional vector on a three-dimensional unit sphere
where is the unit vector along the new w axis. The transformation mapping p to η can be uniquely inverted; for example, the x component of the momentum equals
and similarly for py and pz. In other words, the three-dimensional vector p is a stereographic projection of the four-dimensional vector, scaled by p0 (Figure 8).
Without loss of generality, we may eliminate the normal rotational symmetry by choosing the Cartesian coordinates such that the z axis is aligned with the angular momentum vector L and the momentum hodographs are aligned as they are in Figure 7, with the centers of the circles on the y axis. Since the motion is planar, and p and L are perpendicular, pz = ηz = 0 and attention may be restricted to the three-dimensional vector = (ηw, ηx, ηy). The family of Apollonian circles of momentum hodographs (Figure 7) correspond to a family of great circles on the three-dimensional sphere, all of which intersect the ηx axis at the two foci ηx = ±1, corresponding to the momentum hodograph foci at px = ±p0. These great circles are related by a simple rotation about the ηx-axis (Figure 8). This rotational symmetry transforms all the orbits of the same energy into one another; however, such a rotation is orthogonal to the usual three-dimensional rotations, since it transforms the fourth dimension ηw. This higher symmetry is characteristic of the Kepler problem and corresponds to the conservation of the LRL vector.
An elegant action-angle variables solution for the Kepler problem can be obtained by eliminating the redundant four-dimensional coordinates in favor of elliptic cylindrical coordinates (χ, ψ, φ)[55]
where sn, cn and dn are Jacobi's elliptic functions.
Generalizations to other potentials and relativityEdit
The Laplace–Runge–Lenz vector can also be generalized to identify conserved quantities that apply to other situations.
In the presence of a uniform electric field E, the generalized Laplace–Runge–Lenz vector is[17][56]
where q is the charge of the orbiting particle. Although is not conserved, it gives rise to a conserved quantity, namely .
Further generalizing the Laplace–Runge–Lenz vector to other potentials and special relativity, the most general form can be written as[18]
where u = 1/r and ξ = cos θ, with the angle θ defined by
and γ is the Lorentz factor. As before, we may obtain a conserved binormal vector B by taking the cross product with the conserved angular momentum vector
These two vectors may likewise be combined into a conserved dyadic tensor W,
In illustration, the LRL vector for a non-relativistic, isotropic harmonic oscillator can be calculated.[18] Since the force is central,
the angular momentum vector is conserved and the motion lies in a plane.
The conserved dyadic tensor can be written in a simple form
although p and r are not necessarily perpendicular.
The corresponding Runge–Lenz vector is more complicated,
is the natural oscillation frequency, and
Proofs that the Laplace–Runge–Lenz vector is conserved in Kepler problemsEdit
The following are arguments showing that the LRL vector is conserved under central forces that obey an inverse-square law.
Direct proof of conservationEdit
A central force acting on the particle is
for some function of the radius . Since the angular momentum is conserved under central forces, and
where the momentum and where the triple cross product has been simplified using Lagrange's formula
The identity
yields the equation
For the special case of an inverse-square central force , this equals
Therefore, A is conserved for inverse-square central forces[57]
A shorter proof is obtained by using the relation of angular momentum to angular velocity, , which holds for a particle traveling in a plane perpendicular to . Specifying to inverse-square central forces, the time derivative of is
where the last equality holds because a unit vector can only change by rotation, and is the orbital velocity of the rotating vector. Thus, A is seen to be a difference of two vectors with equal time derivatives.
As described elsewhere in this article, this LRL vector A is a special case of a general conserved vector that can be defined for all central forces.[18][19] However, since most central forces do not produce closed orbits (see Bertrand's theorem), the analogous vector rarely has a simple definition and is generally a multivalued function of the angle θ between r and .
Hamilton–Jacobi equation in parabolic coordinatesEdit
The constancy of the LRL vector can also be derived from the Hamilton–Jacobi equation in parabolic coordinates (ξ, η), which are defined by the equations
where r represents the radius in the plane of the orbit
The inversion of these coordinates is
Separation of the Hamilton–Jacobi equation in these coordinates yields the two equivalent equations[17][58]
where Γ is a constant of motion. Subtraction and re-expression in terms of the Cartesian momenta px and py shows that Γ is equivalent to the LRL vector
Noether's theoremEdit
The connection between the rotational symmetry described above and the conservation of the LRL vector can be made quantitative by way of Noether's theorem. This theorem, which is used for finding constants of motion, states that any infinitesimal variation of the generalized coordinates of a physical system
that causes the Lagrangian to vary to first order by a total time derivative
corresponds to a conserved quantity Γ
In particular, the conserved LRL vector component As corresponds to the variation in the coordinates[59]
where i equals 1, 2 and 3, with xi and pi being the ith components of the position and momentum vectors r and p, respectively; as usual, δis represents the Kronecker delta. The resulting first-order change in the Lagrangian is
Substitution into the general formula for the conserved quantity Γ yields the conserved component As of the LRL vector,
Lie transformationEdit
Figure 9: The Lie transformation from which the conservation of the LRL vector A is derived. As the scaling parameter λ varies, the energy and angular momentum changes, but the eccentricity e and the magnitude and direction of A do not.
The Noether theorem derivation of the conservation of the LRL vector A is elegant, but has one drawback: the coordinate variation δxi involves not only the position r, but also the momentum p or, equivalently, the velocity v.[60] This drawback may be eliminated by instead deriving the conservation of A using an approach pioneered by Sophus Lie.[61][62] Specifically, one may define a Lie transformation[49] in which the coordinates r and the time t are scaled by different powers of a parameter λ (Figure 9),
This transformation changes the total angular momentum L and energy E,
but preserves their product EL2. Therefore, the eccentricity e and the magnitude A are preserved, as may be seen from the equation for A2
The direction of A is preserved as well, since the semiaxes are not altered by a global scaling. This transformation also preserves Kepler's third law, namely, that the semiaxis a and the period T form a constant T2/a3.
Alternative scalings, symbols and formulationsEdit
Unlike the momentum and angular momentum vectors p and L, there is no universally accepted definition of the Laplace–Runge–Lenz vector; several different scaling factors and symbols are used in the scientific literature. The most common definition is given above, but another common alternative is to divide by the constant mk to obtain a dimensionless conserved eccentricity vector
where v is the velocity vector. This scaled vector e has the same direction as A and its magnitude equals the eccentricity of the orbit, and thus vanishes for circular orbits.
Other scaled versions are also possible, e.g., by dividing A by m alone
or by p0
which has the same units as the angular momentum vector L.
In rare cases, the sign of the LRL vector may be reversed, i.e., scaled by −1. Other common symbols for the LRL vector include a, R, F, J and V. However, the choice of scaling and symbol for the LRL vector do not affect its conservation.
Figure 4: The angular momentum vector L, the LRL vector A and Hamilton's vector, the binormal B, are mutually perpendicular; A and B point along the major and minor axes, respectively, of an elliptical orbit of the Kepler problem.
An alternative conserved vector is the binormal vector B studied by William Rowan Hamilton,[16]
which is conserved and points along the minor semiaxis of the ellipse. (It is not defined for vanishing eccentricity.)
The LRL vector A = B × L is the cross product of B and L (Figure 4). On the momentum hodograph in the relevant section above, B is readily seen to connect the origin of momenta with the center of the circular hodograph, and to possess magnitude A/L. At perihelion, it points in the direction of the momentum.
The vector B is denoted as "binormal" since it is perpendicular to both A and L. Similar to the LRL vector itself, the binormal vector can be defined with different scalings and symbols.
The two conserved vectors, A and B can be combined to form a conserved dyadic tensor W,[18]
where α and β are arbitrary scaling constants and represents the tensor product (which is not related to the vector cross product, despite their similar symbol). Written in explicit components, this equation reads
Being perpendicular to each another, the vectors A and B can be viewed as the principal axes of the conserved tensor W, i.e., its scaled eigenvectors. W is perpendicular to L ,
since A and B are both perpendicular to L as well, LA = LB = 0.
More directly, this equation reads, in explicit components,
See alsoEdit
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Further readingEdit |
847de46e82c41f8c | Five Predictions On Watching Movies In 2022
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6b5c350502a07349 | Future tension | Aeon
Future tension
by Anthony Sudbery + BIO
Que sera sera
Whatever will be will be
The future’s not ours to see
Que sera sera.
For a couple of centuries, Newton’s dream seemed to be coming true. More and more of the physical world came under the domain of physics, as matter was analysed into molecules and atoms, and the behaviour of matter, whether chemical, biological, geological or astronomical, was explained in terms of Newtonian forces. The particles of matter that Newton dreamed of had to be supplemented by electromagnetic fields to give the full picture of what the world was made of, but the basic idea remained that they all followed deterministic laws. Capricious events such as storms and floods, formerly seen as unpredictable and attributed to the whims of the gods, became susceptible to weather forecasts; and if some such events, like earthquakes, remain unpredictable, we feel sure that advancing knowledge will make them also subject to being forecast.
This scientific programme has been so successful that we have forgotten there was ever any other way to think about the future. Mark G Alford, a physicist at Washington University, writes:
In ordinary life, and in science up until the advent of quantum mechanics, all the uncertainty that we encounter is presumed to be … uncertainty arising from ignorance.
We have completely forgotten what an uncertain world was inhabited by the human race before the 17th century, and we take Newton’s dream as a natural view of waking reality.
Well, it was a nice dream. But it didn’t work out that way. In the early years of the 20th century, Ernest Rutherford, investigating the recently discovered phenomenon of radioactivity, realised that it showed random events happening at a fundamental level of matter, in the atom and its nucleus. This did not necessarily mean that Newton’s dream had to be abandoned: the nucleus is not the most fundamental level of matter, but is a complicated object made up of protons and neutrons, and – maybe – if we knew exactly how these particles were situated and how they were moving, we would be able to predict when the radioactive decay of the nucleus would happen. But other, stranger discoveries at around the same time led to the radical departure from Newtonian physics represented by quantum mechanics, which strongly reinforced the view that events at the smallest scale are indeed random, and there is no possibility of precisely knowing the future.
Quantum theory is so puzzling, it’s not clear it should be described as an ‘explanation’ of the puzzling facts it subsumes
The discoveries that had to be confronted by the new physics of the 1920s were two-fold. On the one hand, Max Planck’s explanation of the distribution of wavelengths in the radiation emitted by hot matter, and Albert Einstein’s explanation of the photoelectric effect, showed that energy comes in discrete packets, instead of varying continuously as it must do in Newton’s mechanics and James Clerk Maxwell’s electromagnetic theory. On the other hand, experiments on electrons by George Paget Thomson, Clinton Davisson and Lester Germer showed that electrons, which had been firmly established to be particles, also sometimes behaved like waves.
These puzzling facts found a systematic, coherent, unified mathematical description in the theory of quantum mechanics which emerged from the work of theorists after 1926. This theory is itself so puzzling that it is not clear that it should be described as an ‘explanation’ of the puzzling facts it subsumes; but an essential feature of it, which seems inescapable, is that, when applied to give predictions of physical effects, it yields probabilities rather than precise numbers.
This is still not universally accepted. Some people believe that there are finer details to be discovered in the make-up of matter, which, if we knew them, would once again make it possible to predict their future behaviour precisely. This is indeed logically possible, but there would necessarily be aspects of such a theory that would lead most physicists to think it highly unlikely.
The format of quantum theory is quite different from previous physical theories such as Newtonian mechanics or electromagnetism (or both combined). These theories work with a mathematical description of the state of the world, or any part of the world; they have an equation of motion that takes such a mathematical description and tells you what it will change into after a given time. Quantum mechanics also works with a mathematical object that describes a state of the world; it is called a state vector (though it is not a vector in three dimensions like velocity), and is often denoted by the Greek letter Ψ or some similar symbol.
But this is a different kind of mathematical description from that in mechanics or electromagnetism. Each of those theories uses a set of numbers that measure physical quantities such as the velocity of a specified particle, or the electric field at a specified point of space. The quantum state vector, on the other hand, is a more abstruse object whose relation to physical quantities is indirect. From the state vector, you can obtain the values of physical quantities, but only some physical quantities: you can choose which quantities you would like to know, but you are not allowed to choose all of them.
Moreover, once you have chosen which ones you would like to know, the state vector will not give you a definite answer; it will give you only probabilities for the different possible answers. This is where quantum mechanics departs from determinism. Strangely enough, in its treatment of change, quantum mechanics looks like the old deterministic theories. Like them, it has an equation of motion, the Schrödinger equation, which will tell you what a given state vector of the world will become after a given time; but because you can get only probabilities from this state vector, it cannot tell you what you will see after this time.
State vectors, in general, are puzzling things, and it is not at all clear how they describe physical objects. Some of them, however, do correspond (if you don’t look too closely) to descriptions that we can understand. Among the state vectors of a cat, for example, is one describing a cat sitting and contentedly purring; there is another one describing it lying dead, having been poisoned in a diabolical contraption devised by the physicist Erwin Schrödinger.
But there are others, obtained mathematically by ‘superposing’ these two state vectors; such a superposed state vector could be made up of a part describing the cat as alive and a part describing it as dead. These are not two cats; the point of Schrödinger’s story was that one and the same cat seems to be described as both alive and dead, and we do not understand how such states could describe anything that could arise in the real world. How can we believe this theory, generations of physicists have asked, when we never see such alive-and-dead cats?
it follows from quantum mechanics that although cats have states in which they seem to be both alive and dead, we will never see a cat in such a state
There is an answer to this puzzle. If I were to open the box in which Schrödinger has prepared this poor cat, then the ordinary laws of everyday physics would ensure that, if the cat was alive, I would have the image of a living cat on my retina and in my visual cortex, and the system consisting of me and the cat would end up in an understandable state in which the cat is alive and I see a living cat. If the cat was dead, I would have the image of a dead cat, and the system consisting of me and the cat would end up in a state in which the cat is dead and I see a dead cat.
It now follows, according to the laws of quantum mechanics, that if the cat is in a superposition of being alive and being dead, then the system consisting of me and the cat ends up in a superposition of the two final states described above. This superposition does not contain a state of my brain seeing a peculiar alive-and-dead state of a cat; the only states of my brain that occur are the familiar ones of seeing a live cat and seeing a dead cat. This is the answer to the question at the end of the earlier paragraph; it follows from quantum mechanics itself that although cats have states in which they seem to be both alive and dead, we will never see a cat in such a state.
But now the combined system of me and the cat is in one of the strange superposition states introduced by quantum mechanics. It is represented mathematically by the familiar sign +, and called an entangled state of me and the cat. How are we to understand it? Maybe the mathematical sign + just means ‘or’; that would make sense. But unfortunately this meaning, if applied to the states of an electron, is not compatible with the facts of interference observed in the experiments that show the electron behaving like a wave. Some people think that this + should be understood as ‘and’: when the cat and I are in the superposition state, there is a world in which the cat has died and I see a dead cat, and another world in which the cat is still alive and I see a living cat. Others do not find this a helpful picture. Perhaps we should just take it as (in some sense) a true description of the cat and me, whose meaning is beyond us.
Now let us broaden our horizon and consider the whole universe, which contains each one of us considered as a sentient, observing physical system. According to quantum mechanics, this has a description by a state vector in which the sentient system is entangled with the rest of the universe, and several different experiences of the sentient system are involved in this entanglement. The same overall state vector of the whole universe can be seen as such an entangled state for every sentient system inside the universe; these are simply different views of the same universal truth.
But saying that this is the truth about the universe seems to conflict with my knowledge of what I see. To illustrate this, let us again consider a little universe containing just me and a cat. Let us suppose that the cat survived when I did Schrödinger’s experiment. Then I know what my state is: I see a living cat. From this I know what the state of the cat is: it is alive. The entangled state of my little universe that was produced by my experiment also contains a part with a dead cat and my brain full of remorse.
But seeing a live cat, as I do, I reckon that this other picture is not part of the truth; it describes something that might have happened but didn’t. In general, considering the whole universe, I know that I have just one definite experience. But this contradicts what was asserted in the previous paragraph. Which of these is the truth?
This contradiction is of the same type as many familiar contradictions between objective and subjective statements. In The View from Nowhere (1986), Thomas Nagel shows how some of these contradictions can be resolved: we must recognise that there are two positions from which we can make statements of fact or value, and statements made in these two contexts are not commensurable. This applies to the puzzle presented by quantum mechanics as follows. In the external context (the God’s-eye view, or the ‘view from nowhere’) we step outside our own particular situation and talk about the whole universe. In the internal context (the view from now, here), we make statements as physical objects inside the universe.
Thus, in the external view, the entangled universal state vector is the whole truth about the universe; the components describing my different possible experiences, and the corresponding states of the rest of the universe, are (unequal) parts of this truth. But in the internal view, from the perspective of some particular experience that I know I am having, this experience, together with the corresponding state of the rest of the universe, is the actual truth. I might know what the other components are, because I can calculate the universal state vector using the equations of quantum mechanics; but these other components, for me, represent things that might have happened but didn’t.
Since I cannot see the future, none of the worlds of the future are singled out for me
We can now look at what quantum mechanics tells us about the future. As we should now expect, there are two answers, one for each of the two perspectives. From the external perspective, the universe at any one time is described by a universal state vector, and state vectors at different times are related by the Schrödinger equation. Given the state vector at the present time, the Schrödinger equation delivers a unique state vector at any future time: the theory is deterministic, in complete accord with Laplace’s world-view (in a quantum version).
From the internal perspective, however, things are quite different. We now have to specify a particular observer (who has been me in the above discussion, but it could have been you or anyone else, or indeed the whole human race taken together), with respect to which we can carve up the universal state vector as described above; and we have to specify a particular experience state of that observer. From that perspective, it is by definition true that the observer has that definite experience, and that the rest of the universe is in a corresponding definite state.
So quantum mechanics tells us that at this moment there are a number of different worlds, but I know that one of them is singled out, for me, as being the world that I see and whose finer details are revealed to me by experiment. But when we turn to the future the situation is different. Since I cannot see the future, none of the worlds of the future are singled out for me. Even if there is only one world now, and what I see agrees with the universal state vector of quantum mechanics, it might happen that the laws of quantum mechanics produce a superposition of worlds at a future time. For example, if I start with the experience of setting up Schrödinger’s experiment with the cat, then at the end of the experiment the universal state vector will be the superposition that we have already encountered, with one part containing me seeing a living cat and another part containing me seeing a dead cat. Then what can I say about what I will see at that future time?
I found this rather startling when I first encountered it. I was used to thinking that there is something awaiting me in the future, even if I cannot know what it is, and even if there is no law of nature that determines what it is. Whatever will be will be, indeed. But Aristotle already saw that this is wrong. Statements in the future tense do not obey the same logic as present-tense statements: they do not have to be either true or false. Logicians following Aristotle have allowed the possibility of a third truth value, ‘undetermined’ or ‘undecided’, in addition to ‘true’ and ‘false’.
However, Aristotle also pointed out that, although no one statement about the future is actually true, some of them are more likely than others. Similarly, the universal state vector at a future time contains more information, for me, than simply what experiences I might have at that time. These experiences, occurring as components of the universal state vector, contribute to it in different amounts, measured by coefficients that are usually used in quantum mechanics to calculate probabilities. So we can understand the future universal state as giving information, not only about what experiences are possible for me at that future time, but also about how probable each experience is.
Now, truth and falsity can be expressed numerically: a true statement has truth value 1, a false one has truth value 0. If a future event X is very likely to happen, so that the probability of X is close to 1, then the statement ‘X will happen’ is very nearly true; if it is very unlikely to happen, so that its probability is close to 0, then the statement ‘X will happen’ is very nearly false. This suggests that the truth value of a future-tense statement should be a number between 0 and 1. A true statement has truth value 1; a false statement has truth value 0; and if a future-tense statement ‘X will happen’ has a truth value between 0 and 1, that number is the probability that X will happen.
The nature of probability is a long-standing philosophical problem, to which scientists also need an answer. Many scientists take the view that the probability of an event makes sense only when there are many repetitions of the circumstances in which the event might occur, and we work out the proportion of times that it does occur; they hold that the probability of a single, unrepeated event does not make sense. But what we have just outlined does seem to be a calculation of a single event at a time that will come only once. In everyday life, we often talk about the probability that something will happen on just one occasion: that it will rain tomorrow, or that a particular horse will win a race, or that there will be a sea-battle. A standard view of such single-event probability is that it refers to the strength of the belief of the person who is asserting the probability, and can be measured by the betting odds they are prepared to offer on the event happening.
But the probability described above is an objective fact about the universe. It has nothing to do with the beliefs of an individual, not even the individual whose experiences are in question; that individual is being told a fact about his future experiences, whether he believes it or not. The logical theory gives an objective meaning to the probability of a single event: the probability of a future event is the truth value of the future-tense proposition that that event will happen. I explore this view of probability, and the way that quantum mechanics supports the associated many-valued logic of tensed propositions, in ‘The Logic of the Future in Quantum Theory’ (2016).
It has now become clear that the description of the physical world given by quantum mechanics, namely the universal state vector, plays very different roles in the two perspectives, external and internal. From the external perspective, it is a full description of reality; it tells how the universe is constituted at a particular time. This complete reality can be analysed with respect to any given sentient system, yielding a number of components, attached to different experiences of the chosen sentient system, which are all parts of the universal reality.
Those things that might have happened, but didn’t, some of which we don’t even know about, might still affect the future
From the internal perspective of this system, however, reality consists of just one of these experiences; the component attached to this experience is the complete truth about the universe for the sentient system. All the other non-zero components are things that might have happened, but didn’t. The role of the universal state vector at a later time, in this perspective, is not to describe how the universe will be at that time, but to specify how the present state of the universe might change between now and then. It gives a list of possibilities at that later time, with a probability for each of them that it will become the truth.
It might seem that we can at least know these probabilities for the future, being able to calculate them from our certain knowledge of our present experience, using the Schrödinger equation. But even this is uncertain. Our present experience could well be only part of the universal state, and it is the whole universal state vector that must be put into the calculation of future probabilities. Those things that might have happened, but didn’t, some of which we don’t even know about, might still affect the future. However, if those things are sufficiently different from our actual experience on a macroscopic scale, then quantum theory assures us that the effect they might have on the future is so small as to be utterly negligible. This consequence of the theory is known as decoherence.
Knowledge of the future, therefore, is limited in a fundamental way. It is not that there are true facts about the future, but the knowledge of them is not accessible to us; there are no facts out there, and there is simply no certain knowledge to be had. Nevertheless, there are facts about the future with partial degrees of truth. We can attain knowledge of the future, but that knowledge will always be uncertain.
An expanded version of this article will appear in Space, Time and the Limits of Human Understanding, ed. Shyam Wuppuluri and Giancarlo Ghirardi, to be published by Springer
MathematicsLogic and probabilityThe future
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4e2a22cc40ca40ba | Vortex patterns beyond hypergeometric. (English) Zbl 1267.82142
The paper studies the existence of confined vortex loops in a superconducting infinite space, where the magnetic field is generated by a point-like magnetic dipole placed at the origin. First, a theoretical formalism based on the functional of free-energy in the Ginzburg-Landau (GL) theory is introduced. In the approximation of the weak magnetic field, the Euler-Lagrange PDEs arising from the functional reduce to a magnetic Schrödinger equation. The exact solutions are obtained in the dipole coordinates \((a, b)\) using dimensionless variables. In order to obtain a separation of the dipole variables, the solutions are studied for \(a >> b\), which describes points close to the \(z\)-axis. The gradient of the order parameter function is considered to be directed mainly radially and orthogonally to the magnetic dipole field lines. In this case, the coordinate surfaces \(a = \mathrm{const}\) describe the vortex surfaces. The exact solution of the dipole equation is obtained in the form of Heun functions and is sufficient to prove the occurrence of spontaneous vortex phases with a mutual interconnection of vortices at the origin. The analytic solutions of the linearized dipole equation are investigated by mapping it to a double confluent Heun equation and then taking linear combinations of two solutions of the dipole equation. In order to generate physical solutions for the full nonlinear GL problem, the nonlinear dipole equation is introduced in the Gibbs free energy equation and this integral is minimized in the space of unknown parameters. The author proves that it is enough to investigate order parameters constructed using only two linear solutions. It is proved that multi-vortex states are possible even without the presence of an external field and that they are characterized by confined vortex loops. For the minimization of the Gibbs free energy functional, the Hessian matrix method is used. Finally, the symmetries of the confined phase starting from solutions of the linearized Ginzburg-Landau equations are determined.
82D55 Statistical mechanics of superconductors
35Q56 Ginzburg-Landau equations |
40630837e1f5fb1d | PARSEC is a computer code that solves the Kohn-Sham equations by expressing electron wave-functions directly in real space, without the use of explicit basis sets. It uses norm-conserving pseudopotentials (Troullier-Martins and other varieties). It is designed for ab initio quantum-mechanical calculations of the electronic structure of matter, within density-functional theory. PARSEC is optimized for massively parallel computing environment, but it is also compatible with serial machines. A finite-difference approach is used for the calculation of spatial derivatives. Owing to the sparsity of the Hamiltonian matrix, the Kohn-Sham equations are solved by direct diagonalization, with the use of extremely efficient sparse-matrix eigensolvers. Some of its features are: Choice of boundary conditions: periodic (on all three directions), or confined. Structural relaxation. Simulated annealing. Langevin molecular dynamics. Polarizability calculations (confined-system boundary conditions only). Spin-orbit coupling. Non-collinear magnetism.
References in zbMATH (referenced in 20 articles )
Showing results 1 to 20 of 20.
Sorted by year (citations)
2. Li, Ruipeng; Xi, Yuanzhe; Erlandson, Lucas; Saad, Yousef: The eigenvalues slicing library (EVSL): algorithms, implementation, and software (2019)
3. Bodroski, Zarko; Vukmirović, Nenad; Skrbic, Srdjan: Gaussian basis implementation of the charge patching method (2018)
4. Duersch, Jed A.; Shao, Meiyue; Yang, Chao; Gu, Ming: A robust and efficient implementation of LOBPCG (2018)
5. Ghosh, Swarnava; Suryanarayana, Phanish: SPARC: accurate and efficient finite-difference formulation and parallel implementation of density functional theory: extended systems (2017)
6. Banerjee, Amartya S.; Suryanarayana, Phanish: Cyclic density functional theory: a route to the first principles simulation of bending in nanostructures (2016)
7. Li, Ruipeng; Xi, Yuanzhe; Vecharynski, Eugene; Yang, Chao; Saad, Yousef: A thick-restart Lanczos algorithm with polynomial filtering for Hermitian eigenvalue problems (2016)
8. Wen, Zaiwen; Yang, Chao; Liu, Xin; Zhang, Yin: Trace-penalty minimization for large-scale eigenspace computation (2016)
9. Xi, Yuanzhe; Saad, Yousef: Computing partial spectra with least-squares rational filters (2016)
10. Banerjee, Amartya S.; Elliott, Ryan S.; James, Richard D.: A spectral scheme for Kohn-Sham density functional theory of clusters (2015)
11. Liu, Xin; Wen, Zaiwen; Zhang, Yin: An efficient Gauss-Newton algorithm for symmetric low-rank product matrix approximations (2015)
12. Zhou, Yunkai; Chelikowsky, James R.; Saad, Yousef: Chebyshev-filtered subspace iteration method free of sparse diagonalization for solving the Kohn-Sham equation (2014)
13. Di Napoli, Edoardo; Berljafa, Mario: Block iterative eigensolvers for sequences of correlated eigenvalue problems (2013)
14. Fang, Jun; Gao, Xingyu; Zhou, Aihui: A symmetry-based decomposition approach to eigenvalue problems (2013)
15. Fang, Jun; Gao, Xingyu; Zhou, Aihui: A finite element recovery approach to eigenvalue approximations with applications to electronic structure calculations (2013)
16. Fang, Jun; Gao, Xingyu; Zhou, Aihui: A Kohn-Sham equation solver based on hexahedral finite elements (2012)
17. Soba, Alejandro; Bea, Edgar Alejandro; Houzeaux, Guillaume; Calmet, Hadrien; Cela, José María: Real-space density functional theory and time dependent density functional theory using finite/infinite element methods (2012)
18. Sidje, Roger B.; Saad, Yousef: Rational approximation to the Fermi-Dirac function with applications in density functional theory (2011)
19. Rizea, M.; Ledoux, V.; Van Daele, M.; Vanden Berghe, G.; Carjan, N.: Finite difference approach for the two-dimensional Schrödinger equation with application to scission-neutron emission (2008)
20. Zhou, Yunkai; Saad, Yousef: Block Krylov-Schur method for large symmetric eigenvalue problems (2008) |
fb7ac4f4b9b40ac1 | Title: Development of 3d-printed microwave metamaterial absorbers
This project relates to the development of broadband metamaterial absorbers for mm-wavelength radiation. Such absorber technologies are being considered for next-generation telescopes measuring the polarization of the cosmic microwave background (CMB). The project involves the study of potential plastic resin candidates doped with conductive particles to increase mm-wavelength absorption properties as well development of code that generates geometries suitable for 3D printing. As part of the project, the student will develop a familiarity with a resin 3D printer and test fabrication of absorbing geometries. The project will conclude with measurement of material properties and comparison with expectations.
Supervisor: Jon Gudmundsson
Title: Machine learning algorithms and optics design
The project aims to explore and develop machine learning algorithms to design optical instrumentation for astronomical observatories automatically. High-fidelity astronomy requires designing ever-increasing complex optical instruments that meet the researcher’s demands. Traditional optical systems have consisted only of limited pre-machined convex or concave shapes. In recent years, it has become feasible to produce so-called freeform lenses of arbitrary surface shapes, allowing for a wider variety of optical tasks that can be addressed with them.
This opens the opportunity to design more complex optical instruments that satisfy specified criteria. However, the ability to machining such new lenses comes at the expense of an increased design task. This thesis will explore various machine learning approaches to search for optimal optical designs that satisfy predefined criteria and requirements of scientific instrumentation.
Supervisors: Jens Jasche and Jon Gudmundsson
Title: Method of moments approach to optical modeling for mm-wavelength telescopes
In this project, we will develop an algorithm that makes use of method of moments approach for electromagnetic scattering—a concept often employed in stealth technology—to study realistic optical systems and their impact on the overall noise budget of our bolometers. Although the method of moments approach to electromagnetic scattering problems has been around for a few decades, it has not seen much use in modeling of large telescope systems operating at radio and mm wavelengths. Some of the basic concepts are introduced in textbooks by Harrington.
We will review the algorithm and implement it on a few simple electromagnetic scattering problems before developing code that applies the algorithm to realistic telescopes used by CMB experiments.
Contact Jón Gudmundsson for more information
Title: Avoided level crossings in cosmology
There is strong observational evidence that most of the matter in the universe is invisible, or ’dark’. It is not known what particles comprise this dark matter, or how they came into being. An ‟avoided level crossing” is a quantum phenomenon, well-known in atomic physics and neutrino physics, where two eigenvalues of a time-dependent Hamiltonian become almost degenerate, and transitions between the eigenstates can become ‟resonant” and unsuppressed.
This project will explore if avoided level crossings can explain the production of dark matter in the early universe, and if so, what additional predictions this entails. This project requires an active interest in theoretical quantum physics, astroparticle physics, and cosmology.
Contact: David Marsh
Title: Axion-photon conversion and correlation functions
The QCD axion and more general axion-like particles (ALPs) are theoretically very well-motivated extensions of the Standard Model of particle physics. A prediction of this theory is that ALPs mix with photons in the presence of magnetic fields. The mixing equation is classical, but can be written as a Schrödinger equation for a 3-level system. This makes it possible to translate results from quantum mechanics to learn about the predictions of ALP theories.
This project will develop the formalism of axion photon mixing as applied to one of the most promising targets for axion searches: the X-ray emitting intracluster medium of galaxy clusters. This project will include theoretical and numerical work that may contribute to the foundation for future ALP searches by the next generation of satellite missions.
Contact: David Marsh
Title: Geometric destabilisation of cosmological fields
Inflation provides the leading paradigm for the origin of cosmic structure. In this framework, quantum scalar field fluctuations froze into the fabric of space during a hypothetical period of accelerated expansion in the early universe. Inflation can be realised with rather simple ingredients, such as one or more scalar fields slowly ‟rolling” down a potential. Theories with more than one field are theoretically very well-motivated, and includes new features such as non-trivial curvature of the field space itself.
This project will investigate a potential instability of multifield inflationary theories in which the field space curvature induces an instability for certain fluctuations around the inflationary background. After carefully characterising this effect geometrically, this project may investigate whether it can be realised in the type of complex Kähler geometries that are relevant in theories of supergravity. This project involves theoretical work in cosmology and field theory, including aspects of geometry.
Contact: David Marsh
Title: Modified dispersion relations in cosmology
Can we use cosmological observations to observe, e.g. quantum effects of gravity?
If we can measure the gravitational lensing effect and/or the group velocity through the light travel time for single sources at different energies, we are able to investigate modified dispersion relations.
First, review systems that have observations at different wavelengths. Second, use available data to constrain the energy scale of modifications to the dispersion relation. Last, interpret results in terms of modifications to general relativity.
Is it possible that modified dispersion relations can help easing the tension between the inferred lensing of the cosmic microwave background and lensing of galaxies observed in optical light? How do these limits compare to time delay constraints in, e.g., https://arxiv.org/abs/2109.07850?
Contact Edvard Mörtsell for more information
Title: Lensed gravitational waves
Cook up a lens that explains the model of lensed gravitational waves in https://arxiv.org/abs/2007.12709.
Also, is it possible that many other gravitational wave events are lensed and in fact at much higher redshifts than commonly believed, see https://arxiv.org/abs/2006.13219 and https://arxiv.org/abs/2106.06545?
Contact Edvard Mörtsell for more information
Title: Gravitational waves in modified gravity
In extended gravity theories, e.g., massive graviton theories, the gravitational wave velocity will be energy dependent. This means that the higher frequency signal from the later stages of a coalescing event may catch up with the early low frequency part causing a gravitational wave sound bang at the detector, or even an inverted signal. In principle this could be searched for in archival data. The project aims at deriving possible highly distorted gravitational wave signals very different from the ones nominally searched for and therefore possibly missed by common detection pipelines.
Contact Edvard Mörtsell for more information
Title: Micro lensing bias in gravitationally lensed supernovae and quasars
In cases of multiply imaged supernovae and quasars, we expect the magnification of individual images to be affected by micro lensing from stars and possibly compact dark matter in the lens galaxy. In principle, this can be corrected for on a statistical basis. However, the fact that it is easier to detect high magnification events may cause a bias towards such events in the observed data, possibly invalidating such simple corrections. The project aims at quantifying this bias through the use of simulated events.
Contact Edvard Mörtsell for more information
Title: Modified gravity and rotation curves
Can we fit galactic rotation curves if the acceleration scale of Modified Newtonian Dynamics (MOND, or some similar theory) is mass dependent? An example is bimetric theory where the so called Vainshtein radius within which general relativity is restored, . References: e.g. https://arxiv.org/abs/1401.5619 and https://arxiv.org/abs/1705.02366.
Contact Edvard Mörtsell for more information
Title: The prior dependence in Bayseian model selection
When doing model selection using Bayesian statistics, the assumed prior on the parameters of the model is very important. Especially, the range of the prior can have a large impact on the validity of the model in question. As an example, the case for a non-zero cosmological constant, , is expected to beyond doubt given how much better the fit is to observed data compared to the case with . However, given that the theoretical prior on the possible value of is huge, it is not obvious that it will be preferred given a strict Bayesian analysis. This, and simliar cases, should be investigated in the project. See also https://arxiv.org/abs/2102.10671 and https://arxiv.org/abs/2111.04231.
Contact Edvard Mörtsell for more information
Title: Dark matter direct detection project
The XENONnT experiment is one of the world’s most sensitive detectors
for measuring direct interactions between potential dark matter
candidates and ordinary matter. It is situated at LNGS in Italy ca. 1.4
km deep under the Abruzzo mountains and started taking data in 2021.
Our group is involved in the data analysis, specifically the high-end
statistical analysis, the development of a new statistical framework and
the operation of the detector, specifically the photosensors used to
measure the light and charge signals which are expected from a dark
matter interaction with our detector.
In addition we are involved in the development of a completely new type
of photosensor, called ABALONE, for the future DARWIN experiment which
will be even more sensitive than our current XENONnT detector.
Potential projects in our group include the setup and operation of
cryogenic tests (at -100˚C) for these new photosensors in our lab at
AlbaNova as well as the data analysis of these tests.
If you are interested in this topic or other topics our group is
involved in, please don’t hesitate to contact us (contacts: Jörn
and Jan Conrad).
Analysis of Supernovae from the Zwicky Transient Facility
Type Ia Supenovae (SNe Ia) are bright explosions of white-dwarf stars, that can be used to measure cosmological distances. The accelerated expansion of the Universe was discovered using only ~100 SNe Ia in 1998 (Nobel Prize in 2011), and we now have more than 4000 SNe Ia discovered by the Zwicky Transient Facility (ZTF) to analyse.
SNe Ia remain essential for studying the properties of the “dark energy” driving the accelerated expansion of the Universe, but the lack of understanding of the white dwarf star progenitor systems and the standardization corrections to the light-curve shapes and colors represent severe limitations for SNe Ia as cosmological probes. One source of uncertainty comes from cosmic dust particles in along the line-of-sight (e.g. in the Milky way, host galaxies and/or circumstellar environments of the SNe), that affect the observed colours and luminosities of SNe Ia - typically making them redder and fainter.
Related thesis projects: Measuring extinction from dust in the Milky way, SN host galaxies and the circumstellar environment using the most reddened SNe Ia in ZTF; Analyzing light-curves and spectra of extreme and “weird” SNe Ia, and comparing to different explosion scenarios; Correlations between supernova features and host galaxy properties; Sample analysis of SN Ia ”siblings” (SNe sharing the same host galaxy, e.g. Biswas et al. 2021) to self-calibrate the light-curve corrections. (Contacts: Joel Johansson, Ariel Goobar, Steve Schulze)
Cosmology with gravitationally lensed Supernovae and Quasars
For the rare cases of nearly perfect alignment between an observer, an intervening galaxy and a background source, multiple images of a single source can be detected, a phenomenon known as strong gravitational lensing. Multiple images of lensed sources arrive at different times because they travel along different paths and through different gravitational potentials to reach us. For transient phenomena like supernovae (SNe) or quasars (QSOs), strong lensing offers exciting opportunities to directly measure time-delays between the images, which can be used to study the distribution matter in the lensing object and to measure the Hubble constant, H0, which is currently the most hotly contested parameter in cosmology.
The first strongly lensed Type Ia supernova (SN Ia) was recently discovered (iPTF16geu , Goobar et al. 2017), and ZTF is well-suited to search for more of these rare transient phenomena. Gravitationally lensed SNe Ia are particularly interesting due to their “standard candle" nature, i.e., all explosions have nearly identical peak luminosity, intrinsic colors and lightcurve shapes, making them ideal tools for magnification and time-delay measurements, as well as probes of the lensing matter distribution.
Related thesis projects: Implement Machine Learning techniques to detect gravitationally lensed SNe in ZTF; Measure time delays from multiply imaged QSO’s in ZTF; Time delays from simulated spectral time series of Supernovae, see e.g. Johansson et al. 2021. (Contacts: Ana Sagués Carracedo, Remy Joseph, Joel Johansson)
UV data of Superluminous Supernovae
The era of large-scale time-domain astronomical surveys has arrived. Every night, modern all-sky surveys detect hundreds of thousands of extragalactic transients. In less than 5 years, the Vera Rubin Observatory will increase the nightly discovery rate by a factor of 10. It will also push large-scale time-domain astronomy to the young high-redshift Universe. As we look further back in time (=higher redshift), telescopes do not observe the optical but the UV emission of SNe, which got redshifted by the expansion of the Universe. However, little is known about the UV emission of SNe and, therefore, how SNe at high-redshift could look like. In the past years, we have collected UV data of a particular SN class, namely superluminous supernovae (SLSNe). SLSNe are 100-times more luminous than regular core-collapse SNe and Type Ia SNe. They have been a focus of SN science ever since, because of the opportunity they provide to study, for instance, new explosion channels of very massive stars in the distant Universe. A master student will use the UV data of SLSNe to predict the light curves of high-redshift SLSNe and compare them to those of known high-redshift SLSNe. (Contact: Steve Schulze) |
ee799ffaf2b287be | In the book Many-Particle Physics by Gerald D. Mahan, he points out that the Schrödinger equation in the form
$$i\hbar\frac{\partial\psi}{\partial t}~=~\Big[-\frac{\hbar^2\nabla^2}{2m}+U(\textbf{r})\Big]\psi(\textbf{r},t)\tag{1.93}$$
can be obtained as the Euler-Lagrange equation corresponding to a Lagrangian density of the form
I have a discomfort with this derivation. As far as I know a Lagrangian is a classical object. Is it justified in constructing a Lagrangian that has $\hbar$ built into it?
• 1
$\begingroup$ This really seems to be deriving a classical field equation which happens to look like the Schrödinger equation for a quantum particle. The interpretations are completely different. $\endgroup$
– knzhou
Feb 10, 2018 at 10:28
• $\begingroup$ Related : The Lagrangian Density of the Schroedinger equation. $\endgroup$
– Frobenius
Sep 24, 2021 at 13:17
2 Answers 2
1. As user JamalS correctly points out in his answer:
2. However, perhaps OP's discomfort with Mahan's TDSE derivation is spurred by the following deeper question:
How we can get the correct semiclassical limit$^1$ and loop expansion$^2$ of a second-quantized path integral$^3$ $$Z~=~\int\! {\cal D}\frac{\psi_2}{\sqrt{\hbar}}{\cal D}\frac{\psi_2^{\ast}}{\sqrt{\hbar}} ~\exp\left(\frac{i}{\hbar} S\right),\tag{1}$$ if the Schroedinger action $S$ depends on $\hbar$, so that various parts of the actions $S$ scales/are suppressed inhomogeneously in the semiclassical limit $\hbar\to 0$?
That's a good question. The answer is that there are implicit hidden $\hbar$-dependence, i.e. one should rescale the variables $$\psi~=~\frac{\psi_2}{\sqrt{\hbar}},\qquad m~=~\hbar m_2,\qquad U~=~\hbar U_2,\tag{2}$$ to obtain a classical ($\hbar$-independent) action $$\begin{align} S~=~&\int \! \mathrm{d}t ~\mathrm{d}^3r \left( i\hbar \psi^{\ast}\dot{\psi}-\frac{\hbar^2}{2m} |\nabla\psi|^2 -U|\psi|^2 \right)\cr ~\stackrel{(2)}{=}~&\int \! \mathrm{d}t ~\mathrm{d}^3r \left( i \psi_2^{\ast}\dot{\psi}_2-\frac{1}{2m_2} |\nabla\psi_2|^2 -U_2|\psi_2|^2 \right) ,\end{align}\tag{3}$$ and to restore a correction loop expansion.
$^1$ For the semiclassical limit, see e.g. this Phys.SE post.
$^2$ For the $\hbar$/loop-expansion, see e.g. this Phys.SE post.
$^3$ Here the subscript 2 refers to a properly normalized second-quantized formulation.
• $\begingroup$ Notes for later: The coupling constant $\frac{1}{m}$ has negative mass dimension, and hence corresponds to a non-renormalizable coupling, cf. Schwartz, section 22.1, p. 395. $\endgroup$
– Qmechanic
Feb 26 at 18:22
Firstly, one may think of this as a mathematical rather than physical procedure. In the end one is simply constructing a functional,
$$S = \int \mathrm dt \, L$$
whose extremisation, $\delta S = 0$ leads to the Schrodinger equation. However, Lagrangians containing $\hbar$ are not uncommon. In quantum field theory, one can construct effective actions from computing Feynman diagrams, which may have factors of $\hbar$, outside of natural units.
Your Answer
|
14a95fc888c62fd3 | Вы находитесь на странице: 1из 13
nanoMOS 2.5: A two-dimensional simulator for quantum transport in double-
gate MOSFETs
Article in IEEE Transactions on Electron Devices · October 2003
DOI: 10.1109/TED.2003.816524 · Source: IEEE Xplore
241 612
5 authors, including:
Zhibin Ren Rachana Venugopal
IBM CMR Institute of Technology
Sebastien Goasguen M.s. Lundstrom
Clemson University Purdue University
Scientific Software Development for nanoHUB View project
All content following this page was uploaded by M.s. Lundstrom on 21 May 2013.
The user has requested enhancement of the downloaded file.
nanoMOS 2.5: A Two-Dimensional Simulator for
Quantum Transport in Double-Gate MOSFETs
Zhibin Ren, Ramesh Venugopal, Sebastien Goasguen, Member, IEEE, Supriyo Datta, Fellow, IEEE, and
Mark S. Lundstrom, Fellow, IEEE
Abstract—A program to numerically simulate quantum trans- phase [17]. The method has been developed into a practical tool
port in double gate metal oxide semiconductor field effect transis- for simulating one-dimensional resonant tunneling diodes [14],
tors (MOSFETs) is described. The program uses a Green’s func-
and more recently, it has been extended to two-dimensional
tion approach and a simple treatment of scattering based on the
idea of so-called Büttiker probes. The double gate device geometry bulk MOSFETs [13], [31]. It is also now widely used to explore
permits an efficient mode space approach that dramatically lowers conduction at the mesoscopic [6] and molecular scales [32]. In
the computational burden and permits use as a design tool. Also comparison to phenomenological quantum approaches [24],
implemented for comparison are a ballistic solution of the Boltz- [29], the Green’s function method offers the advantage of rigor,
mann transport equation and the drift-diffusion approaches. The
program is described and some examples of the use of nanoMOS but the cost can be a heavy computational burden. The simple
for 10 nm double gate MOSFETs are presented. geometry of the fully-depleted SOI MOSFET (single or double
gate) reduces this computational burden. It also permits the
Index Terms—Boltzmann transport equation, Büttiker probes,
double gate, drift-diffusion, MOSFETs, nanoMOS, quantum use of a mode space approach that, while not suitable for bulk
transport, scattering. MOSFETs, dramatically lowers the computational time for
ultra-thin-body SOI MOSFETs [35].
For ballistic transport, the NEGF formalism is equivalent to
solving the Schrödinger equation, but the NEGF formalism is
A S METAL oxide semiconductor field effect transistor
(MOSFET) channel lengths rapidly shrink below 100 nm,
current research focuses on understanding device physics and
readily extendible to two- and three-dimensions or to the use of
molecular orbitals [22]. It also provides a clear prescription for
treating large contacts attached to an intrinsic device [28]. The
ultimate scaling limits as well as the practical issues that need NEGF formalism also provides a sound conceptual basis upon
to be addressed to achieve channel lengths at the 10 nm scale which additional physics (such as scattering) can be included
[1]. Computational studies can help address these issues, but as needed. See [9], [23] for a tutorial introduction to the NEGF
the methods commonly used in computer-aided design tools method and [7] for an example of its application to conduction
do not include (or do so only phenomenologically) important in molecules.
quantum mechanical effects. What is needed is a full quantum This paper describes nanoMOS2.5, a simulation program that
transport model—one that describes quantum phenomena uses the NEGF approach to simulate fully-depleted, nanoscale,
(especially confinement and tunneling) as well as phase ran- SOI MOSFETs. It begins in Section II with a description of
domizing scattering. Computational efficiency is necessary to the methods focusing on the mode space approach. Section III
permit use for engineering design and to explore numerous
discusses some questions relating to the numerical implementa-
design options. It would be useful for such a program to be
tion. In Section IV, we describe a phenomenological treatment
based on an approach that can be extended to nonconventional
of scattering, and in Section V some results are presented.
devices, such as carbon nanotube FETs [2], [10], [20] and
other molecular transistors. In this paper, we describe such a
computer simulation tool, nanoMOS2.5. II. APPROACH
The program, nanoMOS, uses the nonequilibrium Green’s In this section, we describe the approach used in nanoMOS
function (NEGF) approach, which provides a rigorous de- from a wavefunction perspective, because ultra-thin body DG
scription of quantum transport and interactions that randomize MOSFETs can be treated by a one–dimensional (1-D) approach,
and for 1-D quantum transport with the simple scattering
Manuscript received August 29, 2002; revised June 11, 2003. This work model currently implemented in nanoMOS, the wavefunction
was supported by the Semiconductor Research Corporation under Contract
NJ-99-724 and an ARO Defense University Research Initiatives in Nanotech-
and NEGF approaches are equivalent. We use a wavefunction
nology (DURINT) Grant. The review of this paper was arranged by Editor H. description because it is more familiar to readers of this journal,
Sakaki. but in the Appendix, we translate the key equations into the
Z. Ren is with IBM, Yorktown Heights, NY 10598 USA (e-mail: zhibinr@
us.ibm.com). language of the NEGF formalism. Readers are referred to a
S. Goasguen, S. Datta, and M. S. Lundstrom are with Purdue University, West tutorial introduction [9] or to standard references [15], [18] for
Lafayette, IN 47907-1285 USA. a more extensive discussion of the NEGF approach.
R. Venugopal is with Texas Instruments, Dallas, TX 75235 USA (e-mail:
[email protected]). Fig. 1 sketches the double gate MOSFET geometry that we
Digital Object Identifier 10.1109/TED.2003.816524 assume; we seek a solution for the intrinsic device only. Because
0018-9383/03$17.00 © 2003 IEEE
Fig. 1. Double gate MOSFET structure examined in this work. An oxide
thickness of 1 nm, and a body thickness of 3 nm have been used to highlight
quantum effects within this device geometry. The power supply is 0.4 V and
the gate work function is adjusted to obtain an I (10 A/m) consistent
with the ITRS specifications for a high performance transistor. The S/D doping
is 2 10 cm and the transistor is assumed to be wide (Y-dimension is
Fig. 2. Sketch of the generic subband energy versus position along the channel
treated as infinite).
(x). Also shown are the semi-infinite source-drain contacts (bounded by open
rectangles) and the boundary conditions for injection of a unit amplitude from
we assume that the device is wide in the -direction, the wave- the source end. The nodes within the device are numbered 1 to N and the active
device extends from x = 0 (source) to x = L (drain).
function can be written as ,
where is obtained from
where is the expansion coefficient of with re-
spect to the mode-space eigenvector, .
(1) Equation (4) is the decoupled mode space transformation
of (1); the key assumption is that the shape of the mode does
where is the longitudinal energy. One could directly dis- not change rapidly along . Although the potential, ,
cretize (1) in real space, which is often necessary, but it imposes varies from source to drain, the largest variation is with position,
a heavy computational burden [35]. For thin body, SOI MOS- . For a thin body, the shape of the mode (i.e., its -dependence)
FETs, however, quantum confinement in the -direction intro- changes slowly along the channel. A careful comparison of
duces subbands, and for a thin body, only a few subbands are the decoupled approach with a direct, real-space discretization
occupied. Accordingly, we expand the wavefunction in an or- shows that the decoupled approach produces results that are
thonormal basis as [35] essentially identical to the exact solution for body thicknesses
of up to at least 5 nm [35]. There are, however, conditions for
(2) which the decoupled approach fails [35]. One example is the
transition from the flared out contact to the ultra-thin channel
where are the eigenfunctions (modes) associated (which is not treated in nanoMOS), and another is the bulk
with confinement in the -direction as defined in Fig. 1. These MOSFET.
eigenfunctions and the associated eigenenergies are obtained Equation (4) is a 1-D wave equation that greatly reduces the
by solving a one-dimensional wave equation in the -direction size of the two-dimensional (2-D) problem. In the ballistic limit,
within each vertical slice of the device along the dimension each mode (subband) can be treated separately. Each transverse
mode is also independent, because we assumed there is no po-
tential variation in the -direction. In this 1-D equation, instead
of the bottom of the conduction band, , we have the
(3) bottom of the relevant subband, , which is determined
The energy, , represents the bottom of subband, , which by a solution to (3) in the confinement direction. (Accordingly,
varies with position, , along the channel. The envelope wave- when we plot energy band diagrams, we will plot rather
functions are assumed to be zero at the oxide/Si interfaces if than the bottom of the conduction band, ). We also see
electron penetration into the oxide regions is neglected (other- from (4) that the relevant energy for the 1-D problem is not the
wise, the zero boundary is extended to the gate contact/oxide total energy, , but the longitudinal energy, .
interfaces). Because the decoupled mode space solution reduces the 2-D
By using the orthonormal basis of (2), we can transform (1) problem to a set of 1-D problems, one for each subband and
to a mode-space basis. By retaining only a few occupied modes, transverse energy, the solution procedure is much like that of a
the computational burden can be significantly reduced com- true 1-D problem. Fig. 2 is a sketch of the subband energy versus
pared to a direct discretization in two-dimensional real space position for one subband. Semi-infinite contacts are attached to
[8]. The geometry of the double gate MOSFET offers yet an- the device at the source and drain ends. Because the potential
other simplification. If we assume that the shape of the confined in the contacts is assumed to be uniform, the solutions in the
mode does not change along , i.e., then (2) semi-infinite contacts are plane waves. If a unit amplitude wave
becomes [26], [35] is injected from the left (source) contact, then some portion re-
flects from the device and some transmits across and exits the
perfectly absorbing right (drain) contact
(4) (5a)
and where is the local density of states due to injection
(5b) from the drain, which is computed from the wavefunction with
boundary conditions analogous to (5a) and (5b). Note that
where and are the reflection and transmission coefficients in deriving (9), an equilibrium Fermi–Dirac distribution was
for source injection into mode and is the length of the active assumed to prevail within the S/D contacts only. No assumption
device. was made on the shape of the distribution function within the
By solving (4) subject to the boundary conditions, (5a) and active device. In the ballistic case, the source and drain injected
(5b), we find the wavefunction due to the injection of a unit states are independent, and can be filled by the appropriate
amplitude wave from the source. The corresponding electron Fermi functions as shown in (9). Equation (9) clearly indicates
density versus position for confined mode, , at each , is that under nonequilibrium conditions (the source Fermi level
obtained by summing the contributions from each transverse is different from the drain Fermi level), the net 2-D electron
mode distribution within the device, which is a combination of source
and drain injected streams, cannot be characterized by a single
(6) Fermi level. Its shape is not equilibrium Fermi Dirac.
To obtain the current due to source injection into mode , at
where refers to the -component of the wavevector of an each we evaluate
electron with total energy, , in the source contact and the sub- (10)
script, 1, refers to injection from the first (left or source) contact.
The probability that the state at energy, , in the source contact
is occupied is give by the source Fermi function because we as- where is the current transmission coefficient from contact
sume that scattering maintains thermal equilibrium in the con- 1 to contact 2 for electrons in subband, . The net current due
tacts. to mode is obtained by summing (10) over all the positive
The 2-D charge density for mode at site is obtained by states and by subtracting the corresponding drain injected
summing (6) over all of the positive states (negative states component. On converting the sums over the ’s to integrals
are not injected into the device from the source). Since the width in (10), and by performing the integral over transverse energy
in the transverse direction, , is assumed to be large, we can analytically we find
convert the sum over transverse states to an integral over the
transverse energy. We can also convert the sum over the injected (11a)
states to an integral over longitudinal energy. The integral
over transverse energy can be done analytically and the final where
result expressed as [26]
(7a) (11b)
and is the Fermi function which is analytically integrated over
where we let the top of the band approach infinity and transverse energy (spin degeneracy is included in ). The total
current is obtained by summing the contributions from each sub-
F band and valley.
Finally, the current transmission coefficient for mode,
(7b) is
with (12)
(8) which, from (5a) can be expressed in terms of the computed
wavefunction as
being the local density of states due to injection from the source (13)
(spin degeneracy is included in the square root factor). In (7b)
F is the Fermi–Dirac integral of order [3] and F is
the Fermi function integrated over the transverse energy. To find III. NUMERICAL SOLUTION
the total electron density within the device, we sum the contribu- A. Solving for the Wavefunction
tions from each subband and valley and include the contribution
due to injection from the drain To evaluate the expression for the electron density and current
in the previous section, the wavefunction within the device must
be known. In the confinement direction, (3) is a standard eigen-
value problem solved by finite differences. In the longitudinal
direction, we discretize (4) on a finite difference grid imposing
the boundary conditions, (5a) and (5b), to find
where is the discretized Hamiltonian operator where is the three-dimensional (3-D) effective density-of-
states, and the function, , is the Fermi–Dirac integral of
order . We regard (22) as a mathematical change of variables
with an on-site energy of to a quantity, , whose physical significance under off-equi-
librium conditions is unclear, but unimportant. When (22) is
(16) inserted into (21), a nonlinear Poisson equation results. After
computing from the quantum transport model, a cor-
for a finite difference grid with node spacing, . The energy, responding is computed from (22). This “quasi-Fermi
, in (15) is the effective potential energy for electrons in level” is used in a nonlinear Poisson equation that is obtained by
mode . The “self-energy” matrices, and account inserting (22) in (21) and solved by Newton’s method. The ad-
for the open boundary conditions [(5a) and (5b)] and are vantage of this approach is that it builds a negative feedback into
the iterative process. If the potential increases (conduction band
decreases) during the Poission solution, the subsequent trans-
(17b) port solution will increase as carriers flow to regions of
The vector, is a source term accounting for injection lower energy. This coupling is built into the Poisson equation
from the left contact (source). It has only one nonzero compo- when (22) is used. The approach has proven effective in pre-
nent, the first vious quantum and semiclassical transport simulations [5], [34]
and proved similarly effective here.
(18) C. Boundary Conditions
Boundary conditions must be specified for both the transport
For injection from the drain, we use boundary conditions anal- equation and for Poisson’s equation. For Poisson’s equation,
ogous to (5a) and (5b), and the corresponding vector has a the boundary conditions at the gate electrodes (Dirichlet) and
nonzero component in position, . at the oxide/air interfaces (Neumann) are standard. Boundary
The solution to (14) gives the value of the wavefunction, , conditions at the interface between the S/D extensions and the
at each of the finite difference nodes. The formal solution is flared out contacts are also needed. A more complete solution
(19) would include the flared out region to explore the resistive drops
that may occurs at the wide/narrow transition. In this work, we
where seek simple, upper limit boundary conditions that define our
(20) “ideal” contacts.
External to the intrinsic device being simulated, we assume
is the retarded Green’s function in a discrete basis. Because large source and drain contacts where scattering maintains
the matrix in (14) is tridiagonal, it can be efficiently solved by thermal equilibrium. We solve the wave equation assuming
Gaussian elimination; the carrier density and current are then a unit amplitude injected wave, then weight by the Fermi
evaluated from the computed wavefunctions. Alternatively, as function of the appropriate contact. The Fermi levels in the
shown in the Appendix, we can express all of the results in expressions presented in Section II refer to the Fermi levels
terms of the retarded Green’s function. For this simple, in the equilibrium source and drain contacts that are external
1-D problem, the Green’s function approach may appear to be to the region being simulated. The self-energy matrices are
a complicated way to solve a simple problem. The advantage of expressed in terms of the wavevector, , in the contact from
the NEGF formalism become apparent when extensions to two which the electrons are injected. For a uniform contact with
and three-dimensions are contemplated, when an atomic basis simple bands
is essential, or when a rigorous treatment of scattering is neces-
sary. (23a)
B. Poisson’s Equation but the discrete grid modifies this relation to
The program, nanoMOS, computes a self-consistent solution (23b)
to a quantum transport equation and Poisson’s equation
where is the grid spacing. The two expressions are nearly
(21) equal when . Care must be taken to ensure that the
maximum energy used is well below the top of the band so that
Equation (7) gives the electron density per unit area within the
(23b) approximates (23a). It should also be recognized that the
device; to convert it to a density per unit volume, it is distributed
maximum energy that can be safely used depends on the grid
in the -direction according to the computed eigenfunction in
spacing, .
the confinement direction. Direct use of (21) leads to slow con-
For ballistic transport, the boundary conditions on Poisson’s
vergence. Instead, we solve a nonlinear Poisson equation. The
equation must be carefully specified. Fig. 3 is a sketch of a
electron density evaluated from the wavefunction can be related
generic subband profile from the source to the drain. Under low
to a quasi-Fermi level by
gate bias and high drain bias, the source-to-channel barrier is
(22) high. Therefore most electrons injected from the source (empty
Fig. 3. Illustration of why floating boundary conditions are assumed for the
potential at the contacts. The empty circles on the source side, below the dotted
line (source to channel barrier) represent source injected electrons reflected by
the barrier. The drain injected electrons are represented by filled circles.
circles in Fig. 3) reflect from the source to channel potential bar-
rier, and both positive and negative velocity states ( states) are
occupied in the source extension as a result of source injec-
tion. Under high gate and drain bias, however, fewer electrons
are reflected from the barrier, and the electron density in the
source extension decreases (there is very little contribution
to the overall electron concentration at the source end due to
drain side injection as the drain voltage is high). If the poten-
tial at the source contact is fixed, space-charge neutrality (the
2-D donor doping concentration equals the 2-D electron density
obtained by integrating along ), which is a result of self-con- Fig. 4. Summary of the ballistic solution scheme.
sistent electrostatics, cannot be maintained. (Note that the loca-
tion is the Fermi level is fixed by the large, thermal equilibrium 2) For each position, , along the channel, (3) is solved to
source reservoir.) Therefore, in order to maintain space-charge find the eigenfunctions and eigenenergies versus position.
neutrality in the source/drain (S/D) extensions, the electrostatic A grid spacing of in the -direction is typical.
potential floats relative to the fixed Fermi potentials in the S/D 3) A grid in longitudinal energy is defined (with a typical
contacts. We allow for this to happen by specifying grid spacing of 0.5 meV) and (4) is solved to find the
wavefunction due to injection of a unit amplitude wave at
(24) longitudinal energy, , from both the source and drain
contacts. The electron density for each mode is evalu-
ated from (7b). This step is repeated for each longitudinal
at the idealized source and drain contacts. This boundary con-
energy in the grid and for each occupied confined mode
dition causes the potential at the source end to float down, thus
(as set by the user). The total carrier density is obtained
allowing more carriers to enter into the device from the source
by summing the contributions from each longitudinal en-
reservoir. This allows macroscopic space-charge neutrality to be
ergy, confined mode, and valley as in (9).
maintained (Fig. 8) at the source end irrespective of the biasing
4) The nonlinear Poisson equation is solved to update the
condition. The floating boundary condition eliminates the need
electrostatic potential. The maximum change in potential
to resolve the potential at the point where the thin body couples
is compared to the convergence criteria, and the process
to the large S/D. It represents an ideal upper limit contact. In
continues until convergence is achieved. The minimum
practice, the transition from the flared out S/D to the ultra-thin
change in potential that can be achieved is essentially the
intrinsic device could introduce quantum mechanical parasitic
spacing in the energy grid.
resistances that are not treated within nanoMOS. The validity
5) After convergence is achieved, the transmission coeffi-
of this approach is supported by the results that will be shown
cient and current at each energy are evaluated from (13).
in Section V.
The contributions from each energy are summed to find
the total current from (11). The process is repeated for
D. Solution Procedure each mode, and the results summed to compute the total
The solution procedure is summarized in Fig. 4; it consists of current.
the following steps. Although we have described the solution process from a wave-
1) An initial guess for the 2-D electrostatic potential is de- function perspective, because this approach may be more fa-
fined. (We use a drift-diffusion simulation as our initial miliar to readers of this journal, the actual equations solved in
guess.) nanoMOS are expressed in the Green’s function formalism as
described in the Appendix. The Green’s function formalism is
a generalization of the concepts outlined this far, which can be
naturally extended to model devices in two or three dimensions,
including the effects of scattering.
The NEGF formalism provides a prescription for including
scattering, but scattering greatly increases the computational
burden because it couples the longitudinal and transverse en-
ergies. Instead of treating transverse modes as independent and
integrating over them analytically, we need a grid in transverse
Fig. 5. Illustration of the concept of Büttiker probes. The probe self-energy is
energy too. And instead of treating each longitudinal energy in- adjusted to tune scattering rates and the probe Fermi-levels to obtain zero net
dependently, they must be coupled. Scattering has been treated current at each scatterer.
rigorously in NEGF simulations of MOSFETs [30], [36], but
the resulting computational burden limits the use of such simu- them into the device. Following the prescription for filling states
lations. The advantage of the NEGF approach lies in its ability from the two real contacts, (9), we have
to treat quantum transport using a atomic level Hamiltonian, if
necessary, but we also need a simple way to capture the main
effects of scattering.
We treat scattering in a way that is analogous to the well-
known relaxation time approximation to the collision operator
for the Boltzmann equation
The spectral density, , gives the local density-of-
which describes carriers out-scattering from a state at a rate,
states everywhere along the channel due to injection from the
, and in-scattering from a thermal equilibrium distribution.
probe located at . We have assumed a single scatterer at
(Under nonequilibrium conditions, the magnitude of the in-scat-
with a Fermi level, , but in practice, probes are placed at
tering rate is adjusted to preserve current continuity.) An analo-
each node and each one has a different Fermi level. The electron
gous treatment for the wave equation can be implemented with
density at due to injection from all reservoirs (the source and
an idea due to Büttiker [4]. As shown in Fig. 5, we conceptu-
drain contacts as well as the floating probes used to introduce
ally attach a floating contact to each node. Carriers are removed
scattering) is
from the device and injected into the floating contact where they
are thermalized and re-injected into the device. (29)
Since each scattering center is phenomenologically treated
as an additional contact, nanoMOS models the effect of out-
where the sum runs over all the reservoirs, each with its own
scattering by adding an additional potential to the Schrödinger
Fermi level. The source and drain Fermi levels are fixed by the
equation [9]
applied source-drain bias; the Fermi level of each probe is de-
termined by requiring that current is conserved. The current in
(26) subband, , at reservoir, , is given by an expression analogous
to (11b)
Note that the scattering potential is similar to the “self-energies”
[(17a) and (17b)] used to account for the open boundary condi-
tions [(5a) and (5b)], and that the coupling energy between the
device and the scatterer, , can be smoothly varied to mimic (30)
specific low-field mobilities as described in [36]. Our scattering where the sum over is over the remaining reservoirs. The trans-
model is phenomenological and analogous to the relaxation time mission coefficient between any two reservoirs is computed in a
approximation. It captures the cumulative effects of all types manner analogous to (12). The source and drain contacts carry
of scattering, including phonon scattering. When scattering is current, but the Büttiker probes do not, so for Büttiker probe,
present, the retarded Green’s function in (20) becomes
(27) (31)
Equation (26) does a reasonable job of describing scattering Equation (31) gives a set of constraints on the Fermi levels
in MOSFETs if the scattering strength is calibrated to an ap- of each probe. This set of constraining equations is solved it-
propriate mobility. The diagonal self-energy results because the eratively for the Fermi level of the probe [36]. Note that the
scattering potential is assumed to be a -function at each node. resulting position dependent Fermi level behaves in much the
We have described the process of out-scattering electrons to same way as the quasi-Fermi level in conventional semicon-
a probe where they are thermalized. The next step is to re-inject ductor theory and that the re-injection process (in-scattering)
Fig. 6. Conduction and first two subband minima, as a function of position in Fig. 7. Total 3-D electron density, n(x; z ), in the on-state. The thin silicon
the on-state (V = V = 0:4 V). Note that the conduction band is a function body is volume inverted, and the electron density goes to zero at the top and
bottom oxide/silicon interface (0, 3 nm). Quantum effects due to confinement
of both x and z , while the subband minima are functions of x alone. The lower
subband minima is for the unprimed valleys with two fold degeneracy, while the are accurately captured by nanoMOS.
higher minima is for the primed valleys with four fold degeneracy.
from the probes to the device mixes different subband popula-
tions, thus capturing the effects of inter subband and inter valley
Several studies of nanoscale MOSFET device physics and de-
sign issues that make use of nanoMOS simulations have already
been published [25]–[27], [19]. Our purpose in this section is
to present some simulation results that illustrate the capabili-
ties of the program. The simulated device (recall Fig. 1) is an
idealized structure for which we do not treat the flared-out S/D
contacts that would be present in an actual device. The n-type
source and drain regions are doped at cm , the channel
is intrinsic, and the S/D junctions are abrupt. No gate-to-S/D Fig. 8. Areal electron density (cm ) along the channel is plotted in the
off-state (V = 0; V = 0:4 V) from both the quantum (solid line) and the
overlap is assumed. The oxide thickness is 1.0 nm for both top classical (dashed line) ballistic transport models at room temperature. Quantum
and bottom gates, and the silicon film thickness is 3.0 nm. The mechanical tunneling through the source-to channel barrier results in a higher
gate workfunction was set to 4.22 eV in order to produce an channel charge density in case of the quantum model. Charge neutrality is
achieved in the S/D regions in both cases.
off-current of A/ m, consistent with the 2016 node of
the International Technology Roadmap for Semiconductors [1].
The power supply is V and the gate length (equals quantum mechanical tunneling into the source-channel barrier
the channel length as the S/D junctions are abrupt) is 10 nm. increases the carrier density in the channel. A careful examina-
Fig. 6 is a plot of the first two subband minima versus po- tion of the figure also reveals that the quantum mechanical car-
sition in the on-state. Also shown is a plot of the conduction rier density just outside the channel is slightly reduced, an effect
band minimum versus position. Note that the conduction band first observed by Willander [37]. Fig. 9 compares the
varies in two dimensions (in the -direction along the channel versus characteristics from ballistic quantum and classical
from the source to the drain as well in the -direction across nanoMOS simulations. As expected, quantum mechanical tun-
the thickness of the silicon body). The subband minima, how- neling increases the off-current, but note that it also decreases
ever, are determined by solving the Schrödinger equation in the the on-current. In the on-state, self-consistent gate electrostatics
direction, and they vary only in the -direction. This uncou- tries to maintain a charge density of , at
pled mode space approach is what allows us to treat transport in the top of the source-to-channel barrier irrespective of the so-
each subband with a 1-D Green’s function approach. lution scheme (quantum or classical) used to simulate the de-
Fig. 7, a plot of the electron density within the device under vice. In quantum simulations, some of this charge is due to
on-state conditions, shows the quantum confinement of carriers tunneling electrons (with channel directed energies below the
in the -direction. The profile varies approximately as source-to-channel barrier). Since these electrons are evanescent,
in the S/D regions, which indicates that most electrons reside they carry less current than their thermionic counterparts. There-
in the first subband (primed and unprimed). Classical Boltz- fore, the on-current from classical simulations is higher than
mann and NEGF simulations are compared in Fig. 8, which that from quantum simulations as shown in Fig. 9. Classically,
shows the integrated electron density (per cm ) versus posi- if one views the on-current as a product of a 2-D electron den-
tion in the off-state ( V). As expected, sity multiplied by a velocity, it looks like the electron velocity
space approach for the -direction, the effect of quantum
confinement on the threshold voltage is included in both
simulations. Fig. 10(a) shows that even for a 10 nm channel
length, MOSFETs are expected to behave classically. Quantum
mechanics increases the threshold voltage and decreases the
on-current at a given threshold voltage, but no quantum oscil-
lations are observed. In Fig. 10(b), we compare the common
source characteristics with and without scattering. The Büttiker
probe strength has been adjusted to obtain a low field mobility
of 50 cm /V-s in the heavily doped source drain regions. In
the channel, a doping dependent mobility model is used to
adjust the probe self-energy. It is clear from Fig. 10(b) that
the on-current is significantly reduced due to scattering in the
Fig. 9. I versus V characteristics for the model device from both the
quantum (solid line) and classical (dashed line) ballistic transport models. The heavily doped S/D extensions. This reduction in the on-current
ballistic off-current is higher from the quantum model due to source-to-channel once scattering is turned on is primarily due to
1) source parasitic resistance which degrades the effective
gate to source voltage;
2) change in the shape of the distribution (toward equi-
librium Fermi–Dirac) due to reflections, which causes
a reduction in the injection velocity at the top of the
source-to-channel barrier.
The presence of a parasitic drain resistance does not degrade the
on-current as severely as the source parasitic resistance because
it only affects the drain to source voltage and not the gate to
source voltage.
The conduction bandstructure for silicon is composed of
three sets of ellipsoids. The first set of ellipsoids with their
longitudinal effective mass oriented along the confinement di-
rection, gives rise to the so-called unprimed series of subbands
(a) and the two other sets of ellipsoids that have their transverse
effective masses oriented along the confinement direction
give rise to the primed series of subbands. These subbands,
with their anisotropic effective masses, are independently
treated within nanoMOS-2.5. In the ballistic limit, there is no
mixing between electrons from different subbands, but once
scattering is turned on, electrons populations from different
subbands mix as a result of in-scattering from Büttiker probes.
The ballistic, on-state source to drain transmission coefficient
versus energy is plotted in Fig. 11. The locations of the first
unprimed (dashed line) and the first primed (dash-dot line)
subband minima are also indicated in Fig. 11. Note that the
net transmission coefficient (Fig. 11) includes contributions
from all subbands (both primed and unprimed), at each energy.
Therefore, the net transmission coefficient begins to increase
Fig. 10. (a) I versus V characteristics for the model device from both below the first subband minimum (due to source-to-channel
the quantum (solid line) and classical (dashed line) ballistic transport models.
The ballistic on-current is lower from the quantum model as compared to the tunneling) then smoothly approaches unity above the first
classical model. (b) The I versus V characteristics for the model device subband minimum, and increases once again at higher energies
from both the quantum ballistic (dashed line) and quantum dissipative (solid increases. At high energies, the net transmission coefficient
line) transport models.
approaches three, because we chose to include one subband
from each conduction band valley for our simulations (valley
(which is derived from the current and the 2-D electron density) degeneracy is not accounted for in Fig. 11).
is lower for quantum simulations when compared to classical It is instructive to look at the floating boundary conditions
simulations, in the on-state. more closely. To explain the floating boundary condition, we
The classical and quantum mechanical ballistic common modify our idealized 3 nm body DG MOSFET structure by ap-
source characteristics are compared in Fig. 10(a). Note that the pending heavily doped ( cm ) regions to the left
term classical or quantum refers to the treatment of transport and right ends of the device. The profile of the first subband from
along the channel (the -direction). Because we use a mode the source to the drain, for the modified device structure under
Fig. 11. Ballistic source-to-drain transmission is plotted as a function of
longitudinal energy (E ) in the on-state. The top of the source-to-channel
barrier for both the unprimed (dashed) and the primed (dashed with dots)
subbands is also indicated. Note that the transmission is nonzero for longi-
tudinal energies below the top of the subband barrier (tunneling) and increases
smoothly to unity. The primed subband transmission goes to two instead of one
due to contributions from two sets of valleys.
Fig. 13. (a) Energy resolved Local Density of States (LDOS) is plotted along
the channel in the on-state from the quantum ballistic transport model. Light
areas imply a low density of states, while dark regions indicate a high density of
states. Coherent oscillations in the LDOS are the result of reflections from the
barrier. Non-zero LDOS in the forbidden region (below the subband energy),
(a) leads to tunneling through the source-to-channel barrier. The LDOS from the
primed (blue line) band is higher than the unprimed (red line) due to degeneracy.
(b) The energy resolved LDOS is plotted along the channel in the on-state from
the quantum diffusive transport model. Coherent oscillations in the LDOS are
washed out as a result of scattering. A small, potential drop in the source and
drain regions is also discernible.
barrier gives rise to strong reflections, which maintain a near
equilibrium distribution in the regions even when a large
bias is applied to the drain and gate. Therefore, a fixed poten-
tial boundary condition based on charge neutrality can be used
when solving Poisson’s equation for this modified device. The
regions maintain a near equilibrium distribution even under
On examining the subband profile [Fig. 12(a)] of the modified
(b) device at high gate bias, it is clear that the subband is unchanged
Fig. 12. (a) Subband profile, E (x), for the device in Fig. 1 with an n in the heavily doped regions, but floats to a lower value in
source extension added under low (solid line) and high (dashed line) gate bias the source region of the intrinsic device. This observation
(V = 0 and 0:6 V) at a drain bias of 0.6 V. The potential (subband) floats can be explained by examining Fig. 3. At low gate biases, both
to a lower energy in the n region, but remains unchanged in the n region.
(b) Subband profile, E (x), for the device in Fig. 1 with the n source the positive and the negative halves of the distribution in the
extension and fixed boundary conditions (dashed line) and without the n source are predominantly filled by the source Fermi level. When
extension and floating boundary conditions (solid line) at a high gate voltage. the gate bias is increased to higher and higher values, the number
of source injected electrons reflected off the source-to-channel
a high drain bias and for a high and low gate bias is ( barrier is reduced because the source-to-channel barrier height
and 0 V, V) is shown in Fig. 12(a). The presence of decreases. Although nearly one half of the distribution is un-
the heavily doped regions, creates a large potential barrier occupied at the source, 2-D electrostatics requires that charge
for electrons injected from the source and drain contacts. This neutrality has to be maintained (integrated doping equals the
drain bias, however, confinement is lost near the drain and nu-
merous modes much be treated. This makes a real-space dis-
cretization more suitable [13], [31]. To treat the flared our con-
tact in a FD UTB SOI MOSFET, mode coupling must also be
included. The transition from the thick contact to the thin body
could introduce resistance. For this problem one could use a
coupled mode space approach [8] or a real-space discretization.
The treatment of scattering in nanoMOS is a simple, phe-
nomenological one. A well-defined prescription for including
scattering exists, but the computational burden rapidly gets out
of hand if one resolves scattering between transverse modes
[36]. When using a phenomenological approach, one must
be careful to mimic the correct physics. Our use of Büttiker
probes that relax the energy of scattered carriers mimics the
Fig. 14. Energy resolved electron density is plotted along the channel in the
on-state from the quantum ballistic transport model. In the heavily doped S/D relaxation of longitudinal energy that occurs in MOSFETs
regions, multiple subbands are occupied. However much of the current in the [19]. The strength of the NEGF approach lies in its ability to
channel is from the first subband. The ballistic source injected charge that treat quantum confinement, reflection, and tunneling and our
propagates to the drain without any energy relaxation is clearly observed.
phenomenological approach provides nanoMOS with a way
to include the first order effects of scattering. Monte Carlo
2-D hole density) at every point within the source. To achieve simulations can include a much more detailed treatment of
charge neutrality, the electron density residing in the positive scattering, but it is necessary to include quantum effects phe-
velocity states nearly doubles between the low and high gate nomenologically [33], [29]. The two approaches complement
bias conditions. To accomodate this increased charge the sub- each other; the NEGF approach is preferable when quantum
band (potential) floats to a lower energy as seen in Fig. 12(a) transport is the first order issue and Monte Carlo simulation
(dotted line). when scattering is the key issue.
Carriers injected from the heavily doped region are pre- The nanoMOS program was written in a scripting language
dominantly backscattered by the built in barrier, so the poten- (Matlab) to permit rapid development and debugging. Since
tial in this region is unchanged with gate bias. Now, if we re- much of the computation occurs in compiled matrix routines,
move the regions but use the floating boundary condition the performance penalty is slight. For larger problems, tool-
to terminate the regions, we observe the potential behavior boxes to parallelize the scripts are being developed [11]. When
plotted in Fig. 12(b) (solid line). Fig. 12(b), demonstrates that the time comes for a production CAD program, nanoMOS may
the subband under bias for the device with floating boundaries have to be re-written in C or Fortran, but the scripting language
( - - ) is identical to that of the device with fixed boundaries approach greatly facilitates program development and provides
( - -- - ) within the region of interest ( - - ). adequate performance for use as a research tool.
This indicates that the floating boundary condition does capture
the effect of coupling a ballistic device to a scattering contact.
The energy-resolved local density of states (LDOS) under
ballistic on-state conditions is plotted in Fig. 13(a). The states The program, nanoMOS 2.5, simulates quantum transport
below the barrier are due to tunneling, and the strong influence in fully-depleted, ultra-thin-body SOI MOSFETs with an effi-
of quantum mechanical reflections is apparent. The LDOS for ciency that permits its use on a workstation. We have described
the higher (primed) subband is larger (darker in the grayscale the methods and approaches used in the program as well as the
plot) because of the four-fold degeneracy of the conduction band simplifying approximations that deliver its computational effi-
valleys. Fig. 13(b) shows the same plot in the presence of scat- ciency. The NEGF approach used in nanoMOS provides a solid
tering, which is seen to reduce the contrast of the interference base for simulating electronic devices at the nanoscale which
pattern. Finally, Fig. 14 shows the energy resolved ballistic elec- can be extended to increase the fidelity of the physics (e.g., im-
tron density which results in filling up the LDOS according to proving the bandstructure and the treatment of scattering) or to
the Fermi levels in the source and drain contacts. Carrier tun- treat much different devices such as carbon nanotube transis-
neling below the source barrier is observed as well as oscilla- tors or molecular conductors. The nanoMOS program is a step
tions in the source and drain, which are due to reflections from toward a new generation of simulation tools that will allow de-
the barrier. vice engineers to explore new classes of electronic devices. The
program is available for use through the WWW or for access to
VI. DISCUSSION its source code [21].
Although the nanoMOS program has proven to be a useful
tool for exploring device physics and design, it is necessarily APPENDIX
based on a number of simplifying assumptions. The mode space GREEN’S FUNCTION FORMULATION
approach is particularly efficient for fully-depleted (FD), ultra- Although we described nanoMOS from a wave function per-
thin-body (UTB), SOI MOSFETs, for which only a few, uncou- spective in the text, nanoMOS is actually based on a Green’s
pled modes need to be treated. In a bulk MOSFET under high function formulation of the same problem. The nonequilibrium
Green’s function formalism offers advantages when more com- The velocities at the injection and exit points are
plicated geometries or basis functions are considered and when
scattering needs to be included in a more rigorous manner. In
this Appendix we translate the wavefunction approach of the
text into the NEGF formalism. Several references provide a (A8a)
fuller discussion of the method [9].
To simulate a MOSFET, we need to compute the electron
density and the current. In the wavefunction picture, (9), we
need at each node, where is determined by the (A8b)
Greens function according to (19). To get , consider
the generalization Using these results, we find
(A1) (A9)
where Since
.. (A2)
we have
is an vector giving the value of the wavefunction at each
node, and the superscript, , denotes the Hermetian transpose. which can be expressed as
The matrix product in (A2) is
.. .. (A3) To summarize, after computing the Green’s function, we find
. . the transmission coefficient from (A12), which is then used with
(11b) to find the current. Note that (A12) can be generalized to
By defining an matrix find the transmission between any two probes as
.. (A4) which is useful when Büttiker probes are used to include scat-
. . tering.
[where is the self-energy describing the connection to the
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[14] Standard (2003). [Online]. Available: http://www-hpc.jpl.nasa.gov/ Ramesh Venugopal was born in Chennai, India. received the Ph.D. degree in
PEP/gekco/nemo/nemo.html electrical and computer engineering from Purdue University, West Lafayette,
[15] L. V. Keldysh, “Diagram technique for nonequilibrium processes,” Sov. IN, in 2003.
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an ultrashort channel single-gated n-MOSFET on SOI,” IEEE Trans.
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6670, 1992. Sebastien Goasguen (S’99–M’01) was born in
[18] D. C. Langreth, “Linear and nonlinear electron transport in solids,” in Rennes, France, on March 22, 1974. He received
NATO Advanced Study Instruction Series B. New York: Plenum, 1976, the B.S. degree in electrical engineering from
vol. 17, p. 3. the Polytechnic Institute of Toulouse, France, in
[19] M. S. Lundstrom and Z. Ren, “Essential physics of nanoscale MOS- 1997, the M.S. degree (with honors) in electronics
FETs,” IEEE Trans. Electron Devices, vol. 49, p. 133, Jan. 2002. research, from King’s College of London, London,
[20] P. L. McEuen, M. S. Fuhrer, and H. Park, “Single-walled carbon nan- U.K., in 1998, and the Ph.D. degree in electrical
otube electronics,” IEEE Trans. Nanotechnol., vol. 1, p. 78, Mar. 2002. engineering from Arizona State University, Tucson,
[21] Nanotechnology Simulation Hub (2003). [Online]. Available: in 2001.
http://www.nanohub.purdue.edu His main area of interest was global modeling of
[22] M. Nekovee, B. Geurts, H. M. J. Bootsand, and M. F. H. Schuurmans, microwave active circuits including the use of neural
“Failure of extended-moment-equation approaches to describe ballistic networks and wavelet based numerical methods. In September 2001, he joined
transport in submicrometer structures,” Phys. Rev. B, vol. 45, p. 6643, Purdue University, West Lafayette, IN, as a Post-Doctoral Research Associate
1992. and became a Visiting Professor in August 2002. Since September 2002, he has
[23] M. Paulsson, F. Zahid, and S. Datta, “Resistance of a molecule,” Tech. acted as Technical Director for the Network for Computational Nanotechnology,
Rep., www.arxiv.org/abs/cond-mat/0 208 183, 2002. in charge of high-performance computing solution to nanotechnology compu-
[24] C. S. Rafferty, B. Biegel, Z. Yu, M. G. Acona, J. Bude, and R. W. Dutton, tational challenges and the implementation of a cyber-infrastructure.
“Multidimensional quantum effects simulation using a density gradient
model and script level programming technique,” in Proc. SISPAD’98,
1998, p. 137.
[25] Z. Ren, R. Venugopal, S. Datta, M. S. Lundstrom, D. Jovanovic, and J.
G. Fossum, “The ballistic nanotransistor: A simulation study,” in IEDM Supriyo Datta (F’96) was born on February 2, 1954.
Tech. Dig., 2000, p. 715. He received the B.Tech. degree from the Indian Insti-
[26] Z. Ren, “Nanoscale MOSFETs: Physics, simulation, and design,” Ph.D. tute of Technology, Kharagpur, in 1975 and the Ph.D.
dissertation, Purdue Univ., West Lafayette, IN, Dec. 2001. degree from the Univeristy of Illinois, Urbana-Cham-
[27] Z. Ren, R. Venugopal, S. Datta, and M. S. Lundstrom, “Examination of paign, in 1979.
design and manufacturing issues in a 10 nm double gate MOSFET using In 1981, he joined Purdue University, West
nonequilibrium Green’s function simulation,” in IEDM Tech. Dig., Dec. Lafayette, IN, where he is currently the Thomas
3–5, 2001, p. 107. Duncan Distinguished Professor in the School of
[28] J. H. Rhew and M. S. Lundstrom, “Benchmarking macroscopic transport Electrical and Computer Engineering. He is the
models for nanotransistor TCAD,” J. Computat. Electron., 2002. author of Surface Acoustic Wave Devices (Engle-
[29] L. Shifren, A. Akis, and D. K. Ferry, “Correspondence between quantum wood Cliffs, NJ: Prentice-Hall, 1986), Quantum
and classical motion: Comparing bohmian mechanics with a smoothed Phenomena (Reading, MA: Addison-Wesley, 1989), and Electronic Transport
effective potential,” Phys. Lett. A, vol. 274, p. 75, 2000. in Mesoscopic Systems (Cambridge U.K.: Cambridge, 1995). His current
[30] A. Svizhenko, M. Anantram, and T. Govindan, “The role of scattering in research interests are centered around the physics of nanostructures and
nanotransistors,” IEEE Trans. Electron Devices, vol. 50, pp. 1459–1466, includes molecular electronics, nanoscale device physics, spin electronics and
June 2003. mesoscopic superconductivity.
[31] A. Svizhenko, M. Anantram, T. Govindan, B. Biegel, and R. Venu- Dr. Datta received the NSF Presidential Young Investigator Award and the
gopal, “Nano-transistor modeling: Two dimensional Green’s function IEEE Centennial Key to the Future Award in 1984, the Frederick Emmons
method,” J. Appl. Phys., vol. 91, p. 2343, 2002. Terman Award from the ASEE in 1994, and shared the SRC Technical
[32] W. Tian, S. Datta, S. Hong, R. Riefenberger, J. I. Henderson, and C. P. Excellence Award, 2001 and the IEEE Cledo Brunetti Award, 2002. He is a
Kubiak, “Conductance spectra of molecular wires,” J. Chem. Phys., vol. Fellow of the American Physical Society (APS) and the Institute of Physics
109, p. 2874, 1998. (IOP).
[33] H. Tsuchiya and U. Ravaioli, “Particle Monte-Carlo simulation of
quantum phenomena in semiconductor nanostructures,” J. Appl. Phys.,
vol. 89, p. 4023, 2001.
[34] F. Venturi, R. K. Smith, E. C. Sangiorgi, M. R. Pinto, and B. Ricco,
“A general purpose device simulator coupling Poisson and Monte Carlo Mark S. Lundstrom (F’94) received the B.E.E. and
transport with applications to deep submicron MOSFETs,” IEEE Trans. M.S.E.E. degrees from the University of Minnesota,
Computer-Aided Design, vol. 8, p. 360, Apr. 1989. Minneapolis, in 1973 and 1974, respectively, and
[35] R. Venugopal, Z. Ren, S. Datta, M. S. Lundstrom, and D. Jovanovic, the Ph.D. degree from Purdue University, West
“Simulating quantum transport in nanoscale MOSFETs: Real versus Lafayette, IN, in 1980.
mode space approaches,” J. Appl. Phys., vol. 92, p. 3730, 2002. He is the Scifres Distinguished Professor of
[36] R. Venugopal, M. Paulsson, S. Goasguen, S. Datta, and M. Lundstrom, Electrical and Computer Engineering at Purdue
“A simple quantum mechanical treatment of scattering in nanoscale tran- University, West Lafayette, where he also directs the
sistors,” J. Appl. Phys., vol. 93, p. 5613, 2003. NSF Network for Computational Nanotechnology.
[37] Y. Fu, M. Karlsteen, M. Willander, N. Collaeert, and K. De Meyer, Before attending Purdue, he worked at Hewlett-
“Quantum transport and I-V characteristics of quantum sized field-ef- Packard Corporation, Loveland, CO, on integrated
fect-transistors,” Superlatt. Microstruct., vol. 24, no. 2, p. 111, 1998. circuit process development and manufacturing. His current research interests
center on the physics of semiconductor devices, especially nanoscale transis-
tors. His previous work includes studies of heterostructure devices, solar cells,
heterojunction bipolar transistors, and semiconductor lasers. During the course
of his Purdue career, he has served as Director of the Optoelectronics Research
Zhibin Ren was born in China. He received the Ph.D. degree in electrical en- Center and Assistant Dean of the Schools of Engineering.
gineering from Purdue University, West Lafayette, IN, in 2001. Dr. Lundstrom received several awards for teaching and research, most re-
He is currently with IBM Corporation, Yorktown Heights, NY. His research cently the 2002 IEEE Cledo Brunetti Award and the 2002 Semiconductor Re-
interests are primarily centered on device physics, modeling, and experimental search Corporation Technical Achievement Award for work on nanoscale elec-
characterization of MOSFETs. tronics. He is a Fellow of the American Physical Society.
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6de410518ccf92a4 | Friday, July 26, 2019
Interpreting Quantum Surreality
The universe changes by quantum matter action and so quantum phase, matter, and action are all just the way the universe is and are therefore all useful archetypes for predicting outcomes from precursors. There really is no need to interpret the nature of quantum phase just as there is no need to interpret the natures of matter or action. While people do not often ask about the interpretation of the very intuitive and causal matter and action realities, people do still ask about the interpretation of the somewhat less intuitive and surreal quantum phase. In particular, people ask how does quantum phase surreality connect with the more intuitive macroscopic reality of relativistic gravity matter action.
All matter vibrates or oscillates and so any two particles or bodies can be in phase or out of phase or anywhere in between. Two particles that are in phase can bond in a collision by emitting light and two particles that are out of phase will scatter and not bond. Of course, two people who like each other are in phase and will bond while two people who do not like each other are out of phase and will conflict and therefore not bond. We don’t normally associate the intuitive bonding among people with quantum phase correlation, but quantum phase bonding is a perfect analog for human bonding. Of course, all of reality is made up of quantum phase bonds and conflicts and there does seem to be interference and superposition in relations among people.
Quantum phase bonds and conflicts are a common part of our macroscopic reality and the pure quantum phase of light pulses make up the surreality of phase exchange bonds of matter. Quantum phase is also an important part of the universe matter pulse, but macroscopic gravity relativity on the cosmic scale does not include the bonding of quantum phase even though microscopic charge certainly does. Things happen when one discrete quantum state transitions to another discrete quantum state in a fully reversible process known as wavefunction collapse. This reversibility does create a causal confusion for time direction that irreversible macroscopic reality does not have. Macroscopic things always happen somehow irreversibly and seemingly without regard to quantum phase and in fact our notion of time emerges from the irreversible entropy that results from large numbers of matter actions.
The key to the irreversible nature of macroscopic reality is with the decoherence of quantum phase. Phase decoherence collapses large numbers of reversible wavefunctions into the effectively irreversible entropy of that large causal set of matter actions. The electron motion in a hydrogen atom is the result of a charge bond with negligible gravity. Nevertheless, two hydrogen atoms at 70 nm separation have their charge dipole-induced-dipole or dispersive attraction equal to their gravity attraction. At 70 nm separation, gravity and charge fluctuations are equal as a characteristic and continuous perturbation in both time and space.
Each hydrogen bond has a quantum phase correlated with the photon emission that bonded that hydrogen. This means that the two (or more) photons of these two hydrogen atoms have persistent dispersive attractions that we call gravity. The phase correlation of this biphoton means that there are slight differences in the gravities of atom particles due to each atom’s history.
The universe pulse gives a characteristic quantum gravity noise known as continuous spontaneous localization (CSL), which collapses wavefunctions and makes our macroscopic reality real by dephasing matter actions. Normally, gravity is too small to affect charge at a microscopic scale, but the very slow universe pulse fluctuation frequency of 0.255 ppb/yr at 70 nm is sufficient as the plot below shows.
This plot also shows that it will take another 2-3 orders of magnitude sensitivity with gravity wave detectors to finally confirm the mattertime decay of our universe pulse. However, mattertime decay does show up in a large number of other measurements, but those measurements are invariably complicated by classical noise. Note that it is the very slow quantum fluctuations in the universe pulse, 0.26 ppb/yr, that collapse wavefunctions at 70 nm, but the dephasing of quantum wavefunction collapse occurs everywhere in the universe.
Matter decay and force growth are everywhere and in everything that happens. Here is a plot of the mattertime decay versus frequency for a large number of periodic events. Pulsars are rotating neutron stars that show very characteristic pulsing as well as decay and pulsar decay follows the mattertime decay line. However, pulsars also decay by radiation of light and gravity and so this complicates the interpretation as a universal decay.
The Allan deviation of atomic clock synchronisation also follows the mattertime decay line as well as the earth spin decay and the moon-earth distance, as well as the approach of Andromeda galaxy. Of course, this could all be just a coincidence, but it does mean that the electron charge radius, re, does decay and therefore the electron spin period as well.
The next plot shows the decay of the kilogram standard, IPK, over 130 yrs relative to a number of secondary standards and the IPK decay is 0.51 ppb/yr or twice the mattertime decay. Thus far the IPK decay has no explanation and in mattertime, the frequent careful cleaning of the secondary standards actually adds mass to keep many of the secondary standards constant over time. The IPK cleaning only happened each of the three times it was measured.
The decay of earth’s day in the next plot includes a very much greater annual variation from 1963 to 2015. There are large annual fluctuations of several ms as well as a long term decay that is consistent with 0.26 ppb/yr. However, most of the variations are due to perturbations of the moon and planets along with tidal heating of earth’s oceans also occurs and this complicates the interpretation.
Thus the quantum dephasing decay of the universe pulse makes our macroscopic reality real and yet still consistent with our surreal quantum time confusion. Quantum phase does have macroscopic effects as light polarization and interference, but very large bodies have all dephased and therefore do not show quantum phase effects.
The universe pulse is after all the pilot wave that guides all light and matter action. Pilot wave or de Broglie-Bohm theory is a deterministic quantum mechanics that creates hidden variables as pilot waves to guide all matter particles, not wavefunctions. However, the universe pulse as a pilot wave and so does not introduce any hidden variables since that is just the way the universe is. Thus, the relativistic gravity Hamilton-Jacobi equation becomes the basic equation of motion as a quadratic and relativistic form of the quantum Schrödinger equation. The Klein-Gordon equation is also a quadratic and relativistic form of the Schrödinger equation and is the basis for quantum field theory and the standard model of particle physics. |
85d8b0aa4ac20347 | Free preview of my book available
The Sun in a laboratory container
In addition to quantum physics, I also have of course other interests and fascinations. And sometimes some other than a quantum physics subject is so impressive and important that I want to say something about it on this website, even though it’s not about quantum physics.
SAFIRE project
It’s about the SAFIRE project. The acronym means: Stellar Athmospheric Function in Regulation Experiment. It was started by a group of plasma physicists, astrophysicists and electrical engineers who wanted to test an idea differing from mainstream physics about the forces that play an important role within our solar system and also in interstellar space. This group is called out by RationalWiki as a bunch of garden-variety physicists or pseudo-physicists. Well, they have answered the challenge and started the SAFIRE project. They have implemented their model of how they think the sun works in a laboratory container, a three-year project, to see if their model can be falsified.
Click on the image to download the SAFIRE report as pdf
Their result is truly amazing. View the film they produced, read their 72 page report and think for yourself. Either they are completely fraudulent, or they have discovered something particularly important (and that option is my firm impression) that can have enormous implications for:
• Our knowledge about the real processes that take place in a star, especially in our own nearby sun.
• Insights about the origin of the elements heavier than hydrogen and helium.
• Free energy production: a revolutionary way in which energy can be generated. It seems nuclear fusion is happening, because heavy elements appear to be produced, without any adverse side effects and without the need for an incredibly expensive and complex fusion reactor, which has to enclose the hot plasma in extremely strong magnetic fields.
• Safe processing of radioactive waste.
Energy by transmutation of light elements
If this is true, then this is incredibly good news, especially in the context of our current problems with regard to our global energy needs.
Confirmation by replication
When watching the film and reading their report, I am reminded of the facilities that are available on the most universities, to replicate this and to test it. It is not beyond the capabilities of an academic technician with adequate resources. Physics students, accept the challenge.
Beyond Weird & The Quantum Handshake
To keep up to date with the subjects on my website I have to read quite a bit. And a lot of highly interesting material on quantum physics is being written and published. But occasionally I come across something that impresses me particularly and seems worth of special attention. Especially when it considerably broadens or clarifies my view on quantum physics and its interpretations. Therefore highly recommended stuff for visitors of my website. So, I’ll discuss two books here. The first one I want to discuss is: “Beyond Weird – Why Everything You Thought About Quantum Physics is .. different” by Philip Ball.
Beyond Weird
I am grateful to the student who put this book in my hands. Philip Ball is a science journalist who has been writing about this topic in Nature for many years. You don’t need to be able to solve exotic Schrödinger equations to follow his fascinating and utterly clear explanation of the quantum world and the riddles it presents. Also, he clears some misunderstandings up about this subject. Such as the word quantum, which is actually not the fundamental thing in quantum physics but rather an emerging phenomenon. The state wave is not quantized but fundamentally very continuous. He desctibes how quantum physics in its character and history deviates from all previous physical theories. It is a theory that is not built by extrapolation on the older theories. You can’t imagine what happens in the quantum world as you can do with, for example, gravity, electric currents, gas molecules, etc. The mathematical basis of quantum physics, quantum mechanics was not created by starting from fundamental principles but was the result of particularly happy intuitions that worked well but whose creators could not fundamentally explain what they were based on. Examples are: The matrix mechanics of Heisenberg, the Schrödinger equation, the idea of Born that the state function gives you the probability of finding the particle at a certain place when measured. It was all inspired intuitive guesswork that laid the foundation for an incredibly successful theory we still don’t really understand how and why it works. Ball makes presents a good case for the idea that quantum mechanics seems to be about information. It is a pity, in my opinion, that he ultimately appears to adhere to the decoherence hypothesis. That is the point in his book where the critical reader will notice that what was until then comparably good to follow step by step suddenly loses its strict consistency and that from there one has to do with imperfect metaphors. His account remains interesting but isn’t that convincing anymore. Despite that, the book is highly recommended for anyone who wants to understand more about the quantum world and especially about quantum computers.
The Quantum Handshake
A completely different type of book is “The Quantum Handshake – Entanglement, Nonlocality and Transactions” by John Cramer. His interpretation of quantum physics seems, in my opinion incorrectly, not to be placed on the long list of serious quantum interpretations. Not a big group of supporters. In any case, I had never heard of his interpretation until it was brought forward by someone at a presentation about consilience I attended a short time ago. The subject made me curious because the state wave seems to stretch out backward and forward in time as I see it. Cramers’ hypothesis is that the state wave can also travel back in time, creating a kind of ‘handshake’ between the primary departing state wave and the secondary backwards in time reflected state wave. The reflected state wave traveling back in time arrives at the source thus exactly at the time of departure of the primary wave. This handshake between both waves effects the transfer of energy without the need for the so-called quantum collapse. The measurement problem where the continuous state wave instantaneously changes into an energy-matter transfer would then be explained as the result of a energy transfer by the handshaking state waves. However, in order to finally be able to complete that energy-matter transfer from source to measurement device, Cramer has to assume that the state wave is “somewhat” material-physical. This ephemeral quality of the state wave is considered as a severe weakness in his interpretation. Nevertheless the book provides worthwhile reading for those who want to delve into the various interpretations of quantum physics, also and especially because of Cramer’s discussion of a large number of experiments with amazing implications such as, for example, quantum erasers and delayed choice experiments where retro causality appears to occur. His idea of a state wave that is traveling back in time – which is not forbidden in the formulations of quantum mechanics – remains a fascinating possibility.
500 books sold in one year in The Netherlands
I’m very proud of this success. Within one year 500 copies of “Kwantumfysica, informatie en bewustzijn” sold through the regular bookshops in The Netherlands. Copies sold through my own network of friends, acquaintances en students following my lectures are not counted here. The work was certainly not in vain.
In the meantime I am steadily working on the English version to which a new chapter on consilience is being added. This is going to be the introduction to that chapter:
14 Consilience
From Wikipedia:
In science and history, consilience (also convergence of evidence or concordance of evidence) is the principle that evidence from independent, unrelated sources can “converge” on strong conclusions. That is, when multiple sources of evidence are in agreement, the conclusion can be very strong even when none of the individual sources of evidence is significantly so on its own. Most established scientific knowledge is supported by a convergence of evidence: if not, the evidence is comparatively weak, and there will not likely be a strong scientific consensus.
In this book, starting with the scientific revolutions of de 17th century and, following the threads of its developing history until today, we have arrived at a perhaps baffling and remarkable result, hard science – physics – today is not in conflict with the idea of the existence of an of the body independent consciousness, also called the survival hypothesis. On the contrary, it supports it.
However, should this idea only surface after studying quantum physics and nowhere else in the science domain, this support would be as whacky as a table supported by only one leg. Therefore, the question is, is survival supported by published scientific research in other domains? Indeed, it is. Some of this research was already mentioned in preceding chapters. It is time now to pay a little bit more attention to all published and reviewed evidential material concerning consciousness being independent of the material body.
Quantum physics and time
From Wikipedia: Vlatko Vedral is a Serbian-born (and naturalised British citizen) physicist and Professor of Physics at the University of Oxford and CQT (Centre for Quantum Technologies) at the National University of Singapore and a Fellow of Wolfson College. He is known for his research on the theory of Entanglement and Quantum Information Theory. As of 2017 he has published over 280 research papers in quantum mechanics and quantum information and was awarded the Royal Society Wolfson Research Merit Award in 2007. He has held a Lectureship and Readership at Imperial College, a Professorship at Leeds and visiting professorships in Vienna, Singapore (NUS) and at Perimeter Institute in Canada. As of 2017, there were over 18,000 citations to Vlatko Vedral’s research papers. He is the author of several books, including Decoding Reality.
Watch this movie “Living in a quantum world” from Vlatko Vedral on YouTube: At the end of his presentation a question from the audience about time and quantum physics is asked (at about 1: 10) and in his answer he describes the behavior of a super-accurate clock and what happens to the last digits when you lift that clock half a meter in the gravitational field. And then he wonders what it means when you imagine that clock to be in a quantum superposition at the two different heights in the gravitational field. A superposition of two different timelines. Fascinating.
By the way, the first part of his presentation – about 45 minutes – is actually a very compact version of my quantum physics book. Everything is presented in an almost blazing speed: interference, the Mach-Zehnder interferometer, Schrödinger’s cat, the Copenhagen interpretation against the multiverse interpretation, delayed choice experiments, interference with very large molecules shot through double slits, the orientation of our robin on the earth’s magnetic field in its annual migration, the 100% efficiency of chlorophyll. Highly recommended.
Mass and energy, time and and space, the misconceptions
Quanta Magazine, a web service which often brings interesting articles, published shortly an interesting article where relativity, quantum physics and black holes played an important role. However, in reading it I did hit upon a very common misconception, about which I like to comment here.
Quote from: Einstein, Symmetry and the Future of Physics | Quanta Magazine
Symmetry, the simplifying idea behind the great discoveries of physics ..
The misconception is that mass and energy are different things and that energy is somehow mysteriously converted into mass and vice versa. However, that’s not the message of E=mc2. Energy and mass are, in the opinion of almost all physicists, more like the two sides of the same coin. They are identical. This can be understood by considering what happens when an object is accelerated up to the speed of light.
According to the special relativity, all the energy that you spend into that acceleration is converted into inertial mass. It will cost you more and more energy to keep accelerating it. That is why we can never reach the speed of light itself in this way, the inertial mass would become infinite. This effect has been convincingly demonstrated when accelerating protons in the Large Hadron Collider at CERN. The faster they go, the more mass they get and the stronger the magnetic fields must be to keep them neatly in their circulair loop.
In general relativity, the central basic assumption is that inertial mass and heavy mass are identical or that the acceleration force due to gravity is identical to the acceleration force that you encounter in, for example, a merry-go-round. The implication therefore is that inertial mass, heavy mass and energy are really all the same fundamental thing. This means for instance that a charged battery must also be slightly heavier than when discharged. However, the energy released by nuclear fusion is often explained in popular terms as follows:
The mass of the two fused atomic nuclei is smaller than that of the original fused nuclei together. That mass deficit has become energy and that mass is gone.
Thus it seems as if mass alone is not conserved, mass plus energy should be the conserved property. However, Wikipedia says otherwise: “Mass and energy can be seen as two names (and two measurement units) for the same underlying, conserved physical quantity.[18] Thus, the laws of conservation of energy and conservation of (total) mass are equivalent and both hold true”.
Ponder this. The fused atomic nucleus has received an enormous amount of kinetic energy during the fusion, and that is speed. That kinetic energy is exactly having the same mass as the ‘disappeared’ mass. So, that mass has not disappeared at all. Due to the speed with which the fused core now moves, which means kinetic energy, it also has more mass. That is the message of special relativity. If you could have this fusion taking place in a thermally completely sealed box balanced on a pair of scales, you would find zero difference in weight – and therefore in mass.
Another but related topic. That every observer always observes the same speed of light is a physical observation but goes against our so-called common sense which tells us how adding up speeds normally works. Elsewhere on this website I say something about that in ‘What is light‘.
Can Humans Directly Observe the Quantum World?
In the world of physics, we can see a beginning inclination to research the connection between the consciousness of the observer and the observed. Research has already shown that the human senses work and perceive at the quantum level. Not only the eye which after adaptation appears to be able to observe a single photon, but all our senses seem to function at quantum level and even beyond. Our ears are energywise extremely sensitive organs. Read the article by William C. Bushell Ph.D. and Maureen Seaberg at (SAND).
Can Humans Directly Observe the Quantum World? Part I
SSE Conference 2019 on consilience – Broomfield, Colorado
Dean Radin presenting
The 38th Society for Scientific Exploration (SSE) conference was held from June 5-8 in Broomfield Colorado. The theme was “consilience” whereby evidence from diverse and independent sources can be used as valid support for scientific theories. For example, on the one hand in quantum physics a conscious observer seems to be needed to trigger the so-called quantum collapse, on the other hand in current medical science applying advanced life-saving interventions the growing numbers of validated near-death experiences can no longer be ignored. So, in both very different domains, the idea of non-matter-dependent consciousness is confirmed.
Within three days 34 presentations of approx. 20 minutes were held, whether or not supported with PowerPoint slides, offering also the opportunity for three to five questioners after every presentation, and 17 poster presentations set up in the hall in front of the conference hall, for which one and a half hours had been set aside on day 2. Personally, I thought that part was the most accessible because you could come quickly in direct contact with the poster’s creator.
To be honest, in my opinion there were some poster presentations actually deserving a full presentation and vice versa there were presentations that could have been better scheduled as poster presentations.
To download a more extensive report click here. |
0ab30a15907309cb | Thursday, September 20, 2018
Mermin defends Copenhagen Interpretation
1. "There is only an abstract quantum physical description."
This is basically an admission that QM as presently formulated is not about describing 'reality', it is about using a mathematical 'abstraction' that gives you useful answers with little understanding of how.
This can be also seen in the following sentence:
"Physics concerns what we can say about nature."
You could apply this attitude towards epicycles. Hey, they have nothing to do with actual planets or how they move, but what the hell, the answers are useful. So much for reality.
1. Physics has gone nowhere. It's in the last of its funding hay days. Boring answers and even more boring applications.
2. Quantum mechanics is the least successful theory of all time: "the only exact solutions to the Schrödinger equation found so far are for free-particle motion, the particle in a box, the hydrogen atom, hydrogen-like ions, the hydrogen molecular ion, the rigid rotator, the harmonic oscillator, Morse and modified Morse oscillators, and a few other systems [2,3]. For more complicated systems, however, approximation techniques have to be used (such as the variational method or perturbation theory), which sometimes give poor results compared with experimental ones, and practical calculations with them are usually very difficult, even with the use of powerful computers [4]. The difficulty is that in a system made of N interacting particles (where N can be anywhere from three to infinity), the repeated interactions between particles create quantum correlations. As a consequence, the dimension of the Hilbert space describing the system scales exponentially in N. This makes a direct numerical calculation of the Schrödinger’s equation intractable: Every time an extra particle is added to the system, the computational resources would have to be doubled [5]."
2. Found another fantastic skeptic quote, I'm fond of collecting them as they breath fresh sanity into a mentally constipated world:
"There is no more common error than to assume that because prolonged and accurate mathematical calculations have been made, the application of the result to some fact of nature is absolutely certain."
A.N. Whitehead |
6087b83663376326 | Physics Colloquium
Title: Quantum uncertainty relations: further landscapes
Speaker: Prof. Ujjwal Sen, HRI, Allahabad
Date/Time: 10/11/2017 at 05:00 PM
Abstract: The uncertainty relation forms one of the pillars of our understanding of quantum mechanics, and fires the imagination of non-scientists and scientists alike. We discuss the traditional form of the relation, thereby identifying quantum states which were not hitherto encompassed, and which we propose to rectify. The uncertainty relation has important implications for the quantum-to-classical boundary, and we find that the boundary is also modified by going over to the non-traditional form of the relation.
Title: Imaging and Non imaging: Wavefronts without waves
Speaker: Prof. Rajaram Nityananda, Azim Premji University, Bengaluru
Date/Time: 09/10/2017 at 4:30 pm
Abstract: High school geometrical optics is obsessed with the special case when the rays from a single point on an object making small angles to the axis reach a single point on the image. This talk will give a glimpse of the world beyond this restriction. Forming good images with rays making a large angle is important for both telescopes and microscopes. Rays from a distant quasar (bright centre of a galaxy) get bent by the gravity of an intervening galaxy and do not form an image in the usual sense. And if one wants to concentrate sunlight from a large area onto a smaller one, the best solution is not to image at all! The key to a deeper understanding and use of geometrical optics in such situations is the existence of surfaces perpendicular to a family of rays - wavefronts without waves.
Title: Optical Solitons and Modulational Instability: Ultra Pulse Generation and Supercontiuum Generation
Speaker: Prof. K. Porsezian, Department of Physics, Pondicherry University
Date/Time: 20/04/2017 at 05:00 PM
Abstract: In this talk, I would like to discuss the role of optical solitons and modulational Instability (MI) in nonlinear optical fiber. Considering different types of nonlinear Schrödinger equation with different linear and nonlinear optical effects, I will discuss the role of MI and generation of solitons. In particular, I will discuss the side band generation in nonlinear optical fiber and controlling of these side bands. The theoretical investigation of nonlinear femtosecond pulse propagation in liquidcore photonic crystal fiber will be discussed in detail. Finally, I will discuss supercontinuum generation through MI induced spectral broadening process. The effect of saturable nonlinearity and slow nonlinearity due to reorientation contribution of liquid molecules on broadband supercontinuum generation in the femtosecond regime will be discussed using an appropriately modified nonlinear Schrödinger equation.
Title: The Mysterious Magnetic Personality of Our Sun
Speaker: Prof. Arnab Rai Choudhuri, Dept. of Physics, Indian Institute of Science, Bangalore
Date/Time: 20/10/2016 at 05:00 PM
Abstract: The Sun is the first astronomical object in which magnetic fields were discovered in 1908 by using the Zeeman effect. Even before this discovery of magnetic fields in sunspots, it was known that there is a 11-year cycle of sunspots, which could be identified as the magnetic cycle of the Sun after this discovery. The magnetic field of the Sun is also behind many other phenomena, such as the violent explosions known as solar flares, the corona much hotter than the solar surface and the solar wind. Only within the last few decades, major developments in plasma physics and magnetohydrodynamics (MHD) have at last provided a broad framework for the theoretical understanding of these phenomena connected with the solar magnetic fields. I shall give a general introduction to this field – with some emphasis on the research interests of our group. A more detailed account of this field can be found in my recently published popular science book:
Title: When you need a physicist, not a physician, to cure a disease
Speaker: Prof. Sudipta Maiti, Dept. of Chemical Sciences, TIFR, Mumbai
Date/Time: 03/10/2016 at 12:00 Noon
Abstract: Despite billions of dollars spent on medical research, medicine still has no cure for diseases of the brain such as Alzheimer’s and Parkinson’s. They seem to be caused by our own proteins, and not by external infectious agents. In the patients, these proteins become sticky, aggregate together and start killing neurons for some unknown reason. Perhaps these diseases will only be solved when we really understand the structure and properties of these molecules in different aggregated states. By combining a range of powerful techniques borrowed from physics, starting from single molecule fluorescence to solid state NMR, we are now beginning to unravel the structure of the toxic molecular aggregate.
Title: Energy, Environment and Piezoceramics
Speaker: Prof. Ajit R. Kulkarni, Dept. of Metallurgical Engineering and Materials Science, IIT-Bombay, Mumbai
Date/Time: 28/09/2016 at 12:00 Noon
Abstract: Energy crisis have affected the globe primarily due to sources that supply national electricity grids or those used as fuel in vehicles. Industrial development and population growth have led to a surge in the global demand for energy. To partially overcome stress on these natural resources, renewable sources of energy including waste heat, vibration, electromagnetic waves, wind, flowing water, and solar energy are used. In recent years emphasis is on scavenging vibrational/mechanical energy with piezoelectric materials and apply it to self-power micro-devices instead of batteries. Piezoelectric ceramic is the heart of these energy harvesters. In this talk, I shall speak on the state-of-the-art in piezoelectric energy harvesting, the basic material characteristics, discuss material choices and their forms. Piezoelectric LeadZirconateTitanate (PZT) is widely used. However, due to high toxicity of lead oxide, pollution and environmental problems there are restriction on its use, development and disposal. Hence environmentally friendly lead free materials are of current interest.
At IIT Bombay, our group is actively working on engineering lead free alkali niobates, their modified and a few composites as alternative to Lead based materials. Our strategies to enhance materials characteristics through processing, composition modification and electrical properties will be discussed. Finally, opportunities in this harvesting technique will be presented.
Title: Assisted-hopping models of active-absorbing state transition on a line
Speaker: Prof. Deepak Dhar, Dept. of Theoretical Physics, TIFR, Mumbai
Date/Time: 16/09/2016 at 05:00 PM
Abstract: I will describe a class of assisted hopping models in one dimension in which a particle can move only if it has exactly one occupied neighbour, or if it lies in an otherwise empty interval of length less than or equal to (n+1). This system undergoes a phase transition as a function of the density $\rho$ of particles, from a low-density phase with all particles immobile for density below a critical density, to an active state for densities greater than the threshold value. I will describe the exact solution of this problem, where we can determine exactly the critical density, and the average activity as a function of density in the active phase. There is a mapping to a gas of defects with only on-site interaction. The mean fraction of movable particles in the active steady state varies as $(\rho - \rho c)^{\beta}$, for $\rho$ near $\rho_c$. We show that for the model with range $n$, the order parameter exponent $\beta$ equals $n$, and can thus be made arbitrarily large.
Title: SAR Image Formation
Speaker: Prof. S.K. Patra, Scientist 'G', Group Director, Sensor Data Processing Group, Advanced Data Processing Research Institute (ADRIN), 203, Akbar Road, Manovikas Nagar Post, Secunderabad, Hyderabad
Date/Time: 02/09/2016 at 05: PM
Abstract: Synthetic Aperture Radar (SAR) payload in space is usually a multiresolution, multi-swath, multi-mode, multi-polarization system carrying an active antenna on-board to image the ground in microwave frequencies. SAR transmits pulses of microwave radiation to a target and receives backscatter in the form of amplitude, phase with a time delay in the returned signal. The transmitted linearly frequency modulated long pulse in range or across the flight direction views the target for longer duration whereas SAR’s principle of synthesizing the long aperture to achieve finer resolution in azimuth or flight direction increases target dwell time. Together, target history is spread in across track and along track directions in the received signal. SAR algorithms convert raw signal data to interpretable detected images which requires significant processing after acquisition. The presentation will deal with the principles of SAR imaging and techniques used to form SAR images from raw signal data.
Title: Topological insulators and their aging
Speaker: Prof. Kalobaran Maiti, Dept. of Cond. Mat. Physics and Mat. Science, TIFR, Mumbai
Date/Time: 26/08/2016 at 05:00 PM
Abstract: Insulators are materials having an energy gap between the highest occupied band (valence band) and the lowest unoccupied band (conduction band). Topological insulators are a special type of such materials, which possess gapless states with novel electromagnetic properties protected by time reversal symmetry at the surface of bulk insulators. Enormous research has been carried out on these materials as they are expected to bring immense technological advances and new possibilities in the fields of spintronics, quantum computation, dissipationless charge transfer etc. Since the surface states of topological insulators are time reversal symmetry protected, they are expected to be immune to weak disorder, chemical passivation of the surface or temperature change. However, significant discrepancy from such behaviour has been found experimentally in various materials. We studied the detailed electronic structure and its aging of a topological insulator, Bi2Se3 employing high resolution photoemission spectroscopy. Both the band structure results and high resolution angle resolved photoemission data reveal significantly different surface electronic structure for different surface terminations. Furthermore, oxygen impurity on Se terminated surface exhibits an electron doping scenario, while oxygen on Bi terminated surface corresponds to a hole doping scenario. The intensity of the Dirac states reduces with aging indicating fragility of the topological order due to surface impurities.
Title: States and Phases of Matter and Phase Transition
Speaker: Prof. Y. Singh, Distinguish Professor, Dept. of Physics, Institute of Science, BHU, Varanasi
Date/Time: 17/08/2016 at 02:30 PM
Abstract: The talk will focus on basic concepts associated with states and phases of matter. Through few examples of phase diagram and phase transitions the significance of symmetry breaking, order parameters, emergence of new properties, etc. will be explained. Theories which describe phase transitions may also, at elementary level, be discussed. |
16bfd9a7f308d7ad | Consider this penny on my desc. It is a particular piece of metal, well described by statistical mechanics, which assigns to it a state, namely the density matrix $\rho_0=\frac{1}{Z}e^{-\beta H}$ (in the simplest model). This is an operator in a space of functions depending on the coordinates of a huge number $N$ of particles.
The ignorance interpretation of statistical mechanics, the orthodoxy to which all introductions to statistical mechanics pay lipservice, claims that the density matrix is a description of ignorance, and that the true description should be one in terms of a wave function; any pure state consistent with the density matrix should produce the same macroscopic result.
Howewer, it would be very surprising if Nature would change its behavior depending on how much we ignore. Thus the talk about ignorance must have an objective formalizable basis independent of anyones particular ignorant behavior.
On the other hand, statistical mechanics always works exclusively with the density matrix (except in the very beginning where it is motivated). Nowhere (except there) one makes any use of the assumption that the density matrix expresses ignorance. Thus it seems to me that the whole concept of ignorance is spurious, a relic of the early days of statistical mechanics.
Thus I'd like to invite the defenders of orthodoxy to answer the following questions:
(i) Can the claim be checked experimentally that the density matrix (a canonical ensemble, say, which correctly describes a macroscopic system in equilibrium) describes ignorance? - If yes, how, and whose ignorance? - If not, why is this ignorance interpretation assumed though nothing at all depends on it?
(ii) In a though experiment, suppose Alice and Bob have different amounts of ignorance about a system. Thus Alice's knowledge amounts to a density matrix $\rho_A$, whereas Bob's knowledge amounts to a density matrix $\rho_B$. Given $\rho_A$ and $\rho_B$, how can one check in principle whether Bob's description is consistent with that of Alice?
(iii) How does one decide whether a pure state $\psi$ is adequately represented by a statistical mechanics state $\rho_0$? In terms of (ii), assume that Alice knows the true state of the system (according to the ignorance interpretation of statistical mechanics a pure state $\psi$, corresponding to $\rho_A=\psi\psi^*$), whereas Bob only knows the statistical mechanics description, $\rho_B=\rho_0$.
Presumably, there should be a kind of quantitative measure $M(\rho_A,\rho_B)\ge 0$ that vanishes when $\rho_A=\rho_B)$ and tells how compatible the two descriptions are. Otherwise, what can it mean that two descriptions are consistent? However, the mathematically natural candidate, the relative entropy (= Kullback-Leibler divergence) $M(\rho_A,\rho_B)$, the trace of $\rho_A\log\frac{\rho_A}{\rho_B}$, [edit: I corrected a sign mistake pointed out in the discussion below] does not work. Indeed, in the situation (iii), $M(\rho_A,\rho_B)$ equals the expectation of $\beta H+\log Z$ in the pure state; this is minimal in the ground state of the Hamiltonian. But this would say that the ground state would be most consistent with the density matrix of any temperature, an unacceptable condition.
Edit: After reading the paper http://bayes.wustl.edu/etj/articles/gibbs.paradox.pdf by E.T. Jaynes pointed to in the discussion below, I can make more precise the query in (iii): In the terminology of p.5 there, the density matrix $\rho_0$ represents a macrostate, while each wave function $\psi$ represents a microstate. The question is then: When may (or may not) a microstate $\psi$ be regarded as a macrostate $\rho_0$ without affecting the predictability of the macroscopic observations? In the above case, how do I compute the temperature of the macrostate corresponding to a particular microstate $\psi$ so that the macroscopic behavior is the same - if it is, and which criterion allows me to decide whether (given $\psi$) this approximation is reasonable?
An example where it is not reasonable to regard $\psi$ as a canonical ensemble is if $\psi$ represents a composite system made of two pieces of the penny at different temperature. Clearly no canonical ensemble can describe this situation macroscopically correct. Thus the criterion sought must be able to decide between a state representing such a composite system and the state of a penny of uniform temperature, and in the latter case, must give a recipe how to assign a temperature to $\psi$, namely the temperature that nature allows me to measure.
The temperature of my penny is determined by Nature, hence must be determined by a microstate that claims to be a complete description of the penny.
I have never seen a discussion of such an identification criterion, although they are essential if one want to give the idea - underlying the ignorance interpretation - that a completely specified quantum state must be a pure state.
Part of the discussion on this is now at: http://chat.stackexchange.com/rooms/2712/discussion-between-arnold-neumaier-and-nathaniel
Edit (March 11, 2012): I accepted Nathaniel's answer as satisfying under the given circumstances, though he forgot to mention a fouth possibility that I prefer; namely that the complete knowledge about a quantum system is in fact described by a density matrix, so that microstates are arbitrary density matrces and a macrostate is simply a density matrix of a special form by which an arbitrary microstate (density matrix) can be well approximated when only macroscopic consequences are of interest. These special density matrices have the form $\rho=e^{-S/k_B}$ with a simple operator $S$ - in the equilibrium case a linear combination of 1, $H$ (and vaiious number operators $N_j$ if conserved), defining the canonical or grand canonical ensemble. This is consistent with all of statistical mechanics, and has the advantage of simplicity and completeness, compared to the ignorance interpretation, which needs the additional qualitative concept of ignorance and with it all sorts of questions that are too imprecise or too difficult to answer.
• $\begingroup$ Is this not the same problem as the MaxEnt school "runs into" (scare quotes because they don't really) that physics seems to change depending on how much one chooses to ignore? The resolution there is that ultimately one is doing science, so one needs a condition like "this set of control variables is empirically sufficient to control the outputs". $\endgroup$ – genneth Mar 6 '12 at 15:15
• $\begingroup$ Science must be objective, observer independent, hence it should not depend on choices of an observer. So whatever choices there ar, there should be an objective way of assessing them. - I analyzed Max Entropy in Section 10.7 of my book lanl.arxiv.org/abs/0810.1019 Classical and Quantum Mechanics via Lie algebras, and found it wanting:If you choose to ignore things that you shouldn't (such as the energy content) you get completely wrong results in clear contradiciton to experiment. To get a correct theory you must choose to know at least everything that makes a difference to the system! $\endgroup$ – Arnold Neumaier Mar 6 '12 at 15:24
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$\begingroup$ @ArnoldNeumaier yes, but "everything that makes a difference to the system [as measured by macroscopic instruments]" != everything. MaxEnt is founded precisely on ignoring the microscopic details that do not make any difference to the macroscopic state, while not ignoring anything that does. Ignoring things that don't make any difference is good, because it means you don't have to calculate them! $\endgroup$ – Nathaniel Mar 6 '12 at 18:27
• $\begingroup$ Arnold, perhaps this a minor point, but use of the canonical ensemble implies to me the penny is in thermal equilibrium with an environment. This would mean that the penny is entangled with the environment and therefore it would not be described by a pure state. Your questions do not seem to be as sharp if they are posed to the microcanonical ensemble. $\endgroup$ – BebopButUnsteady Mar 6 '12 at 19:19
• $\begingroup$ @BebopButUnsteady: The penny is by assumption in thermal equilibrium, but need not be in equilibrium with the environment (e.g, if I just opened the window, thereby changing the environment.) - But any macroscopic body (not only a penny, and not only in a canonical ensemble, and even if far from equilibrium) is always entangled with its environment. The consequence is that no macroscopic object can be assigned a pure state, not even in principle. But this flatly contradicts the ignorance interpretation of statistical mechanics. Thus more things to defend for the upholders of orthodoxy! $\endgroup$ – Arnold Neumaier Mar 6 '12 at 19:36
I wouldn't say the ignorance interpretation is a relic of the early days of statistical mechanics. It was first proposed by Edwin Jaynes in 1957 (see http://bayes.wustl.edu/etj/node1.html, papers 9 and 10, and also number 36 for a more detailed version of the argument) and proved controversial up until fairly recently. (Jaynes argued that the ignorance interpretation was implicit in the work of Gibbs, but Gibbs himself never spelt it out.) Until recently, most authors preferred an interpretation in which (for a classical system at least) the probabilities in statistical mechanics represented the fraction of time the system spends in each state, rather than the probability of it being in a particular state at the present time. This old interpretation makes it impossible to reason about transient behaviour using statistical mechanics, and this is ultimately what makes switching to the ignorance interpretation useful.
In response to your numbered points:
(i) I'll answer the "whose ignorance?" part first. The answer to this is "an experimenter with access to macroscopic measuring instruments that can measure, for example, pressure and temperature, but cannot determine the precise microscopic state of the system." If you knew precisely the underlying wavefunction of the system (together with the complete wavefunction of all the particles in the heat bath if there is one, along with the Hamiltonian for the combined system) then there would be no need to use statistical mechanics at all, because you could simply integrate the Schrödinger equation instead. The ignorance interpretation of statistical mechanics does not claim that Nature changes her behaviour depending on our ignorance; rather, it claims that statistical mechanics is a tool that is only useful in those cases where we have some ignorance about the underlying state or its time evolution. Given this, it doesn't really make sense to ask whether the ignorance interpretation can be confirmed experimentally.
(ii) I guess this depends on what you mean by "consistent with." If two people have different knowledge about a system then there's no reason in principle that they should agree on their predictions about its future behaviour. However, I can see one way in which to approach this question. I don't know how to express it in terms of density matrices (quantum mechanics isn't really my thing), so let's switch to a classical system. Alice and Bob both express their knowledge about the system as a probability density function over $x$, the set of possible states of the system (i.e. the vector of positions and velocities of each particle) at some particular time. Now, if there is no value of $x$ for which both Alice and Bob assign a positive probability density then they can be said to be inconsistent, since every state that Alice accepts the system might be in Bob says it is not, and vice versa. If any such value of $x$ does exist then Alice and Bob can both be "correct" in their state of knowledge if the system turns out to be in that particular state. I will continue this idea below.
(iii) Again I don't really know how to convert this into the density matrix formalism, but in the classical version of statistical mechanics, a macroscopic ensemble assigns a probability (or a probability density) to every possible microscopic state, and this is what you use to determine how heavily represented a particular microstate is in a given ensemble. In the density matrix formalism the pure states are analogous to the microscopic states in the classical one. I guess you have to do something with projection operators to get the probability of a particular pure state out of a density matrix (I did learn it once but it was too long ago), and I'm sure the principles are similar in both formalisms.
I agree that the measure you are looking for is $D_\textrm{KL}(A||B) = \sum_i p_A(i) \log \frac{p_A(i)}{p_B(i)}$. (I guess this is $\mathrm{tr}(\rho_A (\log \rho_A - \log \rho_B))$ in the density matrix case, which looks like what you wrote apart from a change of sign.) In the case where A is a pure state, this just gives $-\log p_B(i)$, the negative logarithm of the probability that Bob assigns to that particular pure state. In information theory terms, this can be interpreted as the "surprisal" of state $i$, i.e. the amount of information that must be supplied to Bob in order to convince him that state $i$ is indeed the correct one. If Bob considers state $i$ to be unlikely then he will be very surprised to discover it is the correct one.
If B assigns zero probability to state $i$ then the measure will diverge to infinity, meaning that Bob would take an infinite amount of convincing in order to accept something that he was absolutely certain was false. If A is a mixed state, this will happen as long as A assigns a positive probability to any state to which B assigns zero probability. If A and B are the same then this measure will be 0. Therefore the measure $D_\textrm{KL}(A||B)$ can be seen as a measure of how "incompatible" two states of knowledge are. Since the KL divergence is asymmetric I guess you also have to consider $D_\textrm{KL}(B||A)$, which is something like the degree of implausibility of B from A's perspective.
I'm aware that I've skipped over some things, as there was quite a lot to write and I don't have much time to do it. I'll be happy to expand it if any of it is unclear.
Edit (in reply to the edit at the end of the question): The answer to the question "When may (or may not) a microstate $\phi$ be regarded as a macrostate $\rho_0$ without affecting the predictability of the macroscopic observations?" is "basically never." I will address this is classical mechanics terms because it's easier for me to write in that language. Macrostates are probability distributions over microstates, so the only time a macrostate can behave in the same way as a microstate is if the macrostate happens to be a fully peaked probability distribution (with entropy 0, assigning $p=1$ to one microstate and $p=0$ to the rest), and to remain that way throughout the time evolution.
You write in a comment "if I have a definite penny on my desk with a definite temperature, how can it have several different pure states?" But (at least in Jaynes' version of the MaxEnt interpretation of statistical mechanics), the temperature is not a property of the microstate but of the macrostate. It is the partial differential of the entropy with respect to the internal energy. Essentially what you're doing is (1) finding the macrostate with the maximum (information) entropy compatible with the internal energy being equal to $U$, then (2) finding the macrostate with the maximum entropy compatible with the internal energy being equal to $U+dU$, then (3) taking the difference and dividing by $dU$. When you're talking about microstates instead of macrostates the entropy is always 0 (precisely because you have no ignorance) and so it makes no sense to do this.
Now you might want to say something like "but if my penny does have a definite pure state that I happen to be ignorant of, then surely it would behave in exactly the same way if I did know that pure state." This is true, but if you knew precisely the pure state then you would (in principle) no longer have any need to use temperature in your calculations, because you would (in principle) be able to calculate precisely the fluxes in and out of the penny, and hence you'd be able to give exact answers to the questions that statistical mechanics can only answer statistically.
Of course, you would only be able to calculate the penny's future behaviour over very short time scales, because the penny is in contact with your desk, whose precise quantum state you (presumably) do not know. You will therefore have to replace your pure-state-macrostate of the penny with a mixed one pretty rapidly. The fact that this happens is one reason why you can't in general simply replace the mixed state with a single "most representative" pure state and use the evolution of that pure state to predict the future evolution of the system.
Edit 2: the classical versus quantum cases. (This edit is the result of a long conversation with Arnold Neumaier in chat, linked in the question.)
In most of the above I've been talking about the classical case, in which a microstate is something like a big vector containing the positions and velocities of every particle, and a macrostate is simply a probability distribution over a set of possible microstates. Systems are conceived of as having a definite microstate, but the practicalities of macroscopic measurements mean that for all but the simplest systems we cannot know what it is, and hence we model it statistically.
In this classical case, Jaynes' arguments are (to my mind) pretty much unassailable: if we lived in a classical world, we would have no practical way to know precisely the position and velocity of every particle in a system like a penny on a desk, and so we would need some kind of calculus to allow us to make predictions about the system's behaviour in spite of our ignorance. When one examines what an optimal such calculus would look like, one arrives precisely at the mathematical framework of statistical mechanics (Boltzmann distributions and all the rest). By considering how one's ignorance about a system can change over time one arrives at results that (it seems to me at least) would be impossible to state, let alone derive, in the traditional frequentist interpretation. The fluctuation theorem is an example of such a result.
In a classical world there would be no reason in principle why we couldn't know the precise microstate of a penny (along with that of anything it's in contact with). The only reasons for not knowing it are practical ones. If we could overcome such issues then we could predict the microstate's time-evolution precisely. Such predictions could be made without reference to concepts such as entropy and temperature. In Jaynes' view at least, these are purely macroscopic concepts and don't strictly have meaning on the microscopic level. The temperature of your penny is determined both by Nature and by what you are able to measure about Nature (which depends on the equipment you have available). If you could measure the (classical) microstate in enough detail then you would be able to see which particles had the highest velocities and thus be able to extract work via a Maxwell's demon type of apparatus. Effectively you would be partitioning the penny into two subsystems, one containing the high-energy particles and one containing the lower-energy ones; these two systems would effectively have different temperatures.
My feeling is that all of this should carry over on to the quantum level without difficulty, and indeed Jaynes presented much of his work in terms of the density matrix rather than classical probability distributions. However there is a large and (I think it's fair to say) unresolved subtlety involved in the quantum case, which is the question of what really counts as a microstate for a quantum system.
One possibility is to say that the microstate of a quantum system is a pure state. This has a certain amount of appeal: pure states evolve deterministically like classical microstates, and the density matrix can be derived by considering probability distributions over pure states. However the problem with this is distinguishability: some information is lost when going from a probability distribution over pure states to a density matrix. For example, there is no experimentally distinguishable difference between the mixed states $\frac{1}{2}(\mid \uparrow \rangle \langle \uparrow \mid + \mid \downarrow \rangle \langle \downarrow \mid)$ and $\frac{1}{2}(\mid \leftarrow \rangle \langle \leftarrow \mid + \mid \rightarrow \rangle \langle \rightarrow \mid)$ for a spin-$\frac{1}{2}$ system. If one considers the microstate of a quantum system to be a pure state then one is committed to saying there is a difference between these two states, it's just that it's impossible to measure. This is a philosophically difficult position to maintain, as it's open to being attacked with Occam's razor.
However, this is not the only possibility. Another possibility is to say that even pure quantum states represent our ignorance about some underlying, deeper level of physical reality. If one is willing to sacrifice locality then one can arrive at such a view by interpreting quantum states in terms of a non-local hidden variable theory.
Another possibility is to say that the probabilities one obtains from the density matrix do not represent our ignorance about any underlying microstate at all, but instead they represent our ignorance about the results of future measurements we might make on the system.
I'm not sure which of these possibilities I prefer. The point is just that on the philosophical level the ignorance interpretation is trickier in the quantum case than in the classical one. But in practical terms it makes very little difference - the results derived from the much clearer classical case can almost always be re-stated in terms of the density matrix with very little modification.
• $\begingroup$ Thanks for the clarification on the origins. The problem with your answer to (iii) is that in the particular case mentioned in my edited statement on (iii), the ground state would be the most consistent pure state, irrespective of temperature. Thus the K/L measure doesn't allow me to assess whether treating the pure state $\psi$ as a canonical example (if I am only interested in macroscopic consequences) is or isn't acceptable. $\endgroup$ – Arnold Neumaier Mar 6 '12 at 17:33
• $\begingroup$ The only lesson to draw from this is that it isn't always sensible to try and take a single "most representative" pure state from a probability distribution and expect it to have similar properties. If you're interested in macroscopic properties you should be calculating expectations. If there is a pure state whose properties (or at least the ones you're interested in) behave similarly to expectations calculated from the density matrix then you'd be justified in what you're triyng to do. I agree that the KL measure by itself doesn't tell you this, of course. $\endgroup$ – Nathaniel Mar 6 '12 at 18:09
• $\begingroup$ But if I have a definite penny on my desk with a definite temperature, how can it have several different pure states? Either this penny has a particular wave function $\psi$ which gives its complete quantum mechanical description (even though we are never able to say which one it is), then this state must somehow have an associated temperature , since Nature knows this temperature, and the description is complete. - Or such a unique $\psi$ doesn't exist, in which case the concept of microstates breaks down, and there is only the density matrix to describe the system. $\endgroup$ – Arnold Neumaier Mar 6 '12 at 18:15
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$\begingroup$ In Jaynes' view, the macrostate is a probability distribution over the microstates, and the temperature is a property of the macrostate, not the microstate. $T=\partial S/\partial U$, where $S$ is the entropy of the macrostate. If we completely knew the microstate we would be talking about a probability distribution where one state has $p=1$ and the rest 0. There would be no entropy, and hence no temperature. $\endgroup$ – Nathaniel Mar 6 '12 at 18:30
• $\begingroup$ In ignorance terms, $\partial S/\partial U$ means something like "if I added a little bit more energy to this penny, how much more ignorance would I then have about its microstate?" I will update my answer to make some of this clearer. $\endgroup$ – Nathaniel Mar 6 '12 at 18:32
I'll complete @Natahniel's answer with the fact that 'knowledge' can have physical implication linked with the behaviour of nature. The problem goes back to Maxwell's demon, who converts his knowledge of the system into work. Recent works (like arXiv:0908.0424 The work value of information) shows that the information theoretical entropies defining the knowledge of the system is connected to the work which is extractable in the same way than the physical entropies are.
To sum al this into a few words, "Nature [does not] change its behaviour depending on how much we ignore", but "how much we ignore" changes the amount of work we can extract fro Nature.
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$\begingroup$ Indeed. And to see a really great example of how our knowledge of a natural system can affect our ability to extract work from it, read this paper (by Edwin Jaynes): bayes.wustl.edu/etj/articles/gibbs.paradox.pdf $\endgroup$ – Nathaniel Mar 6 '12 at 16:05
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$\begingroup$ @Frederic: Then you might also be interested in Chapter 10.1 of my book Classical and Quantum Mechanics via Lie algebras lanl.arxiv.org/abs/0810.1019, where I discuss the Gibbs paradox without any reference to anyone's knowledge. $\endgroup$ – Arnold Neumaier Mar 6 '12 at 17:28
• $\begingroup$ @ArnoldNeumaier : Thanks for the reference. I've just read the chapter 10.1. For me (but I'm biased towards information theory), the choice of a description level is precisely what is related to the physicist's knowledge. But I agree that it is a (useful) philosophical debate, and the whole question is linked to the study of the model choice itself. $\endgroup$ – Frédéric Grosshans Mar 7 '12 at 18:21
• $\begingroup$ By the way, the paper linked to in my answer is not directly related to Gibbs paradox, but is a computation of the work which can (probabilistically) be extracted from a system on which we have a partial knowledge (quantified by Shannon/Smooth-Rényi entropies) $\endgroup$ – Frédéric Grosshans Mar 7 '12 at 18:21
• $\begingroup$ @Frederic: I read the paper. - On Chapter 10.1: I think it is a big difference between knowledge (or ignorance, its absence) which is a subjective and very dubious concept, and model choice, which is a necessity in any physical investigation, not special to statistical mechanics. The point of my discussion is that the choice of a description level in statistical mechanics is not really different from that in mechanics - you need to include all observable degrees of freedom, and anything extra doesn't help. $\endgroup$ – Arnold Neumaier Mar 7 '12 at 19:20
When it comes to discussion of these matters, I make a following comment witch starts with the citation fom Landau-Lifshitz, book 5, chapter 5:
The averaging by means of the statisitcal matrix ... has a twofold nature. It comprises both the averaging due to the probalistic nature of the quantum description (even when as complete as possible) and the statistical averaging necessiated by the incompleteness of our information concerning the object considered.... It must be borne in mind, however, that these constituents cannot be separated; the whole averaging procedure is carried out as a single operation, and cannot be represented as the result of succesive averagings, one purely quantum-mechanical and the other purely statistical.
... and the following ...
It must be emphasized that the averaging over various $\psi$ states, which we have used in order to illustrate the transition from a complete to an incomplete quantum-mechanical description has only a very formal significance. In particular, it would be quite incorrect to suppose that the description by means of the density matrix signifies that the subsystem can be in various $\psi$ states with various probabilities and that the average is over these probabilities. Such a treatment would be in conflict with the basic principles of quantum mechanics.
So we have two statements:
Statement A: You cannot "untie" quantum mechanical and statistical uncertainty in density matrix.
(It is just a restatement of the citations above.)
Statement B: Quantum mechanical uncertainty cannot be expressed in terms of mere "ignorance" about a system.
(I'm sure that this is self-evident from all that we know about quantum mechanics.)
Therefore: Uncertainty in density matrix cannot be expressed in terms of mere "ignorance" about a system.
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$\begingroup$ The conclusion does not follow from the premises. I could just as easily say "1. quantum and statistical uncertainty cannot be untied in the density matrix formalism. 2. the uncertainty in a density matrix cannot be expressed as mere 'quantum' uncertainty (otherwise it would be a pure state). Therefore, 3. uncertainty in the density matrix cannot be expressed in terms of mere 'quantum' uncertainty." A much more reasonable conclusion is that some of the uncertainty in the density matrix is quantum and some is statistical; it's just impossible to untie them. $\endgroup$ – Nathaniel Mar 6 '12 at 16:16
• $\begingroup$ @Nathaniel I agree with your statement 3 and see no problem with it. It doesn't contradict anything. And also it doesn't in any way refute my statement. While the "much more reasonable conclusion" is just restatement of statement 1. $\endgroup$ – Kostya Mar 6 '12 at 16:25
• $\begingroup$ @Nathaniel: Why should your point 2 in your comment be true? Surely a density matrix is a quantum object and expresses quanutm uncertainty. The success of statistical mechanics together with the fact that you cannot untie the information in a density matrix rather suggests that the density matrix is the irreducible and objective quantum information, and the pure state is only a very special, rarely realized case. $\endgroup$ – Arnold Neumaier Mar 6 '12 at 17:18
• $\begingroup$ @Kostya, sorry - in that case I misunderstood - I interpreted you as saying that none of the uncertainty in the density matrix can be expressed in terms of ignorance. If you were only saying that some of it can't then no problem. (Though having said that, for someone who supports a non-local hidden variable interpretation, it can all be expressed as ignorance. Some people might find that more palatable than abandoning locality; I'm not sure whether I do or not.) $\endgroup$ – Nathaniel Mar 6 '12 at 17:52
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$\begingroup$ @ArnoldNeumaier consider a machine that mechanically flips a coin, then based on the result prepares an electron in one pure state (call it $|A\rangle$) or another ($|B\rangle$). To model the state of an electron from this machine you would use the density matrix $\frac{1}{2}\left( |A\rangle\langle A| + |B \rangle \langle B| \right)$. Surely this represents both the quantum uncertainty inherent in the pure states and your classical uncertainty about the outcome of a (hidden) coin flip. So at least in some situations some of the density matrix's uncertainty is ignorance. $\endgroup$ – Nathaniel Mar 6 '12 at 18:01
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b5807c83c929ac3a | Optics Research in Qatar Gaining Traction and Entering Collaborative Phase
Lasers running through a medium
Lasers running through a medium
While working for Bloomsbury Qatar Foundation Journals’ QScience media organization from 2011 to 2016, we served QNRF as a publisher of their newsletter. Although credits have not been assigned or retained, I researched, interviewed and wrote this article, and it exists in the QNRF newsletter archives. It is linked out to the archives directly before the following text. Researchers and organizations will attest to my work if contacted.
— Emily Alp
ARCHIVE. Compared to studies in the fields of biology and engineering, nonlinear dynamics might not be so obvious in terms of its worth. In reality, it is an area of physics research that permeates the natural world and a field integral to so many others. Dr. Milivoj Belic won the 2012 QNRF Research Team of the Year Award for his prolific contributions in this field, accounting for more than ten percent of Texas A&M at Qatar’s publications. His team’s specific focus is nonlinear optics, wherein they research the behavior of materials and laser light as they interact.
“What we do is manipulate photons, which are particles of light that can also be considered waves,” Dr. Belic said, “and we consider processes that happen in material when you shine laser light on them. So in essence we play with the wave phenomena. This is under the umbrella of quantum mechanics, but we do not do quantum mechanics; we do nonlinear optics.”
In linear optics photons do not “talk to” each other; however, in nonlinear optics they do, through the medium. Understanding the conversations—through the evolving language of nonlinear equations, i.e., nonlinear dynamics—helps researchers understand the material under study.
“In physics, very few things are done and finished once and for all, at least what has been done within the last century,” Dr. Belic said. “Most of those things are a never-ending story. Bit by bit you discover new things. But the problems and topics of research are there … an immense number of unsolved questions and half-baked answers.”
By running lasers through different types of materials such as gases, photo-refractive crystals, and nematic liquid crystals, Dr. Belic and his team observe the entire system as the light propagates, to get an idea of the material's response and the processes at play. The equations describing these processes are linked to waves and light and also with the response of the material—so it is both the response of the material and the behavior of the laser light, together, that are studied.
“We attack nonlinear equations; so it’s a mathematical physics problem. Now with such equations, it’s not like ‘aha, that’s it, we solved it!’—most often, it cannot be solved, at least not analytically. You have to try something different. Still, for many such equations we found ways to treat them analytically and this is something for which my team is becoming known internationally.”
The mathematical language around many physical phenomena is based on differential equations. A classic example would be the Schrödinger equation, which describes how the state of a quantum system changes over time. This is useful in linear systems and quantum mechanics. However, Dr. Belic explained that nonlinear dynamics is even more challenging than quantum mechanics. Specifically, it involves nonlinear Schrödinger equations and relies heavily on computers to crunch numbers because the responses in nonlinear systems are sometimes so erratic, evading analysis through the equations used in more predictable systems. Interestingly, most natural systems and materials require nonlinear thinking.
“Laws of physics are laws of nature,” Belic explained. “You have to master them and you have to know how to apply them. Mathematics is the language of nature. Things in nature are best explained through mathematics. Physics is essentially applied mathematics. In theoretical physics, you have to reason. But then you have experimental physics, so you have to experiment—to make a model, make predictions and test them. This can also turn out the other way around, where somebody finds something experimentally and then explains it.”
Whereas research in many fields is goal or product oriented, Dr. Belic said his team’s research is often curiosity-driven. A co-evolution of experiment and theory, the research requires a constant striving into the unknown.
“We are always trying to understand things, to contribute to a bank of understanding about nature at the basic level,” Dr. Belic explained. “We don’t produce gadgets—we want to know how they work. Here in Qatar, we had to start from scratch, so we started with theory. Some of our experiments are performed in other places such as Australia, the US, Serbia, France and Germany … we have a lot of collaborators.
“This work could contribute to other fields, not tomorrow, not today but in the foreseeable future,” he continued. “Newton formulated his laws in mathematical terms, and at the time people were asking ‘what is this for?’ It was a hundred years before people realized how useful they were.”
What excites Dr. Belic now is the potential to collaborate with researchers in other fields, enriching findings with the basic knowledge of physics and properties of materials.
“Before, physicians were doing their thing, mathematicians were doing their thing, chemists were doing their thing, and that approach was disjointed. But now we realize that if you want to make progress in brain research you cannot do so by the medical profession alone. For example, one of my collaborators is making a mathematical model of a brain cancer tumor. We all have to work together and that is the idea. And that is really the push nowadays with the funding agencies. Our team would like to go and collaborate with the Qatar Foundation institutes and has begun discussions with many of them.”
Establishing homegrown teams that are capable of producing great research requires a long period of cultivation. Qatar Foundation and TAMUQ have chosen this path and have generously supported the creation of high-quality team-oriented research centers. This turns the spotlight toward Qatar Foundation and TAMUQ as well as the whole Middle Eastern region. We greatly appreciate the strong support we have been given by TAMUQ and QNRF, and look forward to a bright future,” Dr. Belic said.
NPRP 25-6-7-2
Nonlinear Photonics for All-optical Telecommunication and Information Technologies. |
1af4ebd6c5df1532 | Main Optical Fiber Telecommunications. Volume VIB: Systems and Networks
Book cover Optical Fiber Telecommunications. Volume VIB: Systems and Networks
Optical Fiber Telecommunications. Volume VIB: Systems and Networks
, ,
Volume: VIB
Year: 2013
Edition: 6
Language: english
Pages: 1148
ISBN 10: 0123969603
ISBN 13: 9780123972378
ISBN: 012397237X
Series: Optics and photonics
File: EPUB, 19.69 MB
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Chapter 8
Multicarrier Optical Transmission
Xi Chen, Abdullah Al Amin, An Li and William Shieh, Department of Electrical and Electronic Engineering, The University of Melbourne, VIC 3010, Australia
In this chapter, we present an overview of multicarrier transmission and its application to optical communication, which has been in the focus of research for the last few years. Among all the multicarrier communication techniques, orthogonal frequency-division multiplexing (OFDM) is most well known and has been already adopted in many radio-frequency (RF) communication standards. With the advent of coherent detection technologies, optical multicarrier techniques, mainly optical OFDM has become an attractive candidate for high-speed optical transmission, especially at the emerging rates of 100 Gb/s to 1 Tb/s. In the following sections, we highlight some of the historical perspectives in the development of optical multicarrier technologies, followed by basic mathematical formulations for OFDM, the most popular multicarrier technique. We next present different variants of optical multicarrier transmission, including electronic and optical FFT-based realizations. We also highlight the problem of fiber nonlinearity in optical multicarrier transmission systems and present an analysis of fiber capacity under nonlinear impairments. Furthermore, we discuss applications of multicarrier techniques to long-haul systems, access networks, and free-space optical communication systems. Finally, we summarize with some possible research directions in implementing multicarrier technologies in optical transmission.
8.1 Historical perspective of optical multicarrier transmission
The concept of multicarrier transmission is attractive as an effective way of increasing data transmission rate by using many parallel carriers each carrying relatively slow data rate. It is an old concept which began in the form of subcarrier multiplexing (SCM), but with subcarrier spacing at more than multiple of symbol rate. With the invention of orthogonal frequency division multiplexing (OFDM) in the 1960s [1], and subsequent efficient realization by discrete Fourier transforms (DFT) [2], it emerged as an effective modulation format to combat inter-symbol-interference (ISI) from multi-paths or other dispersive effects. Since early 1990s, OFDM and its variant, discrete multi-tone (DMT), have been widely deployed in a number of wireless and cable transmission standards.
Even though the wavelength-division multiplexing (WDM) schemes in optical fiber transmission can also be considered as a form of multicarrier transmission [3], the application of multicarrier modulation within each optical channel to mitigate dispersion and gain high spectral efficiency (SE) is a relatively new trend in the optical communications community. Before the advent of coherent detection, each WDM channel used a single carrier (SC) modulation with simple generation and detection methods, but going to higher data rates such as 40 Gb/s or 100 Gb/s became problematic due to inter-symbol-interference (ISI) from chromatic and polarization mode dispersion (CD/PMD), which required precise dispersion management. A multicarrier technique called subcarrier multiplexing has been proposed to achieve high date rates but with low spectral efficiency and receiver sensitivity [4]. With the arrival of full-field optical signal capture by coherent detection and subsequent digital signal processing (DSP), many choices of advanced modulation formats have been explored. While SC-based optical transmission was quickly developed and has now become a commercial product for 40/100 Gb/s line cards [5], the strengths of multicarrier techniques to overcome linear dispersive effects like CD/PMD and the ability to achieve high SE are also recognized in the optical communications community. The increased interest in multicarrier transmission is evidenced by explosive growth of the number of publications on this topic from 2007 onwards.
8.1.1 Variations of optical multicarrier transmission methods
While some reports focused on the simple implementation of optical OFDM by directly modulating the OFDM waveform on optical signal intensity as early as 1996 [6] and more recently [7–9], others explored the flexibility of complex (intensity and phase combined) OFDM modulation combined with coherent detection [10,11]. These two methods are commonly known as direct detection optical (DDO-) OFDM and coherent optical (CO-) OFDM, respectively [12]. Together, they form an important class of optical multicarrier transmission where the waveform is electronically generated and demodulated by fast Fourier transform (FFT) in the digital domain. Because the signal is generated and detected by electronic digital-to-analog or analog-to-digital converters (DAC/ADC), respectively, the OFDM signal bandwidth is limited by the sampling rate of DAC/ADC.
Another class of optical multicarrier transmission relies on all-optically generated or demodulated orthogonal subcarriers, for which the signal bandwidth becomes the product of DAC/ADC sampling rates times the number of optical carriers employed. In this way, a large data rate (beyond 10s of Tb/s per channel) has been shown to be achievable. Some variations of this method include all-optical OFDM [13,14], no-guard interval OFDM [15,16], and coherent WDM [17].
In terms of their applications, optical multicarrier transmission has been demonstrated to be feasible for both short-reach and long-reach applications. Typically the short-reach, access network applications require a very cost-effective implementation rather than high SE [18,19]. On the other hand, the high-capacity long-haul applications can pay a premium for SE and reach [20,21] (typically over 1000 km). There has also been some study on free-space optical communication by multicarrier techniques [22–24].
8.1.2 Research trends in optical multicarrier transmission
In order to implement very high SE and capacity, a number of multiplexing methods have been proposed. The most straightforward method is to multiplex the multicarrier signal on two orthogonal polarizations of the single-mode fiber (SMF), which enables the doubling of data rate via a channel equalization method known as multiple input, multiple output (MIMO) [25]. MIMO is very well suited for multicarrier methods such as OFDM, where the linear interference from neighboring transmitters on the same frequency is reversed by a simple matrix multiplication [26]. A further extension of MIMO is the use of space-division multiplexing (SDM), whereby the signal travels from N transmitters to N receivers via an optical fiber link that can support N modes (spatial and polarization) [27–32]. Recently, this type of novel fiber has attracted much attention, and optical OFDM transmission is shown to be a very effective modulation format for such SDM systems.
Even though SC coherent optical systems are becoming commercially available in the data rates up to 100 Gb/s in the last few years, no such products have so far been developed for optical OFDM, and as such a question is asked about the future of this method in comparison to SC [33]. With the coherent detection and DSP methods, linear impairments have been shown to be mitigated along with the phase noise [12], which can affect OFDM because of its longer symbol lengths [34]. But another detrimental effect is the fiber nonlinearity which is exacerbated in multicarrier systems due to high peak-to-average ratio (PAPR). Similar to SC systems, such nonlinear effects could be compensated in multicarrier cases, and it has even been shown that for very high data rates, multicarrier systems could outperform SC ones [35].
As the demand is growing for further higher speeds (400 Gb/s or 1 Tb/s) amid lack of such high-bandwidth electronics, it has been accepted in the community that adoption of some form of multicarrier techniques is indispensable for these data rates. This multicarrier format for parallel transmission within a channel is called “superchannel” [36], which may also lead to grid-less or flexible-grid optical networks in the future [37,38]. In this way, multicarrier optical transmission methods continue to attract research attention for their versatility and flexibility to realize software-reconfigurable optical links and optical networks in the coming era.
8.2 OFDM Basics
One of the central features that set OFDM apart from SC modulation is its uniqueness of signal processing. SC technique has been employed in optical communication systems for the last three decades. As a result, OFDM signal processing may seem unfamiliar to an optical engineer at first glance. However, OFDM technology provides an exceptionally scalable pathway for migration to higher data rates. Once the algorithms and hardware designs are developed for the current generation product, it is very likely that these skill sets can be incorporated for the next generation product. In this respect, OFDM is a future-proof technology, and subsequently various aspects of OFDM signal processing deserve careful perusal.
For conventional optical SC systems, as the transmission speed increases, the requirement for optimal timing sampling precision becomes critical. Excessive timing jitter would place the sampling point away from the optimal, incurring severe penalty. On the other hand, for optical OFDM systems, a precise time sampling is not necessary. As long as an appropriate “window” of sampling points is selected containing an uncontaminated OFDM symbol, it is sufficient to remove inter-symbol-interference (ISI). However, this tolerance to sampling point imprecision is traded off against the stringent requirement of frequency offset and phase noise in OFDM systems.
In this section, we will lay out various aspects of OFDM signal processing associated with (i) OFDM basics and mathematical aspects of OFDM, (ii) DFT implementation, and (iii) cyclic prefix for OFDM. Following the description of signal processing, we will discuss the spectral efficiency of optical OFDM.
8.2.1 Mathematical formulation of an OFDM signal
In a generic OFDM system [1], any signal [image: image] can be represented as
[image: image] (8.1)
[image: image] (8.2)
[image: image] (8.3)
where [image: image] is the ith information symbol at the kth subcarrier, [image: image] is the waveform for the kth subcarrier, [image: image] is the number of subcarriers, [image: image] is the frequency of the subcarrier, [image: image] is the symbol period, and [image: image] is the pulse-shaping function. The optimum detector for each subcarrier could use a filter that matches the subcarrier waveform, or a correlator matched to the subcarrier. Therefore, the detected information symbol [image: image] at the output of the correlator is given by
[image: image] (8.4)
where [image: image] is the received time-domain signal. Typical multicarrier modulation uses non-overlapped band-limited signals and can be implemented with a bank of large number of oscillators and filters at both transmit and receive end [39]. The major disadvantage of this implementation is that it requires excessive bandwidth. This is because in order to design the filters and oscillators cost-effectively, the channel spacing has to be multiple of the symbol rate, greatly reducing the spectral efficiency. OFDM was investigated as a novel approach employing spectrally overlapped yet orthogonal signal set [1]. This orthogonality originates from straightforward correlation between any two subcarriers, given by
[image: image] (8.5)
It can be seen that if the following condition
[image: image] (8.6)
is satisfied, then the two subcarriers are orthogonal to each other. This signifies that these orthogonal subcarrier sets, with their frequencies spaced at multiple of inverse of the symbol rate, can be recovered with the matched filters in (8.4) without inter-carrier-interference (ICI), in spite of strong signal spectral overlapping.
8.2.2 Discrete Fourier transform implementation of OFDM
A fundamental challenge with OFDM is that a large number of subcarriers are needed so that the transmission channel affects each subcarrier as a flat channel. This leads to an extremely complex architecture involving many oscillators and filters at both transmit and receive end. Weinstein and Ebert first revealed that OFDM modulation/demodulation can be implemented by using inverse discrete Fourier transform (IDFT)/discrete Fourier transform (DFT) [2]. This is evident by studying OFDM modulation (8.2) and OFDM demodulation (8.4). Let us temporarily omit the index “i,” re-denote [image: image] as N in (8.2) to focus our attention on one OFDM symbol, and assume that we sample s(t) at every interval of [image: image], and the mth sample of s(t) from the expression (8.2) becomes
[image: image] (8.7)
Using orthogonality condition of (8.6), and the convention that
[image: image] (8.8)
and substituting (8.8) into (8.7), we have
[image: image] (8.9)
where F stands for Fourier transform and [image: image]. In a similar fashion, at the receive end, we arrive at
[image: image] (8.10)
where [image: image] is the received signal sampled at every interval of [image: image]. From (8.9) and (8.10) it follows that the discrete value of the transmitted OFDM signal s(t) is merely a simple N-point IDFT of the information symbol [image: image], and received information symbol [image: image] is a simple N-point DFT of the receive sampled signal. It is worth noting that there are two critical devices we have assumed for the DFT/IDFT implementation which are: (i) DAC, needed to convert the discrete value of [image: image] to the continuous analog value of s(t) and (ii) ADC, needed to convert the continuous received signal r(t) to discrete sample [image: image]. There are two fundamental advantages of DFT/IDFT implementation of OFDM. First, because of existence of efficient IFFT/FFT algorithm, the number of complex multiplications for IFFT in (8.9) and FFT in (8.10) is reduced from [image: image] to [image: image], increasing almost linearly with the number of subcarriers, N[40]. Second, a large number of orthogonal subcarriers can be generated and demodulated without resorting to much more complex RF oscillator and filter banks. This leads to a relatively simple architecture for OFDM implementation when large number of subcarriers are required. The corresponding architecture using DFT/IDFT and DAC/ADC is shown in Figure 8.1. At the transmit end, the input serial data bits are first converted into many parallel data pipes, each mapped onto corresponding information symbols for the subcarriers within one OFDM symbol, and the digital time domain signal is obtained by using IDFT, which is subsequently inserted with guard interval and converted into real-time waveform through DAC. The guard interval is inserted to prevent ISI due to channel dispersion. The baseband signal can be up-converted to an appropriate RF passband with an IQ mixer/modulator. At the receive end, the OFDM signal is first down-converted to baseband with an IQ demodulator, and sampled with an ADC, and demodulated by performing DFT and baseband signal processing to recover the data.
[image: image]
Figure 8.1 Conceptual diagram for (a) OFDM transmitter and (b) OFDM receiver.
It is worth noting that from (8.7), the OFDM signal [image: image] is a periodical function of [image: image] with a period of [image: image] Therefore, any discrete subcarrier set with its frequency components spanning one period of [image: image] is equivalent. Namely, in Eqs. (8.7) and (8.8), the subcarrier frequency [image: image] and its index k can be generalized as
[image: image] (8.11)
where [image: image] is an arbitrary integer. However, only two subcarrier index conventions are widely used, which are [image: image] and [image: image].
8.2.3 Cyclic prefix for OFDM
One of the enabling techniques for OFDM is the insertion of cyclic prefix [26,41]. Let us first consider two consecutive OFDM symbols that undergo a dispersive channel with a delay spread of [image: image]. For simplicity, assume each OFDM symbol includes only two subcarriers with the fast delay and slow delay spread at [image: image], represented by “fast subcarrier” and “slow subcarrier,” respectively. Figure 8.2a shows that inside each OFDM symbol, the two subcarriers, “fast subcarrier” and “slow subcarrier,” are aligned upon the transmission. Figure 8.2b shows the same OFDM signals upon the reception where the “slow subcarrier” is delayed by [image: image] against the “fast subcarrier.” We select a DFT window containing a complete OFDM symbol for the “fast subcarrier.” It is apparent that due to the channel dispersion, the “slow subcarrier” has crossed the symbol boundary leading to the interference between neighboring OFDM symbols, which is known as inter-symbol-interference (ISI). Furthermore, because the OFDM waveform in the DFT window for “slow subcarrier” is incomplete, the critical orthogonality condition for the subcarriers (8.5) is lost, resulting in an inter-carrier-interference (ICI) penalty.
[image: image]
Figure 8.2 OFDM signals (a) without cyclic prefix at the transmitter, (b) without cyclic prefix at the receiver, (c) with cyclic prefix at the transmitter, and (d) with cyclic prefix at the receiver.
Cyclic prefix was proposed to resolve the channel dispersion induced ISI and ICI [26]. Figure 8.2c shows insertion of a cyclic prefix by cyclic extension of the OFDM waveform into the guard interval, [image: image]. As shown in Figure 8.2c, the waveform in the guard interval is essentially an identical copy of that in the DFT window, with time-shifted by “[image: image]” forward. Figure 8.2d shows the OFDM signal with the guard interval upon reception. Let us assume that the signal has traversed the same dispersive channel, and the same DFT window is selected containing a complete OFDM symbol for the “fast subcarrier” waveform. It can be seen from Figure 8.2d, a complete OFDM symbol for “slow subcarrier” is also maintained in the DFT window, because a proportion of the cyclic prefix has moved into the DFT window to replace the identical part that has shifted out. As such, the OFDM symbol for “slow subcarrier” is a near-identical copy of the transmitted waveform with an additional phase shift. This phase shift is dealt with through channel estimation and will be subsequently removed before symbol decision. Now we arrive at the important condition for ISI-free OFDM transmission, given by
[image: image] (8.12)
It can be seen that to recover the OFDM information symbol properly, there are two critical procedures that need to be carried out: (i) selection of an appropriate DFT window, called DFT window synchronization and (ii) estimation of the phase shift for each subcarrier, called channel estimation or subcarrier recovery. Both signal processing procedures are actively pursued research topics, and their references can be found in these two books [26,41].
An elegant way to describe the cyclic prefix is to maintain the same expression as (8.2) for the transmitted signal s(t), but to extend the pulse shape function (8.3) to the guard interval, given by
[image: image] (8.13)
The corresponding time-domain OFDM symbol is illustrated in Figure 8.3 that shows one complete OFDM symbol comprised of observation period and cyclic prefix. The waveform within the observation period will be used to recover the frequency-domain information symbols.
[image: image]
Figure 8.3 The time-domain OFDM signal for one complete OFDM symbol.
8.2.4 Spectral efficiency for optical OFDM
In direct-detection optical OFDM (DDO-OFDM) systems, the optical spectrum is usually not a linear replica of the RF spectrum. Therefore, the optical spectral efficiency is dependent on the specific implementation method. We will turn our attention to the optical spectral efficiency for coherent optical OFDM (CO-OFDM) systems. In CO-OFDM systems, [image: image] subcarriers are transmitted in every OFDM symbol period of [image: image]. Thus the total symbol rate R for CO-OFDM systems is given by
[image: image] (8.14)
Figure 8.4a shows the spectrum of wavelength-division-multiplexed (WDM) channels each with CO-OFDM modulation, and Figure 8.4b shows the zoomed-in optical spectrum for each wavelength channel. We use the bandwidth of the first null to denote the boundary of each wavelength channel. The OFDM bandwidth, [image: image], is thus given by
[image: image] (8.15)
where [image: image] is the observation period (Figure 8.3). Assuming a large number of subcarriers are used, the bandwidth efficiency of OFDM η is found to be
[image: image] (8.16)
[image: image]
Figure 8.4 Optical spectra for (a) N wavelength-division-multiplexed CO-OFDM channels and (b) zoomed-in OFDM signal for one wavelength.
The factor of 2 accounts for two polarizations in the fiber. Using a typical value of 8/9 for α, we obtain the optical spectral efficiency factor η of 1.8 Baud/Hz. The optical spectral efficiency becomes 3.6 b/s/Hz if QPSK modulation is used for each subcarrier. The spectral efficiency can be further improved by using higher-order QAM modulation [42,43]. To practically implement CO-OFDM systems, the optical spectral efficiency will be reduced because of the need for a sufficient guardband between WDM channels taking account of laser frequency drifts (about 2 GHz). This guardband can be avoided by ensuring orthogonality across the WDM channels by using frequency locked sources [44].
8.3 Optical Multicarrier Systems Based on Electronic FFT
One of the major strengths of OFDM modulation format is its rich variation and ease of adaptation to a wide range of applications. This rich variation stems from the intrinsic advantages of OFDM modulation including dispersion robustness, ease of dynamic channel estimation and mitigation, high spectral efficiency, and capability of dynamic bit and power loading. Recent progress in optical OFDM is no exception. Despite the fact that OFDM has been extensively studied in the RF domain, it is rather surprising that the first report on optical OFDM in the open literature only appeared in 1996 by Pan et al. [6] where they presented in-depth performance analysis of hybrid AM/OFDM subcarrier-multiplexed (SCM) fiber-optic systems. The lack of interest in optical OFDM in the past is largely due to the fact that the digital signal processing power of CMOS integrated circuit (IC) had not reached the point where sophisticated OFDM signal processing could be performed economically.
In this section, we discuss optical multicarrier systems that rely on electronic FFT to process each subcarrier as opposed to using optical FFT. Based on the detection methods, we further classify such electronic FFT based multicarrier systems into two main categories, CO-OFDM and DDO-OFDM. In the remainder of the section, we will describe the fundamentals of these two optical multicarrier systems.
8.3.1 Coherent optical OFDM
CO-OFDM represents the ultimate performance in receiver sensitivity, spectral efficiency, and robustness against polarization dispersion, but requires high complexity in transceiver design. In the open literature, CO-OFDM was first proposed by Shieh and Athaudage [10], and the concept of the coherent optical MIMO-OFDM was formalized by Shieh et al. in [45]. The early CO-OFDM experiments were carried out by Shieh et al. for a 1000 km SSMF transmission at 8 Gb/s [46], and by Jansen et al. for 4160 km SSMF transmission at 20 Gb/s [47]. The principle and transmitter/receiver design for CO-OFDM are given below. Principle of coherent optical OFDM
The synergies between coherent optical communications and OFDM are twofold. OFDM enables channel and phase estimation for coherent detection in a computationally efficient way. Coherent detection provides linearity in RF-to-optical (RTO) up-conversion and optical-to-RF (OTR) down-conversion, much needed for OFDM. Consequently, CO-OFDM is a natural choice for optical transmission in the linear regime. A generic CO-OFDM system is depicted in Figure 8.5. In general, a CO-OFDM system can be divided into five functional blocks including (i) RF OFDM transmitter, (ii) RTO up-converter, (iii) the optical channel, (iv) the OTR down-converter, and (v) the RF OFDM receiver. The detailed architecture for RF OFDM transmitter/receiver has already been shown in Figure 8.1, which generates/recovers OFDM signals either in baseband or an RF band. Let us assume for now a linear channel where optical fiber nonlinearity is not considered. It is apparent that the challenges for CO-OFDM implementation are to obtain a linear RTO up-converter and linear OTR down-converter. It has been proposed and analyzed that by biasing the Mach-Zehnder modulators (MZMs) at null point, a linear conversion between the RF signal and optical field signal can be achieved [10,48]. It has also been shown that by using coherent detection, a linear transformation from optical field signal to RF (or baseband electrical) signal can be achieved [10,48–50]. Now by putting together such a composite system cross RF and optical domain [10,46,47], a linear channel can be constructed where OFDM can perform its best role of mitigating channel dispersion impairment in both RF domain and optical domain. In this section, we use the term “RF domain” and “electrical domain” interchangeably.
[image: image]
Figure 8.5 A CO-OFDM system in (a) direct up-down-conversion architecture and (b) intermediate frequency (IF) architecture. Coherent detection for linear down-conversion and noise suppression
As shown in Figure 8.6, coherent detection uses a six-port 90° optical hybrid and a pair of balanced photo-detectors. The main purposes of coherent detection are: (i) to linearly recover the I and Q components of the incoming signal and (ii) to suppress or cancel the common mode noise. Using a six-port 90° hybrid for signal detection and analysis has been practiced in RF domain for decades [51, 52], and its application to single-carrier coherent optical systems can be also found in [49, 50]. In what follows, in order to illustrate its working principle, we will perform an analysis of down-conversion via coherent detection assuming ideal conditions for each component shown in Figure 8.6.
[image: image]
Figure 8.6 Coherent detection using an optical hybrid and balanced photo-detection.
The purpose of the four output ports of the 90° optical hybrid is to generate a 90° phase shift between I and Q components, and 180° phase shift between balanced detectors. Ignoring imbalance and loss of the optical hybrid, the output signals [image: image] can be expressed as
[image: image] (8.17)
where [image: image] and [image: image] are, respectively, the incoming signal and local oscillator (LO) signal. We further decompose the incoming signal into two components: (i) the received signal when there is no amplified spontaneous noise (ASE), [image: image] and (ii) the ASE noise, [image: image], namely
[image: image] (8.18)
We first study how the I component of the photo-detected current is generated, and the Q component can be derived accordingly. The I component is obtained by using a pair of photo-detectors, PD1 and PD2, in Figure 8.6, whose photocurrent [image: image] can be described as
[image: image] (8.19)
[image: image] (8.20)
[image: image] (8.21)
[image: image] (8.22)
where [image: image] and [image: image] are the average power and relative intensity noise (RIN) of the LO laser, and “Re” or “Im” denotes the real or imaginary part of a complex signal. For simplicity, the photo-detection responsivity is set to unity. The three terms at the right-hand side of (8.21) represent signal-to-signal beat noise, signal-to-ASE beat noise, and ASE-to-ASE beat noise. Because of the balanced detection, using Eqs. (8.19) and (8.20), the I component of the photocurrent becomes
[image: image] (8.23)
Now the noise suppression mechanism becomes quite clear because the three noise terms in (8.21) and the RIN noise in (8.22) from a single detector are completely canceled via balanced detection. Nevertheless, it has been shown that coherent detection can be performed by using a single photo-detector, but at the cost of reduced dynamic range [53].
In a similar fashion, the Q component from the other pair of balanced detectors can be derived as
[image: image] (8.24)
Using the results of (8.23) and (8.24), the complex detected signal [image: image] consisting of both I and Q components becomes
[image: image] (8.25)
From (8.25), the linear down-conversion process via coherent detection becomes quite clear; the complex photocurrent [image: image] is in essence a linear replica of the incoming complex signal that is frequency down-converted by a local oscillator frequency. Thus with linear coherent detection at receiver and linear generation at transmitter, complex OFDM signals can be readily transmitted over the optical fiber channel.
8.3.2 Direct-detection optical OFDM
A direct-detection optical OFDM (DDO-OFDM) aims for simpler transmitter/receiver than CO-OFDM for lower costs. It has many variants which reflect the different requirements in terms of data rates and costs from a broad range of applications. For instance, the first report of the DDO-OFDM [6] takes advantage of the fact that the OFDM signal is more immune to the impulse clipping noise seen in CATV networks. Another example is single-side-band (SSB)-OFDM which has been recently proposed by Lowery et al. and Djordjevic et al. for long-haul transmission [7,9]. Tang et al. have proposed an adaptively modulated optical OFDM (AMOOFDM) that uses bit and power loading showing promising results for both multimode fiber and short-reach SMF fiber links [54–56]. The common feature for DDO-OFDM is use of a simple square-law photodiode at the receiver. DDO-OFDM can be divided into two categories according to how optical OFDM signal is being generated: (i) linearly mapped DDO-OFDM (LM-DDO-OFDM) where the optical OFDM spectrum is a replica of baseband OFDM, and (ii) nonlinearly mapped DDO-OFDM (NLM-DDO-OFDM) where the optical OFDM spectrum does not display a replica of baseband OFDM. In what follows, we discuss the principles and design choices for these two classes of direct-detection OFDM systems. Linearly mapped DDO-OFDM
As shown in Figure 8.7, the optical spectrum of an LM-DDO-OFDM signal at the output of the O-OFDM transmitter is a linear copy of the RF OFDM spectrum plus an optical carrier that is usually 50% of the overall power. The position of the main optical carrier can be one OFDM spectrum bandwidth away [7,57] or right at the end of the OFDM spectrum [58,59]. Formally, such type of DDO-OFDM can be described as
[image: image] (8.26)
where s(t) is the optical OFDM signal, [image: image] is the main optical carrier frequency, Δf is guardband between the main optical carrier and the OFDM band (Figure 8.7), and α is the scaling coefficient that describes the OFDM band strength related to the main carrier. [image: image] is the baseband OFDM signal given by
[image: image] (8.27)
where [image: image] and [image: image] are, respectively, the OFDM information symbol and the frequency for the kth subcarrier. For explanatory simplicity, only one OFDM symbol is shown in (8.27). After the signal passes through fiber link with chromatic dispersion, the OFDM signal can be approximated as
[image: image] (8.28)
[image: image] (8.29)
where [image: image] is the phase delay due to chromatic dispersion for the kth subcarrier. Dt is the accumulated chromatic dispersion in units of ps/pm, [image: image] is the center frequency of optical OFDM spectrum, and c is the speed of light in a vacuum. At the receiver, the photodetector can be modeled as a square-law detector and the resultant photo-current signal is
[image: image] (8.30)
The first term is a DC component that can be easily filtered out. The second term is the fundamental term consisting of linear OFDM subcarriers that are to be retrieved. The third term is the second-order nonlinearity term that needs to be removed.
[image: image]
Figure 8.7 Illustration of linearly mapped DDO-OFDM (LM-DDO-OFDM) where the optical OFDM spectrum is a replica of the baseband OFDM spectrum.
There are several approaches to minimize the penalty due to the second-order nonlinearity term:
a. Offset SSB-OFDM: Sufficient guardband is allocated such that the second-term and third-term RF spectra are non-overlapping. As such, the third term in Eq. (8.30) can be easily removed using a RF or DSP filter, as proposed by Lowery et al. in [7].
b. Baseband optical SSB OFDM: α coefficient is reduced as much as possible such that the distortion as a result of the third-term is reduced to an acceptable level. This approach has been adopted by Djordjevic et al. [9] and Hewitt et al. [58].
c. Subcarrier interleaving: From Eq. (8.30), it follows that if only odd subcarriers are filled, i.e. [image: image] is nonzero only for the odd subcarriers, the second-order intermodulation will be at even subcarriers, which are orthogonal to the original signal at the odd subcarrier frequencies. Subsequently, the third-term does not produce any interference. This approach has been proposed by Peng et al. [60].
d. Iterative distortion reduction: The basic idea is to go through a number of iterations of estimation of the linear term, and compute the second-order term using the estimated linear term, and removing the second-order term from the right hand side of Eq. (8.30). This approach has been proposed by Peng et al. [59].
There are advantages and disadvantages among all these four approaches. For instance, Approach B has the advantage of better spectral efficiency, but at the cost of sacrificing receiver sensitivity. Approach D has both good spectral efficiency and receiver sensitivity, but has a burden of computational complexity.
Figure 8.8 shows one offset SSB-OFDM proposed by Lowery et al. in [61]. They show that such DDO-OFDM can mitigate an enormous amount of chromatic dispersion up to 5000 km standard SMF (SSMF) fiber. The proof-of-concept experiment was demonstrated by Schmidt et al. from the same group for 400 km DDO-OFDM transmission at 20 Gb/s [57].The simulated system is 10 Gb/s with 4-QAM modulation with a bandwidth of around 5 GHz [61]. In the electrical OFDM transmitter, the OFDM signal is up-converted to an RF carrier at 7.5 GHz generating an OFDM band spanning from 5–10 GHz. The RF OFDM signal is fed into an optical modulator. The output optical spectrum has the two side OFDM bands that are symmetric across the main optical subcarrier. An optical filter is then used to filter out one OFDM side band. This single-side band (SSB) is critical to ensure there is one-to-one mapping between the RF OFDM signal and the optical OFDM signal. The power of main optical carrier is optimized to maximize the sensitivity. At the receiver, only one photo-detector is used. The RF spectrum of the photocurrent is depicted as an inset in Figure 8.8. It can be seen that the second-order intermodulation, the third-term in Eq. (8.30), is from DC to 5 GHz, whereas the OFDM spectrum, the second term in Eq. (8.30), spans from 5 GHz to 10 GHz. As such, the RF spectrum of the intermodulation does not overlap with the OFDM signal, signifying that the intermodulation does not cause detrimental effects after proper electrical filtering.
[image: image]
Figure 8.8 Direct-detection optical OFDM (DDO-OFDM) long-haul optical communication systems. After Ref. [61]. Nonlinearly mapped DDO-OFDM (NLM-DDO-OFDM)
The second class of DDO-OFDM is nonlinearly mapped OFDM, which means that there is no linear mapping between the electric field (baseband OFDM) and the optical field. Instead, NLP-DD-OFDM aims to obtain a linear mapping between baseband OFDM and optical intensity. For simplicity, we assume generation of NLM-DDO-OFDM using direct modulation of a DFB laser, the waveform after the direct modulation can be expressed as [62]
[image: image] (8.31)
[image: image] (8.32)
[image: image] (8.33)
[image: image] (8.34)
where E(t) is the optical OFDM signal, A(t) and P(t) are the instantaneous amplitude and power of the optical OFDM signal, [image: image] is the transmitted information symbol for the kth subcarrier, C is the chirp constant for the direct modulated DFB laser [62], [image: image] is the IF frequency for the electrical OFDM signal for modulation, m is the optical modulation index, [image: image] is a scaling constant to set an appropriate modulation index m to minimize the clipping noise, and [image: image] is the baseband OFDM signal. Assuming that the chromatic dispersion is negligible, the detected current is
[image: image] (8.35)
Equation (8.35) shows that the photocurrent contains a perfect replica of the OFDM signal sB(t) with a DC current. We also assume that modulation index m is small enough that clipping effect is not significant. Equation (8.35) shows that by using NLM-DDO-OFDM with no chromatic dispersion, the OFDM signal can be perfectly recovered. The fundamental difference between the NLM- and LM-DDO-OFDM can be gained by studying their respective optical spectra. Figure 8.9 shows the optical spectra of NLM-DDO-OFDM using (a) direct modulation of a DFB laser with the chirp coefficient C of 1 in (8.31) and modulation index m of 0.3 in (8.34) and (b) offset SSB-OFDM. It can be seen that, in sharp contrast to SSB-OFDM, NLM-DDO-OFDM has a multiple of OFDM bands with significant spectral distortion. Therefore, there is no linear mapping from the baseband OFDM to the optical OFDM. The consequence of this nonlinear mapping is fundamental, because when any type of the dispersion, such as chromatic dispersion, polarization dispersion, or modal dispersion, occurs in the link, the detected photocurrent can no longer recover the linear baseband OFDM signal. Namely, any dispersion will cause the nonlinearity for NLM-DD-OFDM systems. In particular, unlike SSB-OFDM, the channel model for direct-modulated OFDM is no longer linear under any form of optical dispersion. Subsequently, NLM-DD-OFDM is only fit for short-haul application such as multimode fiber for local-area networks (LAN), or short-reach single-mode fiber (SMF) transmission. This class of optical OFDM has attracted attention recently due to its low cost. Some notable works of NLM-DD-OFDM are experimental demonstrations and analysis of optical OFDM over multimode fibers [55,56,63] and compatible SSB-OFDM (CompSSB) proposed by Schuster et al. to achieve higher spectral efficiency than offset SSB-OFDM [64].
[image: image]
Figure 8.9 Comparison of optical spectra between (a) NLM-DDO-OFDM through direct-modulation of DFB laser and (b) externally modulated offset SSB DDO-OFDM. The chirp constant C of 1 and the modulation index m of 0.3 are assumed for direct-modulation in (a). Both OFDM spectrum bandwidths are 5 GHz comprising 256 subcarriers.
8.4 Optical Multicarrier Systems Based on Optical Multiplexing
Multicarrier technique has been recognized as a powerful means of combating channel dispersion as illustrated in Sections 8.2 and 8.3, which is focused on electronic DSP implementation. However, there is an increasing gap between the electronic DSP bandwidth and the required baud rate in order to accommodate theexponential growth of the Internet traffic. It is widely believed that Tb/s-class Ethernet will emerge within the next decade, which cannot be directly realized using electronic DSP alone. Another layer of multiplexing seems to be inevitable to achieve Tb/s Ethernet transport. In the last few years, a wide variety of techniques have been proposed and demonstrated in the area of optical multiplexing for ultrahigh-speed and high spectral-efficiency transmission. They include: (i)all-optical OFDM where FFT is realized using optical circuits instead of electronic ones, (ii)optical superchannel where individual optical carriers are optically multiplexed as a Tb/s-class transport entity, and (iii) optical frequency-division multiplexing where multiple unlocked wavelengths are packed into one ITU wavelength slot. In this section, we capture the recent progress on optical multicarrier systems based on optical multiplexing utilizing these three techniques.
8.4.1 All-optical OFDM
All-optical OFDM attracts attention for its advantages of fast processing and low power consumption [65–68]. As the name of all-optical OFDM suggests, the IFFT/FFT digital signal processing to combine or split OFDM subcarriers is done optically. Figure 8.10 shows a typical architecture for all-optical OFDM transmission. For the transmitter, individual laser sources at equidistant frequencies serve as subcarriers. On each subcarrier an IQ modulator encodes the information for transmission [65]. The modulated subcarriers are then combined with an optical coupler to form the optical OFDM signal. Conversely, at the receiver side, an optical FFT circuit can be used to separate the subcarriers, which performs both serial-to-parallel conversion and FFT in the optical domain using a cascade of delayed interferometers (DIs) with subsequent time gates [65] or using an arrayed-waveguide grating router as reported in [66]. After the optical FFT, the separated subcarriers are optically amplified and coherently detected to perform demodulation and symbol decision.
[image: image]
Figure 8.10 Schematic configuration for all-optical OFDM transmitter and receiver [65]. DI: Delayed interferometer, EAM: Electro-absorption modulator, OMA: Optical modulation analyzer.
By using the optical FFT/IFFT, the required electronic DSP only needs to process relatively narrow bandwidth for each low data-rate subcarrier as opposed to performing FFT across the entire signal spectrum, greatly reducing the bandwidth requirement and complexity of electronics. It has been also claimed that the power consumption of such all-optical OFDM systems is effectively reduced compared with conventional optical OFDM transceivers, since the power-hungry FFT algorithms are implemented in the optical domain.
8.4.2 Optical superchannel
The term “optical superchannel” is first given in [36] although similar techniques have already been proposed and demonstrated for Tb/s-class transport [17,69–72]. Optical superchannel commonly refers to a high data rate signal that originates from a single laser source and consists of multiple frequency-locked carriers which are synchronously modulated. By maintaining a suitable orthogonal condition among the modulated carriers, coherent crosstalk can be eliminated. Figure 8.11 illustrates an optical superchannel transceiver, and it has a similar configuration as the all-optical OFDM generator introduced in Section8.3.1. The frequency locking and orthogonal condition are kept among all the optical tones. On the receiver side, the multiple signal bands can be detected either jointly or separately. After transmission, the superchannel signal is divided by a 1:M coupler (M can either equal to or be smaller than the number of bands). Each of the M parts is coherently detected and corresponding DSP is applied to the down-converted signal for signal recovery. Broadly speaking, coherent WDM (Co-WDM) can be also considered as one type of superchannel where the phase of multiple carriers is locked [17].
[image: image]
Figure 8.11 Schematic diagram of an optical superchannel transmitter and receiver [36]. LO: Local oscillator, PD: Photodiode, ADC: Analog-to-digital converter, DSP: Digital signal processing.
8.4.3 Optical frequency division multiplexing
One could also use multiple frequencies within one ITU grid, provided they do not cause significant interference. It was adopted in a commercial line card product [73], and we call this method optical frequency-division multiplexing. In current systems, the optical bandwidth of a path is determined in part by standards in adherence to the International Telecommunication Union (ITU) Channel Grid [74], and in part by the technologies of filters and wavelength-selective switches used to optically steer channels through a network. Using [image: image] different optical frequencies permits [image: image] times the bit rate to be transmitted within the filtered optical path. The maximum value of [image: image] is limited by the ratio of the optical filter bandwidth of the path to the information bandwidth of the modulated optical carrier. The value of [image: image] has been chosen for the commercially available 100 Gb/s product [73]. The optical spectrum of this 100 Gb/s solution is shown as the center channel (b) in Figure 8.12 accompanied by the spectra of single-carrier 10 Gb/s (a) and dual-polarization 40 Gb/s (c) channels. Each spectrum is centered on a 50 GHz ITU channel. The concept of multiple frequency carriers can be extended to arbitrary values of [image: image], provided that the system designer has freedom to vary the optical filter width and channel positions. The extreme case is a continuum of low symbol-rate carriers, densely filling the spectrum, such as OFDM.
[image: image]
Figure 8.12 Optical frequency-division multiplexing comparison: (a) 10 Gb/s single-polarization single-carrier, (b) 100 Gb/s coherent dual-polarization dual-carrier, and (c) 40 Gb/s coherent dual-polarization single-carrier. All three channels are centered on the 50 GHz ITU grid [73].
8.5 Nonlinearity in Optical Multicarrier Transmission
Optical communication has recently witnessed the trend of the signal bandwidth expansion, high spectral efficiency (SE), and ultra-long haul transmission [71,75]. As a result, fiber nonlinear noise becomes one of the major concerns for optical transmission. Especially, there is a common belief that nonlinear impairments are more prominent in multicarrier systems such as optical OFDM due to its high PAPR than in single-carrier systems. In this section, we begin with the review of the recent progress on high SE transmission using optical multicarrier systems. We then discuss the optimal symbol rate for fiber nonlinearity in multicarrier systems. We also show analytical expressions for fiber nonlinearity noise and information spectral limit in multicarrier systems. Finally, we describe a few approaches for nonlinearity mitigation for multicarrier transmission.
8.5.1 High spectral-efficiency long-haul transmission
In the past few years, there have been impressive advances in experimental demonstrations of high SE multicarrier transmission employing narrow frequency guard interval and higher-order modulation format [20,71,76–78]. Table 8.1 summarizes the most recent records for high SE multicarrier transmission. It can be observed that while there is a steady advance in SE, there exists a trade-off between SE and reach. Generally speaking, high SE transmission systems can be achieved by either shrinking frequency guardband between wavelength channels, or utilizing higher-order modulation formats. However, both of these two approaches lead to increased sensitivity to fiber nonlinearity. Consequently, the nonlinearity impact and its mitigation strategy become a critical problem for SE multicarrier transmission.
Table 8.1
Experimental demonstrations of high spectral efficiency transmission.
[image: Image]
8.5.2 Optimal symbol rate in multicarrier systems
It is known that the PAPR is one of the key characteristics which affect the performance of optical transmission due to fiber nonlinearity. It has been shown that multicarrier signals suffer from excessive nonlinear noise during fiber transmission due to their high PAPR. Though by using special algorithms such as hard-clipping the PAPR can be lowered at the transmitter. During transmission the PAPR can become very high again due to fiber dispersion. It is sensible to use not only PAPR reduction algorithms at the transmitter, but also strategies to maintain the low PAPR during transmission. For instance, for ultrahigh-speed systems such as 100-Gb/s and beyond, the fiber dispersion plays a critical role, inducing fast walk-off between subcarriers [79]. The PAPR of such a signal has a transient value during transmission due to fiber link dispersion, which renders the PAPR reduction at the transmitter ineffective. Nevertheless, if the PAPR mitigation approach is performed on a subband basis, due to the fact each subband has a much narrower bandwidth, the signal within each subband can be relatively undistorted over comparatively longer distances. This results in less inter-band and intra-band nonlinearity. In a nutshell, PAPR reduction on a subband basis will be more effective than on an entire spectrum basis.
Based on the above understanding, it is natural to predict there is a best trade-off of subband bandwidth within which the PAPR mitigation should be performed: On one hand, if the subband bandwidth is too broad, the PAPR reduction will not be effective due to the fiber dispersion. On the other hand, if the subbands are too narrow, the neighboring bands will interact just as narrowly spaced OFDM subcarriers, generating large inter-band crosstalk due to narrow subband spacing and incurring a large penalty. In the following, we describe the optimal subband allocation for optical OFDM signal transmission.
There are two mechanisms that may contribute to the optimal subband bandwidth. It relates to the four-wave mixing (FWM) efficiency which was derived in [80] and [81]. Due to the third-order fiber nonlinearity, the interaction of subcarriers at the frequencies of [image: image], and [image: image] produces a mixing product at the frequency of [image: image]. The magnitude of the FWM product for [image: image] spans of the fiber link is given by [80]
[image: image] (8.36)
where [image: image] is the degeneration factor which equals 6 for non-degenerate FWM and 3 for degenerate FWM. [image: image] is the input power at the frequency of [image: image] and L are, respectively, the loss coefficient and length of the fiber per span, respectively, γ is the third-order nonlinearity coefficient of the fiber, and [image: image] is the effective fiber length given by [image: image]. [image: image] is the FWM coefficient which has a strong dependence on the relative frequency spacing between the FWM components [image: image] and [image: image] are the phase mismatch in the transmission fiber, the subscript 1 stands for the parameters associated with the dispersion compensation fiber (DCF). [image: image] is the intra-span FWM coefficient and [image: image] originates from inter-span nonlinear interference. The derivation of [image: image] and [image: image] are shown in [89,90]. The FWM efficiency η will be discussed later in this section in more detail.
Figure 8.13a shows [image: image] with varying fiber chromatic dispersions (CD) in a [image: image] km fiber link. It can be seen that the 3-dB bandwidth of [image: image] is about 11, 8, 4.8 GHz for CDs of 3, 6, and 17 ps/nm/km, respectively. Figure 8.13b shows the FWM coefficient [image: image] as a function of CD compensation ratio (CR) for a transmission fiber with CD of 17 ps/nm/km. For uncompensated systems (CR = 0%), the FWM 3-dB bandwidth is 1.8 GHz whereas for typical CD compensated systems with CR = 95%, the 3-dB bandwidth increases to 8 GHz. The idea behind the optimization of subband bandwidth is to maintain the FWM efficiency close to its maximum value within each subband while minimizing the intra-band FWM efficiency. Therefore, we could use the 3-dB bandwidth of FWM efficiency as the “ballpark” estimate of optimal subband bandwidth, and in that sense, Figures 8.13a and b give the approximate estimate of the optimal subband bandwidth. The 3-dB bandwidth increases with CD compensation, and therefore we anticipate that the optimal subband bandwidth of CD-uncompensated systems is narrower than CD-compensated systems.
[image: image]
Figure 8.13 Four-wave mixing efficiency coefficients for a 10 × 100 km transmission link, with fiber loss coefficient of 0.2 dB/km. (a) due to the transmission fiber per span for different CD and (b) due to phase array effect. CD of 17 ps/nm/km. CR: chromatic dispersion compensation ratio.
We employ the DFT-Spread OFDM (DFT-S-OFDM) modulation to discuss the optimal symbol rate of optical multi-carrier transmission. The principle of DFT-S-OFDM is addressed in detail in [35,82]. A polarization division multiplexed 107-Gb/s coherent optical multiband DFT-Spread OFDM system is used in the simulation. The simulated transmission parameters are: fiber length of 100 km per span, DSSMF = 16 ps/nm/km, [image: image] = 0.2 dB/km, [image: image], noise figure of optical amplifiers of 6 dB, eight WDM channels with 50-GHz channel spacing, 64 number of subcarriers in each subband when the number of subbands is over 8, and QPSK modulation on each subcarrier. There is no dispersion compensation in this transmission simulation. A cyclic prefix ratio of 1/16 is used for all the cases. We simulate the link performance as a function of the optimal number of subbands, or equivalently, the optimal subband bandwidth for 107-Gb/s multiband CO-OFDM signal. Figure 8.14 shows the Q performance at fiber input powers of 4 dBm and 6 dBm for single-wavelength 107-Gb/s multi-band DFT-S-OFDM transmission. It can be seen that the optimal number of bands is close to 8, corresponding to subband bandwidth of 3.6 GHz in a 107-Gb/s multi-band DFT-S-OFDM transmission case using QPSK.
[image: image]
Figure 8.14 Q factor as a function of number of bands at 4- and 6-dBm launch powers with 107-Gb/s single-channel transmission over 10 × 100 km uncompensated SMF link.
8.5.3 The information spectral limit in multicarrier systems
By introducing the orthogonality among the multiple carriers and reducing the frequency guardbands between wavelength channels, the spectral efficiency can be maximized. In such systems, wavelength channels can be either continuously spaced without frequency guardband [20,83,84], or densely spaced with extremely small frequency guardband [72,76]. We now discuss the limits of information spectral efficiency in such multicarrier systems and summarize the outcome of a few theoretical works [85–88] where nonlinear launch power and information capacity are derived in analytical form. From these concise closed-form solutions, we can grasp the dependence of nonlinear transmission performance on some major system parameters such as chromatic dispersion and dispersion compensation ratio for dual-polarization CO-OFDM systems [89,90]. Derivation of analytical expressions for FWM noise in dual-polarization multicarrier transmission systems
1 FWM noise power density
As shown in [89,90], the nonlinear multiplicative noise spectral density [image: image] is given by
[image: image] (8.37)
[image: image] (8.38)
[image: image] (8.39)
where γ is the fiber nonlinear coefficient, I is the launch power density, [image: image] represents coefficient of chromatic dispersion, and ζ is the dispersion compensation ratio. (L) and [image: image] stand for the length of each fiber span and the number of transmission spans, respectively. α is the fiber loss coefficient. B represents the total width of signal bandwidth. We call [image: image] the noise enhancement factor accounting for the FWM noise interference among different fiber spans. We will discuss this interesting nonlinear enhancement factor [image: image] in more detail in the next section. This nonlinear noise power density [image: image] of Eq. (8.37) can be expressed in a more concise form with the definition of nonlinear characteristic power density [image: image] as follows:
[image: image] (8.40)
2 Signal-to-noise ratio and spectral efficiency limit in the presence of nonlinearity
The signal power in the presence of the nonlinear interference can be expressed as [90]
[image: image] (8.41)
The noise can be considered as the summation of the white optical amplified spontaneous noise (ASE), [image: image] and the FWM noise, and is given by [90]
[image: image] (8.42)
[image: image] (8.43)
where [image: image] is the spontaneous emission noise factor equal to half of the noise figure of the optical amplifier NF in the ideal case, h is the Planck constant, and υ is the light frequency. The factor of 2 in Eq. (8.42) accounts for the unpolarized ASE noise. The signal power and noise power density in Eqs. (8.41) and (8.42) include the contribution from both polarizations. The signal-to-noise ratio (SNR) is thus given by
[image: image] (8.44)
For the SNR values larger than 10, Eq. (8.44) can be approximated as
[image: image] (8.45)
The simplification is generally valid for the case of interests where the signal power density is much smaller than [image: image].
It is verified in [87,88] that the FWM noise is of Gaussian distribution. Under the assumption of Gaussian noise distribution, the information spectral efficiency (defined as the maximum information capacity C normalized to bandwidth B) for dual polarization is readily given by [91]
[image: image] (8.46)
From Eq. (8.46), the maximum spectral efficiency [image: image] in the presence of fiber nonlinearity can be easily shown as
[image: image] (8.47)
3 Optimal launch power density, maximum Q, and nonlinear threshold of launch power density
In Eq. (8.46), the maximum possible spectral efficiency is obtained. However, in practice, the performance is always lower because of the practical implementation of modulation and coding. Next we derive a few important parameters that are important from the system design perspective. The first one is the maximum achievable Q factor. Under the Gaussian noise assumption and QPSK modulation, the Q factor is equal to the SNR given by [89–91]
[image: image] (8.48)
The optimum launch power density is another important parameter and is defined as the launch power density achieving where the maximum Q takes place. By simply differentiating Q of Eq. (8.48) over I, and setting it to zero, we obtain the optimum launch power density [image: image] and the maximum Q as
[image: image] (8.49)
[image: image] (8.50)
One of the inconveniences of using the expression in Eq. (8.49) is that it is dependent on the amplifier noise figure. The other commonly used term is nonlinear threshold launch power density that is defined as the maximum launch power density at which the BER due to the nonlinear noise can no longer be corrected by a certain type of forward-error-correction (FEC) code. For standard Reed-Solomon code RS (255, 239), the threshold Q is 9.8 (dB), or linear [image: image] of 3.09. In Eq. (8.48), setting [image: image] to zero and [image: image], we arrive at the nonlinear threshold power density
[image: image] (8.51)
where [image: image] is the correctable linear Q for a specific FEC. Application of the closed-form expressions
1 System Q factor and optimum launch power
Because the concise closed-form expressions are available, we can readily apply them to identify the system performance as a function of system parameters including fiber dispersion, number of spans, dispersion compensation ratio, and overall bandwidth. In this subsection, we will give examples of estimating the achievable system Q factor, optimum launch power, information spectral efficiency, and multi-span noise enhancement factor.
The significance of having closed-form formulas of Eqs. (8.49) and (8.50) for system Q factor and optimum launch power density is that they provide useful scaling rules for system designing. From Eqs. (8.49) and (8.50), it follows that for every 2 ×(3 dB) increase in fiber dispersion, there is 1 dB increase in the optimal launch power density and the achievable Q; for every 3 dB increase in fiber nonlinear coefficient γ, there is 2 dB decrease in the optimal launch power and achievable Q.
As an illustrative example, we use the analytical expressions to generate the optimum launch power and achievable Q for a number of typical dispersion maps. We assume the following parameters: 16 wavelength channels, each covering 31-GHz bandwidth, giving total bandwidth B of 496 GHz; OFDM subcarrier frequency spacing of 85 MHz; QPSK modulation for each subcarrier; no frequency guardband between wavelength channels; 10-span of 100 km fiber link; fiber loss [image: image] of 0.2 dB/km; nonlinear coefficient [image: image]; noise figure of the amplifier of 6 dB. Three transmission systems are investigated: (i) SSMF-type system with CD of 16 ps/nm/km without any dispersion compensation, abbreviated as “system I,” (ii) CD of 16 ps/nm/km, but with dispersion 95% compensated per span, abbreviated as system II, and (iii) non-zero dispersion-shifted type fiber with CD of 4 ps/nm/km, abbreviated as “system III.” As shown in Figure 8.15a, system I has the best performance due to large local dispersion and no per-span dispersion compensation. The advantage of system I over system II increases with the increase of the number of spans, for instance from 0 dB to 2.4 dB when the reach increases from single-span to 10 spans. The advantage of system I over system III is maintained at 1.7 dB independent of the number of spans. Figure 8.15b shows the optimal launch power versus number of spans. The optimum launch powers for non-compensated systems, systems I and III, are constant. This is because both the linear and nonlinear noises increase linearly with the number of the spans that leads to the optimum power independent of the number of spans. However, for the dispersion compensated system II, the optimum launch power density decreases with the number of spans due to the multi-span noise enhancement effect. Another interesting observation from Eqs. (8.49) and (8.50) is that both the optimal Q factor and launch power have very weak dependence on the overall system bandwidth: proportional to 1/3 power of logarithm of the overall bandwidth. It can be easily shown that for both systems I and III, the Q is decreased by only about 0.7 dB with the 10-fold increase of the overall system bandwidth from 400 GHz to 4000 GHz whereas system II incurs a larger decrease of the Q factor of 0.84 dB with the same bandwidth increase.
[image: image]
Figure 8.15 (a) The maximum Q factor and (b) the optimal launch power density versus number of spans with various dispersion maps. CD: chromatic dispersion. CR: (CD) compensation ratio.
2 Information spectral efficiency
The information spectral efficiency is important as it represents the ultimate bound of what we can achieve by employing all possible modulations (of course not limited to QPSK) and codes. For large SNR, we simplify Eq. (8.46) into
[image: image] (8.52)
Equation (8.52) clearly shows the challenges of improving spectral efficiency by redesigning the fiber system parameters: to increase spectral efficiency by 2 bit/s/Hz, the dispersion needs to be increased by a factor of 8, or the nonlinear coefficient γ be decreased by a factor of 2.8, or number of spans be reduced by a factor of 2, all of which are difficult to achieve. In a nutshell, it is of diminishing return to improve the spectral efficiency by modifying the optical fiber system parameters. The only effective method to substantially improve the spectral efficiency is to add more dimensions such as resorting to polarization multiplexing that leads to almost a factor of 2 improvement as discussed in the paper, or fiber mode multiplexing by at least a factor of 2 or more dependent on the capability of achievable digital signal processing (DSP). Figure 8.16 shows the achievable spectral efficiency for the three systems studied in Section 8.1. The only modification is that we assume 40 nm or 5 THz for the total bandwidth. The spectral efficiency for systems I, II, and III is, respectively, 9.90, 8.38, and 8.63 b/s/Hz. This shows a total capacity of 49.5 Tb/s can be achieved for 10 × 100 SSMF uncompensated EDFA-only dual-polarization systems within C-band.
[image: image]
Figure 8.16 Information spectral efficiency as a function of the number of spans for various dispersion maps. The total bandwidth B is assumed to be 40 nm. The other OFDM and link parameters are the same as those for Figure 8.15.
8.5.4 Nonlinearity mitigation for multicarrier systems
Various techniques of nonlinearity mitigation have been proposed to improve the transmission performance. The commonly studied approaches are as follows: (i) pre- and/or post-compensation where the nonlinear phase noise is compensated at the transmitter or/and receiver [45,92], (ii) joint cross-polarization nonlinearity cancelation, which is similar to approach (i), but broadens the compensation to nonlinear polarization rotation [93], (iii) nonlinear digital back-propagation (DBP) where the nonlinearity is unwrapped by back-propagating the received signal toward the transmitter digitally [94,95], (iv) Volterra nonlinear compensation where the nonlinearity is approximated as the Volterra series and the nonlinearity is compensated iteratively [96], and (v) DFT-spread OFDM at optimal symbol rate [35] as discussed in Section 8.5.2. In this subsection, we will focus on DBP which has drawn much attention for its flexible and comprehensive compensation of both intra- and inter-channel nonlinear effects.
With the exact knowledge of channel parameters, the deterministic nonlinear interactions among signals can be completely removed using DBP with fine enough back-propagation steps, but this requires huge digital processing power [45,95,97]. Signals propagating in generic optical fiber transmission systems can be described by the nonlinear Schrödinger equation (NLSE) given by
[image: image] (8.53)
where [image: image] and [image: image] are linear and nonlinear operators, and α, [image: image], γ represent fiber loss, chromatic dispersion, and nonlinear coefficient, respectively. Equation (8.53) can be numerically solved using the symmetric split-step Fourier method (SSFM) [98] as follows
[image: image] (8.54)
where h is the step size. In the absence of noise, the transmitted signal can be calculated from the inverse NLSE:
[image: image] (8.55)
which is equivalent to passing the received signal through a “virtual” fiber with parameters of the opposite sign to the real fiber. By applying Eq. (8.55) to the received signal with appropriate step size, both linear and nonlinear effects incurred from transmission can be removed.
Although solving the inverse NLSE is computationally expensive, DBP offers good nonlinearity mitigation when transmission link parameters are known. Figure 8.17shows the Q factors against launch power of 3200 km CO-OFDM transmission with DBP (solid round curve) using different number of steps [97]. We can see that more than 2-dB improvement in terms of nonlinear power tolerance can be achieved if the number of steps is larger than 8.
[image: image]
Figure 8.17 Q against launch power for a 56-Gb/s 3200 km transmission system using linear equalization or nonlinear (filtered BP) equalization [97].
8.6 Applications of Optical Multicarrier Transmissions
The various optical multicarrier transmission technologies introduced above have been explored for a wide range of potential applications. In this section, we give specific examples in three distinct areas of application, each of which has uniquerequirements in terms of data rate, spectral efficiency, reach, and complexity. These are: (i) long-reach and high-capacity systems, (ii) optical access systems, and (iii) indoor and free-space communication systems.
8.6.1 Long-reach and high-capacity systems
Long-reach applications have a specific demand to achieve high data capacity and spectral efficiency. As such, much research has been conducted to apply optical multicarrier transmission to long-haul and high-capacity systems in recent years [21,99–103]. Table 8.2 lists some of the recent achievements in long-reach systems (over 1000 km) with high transmission capacity (over 400 Gb/s). It is clear multicarrier technologies are capable of overcoming limitations due to fiber dispersion and nonlinearity, enabling high spectral efficiency and long-reach transmission in the post-100 Gb/s era.
Table 8.2
Recent achievements on long-reach (>1000 km) and high-capacity transmission (>400 Gb/s) using multicarrier optical transmission.
[image: Image]
In order to achieve such high-capacity transmission over longer distances, we need to adopt a few techniques to efficiently use bandwidth and reduce nonlinear impairments. For instance, in [99], a reduced-guard-interval (RGI) CO-OFDM technique is used to improve the spectral efficiency, and ultra large area fiber (ULAF) is used to reduce nonlinear impairments. In optical OFDM transceivers, GI (or the cyclic prefix mentioned in Section 8.2) is used to compensate fiber dispersion, the length of which increases with transmission distance and signal bandwidth. A lengthy GI then causes a large redundancy ratio (overhead) and reduces the channel capacity. The RGI-CO-OFDM method employs digital dispersion compensation and therefore relatively short GI is needed. Another technique proposed for long-haul transmission is DFT-Spread OFDM [35], which improves performance of the long-reach transmission by reducing the PAPR of the signal. It also gives the capability to partition the whole bandwidth into smaller subbands with optimal symbol rate, which can minimize the nonlinearity penalty [21].
8.6.2 Optical access networks
Optical multicarrier techniques, such as optical OFDM, can be also a potential candidate for next-generation high-speed optical access networks [18,19,104]. The passive optical network which uses orthogonal frequency division multi-access is called OFDMA-PON. Figure 8.18 depicts an example of OFDMA-PON architecture. As shown in Figure 8.18[18], at the optical line terminal (OLT), a bandwidth-sharing schedule is formed according to the demand from optical network units (ONU) side, and is distributed to all ONUs over pre-assigned subcarriers and/or timeslots. Different OFDM subcarriers are thus assigned to different ONUs. Since traffic is aggregated and de-aggregated electronically on an optical carrier, the architecture is compatible with the legacy fiber distribution network, which enables reuse of existing PON infrastructure and thus saving on deployment cost. At the ONUs, each ONU recovers its pre-assigned OFDM subcarriers and/or time slots in DSP. An orthogonal OFDM-based schedule for upstream transmission is likewise generated by the OLT and distributed to the ONUs. At the OLT, a complete OFDMA frame is assembled from the incoming sub-frames originating at different ONUs. It can be seen that such OFDMA-PON has the advantages of bandwidth flexibility and high spectral efficiency in addition to multi-user access capabilities, which make it an attractive approach for next-generation high-speed PON.
[image: image]
Figure 8.18 Example of OFDMA-PON architecture [18]. OLT: Optical line terminal, ONU: Optical network units, TDM: Time-division multiplexing.
More recently, demonstration of PON utilizing OFDMA for downstream data transmission and achieving more than 100 Mb/s per-cell has been presented with low latency and relatively low cost [104]. This progress confirms that optical multicarrier could be a promising candidature for future optical access networks.
8.6.3 Indoor and free-space multicarrier optical systems
The current indoor and free-space systems being studied use intensity modulation and direct detection (IM/DD) for the simplicity of implementation. However, when optical waves propagate through air, they suffer from atmospheric turbulence causing fluctuations of both amplitude and phase. Such cases resemble wireless fading channels, and as a result optical multicarrier technologies are well suited for indoor and free-space systems. Because each subcarrier carries low rate data, it provides immunity to burst-errors due to intensity fluctuations [22–24].
The basic free-space OFDM transmitter and receiver configurations are shown in Figure 8.19a and c, respectively [22]. The corresponding free space link is shown in Figure 8.19b. An information bearing stream is first encoded (for instance, by an LDPC code) and then parsed into groups in the demultiplexer (DEMUX). The parsed stream is then mapped into a complex-valued signal constellation. The complex-valued signal points from all subchannels are considered as the values of the DFT of a multicarrier OFDM signal. The modulator and demodulator can be implemented by using the IFFT and FFT algorithms, respectively. After a D/A conversion and RF up-conversion, the OFDM signal is driven to the Mach-Zehnder modulator (MZM) and then transmitted over the free space link. At the receiver side, after photo-detection, RF down-conversion, and carrier synchronization, the received signal is demodulated by computing the FFT. The soft outputs of FFT demodulator are used to estimate the bit reliabilities that are fed into the LDPC decoder.
[image: image]
Figure 8.19 Basic free-space OFDM system: (a) transmitter, (b) free-space transmission link, and (c) receiver [22].
8.7 Future Research Directions for Multicarrier Transmission
Optical multicarrier transmission has become a fast progressing and vibrant research field. It is an exciting event that very advanced communication concepts in coding, modulation, reception, and channel equalization are being applied in the optical fiber communication, which traditionally used much simpler transmission and detection schemes compared to the wireless counterpart owing to the high-speed electronics involved. With the advent of extremely fast silicon DSP chips, it has now become possible to employ many of the advanced modulation schemes, enabling unprecedented data rates approaching Tb/s and beyond. Additionally, the marriage between high-speed electronics and photonics presents some tremendous challenges and opportunities in research. As concluding remarks for this chapter, here we lay out some examples of the research and development ideas that in our view will have significant ramifications in the field of optical multicarrier transmission:
1. As the channel rate goes beyond 1 Tb/s, the achievable capacity per fiber may become a bottleneck due to the fiber nonlinearity constraint [93]. To overcome the bottleneck, space division multiplexing (SDM)-based transmission over multi-core fiber (MCF) [32,105,109] or multi-mode fiber (MMF) [111–113], especially the few-mode fiber (FMF)-based transmission for optical multicarrier systems has been proposed recently [27–31]. FMF transmission, in conjunction with MIMO-OFDM, could be a promising technology to achieve capacity higher than 100 Tb/s per fiber taking advantage of the mode-division multiplexing in the optical fiber. But SDM itself has many challenges, such as development of FMF fiber, few-mode fiber amplifiers, switches, add/drop multiplexers, etc. These engineering challenges are to be solved by innovative approaches in device technologies.
2. The traditional optical networks are rigidly designed to support only a fixed link data rate throughout the operational life. OFDM technique, as one realization of software-defined optical transmission (SDOT), provides many nabling functionalities for the future dynamically reconfigurable networks [37,38], the adaptation of channel rate according to the channel condition being one example. Rate-adaptive LDPC codes and adaptive bit loading can be employed to optimize the link capacity for difference reaches.
3. OFDMA has become an attractive multi-user access technique in which the subsets of subcarriers are assigned to individual users. OFDMA enables the flexible time and frequency domain resource partitioning. Additionally, OFDMA can seamlessly bridge the wireless and optical access networks via radio over fiber (RoF) systems. OFDMA is also shown to be a promising approach in offering resource management for passive-optical-networks (PON).
4. The last decade saw the dramatic resurgence of the research interests in optoelectronic integrated circuit (OEIC). Considering the extensive optical devices and digital signal processing involved in optical OFDM, we envisage the possibility of integration of many digital, RF, and optical components into one silicon IC which can perform all the main functionalities of an optical OFDM transceiver. Without doubt, the success of the OEIC will greatly influence the evolution of the optical multicarrier transmission systems.
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106. J. Sakaguchi, B.J. Puttnam, W. Klaus, Y. Awaji, N. Wada, A. Kanno, T. Kawanishi, K.Imamura, H. Inaba, K. Mukasa, R. Sugizaki, T. Kobayashi, M. Watanabe, 19-core fiber transmission of [image: image]-Gb/s SDM-WDM-PDM-QPSK signals at 305 Tb/s, in: Optical Fiber Communication Conference (OFC), 2012, Paper PDP5C.2.
107. R. Ryf, R. Essiambre, A. Gnauch, S. Randel, M.A. Mestre, C. Schmidl, P. Winzer, R. Delbue, P. Pupalaikis, A. Sureka, T. Hayashi, T. Taru, T. Sasaki, Space-division multiplexed transmission over 4200 km 3-core microstructure fiber, in: Optical Fiber Communication Conference (OFC), 2012, Paper PDP5C.3.
108. B. Zhu, T. Taunay, M. Fishteyn, X. Liu, S. Chandrasekhar, M. Yan, J. Fini, E. Monberg, F. Dimarcello, Space-, wavelength-, polarization-division multiplexed transmission of 56-Tb/s over a 76.8-km seven-core fiber, in: Optical Fiber Communication Conference (OFC), 2011, Paper PDPB7.
109. Zhu B, Taunay T, Fishteyn M, et al. 112-Tb/s space-division multiplexed DWDM transmission with 14-b/s/Hz aggregate spectral efficiency over a 76.8-km seven-core fiber. Opt Express. 2011;19:16665–16671.
110. Berdagué S, Facq P. Mode division multiplexing in optical fibers. Appl Opt. 1982;21:1950–1955.
111. Stuart HR. Dispersive multiplexing in multimode optical fiber. Science. 2000;289:281–283.
112. B.C. Thomsen, MIMO enabled 40 Gb/s transmission using mode division multiplexing in multimode fiber, in: Optical Fiber Communication Conference (OFC), 2010, Paper OThM6.
113. B. Franz, D. Suikat, R. Dischler, F. Buchali, H. Buelow, High speed OFDM data transmission over 5 km GI-multimode fiber using spatial multiplexing with [image: image] MIMO processing, in: European Conference and Exhibition On Optical Communication (ECOC), 2010, Tu3.C.4.
Chapter 25
Modern Undersea Transmission Technology
Jin-xing Cai, Katya Golovchenko and Georg Mohs, TE SubCom, 250 Industrial Way West, Eatontown, NJ 07724, USA
The authors wish to thank all of their colleagues in optical fiber telecommunications around the world for their significant achievements in undersea transmission. We also express our heartfelt thanks to the TE SubCom team for their valuable contributions to this work and for kindly providing the material and fruitful discussions that made this chapter possible.
This chapter provides an overview over the progress in undersea transmission technology since the last edition of Optical Fiber Telecommunications in 2007. Since then, digital coherent transmission has become available, enabling a ten-fold increase in spectral efficiency and system capacity. Channel data rates have increased from 10 Gb/s to 100 Gb/s and beyond. After a brief general introduction to undersea systems with their unique challenges and design constraints in Section 25.1, Section 25.2 outlines the principles of coherent transmission technology as they apply to undersea systems including polarization multiplexing, linear equalizers and high-order modulation formats. Section 25.3 describes the use of strong optical filtering to help to improve spectral efficiency and reviews techniques to limit the effects of inter-symbol interference. The best trade-off between spectral efficiency and transmission performance is achieved at Nyquist carrier spacing and Section 25.4 discusses different transmission techniques for this condition. Section 25.5 then introduces higher order modulation formats to further increase spectral efficiency by increasing the number of bits per symbol and discusses the implications and mitigation techniques of the receiver sensitivity degradation that goes along with it. Future trends are outlined in Section 25.6 before summarizing the chapter in Section 25.7.
25.1 Introduction
Undersea communication systems have been providing low latency and high capacity connectivity between the continents of the world for more than 150 years. The first transoceanic telegraph cable became operational in 1858 with a communication speed of about 2 min per character [19], a vast improvement over the 10 days it took to deliver a message by clipper ship. The cable spanned a 4025 km route between Newfoundland and Ireland and lasted only for about a month but was successfully replaced in 1866 with an increased transmission speed of about 8 words per minute [19].
Steady progress has been made ever since to provide faster and faster communication speed or higher and higher bandwidth with submarine cables. Voice traffic on a transatlantic cable became available in 1956 with Transatlantic No. 1 (TAT-1) carrying 36 simultaneous telephone channels with 4 kHz bandwidth each replacing radio connections that had been in use between the United States and Europe since 1927 but which were unreliable due to their dependence on atmospheric conditions. In 1988 the first fiber optic transatlantic cable was put into service ultimately carrying 40,000 telephone circuits equivalent to 560 Mb/s of aggregate traffic on two fiber pairs [1]. The latest generation of systems in the Atlantic between the United States and Europe can carry multiple Tb/s of aggregate capacity and the next generation is actively being discussed with multiple tens of Tb/s capacity [26,20]. Figure 25.1 shows the lit undersea communication capacity around the globe and used capacity by country as of December 2011. Undersea communication cables are nearly ubiquitous in all international waters connecting the countries and continents of the world |
b28c58d1c1c1f393 | söndag 5 februari 2017
From Meaningless Towards Meaningful QM?
The Schrödinger equation as the basic model of atom physics descended as a heavenly gift to humanity in an act of godly inspiration inside the mind of Erwin Schrödinger in 1926.
But the gift showed to hide poison: Nobody could give the equation a physical meaning understandable to humans, and that unfortunate situation has prevailed into our time as expressed by Nobel Laureate Steven Weinberg (and here):
Weinberg's view is a theme on the educated physics blogosphere of today:
Sabine agrees with Weinberg that "there are serious problems", while Lubos insists that "there are no problems".
There are two approaches to mathematical modelling of the physical world:
1. Pick symbols to form a mathematical expression/equation and then try to give it a meaning.
2. Have a meaningful thought and then try to express it as a mathematical expression/equation.
Schrödinger's equation was formed more according to 1. rather than 2. and has resisted all efforts to be given a physical meaning. Interpreting Schrödinger's equation has shown to be like interpreting the Bible as authored by God rather than human minds.
What makes Schrödinger's equation so difficult to interpret in physical terms, is that it depends on $3N$ spatial variables for an atom with $N$ electrons, while an atom with all its electrons seems to share experience in a common 3-d space. Here is how Weinberg describes the generalisation from $N=1$ in 3 space dimensions to $N>1$ in $3N$ space dimensions as "obvious":
• More than that, Schrödinger’s equation had an obvious generalisation to general systems.
Weinberg takes for granted that what "is obvious" does not have to be explained. But everything in rational physics needs rational argumentation and nothing "is obvious", and so this is where quantum mechanics branches off from rational physics. If what is claimed to be "obvious" in fact lacks rational argument, then it may simply be all wrong. The generalisation of Schrödinger's equation to $N>1$ fell into that trap, and that is the tragedy of modern physics.
There is nothing "obvious" in the sense of "frequently encountered" in the generalisation of Schrödinger's equation from 3 space dimensions to 3N space dimension, since it is a giant leap away from reality and as such utterly "non-obvious" and "never encountered" before.
In realQM I suggest a different form of Schrödinger's equation as a system in 3d with physical meaning.
PS Note how Weinberg describes the foundation of quantum mechanics:
• The first postulate of quantum mechanics is that physical states can be represented as vectors in a sort of abstract space known as Hilbert space.
• According to the second postulate of quantum mechanics, observable physical quantities like position, momentum, energy, etc., are represented as Hermitian operators on Hilbert space.
We see that these postulates are purely formal and devoid of physics. We see that the notion of Hilbert space and Hermitian operator are elevated to have a mystical divine quality, as if Hilbert and Hermite were gods like Zeus (physics of the sky) and Poseidon (physics of the sea)...much of the mystery of quantum mechanics comes from assigning meaning to such formalities without meaning...
The idea that the notion of Hilbert space is central to quantum mechanics was supported by an idea that Hilbert space as a key ingredient in the "modern mathematics" created by Hilbert 1926-32 should be the perfect tool for "modern physics", an idea explored in von Neumann's monumental Mathematical Foundations of Quantum Mechanics. Here the linearity of Schrödinger's equation is instrumental and its many dimensions doesn't matter, but it appears that von Neumann missed the physics:
• I would like to make a confession which may seem immoral: I do not believe absolutely in Hilbert space no more. (von Neumann to Birkhoff 1935)
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2d19d2289f3f7edc | Wave function
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Comparison of classical and quantum harmonic oscillator conceptions for a single spinless particle. The two processes differ greatly. The classical process (A–B) is represented as the motion of a particle along a trajectory. The quantum process (C–H) has no such trajectory. Rather, it is represented as a wave; here, the vertical axis shows the real part (blue) and imaginary part (red) of the wave function. Panels (C–F) show four different standing-wave solutions of the Schrödinger equation. Panels (G–H) further show two different wave functions that are solutions of the Schrödinger equation but not standing waves.
Wavefunctions of the electron of a hydrogen atom at different energies. The brightness at each point represents the probability of finding the electron at that point.
A wave function in quantum physics is a mathematical description of the quantum state of a system. The wave function is a complex-valued probability amplitude, and the probabilities for the possible results of measurements made on the system can be derived from it. The most common symbols for a wave function are the Greek letters ψ or Ψ (lower-case and capital psi, respectively).
The wave function is a function of the degrees of freedom corresponding to some maximal set of commuting observables. Once such a representation is chosen, the wave function can be derived from the quantum state.
For a given system, the choice of which commuting degrees of freedom to use is not unique, and correspondingly the domain of the wave function is also not unique. For instance it may be taken to be a function of all the position coordinates of the particles over position space, or the momenta of all the particles over momentum space; the two are related by a Fourier transform. Some particles, like electrons and photons, have nonzero spin, and the wave function for such particles includes spin as an intrinsic, discrete degree of freedom; other discrete variables can also be included, such as isospin. When a system has internal degrees of freedom, the wave function at each point in the continuous degrees of freedom (e.g., a point in space) assigns a complex number for each possible value of the discrete degrees of freedom (e.g., z-component of spin) -- these values are often displayed in a column matrix (e.g., a 2 × 1 column vector for a non-relativistic electron with spin 12).
According to the superposition principle of quantum mechanics, wave functions can be added together and multiplied by complex numbers to form new wave functions and form a Hilbert space. The inner product between two wave functions is a measure of the overlap between the corresponding physical states, and is used in the foundational probabilistic interpretation of quantum mechanics, the Born rule, relating transition probabilities to inner products. The Schrödinger equation determines how wave functions evolve over time, and a wave function behaves qualitatively like other waves, such as water waves or waves on a string, because the Schrödinger equation is mathematically a type of wave equation. This explains the name "wave function," and gives rise to wave–particle duality. However, the wave function in quantum mechanics describes a kind of physical phenomenon, still open to different interpretations, which fundamentally differs from that of classic mechanical waves.[1][2][3][4][5][6][7]
In Born's statistical interpretation in non-relativistic quantum mechanics, [8][9][10] the squared modulus of the wave function, | ψ |2, is a real number interpreted as the probability density of measuring a particle's being detected at a given place - or having a given momentum - at a given time, and possibly having definite values for discrete degrees of freedom. The integral of this quantity, over all the system's degrees of freedom, must be 1 in accordance with the probability interpretation. This general requirement that a wave function must satisfy is called the normalization condition. Since the wave function is complex valued, only its relative phase and relative magnitude can be measured -- its value does not, in isolation, tell anything about the magnitudes or directions of measurable observables; one has to apply quantum operators, whose eigenvalues correspond to sets of possible results of measurements, to the wave function ψ and calculate the statistical distributions for measurable quantities.
Historical background[edit]
In 1905 Einstein postulated the proportionality between the frequency of a photon and its energy, E = hf,[11] and in 1916 the corresponding relation between photon momentum and wavelength, λ = h/p.[12] In 1923, De Broglie was the first to suggest that the relation λ = h/p, now called the De Broglie relation, holds for massive particles, the chief clue being Lorentz invariance,[13] and this can be viewed as the starting point for the modern development of quantum mechanics. The equations represent wave–particle duality for both massless and massive particles.
In the 1920s and 1930s, quantum mechanics was developed using calculus and linear algebra. Those who used the techniques of calculus included Louis de Broglie, Erwin Schrödinger, and others, developing "wave mechanics". Those who applied the methods of linear algebra included Werner Heisenberg, Max Born, and others, developing "matrix mechanics". Schrödinger subsequently showed that the two approaches were equivalent.[14]
In 1926, Schrödinger published the famous wave equation now named after him, indeed the Schrödinger equation, based on classical Conservation of energy using quantum operators and the de Broglie relations such that the solutions of the equation are the wave functions for the quantum system.[15] However, no one was clear on how to interpret it.[16] At first, Schrödinger and others thought that wave functions represent particles that are spread out with most of the particle being where the wave function is large.[17] This was shown to be incompatible with the elastic scattering of a wave packet (representing a particle) off a target; it spreads out in all directions.[8] While a scattered particle may scatter in any direction, it does not break up and take off in all directions. In 1926, Born provided the perspective of probability amplitude.[8][9][18] This relates calculations of quantum mechanics directly to probabilistic experimental observations. It is accepted as part of the Copenhagen interpretation of quantum mechanics. There are many other interpretations of quantum mechanics. In 1927, Hartree and Fock made the first step in an attempt to solve the N-body wave function, and developed the self-consistency cycle: an iterative algorithm to approximate the solution. Now it is also known as the Hartree–Fock method.[19] The Slater determinant and permanent (of a matrix) was part of the method, provided by John C. Slater.
Schrödinger did encounter an equation for the wave function that satisfied relativistic energy conservation before he published the non-relativistic one, but discarded it as it predicted negative probabilities and negative energies. In 1927, Klein, Gordon and Fock also found it, but incorporated the electromagnetic interaction and proved that it was Lorentz invariant. De Broglie also arrived at the same equation in 1928. This relativistic wave equation is now most commonly known as the Klein–Gordon equation.[20]
In 1927, Pauli phenomenologically found a non-relativistic equation to describe spin-1/2 particles in electromagnetic fields, now called the Pauli equation.[21] Pauli found the wave function was not described by a single complex function of space and time, but needed two complex numbers, which respectively correspond to the spin +1/2 and −1/2 states of the fermion. Soon after in 1928, Dirac found an equation from the first successful unification of special relativity and quantum mechanics applied to the electron, now called the Dirac equation. In this, the wave function is a spinor represented by four complex-valued components:[19] two for the electron and two for the electron's antiparticle, the positron. In the non-relativistic limit, the Dirac wave function resembles the Pauli wave function for the electron. Later, other relativistic wave equations were found.
Wave functions and wave equations in modern theories[edit]
All these wave equations are of enduring importance. The Schrödinger equation and the Pauli equation are under many circumstances excellent approximations of the relativistic variants. They are considerably easier to solve in practical problems than the relativistic counterparts.
The Klein-Gordon equation and the Dirac equation, while being relativistic, do not represent full reconciliation of quantum mechanics and special relativity. The branch of quantum mechanics where these equations are studied the same way as the Schrödinger equation, often called relativistic quantum mechanics, while very successful, has its limitations (see e.g. Lamb shift) and conceptual problems (see e.g. Dirac sea).
Relativity makes it inevitable that the number of particles in a system is not constant. For full reconciliation, quantum field theory is needed.[22] In this theory, the wave equations and the wave functions have their place, but in a somewhat different guise. The main objects of interest are not the wave functions, but rather operators, so called field operators (or just fields where "operator" is understood) on the Hilbert space of states (to be described next section). It turns out that the original relativistic wave equations and their solutions are still needed to build the Hilbert space. Moreover, the free fields operators, i.e. when interactions are assumed not to exist, turn out to (formally) satisfy the same equation as do the fields (wave functions) in many cases.
Thus the Klein-Gordon equation (spin 0) and the Dirac equation (spin 12) in this guise remain in the theory. Higher spin analogues include the Proca equation (spin 1), Rarita–Schwinger equation (spin 32), and, more generally, the Bargmann–Wigner equations. For massless free fields two examples are the free field Maxwell equation (spin 1) and the free field Einstein equation (spin 2) for the field operators.[23] All of them are essentially a direct consequence of the requirement of Lorentz invariance. Their solutions must transform under Lorentz transformation in a prescribed way, i.e. under a particular representation of the Lorentz group and that together with few other reasonable demands, e.g. the cluster decomposition principle,[24] with implications for causality is enough to fix the equations.
It should be emphasized that this applies to free field equations; interactions are not included. If a Lagrangian density (including interactions) is available, then the Lagrangian formalism will yield an equation of motion at the classical level. This equation may be very complex and not amenable to solution. Any solution would refer to a fixed number of particles and would not account for the term "interaction" as referred to in these theories, which involves the creation and annihilation of particles and not external potentials as in ordinary "first quantized" quantum theory.
In string theory, the situation remains analogous. For instance, a wave function in momentum space has the role of Fourier expansion coefficient in a general state of a particle (string) with momentum that is not sharply defined.[25]
Definition (one spinless particle in 1d)[edit]
Travelling waves of a free particle.
The real parts of position wave function Ψ(x) and momentum wave function Φ(p), and corresponding probability densities |Ψ(x)|2 and |Φ(p)|2, for one spin-0 particle in one x or p dimension. The colour opacity of the particles corresponds to the probability density (not the wave function) of finding the particle at position x or momentum p.
For now, consider the simple case of a non-relativistic single particle, without spin, in one spatial dimension. More general cases are discussed below.
Position-space wave functions[edit]
The state of such a particle is completely described by its wave function,
where x is position and t is time. This is a complex-valued function of two real variables x and t.
For one spinless particle in 1d, if the wave function is interpreted as a probability amplitude, the square modulus of the wave function, the positive real number
is interpreted as the probability density that the particle is at x. The asterisk indicates the complex conjugate. If the particle's position is measured, its location cannot be determined from the wave function, but is described by a probability distribution. The probability that its position x will be in the interval axb is the integral of the density over this interval:
where t is the time at which the particle was measured. This leads to the normalization condition:
because if the particle is measured, there is 100% probability that it will be somewhere.
For a given system, the set of all possible normalizable wave functions (at any given time) forms an abstract mathematical vector space, meaning that it is possible to add together different wave functions, and multiply wave functions by complex numbers (see vector space for details). Technically, because of the normalization condition, wave functions form a projective space rather than an ordinary vector space. This vector space is infinite-dimensional, because there is no finite set of functions which can be added together in various combinations to create every possible function. Also, it is a Hilbert space, because the inner product of two wave functions Ψ1 and Ψ2 can be defined as the complex number (at time t)[nb 1]
More details are given below. Although the inner product of two wave functions is a complex number, the inner product of a wave function Ψ with itself,
is always a positive real number. The number ||Ψ|| (not ||Ψ||2) is called the norm of the wave function Ψ, and is not the same as the modulus |Ψ|.
If (Ψ, Ψ) = 1, then Ψ is normalized. If Ψ is not normalized, then dividing by its norm gives the normalized function Ψ/||Ψ||. Two wave functions Ψ1 and Ψ2 are orthogonal if 1, Ψ2) = 0. If they are normalized and orthogonal, they are orthonormal. Orthogonality (hence also orthonormality) of wave functions is not a necessary condition wave functions must satisfy, but is instructive to consider since this guarantees linear independence of the functions. In a linear combination of orthogonal wave functions Ψn we have,
If the wave functions Ψn were nonorthogonal, the coefficients would be less simple to obtain.
In the Copenhagen interpretation, the modulus squared of the inner product (a complex number) gives a real number
which, assuming both wave functions are normalized, is interpreted as the probability of the wave function Ψ2 "collapsing" to the new wave function Ψ1 upon measurement of an observable, whose eigenvalues are the possible results of the measurement, with Ψ1 being an eigenvector of the resulting eigenvalue. This is the Born rule,[8] and is one of the fundamental postulates of quantum mechanics.
At a particular instant of time, all values of the wave function Ψ(x, t) are components of a vector. There are uncountably infinitely many of them and integration is used in place of summation. In Bra–ket notation, this vector is written
and is referred to as a "quantum state vector", or simply "quantum state".There are several advantages to understanding wave functions as representing elements of an abstract vector space:
• All the powerful tools of linear algebra can be used to manipulate and understand wave functions. For example:
• Linear algebra explains how a vector space can be given a basis, and then any vector in the vector space can be expressed in this basis. This explains the relationship between a wave function in position space and a wave function in momentum space, and suggests that there are other possibilities too.
• Bra–ket notation can be used to manipulate wave functions.
• The idea that quantum states are vectors in an abstract vector space is completely general in all aspects of quantum mechanics and quantum field theory, whereas the idea that quantum states are complex-valued "wave" functions of space is only true in certain situations.
The time parameter is often suppressed, and will be in the following. The x coordinate is a continuous index. The |x are the basis vectors, which are orthonormal so their inner product is a delta function;
which illuminates the identity operator
Finding the identity operator in a basis allows the abstract state to be expressed explicitly in a basis, and more (the inner product between two state vectors, and other operators for observables, can be expressed in the basis).
Momentum-space wave functions[edit]
The particle also has a wave function in momentum space:
where p is the momentum in one dimension, which can be any value from −∞ to +∞, and t is time.
Analogous to the position case, the inner product of two wave functions Φ1(p, t) and Φ2(p, t) can be defined as:
One particular solution to the time-independent Schrödinger equation is
a plane wave, which can be used in the description of a particle with momentum exactly p, since it is an eigenfunction of the momentum operator. These functions are not normalizable to unity (they aren't square-integrable), so they are not really elements of physical Hilbert space. The set
forms what is called the momentum basis. This "basis" is not a basis in the usual mathematical sense. For one thing, since the functions aren't normalizable, they are instead normalized to a delta function,
For another thing, though they are linearly independent, there are too many of them (they form an uncountable set) for a basis for physical Hilbert space. They can still be used to express all functions in it using Fourier transforms as described next.
Relations between position and momentum representations[edit]
The x and p representations are
Now take the projection of the state Ψ onto eigenfunctions of momentum using the last expression in the two equations,[26]
Then utilizing the known expression for suitably normalized eigenstates of momentum in the position representation solutions of the free Schrödinger equation
one obtains
Likewise, using eigenfunctions of position,
The position-space and momentum-space wave functions are thus found to be Fourier transforms of each other.[27] The two wave functions contain the same information, and either one alone is sufficient to calculate any property of the particle. As representatives of elements of abstract physical Hilbert space, whose elements are the possible states of the system under consideration, they represent the same state vector, hence identical physical states, but they are not generally equal when viewed as square-integrable functions.
In practice, the position-space wave function is used much more often than the momentum-space wave function. The potential entering the relevant equation (Schrödinger, Dirac, etc.) determines in which basis the description is easiest. For the harmonic oscillator, x and p enter symmetrically, so there it doesn't matter which description one uses. The same equation (modulo constants) results. From this follows, with a little bit of afterthought, a factoid: The solutions to the wave equation of the harmonic oscillator are eigenfunctions of the Fourier transform in L2.[nb 2]
Definitions (other cases)[edit]
Following are the general forms of the wave function for systems in higher dimensions and more particles, as well as including other degrees of freedom than position coordinates or momentum components.
One-particle states in 3d position space[edit]
The position-space wave function of a single particle without spin in three spatial dimensions is similar to the case of one spatial dimension above:
where r is the position vector in three-dimensional space, and t is time. As always Ψ(r, t) is a complex-valued function of real variables. As a single vector in Dirac notation
All the previous remarks on inner products, momentum space wave functions, Fourier transforms, and so on extend to higher dimensions.
For a particle with spin, ignoring the position degrees of freedom, the wave function is a function of spin only (time is a parameter);
where sz is the spin projection quantum number along the z axis. (The z axis is an arbitrary choice; other axes can be used instead if the wave function is transformed appropriately, see below.) The sz parameter, unlike r and t, is a discrete variable. For example, for a spin-1/2 particle, sz can only be +1/2 or −1/2, and not any other value. (In general, for spin s, sz can be s, s − 1, ... , −s + 1, −s). Inserting each quantum number gives a complex valued function of space and time, there are 2s + 1 of them. These can be arranged into a column vector[nb 3]
In bra ket notation, these easily arrange into the components of a vector[nb 4]
The entire vector ξ is a solution of the Schrödinger equation (with a suitable Hamiltonian), which unfolds to a coupled system of 2s + 1 ordinary differential equations with solutions ξ(s, t), ξ(s − 1, t), ..., ξ(−s, t). The term "spin function" instead of "wave function" is used by some authors. This contrasts the solutions to position space wave functions, the position coordinates being continuous degrees of freedom, because then the Schrödinger equation does take the form of a wave equation.
More generally, for a particle in 3d with any spin, the wave function can be written in "position–spin space" as:
and these can also be arranged into a column vector
in which the spin dependence is placed in indexing the entries, and the wave function is a complex vector-valued function of space and time only.
All values of the wave function, not only for discrete but continuous variables also, collect into a single vector
For a single particle, the tensor product of its position state vector |ψ and spin state vector |ξ gives the composite position-spin state vector
with the identifications
The tensor product factorization is only possible if the orbital and spin angular momenta of the particle are separable in the Hamiltonian operator underlying the system's dynamics (in other words, the Hamiltonian can be split into the sum of orbital and spin terms[28]). The time dependence can be placed in either factor, and time evolution of each can be studied separately. The factorization is not possible for those interactions where an external field or any space-dependent quantity couples to the spin; examples include a particle in a magnetic field, and spin-orbit coupling.
The preceding discussion is not limited to spin as a discrete variable, the total angular momentum J may also be used.[29] Other discrete degrees of freedom, like isospin, can expressed similarly to the case of spin above.
Many particle states in 3d position space[edit]
Traveling waves of two free particles, with two of three dimensions suppressed. Top is position space wave function, bottom is momentum space wave function, with corresponding probability densities.
If there are many particles, in general there is only one wave function, not a separate wave function for each particle. The fact that one wave function describes many particles is what makes quantum entanglement and the EPR paradox possible. The position-space wave function for N particles is written:[19]
where ri is the position of the ith particle in three-dimensional space, and t is time. Altogether, this is a complex-valued function of 3N + 1 real variables.
In quantum mechanics there is a fundamental distinction between identical particles and distinguishable particles. For example, any two electrons are identical and fundamentally indistinguishable from each other; the laws of physics make it impossible to "stamp an identification number" on a certain electron to keep track of it.[27] This translates to a requirement on the wave function for a system of identical particles:
where the + sign occurs if the particles are all bosons and sign if they are all fermions. In other words, the wave function is either totally symmetric in the positions of bosons, or totally antisymmetric in the positions of fermions.[30] The physical interchange of particles corresponds to mathematically switching arguments in the wave function. The antisymmetry feature of fermionic wave functions leads to the Pauli principle. Generally, bosonic and fermionic symmetry requirements are the manifestation of particle statistics and are present in other quantum state formalisms.
For N distinguishable particles (no two being identical, i.e. no two having the same set of quantum numbers), there is no requirement for the wave function to be either symmetric or antisymmetric.
For a collection of particles, some identical with coordinates r1, r2, ... and others distinguishable x1, x2, ... (not identical with each other, and not identical to the aforementioned identical particles), the wave function is symmetric or antisymmetric in the identical particle coordinates ri only:
Again, there is no symmetry requirement for the distinguishable particle coordinates xi.
The wave function for N particles each with spin is the complex-valued function
Accumulating all these components into a single vector,
For identical particles, symmetry requirements apply to both position and spin arguments of the wave function so it has the overall correct symmetry.
The formulae for the inner products are integrals over all coordinates or momenta and sums over all spin quantum numbers. For the general case of N particles with spin in 3d,
this is altogether N three-dimensional volume integrals and N sums over the spins. The differential volume elements d3ri are also written "dVi" or "dxi dyi dzi".
The multidimensional Fourier transforms of the position or position–spin space wave functions yields momentum or momentum–spin space wave functions.
Probability interpretation[edit]
For the general case of N particles with spin in 3d, if Ψ is interpreted as a probability amplitude, the probability density is
and the probability that particle 1 is in region R1 with spin sz1 = m1 and particle 2 is in region R2 with spin sz2 = m2 etc. at time t is the integral of the probability density over these regions and evaluated at these spin numbers:
Time dependence[edit]
For systems in time-independent potentials, the wave function can always be written as a function of the degrees of freedom multiplied by a time-dependent phase factor, the form of which is given by the Schrödinger equation. For N particles, considering their positions only and suppressing other degrees of freedom,
where E is the energy eigenvalue of the system corresponding to the eigenstate Ψ. Wave functions of this form are called stationary states.
The time dependence of the quantum state and the operators can be placed according to unitary transformations on the operators and states. For any quantum state |Ψ⟩ and operator O, in the Schrödinger picture |Ψ(t)⟩ changes with time according to the Schrödinger equation while O is constant. In the Heisenberg picture it is the other way round, |Ψ⟩ is constant while O(t) evolves with time according to the Heisenberg equation of motion. The Dirac (or interaction) picture is intermediate, time dependence is places in both operators and states which evolve according to equations of motion. It is useful primarily in computing S-matrix elements.[31]
Non-relativistic examples[edit]
The following are solutions to the Schrödinger equation for one nonrelativistic spinless particle.
Finite potential barrier[edit]
Scattering at a finite potential barrier of height V0. The amplitudes and direction of left and right moving waves are indicated. In red, those waves used for the derivation of the reflection and transmission amplitude. E > V0 for this illustration.
One of most prominent features of the wave mechanics is a possibility for a particle to reach a location with a prohibitive (in classical mechanics) force potential. A common model is the "potential barrier", the one-dimensional case has the potential
and the steady-state solutions to the wave equation have the form (for some constants k, κ)
Note that these wave functions are not normalized; see scattering theory for discussion.
The standard interpretation of this is as a stream of particles being fired at the step from the left (the direction of negative x): setting Ar = 1 corresponds to firing particles singly; the terms containing Ar and Cr signify motion to the right, while Al and Cl – to the left. Under this beam interpretation, put Cl = 0 since no particles are coming from the right. By applying the continuity of wave functions and their derivatives at the boundaries, it is hence possible to determine the constants above.
3D confined electron wave functions in a quantum dot. Here, rectangular and triangular-shaped quantum dots are shown. Energy states in rectangular dots are more s-type and p-type. However, in a triangular dot the wave functions are mixed due to confinement symmetry. (Click for animation)
In a semiconductor crystallite whose radius is smaller than the size of its exciton Bohr radius, the excitons are squeezed, leading to quantum confinement. The energy levels can then be modeled using the particle in a box model in which the energy of different states is dependent on the length of the box.
Quantum harmonic oscillator[edit]
The wave functions for the quantum harmonic oscillator can be expressed in terms of Hermite polynomials Hn, they are
where n = 0,1,2,....
Hydrogen atom[edit]
The electron probability density for the first few hydrogen atom electron orbitals shown as cross-sections. These orbitals form an orthonormal basis for the wave function of the electron. Different orbitals are depicted with different scale.
The wave functions of an electron in a Hydrogen atom are expressed in terms of spherical harmonics and generalized Laguerre polynomials (these are defined differently by different authors—see main article on them and the hydrogen atom).
It is convenient to use spherical coordinates, and the wavefunction can be separated into functions of each coordinate,[32]
where R are radial functions and Ym
(θ, φ)
are spherical harmonics of degree and order m. This is the only atom for which the Schrödinger equation has been solved for exactly. Multi-electron atoms require approximative methods. The family of solutions are:[33]
where a0 = 4πε0ħ2/mee2 is the Bohr radius, L2 + 1
n − 1
are the generalized Laguerre polynomials of degree n − 1, n = 1, 2, ... is the principal quantum number, = 0, 1, ... n − 1 the azimuthal quantum number, m = −, − + 1, ..., − 1, the magnetic quantum number. Hydrogen-like atoms have very similar solutions.
This solution does not take into account the spin of the electron.
In the figure of the hydrogen orbitals, the 19 sub-images are images of wave functions in position space (their norm squared). The wave functions each represent the abstract state characterized by the triple of quantum numbers (n, l, m), in the lower right of each image. These are the principal quantum number, the orbital angular momentum quantum number and the magnetic quantum number. Together with one spin-projection quantum number of the electron, this is a complete set of observables.
The figure can serve to illustrate some further properties of the function spaces of wave functions.
• In this case, the wave functions are square integrable. One can initially take the function space as the space of square integrable functions, usually denoted L2.
• The displayed functions are solutions to the Schrödinger equation. Obviously, not every function in L2 satisfies the Schrödinger equation for the hydrogen atom. The function space is thus a subspace of L2.
• The displayed functions form part of a basis for the function space. To each triple (n, l, m), there corresponds a basis wave function. If spin is taken into account, there are two basis functions for each triple. The function space thus has a countable basis.
• The basis functions are mutually orthonormal.
Wave functions and function spaces[edit]
The concept of function spaces enters naturally in the discussion about wave functions. A function space is a set of functions, usually with some defining requirements on the functions (in the present case that they are square integrable), sometimes with an algebraic structure on the set (in the present case a vector space structure with an inner product), together with a topology on the set. The latter will sparsely be used here, it is only needed to obtain a precise definition of what it means for a subset of a function space to be closed. It will be concluded below that the function space of wave functions is a Hilbert space. This observation is the foundation of the predominant mathematical formulation of quantum mechanics.
Vector space structure[edit]
A wave function is an element of a function space partly characterized by the following concrete and abstract descriptions.
• The Schrödinger equation is linear. This means that the solutions to it, wave functions, can be added and multiplied by scalars to form a new solution. The set of solutions to the Schrödinger equation is a vector space.
• The superposition principle of quantum mechanics. If Ψ and Φ are two states in the abstract space of states of a quantum mechanical system, and a and b are any two complex numbers, then aΨ + bΦ is a valid state as well. (Whether the null vector counts as a valid state ("no system present") is a matter of definition. The null vector does not at any rate describe the vacuum state in quantum field theory.) The set of allowable states is a vector space.
This similarity is of course not accidental. There are also a distinctions between the spaces to keep in mind.
Basic states are characterized by a set of quantum numbers. This is a set of eigenvalues of a maximal set of commuting observables. Physical observables are represented by linear operators, also called observables, on the vectors space. Maximality means that there can be added to the set no further algebraically independent observables that commute with the ones already present. A choice of such a set may be called a choice of representation.
• It is a postulate of quantum mechanics that a physically observable quantity of a system, such as position, momentum, or spin, is represented by a linear Hermitian operator on the state space. The possible outcomes of measurement of the quantity are the eigenvalues of the operator.[17] At a deeper level, most observables, perhaps all, arise as generators of symmetries.[17][34][nb 5]
• The physical interpretation is that such a set represents what can – in theory – be simultaneously be measured with arbitrary precision. The Heisenberg uncertainty relation prohibits simultaneous exact measurements of two non-commuting observables.
• The set is non-unique. It may for a one-particle system, for example, be position and spin z-projection, (x, Sz), or it may be momentum and spin y-projection, (p, Sy). In this case, the operator corresponding to position (a multiplication operator in the position representation) and the operator corresponding to momentum (a differential operator in the position the position representation) do not commute.
• Once a representation is chosen, there is still arbitrariness. It remains to choose a coordinate system. This may, for example, correspond to a choice of x, y- and z-axis, or a choice of curvilinear coordinates as exemplified by the spherical coordinates used for the Hydrogen atomic wave functions. This final choice also fixes a basis in abstract Hilbert space. The basic states are labeled by the quantum numbers corresponding to the maximal set of commuting observables and an appropriate coordinate system.[nb 6]
The abstract states are "abstract" only in that an arbitrary choice necessary for a particular explicit description of it is not given. This is the same as saying that no choice of maximal set of commuting observables has been given. This is analogous to a vector space without a specified basis. Wave functions corresponding to a state are accordingly not unique. This non-uniqueness reflects the non-uniqueness in the choice of a maximal set of commuting observables. For one spin particle in one dimension, to a particular state there corresponds two wave functions, Ψ(x, Sz) and Ψ(p, Sy), both describing the same state.
• For each choice of maximal commuting sets of observables for the abstract state space, there is a corresponding representation that is associated to a function space of wave functions.
• Between all these different function spaces and the abstract state space, there are one-to-one correspondences (here disregarding normalization and unobservable phase factors), the common denominator here being a particular abstract state. The relationship between the momentum and position space wave functions, for instance, describing the same state is the Fourier transform.
Each choice of representation should be thought of as specifying a unique function space in which wave functions corresponding to that choice of representation lives. This distinction is best kept, even if one could argue that two such function spaces are mathematically equal, e.g. being the set of square integrable functions. One can then think of the function spaces as two distinct copies of that set.
Inner product[edit]
There is additional algebraic structure on the vector spaces of wave functions and the abstract state space.
• Physically, different wave functions are interpreted to overlap to some degree. A system in a state Ψ that does not overlap with a state Φ cannot be found to be in the state Φ upon measurement. But if Φ1, Φ2, ... overlap Ψ to some degree, there is a chance that measurement of a system described by Ψ will be found un states Φ1, Φ2, ... . Also selection rules are observed apply. These are usually formulated in the preservation of some quantum numbers. This means that certain processes allowable from some perspectives (e.g. energy and momentum conservation) do not occur because the initial and final total wave functions don't overlap.
• Mathematically, it turns out that solutions to the Schrödinger equation for particular potentials are orthogonal in some manner, this is usually described by an integral
where m, n are (sets of) indices (quantum numbers) labeling different solutions, the strictly positive function w is called a weight function, and δmn is the Kronecker delta. The integration is taken over all of the relevant space.
This motivates the introduction of an inner product on the vector space of abstract quantum states, compatible with the mathematical observations of above when passing to a representation. It is denoted (Ψ, Φ), or in the Bra–ket notation ⟨Ψ|Φ⟩. It yields a complex number. With the inner product, the function space is an inner product space. The explicit appearance of the inner product (usually an integral or a sum of integrals) depends on the choice of representation, but the complex number (Ψ, Φ) does not. Much of the physical interpretation of quantum mechanics stems from the Born rule. It states that the probability p of finding upon measurement the state Φ given the system is in the state Ψ is
where Φ and Ψ are assumed normalized. Consider a scattering experiment. In quantum field theory, if Φout describes a state in the "distant future" (an "out state") after interactions between scattering particles have ceased, and Ψin an "in state" in the "distant past", then the quantities out, Ψin), with Φout and Ψin varying over a complete set of in states and out states respectively, is called the S-matrix or scattering matrix. Knowledge of it is, effectively, having solved the theory at hand, at least as far as predictions go. Measurable quantities such as decay rates and scattering cross sections are calculable from the S-matrix.[35]
Hilbert space[edit]
The above observations encapsulate the essence of the function spaces of which wave functions are elements. However the description is not yet complete. There is a further technical requirement on the function space, that of completeness, that allows one to take limits of sequences in the function space, and be ensured that, if the limit exists, it is an element of the function space. A complete inner product space is called a Hilbert space. The property of completeness is crucial in advanced treatments and applications of quantum mechanics. For instance, the existence of projection operators or orthogonal projections relies on the completeness of the space.[36] These projection operators, in turn, are essential for the statement and proof of many useful theorems, e.g. the spectral theorem. It is not very important in introductory quantum mechanics, and technical details and links may be found in footnotes like the one that follows.[nb 7] The space L2 is a Hilbert space, with inner product presented later. The function space of the example of the figure is a subspace of L2. A subspace of a Hilbert space is a Hilbert space if it is closed.
In summary, the set of all possible normalizable wave functions for a system with a particular choice of basis, together with the null vector, constitute a Hilbert space.
Not all functions of interest are elements of some Hilbert space, say L2. The most glaring example is the set of functions e2πip · xh. These are plane wave solutions of the Schrödinger equation for a free particle, but are not normalizable, hence not in L2. But they are nonetheless fundamental for the description. One can, using them, express functions that are normalizable using wave packets. They are, in a sense, a basis (but not a Hilbert space basis, nor a Hamel basis) in which wave functions of interest can be expressed. There is also the artifact "normalization to a delta function" that is frequently employed for notational convenience, see further down. The delta functions themselves aren't square integrable either.
The above description of the function space containing the wave functions is mostly mathematically motivated. The function spaces are, due to completeness, very large in a certain sense. Not all functions are realistic descriptions of any physical system. For instance, in the function space L2 one can find the function that takes on the value 0 for all rational numbers and -i for the irrationals in the interval [0, 1]. This is square integrable,[nb 8] but can hardly represent a physical state.
Common Hilbert spaces[edit]
While the space of solutions as a whole is a Hilbert space there are many other Hilbert spaces that commonly occur as ingredients.
• Square integrable complex valued functions on the interval [0, 2π]. The set {eint/2π, n ∈ ℤ} is a Hilbert space basis, i.e. a maximal orthonormal set.
• The Fourier transform takes functions in the above space to elements of l2(ℤ), the space of square summable functions ℤ → ℂ. The latter space is a Hilbert space and the Fourier transform is an isomorphism of Hilbert spaces.[nb 9] Its basis is {ei, i ∈ ℤ} with ei(j) = δij, i, j ∈ ℤ.
• The most basic example of spanning polynomials is in the space of square integrable functions on the interval [–1, 1] for which the Legendre polynomials is a Hilbert space basis (complete orthonormal set).
• The square integrable functions on the unit sphere S2 is a Hilbert space. The basis functions in this case are the spherical harmonics. The Legendre polynomials are ingredients in the spherical harmonics. Most problems with rotational symmetry will have "the same" (known) solution with respect to that symmetry, so the original problem is reduced to a problem of lower dimensionality.
• The associated Laguerre polynomials appear in the hydrogenic wave function problem after factoring out the spherical harmonics. These span the Hilbert space of square integrable functions on the semi-infinite interval [0, ∞).
More generally, one may consider a unified treatment of all second order polynomial solutions to the Sturm–Liouville equations in the setting of Hilbert space. These include the Legendre and Laguerre polynomials as well as Chebyshev polynomials, Jacobi polynomials and Hermite polynomials. All of these actually appear in physical problems, the latter ones in the harmonic oscillator, and what is otherwise a bewildering maze of properties of special functions becomes an organized body of facts. For this, see Byron & Fuller (1992, Chapter 5).
There occurs also finite-dimensional Hilbert spaces. The space n is a Hilbert space of dimension n. The inner product is the standard inner product on these spaces. In it, the "spin part" of a single particle wave function resides.
• In the non-relativistic description of an electron one has n = 2 and the total wave function is a solution of the Pauli equation.
• In the corresponding relativistic treatment, n = 4 and the wave function solves the Dirac equation.
With more particles, the situations is more complicated. One has to employ tensor products and use representation theory of the symmetry groups involved (the rotation group and the Lorentz group respectively) to extract from the tensor product the spaces in which the (total) spin wave functions reside. (Further problems arise in the relativistic case unless the particles are free.[37] See the Bethe–Salpeter equation.) Corresponding remarks apply to the concept of isospin, for which the symmetry group is SU(2). The models of the nuclear forces of the sixties (still useful today, see nuclear force) used the symmetry group SU(3). In this case as well, the part of the wave functions corresponding to the inner symmetries reside in some n or subspaces of tensor products of such spaces.
• In quantum field theory the underlying Hilbert space is Fock space. It is built from free single-particle states, i.e. wave functions when a representation is chosen, and can accommodate any finite, not necessarily constant in time, number of particles. The interesting (or rather the tractable) dynamics lies not in the wave functions but in the field operators that are operators acting on Fock space. Thus the Heisenberg picture is the most common choice (constant states, time varying operators).
Due to the infinite-dimensional nature of the system, the appropriate mathematical tools are objects of study in functional analysis.
Simplified description[edit]
Continuity of the wave function and its first spatial derivative (in the x direction, y and z coordinates not shown), at some time t.
Not all introductory textbooks take the long route and introduce the full Hilbert space machinery, but the focus is on the non-relativistic Schrödinger equation in position representation for certain standard potentials. The following constraints on the wave function are sometimes explicitly formulated for the calculations and physical interpretation to make sense:[38][39]
• The wave function must be square integrable. This is motivated by the Copenhagen interpretation of the wave function as a probability amplitude.
• It must be everywhere continuous and everywhere continuously differentiable. This is motivated by the appearance of the Schrödinger equation for most physically reasonable potentials.
It is possible to relax these conditions somewhat for special purposes.[nb 10] If these requirements are not met, it is not possible to interpret the wave function as a probability amplitude.[40]
This does not alter the structure of the Hilbert space that these particular wave functions inhabit, but it should be pointed out that the subspace of the square-integrable functions L2, which is a Hilbert space, satisfying the second requirement is not closed in L2, hence not a Hilbert space in itself.[nb 11] The functions that does not meet the requirements are still needed for both technical and practical reasons.[nb 12][nb 13]
More on wave functions and abstract state space[edit]
As has been demonstrated, the set of all possible wave functions in some representation for a system constitute an in general infinite-dimensional Hilbert space. Due to the multiple possible choices of representation basis, these Hilbert spaces are not unique. One therefore talks about an abstract Hilbert space, state space, where the choice of representation and basis is left undetermined. Specifically, each state is represented as an abstract vector in state space.[41] A quantum state |Ψ⟩ in any representation is generally expressed as a vector
where α = (α1, α2, ..., αn) are (dimensionless) discrete quantum numbers, and ω = (ω1, ω2, ..., ωm) are continuous variables (not necessarily dimensionless). All of them index the components of the vector, and |α, ω are the basis vectors in this representation. All α are in an n-dimensional set A = A1 × A2 × ... An where each Ai is the set of allowed values for αi, likewise all ω are in an m-dimensional "volume" Ω ⊆ ℝm where Ω = Ω1 × Ω2 × ... Ωm and each Ωi ⊆ ℝ is the set of allowed values for ωi, a subset of the real numbers . For generality n and m are not necessarily equal.
For example, for a single particle in 3d with spin s, neglecting other degrees of freedom, using Cartesian coordinates, we could take α = (sz) for the spin quantum number of the particle along the z direction, and ω = (x, y, z) for the particle's position coordinates. Here A = {−s, −s + 1, ..., s − 1, s} is the set of allowed spin quantum numbers and Ω = ℝ3 is the set of all possible particle positions throughout 3d position space. An alternative choice is α = (sy) for the spin quantum number along the y direction and ω = (px, py, pz) for the particle's momentum components. In this case A and Ω are the same.
Then, a component Ψ(α, ω, t) of the vector |Ψ⟩ is referred to as the "wave function" of the system.
When interpreted as a probability amplitude (non-relativistic systems with constant number of particles), the probability density of finding the system at α, ω is
The probability of finding system with α in some or all possible discrete-variable configurations, DA, and ω in some or all possible continuous-variable configurations, C ⊆ Ω, is the sum and integral over the density,[nb 14]
where dmω = 12...m is a "differential volume element" in the continuous degrees of freedom. Since the sum of all probabilities must be 1, the normalization condition
must hold at all times during the evolution of the system.
The normalization condition requires ρ dmω to be dimensionless, by dimensional analysis Ψ must have the same units as (ω1ω2...ωm)−1/2.
Whether the wave function really exists, and what it represents, are major questions in the interpretation of quantum mechanics. Many famous physicists of a previous generation puzzled over this problem, such as Schrödinger, Einstein and Bohr. Some advocate formulations or variants of the Copenhagen interpretation (e.g. Bohr, Wigner and von Neumann) while others, such as Wheeler or Jaynes, take the more classical approach[42] and regard the wave function as representing information in the mind of the observer, i.e. a measure of our knowledge of reality. Some, including Schrödinger, Bohm and Everett and others, argued that the wave function must have an objective, physical existence. Einstein thought that a complete description of physical reality should refer directly to physical space and time, as distinct from the wave function, which refers to an abstract mathematical space.[43]
See also[edit]
1. ^ The functions are here assumed to be elements of L2, the space of square integrable functions. The elements of this space are more precisely equivalence classes of square integrable functions, two functions declared equivalent if they differ on a set of Lebesgue measure 0. This is necessary to obtain an inner product (that is, (Ψ, Ψ) = 0 ⇒ Ψ ≡ 0) as opposed to a semi-inner product. The integral is taken to be the Lebesque integral. This is essential for completeness of the space, thus yielding a complete inner product space = Hilbert space.
2. ^ The Fourier transform viewed as a unitary operator on the space L2 has eigenvalues ±1, ±i. The eigenvectors are "Hermite functions", i.e. Hermite polynomials multiplied by a Gaussian function. See Byron & Fuller (1992) for a description of the Fourier transform as a unitary transformation. For eigenvalues and eigenvalues, refer to Problem 27 Ch. 9.
3. ^ Column vectors can be motivated by the convenience of expressing the spin operator for a given spin as a matrix, for the z-component spin operator (divided by hbar to nondimensionalize)
The eigenvectors of this matrix are the above column vectors, with eigenvalues being the corresponding spin quantum numbers.
4. ^ Each |sz is usually identified as a column vector
but it is a common abuse of notation to write
because the kets |sz are not synonymous or equal to the column vectors. Column vectors simply provide a convenient way to express the spin components.
5. ^ For this statement to make sense, the observables need to be elements of a maximal commuting set. To see this, it is a simple matter to note that, for example, the momentum operator of the i'th particle in a n-particle system is not a generator of any symmetry in nature. On the other hand, the total momentum is a generator of a symmetry in nature; the translational symmetry.
6. ^ The resulting basis may or may not technically be a basis in the mathematical sense of Hilbert spaces. For instance, states of definite position and definite momentum are not square integrable. This may be overcome with the use of wave packets or by enclosing the system in a "box". See further remarks below.
7. ^ In technical terms, this is formulated the following way. The inner product yields a norm. This norm in turn induces a metric. If this metric is complete, then the aforementioned limits will be in the function space. The inner product space is then called complete. A complete inner product space is a Hilbert space. The abstract state space is always taken as a Hilbert space. The matching requirement for the function spaces is a natural one. The Hilbert space property of the abstract state space was originally extracted from the observation that the function spaces forming normalizable solutions to the Schrödinger equation are Hilbert spaces.
8. ^ As is explained in a later footnote, the integral must be taken to be the Lebesgue integral, the Riemann integral is not sufficient.
9. ^ Conway 1990. This means that inner products, hence norms, are preserved and that the mapping is a bounded, hence continuous, linear bijection. The property of completeness is preserved as well. Thus this is the right concept of isomorphism in the category of Hilbert spaces.
10. ^ One such relaxation is that the wave function must belong to the Sobolev space W1,2. It means that it is differentiable in the sense of distributions, and its gradient is square-integrable. This relaxation is necessary for potentials that are not functions but are distributions, such as the Dirac delta function.
11. ^ It is easy to visualize a sequence of functions meeting the requirement that converges to a discontinuous function. For this, modify an example given in Inner product space#Examples. This element though is an element of L2.
12. ^ For instance, in perturbation theory one may construct a sequence of functions approximating the true wave function. This sequence will be guaranteed to converge in a larger space, but without the assumption of a full-fledged Hilbert space, it will not be guaranteed that the convergence is to a function in the relevant space and hence solving the original problem.
13. ^ Some functions not being square-integrable, like the plane-wave free particle solutions are necessary for the description as outlined in a previous note and also further below.
14. ^ Here
is a multiple sum.
1. ^ Born 1927, pp. 354–357
2. ^ Heisenberg 1958, p. 143
3. ^ Heisenberg, W. (1927/1985/2009). Heisenberg is translated by Camilleri 2009, p. 71, (from Bohr 1985, p. 142).
4. ^ Murdoch 1987, p. 43
5. ^ de Broglie 1960, p. 48
6. ^ Landau & Lifshitz, p. 6
7. ^ Newton 2002, pp. 19–21
8. ^ a b c d Born 1926a, translated in Wheeler & Zurek 1983 at pages 52–55.
9. ^ a b Born 1926b, translated in Ludwig 1968, pp. 206–225. Also here.
10. ^ Born, M. (1954).
11. ^ Einstein 1905, pp. 132–148 (in German), Arons & Peppard 1965, p. 367 (in English)
12. ^ Einstein 1916, pp. 47–62 and a nearly identical version Einstein 1917, pp. 121–128 translated in ter Haar 1967, pp. 167–183.
13. ^ de Broglie 1923, pp. 507–510,548,630
14. ^ Hanle 1977, pp. 606–609
15. ^ Schrödinger 1926, pp. 1049–1070
16. ^ Tipler, Mosca & Freeman 2008
17. ^ a b c Weinberg 2013
18. ^ Young & Freedman 2008, p. 1333
19. ^ a b c Atkins 1974
20. ^ Martin & Shaw 2008
21. ^ Pauli 1927, pp. 601–623.
22. ^ Weinberg (2002) takes the standpoint that quantum field theory appears the way it does because it is the only way to reconcile quantum mechanics with special relativity.
23. ^ Weinberg (2002) See especially chapter 5, where some of these results are derived.
24. ^ Weinberg 2002 Chapter 4.
25. ^ Zwiebach 2009
26. ^ Shankar 1994, Ch. 1
27. ^ a b Griffiths 2004
28. ^ Shankar 1994, p. 378–379
29. ^ Landau & Lifshitz 1977
30. ^ Zettili 2009, p. 463
31. ^ Weinberg 2002 Chapter 3, Scattering matrix.
32. ^ Physics for Scientists and Engineers – with Modern Physics (6th Edition), P. A. Tipler, G. Mosca, Freeman, 2008, ISBN 0-7167-8964-7
33. ^ David Griffiths (2008). Introduction to elementary particles. Wiley-VCH. pp. 162–. ISBN 978-3-527-40601-2. Retrieved 27 June 2011.
34. ^ Weinberg 2002
35. ^ Weinberg 2002, Chapter 3
36. ^ Conway 1990
37. ^ Greiner & Reinhardt 2008
38. ^ Eisberg & Resnick 1985
39. ^ Rae 2008
40. ^ Atkins 1974, p. 258
41. ^ Dirac 1982
42. ^ Jaynes 2003
43. ^ Einstein 1998, p. 682
Further reading[edit]
External links[edit] |
87c5bb2367888ed2 | fredag 29 juli 2016
Secret of Laser vs Secret of Piano
There is a connection between the action of a piano as presented in the sequence of posts The Secret of the Piano and a laser (Light Amplification by Stimulated Emission of Radiation), which is remarkable as an expression of a fundamental resonance phenomenon.
To see the connection we start with the following quote from Principles of Lasers by Orazio Svelto:
• There is a fundamental difference between spontaneous and stimulated emission processes.
• In the case of spontaneous emission, the atoms emit e.m waves that has no definite phase relation with that emitted by another atom...
• In the case of stimulated emission, since the process is forced by the incident e.m. wave, the emission of any atom adds in phase to that of the incoming wave...
A laser hus emits coherent light as electromagnetic waves all in-phase, and thereby can transmit intense energy over distance.
The question is how the emission/radiation can be coordinated so that the e.m. waves from many/all atoms are kept in-phase. Without coordination the emission will become more or less out-of-phase resulting in weak radiation.
The Secret of the Piano reveals that the emission from the three strings for each note in the middle register, which may have a frequency spread of about half a Herz, are kept in phase by interacting with a common soundboard through a common bridge in a "breathing mode" with the soundboard/bridge vibrating with half a period phase lag with respect to the strings. The breathing mode is initiated when the hammer feeds energy into the strings by a hard hit.
In the breathing mode strings and soundboard act together to generate an outgoing sound from the soundboard fed by energy from the strings, which has a long sustain/duration in time, as the miracle of the piano.
If we translate the experience from the piano to the laser, we understand that laser emission/radiation is (probably) kept in phase by interaction with a stabilising half a period out-of-phase forcing corresponding to the soundboard, while leaving part of the emission to strong in-phase action on a target.
An alternative to quick hammer initiation is in-phase forcing over time, which requires a switch from input to output by half a period shift of the forcing.
We are also led to the idea that black body radiation, which is partially coherent, is kept in phase by interaction with a receiver/soundboard. Without receiver/soundboard there will be no radiation. It is thus meaningless to speak about black body radiation into some vacuous nothingness, which is often done based on a fiction of "photon" particles being spitted out from a body even without receiver, as physically meaningless as speaking into the desert.
torsdag 28 juli 2016
New Quantum Mechanics 10: Ionisation Energy
Below are sample computations of ground states for Li1+, C1+, Ne1+ and Na1+ showing good agreement with table data of first ionisation energies of 0.2, 0.4, 0.8 and 0.2, respectively.
Note that computation of first ionisation energy is delicate, since it represents a small fraction of total energy.
onsdag 27 juli 2016
New Quantum Mechanics 9: Alkaline (Earth) Metals
The result presentation continues below with alkaline and alkaline earth metals Na (2-8-1), Mg (2-8-2), K (2-8-8-1), Ca (2-8-8-2), Rb (2-8-18-8-1), Sr (2-8-18-8-2), Cs (2-8-18-18-8-1) and Ba (2-8-18-18-8-2):
New Quantum Mechanics 8: Noble Gases Atoms 18, 36, 54 and 86
The presentation of computational results continues below with the noble gases Ar (2-8-8), Kr (2-8-18-8), Xe (2-8-18-18-8) and Rn (2-8-18-32-18-8) with the shell structure indicated.
Again we see good agreement of ground state energy with NIST data, and we notice nearly equal energy in fully filled shells.
Note that the NIST ionization data does not reveal true shell energies since it displays a fixed shell energy distribution independent of ionization level, and thus cannot be used for comparison of shell energies.
New Quantum Mechanics 7: Atoms 1-10
This post presents computations with the model of New Quantum Mechanics 5 for ground states of atoms with N= 2 - 10 electrons in spherical symmetry with 2 electrons in an inner spherical shell and N-2 electrons in an outer shell with the radius of the free boundary as the interface of the shells adjusted to maintain continuity of charge density. The electrons in each shell are smeared to spherical symmetry and the repulsive electron potential is reduced by the factor n-1/n with n the number of electrons in a shell to account for lack of self repulsion.
The ground state is computed by parabolic relaxation in the charge density formulation of New Quantum Mechanics 1 with restoration of total charge after each relaxation and shows good agreement with table data as shown in the figures below.
The graphs show as functions of radius, charge density per unit volume in color, charge density per unit radius in black, kernel potential in green and total electron potential in cadmium red. The homogeneous Neumann condition at the interface of charge density per unit volume is clearly visible.
The shell structure with 2 electrons in the inner shell and N-2 in the outer shell is imposed based on a principle of "electron size" depending on the strength of effective kernel potential, which gives the familiar pattern of 2-8-18-32 of electrons in successively filled shells as a consequence of shell volume of nearly constant thickness scaling quadratically with shell number. This replaces the ad hoc unphysical Pauli exclusion principle with a simple physical principle of size and no overlap.
The electron size principle allows the first shell to house at most 2 electrons, the second shell 8 electrons, the third 18 electrons, et cet.
In the next post similar results for Atoms 11-86 will be presented and it will be noted that a characteristic of a filled shell structure 2-8-18-32- is comparable total energy in each shell, as can be seen for Neon below.
The numbers below show table data of total energy in the first line and computed in second line, while the groups show total energy, kinetic energy, kernel potential energy and electron potential energy in each shell.
måndag 25 juli 2016
New Quantum Mechanics 6: H2 Molecule
Computing with the model of the previous post in polar coordinates with origin at the center of an H2 molecule assuming rotational symmetry around the axis connecting the two kernels, gives the following results (in atomic units) for the ground state using a $50\times 40$ uniform mesh:
• total energy = -1.167 (kernel potential: -4.28, electron potential: 0.587 and kinetic: 1.147)
• kernel distance = 1.44
in close correspondence to table data (-1.1744 and 1.40). Here is a plot of output:
söndag 24 juli 2016
New Quantum Mechanics 5: Model as Schrödinger + Neumann
This sequence of posts presents an alternative Schrödinger equation for an atom with $N$ electrons starting from a wave function Ansatz of the form
• $\psi (x,t) = \sum_{j=1}^N\psi_j(x,t)$ (1)
as a sum of $N$ electronic complex-valued wave functions $\psi_j(x,t)$, depending on a common 3d space coordinate $x$ and a time coordinate $t$, with non-overlapping spatial supports $\Omega_j(t)$ filling 3d space, satisfying for $j=1,...,N$ and all time:
• $i\dot\psi_j + H\psi_j = 0$ in $\Omega_j$, (2a)
• $\frac{\partial\psi_j}{\partial n} = 0$ on $\Gamma_j(t)$, (2b)
where $\Gamma_j(t)$ is the boundary of $\Omega_j(t)$, $\dot\psi =\frac{\partial\psi}{\partial t}$ and $H=H(x,t)$ is the (normalised) Hamiltonian given by
• $H = -\frac{1}{2}\Delta - \frac{N}{\vert x\vert}+\sum_{k\neq j}V_k(x)$ for $x\in\Omega_j(t)$,
with $V_k(x)$ the repulsion potential corresponding to electron $k$ defined by
• $V_k(x)=\int\frac{\psi_k^2(y)}{2\vert x-y\vert}dy$,
and the electron wave functions are normalised to unit charge of each electron:
• $\int_{\Omega_j(t)}\psi_j^2(x,t) dx=1$ for $j=1,..,N$ and all time. (2c)
The differential equation (2a) with homogeneous Neumann boundary condition (2b) is complemented by the following global free boundary condition:
• $\psi (x,t)$ is continuous across inter-electron boundaries $\Gamma_j(t)$. (2d)
The ground state is determined as a the real-valued time-independent minimiser $\psi (x)=\sum_j\psi_j(x)$ of the total energy
• $E(\psi ) = \frac{1}{2}\int\vert\nabla\psi\vert^2\, dx - \int\frac{N\psi^2(x)}{\vert x\vert}dx+\sum_{k\neq j}\int V_k(x)\psi^2(x)\, dx$,
under the normalisation (2c), the homogeneous Neumann boundary condition (2b) and the free boundary condition (2d).
In the next post I will present computational results in the form of energy of ground states for atoms with up to 54 electrons and corresponding time-periodic solutions in spherical symmetry, together with ground state and dissociation energy for H2 and CO2 molecules in rotational symmetry.
In summary, the model is formed as a system of one-electron Schrödinger equations, or electron container model, on a partition of 3d space depending of a common spatial variable and time, supplemented by a homogeneous Neumann condition for each electron on the boundary of its domain of support combined with a free boundary condition asking continuity of charge density across inter-element boundaries.
We shall see that for atoms with spherically symmetric electron partitions in the form of a sequence of shells centered at the kernel, the homogeneous Neumann condition corresponds to vanishing kinetic energy of each electron normal to the boundary of its support as a condition of separation or interface condition between different electrons meeting with continuous charge density.
Here is one example: Argon with 2-8-8 shell structure with NIST Atomic data base ground state energy in first line (526.22), the computed in second line and the total energies in the different shells in three groups with kinetic energy in second row, kernel potential energy in third and repulsive electron energy in the last row. Note that the total energy in the fully filled first (2 electrons) and second shell (8 electrons) are nearly the same, while the partially filled third shell (also 8 electrons out of 18 when fully filled) has lower energy. The color plot shows charge density per unit volume and the black curve charge density per unit radial increment as functions of radius. The green curve is the kernel potential and the cyrano the total electron potential. Note in particular the vanishing derivative of charge density/kinetic energy at shell interfaces.
lördag 2 juli 2016
New Quantum Mechanics 4: Free Boundary Condition
This is a continuation of previous posts presenting an atom model in the form of a free boundary problem for a joint continuously differentiable electron charge density, as a sum of individual electron charge densities with disjoint supports, satisfying a classical Schrödinger wave equation in 3 space dimensions.
The ground state of minimal total energy is computed by parabolic relaxation with the free boundary separating different electrons determined by a condition of zero gradient of charge density. Computations in spherical symmetry show close correspondence with observation, as illustrated by the case of Oxygen with 2 electrons in an inner shell (blue) and 6 electrons in an outer shell (red) as illustrated below in a radial plot of charge density showing in particular the zero gradient of charge density at the boundary separating the shells at minimum total energy (with -74.81 observed and -74.91 computed energy). The green curve shows truncated kernel potential, the magenta the electron potential and the black curve charge density per radial increment.
The new aspect is the free boundary condition as zero gradient of charge density/kinetic energy. |
465e372c52c5f24b | Quantum Mechanics: Hydrogen Atom
By Dragica Vasileska1, Gerhard Klimeck2
1. Arizona State University 2. Purdue University
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The solution of the Schrödinger equation (wave equations) for the hydrogen atom uses the fact that the Coulomb potential produced by the nucleus is isotropic (it is radially symmetric in space and only depends on the distance to the nucleus). Although the resulting energy eigenfunctions (the "orbitals") are not necessarily isotropic themselves, their dependence on the angular coordinates follows completely generally from this isotropy of the underlying potential: The eigenstates of the Hamiltonian (= energy eigenstates) can be chosen as simultaneous eigenstates of the angular momentum operator. This corresponds to the fact that angular momentum is conserved in the orbital motion of the electron around the nucleus. Therefore, the energy eigenstates may be classified by two angular momentum quantum numbers, l and m (integer numbers). The "angular momentum" quantum number l = 0, 1, 2, ... determines the magnitude of the angular momentum. The "magnetic" quantum number m = −l, .., +l determines the projection of the angular momentum on the (arbitrarily chosen) z-axis.
In addition to mathematical expressions for total angular momentum and angular momentum projection of wavefunctions, an expression for the radial dependence of the wave functions must be found. It is only here that the details of the 1/r Coulomb potential enter (leading to Laguerre polynomials in r). This leads to a third quantum number, the principal quantum number n = 1, 2, 3, ... The principal quantum number in hydrogen is related to atom's total energy.
Note that the maximum value of the angular momentum quantum number is limited by the principal quantum number: it can run only up to n − 1, i.e. l = 0, 1, ..., n − 1.
Due to angular momentum conservation, states of the same l but different m have the same energy (this holds for all problems with rotational symmetry). In addition, for the hydrogen atom, states of the same n but different l are also degenerate (i.e. they have the same energy). However, this is a specific property of hydrogen and is no longer true for more complicated atoms which have a (effective) potential differing from the form 1/r (due to the presence of the inner electrons shielding the nucleus potential).
Taking into account the spin of the electron adds a last quantum number, the projection of the electron's spin angular momentum along the z axis, which can take on two values. Therefore, any eigenstate of the electron in the hydrogen atom is described fully by four quantum numbers. According to the usual rules of quantum mechanics, the actual state of the electron may be any superposition of these states. This explains also why the choice of z-axis for the directional quantization of the angular momentum vector is immaterial: An orbital of given l and m' obtained for another preferred axis z' can always be represented as a suitable superposition of the various states of different m (but same l) that have been obtained for z.
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Researchers should cite this work as follows:
• ww.eas.asu.edu/~vasilesk
• Dragica Vasileska; Gerhard Klimeck (2008), "Quantum Mechanics: Hydrogen Atom," http://nanohub.org/resources/4993.
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a3cdc9aa46b5f4b9 | Consciousness Studies/Measurement In Quantum Physics And The Preferred Basis Problem
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The Measurement Problem[edit]
In quantum physics the probability of an event is deduced by taking the square of the amplitude for an event to happen. The term "amplitude for an event" arises because of the way that the Schrödinger equation is derived using the mathematics of ordinary, classical waves where the amplitude over a small area is related to the number of photons hitting the area. In the case of light, the probability of a photon hitting that area will be related to the ratio of the number of photons hitting the area divided by the total number of photons released. The number of photons hitting an area per second is the intensity or amplitude of the light on the area, hence the probability of finding a photon is related to "amplitude".
However, the Schrödinger equation is not a classical wave equation. It does not determine events, it simply tells us the probability of an event. In fact the Schrödinger equation in itself does not tell us that an event occurs at all, it is only when a measurement is made that an event occurs. The measurement is said to cause state vector reduction. This role of measurement in quantum theory is known as the measurement problem. The measurement problem asks how a definite event can arise out of a theory that only predicts a continuous probability for events.
Two broad classes of theory have been advanced to explain the measurement problem. In the first it is proposed that observation produces a sudden change in the quantum system so that a particle becomes localised or has a definite momentum. This type of explanation is known as collapse of the wavefunction. In the second it is proposed that the probabilistic Schrödinger equation is always correct and that, for some reason, the observer only observes one particular outcome for an event. This type of explanation is known as the relative state interpretation. In the past thirty years relative state interpretations, especially Everett's relative state interpretation have become favoured amongst quantum physicists.
The quantum probability problem[edit]
The measurement problem is particularly problematical when a single particle is considered. Quantum theory differs from classical theory because it is found that a single photon seems to be able to interfere with itself. If there are many photons then probabilities can be expressed in terms of the ratio of the number hitting a particular place to the total number released but if there is only one photon then this does not make sense. When only one photon is released from a light source quantum theory still gives us a probability for a photon to hit a particular area but what does this mean at any instant if there is indeed only one photon?
If the Everettian interpretation of quantum mechanics is invoked then it might seem that the probability of the photon hitting an area in your particular universe is related to the occurrences of the photon in all the other universes. But in the Everrettian interpretation even the improbable universes occur. This leads to a problem known as the quantum probability problem:
If the universe splits after a measurement, with every possible
measurement outcome realised in some branch, then how can it make
sense to talk about the probabilities of each outcome? Each
outcome occurs.
This means that if our phenomenal consciousness is a set of events then there would be endless copies of these sets of events, almost all of which are almost entirely improbable to an observer outside the brain but all of which exist according to an Everrettian Interpretation. Which set is you? Why should 'you' conform to what happens in the environment around you?
The preferred basis problem[edit]
It could be held that you assess probabilities in terms of the branch of the universe in which you find yourself but then why do you find yourself in a particular branch? Decoherence Theory is one approach to these questions. In decoherence theory the environment is a complex form that can only interact with particles in particular ways. As a result quantum phenomena are rapidly smoothed out in a series of micro-measurements so that the macro-scale universe appears quasi-classical. The form of the environment is known as the preferred basis for quantum decoherence. This then leads to the preferred basis problem in which it is asked how the environment occurs or whether the state of the environment depends on any other system.
According to most forms of decoherence theory 'you' are a part of the environment and hence determined by the preferred basis. From the viewpoint of phenomenal consciousness this does not seem unreasonable because it has always been understood that the conscious observer does not observe things as quantum superpositions. The conscious observation is a classical observation.
However, the arguments that are used to derive this idea of the classical, conscious observer contain dubious assumptions that may be hindering the progress of quantum physics. The assumption that the conscious observer is simply an information system is particularly dubious:
"Here we are using aware in a down - to - earth sense: Quite simply, observers know what they know. Their information processing machinery (that must underlie higher functions of the mind such as "consciousness") can readily consult the content of their memory. (Zurek 2003).
This assumption is the same as assuming that the conscious observer is a set of measurements rather than an observation. It makes the rest of Zurek's argument about decoherence and the observer into a tautology - given that observations are measurements then observations will be like measurements. However, conscious observation is not simply a change of state in a neuron, a "measurement", it is the entire manifold of conscious experience.
In his 2003 review of this topic Zurek makes clear an important feature of information theory when he states that:
There is no information without representation.
So the contents of conscious observation are states that correspond to states of the environment in the brain (i.e.: measurements). But how do these states in the brain arise? The issue that arises here is whether the representation, the contents of consciousness, is entirely due to the environment or due to some degree to the form of conscious observation. Suppose we make the reasonable assumption that conscious observation is due to some physical field in the dendrites of neurons rather than in the action potentials that transmit the state of the neurons from place to place. This field would not necessarily be constrained by decoherence; there are many possibilities for the field, for instance, it could be a radio frequency field due to impulses or some other electromagnetic field (cf: Anglin & Zurek (1996)) or some quantum state of macromolecules etc.. Such a field might contain many superposed possibilities for the state of the underlying neurons and although these would not affect sensations, they could affect the firing patterns of neurons and create actions in the world that are not determined by the environmental "preferred basis".
Zeh (2000) provides a mature review of the problem of conscious observation. For example he realises that memory is not the same as consciousness:
"The genuine carriers of consciousness ... must not in general be expected to represent memory states, as there do not seem to be permanent contents of consciousness."
and notes of memory states that they must enter some other system to become part of observation:
"To most of these states, however, the true physical carrier of consciousness somewhere in the brain may still represent an external observer system, with whom they have to interact in order to be perceived. Regardless of whether the ultimate observer systems are quasi-classical or possess essential quantum aspects, consciousness can only be related to factor states (of systems assumed to be localized in the brain) that appear in branches (robust components) of the global wave function — provided the Schrodinger equation is exact. Environmental decoherence represents entanglement (but not any “distortion” — of the brain, in this case), while ensembles of wave functions, representing various potential (unpredictable) outcomes, would require a dynamical collapse (that has never been observed)."
However, Zeh (2003) points out that events may be irreversibly determined by decoherence before information from them reaches the observer. This might give rise to a multiple worlds and multiple minds mixture for the universe, the multiple minds being superposed states of the part of the world that is the mind. Such an interpretation would be consistent with the apparently epiphenomenal nature of mind. A mind that interacts only weakly with the consensus physical world, perhaps only approving or rejecting passing actions would be an ideal candidate for a QM multiple minds hypothesis.
Further reading and references[edit]
• Pearl, P. (1997). True collapse and false collapse. Published in Quantum Classical Correspondence: Proceedings of the 4th Drexel Symposium on Quantum Nonintegrability, Philadelphia, PA, USA, September 8-11, 1994, pp. 51-68. Edited by Da Hsuan Feng and Bei Lok Hu. Cambridge, MA: International Press, 1997.
• Zeh, H. D. (1979). Quantum Theory and Time Asymmetry. Foundations of Physics, Vol 9, pp 803-818 (1979).
• Zeh, H.D. (2000) THE PROBLEM OF CONSCIOUS OBSERVATION IN QUANTUM MECHANICAL DESCRIPTION. Epistemological Letters of the Ferdinand-Gonseth Association in Biel (Switzerland) Letter No 63.0.1981, updated 2000.
• Zeh, H.D. (2003). Decoherence and the Appearance of a Classical World in Quantum Theory, second edition, Authors:. E. Joos, H.D. Zeh, C. Kiefer D. Giulini, J. Kupsch, and I.-O. Stamatescu. Chapter 2: Basic Concepts and their Interpretation. |
c8dfabd7aa3173ba | This question is closely related to: What counts as information?
Taking the specific example, again, of the EPR experiment. I think everyone agrees on the following:
The act of measuring the system at one detector collapses the wavefunction, effecting the result measured by the other detector.
This is basically saying that the principal of locality is violated. But I do not understand why it is not also a violation of relativity. I.e. how can locality be violated (implying that an object can effect something outside of its light-cone) but special relativity not be violated (which implies all cause and effect relationships must occur within each other's light cones). To me the violation of one should immediately imply the violation of the other i.e. why does the violation of locality not imply the violation of special relativity?
There are a few cases.
First case, you measure your particle then you write a letter to your friend and tell your result to your friend before your friend measures their stuff. They can be amazed to know the result of their measurements before they do them. Or they can be unamazed since by then your measurement has had time to affect them without violating relativity. Nothing interesting.
Second case, same thing except you get a letter from your friend before you measure your stuff. Again, nothing exciting.
Third case, like the first case above but you don't bother sending the letter, but you could have, so nothing interesting.
Fourth case, like the second case above but your friend didn't bother sending a letter, but they could have, so nothing interesting.
Fifth case, both of you measure your stuff before you could have sent a message. Is this spooky? Well, neither of you caused the specific result you got. You could think your result caused your friend's result and someone else could think your friend's result caused your result. But neither of you could control what you got so you couldn't actually use what you did to affect them. You aren't affecting them, so no violation of relativity. You are both getting results and neither of you controls the results it's like someone else is giving you both results that happen to be correlated. This isn't very different than you and your cousin getting a holiday checks from your grandma for the same amount, it's something that happened to you both and it's something that is correlated, but you aren't affecting each other by getting the same things.
The distant people aren't affecting each other. So no serious violation of relativity. For case five, you might ask since who measured first is frame dependent whether I am saying we can't tell who affected whom. But they don't even have separate states that they can affect.
If you write down the Schrödinger equation for a pair of entangled particles and a pair of measuring devices and evolve it according to the actual Hamiltonian for the actual setup (an equation few people ever setup in their entire life, even amongst physicists) then it is clear what a measurement does. It takes a beam in configuration space and bifurcates it into separate beams while possibly polarizing the separate beams.
The bifurcation happens in the part where the measurement happens. But the beam itself exists in configuration space and always will.
It might help to be very very specific about this, so let's pick a concrete example, you are going to measure spin with a Stern-Gerlach device, the beam to measure the spin will travel in the y direction, have some thickness in the x direction and split into two beams that in the x direction do not overlap. In one dimension you could track he x with of the beam over time as starting out as a single line say from 2-4 and over time it splits like a tree branch into two separate lines one say on the left from 1-2 and another branch from 4-5. If you had time go in the y direction it does look like a branch, of a fork in a river. But the actual beam exists in confirmation space (that's normal for quantum mechanics if people don't tell you thus they are oversimplifying but you need it in this case), which has x, y , and z for each particle so we need to have beams that have two x coordinates one for each particle.
So imagine you have a blob that has each x be in the range 2-4 you could draw it as a square in the xy plane with the x being the x position of particle one and the y being the x position of particle two. What a measurement of particle one does is bifurcate the beam so a single beam with spread of x from 2-4 continuously evolves into two beams. It becomes a beam from 1-2 and another from 4-5. So that blob in the xy plane splits down a vertical line down the middle and becomes two rectangles that move apart to have a rectangle of emptiness between them by having each piece move in the horizontal direction. That is what a measurement of particle one does it splits blobs in configuration space with vertical lines and separates them by moving them horizontally.
What does a measurement of particle two do? It splits blobs by making a horizontal line down the middle and then separating them by having them each move differently in the vertical direction.
And that is what they each do. Its like cutting a square pizza, you van slice it one way first or the other way first but it isn't really affecting each other. Because we know what each does, it separates every piece according to the rules. Measurements of particle one make horizontal lines and move them apart by moving them vertically because we are using the vertical direction to represent the locations of particle one and that is what we are measuring. We don't really care if the beam is already separated in another direction.
If you do both at the same time it just means your pizza rips into fours pieces at once instead of ripping into two and then each of those ripping into two. You aren't really changing how the other one operates. There is a wrinkle I'll get to later about entanglement, right now I'm just addressing mutliparticld measurements
But first there is another important detail for a single particle measurement. For particles with spin, there is a spin state as well as a spatial state. When you measure the spin, the location of that vertical or horizontal line depends on how the spin state compare to the orientation of the Stern-Gerlach device, and by the time the beams are separate each (now separate) beam has that particle's spin state polarized to one of its eigenvalues for that orientation's operator.
The Schrödinger equation is unambiguous about how the wave function evolves, so if you have a device that sends spin up to the 4-5 part and sends spin down to the 1-2 part and the state has more spin up than spin down then the middle part of the 2-4 beam, the part nearest 3, actually deflects right, towards the 4-5 part.
But if you have a device that sends spin up to the 1-2 part and sends spin down to the 4-5 part and the state has more spin up than spin down then the middle part of the 2-4 beam, the part nearest 3, actually deflects left, towards the 1-2 part.
So the Schrödinger equation is utterly clear and unambiguous, but the actual detailed motion already depends on trivial kinds of details like how you calibrated your device.
The wave always separates the positions of something, maybe the locations of electrons in your photodetector, maybe the locations of electrons in your computer or in the ink arranged on your lab book. As long as something is changed based on the outcome of the experiment, then the beam separates, and this is normal.
And it is normal for the details to depend on the details of how you did it. But all you know when you measure one particle is whether it separated and then it can interact with other things based on those different places where it can be, it can interact with photodetectors and such in the 1-2 region or it can instead interact with photodetectors in the 4-5 region instead. Your stuff doesn't really act differently based on whether it separated in the direction corresponding to the other particle.
So that is my point the actual beam is like a series of rectangles in the actual configuration space and all you do is separate along the directions you have access to, and all the other people far away do is separate things along the direction they have access to.
So they aren't really affecting you. You separate by making vertical lines and moving things horizontally because the horizontal direction is your stuff. They separate by making horizontal lines and moving things vertically because the vertical direction is their stuff. You aren't affecting how or what they do.
And that's literally what the Schrödinger equation says.
And each of you can interact with your stuff and your stuff is only sensitive to where you particle is, so it interacts with how the beam has split in the horizontal direction and the other people's stuff interacts with how it split in the vertical direction.
So that is what I mean by you not affecting each other. But remember how I talked about the beam splitting causing the spin state for that particle to polarize in a way related to the orientation of the device? Now we can talk about entanglement of the spins. And the only unmentioned things will be the effective irreversibility of measurements and passing entanglement up the measurement chain.
When the spin states were entangled you didn't have single particle spin states, so when you polarize one, you polarize the other too. You create spin states for both particles.
So if you measured yours first you'd make a vertical line and separate them horizontally and the left one would have say spin up and the right one has spin down. But what do the other people see? The just see a beam that in the vertical direction has not been split and still has equal parts spin up and spin down.
Since they are separsted in the horizontal direction by that vertical line, they are technically now orthogonal. But guess what, different spin states were already orthogonal. So absolutely nothing has changed for the other people, they have a beam in the 2-4 range in their direction that also has orthogonally two different parts for two different spins. Nothing has changed and they know what to do when they measure, they split the beam with a horizontal line and separate them vertically. Where you draw the line depends on how spin you have if it is all the spin you are measuring it goes on one edge. So if they are entangled to have opposite spins you get just two rectangles one in the rectangle 1-2x4-5 the other in the 4-5x1-2 (assuming they oriented both their devices to send spin up to 4-5).
But each just sees two beams, one in 1-2 and one in 4-5. Each sees a full 50% of the area in each section of the beam (so sees the results 50% of the time if you choose to go the level of probability, we aren't and don't need to since I am merely talking about what the Schrödinger equation deterministically predicts for the actual experimental setup including the setup of the devices we use to measure them, which is why I keep mentioning whether the device sends spin up to 4-5 or sends spin up to 1-2 whereas at the probability level I could say you just get a result and ignore how it happened, but then people could argue, I'm sticking to just what the Schrödinger equation says). Each knows what a measurement does, it splits a beam horizontally/vertically based on how much spin is in the direction of the device and then interacts with the separated beam in an entirely local way based solely on how the beam separated in that horizontal/vertical fashion.
They simply don't change how the other one acts. The closest a change you see is whether your line separates a unified blob or whether the blob was already broken even though to you it looks like an unseparated beam, but you are still just pushing each section of the beam according to how the spin agrees with your device its just that the beam really always was in configuration space.
This is also actually why the double slit pattern can be destroyed you are separating the beam in that other-particle direction when you do a which-way measurement and so the beams just don't overlap to interfere anymore.
If this is at all confusing, then you might want to first see what the Schrödinger equation says for a single particle spin measurement when you describe the measurement device and process and track what the Schrödinger equation says about how the wave evolves and bifurcates the beam. The entanglement is throwing a level of complexity on top of that and most people don't learn that in detail.
So when I said you are not affecting the results I'm saying that when you separate horizontally the beam is still unseparated vertically and that their measurements and interactions still do the same things they always did.
There is a still an unseparated beam in your direction of separation. There is still 50% of the beam in the two different spin states. Your stuff still only interacts with your beam. And you never ever notice (or learn) that it did indeed spilt into two squares instead of four rectangles (like the Schrödinger equation predicted) until enough time has elapsed for the separation in one direction to start to effect the dynamics of the other, i.e. when enough time has passed to send a letter.
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• $\begingroup$ Hello. I would like to ask something here: If according to relativity, we can't tell who made a measurement first- because it's frame reference depended- we don't know who is affecting who. Is this your statement? And what do you mean by "you are not affecting the results"? If you are referring to the probabilistic nature of qm, my question would be: We don't cause a specific result, but we cause a result by measuring. Isn't there a change to our knowledge and the particles state because of measurement? Thank you. $\endgroup$ – Constantine Black Jun 24 '15 at 10:04
• $\begingroup$ @ConstantineBlack Answer edited $\endgroup$ – Timaeus Jun 25 '15 at 2:26
Simple. SR requires that you cannot pass information outside of your light cone. Measuring EPR doesn't let you pass information on in a speed faster than light.
If I were to measure, I'd know what measuring the other particle would result in, but I'd now have no way to transfer this information anywhere instantaneously.
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Let's say you have previously agreed with your friend doing the other measurement that if the combination of spins is up, down you go to the cinema and if it is down, up you go eat a pizza. Now you do the measurement and he does, both just before going out of the lab. You now have excluded two possible future states, i.e., not meeting because one went to the cinema and the other to the pizza. That happened without sending any "speed of light" message, and it is new information for both. Anyone care to comment?
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d4c9fd2c5c3e5850 | Engineering chiral and topological orbital magnetism of domain walls and skyrmions
Electrons that are slowly moving through chiral magnetic textures can effectively be described as if they were influenced by electromagnetic fields emerging from the real-space topology. This adiabatic viewpoint has been very successful in predicting physical properties of chiral magnets. Here, based on a rigorous quantum-mechanical approach, we unravel the emergence of chiral and topological orbital magnetism in one- and two-dimensional spin systems. We uncover that the quantized orbital magnetism in the adiabatic limit can be understood as a Landau-Peierls response to the emergent magnetic field. Our central result is that the spin–orbit interaction in interfacial skyrmions and domain walls can be used to tune the orbital magnetism over orders of magnitude by merging the real-space topology with the topology in reciprocal space. Our findings point out the route to experimental engineering of orbital properties of chiral spin systems, thereby paving the way to the field of chiral orbitronics.
The field of magnetism is witnessing a recent spark of interest in Berry phase and transport effects, which originate in non-collinear magnetism and spin chirality1,2,3. On the one hand, the recent outstanding observations made in this field is the generation of large current-induced Hall effects in strongly frustrated metallic antiferromagnets4 and the topological Hall effect (THE) in skyrmions2,5. On the other hand, the physics of the fundamental phenomenon of orbital magnetism has been experiencing a true revival, which can be attributed to the advent of Berry phase concepts in condensed matter6,7. The Berry phase origin of the orbital magnetization (OM) and its close relation to the Hall effect makes us believe that non-collinear spin systems can reveal a rich landscape of orbital magnetism relying on spin chirality rather than spin–orbit interaction (SOI)8,9,10,11. The corresponding phenomenon of topological orbital magnetization (TOM)8,9,10,11 is rooted in the same physical mechanism that drives the emergence of non-trivial transport properties such as the THE in chiral skyrmions or the anomalous Hall effect in chiral antiferromagnets12,13,14.
The promises of topological contribution to the orbital magnetization are seemingly very high, since it offers new prospects in influencing and detecting the chirality of the underlying spin texture by addressing the orbital degree of freedom, which is the central paradigm in the advancing field of orbitronics15. And while the emergence of topological orbital magnetism in several nm-scale chiral systems has been shown from first principles and tight-binding calculations8,9, our understanding of this novel phenomenon is basically absent. In particular, this concerns its conceptually clear definition as well as our ability to tailor the properties of this effect in complex interfacial chiral systems, which often exhibit strong spin–orbit interaction. These are the two central questions we address in this work.
As has been shown in the case of skyrmions, the variety of topological phenomena, which arise intrinsically from the non-trivial magnetization configuration \(\widehat {\bf{n}}(x,y)\) can be attributed to an “emergent” magnetic field \(B_{{\mathrm{eff}}}^z\)1. The occurrence of this field is connected to the gauge-invariant Berry phase the electron’s wavefunction acquires when traversing the texture16,17,18 (see Fig. 1 for an intuitive illustration). In the adiabatic limit, this phase can be attributed to the effect of \(B_{{\mathrm{eff}}}^z\), explicitly given by the expression
$$B_{{\mathrm{eff}}}^z = \pm \frac{\hbar }{{2e}}\widehat {\bf{n}} \cdot \left( {\frac{{\partial \widehat {\bf{n}}}}{{\partial x}} \times \frac{{\partial \widehat {\bf{n}}}}{{\partial y}}} \right),$$
where the sign depends on the spin of the electron. When integrated over an isolated skyrmion, the total flux of \(B_{{\mathrm{eff}}}^z\) is quantized to integer multiples of 2Φ0, where Φ0 ≈ 2 × 103 T nm2 is the magnetic flux quantum, while the integer prefactor can be identified with the topological charge of a skyrmion, Nsk, essentially counting the number of times the spin evolves around the unit sphere when traced along a path enclosing the skyrmion center.
Fig. 1
Schematic depiction of emergent magnetic fields. As electrons (gray spheres) are adiabatically traversing a a Néel spiral or b a Néel skyrmion (small arrows, with color indicating the z-projection), their wavefunction twists just in the same way as it would under the influence of an external magnetic field (the direction is depicted with vertical arrows, the sign and magnitude is illustrated by the colored background). The integrated flux of this emergent topological field over the skyrmion is quantized, while the averaged value of the emergent chiral field for a uniform spin-spiral is zero (although it can be non-zero for a 90° domain wall). The emergent field locally gives rise to persistent currents (depicted with circular arrows) and the corresponding a chiral (for a spiral) and b topological (for a skyrmion) orbital magnetization
Formally, the non-collinear system \(\widehat {\bf{n}}(x,y)\) can, therefore, be portrayed as a collinear one, albeit at the price of introducing the magnetic field \(B_{{\mathrm{eff}}}^z\) into the Schrödinger equation. Just as an ordinary magnetic field would, the emergent magnetic field in chiral systems couples directly to the orbital degree of freedom and provides an intuitive mechanism for the THE of skyrmions19 as well as a possible explanation for the emergence of TOM.
Here, we uncover the emergence of distinct contributions to the orbital magnetization in slowly-varying chiral textures by following the intuition that such contributions should acquire the natural form
$${\bf{M}}_{{\mathrm{orbital}}} \propto \chi _{{\mathrm{oms}}}{\kern 1pt} {\bf{B}}_{{\mathrm{eff}}},$$
where χoms is the orbital magnetic susceptibility of the electronic system20,21. Indeed, we demonstrate that in the limit of vanishing SOI the topological orbital magnetization can be expressed in this way. We also discover that in the limit of small, yet non-zero SOI there is a novel chiral contribution to the orbital magnetization described by (2) with the properly defined chiral emergent field, which can be finite already for one-dimensional systems (see Fig. 1a).
Moreover, by exploiting a rigorous semiclassical framework, we demonstrate that in interfacial chiral systems with finite SOI, the orbital magnetism can be tuned over orders of magnitude by varying the SOI strength within the range of experimentally observed values. We also underpin the crucial role that the topology of the local electronic structure of textures has in shaping the properties of orbital magnetism in chiral magnets. We discuss the bright avenues that our findings open, paving the way to the experimental observation of this phenomenon and to the exploitation of the orbital degree of freedom in chiral systems for the purposes of chiral orbitronics.
The semiclassical formalism we are referring to in our work is based on the Green’s function perturbation theory as presented by Onoda et al.22. We put the orbital magnetism of chiral systems on a firm quantum-mechanical ground, formulating a rigorous theory for the emergence of orbital magnetism in non-collinear systems. The motivation for this approach is twofold. First of all, the expression for Beff arises from the adiabatic limit19,23, a regime where semiclassical approaches have been successfully applied in order to investigate Berry phase physics7. Secondly, this certain type of gradient expansion24 provides a systematic guide through higher orders of perturbation theory where standard methods would be cumbersome.
It is based on an approximation to the single-particle Green’s function and allows us to trace the orders of perturbation theory for chiral magnetic textures, distinguishing corrections to the out-of-plane orbital magnetization25 \({\bf{M}}_{{\mathrm{om}}} = \hbar ^1M(\widehat {\bf{n}}){\bf{e}}_z\) of a locally ferromagnetic system, which appear as powers of the derivatives of the magnetization with respect to real-space coordinates:
$${\bf{M}}_{{\mathrm{com}}} = \hbar ^2M_i^\alpha (\widehat {\bf{n}})\left( {\partial _in_\alpha } \right){\bf{e}}_z$$
$${\bf{M}}_{{\mathrm{tom}}} = \hbar ^3M_{ij}^{\alpha \beta }(\widehat {\bf{n}})\left( {\partial _in_\alpha } \right)\left( {\partial _jn_\beta } \right){\bf{e}}_z,$$
where i = /∂xi. Here and in the following discussion, summation over repeated indices is implied with greek indices α, β {x, y, z} and latin indices i, j {x, y}.
The assignment of Mtom to the second order expansion, Eq. (4), is based on our intuitive expectation, Eq. (2). The question whether or not this term is “topological” will be discussed in the following and is answered by the semiclassical perturbation theory (see Methods). In addition to the effect of TOM we propose a novel contribution to the orbital magnetization, which is linear in the derivatives of the underlying texture, Eq. (3), and thereby generally sensitive to its chirality. We thus refer to it as the chiral orbital magnetization (COM). We will show how this effect can be attributed to a different kind of effective field (see Fig. 1a), which emerges from the interplay of spin–orbit coupling and non-collinearity along one spatial dimension.
While our approach is very general, for the purposes of including into consideration the effect of interfacial spin–orbit coupling and providing realistic numerical estimates, we focus our further analysis on the two-dimensional magnetic Rashba model
$$H = \frac{{{\bf{p}}^2}}{{2m_{{\mathrm{eff}}}^ \ast }} + \alpha _{\mathrm{R}}({\boldsymbol{\sigma }} \times {\bf{p}})_z + {\mathrm{\Delta }}_{{\mathrm{xc}}}{\kern 1pt} {\boldsymbol{\sigma }} \cdot \widehat {\bf{n}}({\bf{x}}),$$
where \(m_{{\mathrm{eff}}}^ \ast\) is the electron’s (effective) mass, σ denotes the vector of Pauli matrices, αR is the Rashba spin–orbit coupling constant, and Δxc is the strength of the local exchange field. This model has been proven to be extremely fruitful in unraveling various phenomena in surface magnetism26 and is known for its pronounced orbital response27.
Emergent fields of spin textures
Before discussing the emergence of orbital magnetism in this model, it is rewarding to discuss the appearance of effective fields in slowly-varying chiral spin textures in the limit of \(\left| {\alpha _{\mathrm{R}}} \right| \ll \left| {{\mathrm{\Delta }}_{{\mathrm{xc}}}} \right|\). In this regime, it can be shown that to linear order in αR, the spin–orbit coupling can be absorbed into a perturbative correction of the canonical momentum \({\bf{p}} \to {\bf{p}} + e{\cal A}^R\), with \({\cal A}^R \equiv m_{{\mathrm{eff}}}^ \ast \alpha _{\mathrm{R}}\epsilon ^{ijz}\sigma _i{\bf{e}}_j{\mathrm{/}}e\). This means that the Hamiltonian can be rewritten as:
$$H = \frac{{\left( {{\bf{p}} + e{\cal A}^R} \right)^2}}{{2m_{{\mathrm{eff}}}^ \ast }} + {\mathrm{\Delta }}_{{\mathrm{xc}}}\widehat {\bf{n}} \cdot {\boldsymbol{\sigma }} + {\cal O}\left( {\alpha _{\mathrm{R}}^2} \right).$$
For \(\left| {\alpha _{\mathrm{R}}} \right| \ll \left| {{\mathrm{\Delta }}_{{\mathrm{xc}}}} \right|\) (to be precise, with correct physical dimensions, one should compare the length scales \(\lambda _{\mathrm{R}} = \hbar {\mathrm{/}}\alpha _{\mathrm{R}}m_{{\mathrm{eff}}}^ \ast\) and \(\lambda _{{\mathrm{xc}}} = \hbar {\mathrm{/}}\sqrt {{\mathrm{\Delta }}_{{\mathrm{xc}}}m_{{\mathrm{eff}}}^ \ast }\) and write \(\lambda _{\mathrm{R}} \gg \lambda _{{\mathrm{xc}}}\) instead) and Δxc → ∞ the spin polarization of the wavefunctions is only weakly altered away from \(\widehat {\bf{n}}\) and we can use an SU(2) gauge field, defined by \({\cal U}^\dagger \left( {{\boldsymbol{\sigma }} \cdot \widehat {\bf{n}}} \right){\cal U} \equiv \sigma _z\), to rotate our Hamiltonian into the local axis specified by \(\widehat {\bf{n}}\) (neglecting the terms of the order \({\cal O}\left( {\alpha _{\mathrm{R}}^2} \right)\))28,29:
$$H \to {\cal U}^\dagger H{\cal U} = \frac{{\left( {{\bf{p}} + e{\cal A}({\bf{X}})} \right)^2}}{{2m_{{\mathrm{eff}}}^ \ast }} + {\mathrm{\Delta }}_{{\mathrm{xc}}}{\kern 1pt} \sigma _z,$$
where the potential \({\cal A}\) now comprises the mixing of two gauge fields: \({\cal A} = {\cal U}^\dagger {\cal A}^{\mathrm{R}}{\cal U} + {\cal A}^{{\mathrm{xc}}}\), with the additional contribution \({\cal A}^{{\mathrm{xc}}} = - i\hbar {\cal U}^\dagger \nabla {\cal U}{\mathrm{/}}e\). The essential idea is now the following: as Δxc → ∞, electrons are confined to the bands, which correspond either to spin-up states \(\left| \uparrow \right\rangle\) or spin-down states \(\left| \downarrow \right\rangle\) depending on sgn(Δxc). This means that we can effectively replace the vector potential by its adiabatic counterpart, i.e.,
$$\begin{array}{*{20}{l}} {{\cal A} \to {\cal A}_{{\mathrm{ad}}}} \hfill & \equiv \hfill & {{\mathrm{sgn}}\left( {{\mathrm{\Delta }}_{{\mathrm{xc}}}} \right)\left\langle \downarrow \right|{\cal A}\left| \downarrow \right\rangle } \hfill \\ {} \hfill & = \hfill &\hskip-2pt {{\cal A}_{{\mathrm{ad}}}^{\mathrm{R}} + {\cal A}_{{\mathrm{ad}}}^{{\mathrm{xc}}},} \hfill \end{array}$$
where \({\cal A}_{{\mathrm{ad}}}^{\mathrm{R}} = \left( {{\cal U}^\dagger {\cal A}^{\mathrm{R}}{\kern 1pt} {\cal U}} \right)_{{\mathrm{ad}}}\). Thus, the effective Hamiltonian for Δxc → ∞ contains the vector potential of a classical magnetic field, which couples only to the orbital degree, accompanying the “ferromagnetic” system23,30,31. It is given by the classical expression \({\bf{B}}_{{\mathrm{eff}}} = \nabla \times {\cal A}_{{\mathrm{ad}}} = {\bf{B}}_{{\mathrm{eff}}}^{\mathrm{R}} + {\bf{B}}_{{\mathrm{eff}}}^{{\mathrm{xc}}}\). By following this procedure, one finds the expressions
$$\left( {{\bf{B}}_{{\mathrm{eff}}}^{{\mathrm{xc}}}} \right)_z = - \frac{\hbar }{{2e}}{\mathrm{sgn}}\left( {{\mathrm{\Delta }}_{{\mathrm{xc}}}} \right)\widehat {\bf{n}} \cdot \left( {\frac{{\partial \widehat {\bf{n}}}}{{\partial x}} \times \frac{{\partial \widehat {\bf{n}}}}{{\partial y}}} \right)$$
$$\left( {{\bf{B}}_{{\mathrm{eff}}}^{\mathrm{R}}} \right)_z = - \frac{{m_{{\mathrm{eff}}}^ \ast \alpha _{\mathrm{R}}}}{e}{\mathrm{sgn}}\left( {{\mathrm{\Delta }}_{{\mathrm{xc}}}} \right){\mathrm{div}}{\kern 1pt} \widehat {\bf{n}}.$$
We thus arrive at the fundamental result that in addition to the field given by Eq. (9) above that can be recognized as the generalization of Eq. (1), there is a contribution to the overall field, which explicitly depends on the chirality of the underlying texture and is non-vanishing already for one-dimensional spin textures. In this context, it makes sense to refer to these co-existing fields as topological and chiral for \({\bf{B}}_{{\mathrm{eff}}}^{{\mathrm{xc}}}\) and \({\bf{B}}_{{\mathrm{eff}}}^{\mathrm{R}}\), respectively, see Fig. 1. Importantly, in contrast to the emergent topological field, \(\left( {B_{{\mathrm{eff}}}^{{\mathrm{xc}}}} \right)_z\), the local magnitude of \(\left( {B_{{\mathrm{eff}}}^{\mathrm{R}}} \right)_z\) is directly proportional to the strength of the spin–orbit interaction as given by αR. This appears to be very promising with respect to achieving a large magnitude of the chiral field in chiral spin textures emerging at surfaces and interfaces. To give a rough estimate, assuming a pitch of the texture on a length scale of L = 20 nm and ħαR = 1 eV Å the amplitude of the local chiral emergent field reaches as much as 2πmeαR/(eL) ≈ 270 T, which is roughly by an order of magnitude larger than the corresponding topological field in a skyrmion of a similar size1.
The emergence of two types of fields in spin textures, appearing in Eqs. (9) and (10), is crucial for a qualitative understanding of the emergence of topological and chiral orbital magnetism, which are discussed in detail below.
Chiral orbital magnetization
To get a first insight into the novel effect of COM, we consider the limit of small SOI, i.e., \(\alpha _{\mathrm{R}} \ll {\mathrm{\Delta }}_{{\mathrm{xc}}}\). In this case, the gradient expansion (see Methods) provides an analytic expression for the local space-dependent orbital moment. Up to \({\cal O}\left( {\alpha _{\mathrm{R}}} \right)\) it is given by
$$M_{{\mathrm{com}}} = - \frac{1}{2}\chi _{{\mathrm{LP}}}^{ \uparrow + \downarrow }\left( {{\bf{B}}_{{\mathrm{eff}}}^{\mathrm{R}}} \right)_zh\left( {\mu {\mathrm{/\Delta }}_{{\mathrm{xc}}}} \right),$$
where the function h(x) ≡ (3x2 − 1)Θ(1 − |x|)/2 describes the energy dependence of COM with Θ representing the Heaviside step-function. The magnitude of COM is thus directly proportional to the strength of spin–orbit interaction and vanishes in the limit of zero αR. Furthermore, Mcom is proportional to the diamagnetic Landau-Peierls susceptibility32 \(\chi _{{\mathrm{LP}}}^{ \uparrow + \downarrow } = - e^2{\mathrm{/}}\left( {12\pi m_{{\mathrm{eff}}}^ \ast } \right)\), which characterizes the orbital response of a free-electron gas. Indeed, this seems reasonable in the limit of \(\alpha _{\mathrm{R}} \ll {\mathrm{\Delta }}_{{\mathrm{xc}}}\) with the chemical potential positioned in the majority band, as the true orbital magnetic susceptibility of the Rashba model (as calculated by Fukuyama’s formula20,21) reduces to \(\chi _{{\mathrm{oms}}}\sim \chi _{{\mathrm{LP}}}^{ \uparrow + \downarrow }{\mathrm{/}}2\) in the same limit. For |μ| ≈ |Δxc| we, therefore, arrive at at the intuitive result guided by Eq. (2) with Beff replaced by the chiral emergent field:
This reflects the fact that in the limit of \(\left| {\alpha _{\mathrm{R}}} \right| \ll \left| {{\mathrm{\Delta }}_{{\mathrm{xc}}}} \right|\) the emergence of chiral orbital magnetization can be understood as the coupling of a mixed SU(2) gauge field to the diamagnetic Landau-Peierls susceptibility.
The behavior of COM becomes complicated and deviates remarkably from the αR-linear expression given by Eq. (12) as the Rashba parameter increases. To demonstrate this, we numerically calculate the value of Mcom at the center of a skyrmion, in a wide range of parameters Δxc and αR of the Rashba Hamiltonian, Eq. (5), while fixing the chemical potential at μ = 0. We parameterize the skyrmion in the polar coordinates (ρ, ϕ) by choosing \(\widehat {\bf{n}}(\rho ,\phi )\) = (sin θ(ρ) cos Φ(ϕ), sin θ(ρ) sin Φ(ϕ), cosθ(ρ))T as the local magnetization vector1. Here, we define Φ(ϕ) = + γ with the vorticity m and the helicity γ. For a Néel skyrmion γ = 0, whereas a Bloch skyrmion is represented by the value γ = π/2. The topological charge of the skyrmion then equals Nsk = \({\int} {\kern 1pt} {\mathrm{d}}x{\mathrm{d}}y{\kern 1pt} \widehat {\bf{n}} \cdot \left( {\partial _x\widehat {\bf{n}}\, \times \partial _y\widehat {\bf{n}}} \right){\mathrm{/}}(4\pi )\) = −m. In order to model the radial dependency refer to Romming et al.33 and choose a 360° domain wall, which is described by two parameters: the domain wall width w and the core size c (see Methods).
The results are presented in Fig. 2 for a Néel skyrmion (γ = 0) with w = 20 nm, c = 0 nm and m = 1. The magnetization is given in units of \(\mu _{\mathrm{B}}^ \ast {\mathrm{/nm}}^2\) with the effective Bohr magneton \(\mu _{\mathrm{B}}^ \ast = e\hbar {\mathrm{/}}\left( {2m_{{\mathrm{eff}}}^ \ast } \right)\). In this plot, we observe that while the gauge field picture is valid in the limit of Δxc/αR → ∞, there exists a pronounced region in the (αR, Δxc)-phase-space where COM exhibits a strong non-linear enhancement. This is in contrast to the case of Bloch skyrmions, where COM vanishes identically for all (αR, Δxc), reflecting the symmetry of the Rashba coupling. It also elucidates our terminology, since already the gauge field description can be used to verify that Mcom cos γ (for vorticity m = 1), thereby making COM explicitly dependent on the helicity.
Fig. 2
The phase diagram of chiral and topological orbital magnetization. The magnitude of a Mcom and b Mtom, Eqs. (3) and (4) respectively, is evaluated at the core of a Néel (Bloch) skyrmion (m = 1, c = 0 nm, w = 20 nm) as a function of the parameters Δxc and αR of the Rashba Hamiltonian Eq. (5) with μ = 0. The limit \({\mathrm{\Delta }}_{{\mathrm{xc}}} \gg \alpha _{\mathrm{R}}\), Δxc 0.51 eV corresponds to the coupling of the emergent magnetic field to the diamagnetic Landau-Peierls susceptibilty (what we refer to as “LP limit”). In an intermediate regime of ΔxcαR orbital magnetism is strongly enhanced
Topological orbital magnetization
The TOM appears as the correction to the OM, which is second order in the gradients of the texture, Eq. (4), and while it vanishes for one-dimensional spin textures, we show that it is finite for 2D textures such as magnetic skyrmions. In contrast to COM, the TOM is non-vanishing even without spin–orbit interaction. To investigate this, we set αR to zero, reducing the effective vector potential to \({\cal A} = {\cal A}^{{\mathrm{xc}}}\) and with the emergent field turning into \(\left( {{\bf{B}}_{{\mathrm{eff}}}^{{\mathrm{xc}}}} \right)_z\), Eq. (9). The gradient expansion (see Methods) now indeed reveals that
$$M_{{\mathrm{tom}}} = - \frac{1}{2}\chi _{{\mathrm{LP}}}^{ \uparrow + \downarrow }\left( {{\bf{B}}_{{\mathrm{eff}}}^{{\mathrm{xc}}}} \right)_z\left( {\mu {\mathrm{/\Delta }}_{{\mathrm{xc}}}} \right),$$
which again confirms the gauge-theoretical expectation. Remarkably, the similarity between Eqs. (12) and (13) underlines the common origin of the COM and TOM in the “effective” magnetic field in the system, generated by a combination of a gradient of \(\widehat {\bf{n}}\) along x with spin–orbit interaction (in case of COM), and by a combination of the gradients of \(\widehat {\bf{n}}\) along x and y (in case of TOM).
To explore the behavior of TOM in the presence of spin–orbit interaction, αR ≠ 0, we numerically compute the value of TOM at the center of the Néel (Bloch) skyrmion with parameters used in the previous section, as function of Δxc and αR (at μ = 0). The corresponding phase diagram, presented in Fig. 2, displays two notable features. The first one is the relative stability of Eq. (13) against a perturbation by a spin–orbit field in the limit of \(\left| {{\mathrm{\Delta }}_{{\mathrm{xc}}}} \right| \gg \left| {\alpha _{\mathrm{R}}} \right|\). The second one is the significant enhancement of TOM in the regime where |αR| > |Δxc|, similar to COM (albeit over a larger part of the parameter space).
Interplay of topologies
The phase diagrams in Fig. 2 have been evaluated at the core of a skyrmion. We now take a more global perspective and analyze the decomposition of the overall orbital magnetization into its constituent parts Mcom and Mtom as a function of ρ, the radial position inside the skyrmion with w = 20 nm and c = 0 (see Fig. 3a)). By fixing ħαR to 2 eV Å, and Δxc to 0.9 eV with μ = 0, we position ourselves precisely in the region of orbital enhancement discussed above in the context of the phase diagrams. When the local direction of the magnetization (parametrized by spherical coordinates θ and ϕ) is close to the z-axis, Mcom and Mtom are rather gently varying, whereas their behavior reveals strong resonance in the vicinity of in-plane directions (θ ≈ π/2). The emergence of this resonance coincides with an occurrence of a band crossing at the critical k-value of ħkc = |Δxc/αR| with the polar coordinate in the Brillouin zone of ϕk = ϕ − π/2 in the local ferromagnetic electronic structure, which corresponds to the given magnetization direction, see Fig. 3b, c.
Fig. 3
An interplay of topologies in the orbital magnetism of skyrmions. a Following the radial direction ρ in a Néel skyrmion (m = 1, c = 0 nm, w = 20 nm, Nsk = −1) one finds a strong resonance in the local magnitude of Mcom and Mtom (evaluated for ħαR = 2 eV Å and Δxc = 0.9 eV) in the vicinity of the in-plane direction of \(\widehat {\bf{n}}\). b This resonance coincides with a critical point in the mixed space that is spanned by the momentum space coordinates k and the polar angle, which \(\widehat {\bf{n}}\) encloses with the z-axis. Here, the two Rashba bands cross, which is further illustrated in figure c, showing the energy levels as a function of ρ, where k is held fixed at its critical value kc. The nature of this crossing—which is a necessary consequence of the topology in real and momentum space—is further studied in the figures d, e which depict Mtom as a function of the chemical potential μ across the bandstructure for two different positions in the skyrmion (indicated by the red arrow in the gray coordinate spheres). The symbolic arrows on the right mark the values of μ, which are used to evaluate the real-space distributions of Mcom + Mtom shown in fh. These exemplify the complex real-space landscape and intricate energy dependence of orbital magnetism in spin–orbit coupled interfacial skyrmions as a consequence of the interplay between real- and reciprocal space topology
It is known that this specific band crossing in the Rashba model leads to a vastly enhanced diamagnetic susceptibility27 and in close analogy, a strong response in Mcom and Mtom can be expected based on Eq. (2). To study the origin of this effect in greater detail, we plot Mtom as a function of μ for two different magnetization directions. The results, presented in Fig. 3d, e reveal the sensitivity of Mtom to the SOI-mediated deformation of the purely parabolic free-electron bands separated by Δxc. The magnitude of TOM is largest and exhibits pronounced oscillations in a narrow energy interval around the band edges. When we turn \(\widehat {\bf{n}}\) into the in-plane direction, it can be seen how the resonances of Mtom are enhanced in magnitude and are carried along by those band extrema, which eventually touch at θ = π/2, pushing the peaks of Mtom through the chemical potential (which was aligned to μ = 0 for Fig. 3b). For three different values of the chemical potential (indicated by the symbolic arrows) the strongly μ-dependent real-space density of the total orbital magnetization M is shown in Fig. 3f–h. This anisotropic behavior cannot be accounted for within the emergent magnetic field picture, which only relies on the real-space texture with its associated topological charge and winding density.
The “critical” metallic point in the Rashba model that we encounter is topologically non-trivial. Indeed, the upper and lower bands of the magnetic Rashba model bare non-zero Chern numbers, \({\cal C}_1\) = ±sgn(Δxc)1/2, with the sign depending on the band34. The Chern number is a topological invariant of energy bands in k-space and can only change when bands are crossing. Since the sign of \({\cal C}_1\) changes under the transformation Δxc → −Δxc, the emergence of the critical metallic point at θ = π/2 is enforced when the direction of the magnetization is changed from θ = 0 to θ = π. This is illustrated in Fig. 3c. In the context of topological metals, such a point is known as a mixed Weyl point35, owing to the quantized flux of the Berry curvature permeating through the mixed space of k and θ (as confirmed explicitly by the calculations for the magnetic Rashba model). These points have recently been shown to give rise to an enhancement of spin–orbit torques and Dzyaloshinskii-Moriya interaction in ferromagnets35. Here, we demonstrate the crucial role that such topological features in the electronic structure could play for the pronounced chirality-driven orbital magnetism of spin textures. Given the observation that TOM simply follows the evolution of the electronic structure in real-space via the direction of the local magnetization (as illustrated in a schematic way in Fig. 3d, e), the close correlation of real- and reciprocal space topologies offer promising design opportunities in skyrmions or domain walls of transition-metals with complex anisotropic electronic structure.
Topological quantization and stability
One of the key properties of Mtom is its origin in the local real-space geometry of the texture. This has drastic consequences for the topological properties of the overall orbital moment of two-dimensional topologically non-trivial spin textures as we reveal below for the case of chiral magnetic skyrmions. We thus turn to the discussion of the total integrated values of the orbital moments in chiral spin textures by defining them as
$$m_{{\mathrm{com/tom}}} = {\int} {\kern 1pt} {\mathrm{d}}{\bf{x}}{\kern 1pt} M_{{\mathrm{com/tom}}}({\bf{x}}).$$
Concerning the total value of the COM-driven orbital moment in one-dimensional uniform 360° or 180° chiral domain walls it always vanishes identically by arguments of symmetry (although it can be finite for example in a 90° wall). In sharp contrast, the TOM-driven total orbital moment of isolated skyrmions generally does not vanish. This can be most clearly shown in the limit when the gauge field approach is valid (i.e., \({\mathrm{\Delta }}_{{\mathrm{xc}}} \gg \alpha _{\mathrm{R}}\)). In this case, the total flux of the emergent topological and chiral fields through an isolated skyrmion is given by
$${\mathrm{\Phi }}^{{\mathrm{xc}}} \equiv {\int}_{{\Bbb R}^2} {\kern 1pt} {\mathrm{d}}{\bf{x}}\left( {{\bf{B}}_{{\mathrm{eff}}}^{{\mathrm{xc}}}} \right)_z = 2{\mathrm{\Phi }}_0N_{{\mathrm{sk}}}$$
$${\mathrm{\Phi }}^{\mathrm{R}} \equiv {\int}_{{\Bbb R}^2} {\kern 1pt} {\mathrm{d}}{\bf{x}}\left( {{\bf{B}}_{{\mathrm{eff}}}^{\mathrm{R}}} \right)_z = 0.$$
It then follows from Eq. (12), that the integrated value of Mcom vanishes while Eq. (13) predicts the quantization of the topological orbital moment mtom to integer multiples of \(\chi _{{\mathrm{LP}}}^{ \uparrow + \downarrow }{\mathrm{\Phi }}_0 = - \mu _{\mathrm{B}}^ \ast {\mathrm{/}}6\) (at |μ| = |Δxc|). In this limit, the skyrmion of non-zero topological charge Nsk ≠ 0 thus behaves as an ensemble of Nsk effective particles which occupy a macroscopic atomic orbital with an associated value of the orbital angular momentum of \(\mu _{\mathrm{B}}^ \ast {\mathrm{/}}6\). This quantization is explicitly confirmed in Fig. 4a), where we present the calculations of mtom for Néel and Bloch-type skyrmions with dimensions c = 0 nm and w = 20 nm at a fixed value of Δxc = 0.9 eV while varying αR for different topological charges Nsk {−1, −2, −3, −4}. The results, presented in units of \(m_0 = - \mu _{\mathrm{B}}^ \ast {\mathrm{/}}12\) (corresponding to the Landau-Peierls limit at μ = 0 and Nsk = −1), reveal a stable plateau, corresponding to the regime of topological quantization, where mtom attains the value Nsk m0.
Fig. 4
The breakdown of topological quantization. a When \(\alpha _{\mathrm{R}} \ll {\mathrm{\Delta }}_{{\mathrm{xc}}}\), the integrated mtom is a topological quantity, which cannot distinguish between topologically equivalent structures such as Néel (black triangles) and Bloch skyrmions (black stars). Tracing the topological charges Nsk = −1, −2, −3, −4 as a function of αR and with Δxc = 0.9 eV, μ = 0, this figure illustrates how mtom (in units of \(m_0 = - \mu _{\mathrm{B}}^ \ast {\mathrm{/}}12\)) passes from its regime of topological quantization (mtom/m0 = −Nsk) to a regime of strong enhancement, with Néel and Bloch structures clearly distinguishable. Intermediate phases form a continuum between the two values (shaded regions). The inset demonstrates for the case of Néel skyrmions, that a level structure is still present at αR = 2 eV. b For the particular case of αR = 2.0 and Δxc = 0.9 eV, μ = 0 the γ phase shift is used to interpolate from a Néel to a Bloch-type Skyrmion of charge Nsk = −1 (equivalent to vorticity m = 1), leading to a drastic loss of mtom. Variations in the shape of the skyrmion (shown in the inset), as quantified by the ratio of c/w, have a very small effect
In the opposite limit of αR > Δxc the magnitude of Mtom can be enhanced drastically with respect to the topologically quantized value. When ħαR reaches a magnitude of about 1 eV Å, the emergent field picture breaks down and we discover a drastic increase in Néel–mtom by as much as one order of magnitude upon increasing αR. And although mtom is not topologically quantized in this regime, it still attains a distinctly different magnitude for different skyrmion charges, and while it is weakly dependent on the c/w ratio (up to a couple of percent), when c = const the variations of w keep mtom strictly constant (see the insets in Fig. 4 with c = 0). The latter robustness can be demonstrated already analytically on the level of Eq. (4) using the transformation of coordinates x → x/w. Remarkably, in the regime of enhanced SOI, the strong dependence of the local TOM on the helicity of the skyrmion (i.e., Néel or Bloch), uncovered in Fig. 2, is translated into a drastic dependence of the overall topological orbital moment on the way that the magnetization rotates from the core towards the outside region, as shown in Fig. 4. Such behavior of the topological orbital moment with respect to deformations of the underlying texture suggests that monitoring the dynamics of the orbital magnetization in skyrmionic systems can be used not only to detect the formation of skyrmions with different charge, but also to distinguish various types of dynamical “breathing” modes of skyrmion dynamics36.
On a fundamental level, COM and TOM arise as a consequence of the changes in the local electronic structure caused by a non-collinear magnetization texture. Since the effective magnetic fields directly couple to the orbital degree of freedom, they lead to the emergence of chiral and topological orbital magnetization. While this intuitive interpretation in terms of real-space gauge fields eventually breaks down at large SOI, it makes room for a regime of strong enhancement in which the intertwined topologies of real- and reciprocal space lead to novel design aspects in the bandstructure engineering of orbital physics. This is possible by exploiting either the spin–orbit interaction or the dispersive behavior of the bands, i.e., their effective mass. In particular, the metallic point in the mixed parameter space of the non-collinear Rashba model reveals its strong impact on COM and TOM. Such critical points will have a pronounced effect on the orbital magnetism even if they are emerging on the background of metallic bands in transition-metal systems. Therefore, our analysis indicates in which materials an experimental detection of orbital magnetism that is originating from non-collinearity is the most feasible. By numerically evaluating the magnitude and real-space behavior of the TOM and COM, we thereby uncover that by tuning the parameters of surface and interfacial systems the orbital magnetism of domain walls and chiral skyrmions can be engineered in a desired way.
Concerning experimental observation of the effects discussed here, Mtom and Mcom could be accessible by techniques such as off-axis electron holography37 (sensitive to local distribution of magnetic moments), or scanning tunneling spectroscopy (sensitive to the local electronic structure) in terms of B-field induced changes in the dI/dU or d2I/dU2 spectra38. An already existing proposal for the detection of non-collinearity driven orbital magnetization of skyrmions by Dias et al.9 relies on X-ray magnetic circular dichroism (XMCD), which is able to distinguish orbital contributions to the magnetization from the spin contributions9.
Further, the emergence of COM and TOM can give a thrust to the field of electron vortex beam microscopy39—where a beam of incident electrons intrinsically carries orbital angular momentum interacting with the magnetic system—into the realm of chiral magnetic systems. For example, we speculate that at sufficient intensities, electron vortex beams could imprint skyrmionic textures possibly by partially transforming its orbital angular momentum into TOM. Since the topological orbital moment is directly proportional to the topological charge of the skyrmions, we also suggest that the interaction of TOM with external magnetic fields could be used to trigger the formation of skyrmions with large topological charge. Ultimately, the currents of skyrmions can be employed for low-dissipation transport of the associated topological orbital momenta over large distances in skyrmionic devices.
While in this work we focus primarily on TOM, the here discovered chiral orbital magnetization has been an overlooked quantity in chiral magnetism so far. Besides the fact that it emerges already in one-dimensional chiral systems and serves as a playground to study the effects of mixed space Berry phases, it can reach very large values depending on the details of the texture as well as strength of SOI. Even in case of skyrmions, where the argument of vanishing effective flux, Eq. (16), might suggests that COM is not of importance, it turns out that beyond the αR → 0 limit the integral effect of Mcom can be substantially enhanced in a way similar to TOM. A prominent example for the importance of COM is given e.g., in Vanadium-doped BiTeI40, which has a large SOC of ħαR = 3.8 eV Å and an exchange gap of Δxc = 45 meV. If this material would host Néel skyrmions (m = 1, w = 20 nm, c = 0 nm), mcom would reach approximately \(12\mu _{\mathrm{B}}^ \ast\), which has magnitudes larger than the corresponding mtom of about \(- 0.7\mu _{\mathrm{B}}^ \ast\). Creating skyrmions and large COM in strong Rashba systems might, therefore, be a promising direction to pursue.
In a wider perspective, the emergence of TOM and COM gives rise to a physical object that is directly connected to the orbital degree of freedom with the advantage that it can be understood from a semiclassical perspective in a way, which is engineerable and controllable. Our findings thus open new vistas for exploiting the orbital magnetism in chiral magnetic systems, thereby opening interesting prospects for the field of “chiral” spintronics and orbitronics.
Gradient expansion
The expansion in exchange field gradients is naturally achieved within the phase-space formulation of quantum mechanics, the Wigner representation17,22. The key quantity in this approach is the retarded single-particle Green’s function GR, implicitly given by the Hamiltonian H via the Dyson equation
$$\left( {\epsilon - H(x,{\boldsymbol{\pi }}) + i0^ + } \right) \star G^{\mathrm{R}}(x,\pi ) = {\mathrm{id}},$$
where xμ = (t, x) and πμ = (\(\epsilon\), π) are the four-vectors of position and canonical momentum, respectively. The latter of the two, in terms of the elementary charge e > 0 and the electromagnetic vector potential A, is related to the zero-field momentum p by the relation πμ(x, p) = pμ + eAμ(x). The -product, formally defined by the operator
$$\star \equiv {\mathrm{exp}}\left\{ {\frac{{i\hbar }}{2}\left( {{\mathop {\partial }\limits^\leftarrow}_{x^\mu }{\mathop {\phantom{.}\partial\phantom{.} }\limits^\rightarrow}_{\pi _\mu }-{\mathop {\partial }\limits^\leftarrow}_{\pi _\mu }{\mathop {\phantom{.}\partial\phantom{.} }\limits^\rightarrow}_{x^\mu }- eF^{\mu \nu }{\mathop {\partial }\limits^\leftarrow}_{\pi ^\mu }{\mathop {\phantom{.}\partial\phantom{.} }\limits^\rightarrow}_{\pi ^\nu }}\right)} \right\}$$
of left- and right-acting derivatives \(\mathop {\partial }\limits^ \leftrightarrow\), allows for an expansion of GR in powers of ħ, gradients of \(\widehat {\bf{n}}\) and external electromagnetic fields, captured in a covariant way by the field tensor \(F^{\mu \nu } = \partial _{x_\mu }A^\nu - \partial _{x_\nu }A^\mu\)22.
In this work, we are after the orbital magnetization (OM) in z-direction. Given the grand canonical potential Ω, the surface density of the orbital moment is given by25,41
$$M(x) = - \partial _B\left\langle {{\mathrm{\Omega }}(x)} \right\rangle ,$$
which requires an expansion of Ω up to at least first order in the magnetic field B = Bez in the collinear case. In the limit of T → 0, the grand potential is asymptotically related to the Green’s function GR via
$$\left\langle {\mathrm{\Omega }} \right\rangle \sim - \frac{1}{\pi }\Im {\int} \frac{{{\mathrm{d}}p}}{{(2\pi \hbar )^2}}f(\epsilon )(\epsilon - \mu ){\mathrm{tr}}{\kern 1pt} G^{\mathrm{R}}(x,p),$$
where denotes the imaginary part, the integral measure is defined as dp = d\(\epsilon\) d2p, f(\(\epsilon\)) represents the Fermi function f(\(\epsilon\)) = (eβ(\(\epsilon\)μ) + 1)−1, μ is the chemical potential and β−1 = kBT. In our approach, deviations from the collinear theory enter the formalism as gradients of \(\widehat {\bf{n}}\) and can be traced systematically in GR and in Ω, finally leading to Eqs. (3) and (4).
Computational details
All calculations were performed with a Green’s function broadening i0+ → iΓ with Γ = 100 meV while we approach the zero-temperature limit by setting kBT = 10 meV. The k-space integrals are then performed on a quadratic 512 × 512 mesh. The effective electron mass was set to \(m_{{\mathrm{eff}}}^ \ast {\mathrm{/}}m_{\mathrm{e}} = 3.81\) everywhere except for the example of V-doped BiTeI, where \(m_{{\mathrm{eff}}}^ \ast {\mathrm{/}}m_{\mathrm{e}} = 0.1\)42.
Skyrmion parametrization
In order to model the skyrmions in this work we choose the profile, which can be described by the parametrization1
$$\widehat {\bf{n}}(\rho ,\phi ) = \left( {\begin{array}{*{20}{c}} {{\mathrm{sin}}(\theta (\rho )){\mathrm{cos}}(\Phi (\phi ))} \\ {{\mathrm{sin}}(\theta (\rho )){\mathrm{sin}}(\Phi (\phi ))} \\ {{\mathrm{cos}}(\theta (\rho ))} \end{array}} \right).$$
The topological charge is then given by
$$\begin{array}{*{20}{l}} {N_{{\mathrm{sk}}}} \hfill & = \hfill & {\frac{1}{{4\pi }}{\int} {\kern 1pt} {\mathrm{d}}x{\int} {\kern 1pt} {\mathrm{d}}y{\kern 1pt} \widehat {\bf{n}} \cdot \left( {\partial _x\widehat {\bf{n}} \times \partial _y\widehat {\bf{n}}} \right)} \hfill \\ {} \hfill & = \hfill & { - \frac{1}{{4\pi }}{\mathrm{\Phi }}\left. {(\phi )} \right|_0^{2\pi }{\mathrm{cos}}\theta \left. {(\rho )} \right|_0^\infty .} \hfill \end{array}$$
Assuming Φ(ϕ) = + γ, with the vorticity \(m \in {\Bbb Z}\) and the helicity \(\gamma \in {\Bbb R}\) (Néel skyrmions correspond to γ = 0, Bloch skyrmions to γ=π/2), as well as the property θ(0) =π and θ(∞) = 0, the integral evaluates to Nsk = −m. A realistic profile satisfying these requirements and which is used in this work is given by33
$$\theta (\rho ) = \mathop {\sum}\limits_ \pm {\kern 1pt} {\mathrm{arcsin}}\left( {{\mathrm{tanh}}\left( { - \frac{{ - \rho \pm c}}{{w{\mathrm{/}}2}}} \right)} \right) + \pi {\mathrm{,}}$$
with the core size c and the domain wall width w.
Data availability
The code and the data that support the findings of this work are available from the corresponding authors on request.
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We thank J.-P. Hanke, M.d.S. Dias and S. Lounis for fruitful discussions, and gratefully acknowledge computing time on the supercomputers JUQUEEN and JURECA at Jülich Supercomputing Center, and at the JARA-HPC cluster of RWTH Aachen. We acknowledge funding under MO 1731/5-1 of Deutsche Forschungsgemeinschaft (DFG) and the European Union’s Horizon 2020 research and innovation program under grant agreement number 665095 (FET-Open project MAGicSky). This work has been also supported by the DFG through the Collaborative Research Center SFB 1238 and the priority program SPP 2137.
Author information
F.R.L. uncovered the emergence of chiral and topological orbital magnetization from the semiclassical expansion. F.R.L. and Y.M. wrote the manuscript. F.R.L., F.F, S.B, and Y.M. discussed the results and reviewed the manuscript.
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Correspondence to Fabian R. Lux.
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Lux, F.R., Freimuth, F., Blügel, S. et al. Engineering chiral and topological orbital magnetism of domain walls and skyrmions. Commun Phys 1, 60 (2018).
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acd82671415b1efe | The 17 Equations That Changed The Course Of Humanity
A great example of the human impact of maths is the financial crisis. Black Scholes, number 17 on this list, is a derivative pricing equation that played a role.
“It’s actually a fairly simple equation, mathematically speaking,” Professor Stewart told Business Insider. “What caused trouble was the complexity of the system the mathematics was intended to model.”
Without the equations on this list, we wouldn’t have GPS, computers, passenger jets, or countless inventions in between.
You can find the book here.
The Pythagorean Theorem
What does it mean: The square of the hypotenuse of a right triangle is equal to the SUM of the squares of its legs.
History: Attributed to Pythagoras, it isn't certain that he first proved it. The first clear proof came from Euclid, and it is possible the concept was known 1000 years before Pythoragas by the Babylonians.
Importance: The equation is at the core of geometry, links it with algebra, and is the foundation of trigonometry. Without it, accurate surveying, mapmaking, and navigation would be impossible.
Source: In Pursuit of the Unknown: 17 Equations That Changed the World
The logarithm and its identities
What does it mean: You can multiply numbers by adding related numbers.
Modern use: Logarithms still inform our understanding of radioactive decay.
The fundamental theorem of calculus
What does it mean?: Allows the calculation of an instantaneous rate of change.
History: Calculus as we currently know it was described around the same in the late 17th century by Isaac Newton and Gottfried Leibniz. There was a lengthy debate over plagiarism and priority which may never be resolved. We use the leaps of logic and parts of the notation of both men today.
Importance: According to Stewart, 'More than any other mathematical technique, it has created the modern world.' Calculus is essential in our understanding of how to measure solids, curves, and areas. It is the foundation of many natural laws, and the source of differential equations.
Modern use: Any mathematical problem where an optimal solution is required. Essential to medicine, economics, and computer science.
Newton's universal law of gravitation
What does it mean?: Calculates the force of gravity between two objects.
Importance: Used techniques of calculus to describe how the world works. Even though it was later supplanted by Einstein's theory of relativity, it is still essential for practical description of how objects interact with each other. We use it to this day to design orbits for satellites and probes.
Value: When we launch space missions, the equation is used to find optimal gravitational 'tubes' or pathways so they can be as energy efficient as possible. Also makes satellite TV possible.
The origin of complex numbers
What does it mean?: The square of an imaginary number is negative.
History: Imaginary numbers were originally posited by famed gambler/mathematician Girolamo Cardano, then expanded by Rafael Bombelli and John Wallis. They still existed as a peculiar, but essential problem in maths until William Hamilton described this definition.
Importance: According to Stewart '.... most modern technology, from electric lighting to digital cameras could not have been invented without them.' Imaginary numbers allow for complex analysis, which allows engineers to solve practical problems working in the plane.
Modern use: Used broadly in electrical engineering and complex mathematic theory.
Euler's formula for polyhedra
What does it mean?: Describes a space's shape or structure regardless of alignment.
History: The relationship was first described by Descartes, then refined, proved, and published by Leonhard Euler in 1750.
Importance: Fundamental to the development of topography, which extends geometry to any continuous surface. An essential tool for engineers and biologists.
Modern use: Topography is used to understand the behaviour and function of DNA.
The normal distribution
What does it mean?: Defines the standard normal distribution, a bell shaped curve in which the probability of observing a point is greatest near the average, and declines rapidly as one moves away.
Modern use: Used to determine whether drugs are sufficiently effective relative to negative side effects in clinical trials.
The wave equation
What does it mean?: A differential equation that describes the behaviour of waves, originally the behaviour of a vibrating violin string.
Importance: The behaviour of waves generalizes to the way sound works, how earthquakes happen, and the behaviour of the ocean.
The Fourier transform
What does it mean?: Describes patterns in time as a function of frequency.
History: Joseph Fourier discovered the equation, which extended from his famous heat flow equation, and the previously described wave equation.
Importance: The equation allows for complex patterns to be broken up, cleaned up, and analysed. This is essential in many types of signal analysis.
The Navier-Stokes equations
What does it mean?: The left side is the acceleration of a small amount of fluid, the right indicates the forces that act upon it.
History: Leonhard Euler made the first attempt at modelling fluid movement, French engineer Claude-Louis Navier and Irish mathematician George Stokes made the leap to the model still used today
Importance: Once computers became powerful enough to solve this equation, it opened up a complex and very useful field of physics. It is particularly useful in making vehicles more aerodynamic.
Maxwell's equations
What does it mean?: Maps out the relationship between electric and magnetic fields.
History: Michael Faraday did pioneering work on the connection between electricity and magnetism, James Clerk Maxwell translated it into equations, fundamentally altering physics.
Importance: Helped predict and aid the understanding of electromagnetic waves, helping to create many technologies we use today.
Modern use: Radar, television, and modern communications.
Second law of thermodynamics
What does it mean?: Energy and heat dissipate over time.
Importance: Essential to our understanding of energy and the universe via the concept of entropy. It helps us realise the limits on extracting work from heat, and helped lead to a better steam engine.
Modern use: Helped prove that matter is made of atoms, which has been somewhat useful.
Einstein's theory of relativity
What does it mean?: Energy equals mass times the speed of light squared.
History: The less known (among non-physicists) genesis of Einstein's equation was an experiment by Albert Michelson and Edward Morley that proved light did not move in a Newtonian manner in comparison to changing frames of reference. Einstein followed up on this insight with his famous papers on special relativity (1905) and general relativity (1915).
The Schrödinger equation
What does it mean?: Models matter as a wave, rather than a particle.
Modern use: Essential to the use of the semiconductor and transistor, and thus, most modern computer technology.
Shannon's information entropy
What does it mean?: Estimates the amount of data in a piece of code by the probabilities of its component symbols.
Importance: According to Stewart, 'It is the equation that ushered in the information age.' By stopping engineers from seeking codes that were too efficient, it established the boundaries that made everything from CDs to digital communication possible.
Modern use: Pretty much anything that involves error detection in coding. Anybody use the internet lately?
The logistic model for population growth
What does it mean?: Estimates the change in a population of creatures across generations with limited resources.
History: Robert May was the first to point out that this model of population growth could produce chaos in 1975. Important work by mathematicians Vladimir Arnold and Stephen Smale helped with the realisation that chaos is a consequence of differential equations.
Modern use: Used to model earthquakes and forecast the weather.
The Black–Scholes model
What does it mean?: Prices a derivative based on the assumption that it is riskless and that there is no arbitrage opportunity when it is priced correctly.
Bonus: Hodgkin-Huxley equations
From an email interview with Dr. Stewart:
'At one stage I planned to include the Hodgkin-Huxley equations, which gave mathematical biology a huge boost by using equations to model the way nerve cells send signals to each other. It formed the basis of theoretical neuroscience, and is still important. But it made the book too long, and in the end I felt that its impact on human history has not yet been quite great enough. However, that is likely to change by the middle of this century, as mathematical methods become a major part of mainstream biology -- which I think they will.'
From an email interview with Dr. Stewart:
'My current candidate for and 18th equation (number 1 in 'Seventeen MORE Equations That Changed the World' -- I joke... I think...) is the basic equation behind Google. This describes how to rate the importance of a website in terms of the links to it, and it's a clever application of basic undergraduate linear algebra. It deserved to be in the book, but I was running out of space -- and worried that my readers' enthusiasm for yet another equation might be drying up.'
Bonus: The Kalman Filter
From an email interview with Dr. Stewart:
'Many people feel that using mathematics (and science) for warfare is itself an abuse. Guided missiles are controlled using fancy versions of the Kalman filter, which basically is an equation. But so are aeroplanes and spacecraft, and other applications include computer vision, signal processing, and econometrics. What counts is not the equation, but what human beings do with it.'
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Top and bottom envelope functions for a modulated sine wave.
See also: Modulation
In physics and engineering, the envelope function of a rapidly varying signal is a smooth curve outlining its extremes in amplitude.[1] The figure illustrates a sine wave varying between an upper and a lower envelope. The envelope function may be a function of time, or of space, or indeed of any variable.
Example: Beating waves
See also: Beat (acoustics)
(PD) Image: John R. Brews
A modulated wave resulting from adding two sine waves of nearly identical wavelength and frequency.
which uses the trigonometric formula for the addition of two sine waves, and the approximation Δλ<<λ:
Phase and group velocity
with subscripts c and e referring to the carrier and the envelope. The same amplitude F of the wave results for the same value of ξc and ξb, even though this value results for different choices of x and t. This invariance means that one can trace these waveforms in space to find how a position of fixed amplitude propagates in time with a speed that keeps ξ fixed; that is, for the carrier:
which determines for a constant amplitude the distance Δx is related to the time interval Δt by the so-called phase velocity vp
so the group velocity can be rewritten as:
In all media, frequency and wavevector are related by a dispersion relation, 2πf ≡ ω = ω(k), and the group velocity can be written:
Here ω is the frequency in radians/s. In a medium such as classical vacuum the dispersion relation for electromagnetic waves is:
where c0 is the speed of light in classical vacuum. For this case, the phase and group velocities both are c0. In so-called dispersive media the dispersion relation can be a complicated function of wavevector, and the phase and group velocities are not the same. In the general case, the phase and group velocities may have different directions.[6]
Example: Envelope function approximation
(PD) Image: John R. Brews
Electron probabilities in lowest two quantum states of a 160Ǻ GaAs quantum well in a GaAs-GaAlAs quantum heterostructure as calculated from envelope functions.[7]
In condensed matter physics the wavefunction for a mobile charge carrier in a crystal can be expressed as a Bloch wave:
where n is the index for the band (for example, conduction or valence band) r is the spatial location of the particle, and k is its wavevector. The exponential is a sinusoidally varying function corresponding to a slowly varying envelope modulating the rapidly varying part of the wavefunction u, describing the behavior of the wavefunction close to the cores of the atoms of the lattice.
In determining the behavior of the carriers using quantum mechanics, the envelope approximation usually is used in which the Schrödinger equation is simplified to refer only to the behavior of the envelope, and boundary conditions are applied to the envelope function directly, rather than to the complete wavefunction.[8] For example, the wavefunction of a carrier trapped near an impurity is governed by an envelope function F that becomes a superposition of Bloch envelope functions:
where the coefficients A(k) are found from the approximate Schrödinger equation, and V is the crystal volume.[9]
Example: Diffraction patterns
(PD) Image: John R. Brews
where q is the number of slits, and g is the grating constant. The first factor, the single-slit result, modulates the more rapidly varying second factor that depends upon the number of slits and their spacing.
Example: Sound envelope
(PD) Image: John R. Brews
The amplitude of a musical note varies in time according to its sound envelope.[11]
A musical instrument playing a musical note produces a tone, which is a superposition of various frequencies with various amplitudes and phases that give each instrument its unique character. The manner of play determines the sound envelope of the frequencies in the tone, and governs the amplitude of the tone in time. By controlling the sound envelope, each musician imparts their particular interpretation of the music.
1. C. Richard Johnson, Jr, William A. Sethares, Andrew G. Klein (2011). “Figure C.1: The envelope of a function outlines its extremes in a smooth manner”, Software Receiver Design: Build Your Own Digital Communication System in Five Easy Steps. Cambridge University Press, p. 417. ISBN 0521189446.
2. 2.0 2.1 Blair Kinsman (2002). Wind Waves: Their Generation and Propagation on the Ocean Surface, Reprint of Prentice-Hall 1965. Courier Dover Publications, p. 186. ISBN 0486495116.
3. Mark W. Denny (1993). Air and Water: The Biology and Physics of Life's Media. Princeton University Press, p. 289. ISBN 0691025185.
4. 4.0 4.1 Paul Allen Tipler, Gene Mosca (2008). Physics for Scientists and Engineers, Volume 1, 6th ed. Macmillan, p. 538. ISBN 142920124X.
5. Peter W. Milonni, Joseph H. Eberly (2010). “§8.3 Group velocity”, Laser Physics, 2nd ed. John Wiley & Sons, p. 336. ISBN 0470387718.
6. V. Cerveny, Vlastislav Červený (2005). “§2.2.9 Relation between the phase and group velocity vectors”, Seismic Ray Theory. Cambridge University Press, p. 35. ISBN 0521018226.
7. G Bastard, JA Brum, R Ferreira (1991). “Figure 10 in Electronic States in Semiconductor Heterostructures”, Henry Ehrenreich, David Turnbull, eds: Solid state physics: Semiconductor Heterostructures and Nanostructures. ISBN 0126077444.
8. Christian Schüller (2006). “§2.4.1 Envelope function approximation (EFA)”, Inelastic Light Scattering of Semiconductor Nanostructures: Fundamentals And Recent Advances. Springer, p. 22. ISBN 3540365257.
9. For example, see Giuseppe Grosso, Giuseppe Pastori Parravicini (2000). Solid State Physics, 6th ed. Academic Press, p. 478. ISBN 012304460X.
10. 10.0 10.1 Kordt Griepenkerl (2002). “Intensity distribution for diffraction by a slit and Intensity pattern for diffraction by a grating”, John W Harris, Walter Benenson, Horst Stöcker, Holger Lutz, editors: Handbook of physics. Springer, pp. 306 ff. ISBN 0387952691. |
8bcf28b309b2ea7e | The Puzzle Of Quantum Reality : 13.7: Cosmos And Culture Despite the incredibly accurate predictions of quantum theory, there's a lot of disagreement over what it says about reality — or even whether it says anything at all about it, says guest Adam Becker.
NPR logo The Puzzle Of Quantum Reality
The Puzzle Of Quantum Reality
Pasieka/Getty Images/Science Photo Library RF
Conceptual computer art of superstrings; the superstring theory is a "theory of everything."
Pasieka/Getty Images/Science Photo Library RF
There's a hole at the heart of quantum physics.
It's a deep hole. Yet it's not a hole that prevents the theory from working. Quantum physics is, by any measure, astonishingly successful. It's the theory that underpins nearly all of modern technology, from the silicon chips buried in your phone to the LEDs in its screen, from the nuclear hearts of the most distant space probes to the lasers in the supermarket checkout scanner. It explains why the sun shines and how your eyes can see. Quantum physics works.
Yet the hole remains: Despite the wild success of the theory, we don't really understand what it says about the world around us. The mathematics of the theory makes incredibly accurate predictions about the outcomes of experiments and natural phenomena. In order to do that so well, the theory must have captured some essential and profound truth about the nature of the world around us. Yet there's a great deal of disagreement over what the theory says about reality — or even whether it says anything at all about it.
Even the simplest possible things become difficult to decipher in quantum physics. Say you want to describe the position of a single tiny object — the location of just one electron, the simplest subatomic particle we know of. There are three dimensions, so you might expect that you need three numbers to describe the electron's location. This is certainly true in everyday life: If you want to know where I am, you need to know my latitude, my longitude, and how high above the ground I am. But in quantum physics, it turns out three numbers isn't enough. Instead, you need an infinity of numbers, scattered across all of space, just to describe the position of a single electron.
This infinite collection of numbers is called a "wave function," because these numbers scattered across space usually change smoothly, undulating like a wave. There's a beautiful equation that describes how wave functions wave about through space, called the Schrödinger equation (after Erwin Schrödinger, the Austrian physicist who first discovered it in 1925). Wave functions mostly obey the Schrödinger equation the same way a falling rock obeys Newton's laws of motion: It's something like a law of nature. And as laws of nature go, it's a pretty simple one, though it can look mathematically forbidding at first.
Yet despite the simplicity and beauty of the Schrödinger equation, wave functions are pretty weird. Why would you need so much information — an infinity of numbers scattered across all of space — just to describe the position of a single object? Maybe this means that the electron is smeared out somehow. But as it turns out, that's not true. When you actually look for the electron, it shows up in only one spot. And when you do find the electron, something even stranger happens: The electron's wave function temporarily stops obeying the Schrödinger equation. Instead, it "collapses," with all of its infinity of numbers turning to zero except in the place where you found the electron.
So what are wave functions? And why do they only obey the Schrödinger equation sometimes? Specifically, why do they only obey the Schrödinger equation when nobody is looking? These unanswered questions circumscribe the hole at the heart of quantum physics. The last question, in particular, is notorious enough that it has been given a special name: the "measurement problem."
The measurement problem seems like it should stop quantum physics in its tracks. What does "looking" or "measurement" mean? There's no generally agreed-upon answer to this. And that means, in turn, that we don't really know when the Schrödinger equation applies and when it doesn't. And if we don't know that — if we don't know when to use this law and when instead to put it aside — how can we use the theory at all?
The pragmatic answer is that when we physicists do quantum physics, we tend to think of it only as the physics of the ultra-tiny. We usually assume that the Schrödinger equation doesn't really apply to sufficiently large objects — objects like tables and chairs and humans, the things in our everyday lives. Instead, as a practical matter, we assume that those objects obey the classical physics of Isaac Newton, and that the Schrödinger equation stops applying when one of these objects interacts with something from the quantum world of the small. This works well enough to get the right answer in most cases. But almost no physicists truly believe this is how the world actually works. Experiments over the past few decades have shown that quantum physics applies to larger and larger objects, and at this point few doubt that it applies to objects of all sizes. Indeed, quantum physics is routinely and successfully used to describe the largest thing there is — the universe itself — in the well-established field of physical cosmology.
But if quantum physics really applies at all scales, what's the true answer to the measurement problem? What's actually going on in the quantum world? Historically, the standard answer was to say that there is no measurement problem, because it's meaningless to ask what's going on when nobody's looking. The things that happen when nobody's looking are unobservable, and it's meaningless to talk about unobservable things. This position is known as the "Copenhagen interpretation" of quantum physics, after the home of the great Danish physicist Niels Bohr. Bohr was the godfather of quantum physics and the primary force behind the Copenhagen interpretation.
Despite its historical status as the default answer to these quantum questions, the Copenhagen interpretation is inadequate. It says nothing about what's going on in the world of quantum physics. In its stubborn silence on the nature of reality, it offers no explanation of why quantum physics works at all, since it can point to no feature of the world that is anything like the mathematical structures at the heart of the theory. There's no compelling logical or philosophical grounds for declaring unobservable things meaningless. And the word "unobservable" isn't much better defined than the word "measurement" anyhow. So declaring unobservable things meaningless is not only a silly position, it's a vague one. That vagueness has plagued the Copenhagen interpretation from the start; today, the "Copenhagen interpretation" has become a collective label for several mutually contradictory ideas about quantum physics.
Despite this host of problems, the Copenhagen interpretation was overwhelmingly dominant within the physics community for much of the 20th century, because it allowed physicists to perform accurate calculations without worrying about the thorny questions at the heart of the theory. But over the past 30 years, support for the Copenhagen interpretation has eroded. Many physicists still voice support for it — surveys suggest that a plurality or majority of physicists subscribe to it — but there are live alternatives that now have significant support.
The best known of these alternatives is the "many-worlds" interpretation of quantum physics, which states that the Schrödinger equation always applies and wave functions never collapse. Instead, the universe continually splits, with every possible outcome of every event occurring somewhere in the "multiverse." Another alternative, pilot-wave theory, states that quantum particles are guided in their motions by waves, and that the particles in turn can exert faster-than-light influences on far-distant waves (though this cannot be used to send energy or signals faster than light).
These two ideas give two very different depictions of reality, but they both line up perfectly with the mathematics of quantum mechanics as we know it. There are also alternative theories that modify the mathematics of quantum physics, such as spontaneous-collapse theories, which suggest that the collapse of the wave function has nothing to do with measurement, and is instead a natural process that happens entirely at random.
There are many, many other alternatives. Quantum foundations, the field that deals in resolving the measurement problem and the other basic questions of quantum theory, is a lively subject brimming with creative ideas. The hole at the heart of quantum physics is still there — there's still an open problem that needs solving — but there are many fascinating theories that have been proposed to solve these problems. These ideas might also point the way forward on other problems in physics, such as a theory of quantum gravity, the "theory of everything" that has been the ultimate goal of physicists since Albert Einstein.
Adam Becker is the author of What Is Real?: The Unfinished Quest For The Meaning Of Quantum Physics, published March 20. He is a visiting scholar in the University of California, Berkeley, Office for History of Science and Technology. Becker holds a PhD in astrophysics from the University of Michigan and a BA in philosophy and physics from Cornell. |
17a152931125997a | By William Roland-Batty, University of Newcastle
There are two ‘forms’ in which you can represent a function- in spatial form (where the variable is in metres, seconds, etc) and in frequency form (per metres, per seconds, etc.). The Fourier transform converts a function from its spatial representation into its frequency representation and the inverse Fourier transform converts a function from its frequency representation into its spatial representation. If you apply the Fourier transform to a function twice, you get the function flipped around the y-axis; three times and you get the inverse Fourier transform; four times and you get the original function back.
The fractional Fourier transform describes the ‘in-between’ states – for example, applying the Fourier transform ‘one-half times’, which, if you do twice, is the Fourier transform. The fractional Fourier transform takes an ‘angle’ parameter: the fractional Fourier transform of angle pi/2 is the Fourier transform, and angle 2pi is the same as four times the Fourier transform, which gives the original function.
The fractional Fourier transform has special applications in optics. A quadratic graded-index medium is a medium where the refractive index varies along its radius according to a certain function. If a function describes the distribution of light on some plane, then as this plane propagates along the medium the distribution of light will become the fractional Fourier transform of the original function with the angle depending on the distance.
My supervisor told me about a book on the topic which I borrowed from the library. Most of my time during the project was spent reading it and trying to understand all of the content. I found out about a lot of unexpected relations throughout the project, like the far-field diffraction pattern of an aperture being its Fourier transform, or the fractional Fourier transform’s relation to solutions of the Schrödinger equation in physics.
Our project focused on finding the behaviour of light as it propagated through a quadratic graded-index medium with regular apertures placed along it. We used some special functions called the Legendre polynomials and the Hermite-Gaussian functions to compute a matrix to model the transmission between two apertures. However, this matrix was infinitely large, so we had to cut it off at some point to actually calculate it. The larger we made it before cutting it off, the more accurate it was.
We checked the accuracy in two ways – we knew analytically that if we set the distance between the apertures to zero, we would get the identity matrix, which is 1s along the diagonal and 0s everywhere else. We did this, and the entries in the matrix approached their appropriate values as the dimension was increased (the entries in the matrix changed as the dimension did since they were calculated by a truncated infinite sum).
The other way was to find the eigenvectors of the matrix for the distance where the function would be exactly once Fourier transformed between the apertures. Analytically, we know the eigenfunctions of this to be the special prolate functions. When we reconstructed the eigenfunctions from the eigenvectors, we found that they were roughly similar to the prolates but not particularly precise
Most of our theoretical background was referenced from the following book:
Ozaktas, HM. et al. 2001, The Fractional Fourier Transform with Applications in Optics and Signal Processing, John WIley & Sons, Chichester
William Roland-Batty was a recipient of a 2018/19 AMSI Vacation Research Scholarship.
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1407a615404c298e | Tag Archives: history
Quantum Physics at the Crossroads
When I was but a young undergrad student, I read some interesting books about the history and foundations of quantum theory. In those books the Solvay conferences played a major role, particularly the 5th conference in 1927. I was informed that the official part of the proceedings was largely insignificant, and that all the action centred around the debates that took place between Bohr and Einstein, in which Einstein repeatedly tried to undermine the uncertainty principle via a series of thought experiments, but Bohr was always quick to respond with a correct analysis of the experiment that showed uncertainty to be triumphant. This always put in my mind a picture similar to da Vinci’s “Last Supper”, with Bohr playing the role of Jesus, regailaing his many disciples with the moral parable of the day over dinner.
Another piece of folklore concerns the Ph.D. thesis of one Prince Louis de Broglie. This contained the famous de Broglie relation that gives the wavelength of the waves to be associated with matter particles. The story goes that the thesis was on the verge of being rejected, but was saved by Einstein’s recommendation, who was the only person to recognize the deep significance of the relation. As the story is told, it is hardly surprising, because the contents of the rest of the thesis is never explained. One is left to imagine a document that could only have been about 10 pages long, which intrroduces the relation and then explains some of its consequences. That may seem stong enough for a very good Phys. Rev. article, but is hardly enough to warrant a Ph.D.
Since those distant days of my youth, I have attended many a physics conference myself. I now recognise that it is the general rule, almost without exception, that the participants regard the discussions they have outside the talks as being much more important and interesting than anything that was said in the talks themselves. This rule holds regardless of the actual inherent interest of the topics under discussion. In fact, it is quite common to find some of the older participants banging on about some Hamiltonian they wrote down in the 1970s, whereas the young guns are talking about something genuinely new and interesting, the significance of which is not understood by the older guys yet. It is also extremely unlikely to find the entire group of conference participants, however small that group may be, listening in rapt attention to the discussion of just two people over dinner (if only because there are simply some groups of people who don’t get on with each other, and others who are more interested in going to the pub), and it is equally unlikely that that conversation represents the only interesting thing going on at the conference.
Also, it goes without saying really that I don’t know of anyone who got their Ph.D. for a 10 page paper, however great the idea contained therein happens to be.
Currently, I am about half way through reading “Quantum Theory at the Crossroads”, the new book by Bacciagaluppi and Valentini about the 1927 Solvay conference. The second half of the book is an English translation of the proceedings, but equally interesting is the new analysis of the conference discussions from a modern point of view, contained in the first half. Here are some things I found particularly interesting.
– The only witnesses to the famous Bohr-Einstein debates were Heisenberg and Ehrenfest. The usuall account of these debates comes directly from an article written by Bohr many years after the conference took place. Heisenberg roughly confirms the account, also in recollections written many years later. The only account written shortly after the conference is a letter written by Ehrenfest, which seems to confirm that Bohr was triumphant in the debates, but gives no details.
– Bacciagaluppi and Valentini argue that it is highly unlikely that Einstein’s main target was the uncertainty relations. This is because, outside of Bohr’s account of the conference discussions, Einstein hardly mentions the uncertainty relations as a point of concern in any of his correspondence or published works. Instead, they argue, it is likely that he was trying to get at the point that the concept of separability was incompatible with quantum theory, which was later crystallized in the EPR argument. In fact, Einstein gives an argument in this direction also in the published general discsussion at the conference. It seems likely that Bohr missed this point, just as he seemed to miss the point years later in his published response to the EPR argument.
– At the time of the conference, the consolidation of quantum theory was far from complete. Three approaches were discussed in the talks: de Broglie’s pilot wave theory, Schrödinger’s wave mechanics and Heisenberg’s matrix mechanics (with additions by Born). Despite the fact that “equivalence proofs” between wave and matrix mechanics had been published at the time of the conference, they were treated as distinct theories, which could potentially make different predictions. This is because, at the time, Schrödinger did not accept Born’s statistical hypothesis for wave mechanics, which was not yet formulated for arbitrary observables in any case. Also, Heisenberg and Born did not accept the fundamental significance of the time-dependent Schrödinger equation, and still clung to a view of matrix meachanics as describing the transition probabilities for systems always to be thought of as being in definite stationary states. In fact, it seems that the only person at the conference who presented something that we would now regard as being empirically equivalent to modern quantum theory was de Broglie.
– This was not recognized at the conference, partly because de Broglie did not realize that one sometimes has to treat the apparatus as a quantum system in pilot wave theory in order to get equivalence with standard quantum theory. Also, there was as yet desciption of spin within de Broglie’s theory, but on the other hand this same objection could be levelled at wave mechanics. Finally, de Broglie himself regarded the theory as provisional, since it was not relativistic and involved waves in configuration space rather than ordinary 3d space. He placed great significance on ideas for a better theory, which were far from complete at the time of the presentation.
– Schrödinger emphasizes that de Broglie’s work was a major inspiration for his wave equation. In particular, de Broglie’s idea of unifying the variational principles of Newtonian mechanics with those of geometrical optics, was used in the derivation of the equation.
– de Broglie presented his pilot-wave theory for multiparticle systems, not just for single particles as is commonly thought.
In light of this and other arguments, Bacciagaluppi and Valentini argue that the time is ripe for a revision of the usual textbook history of quantum mechanics, and in particular of de Broglie’s contribution . Those who believe that the history of science should be written with the same objective standards that we hope to uphold for science itself, rather than simply being written by the victors, are well-advised to read this book.
Why not von Neumann?
Anyone who read the comments on my last post will know that von Neumann is something of a hero of mine. Here’s a question that sometimes bothers me – why didn’t von Neumann think of quantum computing? Compare his profile with that of Feynman, who did think up quantum computing, and then ask yourself which one of them you would have bet on to come up with the idea.
• von Neumann: Worked on a variety of different subjects thoughout his career, including interdisciplinary ones. Was well aware of the work by Turing, Church, Post and others that later became the foundation for computer science and of the role of logic in this work. Is credited with the design of the basic architechture of modern computers. Worked on the mathematical and conceptual foundations of quantum mechanics and is responsible for the separable Hilbert space formulation of quantum theory that we still use today. Finally, at some point he was convinced that the best way to understand quantum theory was as a probability theory over logical structures (lattices) that generalize some of those from classical logic.
• Feynmann: Spent most of his career working on mainstream topics in quantum field theory and high energy physics. Only towards the end of his career did his interests significantly diversify to include the theories of computation, quantum gravity and the foundations of quantum theory. Conceived of quantum theory mainly in the “sum over paths” formalism, where one looks at quantum theory as a rule for attaching amplitudes to possible histories as opposed to the probabilities used in classical theories.
None of this is meant as a slight against Feynman – he was certainly brilliant at everything he did scientifically – but it is clear that von Neumann was better positioned to come up with the idea much earlier on. Here are some possible explanations that I can think of:
• The idea of connecting quantum mechanics to computing just never occurred to von Neumann. They occupied disjoint portions of his brain. Ideas that seem simple in hindsight are really not so obvious, and even the greatest minds miss them all the time.
• von Neumann did think of something like quantum computing, but it was not obvious that it was interesting, since the science of computational complexity had not been developed yet. Without the distinction between exponential and polynomial time, there is no way to identify the potential advantage that quantum computers might offer over their classical counterparts.
• The idea of some sort of difference in computing when quantum mechanics is thrown into the mix did occur to von Neumann, but he was unable to come up with a relevant model of computing because he was working with the wrong concepts. As alluded to in a paper of mine, Birkhoff-von Neumann quantum logic is definitely the wrong logic for thinking about quantum computing because the truth of quantum logic propositions on finite Hilbert spaces may be verified on a classical computer in polynomial time. The basic observation was pointed out to me by Scott Aaronson, but one needs to set up the model quite carefully to make it rigorous. I might write this up at some point, especially if people continue to produce papers that use quantum computing as a motivation for studying concrete BvN quantum logic on Hilbert spaces. Anyway, the point is that if von Neumann thought that replacing classical logic with his notion quantum logic was the way to come up with a model of quantum computing, then he would not have arrived at anything useful.
• As a mathematician, von Neumann was not able to think of any practical problem to do with quantum mechanics that looks hard to do on a classical computer, but could be done efficiently in the quantum world. As a physicist, Feynman was much better placed to realize that simulating quantum dynamics was a useful thing to do, and that it might require exponential resources on a classical computer.
As a von Neumann fan, I’d like to think that something other than the first explanation is true, but I am prepared to admit that he might have missed something that ought to have been obvious to him. Hopefully, someday a historian of science will take it upon themselves to trawl the von Neumann archives looking for the answer. |
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de Broglie-Bohm theory (Bohmian mechanics)
(06-03-2016, 05:33 PM)Ioannis Wrote: a) Is the pilot wave a separate entity? IF it is then it is not an intrinsic property of the charged particle itself.
It is clearly a separate entity.
There is some more uncertainty about its status, but the basic dBB assumption is that it has some real existence. So, there is not only the trajectory, which is what we see around us, but also, additionally, some objectively existing wave function.
I tend to prefer another concept, namely that the wave function describes our knowledge. But this knowledge is, then, knowledge about the environment, in particular about the pointers of the measurement devices used in the preparation procedure. Thus, knowledge about some other, external objects, which are somehow correlated with the trajectory, but now the trajectory itself. So, in above variants is is not some intrinsic property of the particle itself, but something different, external.
(06-03-2016, 05:33 PM)Ioannis Wrote: b) Does the pilot wave travels together with the charged particle?
No, it is something completely different in nature, it is a function of all imaginable configurations.
The closest analogy is in classical theory the energy. It is a function of the trajectory - you can, for every trajectory, compute its energy, the energy of this particular trajectory. And, then, this energy "guides" the trajectory by energy conservation: The trajectory cannot be arbitrary, it has to continue in such a way that the energy remains unchanged in time. But the energy may not change at all, while the position of the particle changes in time. And the energy is defined also for all other configurations, not only the one which we see.
(06-03-2016, 05:33 PM)Ioannis Wrote: c) Is the pilot wave in phase with the mass properties of charged particle or not?
Hm. The pilot wave follows the Schrödinger equation. The mass is a parameter in the Schrödinger equation.
d) IF the pilot wave travels together with the charged particle that means is a Wave-Particle instance at all moments then, the charged particle would be in a Quantum Tunneling state in its entire life after approaching the speed of light. It follows that under such circumstances the charged particle would be then undetectable in our world.
Sorry, I don't understand this text. Quantum theory, as well as dBB theory, are theories with an absolute but unmeasurable time. Every clock is, independent of any relativity, inaccurate, and even goes backward in time with some non-zero probability.
(06-03-2016, 05:33 PM)Ioannis Wrote: e) IF the pilot wave travels together with the charged particle what is the Relativistic Energy of the Wave-Particle entity? Einstein's Relativity predicts the Relativistic Energy for charged particles associated not to pilot waves. How dBB copes with this?
There is a variant of dBB theory for relativistic particles, but I don't like it. I prefer for relativistic theory the field theory. In this case, particles become quite irrelevant, quantum effects without any fundamental importance, like phonons in condensed matter theory (some "sound particles" which are simply quantum effects like the discrete energy levels in an atom).
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RE: de Broglie-Bohm theory (Bohmian mechanics) - by Schmelzer - 06-04-2016, 08:34 AM
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e29e51c14a5399de | Download The Interaction of Radio-Frequency Fields With
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Volume 117 (2012)
Journal of Research of the National Institute of Standards and Technology
The Interaction of Radio-Frequency Fields
With Dielectric Materials at
Macroscopic to Mesoscopic Scales
James Baker-Jarvis and
Sung Kim
Electromagnetics Division,
National Institute of Standards
and Technology,
Boulder, Colorado 80305
[email protected]
[email protected]
The goal of this paper is to overview
radio-frequency (RF) electromagnetic
interactions with solid and liquid materials
from the macroscale to the nanoscale. The
overview is geared toward the general
researcher. Because this area of research is
vast, this paper concentrates on currently
active research areas in the megahertz
(MHz) through gigahertz (GHz)
frequencies, and concentrates on dielectric
response. The paper studies interaction
mechanisms both from phenomenological
and fundamental viewpoints. Relaxation,
resonance, interface phenomena,
plasmons, the concepts of permittivity
and permeability, and relaxation times are
summarized. Topics of current research
interest, such as negative-index behavior,
noise, plasmonic behavior, RF heating,
nanoscale materials, wave cloaking,
polaritonic surface waves, biomaterials,
and other topics are overviewed.
In this paper we will overview electromagnetic interactions with solid and liquid dielectric and magnetic
materials from the macroscale down to the nanoscale.
We will concentrate our effort on radio-frequency (RF)
waves that include microwaves (MW) and millimeterwaves (MMW), as shown in Table 1. Radio frequency
waves encompass frequencies from 3 kHz to 300 GHz.
Microwaves encompass frequencies from 300 MHz to
30 GHz. Extremely high-frequency waves (EHF) and
millimeter waves range from 30 GHz to 300 GHz.
Many devices operate through the interaction of RF
electromagnetic waves with materials. The characterization of the interface and interaction between fields
Relaxation, resonance, and related
relaxation times are overviewed. The
wavelength and material length scales
required to define permittivity in materials
is discussed.
Key words: dielectric; electromagnetic
fields; loss factor; metamaterials;
microwave; millimeter wave; nanoscale;
permeability; permittivity; plasmon;
Accepted: August 25, 2011
Published: February 2, 2012
Table 1. Radio-Frequency Bands [1]
3 – 30 kHz
30 – 300 kHz
0.3 – 3 MHz
3 – 30 MHz
30 – 300 MHz
300 – 3000 MHz
3 – 30 GHz
30 – 300 GHz
300 – 3000 GHz
100 – 10 km
10 – 1 km
1 – 0.1 km
100 – 10 m
10 – 1 m
100 – 10 cm
10 – 1 cm
10 – 1 mm
1 – 0.1 mm
Volume 117 (2012)
Journal of Research of the National Institute of Standards and Technology
and materials is a critical task in any electromagnetic
(EM) device or measurement instrument development,
from nanoscale to larger scales. Electromagnetic waves
in the radio-frequency range have unique properties.
These attributes include the ability to travel in guidedwave structures, the ability of antennas to launch waves
that carry information over long distances, possess
measurable phase and magnitude, the capability for
imaging and memory storage, dielectric heating, and
the ability to penetrate materials.
Some of the applications we will study are related to
areas in microelectronics, bioelectromagnetics, homeland security, nanoscale and macroscale probing,
magnetic memories, dielectric nondestructive sensing,
radiometry, dielectric heating, and microwave-assisted
chemistry. For nanoscale devices the RF wavelengths
are much larger than the device. In many other applications the feature size may be comparable or larger than
the wavelength of the applied field.
We will begin with an introduction of the interaction
of fields with materials and then overview the basic
notations and definitions of EM quantities, then
progress into dielectric and magnetic response, definitions of permittivity and permeability, fields, relaxation
times, surfaces waves, artificial materials, dielectric
and magnetic heating, nanoscale interactions, and field
fluctuations. The paper ends with an overview of
biomaterials in EM fields and metrologic issues.
Because this area is very broad, we limit our analysis to
emphasize solid and liquid dielectrics over magnetic
materials, higher frequencies over low frequencies,
and classical over quantum-mechanical descriptions.
Limited space will be used to overview electrostatic
fields, radiative fields, and terahertz interactions. There
is minimal discussion of EM interactions with nonlinear materials and gases.
molecules, cells, or inorganic materials are subjected to
external electric fields, the molecules can respond in a
number of ways. For example, a single charged particle
will experience a force in an applied electric field. Also,
in response to electric fields, the charges in a neutral
many-body particle may separate to form induced
dipole moments, which tend to align in the field; however this alignment is in competition with thermal
effects. Particles that have permanent dipole moments
will interact with applied dc or high-frequency fields.
In an electric field, particles with permanent dipolemoments will tend to align due to the electrical torque,
but in competition to thermal randomizing effects.
When EM fields are applied to elongated particles with
mobile charges, they tend to align in the field. If the
field is nonuniform, the particle may experience dielectrophoresis forces due to field gradients.
On the microscopic level we know that the electromagnetic field is modeled as a collection of photons
[2]. In theory, the electromagnetic field interactions
with matter may be modeled on a microscopic scale by
solving Schrödinger’s equation, but generally other
approximate approaches are used. At larger scales the
interaction with materials is modeled by macroscopic
Maxwell’s equations together with constitutive relations and boundary conditions. At a courser level of
description, phenomenological and circuit models are
commonly used. Typical scales of various objects are
shown in Fig. 1. The mesoscopic scale is where
classical analysis begins to be modified by quantum
mechanics and is a particularly difficult area to model.
The interaction of the radiation field with atoms
is described by quantum electrodynamics. From a
quantum-mechanical viewpoint the radiation field is
quantized, with the energy of a photon of angular
frequency ω being E = ω. Photons exhibit waveduality and quantization. This quantization also occurs
in mechanical behavior where lattice vibrational
motion is quantized into phonons. Commonly, an atom
is modeled as a harmonic oscillator that absorbs or
emits photons. The field is also quantized, and each
field mode is represented as a harmonic oscillator and
the photon is the quantum particle.
The radiation field is usually assumed to contain a
distribution of various photon frequencies. When the
radiation field interacts with atoms at the appropriate
frequency, there can be absorption or emission of
photons. When an atom emits a photon, the energy of
the atom decreases, but then the field energy increases.
Rigorous studies of the interaction of the molecular
Electromagnetic Interactions From the
Microscale to Macroscale
In this section we want to briefly discuss electromagnetic interaction with materials on the microscale to the
Matter is modeled as being composed of many
uncharged and charged particles including for example,
protons, electrons, and ions. On the other hand, the
electromagnetic field is composed of photons. The
internal electric field in a material is related to the sum
of the fields from all of the charged particles plus
any applied field. When particles such as biological
Volume 117 (2012)
Journal of Research of the National Institute of Standards and Technology
Fig. 1. Scales of objects.
to the energy of the interacting quasiparticles. Brillouin
scattering can be used to probe mesoscopic properties
such as elasticity. Raman scattering is an inelastic
process similar to Brillouin scattering, but where the
scattering is due to molecular or atomic-level transitions. Raman scattering can be used to probe chemical
and molecular structure. Surface-enhanced Raman
scattering (SERS) is due to enhancement of the EM
field by surface-wave excitation [5].
Optically transparent materials such as glass have
atoms with bound electrons whose absorption frequencies are not in the visible spectrum and, therefore, incident light is transmitted through the material. Metallic
materials contain free electrons that have a distribution
of resonant frequencies that either absorb incoming
light or reflect it. Materials that are absorbing in one
frequency band may be transparent in another band.
Polarization in atoms and molecules can be due to
permanent electric moments or induced moments
caused by the applied field, and spins or spin moments.
The response of induced polarization is usually weaker
than that of permanent polarization, because the typical
radii of atoms are on the order of 0.1 nm. On application of a strong external electric field, the electron
cloud will displace the bound electrons only about
10–16 m. This is a consequence of the fact that the
atomic electric fields in the atom are very intense,
field with the radiation field involve quantization of the
radiation field by expressing the potential energy
V(r) and vector potential A(r, t) in terms of creation and
annihilation operators and using these fields in the
Hamiltonian, which is then used in the Schrödinger
equation to obtain the wavefunction (see, for example,
[3]). The static electromagnetic field is sometimes
modeled by virtual photons that can exist for the short
periods allowed by the uncertainty principle. Photons
can interact by depositing all their energy in photoelectric electron interactions, by Compton scattering
processes, where they deposit only a portion of the
energy together with a scattered photon, or by pair
production. When a photon collides with an electron it
deposits its kinetic energy into the surrounding matter
as it moves through the material. Light scattering is a
result of changes in the media caused by the incoming
electromagnetic waves [4]. In Rayleigh elastic light
scattering, the photons of the scattered incident light
are used for imaging material features. Brillouin
scattering is an inelastic collision that may form or
annihilate quasiparticles such as phonons, plasmons,
and magnons. Plasmons relate to plasma oscillations,
often in metals, that mimic a particle and magnons are
the quanta in spin waves. Brillouin scattering occurs
when the frequency of the scattered light shifts in
relation to the incident field. This energy shift relates
Volume 117 (2012)
Journal of Research of the National Institute of Standards and Technology
are~ B (ω) = μ H (ω) + η1 E (ω) and D (ω) =
ε E(ω) + η2 H
(ω). In most
the constitutive
and D (ω) =
ε0 E(ω) + P(ω) are used.
In any complex lossy system, energy is converted
from one form to another, such as the transformation of
EM energy to lattice kinetic energy and thermal energy
through photon-phonon interactions. Some of the
energy in the applied fields that interact with materials
is transfered into thermal energy as infrared phonons.
In a waveguide, there is a constant exchange of energy
between the charge in the guiding conductors and the
fields [6].
When the electromagnetic field interacts with material degrees of freedom, a collective response may be
generated. The term polariton relates to bosonic quasiparticles resulting from the coupling of EM photons or
waves with an electric or magnetic dipole-carrying
excitation [4, 5]. The resonant and nonresonant
coupling of EM fields in phonon scattering is mediated
through the phonon-polariton transverse-wave quasiparticle. Phonon polaritons are formed from photons
interacting with terahertz to optical phonons.
Ensembles of electrons in metals form plasmas and
high-frequency fields applied to these electron gases
produce resonant quasi-particles, commonly called
plasmons. Plasmons are a collective excitation of a
group of electrons or ions that simultaneously oscillate
in the field. An example of a plasmon is the resonant
oscillation of free electrons in metals and semiconductors in response to an applied high-frequency field.
Plasmons may also form at the interface of a dielectric
and a metal and travel as a surface wave with most of
the EM energy confined to the low-loss dielectric. A
surface plasmon polariton is the coupling of a photon
with surface plasmons. Whereas transverse plasmons
can couple to an EM field directly, longitudinal
plasmons couple to the EM field by secondary particle
collisions. In the microwave and millimeter wave
bands artificial structures can be machined in metallic
surfaces to produce plasmons-like excitations due to
geometry. Magnetic coupling is mediated through
magnons and spin waves. A magnon is a quantum of a
spin wave that travels through a spin lattice. A polaron
is an excitation caused by a polarized electron traveling
through a material together with the resultant polarization of adjacent dipoles and lattice distortion [4]. All of
these effects are manifest at the mesoscale through
macroscale in the constitutive relations and the resultant permittivity and permeability.
approximately 1011 V/m. The splitting of spectral lines
due to the interaction of electric fields with atoms and
molecules is called the Stark effect. The Stark effect
occurs when interaction of the electric-dipole moment
of molecules interacts with an applied electric field that
changes the potential energy and promotes rotation and
atomic transitions. Because the rotation of the molecules depends on the frequency of the applied field, the
Stark effect depends on both the frequency and field
strength. The interaction of magnetic fields with molecular dipole moments is called the Zeeman effect. Both
the Stark and Zeeman effects have fine-structure modifications that depend on the molecule’s angular
momentum and spin. On a mesoscopic scale, the interactions are summarized in the Hamiltonian that contains the internal energy of the lattice, electric and magnetic dipole moments, and the applied fields.
In modeling EM interactions at macroscopic scales,
a homogenization process is usually applied and the
classical Maxwell field is treated as an average of the
photon field. There also is a homogenization process
that is used in deriving the macroscopic Maxwell
equations from the microscopic Maxwell equations.
The macroscopic Maxwell’s equations in materials are
formed by averaging the microscopic equations over a
unit cell. In this averaging procedure, the macroscopic
charge and current densities, the magnetic field H, the
magnetization M, the displacement field D, and the
electric polarization field P are formed. At these scales,
the molecule dipole moments are averaged over a unit
cell to form continuous dielectric and magnetic polarizations P and M. The constitutive relations for the
polarization and magnetization are used to define the
permittivity and permeability. At macroscopic to mesoscopic scales the permittivity, permeability, refractive
index, and impedance are used to model the response of
materials to applied fields. We will discuss this in detail
in Sec. 4.5. Quantities such as permittivity, permeability, refractive index, and wave impedance are not microscopic quantities, but are defined through an averaging
procedure. This averaging works well when the wavelength is much larger than the size of the molecules or
atoms and when there are a large number of molecules.
In theoretical formulations for small scales and wavelengths near molecular dimensions, the dipole moment
and polarizability tensor of atoms and molecules can be
used rather than the permittivity or permeability. In some
materials, such as magnetoelectric and chiral materials,
there is a coupling between the electric and magnetic
responses. In such cases the time-harmonic constitutive
Volume 117 (2012)
Journal of Research of the National Institute of Standards and Technology
Responses to Applied RF Fields
or lattice, assuming one can isolate the effects of the
molecules from the measurement apparatus. This
metrology is not always easy because a measurement
contains effects of electrodes, probes, and other
environmental factors. The concepts of atomic polarizability and dipole molecular moment are valid on a
smaller scale than are permittivity and permeability.
In the absence of an applied field, small random
voltages with a zero mean are produced by equilibrium
thermal fluctuations of random charge motion [17].
Fluctuations of these random voltages create electrical
noise power in circuits. Analogously, spin noise is due
to spin fluctuations. Quasi-monochromatic surface
waves can also be excited by random thermal fluctuations. These surface waves are different from blackbody radiation [18]. Various interesting effects are
achieved by random fields interacting with surfaces.
For example, surface waves on two closely spaced
surfaces can cause an enhanced radiative transfer.
Noise in nonequilibrium systems is becoming more
important in nanoscale measurements and in systems
where the temperatures vary in time. The information
obtained from radiometry at a large scale, or microscopic probing of thermal fluctuations of various
material quantities, can produce an abundance of information on the systems under test.
If we immerse a specimen in an applied field and the
response is recorded by a measurement device, the data
obtained are usually in terms of a digital readout or a
needle deflection indicating the phase and magnitude of
a voltage or current, a difference in voltage and
current, power, force, temperature, or an interference
fringe. For example, we deduce electric and magnetic
field strengths and phase through Ampere’s and
Faraday’s laws by means of voltage and current
measurements. The scattering parameters measured on
a network analyzer relates to the phase and magnitude
of a voltage wave. The detection of a photon’s energy is
sensed by an electron cascade current. Cavities and
microwave evanescent probes sense material characteristics through shifts in resonance frequency from the
influence of the specimen under test. The shift in resonance frequency is again determined by voltage and
power measurements on a network analyzer. Magnetic
interactions are also determined through measurements
of current and voltage or forces [4, 7-9]. These measurement results are usually used with theoretical models,
such as Maxwell’s equations, circuit parameters, or the
Drude model, to obtain material properties.
High-frequency electrical responses include the measurement of the phase and magnitude of guided waves in
transmission lines, fields from antennas, resonant frequencies and quality factors (Q) of cavities or dielectric
resonators, voltage waves, movement of charge or spin,
temperature changes, or forces on charge or spins. These
responses are then combined through theoretical models
to obtain approximations to important fundamental
quantities such as: power, impedance, capacitance,
inductance, conductance, resistance, conductivity, resistivity, dipole and spin moments, permittivity, and permeability, resonance frequency, Q, antenna gain, and nearfield response [10-16].
The homogenization procedure used to obtain the
macroscopic Maxwell equations from the microscopic
Maxwell equations is accomplished by averaging the
molecular dipole moments within a unit cell and constructing an averaged continuous charge density function. Then a Taylor series expansion of the averaged
charge density is performed, and, as a consequence, it
is possible to define the averaged polarization vector.
The spatial requirement for the validity of this averaging is that the wavelength must be much larger than the
unit cell dimensions (see Sec. 4.6 ). According to this
analysis, the permittivity of an ensemble of molecules
is valid for applied field wavelengths that are much
larger than the dimensions of an ensemble of molecules
RF Measurements at Various Scales
At RF frequencies the wavelengths are much larger
than molecular dimensions. There are various approaches to obtaining material response with long wavelength
fields to study small-scale particles or systems. These
methods may use very sensitive detectors, such as
single-charge or spin detectors or amplifiers, or average
the response over an ensemble of particles to obtain a
collective response. To make progress in the area of
mesoscale measurement, detector sensitivity may need
to exceed the three or four significant digits obtained
from network analyzer scattering parameter measurements, or one must use large ensembles of cells for a
bulk response and infer the small-scale response.
Increased sensitivity may be obtained by using resonant
methods or evanescent fields.
Material properties such as collective polarization
and loss [19] are commonly obtained by immersing
materials in the fields of EM cavities, dielectric
resonators, free-space methods, or transmission lines.
Some responses relate to intrinsic resonances in a
material, such as polariton or plasmon response,
ferromagnetic and anti-ferromagnetic resonances, and
terahertz molecular resonances.
Volume 117 (2012)
Journal of Research of the National Institute of Standards and Technology
Broadband response is usually obtained by use of
transmission lines or antenna-based systems [12-14,
19, 20]. Thin films are commonly measured with
coplanar waveguides or microstrips [14]. Common
methods used to measure material properties at small
scales include near-field probes, micro-transmission
lines, atomic-force microscopes, and lenses.
In strong fields, biological cells may rotate, deform,
or be destroyed [21]. In addition, when there is more
than one particle in the applied field, the fields between
the particles can be modified by the presence of
nearby particles. In a study by Friend et al. [22], the
response of an amoeba to an applied field was studied
in a capacitor at various voltages, power, and frequencies. They found that at 1 kHz and at 10 V/cm the
amoeba oriented perpendicular to the field. At around
10 kHz and above 15 V/cm the amoeba’s internal
membrane started to fail. Above 100 kHz and a field
strength of above 50 V/cm, thermal effects started to
damage the cells.
Fundamental Electromagnetic
Parameters and Concepts Used in
Material Characterization
Electrical Parameters for High-Frequency
In this section, the basic concepts and tools needed to
study and interpret dielectric and magnetic response
over RF frequencies are reviewed [24].
In the time domain, material properties can be
obtained by analyzing the response to a pulse or impulse;
however most material measurements are performed by
subjecting the material to time-harmonic fields.
The most general causal linear time-domain
relationships between the displacement and electric
fields and induction and magnetic fields are
D(r,t )=ε 0 E( r, t)+ ε 0
f p (τ ) ⋅ E( r, t − τ ) dτ ,
Electromagnetic Measurement Problems
Unique to Microscale and Nanoscale Systems
Usually, the electrical skin depth for field penetration
is much larger than the dimensions of nanoparticles.
Because nanoscale systems are only 10 to 1000 times
larger than the scale of atoms and small molecules,
quantum mechanics plays a role in the transport properties. Below about 10 nm, many of the continuous
quantities in classical electromagnetics take on a quantized aspect. These include charge transport, capacitance, inductance, and conductance. Fluctuations in
voltage and current also become more important than in
macroscopic systems. Electrical conduction at the
10 nanoscale involves movement of a small number of
charge carriers through thin structures and may attain
ballistic transport. For example, if a 1 μA charge
travels through a nanowire of radial dimensions 30 nm,
then the current density is on the order of 3 × 109 A/m2.
Because of these large current densities, electrical
transport in nanoscale systems is usually a nonequilibrium process, and there is a large influence of
electron-electron and electron-ion interactions.
In nanoscale systems, boundary layers and interfaces
strongly influence the electrical properties, and the local
permittivity may vary with position [23]. Measurements
on these scales must model the contact resistance
between the nanoparticle and the probe or transmission
line and deal with noise.
where f p (t) is a polarization impulse-response dyadic,
B(r,t )=μ0 H(r, t)+ μ 0
fm (τ ) ⋅ H( r, t − τ ) dτ ,
where f m (t) is a magnetic impulse-response dyadic.
The permittivity ↔
ε(ω) dyadic is the complex~ parameter ~ in the time-harmonic field relation D(ω) =
ε(ω) . E(ω) and, is defined in terms of the Fourier transform of the impulse-response function. For isotropic
linear media, the scalar complex relative permittivity εr is defined in terms of the absolute permittivity ε and the permittivity of vacuum ε 0 (F/m), as
follows ε (ω) = ε 0ε r (ω), where ε r (ω) = ε r ∞ + χr (ω) =
ε′r (ω) – iε″r (ω), and ε r ∞ is the optical-limit of the
relative permittivity. The value of the permittivity
of free space is ε 0 ≡ 1/μ 0c2ν ≈ 8.854 × 10–12 (F/m), where
the speed of light in vacuum is cν ≡ 299792458 (m/s)
and the exact value of the permeability of free space is
μ 0 = 4π × 10 –7 (H/m). Also, tanδd = ε″r /ε′r is the loss
tangent in the material [25].
Volume 117 (2012)
Journal of Research of the National Institute of Standards and Technology
Note that in the SI system of units the speed of light,
permittivity of vacuum, and permeability of vacuum
are defined constants. All measurements are related to
a frequency standard. Note that the minus sign before
the imaginary part of the permittivity and permeability
is due to the e iωt time dependence. A subscript eff on the
permittivity or permeability releases the quantity from
some of the strict details of electrodynamic analysis.
The permeability in no applied field is: μ(ω) =
μ 0(μ′r (ω) – iμ″r (ω)) and the magnetic loss tangent is
tanδm = μ″r (ω)/μ′r (ω).
For anisotropic and gyrotropic media with an applied
magnetic field, the permittivity and permeability
tensors are hermitian and can be expressed in the
general form
⎛ ε xx
ε (ω) = ⎜ ε xy + ig z
⎝ ε xz − ig y
ε xy − ig z
ε yy
ε yz + igx
ε xz + ig y ⎞
ε yz − igx ⎟ .
ε zz ⎟⎠
c 2
1 + tan 2 δd −1 .
ε ' r μ'r
In dielectric media with low loss, tanδd <<1, and α
reduces in this limit to α → ω ε'r μ'r tan δ d /2 c. The
skin depth is the distance a plane wave travels until it
decays to 1/e of its initial amplitude, and is related to
the attenuation coefficient by δs = 1/α. The concept of
skin depth is useful in modeling lossy dielectrics and
metals. Energy conservation constrains a to be positive.
The skin depth is defined for lossy dielectric materials
δs =
c 2
1 + tan 2 δ d −1
ε ' r μ' r
In Eq. (6), δs reduces in the low-conductivity limit to
to δs → 2c/(ω ε'r μ'r tan δ d ). The depth of penetration
Dp = δs /2 is the depth where the plane-wave energy
drops to 1/e of its value on the surface. In metals, where
the conductivity is large, the skin depth reduces to
For a definition of gyrotropic media see [4]. The offdiagonal elements are due to gyrotropic behavior in an
applied field.
Electric and magnetic fields are attenuated as they
travel through lossy materials. Using time-harmonic
signals the loss can be studied at specific frequencies,
where the time dependence is e iωt. The change in loss
with frequency is related to dispersion.
The propagation coefficient of a plane wave is
γ = α + i β = ik = iω εμ . The plane-wave attenuation
coefficient in an infinitely thick half space, where the
guided wavelength of the applied field is much longer
than the size of the molecules or inclusions, is denoted
by the quantity α and the phase is denoted by β. Due to
losses of a plane wave, the wave amplitude decays as
|E| ∝ exp(– αz). The power in a plane wave of the form
E(z, t) = E0 exp (– αz) exp (iωt – iβz), attenuates as
P ∝ exp (– 2αz). For waves in a guided structure:
δs =
π f μ0 μ'r σ dc
where σdc is the dc conductivity and f is the frequency.
We see that the frequency, conductivity, and permeability of the material determine the skin depth in
The phase coefficient β for a plane wave is given by
β =±
ε'r μ'r (((tan δ d tan δ m −1) 2 +(tan δd +tan δm) 2) 1/2
c 2
− (tan δ d tan δ m −1))1/2 .
In dielectric media, β reduces to
γ = i k 2 − kc2 , where kc = ωc / c = 2π / λc is the cutoff
β =±
wavenumber, and speed of light c . Below cutoff, the
ε'r μ'r
c 2
1 + tan 2 δ d +1 .
propagation coefficient becomes γ = kc2 − k 2. α of a
plane wave is given by
c 2
The imaginary part of the propagation coefficient
defines the phase of an EM wave and is related to the
ε'r μ'r (((tan δ d tan δm −1) 2 +(tan δd +tan δm) 2) 1/2
+ (tan δ d tan δ m −1))
refractive index by n = ± ε 'r μ 'r . In normal dielectrics
the positive square root is taken in Eq. (8). Veselago
[26] developed a theory of negative-index materials
(NIM) where he used negative intrinsic
and has units of Np/m. α is approximated for dielectric
materials as
Volume 117 (2012)
Journal of Research of the National Institute of Standards and Technology
ε′r and μ ′r , and the negative square root in Eq. (8) is
used. There is controversy over the interpretation of
metamaterial NIM electrical behavior since the permeability and permittivity are commonly effective values.
We will use the term NIM to describe materials that
achieve negative effective permittivity and permeability over a band of frequencies.
is defined from
the Fourier
of Maxwell’s
+ σdc E (ω) ≡
iωεt o t E (ω), so that
ε to t = ε'rε 0 − i (ε'r'ε 0 +
σ dc
In plots of RF measurements, the decibel scale is
often used to report power or voltage measurements.
The decibel (dB) is a relative unit and for power is
calculated by 10 log10 (Pout / Pin). Voltages in decibels
are defined as 20 log10 (Vout / Vin). α has units of
Np/m. The attenuation can be converted from
1 Np/m = 8.686 dB/m. dBm is similar to dB, but relative to power in milliwatts 10 log(P/mW).
The wave impedance for a transverse electric
and magnetic mode (TEM) is μ / ε ; for a transverse
electric mode (TE) is iωμ / γ , and for a transverse
magnetic mode (TM) is γ / iωε. The propagating plane
wave wavelength in a material is decreased by a permittivity greater than that of vacuum; for example, for a
TEM mode, λ ≈ cvac /( ε'r μ'r f ). In waveguides the guided wavelength λg depends on the cutoff wavelength λc
and is given by λg =1/ ε' μ'f 2/ c 2 −1/ λc2 = λ / 1 −( λ / λc ) 2.
In the time domain the internal field energy U satisfies: ∂U/∂t = ∂D/∂t · E + ∂B/∂t · H. Using Maxwell’s
equations with a current density J, then produces
Poynting’s Theorem: ∂U/∂t + ∇ · (E × H) = – J · E,
where the time-domain Poynting vector is S(r, t) =
E(r, t) × H(r, t). The complex power flux (W/m2)
is summarized
the complex Poynting vector Sc(ω) =
=(1/2)(E (ω) × H (ω)). The real part of Sc represents
dissipation and is the time average over a complete
cycle. The imaginary part of Sc relates to the reactive
stored energy.
The surface impedance in ohms/square of a conducting material is Zm = (1 + i )σδs . The surface resistance
for highly lossy materials is
Rs =
π f μ0 μ'r
δ sσ dc
σ dc
When the conductors on a substrate are very thin, the
fields can penetrate through the conductors into the
substrate. This increases the resistance of a propagating
field because it is in both the metal and the dielectric.
As a consequence of the skin depth, the internal inductance in a highly-conducting material decreases with
increasing frequency, whereas the surface resistance Rs
increases with frequency in proportion to √f .
Any transmission line will have propagation delay
that relates to the propagation speed in the line. This is
related to the dielectric permittivity and the geometry
of the transmission line. Propagation loss is due to
conductor and material loss.
Some materials exhibit ionic conductivity, so that
when a static electric field is applied, a current is
induced. This behavior is modeled by the dc conductivity σdc , which produces a low-frequency loss (∝ 1/ω) in
addition to polarization loss (ε″r ). In some materials,
such as semiconductors and disordered solids, the
conductivity is complex and depends on frequency.
This is because the free charge is partially bound and
moves by tunneling through potential wells or hops
from well to well.
The total permittivity for linear, isotropic materials
that includes both dielectric loss and dc conductivity
Electromagnetic Power
Quality Factor
The band width of a resonance is usually modeled by
the quality factor (Q) in terms of the decay of the
internal energy. The combined internal energy in a
mechanical system is the kinetic plus the potential
energy; in an electromagnetic system it is the field
stored energy plus the potential energy. In the time
domain the quality factor is related to the decay of the
internal energy for an unforced resonator as as [27]
dU (t )
= − 0 U (t ) .
The EM field is modeled by a damped harmonic
oscillator at frequencies around the lossless resonant
frequency ω0 and frequency pulling factor (the resonant
frequency decreases from ω0 due to material losses),
Δω as [27]
E (t ) = E0 e−ω0t / 2Q0 ei (ω0 +Δω )t .
Volume 117 (2012)
Journal of Research of the National Institute of Standards and Technology
Taking a Fourier transform of Eq. (13), the absolute
value squared becomes
E (ω) =
(ω − ω0 − Δω) 2 + (
ω 2
We + Wh
Maxwell’s Equations in Materials
Maxwell’s Equations From Microscopic to
Macroscopic Scales
Maxwell’s microscopic equations in a media with
charged particles are written in terms of the microscopic fields b, e and sources j, and ρm as
and therefore |E(ω)|2, which is proportional to the
power, is a Lorentzian. This linear model is not exact
for dispersive materials, because Q0 may be dependent
on frequency. The quality factor is calculated from the
frequency at resonance f0 as Q0 = f0 / 2(| f 0 – f 3dB|), or
from a fit of a circle when plotting S11(ω) on the Smith
chart. The quality factor is calculated from Q0 = f 0/Δf ,
where Δf is the frequency difference between 3 dB
points on the S21 curve [28]. For resonant cavity measurements, the permittivity or permeability is determined from measurements of the resonance frequency
and quality factor, as shown in Fig. 2. For timeharmonic fields the Q is related to the stored field
energies We ,Wh , the angular frequency at resonance ωr ,
and the power dissipated Pd at the resonant frequency:
Q = ωr
+ μ 0 j,
ε 0 ∇ ⋅ e = ρm ,
∇ × b = ε 0 μ0
∇×e = −
∇ ⋅ b = 0.
Note, that at this level of description the macroscopic
magnetic field H and the macroscopic displacement
field D are not defined, but can be formed by averaging
dielectric and magnetic moments and expanding the
microscopic charge density in a Taylor series. In
performing the averaging process, the material length
scales allow the dipole moments in the media to be
approximated by continuously varying functions P and
M. Once the averaging is completed, the macroscopic
Maxwell’s equations are (see Sec. 4.6) to obtain
[27, 29, 30]
Resonant frequencies can be measured with high
precision in high-Q systems; however the parasitic
coupling of the fields to fixtures or materials needs to
be modeled in order to make the result meaningful.
Material measurements using resonances have much
higher precision than using nonresonant transmission
The term antiresonance is used when the reactive
part of the impedance of a EM system is very high. This
is in contrast to resonance, where the reactance goes to
zero. In a circuit consisting of a capacitor and inductance in parallel, antiresonance occurs when the voltage
and current are in phase.
∇×H =
+J ,
∇×E = −
∇⋅D = ρ ,
∇⋅B = 0 .
J denotes the current density due to free charge and
source currents. Because there are more unknowns than
equations, constitutive relations for H and D are needed. Even though B and E are the most fundamental
fields, D usually is expressed in terms of E, and B is
usually expressed in terms of H.
Fig. 2. Measuring resonant frequency and Q.
Volume 117 (2012)
Journal of Research of the National Institute of Standards and Technology
Constitutive Relations
fields. For chiral and magneto-electric materials, Eqs.
(24) and (25) must be modified to accommodate crosscoupling behavior between magnetic and dielectric
response. General, linear relations defining polarization
in non-magnetoelectric and non-chiral dielectric and
magnetic materials in terms of the impulse-response
dyadics are given by Eqs. (1) and (2). Using the
Laplace transform L, gives
Linear Constitutive Relations
Since there are more unknowns than macroscopic
Maxwell’s equations, we must specify the constitutive
relationships between the polarization, magnetization,
and current density as functions of the macroscopic
electric and magnetic fields [31, 32]. In order to satisfy
the requirements of linear superposition, any linear
polarization relation must be time invariant, further,
this must also be a causal relationship as given in
Eqs. (1) and (2).
The fields and material-related quantities in
Maxwell’s equations must satisfy underlying symmetries. For example, the dielectric polarization and
electric fields are odd under parity transformations and
even under time-reversal transformations. The magnetization and induction fields are even under parity
transformation and odd under time reversal. These
symmetry relationships place constraints on the nature
of the allowed constitutive relationships and requires
the constitutive relations to manifest related symmetries [29, 33-39]. The evolution equations for the
constitutive relationships need to be causal, and in
linear approximations must satisfy time-invariance
properties. For example, the linear-superposition
requirement is not satisfied if the relaxation time in
Eq. (4) depends on time. This can be remedied by using
an integrodifferential equation with restoring and
driving terms [40, 41].
The macroscopic displacement and induction fields
D and B are related to the macroscopic electric field E
and magnetic fields H, as well as M and P, by
(ω) = χ ( ω) E
( ω) ,
χ er (ω) =
f p (τ ) e−iωt dt = χ'er ( ω) − i χ''er ( ω). (28)
So the real part is the even function of frequency given by
χ'er (ω) =
f p (t) cos( ωt) dt,
and the imaginary part is an odd function of frequency
χ"er (ω) =
f p ( t) sin( ω t) dt,
and therefore
ε r (ω) = I + χ'er ( ω) − i χ''er ( ω);
∞
χ er′ (0) =
f p (t ) dt,
χ er′′ (0) = 0 .
D = ε 0 E + Pd − ∇ ⋅ Q + ≡ ε 0E + P ,
The time-evolution constitutive relations for dielectric materials are generally summarized by generalized
harmonic oscillator equations or Debye-like equations
as overviewed in Sec. 5.2.
B = μ0 H + μ 0 M .
J = J (E, H) ,
Through the methods of nonequilibrium quantumbased statistical-mechanics it is possible to show that
the constitutive relation for the magnetization in ferromagnetic materials is an evolution equation given by
In addition,
Generalized Constitutive Relations
∂M (r, t )
= − γ g M (r, t ) × H eff ( r, t)
t
− d 3 r ′ K m (r, t, r ′, τ ) ⋅ χ 0 H eff ( r ′, τ ) dτ ,
where J is ↔a function of the electric and magnetic
fields, and Q is the macroscopic quadrupole moment
density. Pd is the dipolemoment density, whereas P is
the effective macroscopic polarization that also
includes the effects of the macroscopic quadrupolemoment density [27, 29, 30, 32, 42]. The polarization
and magnetization for time-domain linear response are
expressed as convolutions in terms of the macroscopic
where K m is a kernel that contains of the microstructural interactions given in [43], γg is the gyromagnetic ratio, χ0 is the static susceptibility, and Heff
Volume 117 (2012)
Journal of Research of the National Institute of Standards and Technology
is the effective magnetic field. Special cases of Eq. (34)
reduce to constitutive relations such as the LandauLifshitz, Gilbert, and Bloch equations. The LandauLifshitz equation of motion is useful for ferromagnetic
and ferrite solid materials:
γ g M (r, t ) × Heff ( r, t)
∂M (r, t )
α γg
where α is a damping constant. Another special case of
Eq. (34) reduces to the Gilbert equation
∂M (r, t )
∂M (r, t )
M (r , t ) ×
The Time-Harmonic Field Approximation
Material Response to Applied Fields
When a field is suddenly applied to a material, the
charges, spins, currents, and dipoles in a medium
respond to the local fields to form an average field. If
an EM field is suddenly applied to a semi-infinite
material, the total field will include the effects of both
the applied field, transients, and the particle backreaction fields from charge, spin, and current rearrangement that causes depolarization fields. This will cause
the system to be in nonequilibrium for a period of time.
For example, as shown in Fig. 3, when an applied EM
field interacts with a dielectric material, the dipoles
reorient and charge moves, so that the macroscopic and
local fields in the material are modified by surface
charge dipole depolarization fields that oppose the
applied field. The macroscopic field is approximately
the applied field minus the depolarization field.
Depolarization, demagnetization, thermal expansion,
exchange, nonequilibrium, and anisotropy interactions
can influence the dipole orientations and therefore the
fields and the internal energy. In modeling the constitutive relations in Maxwell’s equations, we must express
the material properties in terms of the macroscopic
field, not the applied or local fields, and therefore we
need to make clear distinctions between the interaction
processes [40].
Materials can be studied by the response of frequency-domain or time-domain fields. When considering
time-domain pulses rather than time-harmonic fields,
where χ
b has only the diagonal elements χb (11) = 1/T2 ,
χb (22) = 1/T2 , χb (33) = 1/T1 , and Ms = Ms z→. An equation
analogous to (34) can be written for the electrical
polarization [46] as [43]
t
∂P(r, t )
= − d 3 r ′ K e (r, t, r ′, τ )
× (P(r ′,τ ) − χ 0 ⋅ E( r′, τ ) dτ ) .
In electron-spin resonance (EPR) and nuclear
magnetic resonance (NMR) measurements, the Bloch
equations with characteristic relaxation times T1 and T2
are used to model relaxation. T1 relates to spin-lattice
relaxation as the paramagnetic material interacts with
the lattice. T2 relates to spin-spin interactions:
∂M (r, t )
≈ − γ g M (r, t ) × H(r, t)
− χ b ⋅ M (r , t ) + s ,
Electromagnetic Fields in Materials
Time-harmonic fields are very useful for solving the
linear Maxwell’s equations when transients are not
important. In the time harmonic field approximation,
the field is assumed to be present without beginning or
end. Periodic signals over − ∞ < t < ∞ are nonphysical since all fields have a beginning where transients
are generated, but are very useful in probing material
Solutions of Maxwell’s equations that include
transients are most easily obtained with the Laplace
transform. Note that the Laplace or Fourier transformed
fields do not have the same units as the time-harmonic
fields due to integration over time. In Eq. (1), causality
is incorporated into the convolution relation for linear
response. D(t) depends only on E(t) at earlier times and
not future times.
M (r, t ) × ( M( r, t) × Heff ( r, t))
The Debye relaxation differential
equation is
t , r′, τ) =
I δ(t – τ)δ(r, – r′)/τe .
Volume 117 (2012)
Journal of Research of the National Institute of Standards and Technology
scopic variables are directly related to the spatial and
temporal detail incorporated in the constitutive material
parameters. Constitutive relations can be exact as in
[40] and Eqs. (34) and (38), but usually, to be useful,
are approximate.
Plane waves are a useful approximation in many
applications. Time-harmonic EM plane waves in materials can be treated either as traveling without attenuation, propagating with attenuation, or evanescent. Plane
waves may propagate in the form of a propagating wave e i(ωt – βz), or a damped propagating wave
e i(ωt – βz) – αz, or an evanescent wave e iωt – αz. Evanescent
fields are exponentially damped waves. In a waveguide, this occurs for frequencies below any transverse
resonance frequencies [24, 48], when k2 – k 2c < 0, where
kc is the cutoff wave number calculated from the
Fig. 3. Fields in materials.
this interaction is more complex. The use of timedomain pulses have the advantage of sampling a
reflected pulse as a function of time, which allows a
determination of the spatial location of the various
Time-harmonic fields are often used to study material properties. These have a specific frequency from
time minus infinity to plus infinity, without transients;
that is, fields with a e iωt time dependence. As a consequence, in the frequency domain, materials can be
studied through the reaction to periodic signals. The
measured response relates to how the dipoles and
charge respond to the time-harmonic signal at each
frequency. If the frequency information is broad
enough, a Fourier transform can be used to study the
corresponding time-domain signal.
The relationships between the applied, macroscopic,
local, and the microscopic fields are important for
constitutive modeling (Fig. 3). The applied field
originates from external charges, whereas the macroscopic fields are averaged quantities in the medium.
The displacement and inductive (or magnetic) macroscopic fields in Maxwell’s equations are implicitly
defined through the constitutive relationships and
boundary conditions. The local field is the averaged
EM field at a particle site due to both the applied field
and the fields from all of the other sources, such as
dipoles, currents, charge, and spin [47]. The microscopic field represents the atomic-level EM field,
where particles interact with the field from discrete
charges. Particles interact with the local EM field that
is formed from the applied field and the microscopic
field. At the next level of homogenization, groups
of particles interact with the macroscopic field. The
spatial and temporal resolution contained in the macro-
transverse geometry and k = ω εμ = ( ω / c) ε 'r μ 'r .
Evanescent and near field EM fields occur at apertures
and in the vicinity of antennas. Evanescent fields can be
detected when they are perturbed and converted into
propagating waves or transformed by dielectric loss.
Electromagnetic waves may convert from near field to
propagating. For example, in coupling to dielectric
resonators the near field at the coupling loops produce
propagating or standing waves in a cavity or dielectric
resonator. Evanescent and near fields in dielectric
measurements are very important. These fields do not
propagate and are used in near-field microwave probes
to measure or image materials at dimensions much less
than λ/2 [49, 50] (see Fig. 17). The term near field usually refers to the waves close to an waveguide, antenna,
or probe and is not necessarily an exponentially
damped plane wave. In near-field problems the goal is
to model the reactive region. Near fields in the reactive
region, (L < λ/2π), contain stored energy and there is
no net energy transport over a cycle unless there are
losses in the medium. By analogy, the far field relates
to radiation. These remove energy from the transmitter
whether they are immediately absorbed or not. There is
a transition region called the radiative near field.
Because electrical measurements can now be performed at very small spatial resolutions, and the
elements of electrical circuits are approaching the
molecular level, we require good models of the macroscopic and local fields. This is particularly important,
because we know that the Lorentz theory of the local
field is not always adequate for predicting polarizabilities [51, 52]. Also, when solving Maxwell’s equations
at the molecular level, definitions of the macroscopic
Volume 117 (2012)
Journal of Research of the National Institute of Standards and Technology
field and constituative relationships are important. A
theoretical analysis of the local EM field is important
in dielectric modeling of single-molecule measurements and thin films. The effective EM fields at this
level are local, but not atomic-scale, fields.
The formation of the local field is a very complex
process whereby the applied electric field polarizes
dipoles in a molecules or lattices and the applied
magnetic field causes current and precession of spins.
Then, the molecule’s dipole field modifies the dipole
orientations of other molecules in close proximity,
which then reacts back to produce a correction to the
molecule’s field in the given region. This process gets
more complicated for behavior that depends on time.
We define the local EM field as the effective, averaged
field at a specific point in a material, not including the
field of the particle itself. This field is a function of
both the applied and multipole fields in the media. The
local field is related to the average macroscopic and
microscopic EM fields in that it is a sum of the macroscopic field and the effects of the near-field. In ferroelectric materials, the local electric field can become
very large and hence there is a need for comprehensive
local field models. In the literature on dielectric materials, a number of specific fields have been introduced to
analyze polarization phenomena. The electric field
acting on a nonpolar dielectric is commonly called the
internal field, whereas the field acting on a permanent
dipole moment is called the directing field. The difference between the internal field and directing fields is
the average reaction field. The reaction field is the
result of a dipole polarizing its environment [53].
Nearly exact classical theories have been developed
for the static local field. Mandel and Mazur developed
a static theory for the local field in terms of the polarization response of a many-body system by use of
the T-matrix formalism [54]. Gubernatis extended the
T-matrix formalism [55]. However, the T-matrix contributions are difficult to calculate. Keller’s review article
[56] on the local field uses an EM propagator approach.
Kubo’s linear-response theory and other theories have
also been used for EM correlation studies [40, 53, 57].
If the applied field has a wavelength that is not much
longer than the typical particle size in a material, an
effective permittivity and permeability is commonly
assigned. The terms effective permittivity and permeability are commonly used in the literature for studies
of composite media. The assumption is that the properties are “effective” if in some sense they do not adhere
to the definitions of the intrinsic material properties. An
effective permittivity is obtained by taking a ratio of
some averaged displacement field to an averaged
electric field. The effective permeability is obtained by
taking a ratio of some averaged induction field to an
averaged magnetic field. This approach is commonly
used in modeling negative-index material properties
when scatterers are designed in such a manner such that
the scatterers themselves resonate. In these situations
the wavelength may approach the dimensions of the
Macroscopic and Local Electromagnetic Fields
in Materials
The mesoscopic description of the EM fields in a
material is complicated. As a field is applied to a
material, charges reorient to form new fields that
oppose the applied field. In addition, a dipole tends to
polarize its immediate environment, which modifies
the field the dipole experiences. The field that polarizes
a molecule is the local field El and the induced dipole
moment is p = α
El , where α
is the polarizability. In
order to use this expression in Maxwell’s equations, the
local field needs to be expressed in terms of the macroscopic field. Calculation of this relationship is not
always simple.
To first approximation, the macroscopic field is
related to the external or applied field (Ea ), and the
depolarization field by
E = Ea −
The local field is composed of the macroscopic field
and a material-related field. In the literature, the effective local field is commonly modeled by the Lorentz
field, which is defined as the field in a small cavity that
is carved out of a material around a specific site, but
excludes the field of the observation dipole. A wellknown example of the relationship between the
applied, macroscopic, and local fields is given by an
analysis of the Lorentz spherical cavity in a static
electric field. For a Lorentz sphere the local field is the
sum of applied, depolarization, Lorentz, and atomic
fields [4, 56, 58]:
Et = Ea + Edepol + ELorentz + Eatom .
For cubic lattices in a spherical cavity, the Lorentz local
field is related to the macroscopic field and polarization
El = E +
Volume 117 (2012)
Journal of Research of the National Institute of Standards and Technology
In the case of a sphere, the local field in Eq. (39) equals
the applied field.
For induced dipoles,
P = Nα El ,
relaxation are related to time dependent polarization
correlation functions. The polarization P(t) is related to
the response dyadic φ (t) and the driving field E(t) by
[53, 60]
P (t ) =
where N is the density of dipoles, and Eq. (41) yields
El = E/(1 – Nα/3ε0 ) = P/Nα.
Onsager [53] generalized the Lorentz theory by
distinguishing between the internal field that acts on
induced dipoles and the directing field that acts on
permanent dipoles. If we use P = ε0 (εr – 1)E in
Eq. (41), we find El = ((εr + 2)/3)E. Therefore, for
normal materials the Lorentz field exceeds the macroscopic field. For a material where the permittivity is
negative we can have El ≤ E. In principle, we can null
out the Lorentz field when εr = – 2. Some of the essential problems encountered in microscopic constitutive
theory center around the local field. Note that for some
materials, recent research indicates that the Lorentz
local field does not always lead to the correct polarizabilities [51]. We expect the Lorentz local field expression to break down near interfaces. For nanoparticles, a
more complicated theory needs to be used for the local
A rigorous expression for the static local field created by a group of induced dipoles can be obtained by an
iterative procedure [53, 59] using pi = αiEl (ri) and
Eij ( rj ) ,
φ( t − τ) ⋅ E( τ) d τ,
where φ (t – τ) = 0 for t – τ < 0. The susceptibility is
defined as
χ(ω) =
φ (τ)e−iωτ d τ = χ' (ω) − i χ'' (ω) ,
where the response in volume V is related to the correlation function for stationary processes in terms of the
microscopic polarization
d < p(0)p(τ) >0
φ(τ) = −V
kB T
and therefore for microscopic polarizations
χ(ω) =
e −iωτ φ(τ)d τ =
Vω ⎡
k BT ⎢⎣
Ei (r j ) = Ea +
d < p (0)p(τ) >0
kB T
< p(0)p( τ) >0 sin( ωτ) d τ −
< p(0)p( τ) >0 cos( ωτ) d τ⎥ .
i =l ,i ≠ j
Eij (r j ) ≈
p(ri ) ⎥
1 ⎢ 3(r j − ri )p(ri )
4πε0 ⎢ r − r 5
i ⎦
Once the correlation functions are determined then
the susceptibility can be found. An approach that
models relaxation beyond linear response is given in
[40, 43, 44, 61]. The method of linear response has
exceeded expectations and has been a cornerstone of
statistical mechanics.
If there are also permanent dipoles, they need to be
included as p(ri) = pperm(ri) + αiEl (ri ).
Averaging to Obtain Macroscopic Field
If we consider modeling of EM wave propagation
from macroscopic through molecular and sub-molecular to atomic scales, the effective response at each level
is related to different degrees of homogenization. At
wavelengths short relative to particle size the EM propagation is dominated by scattering, whereas at long
wavelengths it is dominated by traveling waves. In
microelectrodynamics, there have been many types of
ensemble and volumetric averaging methods used to
Overview of Linear-Response Theory
Models of relaxation that are based on statistical
mechanics can be developed from linear-response
theory. Linear-response theory uses an approximate
solution of Liouville’s equation and a Hamiltonian that
contains a time-dependent relationship of the field
parameters based on a perturbation expansion. This
approach shows how the response functions and
Volume 117 (2012)
Journal of Research of the National Institute of Standards and Technology
define the macroscopic fields obtained from the microscopic fields [27, 29, 30, 40, 54]. For example, in the
most commonly used theory of microelectromagnetics,
materials are averaged at a molecular level to produce
effective molecular dipole moments. The microscopic
EM theories developed by Jackson, Mazur, and
Robinson [27, 29, 30] average multipoles at a molecular level and replace the molecular multipoles, with
averaged point multipoles usually located at the centerof-mass position. This approach works well down to
near molecular level, but breaks down below the
molecular to submolecular level.
In the various approaches, the homogenization of the
fields are formed in different ways. The averaging is
always volumetric rather than a time average. Jackson
uses a truncated averaging test function to proceed
from microscale to the macroscale fields [27]. Robinson
and Mazur use ensemble averaging [29, 30] and statistical mechanics. Ensemble averaging assumes there is a
distribution of states. In the volumetric averaging
approach, the averaging function is not explicitly determined, but the function is assumed to be such that the
averaged quantities vary in a manner smooth enough to
allow a Taylor-series expansion to be performed. In the
approach of Mazur, Robinson, and Jackson [27, 29, 30]
the charge density is expanded in a Taylor series and
the multipole moments are identified as in Eq. (49).
The microscopic charge density can be related to the
macroscopic charge density, polarization, and quadrupole density by a Taylor-series expansion [27]
< ρmicro (r , t ) >≈ ρmacro ( r, t)−∇ ⋅ P( r, t)−∇ ⋅ (∇ ⋅ Q)( r, t),
In NIM materials, effective properties are obtained
by use of electric and magnetic resonances of embedded structures that produce negative effective ε′ef f [62].
In Sec. 4.6 the issue of whether this response can be
summarized in terms of material parameters is discussed. Defining permittivity and permeability on these
scales of periodic media can be confusing. The field
averaging used in NIM analysis is based on a unit cell
consisting of split-ring resonators, wires, and ferrite or
dielectric spheres [62, 63].
In order to obtain a negative effective permeability in
NIM applications, researchers have used circuits that
are resonant, which can be achieved by the introduction
of a capacitance into an inductive system. Pendry et al.
[63-65] obtained the required capacitance through gaps
in split-ring resonators. The details of the calculation of
effective permeability are discussed in Reference [63].
Many passive and/or active microwave resonant
devices can be used as sources of effective permeability in the periodic structure designed for NIM
applications [66]. We should note that the composite
materials used in NIM are usually anisotropic. Also, the
use of resonances in NIM applications produce effective material parameters that are spatially varying and
frequency dispersive.
The goal of this section is to study the electrical
permittivity and permeability in materials starting from
microscopic concepts and then progressing to macroscopic concepts. We will study the limitations of the
concept of permittivity in describing material behavior
when wavelengths of the applied field approach the
dimensions of the spaces between inclusions or
inclusion sizes. When high-frequency fields are used in
the measurement of composite and artificial structures,
these length-scale constraints are important. We will
also examine alternative quantities, such as dipole
moment and polarizability, that characterize dielectric
and magnetic interactions of molecules, atoms, and that
are still valid even when the concepts of permittivity
and permeability are fuzzy.
The concepts of polarizability and dipole moment
p in p = αEl are valid down to the atomic and molecular levels. Permittivity and permeability are frequencydomain concepts that result from the microscopic timeharmonic form of Maxwell’s equations averaged over a
unit cell. They are also related to the Fourier transform
of the impulse-response function. The most common
where Q (r, t) is the quadrupole tensor. In this interpretation, the concepts of P and ρmacro are valid at length
scales where a Taylor-series expansion is valid. These
moments are calculated about each molecular center of
mass and are treated as point multipoles. However, this
type of molecular averaging limits the scales of the
theory to larger than the molecular level and limits the
modeling of induced-dipole molecular moments [40].
Usually, the averaging approach uses a test function fa
and microscopic field e given by
E = dr' e(r − r' ) fa ( r') .
Averaging to Obtain Permittivity and
Permeability in Materials
However, the distribution function is seldom explicitly needed or determined in the analysis. The macroscopic magnetic polarization is found through an analogous expansion of the microscopic current density.
Volume 117 (2012)
Journal of Research of the National Institute of Standards and Technology
way ↔to define ↔
ε is through the impulse-response function f p (t).
Statistical mechanics yields an expression for the
impulse-response function in terms of correlation functions of the microscopic polarizations p. For linear
response [53]
dt k BT
< p eiL0 (t −τ ) p >0 ⋅E p ( τ) d τ ,
When the ratio of the dipole length scale to wavelength is not very small, the Taylor’s series expansion
is not valid and the homogenization procedure breaks
down. When this criteria is not satisfied for metafilms,
some researchers use generalized sheet transition conditions (GSTC’s) [67-70] at the material boundaries;
however, the concept of permittivity for these structures, at these frequencies, is still in question and is
commonly assigned an effective value. Drude and
others [67, 68] compensated for this by introducing
boundary layers. In such cases, it is not clear whether
mapping complicated field behavior onto effective
permittivity and permeability is useful, since at these
scales, the results can just as well be thought of as
scattering behavior.
When modeling the permittivity or permeability in a
macroscopic medium in a cavity or transmission line,
the artifacts of the measurement fixture must be
separated from the material properties by solving a
relevant macroscopic boundary-value problem. At
microwave and millimeter frequencies a low-loss
macroscopic material can be made to resonate as a
dielectric resonator. In such cases, if the appropriate
boundary-value problem is solved, the intrinsic permittivity and permeability of the material can be extracted
because the wavelengths are larger than the constituent
molecule sizes, and as a result, the polarization vector
is well defined. However, many modern applications
are based on artificial structures that produce an EM
response where the wavelength in the material is only
slightly larger than the feature or inclusion size. In such
cases, mapping the EM response onto a permittivity
and permeability must be scrutinized. In general, the
permittivity is well defined in materials where wave
propagation through the material is not dominated by
multiple scattering events.
where V is the volume, L0 is Liouville’s operator,
p denotes iL0p, and < >0 denotes averaging over phase.
From this equation,
we can identify the impulse↔
response dyadic f p from P(t) = V ∫ 0< p (t)p(τ) >0 ↔
· E(τ)
dτ / kBT , and for a stationary system, f p(t) =
V < p (0)p(– t) >0 / kBT [53].
Ensemble and volumetric averaging methods are
used to obtain the macroscopic fields from the microscopic fields (see Jackson [27] and the references
therein). For example, in the most commonly used
theory, materials are averaged at a molecular level to
produce effective molecular dipole moments. When
deriving the macroscopic Maxwell’s equations from
the microscopic equations, the electric and magnetic
multipoles within a molecule are replaced with averaged point multipoles usually located at the molecular
center-of-mass positions. Then these effective moments
are assumed to form a continuum, which then forms the
basis of the macroscopic polarizations. The procedure
assumes that the wavelength in the material is much
larger than the individual particle sizes. As Jackson
[27] notes, the macroscopic Maxwell’s equations can
model refraction and reflection of visible light, but are
not as useful for modeling x-ray diffraction. He states
that the length scale L0 of 10 nanometers is effectively
the lower limit for the validity of the macroscopic
equations. Of course, this limit can be decreased with
improved constitutive relationships.
For macroscopic heterogeneous materials the wavelengths of the applied fields must be much longer
than individual particle or molecule dimensions that
constitute the material. When this criterion does not
hold, then the spatial derivative in the macroscopic
Maxwell’s equations, for example, (∇ × H), and the
displacement field loses its meaning. Associated with
this homogenization process at a given frequency is the
number of molecules or inclusions that are required to
define a displacement field and thereby the related
Overview of the Dielectric Response to
Applied Fields
Modeling Dielectric Response Upon
Application of an External Field
Dielectric parameters play a critical role in many
technological areas. These areas include electronics,
microelectronics, remote sensing, radiometry, dielectric
heating, and EM-assisted chemistry [20]. At RF
frequencies dielectrics exhibit behavior that metals
cannot achieve because dielectrics allow field penetration and can have low-to-medium loss characteristics.
Volume 117 (2012)
Journal of Research of the National Institute of Standards and Technology
Using dielectric spectroscopy as functions of both
frequency and temperature we can obtain some, but not
all of the information on a material’s molecular or
lattice structure. For example, measurements of the
polarization and conductivity indicate the polarizability
and free charge of a material and polymer mobility of
side chains can be studied with dielectric spectroscopy.
Also, when a polymer approaches a glass transition
temperature the relaxation times change abruptly.
This is observable with dielectric spectroscopy. In
addition, the loss peaks of many liquids change with
When an EM field is applied to a material, the atoms,
molecules, free charge, and defects adjust positions. If
the applied field is static, then the system will eventually reach an equilibrium state. However, if the applied
field is time dependent then the material will continuously relax in the applied field, but with a time lag. The
time lag is due to screening, coupling, friction, and
inertia. An abundance of processes are occurring during
relaxation, such as heat conversion processes, latticephonon, and photon phonon coupling. Dielectric relaxation can be a result of dipolar and induced polarization, lattice-phonon interactions, defect diffusion,
higher multipole interactions, or the motion of free
charges. Time-dependent fields produce nonequilibrium behavior in the materials due both to the heat
generated in the process and the constant response to
the applied field. However, for linear materials and
time-harmonic fields, when the response is averaged
over a cycle, if heating is appreciable, nonequilibrium
effects such as entropy production relate more to
temperature effects than the driving field stimulus. The
dynamic readjustment of the molecules in response
to the field is called relaxation and is distinct from
resonance. For example, if a dc electric field is applied
to a polarizable dielectric and then the field is suddenly turned off, then the dipoles will relax over a characteristic relaxation time into a more random state.
The response of materials depends strongly on
material composition and lattice structure. In many
solids, such as solid polyethylene, the molecules are not
able to appreciably rotate or polarize in response to
applied fields, indicating a low permittivity and small
dispersion. The degree of crystallinity, existence of
permanent dipoles, dipole-constraining forces, mobility
of free charge, and defects all contribute to dielectric
response. Typical responses for high-loss and low-loss
dielectrics are shown in Figs. 4, 5, and 6.
Fig. 4. Broadband permittivity variation for materials [71].
Fig. 5. Typical frequency dependence of ε′r of low-loss fused silica
as measured by many methods.
Fig. 6. Typical frequency dependence of the loss tangent in low-loss
materials such as fused silica.
Volume 117 (2012)
Journal of Research of the National Institute of Standards and Technology
A material does not respond instantaneously to an
applied field. As shown in Fig. 4, the real part of the
permittivity is a monotonically decreasing function of
frequency in the relaxation part of the spectrum, far
away from intrinsic resonances. At low frequencies, the
dipoles generally follow the field, but thermal agitation
also tends to randomize the dipoles. As the frequency
increases to the MMW band, the response to the driving field generally becomes more incoherent. At higher
frequencies, in the terahertz or infrared spectrum, the
dipoles may resonate, and therefore the permittivity
rises until it becomes out of phase with the field and
then drops. At RF frequencies, materials with low loss
respond differently from materials with high loss
(compare Fig. 4 for a high-loss material versus a lowloss material in Figs. 5 and 6). For some materials, at
frequencies at the low to middle part of the THz band,
ε′r may start to contain some of the effects of resonances
that occur at higher frequencies, and may start to
slowly increase with frequency, until resonance, and
then decreases again.
The local and applied fields in a dielectric are
usually not the same. As the applied field interacts with
a material it is modified by the fields of the molecules
in the substance. Due to screening, the local electric
field differs from the applied field and therefore
theories of relaxation must model the local field (see
Sec. 4.3).
Over the years, many models of polar and nonpolarmaterials have been developed that use different
approximations to the local field. The ClausiusMossotti equation was developed for noninteracting,
nonpolar molecules governed by the Lorentz equation
for the internal field. This equation works well for nonpolar gases and liquids. Debye introduced a generalization of the Clausius-Mossotti equation for the case of
polar molecules. Onsager developed an extension of
Debye’s theory by including the reaction field and a
more comprehensive local field expression [53]. For a
dielectric composed of permanent dipoles, the polarization is written in terms of the local field as Eq. (42)
There are electronic, ionic, and permanent dipole
polarizability contributions, so that μ→d = (αel + αion +
αperm )El , αel = 4πε0R3 / 3, αion = e2 / Yd0 . Here, Y is
Young’s modulus, R is the radius of the ions, d0 is
the equilibrium separation of the ions, and αperm =
|μ→e |2 / 3kBT, where μ→e is the permanent dipole moment.
There may also be a contribution to the polarizability
due to excess charge at microscopic interfaces. Using
the Lorentz expression for the local field, the polarization can be written as
P = Nα ⎜ E +
Nα E
= ε 0 (ε r −1) E .
1 − Nα / 3ε 0
This is the Clausius-Mossotti relation that is commonly used to estimate the permittivity of nonpolar
materials from atomic polarizabilities:
εr − 1 Nα
εr + 2 3ε0
εr =
3ε0 + 2 Nα
3ε0 − Nα
The Clausius-Mossotti relation relates the permittivity
to the polarizability. The polarizability is related to the
vector dipole moment μ→d of a molecule or atom and
the local field El , μ→d = αEl . In principle, once the
polarizability is determined for a group of molecules,
then the permittivity of the ensemble can be calculated
with the implicit assumption that there are many molecules located over the distance of a wavelength. Typical
polarizabilities of atoms are between 0.1 and 100 Fm2
[72]. Polarizabilities of molecules can be higher than
for atoms. The local field for a sphere is related to the
polarization by Eq. (41).
A generalization of the Clausius-Mossotti equation
to include a permanent moment μ→e is summarized in
what is called the Debye equation that is valid for gases
and dilute solutions:
2
εr − 1
N ⎛
⎜α + e
εr + 2 3ε0 ⎜
3k B T
The Debye equation could be used to estimate the
permittivity of a gas if both the polarizability and the
dipole moment were known from experiment.
Volume 117 (2012)
Journal of Research of the National Institute of Standards and Technology
For a specific dipole immersed in an environment of
surrounding dipoles, the dipole will tend to polarize the
surrounding dipoles and thereby create a reaction field.
Onsager included the effects of the reaction field into
the local field and obtained the following relationship
for the static field that, and unlike the Debye equation,
can be used to model the dipole moment of some pure
9kΒΤε0 (ε s − ε∞ )(2ε s − ε ∞ )
ε s (ε∞ + 2ε0 ) 2
A resonance example is shown in Fig. 7. Intrinsic
material resonances in ionic solids can occur at high
frequencies due to driving at phonon normal-mode
frequencies and relate to the mass inertial aspect in
ω0 =
where ε∞ is the optical limit of the permittivity. The
Onsager equation is often used to calculate dipole
moments of gases. Both atoms and molecules can
polarize when immersed in a field. Note that Eq. (57)
uses the permittivity of the liquid, which is a macroscopic quantity to estimate the microscopic dipole
k / m of the positive and negative charges of
Fig. 7. Theoretical resonance in the real part of the permittivity from
Eq. (59) and associated loss factor.
Dielectric Relaxation and Resonance
Simple Differential Equations for Relaxation
and Resonance
If we eliminate the inertial interaction when
ω20 >> ω2, we have the time-domain Debye differential
equation for pure relaxation:
A very general, but simplistic equation, for modeling
polarization response that depends on time is given by
a harmonic-oscillator relation:
1 d P
ω02 dt 2
+ P = χ0 E ,
P (ω) =
is the natural frequency ( ω0 = k / m), and χ0 = εs − ε∞.
Various special cases of Eq. (58) serve as simple, naive
models of relaxation, resonance, and plasmonic
response. The first term relates to the effects of inertia,
the second to dissipation, the third to restoring forces,
and the RHS represents the driving forces. A weakness
of Eq.(58) is that the simple harmonic oscillator model
assumes only a single relaxation time, and resonance
frequency. This equation can be generalized to include
interactions, (see Eq. (117)). In most materials, the
molecules are coupled and have a broad range of relaxation frequencies that widens the dielectric response.
For time-harmonic fields Eq. (58) is
1 − 2 + iωτ
( ω).
For time-harmonic fields, the Debye response is
where P is polarization, τ is the relaxation time, ω0
P (ω) =
+ P = χ0 E .
χ0
E( ω).
1 + iωτ
Except for liquids like water, dielectrics rarely exhibit
the response of Eq. (61) since there is no single
relaxation time over RF frequencies.
We generally assume that dipoles reorient in an
applied field in discrete jumps as the molecule makes
transitions from one potential well minimum to
another with the accompanied movement of a polaron
or defect in the lattice. The Debye model of relaxation
assumes that dipoles relax individually with no interaction between dipoles and with no inertia, but includes
frictional forces. The real part of the permittivity for
dipolar systems generally does not exhibit single-pole
Debye response, but rather a power-law dependence.
The origin of this difference can be attributed to manybody effects that tend to smear the response over a
frequency band.
Volume 117 (2012)
Journal of Research of the National Institute of Standards and Technology
If we eliminate the restoring force term in Eq. (59),
we have an equation of motion for charged plasmas,
1 d 2P
= χ0E .
where 0 < n, m < 1. In this model, a short time scale
corresponds to frequencies in the microwave region
(τ ∝ 1 / f < 1 × 10–9 s) and long relaxation times refer to
frequencies less than 10 kHz (τ ∝ 1 / f < 1 × 10–4 s). In
order to satisfy theoretical constraints at very short
periods the current must depart from Eq. (64). There
are exceptions to the behavior given in Eqs. (64) and
(65) in dipolar glasses, polycrystalline materials, and
other materials [77]. The susceptibility of many lossy
disordered solids typically behave at high frequencies
as a power law
For time-harmonic fields, this becomes
P (ω) =
− 2 + iωτ
( ω).
χ ′(ω) ∝ χ ′′( ω) ∝ ω(n −1) .
This implies χ″ / χ′ is independent of frequency. On the
other hand, measurements of many ceramics, glasses,
and polymers exhibit a loss tangent that increases
approximately linearly with frequency as shown in
Fig. 6.
Dissado and Hill conclude that nonexponential
relaxation is related to cluster response [75]. In their
model, molecules within a correlated region react to the
applied field with a time delay. The crux of this
approach is that in most condensed-matter systems the
relaxation is due not to independently relaxing dipoles,
but rather that the relaxation of a single dipole depends
on the state of other dipoles in a cluster. Therefore their
model includes dipole-dipole coupling. This theory of
disordered solids is based on charge hopping and
dipolar transitions within regions surrounding a defect
and between clusters [75]. The effect is to spread out
the response over time and therefore to produce nonexponential behavior. Dissado and Hill developed a
representation of a correlation function that includes
cluster interaction. According to this theory, the
time-domain response for short time scales is Gaussian
2 2
e– t / τ .
At longer periods there are intra-cluster transitions
that follow a power law of the form t –n. At still longer
periods there are inter-cluster transitions with a Debyetype response e– t / τ, and finally at very long periods
there is response of the form t – m – 1 [75].
Jonscher, Dissado, and Hill have developed theories
of relaxation based on fractal self-similarity [78, 79].
Jonscher’s approach is based on a screened-hopping
model where response is modified due to many-body
charge screening [80]. In the limit of weak screening,
the Debye model is recovered.
Nonexponential response has been obtained with
many models. In any materials where the dipoles do not
rotate independently, the relaxation is nonexponential.
Modeling Relaxation in Dielectrics
The polarization of a material in an applied field
depends on the permanent and induced dipole
moments, the local field, and their ability to rotate with
the field. Dielectric loss in polar materials is due
primarily to the friction caused by rotation, free charge
movement, and out-of-phase dipole coupling. Losses in
nonpolar materials originate mainly from the interaction with neighboring permanent and induced
dipoles, intrinsic photon-phonon interactions with the
EM field, and extrinsic loss mechanisms caused by
defects, dislocations, and grain structure. Loss in many
high-purity crystals is primarily intrinsic in that a
crystal will vibrate nearly harmonically; however,
anharmonic coupling to the electric field and the
presence of defects modifies this behavior. The anharmonic interaction allows photon-phonon interaction
and thereby introduces loss [73]. High-purity centrosymmetric dielectric crystals, that is, crystals with
reflection symmetry, such as crystalline sapphire,
strontium titanate, or quartz, have generally been found
to have lower loss than crystals with noncentrosymmetry [74].
A transient current may be induced if an electric field
is applied, removed, or heated. This can be related to
the dielectric response. The depolarization current for
many lossy disordered solids is nonexponential and, at
time scales short relative to the relaxation time of the
media, can satisfy a power law of the form [75, 76]
I (t ) ∝ t − n ,
and satisfy a power law at long times of the form
I (t ) ∝ t − (1+ m ) ,
Volume 117 (2012)
Journal of Research of the National Institute of Standards and Technology
dielectric and magnetic relaxation response and the
associated entropy production [19, 40, 41, 43, 44].
Nonexponential response has also been reproduced in
computer simulations for chains of dipoles by means of
a correlation-function approach with coupled rate
Eqs. [81-83].
Note that nonexponential time-domain response is
actually required for over some bands in order to have a
causal-function response over all frequencies. This is a
consequence of the Paley-Wiener theorem [84].
According to this theorem, the correlation or decay function cannot be a purely damped exponential function for large times. If C(t) is the decay function then
log C ( τ) ||
1 + τ2
There are many models used to fit measured frequency-dependent dielectric relaxation data for homogeneous materials. These models are usually general
enough to fit many types of response. When dealing
with heterogeneous materials, mixture equations are
commonly used (Sec. 22). The DRT model is restricted
to relaxation, and it assumes there is a probability
distribution y (t) that underpins the relaxation response
with a relaxation time τ. In this model, the permittivity
can be written as
must be finite. This requires the decay function to
vanish less fast than a pure exponential at large times,
C(t) ≈ exp (– ct q ) where q < 1 and c is a constant. We
can show that at short times, decay occurs faster than
exponential [85].
Nigmatullian et al. [86, 87] used the Mori-Zwanzig
formalism to express the permittivity in a very general
ε −ε
ε(iω) = ε∞ + s ∞ ,
1 + R± (iω)
The Distribution of Relaxation Times
(DRT) Model for Homogeneous
ε(ω) = ε∞ + ( ε s − ε∞ )
y (τ)
1 + (iωτ)
y (τ) d τ = 1.
Note that DRT is a single-pole model and cannot be
used for resonances. We see that in the DRT, Debye
relaxations are weighted by a probability-density
function. Equation (69) can be inverted by the Laplace
transform as shown in the Appendix of Böttcher [53].
The DRT approach is sufficiently general that most
causal, relaxation dielectric-response phenomena can
be described by the model for Debye and power-law
response. In the DRT the slope of ε′r (ω) is always
negative [90]. This is consistent with causality. It also
indicates that the model is only valid for relaxation and
not resonance. Around resonance ε′r (ω) can increase
with frequency and become negative as indicated in
Fig. 7.
Equation (69) can fit the relaxation response of
many dielectrics because the Debye equation originates
from a rate equation based on thermodynamics containing the essential physics, and Eq. (69) is a distribution
of Debye relaxations. The DRT then extends this into a
multi-relaxation period rate equation. We consider
various special cases of Eq. (69) below. For other
special cases please see Böttcher [53]. In any complex
dielectric material, we would expect there would be a
broadening of relaxation times due to heterogeneity of
the molecular response, and in this context the DRT
model makes sense. This approach is often criticized,
and concluded that for most disordered materials, the
response is similar to that of a distributed circuit with
R±(iω) = [(iωτ1)± ν 1 + (iωτ2)± ν 2]±, where νi are constants
determined by numerical fits. In the formulation of
Baker-Jarvis et al. [88], R± corresponds to the complex
relaxation times τ(ω) as R+(iω) = iωτ(ω) (see Sec. 11).
A (iωτ)(n–1) frequency dependence of the complex
relaxation periods corresponds to a impulse-response
function of the form t – n.
In addition, in analyzing dielectric data the electric
modulus approach is sometimes used where M(ω) =
M′ (ω)+iM ″ (ω) = 1 / εr = ε′r / (εr′2 + ε r″2 )+iε r″/ (εr′2 + ε r″2 ).
Dielectric relaxation has also been described by
Kubo’s linear-response theory that is based on correlation functions. This is an example of a relaxation
theory derived from Liouville’s equation. The main
difficulty with these approaches is that the correlation
functions are difficult to approximate to highlight the
essential physics, and gross approximations are usually
made in numerical calculations. The linear expansion
of the probability-density function in Kubo’s theory
also limits its usefulness for highly nonequilibrium
problems. Baker-Jarvis et al. have recently used a
statistical-mechanical projection-operator method
developed by Zwanzig and Robertson [89] to model
Volume 117 (2012)
Journal of Research of the National Institute of Standards and Technology
because it is not always possible obtain a physical interpretation of the distribution function [75].
ε(ω) = ε∞ +
ε s − ε∞
1 + (iωτ0 )1−α
where α < 1. The pulse response function is fp (t) =
Debye Model
(1/ τ0 )
The simplest case of the distribution function in
Eq. (69) is an uncorrelated approximation where
y (t ) = δ(t − τ) ,
( −1) m / Γ(( m +1)(1 − α))( t / τ 0) m (1−α )−α. The
ε'r (ω) =ε∞ + (ε s −ε ∞ )
(ε s − ε∞ )
1 + iωτ
ε''r (ω) =(ε s −ε ∞ )
f p (t ) = exp( −t / τ) / τ .
1 + ω2 τ2
1+ (ωτ0 )2(1−α) + 2( ωτ0 )1−α sin( πα/2)
(ωτ0 )
1+ (ωτ0 )
cos( πα/2)
A plot of εr′(ω) versus ε r″(ω) yields a circle, where the
center is below the vertical axis.
In terms of components,
ε s − ε∞
1+ (ωτ0 )1−α sin( πα/2)
In this case, the pulse response function is
ε'r (ω) = ε∞ +
real and imaginary parts of the permittivity can be
separated into
which yields the Debye response
ε = ε∞ +
6.3 Cole-Davidson Model
ε''r (ω) =
(ε s − ε ∞ )ωτ
1 + ω2 τ2
The Cole-Davidson model has also been found
useful for modeling many liquids, semisolids, and other
materials [53]. If we consider the case τ ≤ τ0 :
If ωτ is eliminated in the Debye model, and the
equations for εr′(ω) and ε r″ (ω) are plotted against each
other, we obtain the equation for a circle:
y (τ) =
and zero otherwise. The permittivity is
ε s − ε∞ ⎤
⎡ ε s − ε∞ ⎤
⎢ε′(ω) − 2 ⎥ + ε ′′ (ω) = ⎢ 2 ⎥ .
1⎡ τ ⎤
⎥ sin πβ
π ⎣ τ0 − τ ⎦
ε r (ω) = ε∞ +
The center of the circle is on the horizontal axis.
The reasons why the Debye equation is a paradigm
in dielectric relaxation theory is because it is simple
and contains the essential physics and thermodynamics
in relaxation. That is, it models idealized relaxation,
and it yields predictions on the temperature dependence
of the relaxation time τ = A exp (Ea / RT), where Ea is
the activation energy.
(ε s − ε∞ )
(1 + iωτ0 )β
where β < 1. The pulse response function is
f p (t ) = (1/ τ0 Γ( β))( t / τ 0) β−1 exp( − t/ τ 0) .
The real and imaginary parts of the permittivity can be
separated into
ε'r (ω) = ε ∞ + (ε s − ε ∞ )(1+ω2 τ 02 ) −β/2 cos ( β Arg[1 + iωτ0 ]),
Cole-Cole Model
The Cole-Cole model has been found useful for
modeling many liquids, semisolids, and other materials
[53]. In this case,
y ( τ) =
sin πα
2π cosh[(1 − α)In τ/τ0 − cos πα]
ε''r (ω) = (ε s − ε ∞ )(1+ω 2τ 02 ) −β/2 sin (β Arg[1 + iωτ 0 ]) .
The plot of εr′(ω) versus ε r″(ω) maps out a skewed arc
rather than a circle.
Volume 117 (2012)
Journal of Research of the National Institute of Standards and Technology
Havrilak-Negami Model
energy is high, but as the frequency increases, the
dipole response tends to fall behind the applied field
and, therefore, the loss usually increases and the stored
energy decreases. This is related to the phasing between
the current and voltage waves, in analogy to the heating
an electric motor encounters when the phase between
the voltage and current changes.
Ionic conduction in insulating dielectrics is due to
the migration of charged ions. The migration takes
place through tunneling or jumps induced by the
applied field, or by slow migration under the applied
field. In solid polymers it may proceed by jumps from
one vacancy to another or by electronic conduction. In
oxide glasses it is the movement of positively charged
alkali ions in the applied field. In many materials, the
dielectric losses originate in vacancy-vacancy and
vacancy-impurity relaxations.
At high frequencies, lossy semiconductors, superconductors, and metals have a complex free-charge ac
conductivity that is explained by the Drude model. This
can cause the effective permittivity to become negative
[27]. To understand this, consider Maxwell’s equation,
The Havrilak-Negami distribution has two parameters to fit data and is very general. It can be used to
fit the response of many liquids and non-Debye solid
materials [53]. In the special case α = 0 it reverts to the
Cole-Davidson model. The distribution function is
( )β(1−α) sin βθ
y ( τ) =
π [( τ )]2(1−α) + 2( τ )(1−α) cos( π(1− α)) +1]β/2
and θ = tan–1{sin π (1 – α) / (τ / τ0 + cos π (1 – α))}
ε r (ω) = ε∞ +
(ε s − ε∞ )
(1 + (iωτ0 )1−α )β
where 0 < α ≤ 1 and 0 < β ≤ 1, and
cos βθ
[1+ 2( ωτ0 )(1−α)sin ( πα) +( ωτ0 ) 2(1−α))]β / 2
+ J = ∇×H .
We can define an effective charge current as
ε''r (ω) = ε∞ + ( ε s − ε∞ )
sin βθ
J e f f (t ) =
σ( t − τ) E( τ) d τ +
or for time-harmonic fields
( ω) + iω D
( ω) .
J e f f (ω) = σ( ω)E
Loss and Conductivity
Combining ac J with the displacement field produces
an effective real part of the permittivity that can be
negative over a region of frequencies. For example in
plasmas and superconductors, the effective conduct∼
ivity satisfies iωD (ω) + J (ω) = [iω(ε′(ω) – iε′′(ω)) +
σ′(ω) – iσ′′(ω)] E (ω), yielding
Loss originates from the conversion of EM field
energy into heat and radiation through photon-phonon
interactions. In dielectrics the heating is caused by the
transformation of electromagnetic energy into lattice
kinetic energy, which is seen as frictional forces on
dipoles and the motion and resulting friction of free
charges in materials. Major mechanisms of conduction
in dielectrics in the RF band are ionic or electrolytic
migration of free ions, impurities or vacancies, electrophoretic migration of charged molecules, and electronic conduction of semi-free electrons that originate
from jump processes of polarons. At low frequencies,
dipoles can respond to the changes in the applied field,
so dielectric losses usually are low and the stored
εe f
f (r ) ( ω) = εr′ ( ω) −
σ′′(ω) ⎛
σ′( ω) ⎞
− i ⎜ εr′′( ω) +
ε0 ω
ε0 ω ⎠ (93)
where σ′ ≈ σdc and σ″ relates to the reactive part of the
surface impedance. A large σ″ can produce a negative
real part of the total permittivity such as what occurs in
superconductors [91].
Volume 117 (2012)
Journal of Research of the National Institute of Standards and Technology
A total conductivity has been used in the literature to
model either the ac effects of the free charge and
partially bound free charge in hopping and tunneling
conduction, or as another way of re-expressing the
complex permittivity. Because some charge is only
partially bound, the distinction between conductivity
and permittivity can, at times, get blurred. This blurring
points out the mesoscopic property of the permittivity.
Most models of ac conductivity are based on charged
particles in potential wells where energy fluctuations
determine whether the particle can surmount a potential
barrier and thereby contribute to the conductivity. In
conducting liquids, human tissue, and water-based
semisolids the conductivity is generally flat with
increasing frequency until megahertz frequencies, and
then it increases, often in a nearly linear fashion.
There are a number of distinct models for σtot . The
Drude model of the complex conductivity of electrons
or ions in a metal is approximately modeled as
σtot = Ne2 /( m(γ 0 + iω)) = σ′ − iσ′′ ,
For disordered solids, where hopping and tunneling
conduction takes place with a relaxation time τe , the ac
conductivity can be expressed as [93, 94]
σtot (ω) = σ0 iωτe / ln(1 + iωτe )
ωτe arctan ( ωτe )
1 2
ln (1 + ω2 τ2e ) + arctan 2 (ωτe )
ωτe ln(l + ( ωτe ) 2 )
1 2
ln (1 + ω2 τ2e ) + 2 arctan 2 (ωτe )
= σ0 [
Double Layers and Conducting
Materials Near Metal Interfaces
Conducting and semiconducting dielectric materials
at interfaces or metallic contacts can be influenced by
the effects of double layers. Measurements on conducting liquids are complicated by the effects of electrode
polarization, which are the direct result of the double
layers [95]. Double layers and electrode polarization
are due to the build up of anions and cations at the
interface of electrodes and conducting materials,
as shown in Fig. 8. Modeling ionic solutions near
electrodes is complicated, because the charge is mobile
and depends on the potential.
where γ0 is the collision frequency, N is the electron
density, m is the ion mass, and e is the electronic charge
[27]. Note that the dc conductivity is σdc = Ne2 / mγ0 .
The net dielectric response is a sum of the dipolar
contribution and that due to the ions, where
ε′e f f = ε′d – Ne2 / m (γ20 + ω2) and ε″(ω) = Ne2γ0 / mω (γ20
+ ω2) + ε″d (ω). Therefore, for metals, the real part of the
permittivity is negative for frequencies near the plasma
frequency, ω p =
Ne 2 / ε0 m . The plasma frequency in
metals is usually well above 100 GHz. The conductivity is thermally active and can be modeled for some
ionic materials as [92]
σ dc =
nc e2 b 2ν 0
exp ( −
k BT
kB T
where nc is the ion vacancy, b is the ion jump distance,
ν0 is a characteristic ion frequency, and ΔG is the
Gibb’s free energy.
For plasmas at high frequencies
ε′(ω) → ε 0 ⎜1 − 2
⎜ γ 0 + ω2
Fig. 8. Electrolyte charges near an electrode.
Volume 117 (2012)
Journal of Research of the National Institute of Standards and Technology
Two conducting dissimilar materials can have different electronic affinities. When these dissimilar materials are in contact, a potential gradient frequently
develops between the materials. As a result an electrical double layer forms at a material interface. This
interface could be between liquid and metal electrodes
or the layer between a biomolecule and a liquid. The
potential difference will attract ions of opposite charge
to the surface and repel like charges. For a double
layer, the charge density depends nonlinearly on the
applied potential and is modeled at low frequencies
by the Poisson-Boltzmann equation for the
potential [96, 97] (∇2ψ = −ρ(ψ)/ε). The potential
decreases roughly exponentially from the surface as
ψ(x) = ψ0 exp (– x / λD), where λD is the Debye screening length or skin depth. The region near the electrode
consists of the Stern layer and a diffuse region beyond
the Stern layer where the potential decays less rapidly.
It is known that the Poisson-Boltzmann equation is of
limited use for calculating the potential around many
biomolecules due to molecular interactions and the
effects of excluded volume [97].
At the interface of conductive materials and electrodes, electrode polarization produces a capacitive
double-layer region in series with the specimen under
test. The presence of electrode polarization results in
ε′e f f being much greater than the value for the liquid by
itself. Because the electrode capacitance is not a
property of the material under test, but rather the interface, it can be treated as a systematic uncertainty and
methods to remove it from the measurement can be
applied. Double layers also form at the metal interface
with semiconducting materials where the conductivity
is a function of applied voltage.
The effects of electrode polarization can strongly
affect dielectric measurements up to around 1 MHz, but
the effects can be measurable up into the low gigahertz
frequencies. Any electrode influencing the calculated
permittivity should be treated as a systematic source
of uncertainty. Alternatively, the permittivity with
the electrode effects could be called the effective
The effects of electrode polarization capacitance as
commonly analyzed with the following model [98]
C = Cs +
ω R 2C p
R = Rs (1 + ω2 R2 C 2 ) + Rp ,
where C and R are the measured capacitance and resistance, Cp and Rp are the electrode double-layer capacitance and resistance, and Cs and Rs are the specimen
capacitance and resistance. A way to partially eliminate
electrode polarization is to measure the capacitances C1
and C2 and resistances R1 and R2 at two separations d1
and d2 . Because Cp is the same for each measurement
and Cs can be scaled as Cs2 = (d1 / d2 )Cs1 , we can obtain
the specimen capacitance. Another way of minimizing
the effects of electrode polarization is to coat the
capacitor plates with platinum black [99]. This lessens
the influence of electrode polarization by decreasing
the second term on the right hand side of Eq. (98).
However, both the coating and two-distance methods
schemes do not completely solve this problem. For
biological liquids, often the buffer solution is first
measured by itself and then again with the added
biological material and the difference between the
measurements is reported.
For dielectric measurements, probably the best
approach is to bypass much of the electrode-polarization problem altogether and use a four-probe capacitor
system as shown in Fig. 9. The four-probe capacitance
technique overcomes electrode problems by measuring
the voltage drop away from the plates and thereby
avoiding the double layer [100].
Fig. 9. Four probe measurement.
Volume 117 (2012)
Journal of Research of the National Institute of Standards and Technology
Relationships of the Permittivity
Components: Causality and
Kramers-Kronig Equations
The average moment for N dipoles is therefore
∫ π p (θ) sin θ cos θd θ
P = N μe 0 π
∫ 0 p(θ) sin θd θ
Kramers-Kronig relations relate the real and imaginary parts of the permittivity. These equations are a
result of causality and analytic functions. There are
many forms of the Kramers-Kronig conditions [101],
below are standard relationships
ε′r (ω0 ) − ε∞ =
ε′′r (ω0 ) =
ε′′r (ω)ω − ε′′r ( ω0 ) ω0
ω2 − ω20
d ω,
∞ [ε′ ( ω) − ε′(ω )]
d ω.
ω2 − ω20
μ E k BT
⎡
−
P = N μe ⎢ coth e
μe E
k BT
ε′′( x)
dx .
where L(x) = coth(x) – 1/x ≈ x/3 – x3/45 + x5/945... is
the Langevin function. At high temperatures or weak
fields, the Langevin function is approximated as
μ e2 E
⎛ μe E ⎞
P = N μe L ⎜
⎟≈ N
3 kB T
⎝ kBT ⎠
Static and Dipolar Polarization
Static Polarization
< M h >= N
The total kinetic energy (K) plus potential energy of
a dipole in a static applied field is approximately
U = K − μd ⋅ E .
μh2 μ0 H
3k BT
Deriving Relaxation Equations by
Analyzing Dipolar Orientation in
an Applied Field
Upon application of an electric field, dipole moments,
impurities, and vacancies can change positions in the
lattice potential wells. This is the origin of rotation,
conduction, and jump reorientation [53, 102].
Consider the density of N± molecules where there
are N± dipole moments that are aligned either parallel
(+) or antiparallel (–) to the applied field. The time
evolution of the numbers of dipoles is described by the
number of dipoles flipping one direction minus the
The probability that a dipole is aligned at angle θ to the
directing electric field is
μe ⋅ E
p(θ) = A exp(
3k BT
and in the approximation we assume | μ→e ||E|/kBT < 0.1.
Note that the model shows that the polarizing effect of
the applied field affects < cosθ >, and there is a lesser
effect on the direction of the individual dipole
moments. At room temperature this corresponds to an
electric field of about 3 × 107 (V / m), which is a very
strong field. In intense fields or low temperatures, higherorder terms in the Langevin function must be included
Using a similar analysis, the magnetic moment for
noninteracting paramagnetic materials has the same
form as Eq. (107)
We should note that σdc is not causally related to the
permittivity and, therefore, before Kramers-Kronig
analysis is performed, the contribution of conductivity
to the loss should be subtracted.
As a consequence of causality, the permittivity
satisfies the condition ε∗(ω) = ε(– ω). Causality and
second law of thermodynamics requires that when
the response is averaged over a cycle, for a passive
system ε″(ω) > 0 and μ″(ω) > 0. However, ε′(ω) or
μ′ (ω) can be greater or less than zero. Also, the real part
of the characteristic impedance must be greater than
⎛ μe E ⎞
⎥ = N μe L⎜
⎝ kB T ⎠
For example, if we neglect any dc conductivity, the
dc permittivity must satisfy
ε s − ε∞ =
Volume 117 (2012)
Journal of Research of the National Institute of Standards and Technology
number flipping the other direction characterized by the
transition rates ν± , where ν+ denotes the rate of going
from a + state to a – state
dN ±
= ν N − ν ± N ± .
where ν∞ is the maximum transition rate and the factor
3 is related to isotropic polarization pE → | μ→e ⋅ Ε | / 3. At
high temperatures and ν0 = ν∞ exp (– U0 / kBT )
μ E
ν ± ≈ ν 0 ⎜1 e
k BT
Therefore, for molecules that each have a permanent
electric dipole moment μ→e , the net polarization is
P(t) = | μ→e |(N+ – N–) = | μ→e |(2N+ – N), and
2
2ν μ NE
+ 2ν 0 P = 0 e
3 kB T
The relaxation time is τ = 1/2ν0 = (1/2ν∞ ) exp (U0 / kBT).
In this model the susceptibility is
χs =
N μe2
3k BT
Therefore Eq. (112) reduces to the Debye equation
+ P / τ = χs E / τ .
Relaxation Times
When a field is applied to a material, the material
responds by re-arranging charge, causing spin precession, and currents. The characteristic time it takes
for the response is called a relaxation time. Relaxation
times are parameters used to characterize both dielectric and magnetic materials. Dielectric relaxation times
are correlated with mechanical relaxation times [103].
Magnetic relaxation in NMR and ESR is modeled by
spin-spin (T2) and spin-lattice (T1) relaxation times.
In the literature, dielectric relaxation times have been
identified for molecules and bulk materials. The first is
a single molecule relaxation time τs and the other is a
Debye mesoscopic relaxation time τD . For magnetic
nanoparticles in a fluid, where the magnetic moment is
locked in place in the lattice, the Brownian time
constant is defined as τB = 3νVH / kBT, where ν is the
fluid viscosity and VH is the hydrodynamic volume of
the particle [104]. The Neel relaxation time is for
crystals where the magnetic moment is free to rotate in
the field. Dielectric relaxation times are related to how
the dipole moments and charge are constrained by the
surrounding material. The characteristic relaxation time
for a polarized material that was in an applied field at
t = 0 to decay to a steady state is related to the coupling
between dipoles and details of the lattice. At high
frequencies, the electric response of a material lags
behind the applied field when the field changes faster
than the relaxation response of the molecules. This lag
is due to long and short-range forces and inertia. The
characteristic Debye relaxation time τD can be obtained
from the maximum of the loss peak in Eq. (61).
Relaxation times are usually defined through the decay
of the impulse-response function that is approximated
by a Debye response exp (– t / τ). Debye used Onsager’s
cavity model to show that τD / τs = (εs + 2) / (ε∞ + 2)
[105, 106]. Arkhipov and Agmon [105] showed that
τD / τs = (3kBT / μ2d ρc )(εs – ε∞)(2εs + ε∞)/εs , where ρc is
the density of molecules, and μd is the dipole moment.
In their review, Arkhipov and Agmon also discuss the
relationship between macroscopic and microscopic
relaxation times from various perspectives [105]. This
theory predicts that the macroscopic and microscopic
relaxation times are related by τD / τs ≈ (2εs + ε∞) / 3εs.
Debye showed that the microscopic relaxation time for
molecules of radius a is related to the viscosity η and
the friction constant ζ by τs = 4πa3η / kBT = ζ / 2kBT .
In equilibrium and in the absence of an electric field,
the number of transitions in either direction is the same
so that ν+ N+ = ν– N– , where N+ + N– = N. In an electric
field, the transition rates are given by
ν ± = ν∞ exp( −(U 0 / k BT ± N μe E / 3 kB T )) ,
Note that such a simple model can describe to a
remarkable degree the polarization and yields a
relaxation time with a reasonable dependence on
temperature. This indicates the basic physics is correct.
Volume 117 (2012)
Journal of Research of the National Institute of Standards and Technology
The Arrenhius relaxation rate is modeled as
τ = τ0 exp (U / kBT ) . The Vogel-Fulcher relaxation time
is used to model relaxation near polymer glass transition temperatures as τ = τ0 exp (U / kB( T – T0 )). The
relaxation time can also be related to changes in the
activation entropy ΔS, Helmholtz energy of activation
ΔH, and free energy ΔF as τ = ( / kBT ) exp (ΔF / RT),
and the entropy of activation is related by ΔS =
(ΔH – ΔF) / T. Therefore, we have τ = (h / kBT ) exp
(ΔH / RT – ΔS / R). So, by fitting the relaxation times
obtained by dielectric measurements as a function of
temperature we can extract changes in the entropy ΔS
and the Helmholtz free energy ΔH for an activation
The typical relaxation time T1 in NMR experiments
is longer than in EPR [107]. In EPR experiments, relaxation times are generally less than milliseconds. In
dielectrics, the relaxation times of liquids can be
picoseconds, as indicated in Table 2, but in some
glasses they can be seconds and longer. The characteristic relaxation times have been found to change with
the frequency of the applied field [88]. This is due to
the restoring and frictional forces acting differently
under different field conditions. In the past researchers
have realized this and resorted to using phenomenological DRT models as in Eq. (69).
ε′′r (ω) = ( εrs − εr ∞ )
Table 2. Relaxation Times of Common Liquids [105]
water (22 °C)
methanol (22 °C)
ethanol (22 °C)
1-propanol (22 °C)
2-propanol (22 °C)
τD (ps)
Relaxation Time Based Model in Fields of
Varying Frequency
A very general approach to modeling the susceptibility can be obtained by the Laplace transform of the
time-invariant approximation to Eq. (38). This yields
a permittivity in terms of complex relaxation times
τ(ω) = τ′(ω) – iτ″(ω) [46]:
ε′r (ω) = εr ∞ +
(ε rs − ε r∞ )
1 − ωτ′′(ω)
(ωτ′(ω))2 + (1 − ωτ ′′(ω)) 2
τ(t − θ)
d P(θ)
d θ + P(t ) = χs Ε ( t) .
Equation (117) highlights the physics of the interaction with materials and is useful in determining the
underlying differential equation related to phenomenological models. For this equation the Debye model is
obtained if τ (t) = τ0δ(t). Relaxation phenomenological
models such as Cole-Davidson can be related τ(ω).
Therefore the underlying differential equations can be
cast into the form of Eq. (117). Because they are
complex pairs, it is not possible to extract the timedomain functions of τ′(ω) and τ″(ω) independently.
It is important to study the origin of the frequencydomain components. Whereas τ′(ω) models the out-ofphase behavior and loss, τ″(ω) models the effects of the
local field on the restoring forces. If τ″(ω) is positive it
is related to inertial effects. If τ″(ω) is negative, it is
related to the local field interaction that tends to
decrease the polarization through depolarization. The
relaxation times are
The assumption of this model is that at RF frequencies
the relaxation has a dependence on the frequency of the
driving field. This frequency dependence originates from
the applied field acting on the molecules in the material
that keeps the molecules in a nonequilibrium electromagnetic state. Equations (115) and (116) have the same
form as the Laplace transform of a linear harmonic oscillator equation of motion. However, this model contains
additional information through the frequency dependence of the relaxation times. For a real, frequency-independent relaxation time (τ′ constant and τ″ = 0), Eq. (38)
is the Debye equation. In the special case where τ′ is constant, the ensemble response function is of the form exp
(–t / τ′) and we have classical Debye relaxation. This can
be traced to the fact that the Debye model assumes there
is no inertia, and therefore, a purely damped motion of
dipoles. Performing the inverse Laplace transform of the
time-invariant approximation to Eq. (38) we obtain
another form for the polarization equation,
+ s .
(ωτ (ω)) + (1 − ωτ (ω))
τ′(ω)ω = (ε ′′r (ω) −
ε0 ω
(ε s − ε r ∞ )
(ε′r (ω) − ε r ∞ ) 2 + (εr′′ (ω) −
ε0 ω
) 2,
Volume 117 (2012)
Journal of Research of the National Institute of Standards and Technology
τ′′(ω)ω = −
(ε′r (ω) − ε r ∞ )(ε s − εr′ (ω)) − (εr′′ (ω) −
ε0 ω
ε0 ω
mass-related inertial interactions are important,
τ″(ω) > 0. This occurs in polaritonic resonances at
terahertz to infrared frequencies and in negative-index
materials. In this case the local field tends to enhance
the polarization through the effects of inertia that counteract restoring forces [5]. When τ″ω = 1, the real part
of the susceptibility goes to zero, indicating the system
is going through resonance. In general, just as in
the Debye and other phenomenological models, the
relaxation times can depend on temperature
(A exp (U0 / kBT )).
In Fig. 10 we plot the relaxation times extracted from
dielectric measurements as well as measurements given
in Reference [108]. We see that the measured τ″(ω)
values are all negative. We see that for ethanediol, τ″ is
very small and τ′ is nearly frequency-independent.
Therefore ethanediol, is well modeled by the Debye
equation. The physical significance of τ′(ω) relates to
the effective time for the material to respond to an
applied electric field. τ″(ω) > 0 at resonance corresponds to an effective ensemble period of oscillation
and τ″(ω) < 0 corresponds to a characteristic time scale
for charge depolarization and screening effects. An
interpretation is that in relaxation the effects of the
local field on the short-range restoring forces and
screening may have a frequency dependence. This
frequency dependence can manifest itself as the
commonly observed frequency shift in the loss peak
relative to the Debye model. We also see that τ″ < 0 can
be interpreted as the effects of the local field on the
short-range electric restoring forces, which tend to
reduce the permittivity and modify the position of
the maximum in the loss curve relative to the Debye
maximum condition (ωτ′ = 1). The behavior for
τ″(ω) < 0 is analogous to what is seen in longitudinal
optical-phonon behavior that yields a local field that
tends to reduce polarization. Over frequencies where
Surface Waves
Electromagnetic surface waves occur in many applications. Surface waves can be supported at the interface
between dielectrics and conductors. These waves
travel on the interface, but decay approximately
exponentially away from the surface. There are many
types of surface waves, including ground waves and
surface plasmons polaritons (SPP) that travel at the
interface between a dielectric and conductor, surface
plasmons on metals, and Sommerfeld and Goubau
waves that travel on coated or uncoated wires. SPP’s
require the real part of the permittivity of the metal to
be negative [109]. A Goubau line guides a surface wave
and consists of a single conductor coated with dielectric
material [110]. A Sommerfeld surface wave propagates
as a TM mode around a finitely conductive single bare
conductor. Plasmonic-like surface waves can form
from incident microwave electromagnetic energy on
subwavelength holes in metal plates. We will examine
plasmonic surface waves in Sec. 14.2.
Electromagnetic Radiation
Classical electrodynamics predicts that accelerated
charged particles generate EM waves. This occurs in
antennas where charged particles oscillate to produce
radiation. Linearly or elliptically polarized radiation
waves are determined by the type of acceleration the
source charged particles undergo. If the charge particle
undergoes oscillation from a nonlinear restoring force,
the emitted radiation may not be monochromatic.
Thermal Noise and Blackbody Fields
Due to the continual Brownian motion of microscopic charges, thermal Johnson noise fields are
produced over a broad distribution of frequencies [111,
112]. There are also many other sources of noise such
Fig. 10. The real (τr ) and imaginary component (τi ) parts of the
relaxation times for various alcohols.
Volume 117 (2012)
Journal of Research of the National Institute of Standards and Technology
speculative research where NIM materials are used for
the microscopic plates to produce levitation of nanoparticles [116]. Casimir effects may play a role in
future modeling of microelectronics because the
electrode separations are close to where these effects
become important.
as phase noise and shot noise. Thermal movement and
blackbody radiation are a source of electrical noise and
was described theoretically by Nyquist [112]. This
theory was expanded by Callen [113]. A blackbody has
an emissivity of near unity and is an excellent absorber
and emitter of radiation. The spectral distribution of
blackbody radiation follows the Planck distribution
for the energy density u (T, f ) = (8πh f 3 / c3 ) /
(exp (h f / kBT – 1)). Examples of blackbody radiation
include radiation from intergalactic space, as well as
black cavities with an aperture. Typical blackbody
materials have some free electrons and a distribution of
molecular resonant frequencies and, as a result, are
useful in converting optical energy into heat energy.
They are also good radiators of infrared thermal
energy. Most materials only partially reflect any
incident energy. Therefore, they do not radiate as much
power as a blackbody at the same temperature. The
ratio of the energy radiated by a material relative to that
of an ideal black body is the emissivity. In a frequency
band Δ f , the emissivity is defined as e = P / (kBTΔ f ).
The emissivity satisfies 0 ≤ e ≤ 1. The brightness
temperature is TB = eT , where T is the physical
temperature. Nyquist/Johnson noise in the RF band
has only a weak frequency dependence. It is modeled
for voltage fluctuations in a transmission line terminated by resistors R over a frequency band Δ f by
< ν2 > / R = 4kBTΔ f [112].
Radiometers in the RF band are usually receiving
antennas that collect noise power from the direction
they are pointed and infer the brightness temperature.
The goal of radiometry is to infer information about
the remote source of noise from the brightness temperature [111].
Quantum-field theory models the vacuum as filled
by quantum fluctuations that contain a spectrum of
frequencies having energy (1/2) ω. In this model,
fluctuations give rise to virtual photons and spontaneous emission of short-lived particles. Virtual photons
and short-lived particles are allowed by the uncertainty
principle between energy and time: Δ EΔ t ≈ .
Vacuum fluctuations can produce attractive forces
between nanometer-spaced parallel electrodes. This
Casimir effect is commonly explained classically by
the cutoff of EM modes between the plates so that the
external radiation pressure exceeds the pressure
between the plates [114]. A more complete and satisfactory description can be derived with quantum
mechanics. The force is extremely short range. It has
also been shown that the force can be made repulsive
by changing one of the plates from a metal to a dielectric such as silica [115]. In addition, there has been
Magnetic Response
Overview of Magnetism
In this section, we will very briefly overview the
basic elements of magnetic phenomena needed in our
applications to RF interactions. Magnetism has a
quantum-mechanical origin intimately related to the
spin and angular momentum and currents of electrons,
nuclei, and other particles. Stern and Gerlach [4]
proved the existence of discrete magnetic moments by
observing the quantized deflection of silver atoms
passing through a spatially varying magnetic field.
Electrons orbiting a nucleus form a magnetic moment
as well as the intrinsic spin of the electron. Magnetic
moments are caused either by intrinsic quantummechanical spin or by currents flowing in closed loops
m ∝ (current)(area).
Spins react to a magnetic field by precessing around
the applied field with damping [117]. For spins of the
nucleus, this precession forms the study of nuclear
magnetic resonance (NMR); for paramagnetic
materials it is called electron-spin or ESR or
electron-paramagnetic resonance or EPR; and for
ferromagnetic materials it is called ferromagnetic
resonance or FMR . The dynamics in spin systems are
tied phenomena such as spin precession, relaxation,
eddy currents, spin waves, and voltages induced by
domain-wall movements [7-9, 118].
Paramagnetism originates from spin alignment in an
applied magnetic field and relates to the competition
between thermal versus magnetic energy (m . B / kBT )
(see A in Fig. 11). Paramagnets do not retain significant
magnetization in the absence of an applied magnetic
field, since thermal motion tend to randomize the spin
The origin of diamagnetism in materials is the orbital
angular momentum of the electrons in applied fields.
Diamagnetic materials usually do not have a strong
magnetic response, although there are exceptions. In
ferromagnetic materials, exchange coupling allows
regions of aligned spins to be formed [119]. Ferromagnetic and ferrimagnetic materials may have spin
resonances in microwave to millimeter wave frequencies [120]. Ferrimagnetic materials consist of two
Volume 117 (2012)
Journal of Research of the National Institute of Standards and Technology
p+ =
e − μ ⋅ B / k BT
e μ ⋅B / kBT + e − μ ⋅ B / kB T
→ .
Therefore, for N spins and when μ
B / kBT << 1
< m >=
e μ ⋅B / kBT − e − μ ⋅ B / kB T
μ ⋅ B / k BT
− μ ⋅ B / kB T
= Nμ tanh
μ ⋅B
μ ⋅B
≈ Nμ
k BT
kB T
In the case of isotropy < m > = Nμ 2B / 3kBT . So we
obtain the same form as in the case of noninteracting
dielectrics in Eq. (107).
Fig. 11. Simplistic summary of spin orientations for A) paramagnetic B) ferromagnetic C) ferrimagnetic D) antiferromagnetic
overlapping lattices whose spins are oppositely directed, but with a larger magnetic moment in one lattice
than the other. Antiferromagnetism is a property of
many transition elements and some metals. In these
materials the atoms form an array with alternating spin
moments, so the average spin and magnetic moment
are zero. Antiferromagnetic materials are composed of
two interpenetrating lattices. Each lattice has all spins
more or less aligned, but the lattices, as a whole, are
inverse. Resonances in antiferromagnetic materials
may occur at millimeter wave frequencies and above.
Antiferromagnetic materials are paramagnetic above
the Neel temperature.
For atoms with angular momentum J with 2 J + 1
discrete energy levels, the average magnetization can
be expressed in terms of the Brillioun equation BJ [4]
< m > = NgJ μ B BJ ( x) ,
i i
where the probabilities of being in the low energy (–) or
high (+) energy states are
p− =
e μ ⋅B / k BT
e μ ⋅B / k BT + e − μ ⋅ B / kB T
Magneto-Dielectric Response: MagnetoElectric, Ferroelectric, Ferroic, and Chiral
Researchers have found that in magneto-electric,
ferroic, and chiral materials the application of magnetic fields can produce a dielectric response and the
application of an electric field can produce a magnetic
response (see for example [121]). These cross coupling
behaviors can be found to occur in specific material
lattices, layered thin films, or by constructing composite materials. An origin of the intrinsic magneto-electric
effect is from the strain-induced distortion of the spin
lattice upon the application of an electric field. When a
strong electric field is applied to a magneto-electric
material such as chromium oxide, the lattice is slightly
distorted, which changes the magnetic moment and
therefore the magnetic response. Extrinsic effects can
be produced by layering appropriate magnetic, ferroelectric, and dielectric materials in such a way that an
applied electric field modifies the magnetic response
and a magnetic field modifies the electric response.
Chiral materials can be constructed by embedding
In order to contrast decoupled spin response with
dielectric dipole response in Sec. 10.2, we will develop
the well-known statistical approach of noninteracting
paramagnetism. In a paramagnetic material, the net magnetic moment is the sum of individual moments in an
applied field. If the spin moments are σ± = ± μ and the
probability density of the spin being up or down in an
applied field is pi , then the net magnetic moment is [4]
∑ σ p (σ) ,
where x = g J μBB / kBT and BJ (x) = (2 J + 1) / 2 J
coth ((2J + 1)x / 2 J ) – (1 / 2 J ) coth(x / 2 J ) and g is the
g-factor given by the Landé equation.
Two-State Spin System
< m >= −
Paramagnetic Response With Angular
Momentum J
Volume 117 (2012)
Journal of Research of the National Institute of Standards and Technology
conducting spirals into a dielectric matrix. In artificial
magneto-electric materials the calculated permittivity
and permeability may be effective rather than intrinsic
properties. The constitutive relations for the induction
and displacement fields are not always simple and can
contain cross coupling between fields. For example,
D (ω) = α1 . E (ω) + α2 . B (ω) , where αi are constitutive parameters.
rotational transition at 22 GHz, and a stronger
transition occurs at around 183 GHz [123].
If high-frequency fields are applied to ferrite materials, there are relaxations in the megahertz frequencies,
and in the megahertz to MMW frequencies there are
spin resonances [119, 121, 124, 125].
14. Artificial Materials: Plasmons,
Super-Lensing, NIM, and Cloaking
Electromagnetic-Driven Material
Resonances in Materials at RF
The term metamaterial refers to artificial structures
that can achieve behaviors not observed in nature.
NIMs are a class of metamaterials where there are
simultaneous resonances in the permittivity and permeability. Many artificial materials are formed from
arrays of periodic unit cells formed from dielectric,
magnetic, and metal components, and when subjected
to applied fields, achieve interesting EM response.
Examples of periodic structures are NIM that utilize
simultaneous electric and magnetic resonances [126].
Metafilms, band filters, cloaking devices, and photonic
structures all use artificial materials. Artificial materials
are also used to obtain enhanced lensing and anomalous
refraction and other behaviors [65, 126-131]. A very
good overview is given in [128]. In the literature NIM
materials are commonly assumed to possess an intrinsic negative permittivity and permeability. However,
the resonator dimensions and relevant length scales
used to achieve this behavior may not be very much
smaller than a wavelength of the applied field [132].
Therefore, the continuous media requirement for
defining the permittivity and permeability becomes
blurred. The mapping of continuous media properties
onto metamaterial behavior can at times cause paradoxes and inconsistencies [69, 133-137]. However, the
measured EM scattering response in NIM is achieved,
whether or not an effective permittivity and permeability can be consistently defined. Because of the
inhomogeneity in the media, the permittivity and permeability in some of these applications are effective parameters and spatially dispersive and not the intrinsic
properties that Veselago assumed for a material [26,
138]. In some metamaterials and metafilms where the
ratio of the particle size to the wavelength is not small,
boundary transition layers are typically included in the
model so that the terminology of effective permittivity
and permeability can be used. In Sec. 4.6, we described
the criterion of defining a polarization by a Taylor
series expansion of the charge density. The problem
At the relatively longer wavelengths of RF frequencies, (1 × 104 m to 1 mm), only a few classes of intrinsic resonances can be observed. Bulk geometric
resonances, standing waves, and higher-mode resonances can occur at any frequency when an inclusion
has a dimension that is approximately equal to an
integral multiple of one-half wavelength in the material. These geometrical resonances are sometimes misinterpreted as intrinsic material resonances. Most of the
intrinsic resonant behavior in the microwave through
millimeter frequency bands are due to cooperative
ferromagnetic and ferrite spin-related resonances,
antiferromagnetic resonances, microwave atomic
transitions, plasmons and plasmon-like resonances, and
polaritons at metal-dielectric interfaces. Atoms such as
cesium have transition resonances in the microwave
band. Large molecules can also be made to resonate
under the application of high RF frequencies and
THz frequencies. NIM commonly use non-intrinsic splitring structure resonances together with plasma
resonances to achieve unique electromagnetic
response. At optical frequencies, individual molecules
or nanoparticles can sometimes be resonated directly or
through the use of plasmons.
Water has a strong relaxation in the gigahertz
frequency range and water vapor has an absorption
peak in the gigahertz range, liquid water has no dielectric resonances in the microwave range. The resonances
of the water molecule occur at infrared frequencies at a
wavelength around 9 μm. In magnetic materials,
ferromagnetic spin resonances occur in the megahertz
to gigahertz to yielding MMW bands. Antiferromagnetic resonances can occur at millimeter frequencies. Gases such as oxygen with a permanent magnetic
moment can absorb millimeter waves [122]. In the
frequency region from 22 to 180 GHz, water-vapor
absorption is caused by the weak electric dipole
Volume 117 (2012)
Journal of Research of the National Institute of Standards and Technology
of whether these composite materials can be described
in terms of a negative index is complicated by the
issues described above. The measured permittivity
tensor is an intrinsic property and should not depend on
the field application or the sample boundaries, if the
electrodynamic problem is modeled correctly.
Pendry [127] introduced the idea of constructing a
lens from metamaterials that could achieve enhanced
imaging that is not constrained by the diffraction limit.
It should be noted that microwave near-field probes
also have the capability of subwavelength imaging by
using the near field around a probe tip (see Sec. 16).
The real part of the susceptibility can be negative
around the resonance frequency (see Fig. 7). A similar
equation can apply for a resonance in a split-ring or
other resonator to obtain a negative real part of the
In most electromagnetic material applications the
plane-wave propagation vector and group velocities are
in the same direction. Backward waves are formed
when the group velocity and phase velocity are in opposite directions. This can be produced when the real parts
of the permittivity and permeability are simultaneously
negative. When this occurs, the refractive index is
negative, n = − ε ′r
Veselago’s Argument for NIM Materials With
Both ε′r(eff ) < 0 and μ′r(eff ) < 0
Because of this result, researchers have argued that this
accounts for the anomalous refraction of waves through
NIMs, reverse Cherenkov radiation, and reverse
Doppler effect, etc.
Snell’s law for the reflection of an interface between
a normal dielectric and an NIM satisfies θinc = θreflection,
but the refracted angle in NIM is θtrans = sgn (nNIM)
sin −1 ( norm sin θinc ) [140]. In addition, the TEM
In this section we overview the theory behind NIM
[26]. The real parts of the permittivity and permeability can be negative over a band of frequencies
during resonances. Of course, to maintain energy
conservation in any passive material, the loss-factor
part of the permittivity and permeability must always
be positive. This behavior has only been recently
exploited to achieve complex field behavior [26, 62,
67, 88, 127, 139].
wave impedance of plane waves for NIM is Z =
Polarization resonance is usually modeled by a
damped harmonic-oscillator equation. The simple
harmonic-oscillator equation for the polarization P (ω)
for single-pole relaxation can be written as Eq. (58).
For a time-harmonic-field approximation, the effective
dielectric susceptibility has the form
+ iωτ +1
− μr′ − i μr′′
. If only the real part of the
− ε′r − i ε′′r
the permittivity or the permeability are negative, then
damped field behavior is attained.
These periodic artificial materials do produce interesting and potentially useful scattering behavior; however since they often involve resonances in structures
that contain metals, they are lossy [62]. There has
been debate in the literature over how to interpret the
observed NIM behavior, and some researchers believe
the results can be explained in terms of surface waves
rather than invoking NIM concepts [137].
The approach used to realize a negative effective
magnetic permeability is different from that for
obtaining a negative effective ε′r. Generally, split-ring
resonators are used to obtain negative μ′r , but recently
there has been research into the use of TM and TE
resonant modes in dielectric cubes [69] or ferrite
spheres to achieve negative properties [62, 141].
Dielectric, metallic, ferrite, or layered dielectricmetallic inclusions such as spheres can be used to
achieve geometric or coupled resonances and therefore
simultaneous negative effective ε′ and μ′ [62]. A commonly used approach to obtain a negative permittivity
is to drive the charges in a wire or free charge in a
Fig. 12. The regions of the permittivity-permeability space for
different metamaterial behaviors.
χ d (ω) =
− μ r′ = i ε ′r i μ ′r = − ε ′r μ ′r .
Volume 117 (2012)
Journal of Research of the National Institute of Standards and Technology
semiconductor or plasma near resonance. Dielectric
resonance response occurs in semiconductors in the
terahertz to infrared range and in superconductors in
the millimeter range. The real part of the permittivity
for a plasma, according to the high-frequency Drude
model, can be negative (ε = ε0(1 – ω2p / ω2)).
There are a number of metrology issues related to
NIM. These include the problem of whether the field
behavior should be modeled as the result of negative
intrinsic permittivity and permeability and negative
index or instead be treated as a scattering problem.
This problem is related to the wave length of the
applied fields versus the parameters of the embedded
resonators. Although the scatterers are generally
smaller than a wavelength of the applied field, they
are not always significantly smaller. When the lattice
spacing a between particles satisfies [142]
defined [62]. Even within these bounds the properties
are not intrinsic permittivity and permeability as
defined previously and are spatially dispersive. A
second issue is the determination of the NIM specimen
length and boundaries to be used to model the array of
macroscopic scatterers (see [69] and references therein
for an analysis of this problem). Another area of debate
is where in the resonance region is a permittivity and permeability well defined.
interface between a dielectric and a conductor,
analogous to the propagation of the Sommerfeld
surface wave on a conductor/dielectric interface.
Plasma polaritons decay exponentially away from the
surface. The effective wavelengths of plasmons are
much shorter than that of the incident EM field and
therefore plasmons can propagate through structures
where the incident radiation could not propagate
through. This effect has been used in photonics and in
microwave circuits through the use of metamaterials.
For example, thin metal films can be embedded in
dielectrics to form dielectric waveguides. Plasmonics is
commonly used for imaging where the fields are used
to obtain a sub-wavelength increase in resolution of
10 to 100 times. Colors in stained glass and metals are
related to the plasma resonance frequency, due to the
preferred reflection and absorption of specific wavelengths. High-temperature superconductors also have
plasmonic behavior and a negative ε′r due to the
complex conductivity [91]. If small metallic particles
are subjected to EM radiation of the proper wavelength,
they can confine EM energy and resonate as surface
plasma resonators. Plasmonic resonances have also
been used to clean carbon nanotubes and enhance other
chemical reactions by thermal or nonthermal activation. Plasmons have been excited in metamaterials by
use of a negative permeability rather than negative
permittivity [143].
Bulk Plasmons
ε′r μr′ ωa / c ≤ 1 , then effective properties can be
Plasmonic Behavior
At the interface between a dielectric and metal an
EM wave can excite a quasiparticle called a surface
polariton (see Fig. 13). Plasmons are charge-density
waves of electron gases in plasmas, metals, or semiconductors. Surface polariton plasmons travel on the
Maxwell’s equations with no source-current densities can be used to obtain
− μ0
∂t 2
= ∇×∇×E .
If E ∝ e iωt−ikz , the dispersion relation is
k (k ⋅ E) − k 2 E = −ε r ( k, ω)
For transverse plane waves k . E = 0, and therefore
k = εr (k, ω)ω2 / c2. For longitudinal waves εr (k, ω) = 0
[144] (this condition ε(ω) = 0) also implies the
Lyddane-Sachs-Teller relation [102] for the ratio of the
longitudinal to transverse phonon frequencies that
Fig. 13. Plasmon resonance.
= s .
Volume 117 (2012)
Journal of Research of the National Institute of Standards and Technology
From Eq.(62), in the time domain for the case of
no loss, and P(t) = – Nex (t), where N is the density
of electrons, we obtain the equation for a harmonic
oscillator for bulk longitudinal plasmon oscillations,
d 2x / dt 2 = – ω2p x . The permittivity of a plasmon can be
modeled as
⎛ ω2p
ε(ω) = ε0 ⎜ 1 − 2
⎜ ω
Transmission Through Subwavelength
Under certain conditions, electromagnetic radiation
has been observed to pass through subwavelength
apertures [145-147]. In extraordinary optical (EOT) or
millimeter wave (EMT) transmission, free-space EM
waves impinging on a metal plate with small holes
transmits more energy than would be expected by a
traditional analysis [148]. At optical frequencies, this
transmission is mediated by surface plasmons. At MW
and MMW frequencies, plasmons are not formed on
homogeneous conducting metal plates. However,
plasmon-like behavior can be formed by an appropriate
selection of holes, metal plate thickness, or corrugations to produce a behavior that simulates surface
plasmons. These plasmons-like features that are sometimes referred by the jargon “spoof plasmons”, can be
the origin of extraordinary transmission through the
holes in metal plates at MW to MMW frequencies.
Below the plasmon frequency ω p = Ne / ε0 m , the
plasma is attenuative and follows the skin-depth
formulas of a metal.
Above the plasma frequency, the real part of the
permittivity becomes negative.
Surface Plasmons
Surface plasmon polaritons [144] can travel at the
interface of a metal and dielectric to produce surface
wave guiding. Plasmonic surface waves have fields that
decay rapidly from the surface interface. For example,
for a 1 μm excitation wavelength, the waves can travel
over 1 cm, leading to the possibility of applications in
microelectronics. Surface plasmonic EM waves can be
squeezed into regions much smaller than allowed by
the diffraction limit. Obtaining the negative effective
ε′rp for plasmons in the megahertz through MMW range
would require the use of NIM. Some applications of
plasmonic behavior can also be tuned by a dc external
magnetic field, and the applied magnetic field produces
a plasmon with a tensorial permittivity.
For surface plasmons, the effective wavelengths of
the plasmons can be much less than that of the exciting
EM fields due to the difference in sign of the permittivities in a metal and dielectric. For example, for a
plasmon at an interface between a metal and a dielectric substrate, if the permittivity of the plasmon is ε′rp
and that of the substrate is ε′rd , then the dispersion
Behaviors in Structures Where ε′r(e f f ) → 0
There are applications where a material is constructed in such a way so that the real part of the
“effective” permittivity is close to 0 (ENZ) (see Fig. 7
as ε′r → 0). This is closely related to plasmon-like
behavior. In this case, the EM behavior simulates static
behavior in that ∇ × H = 0 and ∇ × E = iωμH, which
implies ∇2E = 0. In this case, the phase velocity
approaches infinity and the guided wavelength
becomes infinite, which is analogous to cutoff in a
waveguide (λc ) [47]. This type of behavior can be
achieved for a waveguide near cutoff. The equation
for the guided wavelength in a waveguide is
λg =
εr μr
ω ⎛ 2π ⎞
−⎜ ⎟
c2 ⎝ λc ⎠
⎛ λ ⎞
1−⎜ ⎟
⎝ λc ⎠
where λc is the cutoff wavelength of the guide. Due to
the long effective wavelength near cutoff, the phase of
the wavefront changes minimally. Because the effective
permittivity goes through zero near resonance, we
can think of ENZ as a resonance condition similar to
the propagation cutoff in a waveguide when there is
resonance in the transverse plane. This type of behavior
is achieved, for example, if we have a low-loss dielectric
relation is k = 2 π / λ = ( ω / c) ε rd ε rp /( εrd + εrp ) [144].
When R (ε′rp ) < 0 and ⏐R (ε′rp )⏐ is slightly larger than
ε′rd , then we see that the wavelength becomes very
short in comparison to that of the applied field. This is
also attained by application of laser light to nanoparticles to obtain a resonant state. However, this can also
happen in coupled microwave resonant structures.
Volume 117 (2012)
Journal of Research of the National Institute of Standards and Technology
of length L that completely fills the cross section of a
waveguide (see Fig. 24). Near the cutoff frequency the
material could be thought of as having an effective permittivity ε′r (e f f ) ≈ 0. This same behavior is reminiscent of
a cavity, because as the transmission attains a maximum,
reflection is a minimum, and the reactance goes to 0 near
resonance. The ε′(e f f ) in this model violates the condition
for an intrinsic permittivity since the applied field
wavelength (λ = 1 / f √ εμ ), must be much larger than the
feature size. It has been argued that in ENZ, unlike in a
normal wire, the displacement current dominates over
the charge current in transporting the EM waves [146].
There could be analogous effective permeability going to
zero μ ′r (e f f ) → 0 (MNZ) behavior.
Macroscopic to Mesoscopic Heating
and Electromagnetic-Assisted
15.1 Overview of EM Heating
15.1.1 Dielectric and Magnetic Heating
In EM wave interactions with materials, some of the
applied energy is converted into heat. The heating that
takes place with the application of high-frequency
fields is due to photon-phonon processes modeled by
the friction caused by particle collisions and resistance
to dipole rotation. Over the RF spectrum, heating may
be volumetric at low frequencies and confined to
surfaces at high frequencies. Volumetric heating is due
to the field that penetrates into the material producing
dissipation through the movement of free ions and the
rotation of dipolar molecules. Nanocomposites can be
heated volumetrically by RF EM fields, lasers, and
terahertz applicators. Since the skin depth is long at
low frequencies, the heating of nanoparticles is not
efficient. In the microwave band the heating of very
small particles in a host material is limited by the loss
and density of particles in the material, the power level
of the source, and the diffusion of heat to the surroundings. Plasmon resonances in the infrared to visible
frequencies can be used to locally heat particles [153].
At high frequencies, heat may be absorbed locally in
particles in slow modes where there may be a time lag
for heat to dissipate into the phonon bath when the
fields are removed.
The history of practical RF heating started in the era
when radar was being developed. There are stories of
where engineers sometimes heated their coffee by
placing it near antennas. Also there are reports a
researcher working on a magnetron that noticed that the
candy bar in his pocket had melted when he was near
the high-frequency source.
In a microwave oven, water and bound water are
heated by the movement of free charge and nonresonant rotation [154]. Because the water molecules at
these frequencies cannot react in concert with the field,
energy is transferred from the field energy into kinetic
energy of the molecules in the material. In dielectric
materials at low frequencies, as frequencies increase
into the HF band, the rotations of the molecules tend to
lag the electric field, and this causes the electric field to
have a component in phase with the current. This is
especially true in liquids with hydrogen bonding, where
the rotational motion of the bonding is retarded by the
interconnections to other molecules. This causes
energy in the electric and magnetic fields to be converted into thermal energy [155]. Some polymer molecules
Modeling Electrical Properties to Produce
Cloaking Behavior
Recently, there have been many research papers that
examine the possibility of using the electrical properties
of artificial materials to control the scattering from an
object in such a way as to make the object appear invisible to the applied EM field [129, 130, 149]. This is distinct from radar-absorbing materials, where the applied
field is absorbed by ferrites or layered, lossy materials.
Research in this area uses the method of transformation
optics [149, 150] to determine the material properties
that produce the desired field behavior. In order to exhibit a typical cloaking property, Shivola [151] derived
simple equations for a dielectric-layered sphere that are
assigned permittivities to produce a nearly zero effective
polarizability. Recently, complex arrangements of nonresonant metamaterials have been designed by inverse
optical modeling to fabricate broadband electromagnetic
cloaks [129, 152].
Fig. 14. Cartoon of waves around a cloaked sphere.
Volume 117 (2012)
Journal of Research of the National Institute of Standards and Technology
where κ is the thermal conductivity dyadic. The mass
density is ρd and the specific heat is cp . For nanosystems, the heat transfer is more complicated and may
require modeling phonon interactions. Also, the above
heat transfer expression is only approximate for
nanoscale materials. The temperature rise obtained by
application of EM energy to a material can be estimated by use of the power dissipation relation in Eq. (130).
When the temperature is changed by ΔT , the thermal
energy-density increase is Qh = ρdcp ΔT . The power
dissipated per unit volume by an electric field interacting with a lossy dielectric material is Pd = (1/2)σ|E|2,
where σ is the conductivity. Therefore, the temperature
rise in a specimen with density ρm through heating with
a power Pd for a time Δt is
that have low friction, such as glycerol in solution, tend
to rotate without significant molecule-molecule interactions and therefore produce little thermal energy.
The power dissipated in a bulk lossy material in a
time-harmonic field is
P(ω, T ) =
(ω) 2 + μ ′′(ω, T ) Η
(ω) 2 ) dV.
ω ( ε′′(ω, T ) E
The total entropy produced per unit time at a temperature T is P(ω,T ) / T. Equation (130) is modified for
very frequency-dispersive materials [116]. Dielectric
losses in ohmic conduction and Joule heating originate
in the frictional energy created by charges and dipoles
that are doing work against nonconservative restoring
forces. Magnetic losses include eddy currents,
hysteresis losses, and spin-lattice relaxation. Some of
the allocated heating frequencies are given in Table 3.
ΔT =
Table 3. Heating Frequencies
Frequency (MHz)
Wavelength (cm)
the skin depth δ s ≈ (1/ ω) μ ′ε′/ 2
(1 +tan 2 δ) −1, we
see that fields at lower frequencies will penetrate more
deeply (δ s → 2 c ε′ / ω μ r′ ε′′r ) . In order to obtain the
same dissipative power densities as those at higher
frequencies, the electric field strength at a lower frequency would have to increase. For example, to obtain
the same power densities at two different frequencies
we must have (ε 1″ (ω1)ω1 / ε 2″ (ω2)ω2 = |E2|2 / |E1|2).
The unique volumetric heating capability by EM
fields over broader ranges of frequencies should
stimulate further applications in areas such as
recycling, enhanced oil recovery, and as an aid to
Heating originates from dielectric and magnetic loss
and the strength of the fields. For magnetic materials
the losses relate to μ″(ω) and σdc . In high-frequency
fields, magnetic materials will be heated by both
dielectric and magnetic mechanisms [104, 156]. If
applicators are designed to subject the material to
only magnetic or electric fields, then the heating will
be related only to magnetic or dielectric effects,
When studying dielectric heating we need to also
model the heat transport during the heating process.
This is accomplished by use of the power dissipated as
a source in the heat equation [157]. The transport of
heat through a material is modeled by the thermal
diffusivity αh = κ / ρd cp , where ρd is the density and κ is
the thermal conductivity. In order to model localized
heating, it is necessary to solve the Fourier heat
equation and Maxwell’s equations with appropriate
boundary conditions. The macroscopic heat transfer
equation is
= ∇ ⋅ κ ⋅∇ T + ωε ′′ Ε ,
2ρm c p
The heating rate is determined by the field strength,
frequency, and the loss factor. From the equations for
ρd c p
σ E Δt
Table 4. Radiation Classes and Approximate Photon Energies
Frequency (Hz)
Photon energy (J)
3 × 1020
1.9 × 10– 13
3 × 10
1.9 × 10– 14
1 × 1015
6.4 × 10– 19
Visible light
6 × 10
4.0 × 10– 19
Infrared light
3 × 1012
2.0 × 10– 22
2 × 10
6.0 × 10– 25
High frequency (HF)
1 × 106
6.4 × 10– 28
Volume 117 (2012)
Journal of Research of the National Institute of Standards and Technology
Electromagnetic-Assisted Reactions
as the generation of intense local fields that produce
localized dielectric breakdown or possibly EM-induced
changes in the entropy. Most of the effects seen in
microwave heating are thermal effects due to the
volumetric heating of high-frequency fields [160].
Microwave heating can result in superheating where
the liquid can become heated above the typical boiling
point. For example, in microwave heating, water can be
heated above its boiling temperature. This is due to the
fact that in traditional heating, bubbles form to produce
boiling, whereas in microwave heating the water may
become superheated before it boils.
When RF waves are applied to assist a chemical
reaction, or polymer curing, the observed rate enhancement is due primarily to the effects of microscopic and
volumetric heating. Because chemical reaction rates
proceed in an Arrenhius form τ ∝ exp (E / kBT ), small
temperature increases can produce large reductions in
reaction times. The kinetics of chemical reaction rates
is commonly modeled by the Eyring equation,
keyr =
k BT −ΔG / RT
where h is Planck’s constant, ΔG = ΔH – TΔS is the
Gibb’s free energy, H is Helmholtz’s free energy, ΔS
denotes changes in entropy, and R is the gas constant.
A plot of ln (keyr / T ) = – ΔH / RT + ΔS / R + ln (kB / h)
versus 1 / T can yield ΔS, ΔH, and possibly kB / h.
One would expect that heat transfer by conduction
would have the same effect on reactions as microwave
heating, but this is not always found to be true. Part of
the reason for this is that thermal conduction requires
strong temperature gradients, whereas volumetric
heating does not require temperature gradients.
Because it does not depend on thermal conduction, an
entire volume can obtain nearly the same temperature
simultaneously without appreciable temperature
gradients. In addition, some researchers speculate on
non-thermal microwave effects that are due to the
electric field interacting with molecules in specific
ways that modify the activation energy through
changes in the entropy [158, 159]. Avenues that have
been proposed for nonthermal reactions may be related
to dielectric breakdown that causes plasma of photons
to be emitted, causing photo-reactions. Another avenue
is related to the intense local fields that can develop
near corners or sharp bends in materials or molecules
that cause dielectric breakdown.
Typical energies of microwave through x-ray
photons are summarized in Table 4. Covalent bonds
such as C-C and C-O bonds have activation energies of
nearly 360 kJ / mol, C-C and O-H bonds are in the
vicinity of 400 kJ / mol, and hydrogen bonds are around
4 to 42 kJ / mol. Microwaves are from 300 MHz to
30 GHz and have photon energies from 0.0001 to
0.11 kJ/mol. Therefore, microwave photon bondbreaking events are rare. Nonthermal microwave
effects, therefore, are not likely due to the direct interaction of microwave photons with molecules and, if
they occur at all, and must have secondary origins such
Heat Transfer in Nanoscale Circuits
In microelectronic circuits, higher current densities
can cause phonon heating of thin interconnects that can
cause circuit failure. This heating is related to both the
broad phonon thermal bath and possibly slow thermal
modes where thermal energy can be localized to
nanoscale regions [161, 162]. New transistors will have
an increased surface-to-volume ratio and, therefore, the
power densities could increase. This, combined
with the reduced thermal conductance of the low
conductivity materials and thermal contact resistance
at material interfaces, could lead to heat transport
limitations [162, 163].
Heating of Nanoparticles
When a large number of metallic, dielectric, or
magnetic micrometer or nanometer particles in a host
media are subjected to high-strength RF EM fields,
energy is dissipated. This type of EM heating has been
utilized in applications that use small metallic particles,
carbon black, or palladium dispersed in a material to act
as chemical-reaction initiators and for selective heating
in enhanced drug delivery or tumor suppression [164,
165]. Understanding the total heat-transfer process in
the EM heating of microscopic particles is important. A
number of researchers have found that, due to the
thermal conduction of heat from nanoparticles and the
small volumes involved and the large skin depths of RF
fields, the nanoparticles rapidly thermalize with the
phonon bath and do not achieve temperatures that deviate drastically from the rest of the medium [166]. Only
when there is an appropriate density of particles, is
heating enhanced. There have recently been reports
that thermal energy can accumulate in nanoscale to
molecular regions in slow modes, and it can take
seconds to thermalize with the surrounding heat bath
[166-169]. In such situations, regions may be unevenly
Volume 117 (2012)
Journal of Research of the National Institute of Standards and Technology
heated by field application. However, thermal conduction will tend to smooth the temperature profile within
a characteristic relaxation time. Lasers can selectively
heat micrometer-size particles and by use of plasmonics lasers can heat conducting nanoscale
Due to symmetry and charge neutrality, a polarizable
particle in a uniform electric field will experience no
net force. If a material with a permanent or induced
dipole is immersed in an electric field gradient, then a
dielectrophoretic force on the dipole is formed, as
indicated in Fig. 15 [176]. In a nonuniformelectric
field, the force on a dipole moment p is F = (p . ∇)E.
Macroscopic and Microscopic
High-Frequency Thermal Run-Away
The dielectric loss and thermal conductivity of a
material may possess a temperature dependence so that
the loss increases as temperature increases [170]. This
is due to material decomposition that produces ions as
the temperature increases and results in more loss.
Thermal run away can lead quickly to intense heating
of materials and dielectric breakdown. The temperature
dependence of thermal run away has been modeled
with the dielectric loss factor as εr″ = α0 + α1 (T – T0 )
+ α2(T – T0 )2, where T0 is a reference temperature and
αi are constants [171].
Overview of High-Frequency
Nanoscale Measurement Methods
In the past few decades, a number of methods have
been developed to manipulate single molecules and
dipoles. Methods have been implemented to move,
orient, and manipulate nanowires, viruses, and proteins
that are several orders of magnitude smaller than cells.
These methods allow the researcher to study the electrical and mechanical properties of biological components in isolation. Molecules and cells can be manipulated and measured in applied fields using dielectrophoresis, microwave scanning probes, atomic force
microscopy, acoustic devices, and optical and magnetic tweezers. Some of the methods use magnetic or
electric fields or acoustic fields, others use the EM field
radiation pressure, and others use electrostatic and van
der Waals forces of attraction [139, 172]. Microfluidic
cells together with dc to terahertz EM fields are
commonly used to study microliter to picoliter volumes
of fluids that contain nanoparticles [173-175]. Surface
acoustic waves (SAW) and bulk acoustic waves (BAW)
can be used to drive and enhance microfluidic processes. Since there is a difference of wave velocities in a
SAW substrate and the fluid, acoustic waves can be
transferred into the fluid, to obtain high fluid velocities
for separation, pumping, and mixing.
Fig. 15. Dielectrophoretic Force.
From this the following equation for the dielectrophoresis force on a small sphere of radius r of permittivity εp in a background with permittivity εm has be
derived [177, 178]
⎛ ε p − εm
FDEP = 2πεm r 3R ⎜
⎜ ε p + 2ε m
⎟∇ E .
This force tends to align the molecule along the field
gradient. The force is positive if εp > εm. For dispersive
materials, the attraction or repulsive force can be varied
by the frequency. Dielectrophoresis is commonly used
to stretch, align, move, and determine force constants
of biomolecules such as single-stranded and doublestranded DNA and proteins [179]. Dielectrophoresis
can also be used to separate cells or molecules in a
stream of particles in solution. Usually, dielectrophoretic manipulation is achieved through microfabricated electrodes deposited on chips. For dispersive
Volume 117 (2012)
Journal of Research of the National Institute of Standards and Technology
materials, where the permittivity changes over the
frequency band of interest, there is a cross-over
frequency where there is no force on the molecule.
The approximate force, due to diffusion forces from
particle gradients on a particle with a dimension d, is
Fb = kBT / d. For micrometer particles, the dielectrophoretic field gradients required to overcome this force
is not large. However, for nanoscale particles this field
gradient is much larger.
Spherical particles can be made to rotate through
electrorotation methods [177]. This motion is produced
by a rotating electric field phase around a particle. The
dipole induced in the particle experiences a net torque
due to the dielectric loss that allows the dipole formation to lag the rotating field, as shown in Fig. 16. The
net torque is given by N = p × E. For particles εp in a
matrix εm the torque is [177]
⎛ ε p − εm
N = 4πε′rm r 3R ⎜
E( ω) × E * ( ω) ⎟ .
⎜ ε p + 2ε m
of the particles and the wavelength of the laser light
[180]. Molecules can also be studied by magnetic tweezers with magnetic-field gradients. By attaching magnetic
particles to molecules it is possible to stretch molecules
and determine force constants. Opto-plasmonic tweezers
use radiation from resonant electrons to create patterned
electric fields that can be used through dielectrophoresis
to orient nanoscale objects.
Atomic force microscopy (AFM) is based on
cantilevers. In AFM the force between the probe tip and
the specimen is used to measure forces in the
micronewton range. An AFM probe typically has cantilever lengths of 0.2 mm and a width of around 50 μm.
An AFM can operate in the contact mode, noncontact,
or tapping mode. Force information of the interaction
of the tip with a material is obtained by means of
cantilever bending, twisting, and, in the noncontact
mode, by resonance of the cantilever.
In the microwave range, near-field microwave
scanning probes are commonly used. These probes
have proved valuable to measure the permittivity and
imaging on a surface of a thin film at subwavelength
resolution. These needle probes usually use near-field
microwaves that are created by a resonator above the
probe, as shown in Fig. 17. A shift in resonance
frequency is then related to the material properties
Fig. 16. Electrorotation with probes 90° out of phase.
Fig. 17. Microwave scanning probe system.
under test through software based on a theoretical
model. Therefore, most of these probes are limited to
resonant frequencies of the cavity. Continuous-wave
methods based on microstrip tips have also been
Optical tweezing originates from the EM field gradient obtained from a laser source that produces a field
differential and results in a force on particles. This
effect is similar to dielectrophoresis. The strength of the
radiation pressure on particles is a function of the size
Volume 117 (2012)
Journal of Research of the National Institute of Standards and Technology
Properties and Measurement of Dielectric
inorganic and organic strings of molecules such as
DNA [183-188]. Because they are effectively ordered
in one dimension, they can form a variety of structures
such as rigid lines, spirals, or zigzag pattern. Carbon
nanotubes that have lengths in the millimeters have
been constructed [189].
At these dimensions, quantum-mechanical effects
cannot be totally neglected. For example, the electrons
are confined laterally, which influences the available
energy states like a particle in a one-dimensional box.
This causes the electron transport to be quantized and
therefore the conductance is also quantized (2e2 / h).
The impedance of nanoconductors is on the order of the
quantum resistance h / e2, which is 25 kΩ. For SWNTs,
due to band-structure degeneracy and spin, this is
reduced to 6 kΩ. The ratio of the free-space impedance
to the quantum impedance is two times the fine
structure constant 2α . This high impedance is
difficult to probe with 50 Ω systems [190], and depositing a number of them in parallel has been used to
minimize the mismatch [191].
The resistance of a SWNT depends on the diameter
and chirality. The chirality is related to the tube having
either metallic or semiconducting properties. For
device applications such as nanotransistors, the
nanowires need to be either doped or intrinsic semiconductors. Semiconducting nanowires can be connected to form p-n junctions and transistors [192].
Many nanowires have a permanent dipole moment.
Due to the torque in an electric field, the dipole will
tend to align with the field, particularly for metallic and
semiconducting nanotubes [193].
Nanomaterials could consist of composites of
nanoparticles dispersed in a matrix or isolated particles.
A mixture of conducting nanoparticles dispersed into a
matrix sometimes yields interesting dielectric behavior
[23, 181]. Lewis has noted that the interface between the
nanoparticle and matrix produces unique properties in
nanocomposites [23]. Interfaces and surface charges are
a dominant parameter governing the permittivity and
loss in nanocomposites [23, 181, 182]. Double layers
(Sec. 8) near the particle surface can strongly influence
the properties [23]. In addition, conductivity in some
nanoparticles can achieve ballistic transport.
In order to model a single dielectric nanoparticle in an
applied field the local field can be calculated, as summarized in Sec. 4.3. Kühn et al. [59] studied the local field
around nanoparticles, and they found that use of the
macroscopic field for modeling of a sphere containing
nanoparticles was not valid at below 100 nm. In order to
model small groups of nanoparticles, they found that the
effects of the interface required the use of local fields
rather than the macroscopic field.
When individual nanoparticles are subjected to EM
fields, the question arises of whether it is possible to
define a permittivity of the nanoparticle or whether an
ensemble of particles is required. Whether permittivity
of a nanoparticle is well defined depends on the
number of dipole moments within the particle. If we use
the analogy of a gas, we assume that the large number of
gas molecules together with the vacuum around the particles constitutes a bulk permittivity. This permittivity
does not apply to the individual gas molecules, but rather
to the bulk volume. When individual nanoparticles contain thousands of dipoles, according to criteria of permittivity developed in Sec. 4.6, long-wavelength fields
would allow defining a permittivity of the particle and a
macroscopic field. However, such a permittivity would
be spatially varying due to interfacial effects, and the
definition would break down when there are insufficient
particles to perform an ensemble average [59].
Charge Transport and Length Scales
Electrical conduction through nanowires is strongly
influenced by their small diameter. This constriction
limits the mean free path of conduction electrons [88,
194]. For example in bulk copper the mean free path is
40 nm, but nanowires may be only 1 to 10 nm in
diameter, which is much less than a mean free path and
results in constriction of the current flow.
Carbon nanotubes can obtain ballistic charge transport. Ballistic transport is associated with carrier flow
without scattering. This occurs in metallic nanowires
when the diameter becomes close to the Fermi wavelength in the metal. The electron mean-free path for a
relaxation time τe is le = ντe , and if le is much larger
than the length of the wire, then it is said to exhibit
ballistic transport. Carbon nanotubes can act as antennas and can have plasmonic resonances in the low
terahertz range.
Electrical Properties and the Measurement of
Nanowires are effectively one-dimensional entities
that consist of a string of atoms or molecules with a
diameter of approximately 10–9 meters. Nanowires may
be made of TiO2 , SiO2 , platinum, semiconducting compounds such as gallium nitride and silicon, single
(SWNT) or multi-wall (MWNT) carbon nanotubes, and
Volume 117 (2012)
Journal of Research of the National Institute of Standards and Technology
The Landauer-Buttiker model of ballistic transport
was developed for one-dimensional conduction of
spinless/noninteracting electrons [195, 196]. This
model has been applied to nanowires.
Graphene has shown promise for construction of
transistors due to its high conductivity, but is hampered
by defects. The very high carrier mobility of graphene
makes it a candidate for very high speed radiofrequency electronics [197].
If the noninteracting electrostatic and quantum
impedance are combined, we have
⎛ 1
Z = (L K + L M ) ⎜
A high-frequency nanocircuit model may need to
include the quantum capacitance and kinetic and
magnetic inductance in addition to the classical parameters. The magnetic inductance per unit length
for a nanowire of permeability μ of diameter d and a
distance s over a ground plane is given by [189]
LM =
cosh −1 (2 s/ d ) ≈ ln , typically1(pH/ μm) .
2π d
The kinetic inductance due to quantum effects is
is related to the Fermi velocity ν F , L K = 2 , typically,
2e ν F
16 (nH / μm). At gigahertz frequencies, the kinetic
inductance is not a dominant contribution to the transmission line properties [189]. The electrostatic capacitance between a wire and ground plane in a medium
, typically,
cosh -1 (2 s/d )
8e 2
50 (aF/ μm) . The quantum capacitance is CQ =
hν F
typically, 400 (aF/μm). The electrostatic capacitance is
found to dominate over the quantum capacitance at
gigahertz frequencies. At terahertz frequencies and
above they are of the same order of magnitude, and
both should be included in calculations for nanowires.
Burke notes that the resistance and classical capacitance dominates over the quantum inductance and
capacitance and are not important contributions at gigahertz frequencies, but may be important at terahertz
frequencies [189]. The wave velocity in nanowires is
approximated by
with permittivity ε is CES =
= 2 .
C 2e
Random Fields, Noise, and
Fluctuation-Dissipation Relations
17.1 Electric Polarization and Thermal Fluctuations
As transmission lines approach dimensions of tens of
nanometers with smaller currents, thermal fluctuations
in charge motion can produce small voltages that can
become a significant source of noise [199]. The random
components of charge currents, due to brownian
motion of charges, produce persistent weak random
EM fields in materials and produces a flow of noise
power in transmission lines. These fields contribute to
the field felt by the device. Random fields also are
important in radiative transfer in blackbody and nonblackbody processes.
The quantum characteristic impedance is
Whereas the free-space impedance is 377 Ω, the quantum capacitance and inductance of carbon nanotubes
yields an impedance of approximately 12.5 kΩ.
The resistivity of nanowires and copper are generally
of the same order of magnitude. The ballistic transport
properties at small scales represents an advantage; however, the resistance is still quite high. Copper interconnects have less resistance until the conductor sizes drop
below about 100 nm; currently the microelectronic
industry uses conductors of smaller size. This is an
origin of heating [14, 198]. Because the classical resistance is calculated from R / L = ρ / A, where ρ is resistivity, L is length, and A is the cross-sectional area, the small
area of a SWNT limits the current and increases the
resistance per unit length and the impedance. Due to the
high impedance of nanowires, single nanowires have
distinct disadvantages; for example, carbon nanotubes
may have impedances on the order of 104 Ω. Bundles of
parallel nanowires could form an interconnect [191].
Tselev et al. [191] performed measurements on bundles
of carbon nanotubes that were attached to sharp metal
tips by dielectrophoresis on silicon substrates. Electronbeam lithography was used to attach conductors to the
tubes. High-frequency inductance measurements from
10 MHz to 67 GHz showed that the inductance was
nearly independent of frequency. In modeling nanoscale
antennas made from nanowires, the skin depth as well as
the resistance are important parameters [189].
Distributed Parameters and Quantized
νF ≈
Volume 117 (2012)
Journal of Research of the National Institute of Standards and Technology
Thermal fluctuations in the dipole moments in dielectric and magnetic materials influence the polarization
and are summarized in the well known fluctuation-dissipation relationships. These relationships are satisfied
for equilibrium situations. Equilibrium is a state where
the entropy is a maximum and macroscopic quantities
such as temperature, pressure, and local fields are well
defined. Fluctuation-dissipation relationships can be
obtained from the linear-response formalism (Sec. 4.4)
that yields the susceptibility in terms of the Fourier
transform of the associated correlation functions. By
use of Eq. (30), an expression can be written for the
susceptibility in terms of the polarization
χe′′ (ω) =
relation for the magnetic loss component can be
derived in a way similar to the electric response:
χ′′m (ω) =
k BT
2k B T
(< M(0) M(t) >)sin( ωt) dt
< M(0) M(t) > cos( ω t) dt .
Thermal Fields and Noise
Due to thermal fluctuations, brownian motion of
charges produce random EM fields and noise. In noise
processes the induced current
density can be related
microscopic displacement D and induction fields B.
The cross-spectral density of random fields is
defined as [18]
(< P(0) P(t) >)sin( ωt) dt
< P(0) P(t) > cos( ω t) dt .
f m (t )sin( ωt) dt
V μ0
k BT
ωV μ0
2k B T
f e (t )sin( ωt) dt
S Ekl (r, r ′, ω) =
<Ek ( r, t) El ( r′, t′)> e−iω( t −t ′) d( t − t′) .
Equation (139) is a fluctuation-dissipation relationship
that is independent of the applied field. In this
approach, if the correlation function is known, then the
material properties can be calculated. However, in
practice most material properties are measured through
applied fields. The interpretation of this relationship
is that the random microscopic electric fields in a
polarizable lossy medium produce fluctuations in the
polarization and thereby induces loss in the decay to
equilibrium. These fluctuations can be related to
entropy production [44, 61]. We can obtain an
analogous relation for the real part of the susceptibility
by use of Eq. (29).This relation relates the real part of
the susceptibility to fluctuations
χ′e (ω) =
k BT
The relationship to the time-harmonic correlation
function for the field components is
< Ek (r, ω) El ∗ ( r′, ω′)> = 2 πSEkl ( r, r′, ω) δ( ω − ω′) .
Thermally induced fields can be spatially correlated
[17] and can be modeled to first order as
2iΘ(ω, T ) ∗
< D (ω, r )D∗ ( ω, r ′) >=
( ε − ε ) δ( r - r′) ,
2iΘ(ω, T ) ∗
< B (ω, r )B ∗ ( ω, r ′) >=
( μ − μ ) δ( r - r ′) ,
< B (ω, r )D ( ω, r ′) > = 0 ,
f e (t )cos( ωt) dt
(< P(0) P( t) >)cos( ωt) dt
< P(0) P(t) > sin( ωt) dt .
where Θ(ω,Τ ) = (ω / 2)coth(ω / 2kBT ). Θ → kBT for
kBT >> ω .
The voltage V and current I in a microscopic transmission line with distributed noise sources νn and in that
are caused by random fields can be modeled by coupled
differential equations as shown in [199].
A special case of Eq. (144) is the well-known
Nyquist noise relation for voltage fluctuations from a
resistance R over a bandwidth Δf is
Magnetic Moment Thermal Fluctuations
Magnetic-moment fluctuations with respect to
signal-to-noise limitations are important to magneticstorage technology [200]. This noise can also be
modeled by fluctuation-dissipation relations for
magnetic response. The linear fluctuation-dissipation
< ν 2 > = 4 k BTR Δf .
Volume 117 (2012)
Journal of Research of the National Institute of Standards and Technology
ΔS = ΔQh / T, or by spontaneous processes in the
relaxation of a system from nonequilibrium to an
equilibrium state. In EM interaction with materials, we
can produce entropy either through the dissipation of
the fields in the material or by relaxation processes.
Relaxation processes are usually spontaneous processes from nonequilibrium into an equilibrium state.
The entropy is defined as S = kB ln(W), where W is
the number of accessible states. Entropy is a cornerstone of thermodynamics and non-equilibrium thermodynamics. In thermodynamics the free energy is defined
in terms of the internal energy U as F = – kBT lnZ, where
Z is the partition function. The entropy is also defined in
terms of the free energy as
Fluctuations and Entropy
In thermal equilibrium macroscopic objects have a
well-defined temperature, but in addition there are
equilibrium temperature fluctuations. When the particle
numbers in a system decrease, the thermodynamic
quantities such as temperature and internal energy, have
a less precise meaning than in a large-scale system [61,
201]. In nanosystems, fluctuations in particle energy,
momentum, and local EM fields can be large enough to
affect measurements. These fluctuations translate into
fluctuations in the measured EM fields, internal energy,
temperature, and heat transfer. A system that is far from
thermal equilibrium or very small may not have a welldefined temperature, macroscopic internal energy, or
specific heat [199, 202, 203]. When the applied driving
fields are removed, some polymers and some spin
systems have relaxation times of seconds to hours until
they decay from a nonequilibrium state to an equilibrium state. In these types of nonequilibrium relaxation
processes, equilibrium parameters such as temperature
have only a fuzzy meaning. Fluctuation-dissipation
relations that are used to define transport coefficients in
equilibrium do not apply out of equilibrium.
Nanosystems operate in the region between quantummechanical and macroscopic description and between
equilibrium and nonequilibrium states. Whereas
Johnson noise is related to fluctuations in equilibrium
voltages, there is a need for theoretical work that yields
results that compare well to measurements in this
transition region. As an example, Hanggai et al. showed
that the theoretical bulk definitions for specific heat and
entropy in some nanosystems break down in the high or
low temperature limits [204]. Noise also occurs in nonequilibrium systems and the theoretical foundations are
not as well developed as in thermal equilibrium.
S (T ) = −
In thermodynamics, temperature is defined as
δS (T ) 1
= .
A very general evolution relation for the macroscopic entropy production rate Σ(t) in terms of microscopic entropy production rate s. (t) was derived from
first principles by use of a statistical-mechanical theory
[19, 44, 61, 89, 206]:
Σ(t ) =
< s( t)T ( t, τ)(1 − P( τ)) s( τ) > d τ , (150)
where s. (t) satisfies (< s. (t) > = 0), Σ(t) is the net macroscopic entropy production in the system, and T and P
are evolution operators and projection operators,
respectively. The Johnson noise formula is a special
case of Eq. (150) near equilibrium, when Σ(t) = I2R / T
and s. (t) = (1/2)Iν(t) / T, (with < ν(t) > = 0) is a
fluctuating voltage variable, and I is a bias current.
Fluctuations and Entropy Production
For reliable operation, microelectronic interconnects
require a stable thermal environment because thermal
fluctuations could potentially damage an interconnect
or nano-transistor [205]. An understanding of thermodynamics at the nanoscale and the merging of electromagnetism and non-equilibrium thermodynamics is
important for modeling small systems of molecules.
Modeling of thermal fluctuations can be achieved by
relating Nyquist noise to fluctuations in thermal energy.
Another approach away from equilibrium is to use the
concept of entropy production [44]. Entropy can be
increased either by adding heat to a material,
Dielectric Response of Crystalline,
Semiconductors, and Polymer
Losses in Classes of Single Crystals and
Amorphous Materials
A class of dielectric single-crystal materials have
very low loss, especially at low temperatures. The low
loss is related to the crystal order, lack of free charge,
Volume 117 (2012)
Journal of Research of the National Institute of Standards and Technology
and the low number of defects. Anomalously low
values of the dielectric loss in single-crystal alumina at
low temperatures were reported in 1981 [14, 207]. In
this study, dielectric resonators were used to measure
the loss tangent because cavity resonators do not have
the required precision for very-low-loss materials.
Since then, there has been a large body of research
[208, 209] performed with dielectric resonators that
supports these results. Braginsky et al. [207] showed
that the upper bound for loss in high-quality sapphire
was 1.5 × 10–9 at 3 GHz and at T = 2 K. These reports
were supported by Strayer et al. [210]. These results are
also consistent with the measurements by Krupka et al.
[209], who used a whispering-gallery mode device to
measure losses. Very low loss is obtained in sapphire,
diamond, single-crystal quartz, MGO, and silicon. Low
loss resonators have been studied at candidates for
frequency standards.
The whispering-gallery mode technique is a particularly accurate way of measuring the loss tangent of
materials with low loss [14]. These researchers claim
that the loss tangent for many crystals follows roughly
a f 2 dependence at low temperatures.
In nonpolar materials, dielectric loss originates from
the interaction of phonons or crystal oscillations with
the applied electric field. In the absence of an applied
electric field, the lattice vibrates nearly harmonically
and there is little phonon-phonon interaction. The
electric-field interaction modifies the harmonic elastic
constant and thereby introduces an anharmonic
potential term. The anharmonic interaction allows
phonon-phonon interaction and thereby introduces loss
[73]. Some of the scattering of phonons by other
phonons is manifested as loss.
The loss in many crystals is due to photon quanta of
the electric field interacting with phonons vibrating in
the lattice, thereby creating a phonon in another branch.
Dielectric losses originate from the electric field
interaction with phonons together with two-, three-, and
four-phonon scattering and Umklapp process [73]. The
three- and four-quantum loss corresponds to transitions
between states of the different branches. Crystals with
a center of symmetry have been found to generally
have lower loss than ones with noncentrosymmetry.
The temperature dependence also depends on the
crystal symmetry. For example, a symmetric molecule
such as sapphire has much lower loss than noncentrosymmetric ferroelectric crystals such as strontium
barium titanate. Quasi-Debye losses correspond to
transitions, which take place between the same branch.
In centro-symmetric crystals three- and four-quantum
processes are dominant. In noncentro-symmetric
crystals the three-quantum and quasi-Debye processes
Gurevich and Tagantsev [73] studied the loss tangent
for cubic and rhombohedric symmetries for temperatures far below the Debye temperature TD = 1047 K.
For these materials, the loss tangent can be modeled as
tan δ =
ω2 (k BT ) 4
ερν 5 (k BTD ) 2
where ε is permittivity, ρ is density, ν is speed of sound
in air. For hexagonal crystals, without a center of
ω(k BT )3
ω(k BT )5
εμν5 2 (k BTD ) 2
tan δ =
and with symmetry,
tan δ =
For many dielectric materials with low loss, Gurevich
showed that there is a universal frequency response of
the form tan δ ∝ ω.
The loss tangent in the microwave band of many
low-loss ceramics, fused silica, and many plastics and
some glasses increases nearly linearly as frequency
increases [211]. For materials where the loss tangent
increases linearly with frequency, we can interpolate
and possibly extrapolate microwave loss-tangent
measurement data from one frequency range to another
(Fig. 6). This approach is, of course, limited. This
behavior can be understood in terms of Gurevich’s
relaxation models [73] or by the moment expansion in
This behavior is in contrast to the model of Jonscher
[213] who has stated that χ″ / χ′ is nearly constant with
frequency in many disordered solids.
Electric Properties of Semiconductors
Excellent reviews of the dielectric properties of
semiconductors in the microwave range have been
given by Jonscher and others [14, 213-217]. The dc
conductivities of semiconductors are related to holes
Volume 117 (2012)
Journal of Research of the National Institute of Standards and Technology
and free charge. In the gigahertz region, the total loss in
most semiconductors decreases significantly since the
effects of the dc conductivity decreases; however, the
dielectric component of loss increases. For gallium
arsenide and gallium nitride the conductivity is
relatively low. Figure 18 shows measurement results on
the permittivity of high-resistivity gallium arsenide as a
function of frequency. These measurements were made
by a mode-filtered TE01 X-band cavity. Silicon
semiconductors can exhibit low to high loss depending
on the level of dopants in the material. There are
Schottky barriers at the interface between semiconductors and metals and at p-n junctions that produce
doping the material with small amounts of impurity
atoms. These additional carriers require much less
thermal energy in order to contribute to σdc . This
results in more carriers becoming available as the
temperature increases, until ionization of all the
impurity atoms is complete.
For temperatures above the full ionization range of
the dopants, σdc is increasingly dominated by μn and μp.
In semiconductors such as silicon, the mobility of the
charge carriers decreases as the temperature increases,
due primarily to the incoherent scattering of the
carriers with the vibrating lattice. At a temperature Ti ,
intrinsic effects begin to contribute additional charge
carriers beyond the maximum contributions of the
impurity atoms, and σdc begins to increase again [215,
216, 219-222].
Overview of the Interaction of RF
Fields With BiologicalMaterials
RF Electrical Properties of Cells, Amino
Acids, Peptides, and Proteins
In this section, we will overview the dielectric relaxation of cells, membranes, proteins, amino acids, and
peptides [97, 223-229]. This research area is very large
and we summarize only the most basic concepts as they
relate to RF fields.
Dielectric response of biological tissues to applied
RF fields is related to membrane and cell boundaries,
molecular dipoles, together with associated ionic fluids
and counterions [230]. The ionic solution produces
low-frequency losses that are very high. As a consequence of these mobile charge carriers, counterions
adhere to molecular surfaces, interface charge causes
Maxwell-Wagner capacitances, and electrode polarization is formed at electrode interfaces. All of these
processes can yield a very high effective ε′r at low
frequencies. Some of the effects of the electrodes can
be corrected for by use of standard techniques [230,
231] (Sec. 8).
Some biological tissues exhibit an α relaxation in the
100 Hz to 1 kHz region due to dipoles and MaxwellWagner interface polarization, another β relaxation in
the megahertz region due to bound water, and γ relaxation in the microwave region due to the relaxation of
water and water that is weakly bond.
Amino acids contain carboxyl (COOH) groups,
amide (NH2 ) groups, and side groups. The side groups
and the dipole moment of the amino and carboxyl
groups determine most of the low-frequency dielectric
properties of the acid. Some of the side groups
Fig. 18. Relative permittivity ε′r of gallium arsenide measured by an
X-band cavity [218]. Start, middle, and terminus refer to different
specimens taken from the same boule. For these measurements
the Type B expanded relative uncertainty at 10 GHz in ε′r was
U = kuc = 0.02 (k = 2), where k is coverage factor.
The conductivities of semiconductors at low
frequencies fall between those of metals and
dielectrics. The theory of conductivity of semiconductors begins with an examination of the phenomena in
intrinsic (undoped) samples. At temperatures above
0 K, the kinetic (thermal) energy becomes sufficient to
excite valence band electrons into the conduction band,
where an applied field can act upon them to produce a
current. As these electrons move into the conduction
band, holes are created in the valence band that effectively become another source of current. The total
expression for the conductivity includes contributions
from both electrons and holes and is given by
σdc = q (nμn + pμp ), where q is charge, n is the electron
density, p is the hole density, and μn and μp are the
electron mobility and hole mobility, respectively.
In intrinsic semiconductors, the number of charge
carriers produced through thermal excitation is relatively small, but σdc can be significantly increased by
Volume 117 (2012)
Journal of Research of the National Institute of Standards and Technology
are polar, while others are nonpolar. When ionized, the
amino and carboxyl groups have positive and negative
charges, respectively. This charge separation forms a
permanent dipole (Fig. 5). α amino acids have an
amino group and carboxyl group on the same carbon
denoted Cα and α-amino acids have a dipole moment
of 15 to 17 debyes (D) (1 debye equals 3.33 × 10–30
coulomb-meter). β amino acids have a CH2 group
between the amino and carboxyl groups, which
produces a large charge separation and therefore a
dipole moment on the order of 20 D. For a very good
overview see Pethig [223]. Peptides are formed from
condensed amino acids. A peptide consists of a collection of amino acids connected by peptide bonds.
Peptide bonds provide connections to amino acids
through the CO-NH bond by means of the water molecule as a bridge. The peptide unit has a dipole moment
on the order of 3.7 D. Chains of amino acids are called
polyamino acids or polypeptides. These are terminated
by an amide group on one end and a carboxyl group on
the other side. Typical dipole moments for polypeptides
are on the order of 1000 D.
movement of these nearly free protons in the applied
field or the polarization of counterion sheaths around
molecules [233]. Strong protonic conductivity has also
been observed in DNA. At present, the consensus is
that polar side chains and both permanent dipoles and
the proton-induced polarization contribute to dielectric
relaxation of proteins.
In the literature three dielectric relaxations in
proteins have been identified [231]. These are similar
to that in DNA. The first is the α relaxation in the
10 kHz to 1 MHz region and is due to rotation of the
protein side chains. The second minor β relaxation
occurs in the 100 MHz to 5 GHz range and is thought
to be due to bound water. The third γ relaxation is
around 5 GHz to 25 GHz and is due to semi-free water.
Nucleic acids are high-molecular mass polymers
formed of pyrimidine and purine bases, a sugar, and
phosphoric-acid backbone. Nucleic acids are built up of
nucleotide units, which are composed of sugar, base,
and phosphate groups in helical conformation.
Nucleotides are linked by three phosphates groups,
which are designated α, β, and γ . The phosphate groups
are linked through the pyrophosphate bond. The
individual nucleotides are joined together by groups of
phosphates that form the phosphodiester bond between
the 3′ and 5′ carbon atoms of sugars. These phosphate
groups are acidic. Polynucleotides have a hydroxyl
group at one end and a phosphate group on the other
end. Nucleosides are subunits of nucleotides and
contain a base and a sugar. The bond between the sugar
and base is called the glycosidic bond. The base can
rotate only in the possible orientations about the
glycosidic bond.
Watson and Crick concluded through x-ray diffraction studies that the structure of DNA is in the form of
a double-stranded helix. In addition to x-ray structure
experiments on DNA, information has been gleaned
through nuclear magnetic resonance (NMR) experiments. Types A and B DNA are in the form of righthanded helices. Type Z DNA is in a left-handed conformation. There is a Type B to Z transition between
conformations. A transition from Type A to Type B
DNA occurs when DNA is dissolved in a solvent [234].
The Watson-Crick conception of DNA as a uniform
helix is an approximation. In reality, DNA exists in
many conformations and may contain inhomogeneities
such as attached proteins. In general, double-stranded
DNA is not a rigid rod, but rather a meandering chain.
Once formed, even though the individual bonds
composing DNA are weak, the molecule as a whole is
very stable. The helical form of the DNA molecule
produces major and minor grooves in the outer
Table 5. Approximate dipole moments
Dipole Moment (D)
Typical protein
Amino acid
Polyamino acids can be either in the helical or
random-coil phase. In the helical state, C = O bonds are
linked by hydrogen bonds to NH groups. The helix can
either be right-handed or left-handed; however, the
right-handed helix is more stable. Generally, polyamino
acids have permanent dipole moments and dielectric
relaxation frequencies in the kilohertz region [232].
The origin of relaxation in proteins has been debated
over the years. Proteins are known to be composed of
polyamino acids with permanent dipole moments, but
they also have free and loosely bound protons. These
protons bind loosely to the carboxyl and amino groups.
Kirkwood et al. hypothesized that much of the
observed relaxation behavior of proteins is due to
Volume 117 (2012)
Journal of Research of the National Institute of Standards and Technology
outer surface of the molecule. There are also boundwater molecules in the grooves. Many interactions
between proteins or protons with DNA occur in these
The helix is formed from two strands. The bases in
adjacent strands combine by hydrogen bonding, an
electrostatic interaction with a pyrimidine on one side
and purine on the other. In DNA, the purine adenine
(A) pairs with the pyrimidine thymine (T). The purine
guanine (G) pairs with the pyrimidine cytosine (C). A
hydrogen bond is formed between a covalently bonded
donor hydrogen atom that is positively charged and a
negatively charged acceptor atom. The A-T base pair
associates by two hydrogen bonds, whereas C-G base
pairs associate by three hydrogen bonds. The base-pair
sequence is the carrier of genetic information. The
genetic code is formed of a sequence of three base
pairs, which determine a type of amino acid. For
example, the sequence of TTT AAA AAG GCT
bound. Double-stranded DNA possesses a large
induced dipole moment on the order of thousands of
debye, due to the counterion atmosphere. This fact
is gleaned from dielectric relaxation studies, birefringence, and dichroism experiments [236], and other
light-scattering experiments [237]. The induced dipole
moment μ in an electric field E is defined in terms of
the polarizability μ→ = αE .
Because the individual strands of double-stranded
DNA are antiparallel and the molecule is symmetrical,
the transverse dipole moments should cancel. However,
a number of researchers have measured a small permanent dipole moment for DNA [238]. In alternating
fields, the symmetry of the molecule may be deformed
slightly to produce a small permanent dipole moment
[231]. Another origin of the small permanent dipole
moment is attached charged ligands such as proteins or
multivalent cations [239]. These ligands produce a net
dipole moment on the DNA molecule by breaking the
symmetry. The question of how much of the relaxation
of the DNA molecule is due to induced dipolemoment
versus permanent moment has been studied by Hogan
et al. [236]. The response of permanent vs. induced
dipole moment differs in terms of field strength. The
potential energy of a permanent dipole moment at
an angle θ to the electric field is U = – μ E cos θ,
whereas the induced dipole moment in the electric field
is quadratic, U = – (Δα / 2)E 2 cos2 θ , where Δα is the
difference in polarizability along anisotropy axes of the
molecule. Experiments indicate that the majority of the
moment was induced rather than permanent. Charge
transport through DNA can be ballistic.
determines an amino acid sequence of phenylalaninelysine-lysine-alanine.
The DNA molecule has a net negative charge due to
the phosphate backbone. When dissolved in a cation
solution, some of the charge of the molecule is neutralized by cations. The double-stranded DNA molecule is
generally thought to have little intrinsic permanent
dipole moment. This is because the two strands that
compose the helix are oriented so that the dipole
moment of one strand cancels the other. However,
when DNA is dissolved in a solvent, such as saline
solution, an induced dipole moment forms due to
reorganization of charge into a layer around the
molecule called the counterion sheath.
The interaction of the counterions with biomolecules
has been a subject of intensive research over the years.
Some of the counterions bind to the phosphate backbone with a weak covalent bond. Other counterions are
more loosely bound and some may penetrate into the
major and minor grooves of DNA [235]. Ions are
assumed to be bound near charges in the DNA molecule, so that a double layer forms. The ions attracted to
the charged DNA molecule forms a counterion sheath
that shields some of the charge of the DNA. The
counterion sheath around a DNA molecule is composed
of cations such as Na or Mg, which are attracted to the
backbone negative phosphate charges. These charges
are somewhat mobile and oscillate about phosphate
charge centers in an applied electric field. A portion of
these counterions is condensed near the surface of the
molecule, whereas the vast majority are diffusely
Dielectric Properties of BoundWater and
Knowledge of the permittivity of the water near the
surface of a biomolecule is useful for modeling. The
region close to a biomolecule in water has a relatively
low real part of permittivity and a fixed charge. The
region far from the molecule has a permittivity close
to that of water. Lamm and Pack [240] studied the
variation of permittivity in the grooves, near the surface, and far away from the DNA molecule. The effective permittivity depends on solvent concentration,
distance from the molecule, the effects of the boundary,
and dielectric field-saturation. The variation of permittivity with position significantly alters the predictions
for the electric potential in the groove regions. Model
predictions depend crucially on knowing the dielectric
constant of water. Numerical modeling of the DNA
molecule depends critically on the permittivity of
Volume 117 (2012)
Journal of Research of the National Institute of Standards and Technology
water. When the permittivity of water varies in space,
numerical models indicate that small ions such as hydrogen can penetrate into the minor and major grooves [235,
241]. These predictions are not obtained for models that
use spatially independent permittivity for water. From
modeling results it was found that the real part of the
effective permittivity around the DNA molecule varies
as a function of distance from the center of the molecule and as a function of solvent concentration in moles
per liter (mol / l) [240].
The molecular structure of water is not simple.
Besides the basic H2O triad structure of the water
molecule, there are also complicated hydrogen-bonded
networks created by dipole-dipole interactions that
form hydroxyl OH– and hydronium H3O+ ions. The
dielectric constant of water at low frequencies is about
80, whereas biological water contains ions, which
affect both the real and imaginary parts of the permittivity. Water bound in proteins and DNA has a
decreased permittivity. This is due to constraints on the
movement of the molecules when they are attached to
Fig. 19. Measurements of the relative permittivity of various body
tissues by Gabriel et al. [242] (no uncertainties assigned).
Response of DNA and Other Biomolecules in
Electric Driving Fields
The low-frequency response of DNA is due primarily
to longitudinal polarization of the diffuse counterion
sheath that surrounds the molecule. This occurs at
frequencies in the range of 1 to 100 Hz. Another relaxation occurs in the megahertz region due to movement
of condensed counterions bound to individual phosphate groups. Dielectric data on human tissue is given
in Figs. 19 and 20. A number of researchers have
studied dielectric relaxation of both denatured and
helical conformation DNA molecules in electrolyte
solutions both as a function of frequency and applied
field strength. Single-stranded DNA exhibited less
dielectric relaxation than double-stranded DNA [98,
243-246]. Takashima concluded that denatured DNA
tended to coil and thereby decrease the effective length
and therefore the dipole moment. Furthermore, a high
electric field strength affects DNA conductivity in two
ways [244]. First, it promotes an increased dissociation
of the molecule and thereby increases conductivity.
Second, it promotes an orientation field effect where
alignment of polyions increases conductivity.
There are many other types of motion of the DNA
molecule when subjected to mechanical or millimeter
or terahertz electrical driving fields. For example,
propeller twist occurs when two adjacent bases in a pair
twist in opposite directions. Another motion is the
Fig. 20. Loss tangent of human tissues by Gabriel et al. [242].
breather mode where two bases oscillate in opposition
as hydrogen bonds are compressed and expanded. The
Lippincott-Schröder and Lennard-Jones potentials are
commonly used for modeling these motions. These
modes resonate at wavelengths in the millimeter
region; however, relaxation damping prevents direct
observation. Other static or dynamic motions of the
base pairs of the DNA molecule are roll, twist, and
Single-stranded DNA, in its stretched state, possesses a dipole moment oriented more or less transverse to
the axis. The phosphate group produces a permanent
transverse dipolemoment of about 20 D per 0.34 nm
base-pair section. The Debye (D) is a unit of dipole
moment and has a value of 3.336 × 10–30 C . m.
Because the typical DNA molecule contains thousands
of base pairs, the net dipole moment can be significant.
However, as the molecule coils or the base pairs twist,
the dipole moment decreases. If single strands of
DNA were rigid, since there is a transverse dipole
moment, and relaxation would occur in the megahertz
to gigahertz frequencies.
Volume 117 (2012)
Journal of Research of the National Institute of Standards and Technology
Dynamics of Polarization Relaxation in
trations of divalent metal cations destabilize the DNA
helix [254]. Sakamoto et al. [252] found that the dielectric increment decreased for divalent cations.
On the other hand, histones and protamines tightly
bind in the major groove of the DNA molecule. They
produce stability in the double helix by neutralizing
some of the phosphate charge. Dyes can attach to DNA,
neutralize charge, and thereby decrease dielectric
In order to study relaxation of polypeptides and
DNA in solution, we first consider the simplest model
of a dipolar rigid rod.
The torque on an electric dipole moment p is
N= p×E .
For cases where the dipole moment is perpendicular
to the rod axis, rotations about the major axis can occur.
The longitudinal rotation relaxation time for a molecule
of length L is given in [247]. The relaxation time varies
with the molecule length. Major axis rotation could
occur if the molecule had a transverse dipole moment;
for example, in a single strand of DNA.
When the dipole moment is parallel to the major
axis, end-over-end rotation may occur. This is the type
of relaxation at low frequencies that occurs with the
induced dipole moment in the counterion sheath or a
permanent dipole moment parallel to the longitudinal
axis of the molecule. The relaxation time varies as L3.
Because length of the molecule and molecular mass are
related, the responses for the two relaxations depend on
molecular mass. Also, the model presented in this
section assumes the rod is rigid. In reality, DNA is not
rigid, so a statistical theory of relaxation needs to be
applied [247-249].
Takashima [98] and Sakamoto et al. [243] have
derived a more comprehensive theory for counterion
relaxation and found that the relaxation time varies in
proportion to the square of the length of the molecule
[249, 250]. Most experimental evidence indicates a L2
dependence. This is in contrast to the rigid-rod model
where the relaxation time varies as L3.
Methods for Modeling Electromagnetic Interactions With Biomolecules, Nanoprobes, and
Modeling methods for EM interactions with materials include solving mode-match solutions to Maxwell’s
equations, finite-element and molecular dynamics
simulations, and finite-difference time-domain models.
Finite-element modeling software can solve Maxwell’s
equations for complicated geometries and small-scale
Traditionally, mode-match solutions to Maxwell’s
equations meant solving Maxwell’s equations in each
region and then matching the modal field components
at the interfaces and requiring, by the boundary
conditions, all the tangential electric fields go to zero
on conductors. On the nanoscale, the microwave and
millimeter wavelengths are much larger than the
feature size; the skin depths are usually larger than the
device being measured. Therefore modes must be
defined both outside the nanowire and inside the wire
and matched at the interface. Also, the role of the near
field is more important.
The EM model for a specific problem must capture the
important physics such as skin depth, ballistic
transport, conductor resistance, and quantized capacitance, without including all of the microstructural
content. Modeling nanoscale electromagnetics is
particularly difficult in that quantum effects cannot
always be neglected; however, the EM field in these
models is usually treated classically. In the case of nearfield probes the skin depths are usually larger than the
wire dimensions, and therefore the fields then need to be
determined in both the wire and in the space surrounding
the wire. Sommerfeld and Goubau surface waves and
plasmons propagate at the interface of dielectric and
finite conductivity metals and need to be taken into
account in modeling probe interactions. The probematerial EM communication is often transmitted by the
near field.
Counterion Interaction With DNA and Proteins
The real and imaginary parts of permittivity depend
on the concentration and type of cations [250]. As the
concentration of the solvent increases, more of the
phosphate charge is neutralized and the dielectric
increment (difference between the permittivity of the
mixture and solvent by itself) decreases.
Many types of cations compounds have been used
in DNA solvents; for example, NaCl, LiCl, AgNO2 ,
CuCl2 , MnCl2 , MgCl2 , arginines, protamine, dyes,
lysine, histones, and divalent metals such as Pb, Cd, Ni,
Zn, and Hg [243, 251-253]. The simple inorganicmonovalent cations bind to the DNA molecule near
the phosphate backbone to form both a condensed and
diffuse sheath. There is evidence that strong concen50
Volume 117 (2012)
Journal of Research of the National Institute of Standards and Technology
scattering data is plotted as a function of frequency. The
intrinsic relative permittivity is seen to be roughly
2.05, the commonly accepted value. However, when
dimensional or Fabry-Perot resonances (see example in
Fig. 22) across the sample occur at multiples of onehalf wavelength, the specimen exhibits a geometrical
standing-wave behavior at frequencies corresponding
to nλ / 2 across the sample. So if the sample is treated as
a single particle at these standing wave frequencies,
then the “effective” permittivity from this algorithm is
no longer the intrinsic property of the material, but
rather an artifact of geometric resonances across the
sample. Geometrical resonances are sometimes used by
metamaterial researchers to obtain effective negative
permittivities and permeabilities that produce negative
index response.
Homogeneous solid or liquid dielectric and magnetic materials have few intrinsic material resonances
in the RF frequencies. The intrinsic resonances that do
occur are primarily antiferromagnetic, ferromagnetic,
water vapor and oxygen absorption bands, surface
wave and plasma resonances, and atomic transitions.
Dielectric resonances or standing waves that occur in
solid and liquid dielectrics in RF frequencies are
usually either a) geometric resonances of the fundamental mode across the specimen, b) an artifact of a
higher mode that resonates across the length of the
specimen, c) resonances or standing waves across the
measurement fixture, or d) due to surface waves near
interfaces between materials.
In the measurement of inhomogeneous materials in a
transmission line or samples with a small air gap
between the material and the fixture, higher modes may
be produced and resonate across the specimen length
in the measurement fixture. For example, in a coaxial
line, the TE0n or TE11 mode may resonate across the
specimen in a coaxial line measurement. These higher
modes do not propagate in the air-filled waveguide
since they are evanescent, but may propagate in the
material-filled guide. Because these modes are not
generally included in the field model, they produce a
nonphysical geometric-based resonance in the reduced
permittivity data, as shown in Fig. 25. These higher
modes usually have low power and are caused by slight
material or machining inhomogeneities. When these
modes do propagate and resonate across the length of
the specimen, it may appear as if the molecules in the
material are under going intrinsic resonance, but this is
not happening. In such cases, if the numerical model
used for the data reduction uses only the fundamental
mode, then the results obtained do not represent the
permittivity of the material, but rather a related fixture
Recently, simulators for molecular dynamics have
advanced to the stage where bonding, electrostatic
interactions, and heat transfer can be modeled, and
some now are beginning to include EM interactions.
Metrology Issues
Effects of Higher Modes in Transmission-Line
In this section we describe various common difficulties encountered in measurements of permittivity and
permeability using transmission lines.
The definition of dielectric permittivity becomes
blurred when the particle size in a material is no longer
much smaller than a wavelength. To illustrate this
problem, consider the permittivity from a transmissionline measurement of a PTFE specimen, which was
reduced using the common Nicolson-Ross method [13]
as shown in Fig. 21. Typical scattering parameters are
shown in Fig. 22. The permittivity obtained from the
Fig. 21. Permittivity calculation on a polytetrafluoroethylene
(PTFE) material in a coaxial line that exhibits geometric resonance.
Fig. 22. Scattering parameters |S11| and |S21| as a function of
frequency for nylon in a coaxial line showing one-half wavelength
standing waves.
Volume 117 (2012)
Journal of Research of the National Institute of Standards and Technology
However, in magnetic materials where there are both a
permeability and permittivity, half wave geometric
resonances and produce instabilities in the reduction
algorithms [12].
Behavior of the Real Part of the Permittivity
in Relaxation Response
For linear, homogeneous materials that are relaxing
at RF frequencies, the permittivity decreases as
frequency increases. The permittivity increases only
near tails of intrinsic material resonances that only
occur for frequencies in the high gigahertz region and
above. To show this, we will analyze the prediction of
the DRT permittivity model [90, 212].
We know that the behavior of the orientational polarization of most materials in time-dependent fields can,
as a good approximation at low frequencies, be characterized with a distribution of relaxation times [53].
Typical numerical values of dielectric relaxation times
in liquids are from 0.1 μs to 1 ps.
We consider a description that has a distribution
function y(τ), giving the probability distribution of
relaxation times in the interval (τ, τ + dτ). The DRT
model is summarized in Eq. (69). There are fundamental constraints on the distribution y(τ). It is nonnegative everywhere, y(τ) ≥ 0 on τ ∈ [0,∞), and it is
Fig. 23. A typical coaxial line with a specimen inserted.
Fig. 24 Cross-sectional view of a specimen in a coaxial line.
specific geometric resonance of a higher mode
(Fig. 25). These resonances are distinct from the
fundamental-mode resonances obtained when the
Nicolson-Ross-Weir reduction method is used [11] in
transmission lines for materials at frequencies corresponding to nλg / 2, where n is an integer and λg is the
guided wavelength, as indicated in Fig. 21. The fundamental-mode resonances are modeled in the transmission-line theory and do not produce undue problems.
y( τ)d τ = 1 .
From Eq. (69) we have
d ε′(ω)
= −2(ε s − ε∞ )ω
τ2 y( τ)
d τ < 0 . (156)
(1 + ω2 τ2 )
This shows that ε′ is a decreasing function for all
positive ω where the DRT model is valid (low RF
frequencies), with a maximum only at ω = 0. The result
of Eq. (156) holds for any distribution function y(τ).
This model assumes there is only a relaxational
response. If resonant behavior occurs at millimeter to
terahertz frequencies, then the real part of the permittivity will show a slow increase as it approaches the
resonance. In the regions of relaxation response, the
real part of the permittivity is a decreasing function of
frequency. Therefore, ε′(ω) attains a minimum at some
frequency between relaxation and the beginning of
Fig. 25 Higher non-TEM resonant modes in a coaxial fixture and
anomalous behavior of the permittivity.
Volume 117 (2012)
Journal of Research of the National Institute of Standards and Technology
Permittivity Mixture Equations
nanoscale materials, wave cloaking, polariton surface
waves, biomaterials, and other topics were covered.
The definition and limitations of the concept of permittivity in materials was discussed. We emphasized that
the permittivity and permeability are well defined when
the applied field has a wavelength much longer than the
effective particle size in the material and when multiple
scattering between inclusions is minimal as the wave
propagates through the material. In addition, the use of
the concept of permittivity requires an ensemble of
particles that each have dielectric response.
We can readily estimate the permittivity of a mixture
of a number of distinct materials. The effective permittivity of a mixture εe f f of constituents with permittivities ei and volume fractions θi can be approximated in
various ways. The Bruggeman equation [256] is useful
for binary mixtures:
ε′e f
− ε1′
ε1′ + 2ε′e f
= θ2
ε′2 − εe′ f
ε′2 + 2εe′ f
or the Maxwell-Garnett mixture equation [256] can be
ε′e f
ε′e f
− ε′2
+ 2ε′2
= θ1
ε1′ − ε′2
ε1′ + 2ε′2
We acknowledge discussions with team members of
the Innovation Measurements Science Program at
NIST: Detection of Corrosion in Steel-Reinforced
Concrete by Antiferromagnetic Resonance, discussions
with Nick Paulter of OLES, and Pavel Kabos and many
other over the years.
where ε′1 is the permittivity of the matrix and ε′2 is
the permittivity of the filler [257]. The formula by
Lichtenecker is for a powerlaw dependence of the
real part of the permittivity for –1 ≤ k ≤ 1, and where
the volume fractions of the inclusions and host are
νp and νm :
ε k = ν p ε kp + ν m εmk .
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[234] M. Gueron and J. P. Demaret, A simple explanation of the electrostatics of the B to Z transition of DNA, Biophys. 89, 57405743 (1992).
[235] G. R. Pack, G. A. Garrett, L. Wong, and G. Lamm, The effect
of a variable dielectric coefficient and finite ion size on
Poisson-Boltzmann calculations of DNA-electrolyte systems,
Biophys. J. 65, 1363-1370 (1993).
[236] M. Hogan, N. Dattagupta, and D. M. Crothers, Transient electric dichroism of rod-like DNA molecules, Proc. Natl. Acad.
Sci. 75, 195-199 (1978).
[237] M. Sakamoto, T. Fujikado, R. Hayakawa, and Y. Wada, Low
frequency dielectric relaxation and light scattering under AC
electric field of DNA solutions, Biophys. Chem. 11, 309-316
[238] M. Hanss and J. C. Bernengo, Dielectric relaxation and orientation of DNA molecules, Biopolymers 12, 2151-2159 (1973).
[239] G. E. Plum and V. A. Bloomfield, Contribution of asymmetric
ligand binding to the apparent permanent dipole moment of
[240] G. Lamm and G. R. Pack, Local dielectric constants and
Poisson-Boltzmann calculations of DNA counterion distributions, Int. J. Quant. Chem. 65, 1087-1093 (1997).<
[241] B. Jayaram, K. A. Sharp, and B. Honig, The electrostatic
potential of B-DNA, Biopolymers 28, 975-993 (1989).
[242] C. Gabriel, E. H. Grant, R. Tata, P. R. Brown, B. Gestblom,
and E. Noreland, Microwave absorption in aqueous solutions
[243] M. Sakamoto, R. Hayakawa, andY. Wada, Dielectric relaxation of DNA solutions. III. Effects of DNA concentration,
protein contamination, and mixed solvents, Biopolymers 18,
2769-2782 (1979).
[244] N. Ise, M. Eigen, and G. Schwarz, The orientation and dissociation field effect of DNA in solution, Biopolymers 1, 343-352
[245] K. R. Foster and H. P. Schwan, Dielectric properties of tissues
and biological materials: A critical review, Vol. 17, CRC Press,
NY, pp. 25-104 (1989).
[246] O. Martinsen, S. Grimmes, and H. Schwan, Interface phenomena and dielectric properties of biological tissues,
Encyclopedia of Surface and Colloid Science, 2643-2652
[247] A. J. Bur and D. E. Roberts, Rodlike and random-coil behavior of poly(n-butyl isocyanate) in dilute solution, J. Chem.
Phys. 51, 406-420 (1969).
[248] J. G. Kirkwood, The visco-elastic properties of solutions of
rod-like macromolecules, J. Chem. Phys. 44, 281-283 (1951).
[249] H. Yu, A. J. Bur, and L. J. Fetters, Rodlike behavior of poly(nbutyl) isocyanate from dielectric measurements, J. Chem.
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Dielectric relaxation of DNA in aqueous solutions,
Biopolymers 15, 879-892 (1976).
[251] M. Sakamoto, R. Hayakawa, and Y. Wada, Dielectric relaxation of DNA solutions. II., Biopolymers 17, 1507-1512
[252] M. Sakamoto, R. Hayakawa, and Y. Wada, Dielectric relaxation of DNA solutions. IV. Effects of salts and dyes,
Biopolymers 19, 1039-1047 (1980).
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About the authors: James Baker-Jarvis is a physicist
and a Project Leader and Sung Kim is an electrical
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Electromagnetics Division of the NIST Physical
Measurement Laboratory. The National Institute of
Standards and Technology is an agency of the U.S.
Department of Commerce. |
3bd64c498fde4367 | Solid-state physics
Solid materials are formed from densely packed atoms, which interact intensely. These interactions produce the mechanical (e.g. hardness and elasticity), thermal, electrical, magnetic and optical properties of solids. Depending on the material involved and the conditions in which it was formed, the atoms may be arranged in a regular, geometric pattern (crystalline solids, which include metals and ordinary water ice) or irregularly (an amorphous solid such as common window glass).
The bulk of solid-state physics, as a general theory, is focused on crystals. Primarily, this is because the periodicity of atoms in a crystal — its defining characteristic — facilitates mathematical modeling. Likewise, crystalline materials often have electrical, magnetic, optical, or mechanical properties that can be exploited for engineering purposes.
The forces between the atoms in a crystal can take a variety of forms. For example, in a crystal of sodium chloride (common salt), the crystal is made up of ionic sodium and chlorine, and held together with ionic bonds. In others, the atoms share electrons and form covalent bonds. In metals, electrons are shared amongst the whole crystal in metallic bonding. Finally, the noble gases do not undergo any of these types of bonding. In solid form, the noble gases are held together with van der Waals forces resulting from the polarisation of the electronic charge cloud on each atom. The differences between the types of solid result from the differences between their bonding.
The physical properties of solids have been common subjects of scientific inquiry for centuries, but a separate field going by the name of solid-state physics did not emerge until the 1940s, in particular with the establishment of the Division of Solid State Physics (DSSP) within the American Physical Society. The DSSP catered to industrial physicists, and solid-state physics became associated with the technological applications made possible by research on solids. By the early 1960s, the DSSP was the largest division of the American Physical Society.[1][2]
Large communities of solid state physicists also emerged in Europe after World War II, in particular in England, Germany, and the Soviet Union.[3] In the United States and Europe, solid state became a prominent field through its investigations into semiconductors, superconductivity, nuclear magnetic resonance, and diverse other phenomena. During the early Cold War, research in solid state physics was often not restricted to solids, which led some physicists in the 1970s and 1980s to found the field of condensed matter physics, which organized around common techniques used to investigate solids, liquids, plasmas, and other complex matter.[1] Today, solid-state physics is broadly considered to be the subfield of condensed matter physics, often referred to as hard condensed matter, that focuses on the properties of solids with regular crystal lattices.
Crystal structure and properties
An example of a cubic lattice
Electronic properties
Properties of materials such as electrical conduction and heat capacity are investigated by solid state physics. An early model of electrical conduction was the Drude model, which applied kinetic theory to the electrons in a solid. By assuming that the material contains immobile positive ions and an "electron gas" of classical, non-interacting electrons, the Drude model was able to explain electrical and thermal conductivity and the Hall effect in metals, although it greatly overestimated the electronic heat capacity.
Arnold Sommerfeld combined the classical Drude model with quantum mechanics in the free electron model (or Drude-Sommerfeld model). Here, the electrons are modelled as a Fermi gas, a gas of particles which obey the quantum mechanical Fermi–Dirac statistics. The free electron model gave improved predictions for the heat capacity of metals, however, it was unable to explain the existence of insulators.
The nearly free electron model is a modification of the free electron model which includes a weak periodic perturbation meant to model the interaction between the conduction electrons and the ions in a crystalline solid. By introducing the idea of electronic bands, the theory explains the existence of conductors, semiconductors and insulators.
The nearly free electron model rewrites the Schrödinger equation for the case of a periodic potential. The solutions in this case are known as Bloch states. Since Bloch's theorem applies only to periodic potentials, and since unceasing random movements of atoms in a crystal disrupt periodicity, this use of Bloch's theorem is only an approximation, but it has proven to be a tremendously valuable approximation, without which most solid-state physics analysis would be intractable. Deviations from periodicity are treated by quantum mechanical perturbation theory.
Modern research
Modern research topics in solid-state physics include:
See also
2. ^ Hoddeson, Lillian; et al. (1992). Out of the Crystal Maze: Chapters from The History of Solid State Physics. Oxford University Press. ISBN 9780195053296.
3. ^ Hoffmann, Dieter (2013). "Fifty Years of Physica Status Solidi in Historical Perspective". Physica Status Solidi B. 250 (4): 871–887. Bibcode:2013PSSBR.250..871H. doi:10.1002/pssb.201340126.
Further reading
• Neil W. Ashcroft and N. David Mermin, Solid State Physics (Harcourt: Orlando, 1976).
• Charles Kittel, Introduction to Solid State Physics (Wiley: New York, 2004).
• H. M. Rosenberg, The Solid State (Oxford University Press: Oxford, 1995).
• Steven H. Simon, The Oxford Solid State Basics (Oxford University Press: Oxford, 2013).
• Out of the Crystal Maze. Chapters from the History of Solid State Physics, ed. Lillian Hoddeson, Ernest Braun, Jürgen Teichmann, Spencer Weart (Oxford: Oxford University Press, 1992).
• M. A. Omar, Elementary Solid State Physics (Revised Printing, Addison-Wesley, 1993).
• Hofmann, Philip (2015-05-26). Solid State Physics (2 ed.). Wiley-VCH. ISBN 978-3527412822. |
69c8d553f217dfad | #33: July 31st – August 6th
New features
Added a new optional extra experiments that installs the
qiskit-experiments package and also included it in the all target.
You can now install qiskit-experiments with qiskit using
pip install “qiskit[experiments]” or pip install “qiskit[all]”.
Documentation Changes
Modify README.md to include update instructions
Large-scale quantum machine learning
Tobias Haug,Chris N. Self,M. S. KimAug 03 2021 quant-phcs.LGstat.ML arXiv:2108.01039v1
Quantum computers promise to enhance machine learning for practical applications. Quantum machine learning for real-world data has to handle extensive amounts of high-dimensional data. However, conventional methods for measuring quantum kernels are impractical for large datasets as they scale with the square of the dataset size. Here, we measure quantum kernels using randomized measurements to gain a quadratic speedup in computation time and quickly process large datasets. Further, we efficiently encode high-dimensional data into quantum computers with the number of features scaling linearly with the circuit depth. The encoding is characterized by the quantum Fisher information metric and is related to the radial basis function kernel. We demonstrate the advantages and speedups of our methods by classifying images with the IBM quantum computer. Our approach is exceptionally robust to noise via a complementary error mitigation scheme. Using currently available quantum computers, the MNIST database can be processed within 220 hours instead of 10 years which opens up industrial applications of quantum machine learning.
Quantum convolutional neural network for classical data classification
Tak Hur,Leeseok Kim,Daniel K. ParkAug 03 2021 quant-ph arXiv:2108.00661v1
With the rapid advance of quantum machine learning, several proposals for the quantum-analogue of convolutional neural network (CNN) have emerged. In this work, we benchmark fully parametrized quantum convolutional neural networks (QCNNs) for classical data classification. In particular, we propose a quantum neural network model inspired by CNN that only uses two-qubit interactions throughout the entire algorithm. We investigate the performance of various QCNN models differentiated by structures of parameterized quantum circuits, quantum data encoding methods, classical data pre-processing methods, cost functions and optimizers on MNIST and Fashion MNIST datasets. In most instances, QCNN achieved excellent classification accuracy despite having a small number of free parameters. The QCNN models performed noticeably better than CNN models under the similar training conditions. Since the QCNN algorithm presented in this work utilizes fully parameterized and shallow-depth quantum circuits, it is suitable for Noisy Intermediate-Scale Quantum (NISQ) devices.
Spacetime Neural Network for High Dimensional Quantum Dynamics
Jiangran Wang,Zhuo Chen,Di Luo,Zhizhen Zhao,Vera Mikyoung Hur,Bryan K. ClarkAug 05 2021 cond-mat.dis-nncs.LGphysics.comp-phquant-ph arXiv:2108.02200
We develop a spacetime neural network method with second order optimization for solving quantum dynamics from the high dimensional Schrödinger equation. In contrast to the standard iterative first order optimization and the time-dependent variational principle, our approach utilizes the implicit mid-point method and generates the solution for all spatial and temporal values simultaneously after optimization. We demonstrate the method in the Schrödinger equation with a self-normalized autoregressive spacetime neural network construction. Future explorations for solving different high dimensional differential equations are discussed.
Variational quantum eigensolver for the Heisenberg antiferromagnet on the kagome lattice
Joris Kattemölle,Jasper van WezelAug 05 2021 quant-phcond-mat.str-el arXiv:2108.02175v1
Establishing the nature of the ground state of the Heisenberg antiferromagnet (HAFM) on the kagome lattice is well known to be a prohibitively difficult problem for classical computers. Here, we give a detailed proposal for a Variational Quantum Eigensolver (VQE) with the aim of solving this physical problem on a quantum computer. At the same time, this VQE constitutes an explicit proposal for showing a useful quantum advantage on Noisy Intermediate-Scale Quantum (NISQ) devices because of its natural hardware compatibility. We classically emulate a noiseless quantum computer with the connectivity of a 2D square lattice and show how the ground state energy of a 20-site patch of the kagome HAFM, as found by the VQE, approaches the true ground state energy exponentially as a function of the circuit depth. Besides indicating the potential of quantum computers to solve for the ground state of the kagome HAFM, the classical emulation of the VQE serves as a benchmark for real quantum devices on the way towards a useful quantum advantage.
Machine learning for secure key rate in continuous-variable quantum key distribution
Min-Gang Zhou,Zhi-Ping Liu,Wen-Bo Liu,Chen-Long Li,Jun-Lin Bai,Yi-Ran Xue,Yao Fu,Hua-Lei Yin,Zeng-Bing ChenAug 06 2021 quant-ph arXiv:2108.02578
Continuous-variable quantum key distribution (CV-QKD) with discrete modulation has received widespread attentions because of its experimental simplicity, lower-cost implementation and ease to multiplex with classical optical communication. Recently, some inspiring numerical methods have been applied to analyse the security of discrete-modulated CV-QKD against collective attacks, which promises to obtain considerable key rate over one hundred kilometers of fiber distance. However, numerical methods require up to ten minutes to calculate a secure key rate one time using a high-performance personal computer, which means that extracting the real-time secure key rate is impossible for discrete-modulated CV-QKD system. Here, we present a neural network model to quickly predict the secure key rate of homodyne detection discrete-modulated CV-QKD with good accuracy based on experimental parameters and experimental results. With the excess noise of about 0.010.01, the speed of our method is improved by about seven orders of magnitude compared to that of the conventional numerical method. Our method can be extended to quickly solve complex security key rate calculation of a variety of other unstructured quantum key distribution protocols.
Simulation of Open Quantum Dynamics with Bootstrap-Based Long Short-Term Memory Recurrent Neural Network
Kunni Lin,Jiawei Peng,Feng Long Gu,Zhenggang LanAug 04 2021 physics.chem-phquant-ph arXiv:2108.01310
The recurrent neural network with the long short-term memory cell (LSTM-NN) is employed to simulate the long-time dynamics of open quantum system. Particularly, the bootstrap resampling method is applied in the LSTM-NN construction and prediction, which provides a Monte-Carlo approach in the estimation of forecasting confidence interval. In this bootstrap-based LSTM-NN approach, a large number of LSTM-NNs are constructed under the resampling of time-series data sequences that were obtained from the early-stage quantum evolution given by numerically-exact multilayer multiconfigurational time-dependent Hartree method. The built LSTM-NN ensemble is used for the reliable propagation of the long-time quantum dynamics, and the forecasting uncertainty that partially reflects the reliability of the LSTM-NN prediction is given at the same time. The long-time quantum dissipative dynamics simulated by the current bootstrap-based LSTM-NN approach is highly consistent with the exact quantum dynamics results. This demonstrates that the LSTM-NN prediction combined with the bootstrap approach is a practical and powerful tool to propagate the long-time quantum dynamics of open systems with high accuracy and low computational cost.
Hybrid Quantum-Classical Neural Network for Incident Detection
Zadid Khan,Sakib Mahmud Khan,Jean Michel Tine,Ayse Turhan Comert,Diamon Rice,Gurcan Comert,Dimitra Michalaka,Judith Mwakalonge,Reek Majumdar,Mashrur ChowdhuryAug 04 2021 cs.LGcs.SYeess.SYquant-ph arXiv:2108.01127
The efficiency and reliability of real-time incident detection models directly impact the affected corridors’ traffic safety and operational conditions. The recent emergence of cloud-based quantum computing infrastructure and innovations in noisy intermediate-scale quantum devices have revealed a new era of quantum-enhanced algorithms that can be leveraged to improve real-time incident detection accuracy. In this research, a hybrid machine learning model, which includes classical and quantum machine learning (ML) models, is developed to identify incidents using the connected vehicle (CV) data. The incident detection performance of the hybrid model is evaluated against baseline classical ML models. The framework is evaluated using data from a microsimulation tool for different incident scenarios. The results indicate that a hybrid neural network containing a 4-qubit quantum layer outperforms all other baseline models when there is a lack of training data. We have created three datasets; DS-1 with sufficient training data, and DS-2 and DS-3 with insufficient training data. The hybrid model achieves a recall of 98.9%, 98.3%, and 96.6% for DS-1, DS-2, and DS-3, respectively. For DS-2 and DS-3, the average improvement in F2-score (measures model’s performance to correctly identify incidents) achieved by the hybrid model is 1.9% and 7.8%, respectively, compared to the classical models. It shows that with insufficient data, which may be common for CVs, the hybrid ML model will perform better than the classical models. With the continuing improvements of quantum computing infrastructure, the quantum ML models could be a promising alternative for CV-related applications when the available data is insufficient.
Hybrid Classical-Quantum Deep Learning Models for Autonomous Vehicle Traffic Image Classification Under Adversarial Attack
Reek Majumder,Sakib Mahmud Khan,Fahim Ahmed,Zadid Khan,Frank Ngeni,Gurcan Comert,Judith Mwakalonge,Dimitra Michalaka,Mashrur ChowdhuryAug 04 2021 quant-phcs.CRcs.LG arXiv:2108.01125v1
Image classification must work for autonomous vehicles (AV) operating on public roads, and actions performed based on image misclassification can have serious consequences. Traffic sign images can be misclassified by an adversarial attack on machine learning models used by AVs for traffic sign recognition. To make classification models resilient against adversarial attacks, we used a hybrid deep-learning model with both the quantum and classical layers. Our goal is to study the hybrid deep-learning architecture for classical-quantum transfer learning models to support the current era of intermediate-scale quantum technology. We have evaluated the impacts of various white box adversarial attacks on these hybrid models. The classical part of hybrid models includes a convolution network from the pre-trained Resnet18 model, which extracts informative features from a high dimensional LISA traffic sign image dataset. The output from the classical processor is processed further through the quantum layer, which is composed of various quantum gates and provides support to various quantum mechanical features like entanglement and superposition. We have tested multiple combinations of quantum circuits to provide better classification accuracy with decreasing training data and found better resiliency for our hybrid classical-quantum deep learning model during attacks compared to the classical-only machine learning models.
Quantum Neural Networks: Concepts, Applications, and Challenges
Yunseok Kwak,Won Joon Yun,Soyi Jung,Joongheon KimAug 04 2021 quant-phcs.LG arXiv:2108.01468v1
Quantum deep learning is a research field for the use of quantum computing techniques for training deep neural networks. The research topics and directions of deep learning and quantum computing have been separated for long time, however by discovering that quantum circuits can act like artificial neural networks, quantum deep learning research is widely adopted. This paper explains the backgrounds and basic principles of quantum deep learning and also introduces major achievements. After that, this paper discusses the challenges of quantum deep learning research in multiple perspectives. Lastly, this paper presents various future research directions and application fields of quantum deep learning.
Categories: Week-in-QML
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0b423e5744a20248 | Supercomputing the Building Blocks of the Universe
by Oct 13, 2019Uncategorized0 comments
Article Originally Published from Inside HPC.
Above: Gaute Hagen uses ORNL’s Summit supercomputer to model scientifically interesting atomic nuclei. To validate models, he and other physicists compare computations with experimental observations. Credit: Carlos Jones/ORNL
In this special guest feature, ORNL profiles researcher Gaute Hagen, who uses the Summit supercomputer to model scientifically interesting atomic nuclei.
At the nexus of theory and computation, physicist Gaute Hagen of the Department of Energy’s Oak Ridge National Laboratory runs advanced models on powerful supercomputers to explore how protons and neutrons interact to “build” an atomic nucleus from scratch. His fundamental research improves predictions about nuclear energy, nuclear security and astrophysics.
How did matter that forms our universe come to be?” asked Hagen. “How does matter organize itself based on what we know about elementary particles and their interactions? Do we fully understand how these particles interact?”
The lightest nuclei, hydrogen and helium, formed during the Big Bang. Heavier elements, up to iron, are made in stars by progressively fusing those lighter nuclei. The heaviest nuclei form in extreme environments when lighter nuclei rapidly capture neutrons and undergo beta decays.
For example, building nickel-78, a neutron-rich nucleus that is especially strongly bound, or “doubly magic,” requires 28 protons and 50 neutrons interacting through the strong force. “To solve the Schrödinger equation for such a huge system is a tremendous challenge,” Hagen said. “It is only possible using advanced quantum mechanical models and serious computing power.”
Through DOE’s Scientific Discovery Through Advanced Computing program, Hagen participates in the NUCLEI project to calculate nuclear structure and reactions from first principles; its collaborators represent 7 universities and 5 national labs. Moreover, he is the lead principal investigator of a DOE Innovative and Novel Computational Impact on Theory and Experiment award of time on supercomputers at Argonne and Oak Ridge National Laboratories for computations that complement part of the physics addressed under NUCLEI.
Theoretical physicists build models and run them on supercomputers to simulate the formation of atomic nuclei and study their structures and interactions. Theoretical predictions can then be compared with data from experiments at new facilities producing increasingly neutron-rich nuclei. If the observations are close to the predictions, the models are validated.
‘Random walk’
“I never planned to become a physicist or end up at Oak Ridge,” said Hagen, who hails from Norway. “That was a random walk.”
Graduating from high school in 1994, he planned to follow in the footsteps of his father, an economics professor, but his grades were not good enough to get into the top-ranked Norwegian School of Economics in Bergen. A year of mandatory military service in the King’s Guard gave Hagen fresh perspective on his life. At 20, he entered the University of Bergen and earned a bachelor’s degree in the philosophy of science. Wanting to continue for a doctorate, but realizing he lacked math and science backgrounds that would aid his dissertation, he signed up for classes in those fields—and a scientist was born. He went on to earn a master’s degree in nuclear physics.
Entering a PhD program, he used pen and paper or simple computer codes for calculations of the Schrödinger equation pertaining to two or three particles. One day his advisor introduced him to University of Oslo professor Morten Hjorth-Jensen, who used advanced computing to solve physics problems
The fact that you could use large clusters of computers in parallel to solve for several tens of particles was intriguing to me,” Hagen said. “That changed my whole perspective on what you can do if you have the right resources and employ the right methods.”
Hagen finished his graduate studies in Oslo, working with Hjorth-Jensen and taking his computing class. In 2005, collaborators of his new mentor—ORNL’s David Dean and the University of Tennessee’s Thomas Papenbrock—sought a postdoctoral fellow. A week after receiving his doctorate, Hagen found himself on a plane to Tennessee.
For his work at ORNL, Hagen used a numerical technique to describe systems of many interacting particles, such as atomic nuclei containing protons and neutrons. He collaborated with experts worldwide who were specializing in different aspects of the challenge and ran his calculations on some of the world’s most powerful supercomputers.
Computing had taken such an important role in the work I did that having that available made a big difference,” he said. In 2008, he accepted a staff job at ORNL.
That year Hagen found another reason to stay in Tennessee—he met the woman who became his wife. She works in TV production and manages a vintage boutique in downtown Knoxville.
Hagen, his wife and stepson spend some vacations at his father’s farm by the sea in northern Norway. There the physicist enjoys snowboarding, fishing and backpacking, “getting lost in remote areas, away from people, where it’s quiet and peaceful. Back to the basics.”
Hagen won a DOE early career award in 2013. Today, his research employs applied mathematics, computer science and physics, and the resulting descriptions of atomic nuclei enable predictions that guide earthly experiments and improve understanding of astronomical phenomena.
A central question he is trying to answer is: what is the size of a nucleus? The difference between the radii of neutron and proton distributions—called the “neutron skin”— has implications for the equation-of-state of neutron matter and neutron stars.
In 2015, a team led by Hagen predicted properties of the neutron skin of the calcium-48 nucleus; the results were published in Nature Physics. In progress or planned are experiments by others to measure various neutron skins. The COHERENT experiment at ORNL’s Spallation Neutron Source did so for argon-40 by measuring how neutrinos—particles that interact only weakly with nuclei—scatter off of this nucleus. Studies of parity-violating electron scattering on lead-208 and calcium-48—topics of the PREX2 and CREX experiments, respectively—are planned at Thomas Jefferson National Accelerator Facility.
One recent calculation in a study Hagen led solved a 50-year-old puzzle about why beta decays of atomic nuclei are slower than expected based on the beta decays of free neutrons. Other calculations explore isotopes to be made and measured at DOE’s Facility for Rare Isotope Beams, under construction at Michigan State University, when it opens in 2022.
Hagen’s team has made several predictions about neutron-rich nuclei observed at experimental facilities worldwide. For example, 2016 predictions for the magicity of nickel-78 were confirmed at RIKEN in Japan and published in Nature this year. Now the team is developing methods to predict behavior of neutron-rich isotopes beyond nickel-78 to find out how many neutrons can be added before a nucleus falls apart.
Progress has exploded in recent years because we have methods that scale more favorably with the complexity of the system, and we have ever-increasing computing power,” Hagen said. At the Oak Ridge Leadership Computing Facility, he has worked on Jaguar (1.75 peak petaflops), Titan (27 peak petaflops) and Summit (200 peak petaflops) supercomputers. “That’s changed the way that we solve problems.”
His team currently calculates the probability of a process called neutrino-less double-beta decay in calcium-48 and germanium-76. This process has yet to be observed but if seen would imply the neutrino is its own anti-particle and open a path to physics beyond the Standard Model of Particle Physics.
Looking to the future, Hagen eyes “superheavy” elements—lead-208 and beyond. Superheavies have never been simulated from first principles.
Lead-208 pushes everything to the limits—computing power and methods,” he said. “With this next generation computer, I think simulating it will be possible.”
Source: ORNL
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d5b54ac5e887995a | A not so easy piece: introducing the wave equation (and the Schrödinger equation)
Original post:
The title above refers to a previous post: An Easy Piece: Introducing the wave function.
Indeed, I may have been sloppy here and there – I hope not – and so that’s why it’s probably good to clarify that the wave function (usually represented as Ψ – the psi function) and the wave equation (Schrödinger’s equation, for example – but there are other types of wave equations as well) are two related but different concepts: wave equations are differential equations, and wave functions are their solutions.
Indeed, from a mathematical point of view, a differential equation (such as a wave equation) relates a function (such as a wave function) with its derivatives, and its solution is that function or – more generally – the set (or family) of functions that satisfies this equation.
The function can be real-valued or complex-valued, and it can be a function involving only one variable (such as y = y(x), for example) or more (such as u = u(x, t) for example). In the first case, it’s a so-called ordinary differential equation. In the second case, the equation is referred to as a partial differential equation, even if there’s nothing ‘partial’ about it: it’s as ‘complete’ as an ordinary differential equation (the name just refers to the presence of partial derivatives in the equation). Hence, in an ordinary differential equation, we will have terms involving dy/dx and/or d2y/dx2, i.e. the first and second derivative of y respectively (and/or higher-order derivatives, depending on the degree of the differential equation), while in partial differential equations, we will see terms involving ∂u/∂t and/or ∂u2/∂x(and/or higher-order partial derivatives), with ∂ replacing d as a symbol for the derivative.
The independent variables could also be complex-valued but, in physics, they will usually be real variables (or scalars as real numbers are also being referred to – as opposed to vectors, which are nothing but two-, three- or more-dimensional numbers really). In physics, the independent variables will usually be x – or let’s use r = (x, y, z) for a change, i.e. the three-dimensional space vector – and the time variable t. An example is that wave function which we introduced in our ‘easy piece’.
Ψ(r, t) = Aei(p·r – Et)ħ
[If you read the Easy Piece, then you might object that this is not quite what I wrote there, and you are right: I wrote Ψ(r, t) = Aei(p/ħr – ωt). However, here I am just introducing the other de Broglie relation (i.e. the one relating energy and frequency): E = hf =ħω and, hence, ω = E/ħ. Just re-arrange a bit and you’ll see it’s the same.]
From a physics point of view, a differential equation represents a system subject to constraints, such as the energy conservation law (the sum of the potential and kinetic energy remains constant), and Newton’s law of course: F = d(mv)/dt. A differential equation will usually also be given with one or more initial conditions, such as the value of the function at point t = 0, i.e. the initial value of the function. To use Wikipedia’s definition: “Differential equations arise whenever a relation involving some continuously varying quantities (modeled by functions) and their rates of change in space and/or time (expressed as derivatives) is known or postulated.”
That sounds a bit more complicated, perhaps, but it means the same: once you have a good mathematical model of a physical problem, you will often end up with a differential equation representing the system you’re looking at, and then you can do all kinds of things, such as analyzing whether or not the actual system is in an equilibrium and, if not, whether it will tend to equilibrium or, if not, what the equilibrium conditions would be. But here I’ll refer to my previous posts on the topic of differential equations, because I don’t want to get into these details – as I don’t need them here.
The one thing I do need to introduce is an operator referred to as the gradient (it’s also known as the del operator, but I don’t like that word because it does not convey what it is). The gradient – denoted by ∇ – is a shorthand for the partial derivatives of our function u or Ψ with respect to space, so we write:
You should note that, in physics, we apply the gradient only to the spatial variables, not to time. For the derivative in regard to time, we just write ∂u/∂t or ∂Ψ/∂t.
Of course, an operator means nothing until you apply it to a (real- or complex-valued) function, such as our u(x, t) or our Ψ(r, t):
∇u = ∂u/∂x and ∇Ψ = (∂Ψ/∂x, ∂Ψ/∂y, ∂Ψ/∂z)
As you can see, the gradient operator returns a vector with three components if we apply it to a real- or complex-valued function of r, and so we can do all kinds of funny things with it combining it with the scalar or vector product, or with both. Here I need to remind you that, in a vector space, we can multiply vectors using either (i) the scalar product, aka the dot product (because of the dot in its notation: ab) or (ii) the vector product, aka as the cross product (yes, because of the cross in its notation: b).
So we can define a whole range of new operators using the gradient and these two products, such as the divergence and the curl of a vector field. For example, if E is the electric field vector (I am using an italic bold-type E so you should not confuse E with the energy E, which is a scalar quantity), then div E = ∇•E, and curl E =∇×E. Taking the divergence of a vector will yield some number (so that’s a scalar), while taking the curl will yield another vector.
I am mentioning these operators because you will often see them. A famous example is the set of equations known as Maxwell’s equations, which integrate all of the laws of electromagnetism and from which we can derive the electromagnetic wave equation:
(1) ∇•E = ρ/ε(Gauss’ law)
(2) ∇×E = –∂B/∂t (Faraday’s law)
(3) ∇•B = 0
(4) c2∇×B = j+ ∂E/∂t
I should not explain these but let me just remind you of the essentials:
1. The first equation (Gauss’ law) can be derived from the equations for Coulomb’s law and the forces acting upon a charge q in an electromagnetic field: F = q(E + v×B) – with B the magnetic field vector (F is also referred to as the Lorentz force: it’s the combined force on a charged particle caused by the electric and magnetic fields; v the velocity of the (moving) charge; ρ the charge density (so charge is thought of as being distributed in space, rather than being packed into points, and that’s OK because our scale is not the quantum-mechanical one here); and, finally, ε0 the electric constant (some 8.854×10−12 farads per meter).
2. The second equation (Faraday’s law) gives the electric field associated with a changing magnetic field.
3. The third equation basically states that there is no such thing as a magnetic charge: there are only electric charges.
4. Finally, in the last equation, we have a vector j representing the current density: indeed, remember than magnetism only appears when (electric) charges are moving, so if there’s an electric current. As for the equation itself, well… That’s a more complicated story so I will leave that for the post scriptum.
We can do many more things: we can also take the curl of the gradient of some scalar, or the divergence of the curl of some vector (both have the interesting property that they are zero), and there are many more possible combinations – some of them useful, others not so useful. However, this is not the place to introduce differential calculus of vector fields (because that’s what it is).
The only other thing I need to mention here is what happens when we apply this gradient operator twice. Then we have an new operator ∇•∇ = ∇which is referred to as the Laplacian. In fact, when we say ‘apply ∇ twice’, we are actually doing a dot product. Indeed, ∇ returns a vector, and so we are going to multiply this vector once again with a vector using the dot product rule: a= ∑aib(so we multiply the individual vector components and then add them). In the case of our functions u and Ψ, we get:
∇•(∇u) =∇•∇u = (∇•∇)u = ∇u =∂2u/∂x2
∇•(∇Ψ) = ∇Ψ = ∂2Ψ/∂x+ ∂2Ψ/∂y+ ∂2Ψ/∂z2
Now, you may wonder what it means to take the derivative (or partial derivative) of a complex-valued function (which is what we are doing in the case of Ψ) but don’t worry about that: a complex-valued function of one or more real variables, such as our Ψ(x, t), can be decomposed as Ψ(x, t) =ΨRe(x, t) + iΨIm(x, t), with ΨRe and ΨRe two real-valued functions representing the real and imaginary part of Ψ(x, t) respectively. In addition, the rules for integrating complex-valued functions are, to a large extent, the same as for real-valued functions. For example, if z is a complex number, then dez/dz = ez and, hence, using this and other very straightforward rules, we can indeed find the partial derivatives of a function such as Ψ(r, t) = Aei(p·r – Et)ħ with respect to all the (real-valued) variables in the argument.
The electromagnetic wave equation
OK. That’s enough math now. We are ready now to look at – and to understand – a real wave equation – I mean one that actually represents something in physics. Let’s take Maxwell’s equations as a start. To make it easy – and also to ensure that you have easy access to the full derivation – we’ll take the so-called Heaviside form of these equations:
Heaviside form of Maxwell's equations
This Heaviside form assumes a charge-free vacuum space, so there are no external forces acting upon our electromagnetic wave. There are also no other complications such as electric currents. Also, the c2 (i.e. the square of the speed of light) is written here c2 = 1/με, with μ and ε the so-called permeability (μ) and permittivity (ε) respectively (c0, μand ε0 are the values in a vacuum space: indeed, light travels slower elsewhere (e.g. in glass) – if at all).
Now, these four equations can be replaced by just two, and it’s these two equations that are referred to as the electromagnetic wave equation(s):
electromagnetic wave equation
The derivation is not difficult. In fact, it’s much easier than the derivation for the Schrödinger equation which I will present in a moment. But, even if it is very short, I will just refer to Wikipedia in case you would be interested in the details (see the article on the electromagnetic wave equation). The point here is just to illustrate what is being done with these wave equations and why – not so much howIndeed, you may wonder what we have gained with this ‘reduction’.
The answer to this very legitimate question is easy: the two equations above are second-order partial differential equations which are relatively easy to solve. In other words, we can find a general solution, i.e. a set or family of functions that satisfy the equation and, hence, can represent the wave itself. Why a set of functions? If it’s a specific wave, then there should only be one wave function, right? Right. But to narrow our general solution down to a specific solution, we will need extra information, which are referred to as initial conditions, boundary conditions or, in general, constraints. [And if these constraints are not sufficiently specific, then we may still end up with a whole bunch of possibilities, even if they narrowed down the choice.]
Let’s give an example by re-writing the above wave equation and using our function u(x, t) or, to simplify the analysis, u(x, t) – so we’re looking at a plane wave traveling in one dimension only:
Wave equation for u
There are many functional forms for u that satisfy this equation. One of them is the following:
general solution for wave equation
This resembles the one I introduced when presenting the de Broglie equations, except that – this time around – we are talking a real electromagnetic wave, not some probability amplitude. Another difference is that we allow a composite wave with two components: one traveling in the positive x-direction, and one traveling in the negative x-direction. Now, if you read the post in which I introduced the de Broglie wave, you will remember that these Aei(kx–ωt) or Be–i(kx+ωt) waves give strange probabilities. However, because we are not looking at some probability amplitude here – so it’s not a de Broglie wave but a real wave (so we use complex number notation only because it’s convenient but, in practice, we’re only considering the real part), this functional form is quite OK.
That being said, the following functional form, representing a wave packet (aka a wave train) is also a solution (or a set of solutions better):
Wave packet equation
Huh? Well… Yes. If you really can’t follow here, I can only refer you to my post on Fourier analysis and Fourier transforms: I cannot reproduce that one here because that would make this post totally unreadable. We have a wave packet here, and so that’s the sum of an infinite number of component waves that interfere constructively in the region of the envelope (so that’s the location of the packet) and destructively outside. The integral is just the continuum limit of a summation of n such waves. So this integral will yield a function u with x and t as independent variables… If we know A(k) that is. Now that’s the beauty of these Fourier integrals (because that’s what this integral is).
Indeed, in my post on Fourier transforms I also explained how these amplitudes A(k) in the equation above can be expressed as a function of u(x, t) through the inverse Fourier transform. In fact, I actually presented the Fourier transform pair Ψ(x) and Φ(p) in that post, but the logic is same – except that we’re inserting the time variable t once again (but with its value fixed at t=0):
Fourier transformOK, you’ll say, but where is all of this going? Be patient. We’re almost done. Let’s now introduce a specific initial condition. Let’s assume that we have the following functional form for u at time t = 0:
u at time 0
You’ll wonder where this comes from. Well… I don’t know. It’s just an example from Wikipedia. It’s random but it fits the bill: it’s a localized wave (so that’s a a wave packet) because of the very particular form of the phase (θ = –x2+ ik0x). The point to note is that we can calculate A(k) when inserting this initial condition in the equation above, and then – finally, you’ll say – we also get a specific solution for our u(x, t) function by inserting the value for A(k) in our general solution. In short, we get:
u final form
As mentioned above, we are actually only interested in the real part of this equation (so that’s the e with the exponent factor (note there is no in it, so it’s just some real number) multiplied with the cosine term).
However, the example above shows how easy it is to extend the analysis to a complex-valued wave function, i.e. a wave function describing a probability amplitude. We will actually do that now for Schrödinger’s equation. [Note that the example comes from Wikipedia’s article on wave packets, and so there is a nice animation which shows how this wave packet (be it the real or imaginary part of it) travels through space. Do watch it!]
Schrödinger’s equation
Let me just write it down:
Schrodinger's equation
That’s it. This is the Schrödinger equation – in a somewhat simplified form but it’s OK.
[…] You’ll find that equation above either very simple or, else, very difficult depending on whether or not you understood most or nothing at all of what I wrote above it. If you understood something, then it should be fairly simple, because it hardly differs from the other wave equation.
Indeed, we have that imaginary unit (i) in front of the left term, but then you should not panic over that: when everything is said and done, we are working here with the derivative (or partial derivative) of a complex-valued function, and so it should not surprise us that we have an i here and there. It’s nothing special. In fact, we had them in the equation above too, but they just weren’t explicit. The second difference with the electromagnetic wave equation is that we have a first-order derivative of time only (in the electromagnetic wave equation we had 2u/∂t2, so that’s a second-order derivative). Finally, we have a -1/2 factor in front of the right-hand term, instead of c2. OK, so what? It’s a different thing – but that should not surprise us: when everything is said and done, it is a different wave equation because it describes something else (not an electromagnetic wave but a quantum-mechanical system).
To understand why it’s different, I’d need to give you the equivalent of Maxwell’s set of equations for quantum mechanics, and then show how this wave equation is derived from them. I could do that. The derivation is somewhat lengthier than for our electromagnetic wave equation but not all that much. The problem is that it involves some new concepts which we haven’t introduced as yet – mainly some new operators. But then we have introduced a lot of new operators already (such as the gradient and the curl and the divergence) so you might be ready for this. Well… Maybe. The treatment is a bit lengthy, and so I’d rather do in a separate post. Why? […] OK. Let me say a few things about it then. Here we go:
• These new operators involve matrix algebra. Fine, you’ll say. Let’s get on with it. Well… It’s matrix algebra with matrices with complex elements, so if we write a n×m matrix A as A = (aiaj), then the elements aiaj (i = 1, 2,… n and j = 1, 2,… m) will be complex numbers.
• That allows us to define Hermitian matrices: a Hermitian matrix is a square matrix A which is the same as the complex conjugate of its transpose.
• We can use such matrices as operators indeed: transformations acting on a column vector X to produce another column vector AX.
• Now, you’ll remember – from your course on matrix algebra with real (as opposed to complex) matrices, I hope – that we have this very particular matrix equation AX = λX which has non-trivial solutions (i.e. solutions X ≠ 0) if and only if the determinant of A-λI is equal to zero. This condition (det(A-λI) = 0) is referred to as the characteristic equation.
• This characteristic equation is a polynomial of degree n in λ and its roots are called eigenvalues or characteristic values of the matrix A. The non-trivial solutions X ≠ 0 corresponding to each eigenvalue are called eigenvectors or characteristic vectors.
Now – just in case you’re still with me – it’s quite simple: in quantum mechanics, we have the so-called Hamiltonian operator. The Hamiltonian in classical mechanics represents the total energy of the system: H = T + V (total energy H = kinetic energy T + potential energy V). Here we have got something similar but different. 🙂 The Hamiltonian operator is written as H-hat, i.e. an H with an accent circonflexe (as they say in French). Now, we need to let this Hamiltonian operator act on the wave function Ψ and if the result is proportional to the same wave function Ψ, then Ψ is a so-called stationary state, and the proportionality constant will be equal to the energy E of the state Ψ. These stationary states correspond to standing waves, or ‘orbitals’, such as in atomic orbitals or molecular orbitals. So we have:
E\Psi=\hat H \Psi
I am sure you are no longer there but, in fact, that’s it. We’re done with the derivation. The equation above is the so-called time-independent Schrödinger equation. It’s called like that not because the wave function is time-independent (it is), but because the Hamiltonian operator is time-independent: that obviously makes sense because stationary states are associated with specific energy levels indeed. However, if we do allow the energy level to vary in time (which we should do – if only because of the uncertainty principle: there is no such thing as a fixed energy level in quantum mechanics), then we cannot use some constant for E, but we need a so-called energy operator. Fortunately, this energy operator has a remarkably simple functional form:
\hat{E} \Psi = i\hbar\dfrac{\partial}{\partial t}\Psi = E\Psi Now if we plug that in the equation above, we get our time-dependent Schrödinger equation
OK. You probably did not understand one iota of this but, even then, you will object that this does not resemble the equation I wrote at the very beginning: i(u/∂t) = (-1/2)2u.
You’re right, but we only need one more step for that. If we leave out potential energy (so we assume a particle moving in free space), then the Hamiltonian can be written as:
You’ll ask me how this is done but I will be short on that: the relationship between energy and momentum is being used here (and so that’s where the 2m factor in the denominator comes from). However, I won’t say more about it because this post would become way too lengthy if I would include each and every derivation and, remember, I just want to get to the result because the derivations here are not the point: I want you to understand the functional form of the wave equation only. So, using the above identity and, OK, let’s be somewhat more complete and include potential energy once again, we can write the time-dependent wave equation as:
Now, how is the equation above related to i(u/∂t) = (-1/2)2u? It’s a very simplified version of it: potential energy is, once again, assumed to be not relevant (so we’re talking a free particle again, with no external forces acting on it) but the real simplification is that we give m and ħ the value 1, so m = ħ = 1. Why?
Well… My initial idea was to do something similar as I did above and, hence, actually use a specific example with an actual functional form, just like we did for that the real-valued u(x, t) function. However, when I look at how long this post has become already, I realize I should not do that. In fact, I would just copy an example from somewhere else – probably Wikipedia once again, if only because their examples are usually nicely illustrated with graphs (and often animated graphs). So let me just refer you here to the other example given in the Wikipedia article on wave packets: that example uses that simplified i(u/∂t) = (-1/2)2u equation indeed. It actually uses the same initial condition:
u at time 0
However, because the wave equation is different, the wave packet behaves differently. It’s a so-called dispersive wave packet: it delocalizes. Its width increases over time and so, after a while, it just vanishes because it diffuses all over space. So there’s a solution to the wave equation, given this initial condition, but it’s just not stable – as a description of some particle that is (from a mathematical point of view – or even a physical point of view – there is no issue).
In any case, this probably all sounds like Chinese – or Greek if you understand Chinese :-). I actually haven’t worked with these Hermitian operators yet, and so it’s pretty shaky territory for me myself. However, I felt like I had picked up enough math and physics on this long and winding Road to Reality (I don’t think I am even halfway) to give it a try. I hope I succeeded in passing the message, which I’ll summarize as follows:
1. Schrödinger’s equation is just like any other differential equation used in physics, in the sense that it represents a system subject to constraints, such as the relationship between energy and momentum.
2. It will have many general solutions. In other words, the wave function – which describes a probability amplitude as a function in space and time – will have many general solutions, and a specific solution will depend on the initial conditions.
3. The solution(s) can represent stationary states, but not necessary so: a wave (or a wave packet) can be non-dispersive or dispersive. However, when we plug the wave function into the wave equation, it will satisfy that equation.
That’s neither spectacular nor difficult, is it? But, perhaps, it helps you to ‘understand’ wave equations, including the Schrödinger equation. But what is understanding? Dirac once famously said: “I consider that I understand an equation when I can predict the properties of its solutions, without actually solving it.”
Hmm… I am not quite there yet, but I am sure some more practice with it will help. 🙂
Post scriptum: On Maxwell’s equations
First, we should say something more about these two other operators which I introduced above: the divergence and the curl. First on the divergence.
The divergence of a field vector E (or B) at some point r represents the so-called flux of E, i.e. the ‘flow’ of E per unit volume. So flux and divergence both deal with the ‘flow’ of electric field lines away from (positive) charges. [The ‘away from’ is from positive charges indeed – as per the convention: Maxwell himself used the term ‘convergence’ to describe flow towards negative charges, but so his ‘convention’ did not survive. Too bad, because I think convergence would be much easier to remember.]
So if we write that ∇•ρ/ε0, then it means that we have some constant flux of E because of some (fixed) distribution of charges.
Now, we already mentioned that equation (2) in Maxwell’s set meant that there is no such thing as a ‘magnetic’ charge: indeed, ∇•B = 0 means there is no magnetic flux. But, of course, magnetic fields do exist, don’t they? They do. A current in a wire, for example, i.e. a bunch of steadily moving electric charges, will induce a magnetic field according to Ampère’s law, which is part of equation (4) in Maxwell’s set: c2∇×B = j0, with j representing the current density and ε0 the electric constant.
Now, at this point, we have this curl: ∇×B. Just like divergence (or convergence as Maxwell called it – but then with the sign reversed), curl also means something in physics: it’s the amount of ‘rotation’, or ‘circulation’ as Feynman calls it, around some loop.
So, to summarize the above, we have (1) flux (divergence) and (2) circulation (curl) and, of course, the two must be related. And, while we do not have any magnetic charges and, hence, no flux for B, the current in that wire will cause some circulation of B, and so we do have a magnetic field. However, that magnetic field will be static, i.e. it will not change. Hence, the time derivative ∂B/∂t will be zero and, hence, from equation (2) we get that ∇×E = 0, so our electric field will be static too. The time derivative ∂E/∂t which appears in equation (4) also disappears and we just have c2∇×B = j0. This situation – of a constant magnetic and electric field – is described as electrostatics and magnetostatics respectively. It implies a neat separation of the four equations, and it makes magnetism and electricity appear as distinct phenomena. Indeed, as long as charges and currents are static, we have:
[I] Electrostatics: (1) ∇•E = ρ/εand (2) ∇×E = 0
[II] Magnetostatics: (3) c2∇×B = jand (4) ∇•B = 0
The first two equations describe a vector field with zero curl and a given divergence (i.e. the electric field) while the third and fourth equations second describe a seemingly separate vector field with a given curl but zero divergence. Now, I am not writing this post scriptum to reproduce Feynman’s Lectures on Electromagnetism, and so I won’t say much more about this. I just want to note two points:
1. The first point to note is that factor cin the c2∇×B = jequation. That’s something which you don’t have in the ∇•E = ρ/εequation. Of course, you’ll say: So what? Well… It’s weird. And if you bring it to the other side of the equation, it becomes clear that you need an awful lot of current for a tiny little bit of magnetic circulation (because you’re dividing by c , so that’s a factor 9 with 16 zeroes after it (9×1016): an awfully big number in other words). Truth be said, it reveals something very deep. Hmm? Take a wild guess. […] Relativity perhaps? Well… Yes!
It’s obvious that we buried v somewhere in this equation, the velocity of the moving charges. But then it’s part of j of course: the rate at which charge flows through a unit area per second. But – Hey! – velocity as compared to what? What’s the frame of reference? The frame of reference is us obviously or – somewhat less subjective – the stationary charges determining the electric field according to equation (1) in the set above: ∇•E = ρ/ε0. But so here we can ask the same question: stationary in what reference frame? As compared to the moving charges? Hmm… But so how does it work with relativity? I won’t copy Feynman’s 13th Lecture here, but so, in that lecture, he analyzes what happens to the electric and magnetic force when we look at the scene from another coordinate system – let’s say one that moves parallel to the wire at the same speed as the moving electrons, so – because of our new reference frame – the ‘moving electrons’ now appear to have no speed at all but, of course, our stationary charges will now seem to move.
What Feynman finds – and his calculations are very easy and straightforward – is that, while we will obviously insert different input values into Maxwell’s set of equations and, hence, get different values for the E and B fields, the actual physical effect – i.e. the final Lorentz force on a (charged) particle – will be the same. To be very specific, in a coordinate system at rest with respect to the wire (so we see charges move in the wire), we find a ‘magnetic’ force indeed, but in a coordinate system moving at the same speed of those charges, we will find an ‘electric’ force only. And from yet another reference frame, we will find a mixture of E and B fields. However, the physical result is the same: there is only one combined force in the end – the Lorentz force F = q(E + v×B) – and it’s always the same, regardless of the reference frame (inertial or moving at whatever speed – relativistic (i.e. close to c) or not).
In other words, Maxwell’s description of electromagnetism is invariant or, to say exactly the same in yet other words, electricity and magnetism taken together are consistent with relativity: they are part of one physical phenomenon: the electromagnetic interaction between (charged) particles. So electric and magnetic fields appear in different ‘mixtures’ if we change our frame of reference, and so that’s why magnetism is often described as a ‘relativistic’ effect – although that’s not very accurate. However, it does explain that cfactor in the equation for the curl of B. [How exactly? Well… If you’re smart enough to ask that kind of question, you will be smart enough to find the derivation on the Web. :-)]
Note: Don’t think we’re talking astronomical speeds here when comparing the two reference frames. It would also work for astronomical speeds but, in this case, we are talking the speed of the electrons moving through a wire. Now, the so-called drift velocity of electrons – which is the one we have to use here – in a copper wire of radius 1 mm carrying a steady current of 3 Amps is only about 1 m per hour! So the relativistic effect is tiny – but still measurable !
2. The second thing I want to note is that Maxwell’s set of equations with non-zero time derivatives for E and B clearly show that it’s changing electric and magnetic fields that sort of create each other, and it’s this that’s behind electromagnetic waves moving through space without losing energy. They just travel on and on. The math behind this is beautiful (and the animations in the related Wikipedia articles are equally beautiful – and probably easier to understand than the equations), but that’s stuff for another post. As the electric field changes, it induces a magnetic field, which then induces a new electric field, etc., allowing the wave to propagate itself through space. I should also note here that the energy is in the field and so, when electromagnetic waves, such as light, or radiowaves, travel through space, they carry their energy with them.
Let me be fully complete here, and note that there’s energy in electrostatic fields as well, and the formula for it is remarkably beautiful. The total (electrostatic) energy U in an electrostatic field generated by charges located within some finite distance is equal to:
Energy of electrostatic field
This equation introduces the electrostatic potential. This is a scalar field Φ from which we can derive the electric field vector just by applying the gradient operator. In fact, all curl-free fields (such as the electric field in this case) can be written as the gradient of some scalar field. That’s a universal truth. See how beautiful math is? 🙂
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Angular momentum coupling
Coupling in science
Classical coupling
Rotational-vibrational coupling
Quantum mechanical coupling
Rovibrational coupling
Vibronic coupling
Rovibronic coupling
Angular momentum coupling
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In quantum mechanics, the procedure of constructing eigenstates of total angular momentum out of eigenstates of separate angular momenta is called angular momentum coupling. For instance, the orbit and spin of a single particle can interact through spin-orbit interaction, in which case it is useful to couple the spin and orbit angular momentum of the particle. Or two charged particles, each with a well-defined angular momentum, may interact by Coulomb forces, in which case coupling of the two one-particle angular momenta to a total angular momentum is a useful step in the solution of the two-particle Schrödinger equation. In both cases the separate angular momenta are no longer constants of motion, but the sum of the two angular momenta usually still is. Angular momentum coupling in atoms is of importance in atomic spectroscopy. Angular momentum coupling of electron spins is of importance in quantum chemistry. Also in the nuclear shell model angular momentum coupling is ubiquitous.
spin-orbit coupling in astronomy reflects the general law of conservation of angular momentum, which holds for celestial systems as well. In simple cases, the direction of the angular momentum vector is neglected, and the spin-orbit coupling is the ratio between the frequency with which a planet or other celestial body spins about its own axis to that with which it orbits another body. This is more commonly known as orbital resonance. Often, the underlying physical effects are tidal forces.
General theory and detailed origin
Angular momentum is a property of a physical system that is a constant of motion[1] (is time-independent and well-defined) in two situations: (i) The system experiences a spherical symmetric potential field. (ii) The system moves (in quantum mechanical sense) in isotropic space. In both cases the angular momentum operator commutes with the Hamiltonian of the system. By Heisenberg's uncertainty relation this means that the angular momentum can assume a sharp value simultaneously with the energy (eigenvalue of the Hamiltonian).
An example of the first situation is an atom whose electrons only feel the Coulomb field of its nucleus. If we ignore the electron-electron interaction (and other small interactions such as spin-orbit coupling), the orbital angular momentum l of each electron commutes with the total Hamiltonian. In this model the atomic Hamiltonian is a sum of kinetic energies of the electrons and the spherical symmetric electron-nucleus interactions. The individual electron angular momenta l(i) commute with this Hamiltonian. That is, they are conserved properties of this approximate model of the atom.
An example of the second situation is a rigid rotor moving in field-free space. A rigid rotor has a well-defined, time-independent, angular momentum.
These two situations originate in classical mechanics. The third kind of conserved angular momentum, associated with spin, does not have a classical counterpart. However, all rules of angular momentum coupling apply to spin as well.
In general the conservation of angular momentum implies full rotational symmetry (described by the groups SO(3) and SU(2)) and, conversely, spherical symmetry implies conservation of angular momentum. If two or more physical systems have conserved angular momenta, it can be useful to add these momenta to a total angular momentum of the combined system—a conserved property of the total system. The building of eigenstates of the total conserved angular momentum from the angular momentum eigenstates of the individual subsystems is referred to as angular momentum coupling.
Application of angular momentum coupling is useful when there is an interaction between subsystems that, without interaction, would have conserved angular momentum. By the very interaction the spherical symmetry of the subsystems is broken, but the angular momentum of the total system remains a constant of motion. Use of the latter fact is helpful in the solution of the Schrödinger equation.
As an example we consider two electrons, 1 and 2, in an atom (say the helium atom). If there is no electron-electron interaction, but only electron nucleus interaction, the two electrons can be rotated around the nucleus independently of each other; nothing happens to their energy. Both operators, l(1) and l(2), are conserved. However, if we switch on the electron-electron interaction depending on the distance d(1,2) between the electrons, then only a simultaneous and equal rotation of the two electrons will leave d(1,2) invariant. In such a case neither l(1) nor l(2) is a constant of motion but L = l(1) + l(2) is. Given eigenstates of l(1) and l(2), the construction of eigenstates of L (which still is conserved) is the coupling of the angular momenta of electron 1 and 2.
In quantum mechanics, coupling also exists between angular momenta belonging to different Hilbert spaces of a single object, e.g. its spin and its orbital angular momentum.
Reiterating slightly differently the above: one expands the quantum states of composed systems (i.e. made of subunits like two hydrogen atoms or two electrons) in basis sets which are made of direct products of quantum states which in turn describe the subsystems individually. We assume that the states of the subsystems can be chosen as eigenstates of their angular momentum operators (and of their component along any arbitrary z axis). The subsystems are therefore correctly described by a set of l, m quantum numbers (see angular momentum for details). When there is interaction between the subsystems, the total Hamiltonian contains terms that do not commute with the angular operators acting on the subsystems only. However, these terms do commute with the total angular momentum operator. Sometimes one refers to the non-commuting interaction terms in the Hamiltonian as angular momentum coupling terms, because they necessitate the angular momentum coupling.
1. ^ Also referred to as a conserved property
Spin-orbit coupling
The behavior of atoms and smaller particles is well described by the theory of quantum mechanics, in which each particle has an intrinsic angular momentum called spin and specific configurations (of e.g. electrons in an atom) are described by a set of quantum numbers. Collections of particles also have angular momenta and corresponding quantum numbers, and under different circumstances the angular momenta of the parts add in different ways to form the angular momentum of the whole. Angular momentum coupling is a category including some of the ways that subatomic particles can interact with each other.
In atomic physics, spin-orbit coupling also known as spin-pairing describes a weak magnetic interaction, or coupling, of the particle spin and the orbital motion of this particle, e.g. the electron spin and its motion around an atomic nucleus. One of its effects is to separate the energy of internal states of the atom, e.g. spin-aligned and spin-antialigned that would otherwise be identical in energy. This interaction is responsible for many of the details of atomic structure.
In the macroscopic world of orbital mechanics, the term spin-orbit coupling is sometimes used in the same sense as spin-orbital resonance.
LS coupling
In light atoms (generally Z<30), electron spins si interact among themselves so they combine to form a total spin angular momentum S. The same happens with orbital angular momenta li, forming a single orbital angular momentum L. The interaction between the quantum numbers L and S is called Russell-Saunders coupling or LS coupling. Then S and L add together and form a total angular momentum J:
\mathbf J = \mathbf L + \mathbf S where \mathbf L = \sum_i \mathbf{l}_i and \mathbf S = \sum_i \mathbf{s}_i.
This is an approximation which is good as long as any external magnetic fields are weak. In larger magnetic fields, these two momenta decouple, giving rise to a different splitting pattern in the energy levels (the Paschen-Back effect.), and the size of LS coupling term becomes small.
For an extensive example on how LS-coupling is practically applied, see the article on Term symbols.
jj coupling
In heavier atoms the situation is different. In atoms with bigger nuclear charges, spin-orbit interactions are frequently as large or larger than spin-spin interactions or orbit-orbit interactions. In this situation, each orbital angular momentum li tends to combine with each individual spin angular momentum si, originating individual total angular momenta ji. These then add up to form the total angular momentum J
\mathbf J = \sum_i \mathbf j_i = \sum_i (\mathbf{l}_i + \mathbf{s}_i).
This description, facilitating calculation of this kind of interaction, is known as jj coupling.
Spin-spin coupling
See also: J-coupling and Dipolar coupling in NMR spectroscopy
Spin-spin coupling is the coupling of the intrinsic angular momentum (spin) of different particles. Such coupling between pairs of nuclear spins is an important feature of Nuclear Magnetic Resonance spectroscopy as it can provide detailed information about the structure and conformation of molecules. Spin-spin coupling between nuclear spin and electronic spin is responsible for hyperfine structure in atomic spectra.
Term symbols
Term symbols are used to represent the states and spectral transitions of atoms, they are found from coupling of angular momenta mentioned above. When the state of an atom has been specified with a term symbol, the allowed transitions can be found through selection rules by considering which transitions would conserve angular momentum. A photon has spin 1, and when there is a transition with emission or absorption of a photon the atom will need to change state to conserve angular momentum. The term symbol selection rules are. ΔS=0, ΔL=0,±1, Δl=±1, ΔJ=0,±1
Relativistic effects
In very heavy atoms, relativistic shifting of the energies of the electron energy levels accentuates spin-orbit coupling effect. Thus, for example, uranium molecular orbital diagrams must directly incorporate relativistic symbols when considering interactions with other atoms.
Nuclear coupling
In atomic nuclei, the spin-orbit interaction is much stronger than for atomic electrons, and is incorporated directly into the nuclear shell model. In addition, unlike atomic-electron term symbols, the lowest energy state is not L - S, but rather, l + s. All nuclear levels whose l value (orbital angular momentum) is greater than zero are thus split in the shell model to create states designated by l + s and l - s. Due to the nature of the shell model, which assumes an average potential rather than a central Coulombic potential, the nucleons that go into the l + s and l - s nuclear states are considered degenerate within each orbital (e.g. The 2p3/2 contains four nucleons, all of the same energy. Higher in energy is the 2p1/2 which contains two equal-energy nucleons).
See also
Clebsch-Gordan coefficients
This article is licensed under the GNU Free Documentation License. It uses material from the Wikipedia article "Angular_momentum_coupling". A list of authors is available in Wikipedia. |
d2ea24b492a2a06f | Foundations for Guided-Wave Optics (Hardcover)
Chin-Lin Chen
• 出版商: Wiley
• 出版日期: 2006-11-01
• 售價: $1,450
• 貴賓價: 9.8$1,421
• 語言: 英文
• 頁數: 480
• 裝訂: Hardcover
• ISBN: 0471756873
• ISBN-13: 9780471756873
• 相關分類: 光學 Optics
A classroom-tested introduction to integrated and fiber optics
This text offers an in-depth treatment of integrated and fiber optics, providing graduate students, engineers, and scientists with a solid foundation of the principles, capabilities, uses, and limitations of guided-wave optic devices and systems. In addition to the transmission properties of dielectric waveguides and optical fibers, this book covers the principles of directional couplers, guided-wave gratings, arrayed-waveguide gratings, and fiber optic polarization components.
The material is fully classroom-tested and carefully structured to help readers grasp concepts quickly and apply their knowledge to solving problems. Following an overview, including important nomenclature and notations, the text investigates three major topics:
• Integrated optics
• Fiber optics
• Pulse evolution and broadening in optical waveguides
Each chapter starts with basic principles and gradually builds to more advanced concepts and applications. Compelling reasons for including each topic are given, detailed explanations of each concept are provided, and steps for each derivation are carefully set forth. Readers learn how to solve complex problems using physical concepts and simplified mathematics.
Illustrations throughout the text aid in understanding key concepts, while problems at the end of each chapter test the readers' grasp of the material.
The author has designed the text for upper-level undergraduates, graduate students in physics and electrical and computer engineering, and scientists. Each chapter is self-contained, enabling instructors to choose a subset of topics to match their particular course needs. Researchers and practitioners can also use the text as a self-study guide to gain a better understanding of photonic and fiber optic devices and systems.
Table of Contents
1. Brief review of Electromagnetics and Guided Waves.
1.1 Introduction.
1.2 Maxwell's equations.
1.3 Uniform plane waves in isotropic media.
1.4 State of polarization.
1.5 Reflection and refraction by a planar boundary between two dielectric media.
1.5.1. Perpendicular polarization. Reflection and refraction. Total internal reflection.
1.5.2. Parallel polarization. Reflection and refraction. Total internal reflection.
1.6 Guided waves.
1.6.1 TE modes.
1.6.2 TM modes.
1.6.3 Waveguides with constant index regions.
List of Figures.
2. Step-index Thin-film Waveguides.
2.1 Introduction.
2.2 Dispersion of step-index thin-film waveguides.
2.2.1 TE modes.
2.2.2 TM modes.
2.3 Generalized parameters.
2.3.1 a, b, c, d and V.
2.3.2 bV diagram.
2.3.3 Cutoff thickness and cutoff frequencies.
2.3.4 Number of guided modes.
2.3.5 Birefringence in thin-film waveguides.
2.4 Fields of step-index thin-film waveguides.
2.4.1 TE modes.
2.4.2 TM modes.
2.5 Cover and substrate modes.
2.6 Time-average power and confinement factors.
2.6.1 Time-average power transported by TE modes.
2.6.2 Confinement factor of TE modes.
2.6.3 Time-average power transported by TM modes.
2.7 Phase and group velocities.
List of figures.
3. Graded-index Thin-film waveguides.
3.1 Introduction.
3.2 TE modes guided by linearly graded dielectric waveguides.
3.3 Exponentially graded dielectric waveguides.
3.3.1 TE modes.
3.3.2 TM modes.
3.4 WKB method.
3.4.1 Auxiliary function.
3.4.2 Fields in the R Zone.
3.4.3 Fields in the L Zone.
3.4.4 Fields in the transition zone.
3.4.5 The constants.
3.4.6 The dispersion relation.
3.4.7 An example.
3.5 Hocker and Burns’ numerical method.
3.5.1 TE modes.
3.5.2 TM modes.
3.6 Step-index thin-film waveguides vs. graded-index dielectric waveguides.
List of figures.
4. Propagation Loss in Thin-film Waveguides.
4.1 Introduction.
4.2 Complex relative dielectric constant and complex refractive index.
4.3 Propagation loss in step-index waveguides.
4.3.1 Waveguides having weakly absorbing materials.
4.3.2 Metal-clad waveguides.
4.4 Attenuation in thick waveguides with step-index profiles.
4.5 Loss in TM0 mode.
4.6 Metal-clad waveguides with graded index profiles.
List of Figures.
5. Three-dimensional Waveguides with Rectangular Boundaries.
5.1 Fields and modes guided by rectangular waveguides.
5.2 Orders of magnitude of fields.
5.2.1 modes.
5.2.2 modes.
5.3 Marcatili's method.
5.3.1 modes. Expressions for Hx. Boundary conditions along horizontal boundaries, y = ±h/2, |x| Boundary conditions along vertical boundaries, x = ±w/2, |y| Transverse wave vector K,sub>x. Transverse wave vector Ky. Approximate dispersion relation.
5.3.2 modes.
5.3.3 Discussions.
5.3.4 Generalized guide index.
5.4 Effective index method.
5.4.1 A pseudo waveguide.
5.4.2 An alternate pseudo waveguide.
5.4.3 Generalized guide index.
5.5 Comparison of methods.
List of figures.
6. Optical directional couplers and their applications.
6.1 Introduction.
6.2 Qualitative description of the operation of directional couplers.
6.3 Marcatili’s improved coupled mode equations.
6.3.1 Fields of isolated waveguides.
6.3.2 Normal mode fields of the composite waveguide.
6.3.3 Marcatili’s relation.
6.3.4 Approximate normal mode fields.
6.3.5 Improved coupled mode equations.
6.3.6 Coupled mode equation in an equivalent form.
6.3.7 Coupled mode equation in an alternate form.
6.4 Directional couplers with uniform cross section and constant spacing.
6.4.1 Transfer matrix.
6.4.2 Essential characteristics of couplers with K1 = K2 = K.
6.4.3 3 dB directional couplers.
6.4.4 Directional couplers as electrically controlled optical switches.
6.4.5. Switching diagram.
6.5 Switched δβ directional couplers.
6.6 Optical directional couplers filters.
6.6.1 Directional coupler filters with identical waveguides and uniform spacing.
6.6.2 Directional coupler filters with non-identical waveguides and uniform spacing.
6.6.3 Tapered directional coupler filters.
6.7 Intensity modulators based on directional couplers.
6.7.1 Electrooptic properties of lithium niobate.
6.7.2 Dielectric waveguide with an electrooptic layer.
6.7.3 Directional coupler modulator built on a Z-cut LiNbO3 plate.
6.8 Normal mode theory of directional couplers with two waveguides.
6.9 Normal mode theory of directional couplers with three or more waveguides.
List of Figures.
7. Guided-wave Gratings.
7.1 Introduction.
7.1.1 Types of guided-wave gratings. Static gratings. Programmable gratings. Moving grating.
7.1.2 Applications of guided-wave gratings.
7.1.3. Two methods for analyzing guided-wave grating problems.
7.2 Perturbation theory.
7.2.1 Waveguide perturbation.
7.2.2 Fields of perturbed waveguide.
7.2.3 Coupled mode equations and coupling coefficients.
7.2.4 Co-directional coupling.
7.2.5 Contra-directional coupling.
7.3 Coupling coefficient of a rectangular grating-an example.
7.4 Graphical representation of grating equation.
7.5 Grating reflectors.
7.5.1 Coupled mode equations.
7.5.2 Filter response of grating reflectors.
7.5.3 Bandwidth of grating reflectors.
7.6 Distributed feedback lasers.
7.6.1 Coupled mode equations with optical gain.
7.6.2 Boundary conditions and symmetric condition.
7.6.3 Eigen value equations.
7.6.4 Mode patterns.
7.6.5 Oscillation frequency and threshold gain.
List of Figures.
8. Arrayed-waveguide Gratings.
8.1 Introduction.
8.2 Arrays of isotropic radiators.
8.3 Two examples.
8.3.1 Arrayed-waveguide gratings as dispersive components.
8.3.2 Arrayed-waveguide gratings as focusing components.
8.4 1x2 arrayed-waveguide grating multiplexers and demultiplexers.
8.4.1 Waveguide grating elements.
8.4.2 Output waveguides.
8.4.3 Spectral response.
8.5 NxN arrayed-waveguide grating multiplexers and demultiplexers.
8.6 Applications in WDM communications.
List of Figures.
9. Transmission characteristics of step-index optical fibers.
9.1. Introduction.
9.2. Fields and propagation characteristic of modes guided by step-index fibers.
9.2.1 Electromagnetic fields.
9.2.2 Characteristic equation.
9.2.3 Traditional mode designation and fields.
9.3. Linearly polarized modes guided by weakly guiding step-index fibers.
9.3.1 Basic properties of fields of weakly guiding fibers..
9.3.2 Fields and boundary conditions.
9.3.3 Characteristic equation and mode designation.
9.3.4 Fields of x-polarized LP0m modes.
9.3.5 Time-average power.
9.3.6 Single mode operation.
9.4. Phase velocity, group velocity and dispersion of linearly polarized modes.
9.4.1 Phase velocity and group velocity.
9.4.2 Dispersion. Intermodal dispersion. Intramodal dispersion. Zero dispersion wavelengths.
List of Figures.
10. Input and output characteristics of weakly guiding step-index fibers.
10.1 Radiation of LP modes.
10.1.1 Radiated fields in the Fraunhofer zone.
10.1.2 Radiation by a Gaussian aperture field.
10.1.3 Experimental determination of ka and V.
10.2 Excitation of LP modes.
10.2.1 Power coupled to LP mode .
10.2.2 Gaussian beam excitation.
List of Figures.
11. Birefringence in Single-mode Fibers.
11.1 Introduction.
11.2 Geometrical birefringence.
11.3 Birefringence due to build-in stress.
11.4 Birefringence due to externally applied mechanical stress.
11.4.1 Lateral stress.
11.4.2 Bending. Pure bending. Bending under tension.
11.4.3 Mechanical twisting.
11.5 Birefringence due to externally applied electric and magnetic fields.
11.5.1 Strong transverse electric fields.
11.5.2 Strong axis magnetic fields.
11.6 Jones matrices of birefringent fibers.
11.6.1 Linearly birefringent fibers with stationary birefringent axes.
11.6.2 Linearly birefringent fiber with a continuous rotating axis.
11.6.3 Circularly birefringent fibers.
11.6.4 Linearly and circularly birefringent fibers.
11.6.5 Fibers with linear and circular birefringence and axis rotation.
12. Manufactured fibers.
12.1 Introduction.
12.2 Power-law index fibers.
12.3 Key propagation and dispersion parameters of graded index fibers.
12.3.1 Generalized guide index b.
12.3.2 Normalized group delay.
12.3.3 Group delay and the confinement factor.
12.3.4 Normalized waveguide dispersion.
12.3.5 An example.
12.4 Radiation and excitation characteristics of graded index fibers.
12.4.1 Radiation.
12.4.2 Excitation by a linearly polarized Gaussian beam.
12.5 Mode field radius.
12.5.1 Marcuse?s mode field radius.
12.5.2 First Petermann?s mode field radius.
12.5.3 Second Petermann?s mode field radius.
12.5.4 Comparison of three mode field radii.
12.6 Mode field radius and key propagation and dispersion parameters.
List of Figures.
13. Propagation of pulses in single-mode fibers.
13.1 Introduction.
13.2 Dispersion and group velocity dispersion.
13.3 Fourier transform method.
13.4 Propagation of Gaussian pulses in fibers.
13.4.1 Effects of? the first order group dispersion.
13.4.2 Effects of the second order group dispersion.
13.5 Impulse response.
13.5.1 Approximate impulse response function with β" ignored.
13.5.2 Approximate impulse response function with β" ignored.
13.6 Propagation of rectangular pulses in fibers.
13.7 Envelope equation.
13.7.1 Monochromatic waves.
13.7.2 Envelop equation.
13.7.3 Pulse envelop in non-dispersive media.
13.7.4 Effect of the first order group velocity dispersion.
13.7.5 Effect of the second order group velocity dispersion.
13.8 Dispersion compensation.
List of Figures.
14. Optical Solitons in Optical Fibers.
14.1 Introduction.
14.2 Optical Kerr effect in isotropic media.
14.2.1 Electric susceptibility tensor.
14.2.2 Refractive index.
14.3 Nonlinear envelope equation.
14.3.1 Linear and third-order polarizations.
14.3.2 Nonlinear envelope equation for nonlinear media.
14.3.3 Self-phase modulation.
14.3.4 Nonlinear envelope equation for nonlinear fibers.
14.3.5 Nonlinear Schrödinger equation.
14.4 Qualitative description of solitons.
14.5 Fundamental solitons.
14.5.1 Canonical expression.
14.5.2 General expression.
14.5.3 Basic soliton parameters.
14.5.4 Basic soliton properties.
14.6 Higher-order solitons.
14.6.1 Second-order solitons.
14.6.2 Third-order solitons.
14.7 Generation of solitons.
14.7.1 Integer A.
14.7.2 Non-integer A.
14.8 Soliton units of time, distance and power.
14.9 Interaction of solitons.
List of Figures.
Appendix A: Brown Identity.
A.1 Wave equations for inhomogeneous media.
A.2 Brown identity.
A.3 Two special cases.
A.4 Effect of material dispersion.
Appendix B: Two-dimensional Divergence Theorem and Green’s Theorem.
Appendix C. Orthogonality and Orthonormality of Guided Modes.
C.1 Lorentz’ reciprocity.
C.2 Orthogonality of guided modes.
C.3 Orthonormality of guided modes.
Appendix D: Elasticity, Photoelasticity and Electrooptic Effects.
D1 Strain tensors.
D1.1 Strain tensors in one-dimensional objects.
D1.2 Strain tensors in two-dimensional objects.
D1.3 Strain tensors in three-dimensional objects.
D2 Stress tensors.
D3 Hook’s law in isotropic materials.
D4 Strain and stress tensors in abbreviated indices.
D5 Relative dielectric constant tensors and relative dielectric impermeability tensors.
D6 Photoelastic effect and photoelastic constant tensors.
D7 Index change in isotropic solids: an example.
D8 Linear electrooptic effects.
D9 Quadratic electrooptic effects.
List of Figures.
Appendix E: Effect of mechanical twisting on fiber birefringence.
E1. Relative dielectric constant tensor of a twisted medium.
E2. LP modes in weakly guiding, untwisted fibers.
E3. Eigen polarization modes in twisted fibers.
Appendix F: Derivation of (12.7), (12.8) and (12.9).
Appendix G: Two Hankel transform relations. |
7356b0dc0a22b4a2 | Quantum Chemistry
Published on Jun 1, 2016
Tags:Keywords:, , , ,
Quantum Chemistry Thumbnail
When scientists conducted various kinds of experiments with atoms, they found that atoms actually behave unlike the bigger particles of matter. Scientists found that the explanations that were suggested by classical theory were inadequate and often wrong if the same principles were applied to the atoms. It was agreed that the behavior of the atom has to be studied differently and quantum chemistry was born. It is a relatively new branch of chemistry which is in particular concerned with the behavior of microscopic matter instead of macroscopic objects.
Schrödinger equation
Quantum chemistry was born when chemists tried to apply quantum mechanics to explain the chemical systems. The most crucial step is solving the equation that was put forth by the famous physicist Erwin Schrödinger. The solution of this equation helps in understanding how the electrons are distributed in the molecule. Unfortunately, the equation can only be accurately solved for the hydrogen atom since it involves only a single electron. For more complex systems, the chemists try to obtain only approximate solutions and then try to provide answers based on those solutions. When the electronic structure of a molecule is understood, it can be used to explain the chemical properties that are exhibited by the molecule. Quantum chemistry aims to define the principles which govern the nature of atoms and form a model which can help us provide a reason for all the observations concerning it.
Atomic behavior
Under quantum chemistry, a model was sought so that it was easier to predict as well as provide a rationale for the bizarre behavior of atoms. It was found that matter exhibits dual nature of a particle and a wave and acts as per the requirement. This model is referred to as the wave model because it is based on the wave-like properties of matter. The model states that the location of an electron inside an atom cannot be predicted accurately. There is always some uncertainty that is incorporated in the calculation. This is because the electrons exist not entirely as particles but in the form of “clouds” which have no fixed orbital rotation around the nucleus as it was thought before. According to quantum chemistry, everything has a value of probability associated with it and so it also dictates the probability of finding an electron in a particular part of space around the nucleus.
Quantum numbers
When the Schrodinger equation is solved, it provides a solution which depends on certain numbers called quantum numbers. These quantum numbers help define the space in which finding the probability of an electron will be high. Directly translated, these quantum numbers actually determine the orbital and sub-orbital that an electron would occupy. The principal quantum number is n, the secondary quantum number is l and the spin quantum number is s. n tells the orbital in which the electron is present, l provides information about the sub-orbital. It also helps in determining the angular momentum associated with the electron. The spin quantum number gives information about the orientation or spin of the electron.
Cool Natural Gifts! |
dbdcf0e46caadfa3 |
A classical analog for the electron spin state
K.B. Wharton, R.A. Linck and C.H. Salazar-Lazaro San José State University, Department of Physics and Astronomy, San José, CA 95192-0106
August 18, 2020
Despite conventional wisdom that spin-1/2 systems have no classical analog, we introduce a set of classical coupled oscillators with solutions that exactly map onto the dynamics of an unmeasured electron spin state in an arbitrary, time-varying, magnetic field. While not addressing the quantum measurement problem (discrete outcomes and their associated probabilities), this new classical analog yields a classical, physical interpretation of Zeeman splitting, geometric phase, the electron’s doubled gyromagnetic ratio, and other quantum phenomena. This Lagrangian-based model can be used to clarify the division between classical and quantum systems, and might also serve as a guidepost for certain approaches to quantum foundations.
I Introduction
Despite the conventional view of quantum spin as being an inherently non-classical phenomenonLL , there is a rich history of exploring classical analogs for spin-1/2 systems in particular. For example, there exists a well-developed classical analog to a two-level quantum system, based upon the classical polarization (CP) of a plane electromagnetic waveMcmaster ; HnS ; Klyshko ; Malykin ; Zap . Although this CP-analog has been used to motivate introductory quantum mechanics textsBaym ; Sakurai , the power and depth of the analogy is not widely appreciated. For example, the CP-analog contains a straightforward classical picture for a geometric phase shift resulting from a full rotation of the spin angular momentum, but this fact is rarely given more than a casual mention (with at least one notable exceptionKlyshko ). Still, the CP-analog contains certain drawbacks, especially when the analogy is applied to an electron spin state in an arbitrary, time-varying magnetic field. These drawbacks, along with complications involving quantum measurement outcomes, have prevented a general agreement on exactly which aspects of quantum spin are inherently non-classical.
In this paper, we extend the CP-analog to a system of four coupled oscillators, and prove that this classical system exactly reproduces the quantum dynamics of an unmeasured electron spin state in an arbitrary magnetic field. This result demonstrates, by explicit construction, that if there are any aspects of an electron spin state that cannot be described in a classical context, those aspects must lie entirely in the domain of quantum measurement theory, not the dynamics. In order to accomplish this feat, it turns out there must necessarily be a many-to-one map from the classical system to the quantum state. In other words, the classical system contains a natural set of “hidden variables”, accessible to the classical analog, but hidden to a complete specification of the quantum state.
Some might argue that no classical analog is needed to discuss quantum spin dynamics because an unmeasured quantum state governed by the Schrödinger-Pauli Equation (SPE) could be interpreted as a set of classical quantities coupled by first-order differential equations. One can even analyze the classical Dirac field and deduce quantities which map nicely onto quantum spin concepts Ohanian . But simply reinterpreting quantum wave equations as classical fields is not a very enlightening “analog”, especially if the spin state is considered separately from the spatial state. For example, the use of complex numbers in these equations is significantly different from how they are used to encode phases in classical physics, and therefore have no obvious classical interpretation. And if a system of first-order differential equations cannot be directly transformed into a set of second-order differential equations, it is unclear how certain classical physics concepts (e.g. generalized forces) can be applied. As we will demonstrate below, the full SPE can be expanded to a system of second-order equations, but only by adding additional “hidden variables” along with new overall constraints. The classical analog presented here arrives at this result from a different direction, starting with a simple Lagrangian.
Apart from clarifying how quantum spin might best be presented to students, the question of which aspects of quantum theory are truly “non-classical” is of deep importance for framing our understanding of quantum foundations. For example, Spekkens has recently demonstrated a simple classical toy theory that very roughly maps onto two-level quantum systems, showing several examples of purportedly-quantum phenomena that have a strong classical analogSpekkens . Still, neither Spekkens nor other prominent classical-hidden-variable approaches to two-level quantum systemsBell ; KS have concerned themselves with classical analogies to the curious dynamics of such systems. Our result demonstrates that starting with the dynamics can naturally motivate particular foundational approaches, such as a natural hidden variable space on which classical analogies to quantum measurement theory might be pursued. And because this classical analog derives from a simple Lagrangian, it is potentially a useful test bed for approaches where the action governs the transition probabilities, as in quantum field theory.
The plan for the paper is as follows: After summarizing the CP-analog in Section II, a related two-oscillator analog (similar to a Foucault Pendulum) is presented in Section III. This two-oscillator analog is shown to be identical to a quantum spin state in a one-dimensional magnetic field; a three-dimensional field requires an extension to four oscillators, as shown in Section IV. The final section discusses and summarizes these results – the most notable of which is that a classical system can encompass all the dynamics of a quantum spin-1/2 state.
Ii The Classical Polarization Analog
For a classical plane electromagnetic (EM) wave moving in the z-direction with frequency , the transverse electric fields and in the plane can always be expressed in two-vector notation as the real part of
Here and are complex coefficients, encoding the amplitude and phase of two orthogonal polarizations.
A strong analogy can be drawn between the two-vector on the right side of (1) – the well-known “Jones vector” – and the spinor that defines a spin-1/2 state in quantum mechanics. The quantum normalization condition maps to a normalization of the energy density of the EM wave, and the global phase transformation
This equivalence between a spinor and a Jones vector can be made more explicit by projecting them both onto the surface of a unit sphere in an abstract space (the “Bloch sphere” and the “Poincaré sphere” respectively). Each spinor/Jones vector maps to a unit vector in the angular direction , according to the usual convention , . This is more familiarly described in terms of the sphere’s six intersections with a centered cartesian axis
The CP-analog therefore maps linear x-polarized light to a spin-up electron and linear y-polarized light to a spin-down electron . Electrons with measured spins correspond to xy-diagonal linear polarizations, while correspond to the two circular-polarization modes. In this framework, note that quantum superpositions are no different than ordinary EM superpositions.
The analogy extends further, but this is already sufficient background to classically motivate some of the strange features of spin-1/2 systems. Consider the rotation of a Jones vector around the equator of a Poincaré sphere, corresponding to a continuous rotation of the direction of linear polarization – from horizontal, through vertical, and back to the original horizontal state. Any transformation that leads to this rotation (say, a physical rotation of the wave medium) will then be analogous to a magnetic-field induced precession of a spin state around the corresponding equator of the Bloch sphere.
The key point is that the above-described rotation around the Poincaré sphere merely corresponds to a rotation of the EM polarization in physical space. And this is equivalent to a phase shift in the resulting wave; it would now interfere destructively with an identical unrotated wave. Of course, this is also the observed effect for a rotation of a quantum spin state around the Bloch sphere, although in the latter case the net geometric phase shift is generally thought to be inexplicable from a classical perspective.
What the CP-analog accomplishes is to demonstrate that such behavior does indeed have a straightforward classical interpretation, because the geometrical phase of the spin state is directly analogous to the overall phase of the physical EM wave Klyshko . The key is that the Poincaré sphere does not map to physical space, so a rotation need not return the EM wave to its original state. The CP-analog therefore advocates the viewpoint that the Bloch sphere should not map to physical space, even for an electron spin state. This viewpoint will be implemented below in a fully consistent fashion.
To our knowledge, it has not been explicitly noted that this classical analogy naturally motivates an apparently-doubled gyromagnetic ratio for the electron. In the above-described Poincaré sphere rotation, as the EM wave is being rotated around its propagation axis, suppose an observer had reference to another system (say, a gyroscope) that truly recorded rotation in physical space. As compared to the gyroscope, the Jones vector would seem to complete a full rotation in half the time. If one interpreted the Poincaré sphere as corresponding to real space, the natural conclusion would be that the Jones vector was coupled to the physical rotation at double its “classical” value. Misinterpreting the Bloch sphere as corresponding to physical space would therefore lead to exactly the same conclusion for the electron’s gyromagnetic ratio.
The classical polarization analog can be pursued much further than is summarized here, mapping the quantum dynamics induced by a magnetic field to the effects of different birefringent materials Klyshko ; Malykin ; Zap ; Baym ; Kubo . The two EM modes in such materials then map to the two energy eigenstates, and generic rotations around the Poincaré sphere can be given a physical implementation. Still, this analog becomes quite abstract; there is no easy-to-describe vector quantity of a birefringent material that corresponds to the magnetic field, and the situation is even more convoluted for time-dependent field analogs.
Another disanalogy is the relation between the magnitude of the Zeeman energy splitting and the difference in wavenumber of the two EM modes. A more natural analogy would relate energy to a classical frequency, but the two EM modes always have identical frequencies. And of course, an electromagnetic plane wave cannot be pictured as a confined system with internal spin-like properties. In the next sections, we develop a novel classical analog that alleviates all of these problems.
Iii A Foucault-Pendulum-Like Analog
The central success of the CP-analog stems from its use of two physical oscillators, which need not be electromagnetic. For any two uncoupled classical oscillators with the same natural frequency , their solution can also be encoded by two complex numbers and , representing the amplitude and phase of each oscillator. Therefore the use of Jones vectors and the Poincaré sphere does not only pertain to EM waves.
As an intermediate step towards our proposed classical analog for an electron spin state, consider this classical Lagrangian:
Here the are all purely real quantities, and is some coupling constant that may be time-dependent. (As , this becomes two uncoupled classical oscillators). Equation (4) can be rewritten as , where the conjugate momenta and form the column vector , etc. Note that squaring the matrix yields . In this notation, .
First, consider the case of a constant . The Euler-Lagrange equations of motion for are then
These equations happen to describe the projection of a Foucault pendulum into a horizontal plane (with orthogonal axes and ) in the small-angle limit. Specifically, , where is the rotation frequency of the Earth and is the latitude of the pendulum. (The natural frequency of such a pendulum is actually , because of a term in that does not appear in the Foucault pendulum Lagrangian, but for a constant this is just a renormalization of ).
The precession of the Foucault pendulum therefore provides a qualitative way to understand the effect of a constant on the unnormalized Jones vector . Given a non-zero , it is well-known that linear oscillation in (mapping to on the Poincaré sphere) precesses into a linear oscillation in (mapping to ) and then back to (). But this rotation of around the Poincaré sphere merely corresponds to a rotation of the pendulum’s oscillation axis in physical space, leaving the overall phase of the pendulum shifted by , exactly as was described for the CP-analog.
Quantitatively, solutions to (5) are of the form , where . The generic solution can always be expressed as the real component of
Here and are arbitrary complex parameters (although again note that and are purely real).
One notable feature of this result is that the coupling constant has the effect of producing two solutions with well-defined frequencies equally spaced above and below the natural frequency – just like Zeeman splitting of an electron’s energy levels in a magnetic field. Furthermore, the modes that correspond to these two pure frequencies happen to be right- and left-hand circular motion of the pendulum, directly analogous to and . A comparison of (6) with standard results from quantum mechanics reveals that produces exactly the same dynamics on as does a constant magnetic field in the direction on an electron spin state (apart from an overall global phase).
Given the strong analogy between a constant and a constant (one-component) magnetic field, one can ask whether this correspondence continues to hold for a time-varying . In this case the strict analogy with the Foucault pendulum fails (thanks to the terms in ) and comparing the exact solutions becomes quite difficult. But starting from the Euler-Lagrange equations for a time-varying ,
one can compare them directly to the relevant Schrödinger-Pauli Equation (SPE). Using a -directed magnetic field (where is the gyromagnetic ratio) and an overall phase oscillation corresponding to a rest mass , this yields
Taking an additional time-derivative of (8), and simplifying the result using (8) itself, it is possible to derive the following second-order differential equations:
While and are still complex, the real and imaginary parts have naturally split into separate coupled equations that are formally identical to (7). So every solution to the SPE (8) must therefore have a real component which solves (7).
At first glance it may seem that the imaginary part of (9) contains another set of solutions not encoded in the real part of (9), but these solutions are not independent because they also solve (8). The solution space of (8) is a complex vector space of dimension 2 over the complex numbers. It can be verified that the SPE with a rest-mass oscillation cannot admit purely real solutions. Also, it is an elementary exercise to show that if a vector space over the complex numbers has a function basis given by and there is no complex linear combination of that yields a purely real function, then the set is a linearly independent set of real functions where linear independence is taken over the reals instead of the complex numbers. From this elementary result, it follows that if is a basis for the solution space of (8) over the complex numbers, then the set of functions spans a 4-d real subspace of the solution space of (7). Since (7) indeed has a 4-d solution space over the reals, it follows that the subspace spanned by is indeed the full solution space of (7). In summary, the solutions to the real, second-order differential equations (7) exactly correspond to the solutions to the complex, first-order differential equations (8).
For a one-dimensional magnetic field, these results explicitly contradict the conventional wisdom concerning the inherent complexity of the spin-1/2 algebra. By moving to real second-order differential equations – a natural fit for classical systems – it is possible to retain exactly the same dynamics, even for a time-varying magnetic field. The resulting equations not only account for a Zeeman-like frequency splitting, but demonstrate that the quantum geometric phase can be accounted for as the classical phase of an underlying, high-frequency oscillation (a strict analog to the usually-ignored rest mass oscillation at the Compton frequency).
Despite the breadth of the above conclusions, this coupled-oscillator analog has a major drawback as an analog to an electron spin state. It is limited by is the lack of coupling parameters that correspond to magnetic fields in the or directions, associated with the appropriate rotations around the Poincaré sphere. The classical model in the next section solves this problem, although it comes at the expense of the Foucault pendulum’s easily-visualized oscillations.
Iv The Full Analog: Four Coupled Oscillators
In order to expand the above example to contain an analog of an arbitrarily-directed magnetic field, two more coupling parameters must enter the classical Lagrangian. But with only two oscillators, there are no more terms to couple. With this in mind, one might be tempted to extend the above example to three coupled oscillators, but in that case the odd number of eigenmodes makes the dynamics unlike that of a spin-1/2 system.
It turns out that four coupled oscillators can solve this problem, so long as the eigenmodes come in degenerate pairs. By extending to a real 4-component vector (as opposed to the 2-component vector in the previous section), one can retain the same general form of the earlier Lagrangian:
Here we are still using the definition , but now with a 4x4 matrix encoding three independent coupling coefficients,
Again, note that squaring the matrix yields , where now .
iv.1 Constant Magnetic Fields
The four corresponding Euler-Lagrange equations of motion (for constant ’s) can be written as
Solving (12) for the eigenmodes via the replacement yields only two solutions, as the eigenvalues are doubly degenerate. They are of the same form as in the previous section: .
Because of the degeneracy, the full classical solutions can be expressed in a variety of ways. It is convenient to consider a vector with the cartesian components , and then to transform it into spherical coordinates . Using the two-spinors and defined in (II), the general solutions to (12) can then be written as the real part of
Here the global dependence has been suppressed; one multiplies by this factor and takes the real part to get the actual coordinate values. Having doubled the number of classical oscillators, the solution here is parameterized by four complex numbers ().
This solution bears a striking similarity to the known dynamics of an electron spin state in an arbitrary uniform magnetic field with components . In the basis defined above in (II), those solutions to the SPE are known to be
where the left side of this equation is the spinor . Here and are two complex constants subject to the normalization condition .
It is not difficult to see how all possible SPE solutions (14) have corresponding classical solutions (IV.1). Equating , adding the quantum-irrelevant global phase dependence to , and setting in (IV.1) makes the two expressions appear almost identical if and . (The ’s appear in the definition of ). The final step is to map the fully-real to the complex according to
This mapping turns out to be just one of many possible ways to convert a solution of the form (14) into the form (IV.1). For example, setting , and corresponds to the alternate map
More generally, one can linearly combine the above two maps by introducing two complex parameters and . Under the assignment , , and (which can always be done if ) then the connection between the above equations (IV.1) and (14) corresponds to
This shows that for any solution (IV.1) that obeys the condition, it will always encode a particular quantum solution to (14) via the map (IV.1), albeit with extra parameters , , and a specified global phase. Remarkably, this condition happens to be equivalent to the simple classical constraint . Imposing such a constraint on (12) therefore yields a classical system where all solutions can be mapped to the dynamics of a spin-1/2 quantum state in an arbitrary, constant, magnetic field – along with a number of “hidden variables” not encoded in the quantum state.
iv.2 Time-varying Magnetic Fields
As in Section III, a generalization to time-varying magnetic fields is best accomplished at the level of differential equations, not solutions. Allowing to vary with time again adds a new term to the Euler-Lagrange equations, such that they now read:
Here is given by (11) with time-dependent , , and . This must be again compared with the SPE with an explicit rest mass oscillation :
where again we have used to relate the coupling parameters in with the magnetic field . (Here is the standard vector of Pauli matrices).
While it is possible to use the map (IV.1) to derive (18) from (19) (and its time-derivative) via brute force, it is more elegant to use the quaternion algebra, as it is closely linked to both of the above equations. Defining the two quaternions , and , allows one to rewrite (18) as the quaternionic equation
Note that while operates on from the left, the acts as a right-multiplication on because (11) is of the form of a right-isoclinic rotation in SO(4).
While it is well-known that the components of act like purely imaginary quaternions, the precise mapping of to depends on how one maps to a quaternion . Using the above map (15), combined with the above definition of , it happens that , where is the quaternionic version of (as defined by the combination of (15) and ). This allows one to write the SPE (19) as
This equation uses a quaternionic , not a complex , acting as a left-multiplication (again because of the particular mapping from to defined by (15)). While the SPE would look more complicated under the more general map (IV.1) as applied to , this is equivalent to applying the simpler map (15) along with
so long as is a constant unit quaternion (linking the normalization of and ).
Keeping the SPE in the form (21), we want to show that for any solution to (21), there is a family of solutions to the classical oscillators (20). The time-derivative of (21) can be expanded as
Using (21) to eliminate the ’s on the right side of (23) then yields
If solves (21), it must solve (24), but this is exactly the same equation as (20). And because is multiplied from the left, must then also solve (20). This concludes the proof that all solutions to the SPE (19) – even for a time-varying magnetic field – have an exact classical analog in the solutions to (18).
The question remains as to which subset of solutions to (18) has this quantum analog. If the above connection exists between and , then by definition , where is a unit quaternion. This substitution transforms the left side of (21) into , where is the quaternionic version of the canonical momentum. Therefore, from (21), . As is a unit quaternion, this yields a zero Lagrangian density , consistent with the constant-field case.
V Discussion
The Foucault pendulum is often discussed in the context of classical analogs to quantum spin statesKlyshko , but the discussion is typically restricted to geometric phase. Section III demonstrated that the analogy runs much deeper, as the Coriolis coupling between the two oscillatory modes is exactly analogous to a one-dimensional magnetic field acting on an electron spin state. The analog also extends to the dynamics, and provides a classical description of Zeeman energy splitting, geometric phase shifts, and the appearance of a doubled gyromagnetic ratio. Apart from a global phase, there were no additional classical parameters needed to complete the Section III analog.
In Section IV, we demonstrated that it is possible to take four classical oscillators and physically couple them together in a particular manner (where the three coupling coefficients correspond to the three components of a magnetic field), yielding the equations of motion given in (18). Imposing a global physical constraint () on this equation forces the solutions to have an exact map (IV.1) to solutions of the Schrödinger-Pauli equation for a two-level quantum system with a rest-mass oscillation. This is a many-to-one map, in that there are additional parameters in the solution to (18) that can be altered without affecting the corresponding quantum solution, including an overall phase. From a quantum perspective, these additional parameters would be “hidden variables”.
Perhaps one reason this analog has not been noticed before is that many prior efforts to find classical analogs for the spin-1/2 state have started with a physical angular momentum vector, in real space. Rotating such a physical vector by , it is impossible to explain a geometric phase shift without reference to additional elements outside the state itself, such as in Feynman’s coffee cup demonstration Feynman . In the four-oscillator analog, however, the expectation value of the spin angular momentum merely corresponds to an unusual combination of physical oscillator parameters:
Here is an arbitrary unit vector, and the above definition of in (11) is used to define . Note, for example, that a sign change of both and leaves unchanged. This is indicative of the fact that the overall phase of the oscillators are shifted by under a rotation of , exactly as in the CP-analog and the Foucault pendulum.
This result explicitly demonstrates that if there is any inherently non-classical aspect to a quantum spin-1/2 state, such an aspect need not reside in the dynamics. On the other hand, if the system is measured, this classical analog cannot explain why superpositions of eigenmodes are never observed, or indeed what the probability distribution of measurements should be. That analysis resides in the domain of quantum measurement theory, and these results do not indicate whether or not that domain can be considered to have a classical analog.
With this in mind, these results should still be of interest to approaches where the usual quantum state is not treated as a complete description of reality. The hidden variables that naturally emerge from the above analysis are the complex parameters and (or equivalently, the unit quaternion ). These parameters effectively resulted from the doubling of the parameter space (from two to four oscillators), but do not seem to have any quantitative links to prior hidden-variable approaches. Still, they are loosely aligned with the doubling of the ontological state space in Spekkens’s toy model Spekkens , as well as with the doubling of the parameter space introduced when moving from the first-order Schrödinger equation to the second-order Klein Gordon equation KGE . Another point of interest is that this analog stems from a relatively simple Lagrangian, , and there is good reason to believe than any realistic model of quantum phenomena should have the same symmetries as a Lagrangian density WMP .
One final question raised by these results is whether or not it is possible to construct a working mechanical or electrical version of the classical oscillators described in Section IV. If this were possible, it would make a valuable demonstration concerning the dynamics of an unmeasured electron spin state. Even if it were not possible, some discussion of these results in a quantum mechanics course might enable students to utilize some of their classical intuition in a quantum context.
The authors are indebted to Patrick Hamill for recognizing (3) as the Foucault pendulum Lagrangian; further thanks are due to Ian Durham, David Miller, and William Wharton. An early version of this work was completed when KW was a Visiting Fellow at the Centre for Time in the Sydney Centre for Foundations of Science, University of Sydney.
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• (6) V.S. Zapasskii and G.G. Kozlov, “Polarized light in an anisotropic medium versus spin in a magnetic field,” Phys. Uspekhi 42 817–822 (1999).
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87a32235591111d9 | Whirlpool model of Gravity
I have explained elsewhere how the double slit experiment provides a strong proof of Ether.
Here I will explain how the same Ether model solves another great mystery i.e. gravity, without resorting to absurd concepts like bending of space or warping of space as proposed by General Relativity. It’s a common observation that a spinning body in a pool of water draws near by objects towards it. As a body spins in water, it creates a whirlpool around it, and into which near by objects ‘fall’. Similarly as our earth spins in the ocean of Ether, it creates a whirlpool around it and draws objects in its vicinity. Thus gravity is no more a mystery.
And the faster the body spins, the greater the whirlpool effect or the attractive force. This whirlpool effect or attraction force obviously becomes weaker as we go farther from the body. Thus we can explain all the phenomena of gravity using the whirlpool model.
But how do a whirlpool drags near by objects? Or how does a spinning body attracts near by objects?
To understand this we need to study Bernoulli Principle.
Scientists utilise Bernoulli principle to ‘lift’ aircrafts against the Earth’s gravity but I am sure they don’t really understand how this principle works. If they had, they would have realised long ago that it is the same principle that underlies the mystery of gravity. And Bernoulli principle would have become much more famous than Newton’s laws and wouldn’t have let Einstein’s theories distort our understanding of Gravity.
Bernoulli principle as understood by physicists states that ‘the pressure exerted by a fluid decreases as its velocity increases’. In other words, as a fluid moves faster, it exerts less pressure. Some physicists think that it is the law of conservation of energy that underlies the Bernoulli principle; while others attribute it to Newton’s 2nd law. That just highlights the physicist’s ignorance on not just Bernoulli Effect but also on the laws which they try to make use of to explain Bernoulli Effect. The fact is that we need neither of them to understand how Bernoulli principle works. What we need is just common sense.
To correctly explain Bernoulli’s effect we must first correctly understand about pressure. Pressure is defined as force per unit area. We know that force is a vector which means that a force is not just a quantity but also has a direction. For example if someone says ‘‘a force of 1Newton is applied on the ball’’, it conveys little meaning because we need to mention in what direction that said force is applied to make sense. There could be a number of forces acting simultaneously on a body from many directions, but the sum total of all the forces is what decides the final force vector and hence the direction of work. Because pressure is nothing but force, it implies that pressure is also a vector. So whenever we talk about pressure, it makes again no sense if we just say 1 Pascal or 2 Pascals and not mention the direction of pressure. This fact is often ignored or forgotten when physicists talk about pressure. Pressure i.e. the force exerted by a body, can be different in different directions. For example a book lying on a table may exert a downward pressure of 1pascal but it exerts no pressure in the upward direction or laterally. And we all know that the pressure exerted by water inside a container on the earth is not same in all directions.
Having realised that pressure is a vector; now we will go on to understand what pressure means at a deeper level. We know that a gas or a liquid exerts pressure on the walls of its container. But what is the fundamental mechanism that underlies the phenomenon of pressure? In other words from where does that force which we feel as pressure come? For this we will have to go to the kinetic theory of gases which states that the pressure of a gas is caused by collisions of its molecules against the walls of the container. The sum of the impacts per unit area of a wall is what we measure as the pressure applied upon that wall or in that direction.
We ‘know’ that the molecules or the atoms of a gas are in a state of random motion and collide with each other and with the walls of the container. Random motion implies that the molecules of a gas move equally in all directions (or in other words there is no net movement) and hence collide equally with all the walls and exert equal pressure in all directions. This is probably the reason why physicists ignore direction when they talk about pressure.
It is may be true that a gas inside a balloon exerts equal pressure in all directions in some situations, for example in the outer space and away from the celestial bodies when there is no ‘external influence’ upon the gas particles. But in the vicinity of earth, the effect of gravity can make the molecules move faster toward the bottom wall of a container and hence we may expect a little more pressure exerted upon that wall. (More over the term ‘random motion’ is only true at a gross level. If we magnify things and look deeply into the microcosm we would probably appreciate a highly ordered motion of the molecules and will be able to appreciate the slight differences in pressure in different directions)
In summary,
1) Pressure is nothing but force exerted per unit area of a surface
2) Pressure is a vector quantity
3) It is collisions of particles against a surface which manifests as pressure upon that surface.
Now imagine a container ‘filled’ with some gas. The gas molecules or particles move randomly and collide with the walls of the container. As discussed earlier, the sum of the impacts per unit area of a wall is what we measure as pressure upon that wall. If we ignore gravity and other external influences, the gas molecules collide equally against all the walls and hence exert equal pressure in all the directions i.e. on all the walls of the container. Now let’s remove the left and right walls of the container and make the gas to flow through the box in the rightward direction. Obviously the gas particles no longer move ‘randomly’ in all directions but move ‘preferentially’ towards the right. So the number of collisions against the top, bottom and other remaining walls of the container diminish. The result is that we measure less pressure being exerted by the gas on these remaining walls of the container. And the faster a gas flows in a given direction, the lesser the number of collisions on the side walls and hence the lesser the sideward pressure.
The gas particles collide equally against all the walls and hence exert equal pressure in all directions
The gas particles are no longer in random motion but are moving preferentially toward the right. So they impinge less often upon the sidewalls and exert hence exert less pressure sideward. The particles obviously exert more pressure towards the right.
The gas particles are no longer in random motion but are moving preferentially toward the right. So they impinge less often upon the sidewalls and hence exert less pressure sideward. The particles obviously exert more pressure towards the right.
The statement that a fast moving fluid exerts less pressure makes no sense. The truth is that it exerts less pressure only on the side walls (i.e. in the perpendicular direction). If we place a pressure gauge just opposite to the flow of gas, we will realise that it actually exerts much higher pressure in the direction of flow. (And obviously much lower pressure in the opposite direction)
Now imagine a body suspended in a tank of still water. Obviously the water particles keep colliding with the body on all its sides with equal force. In other words the water exerts equal pressure on all the sides of the body. And because there is no net force acting upon it, the body remains still and suspended inside the water.
Now imagine that there exists another body in the vicinity and which starts spinning vigorously. The body obviously stirs the water around it and induces circular currents in the water tank. Obviously the water particles that are closer to the spinning body get stirred faster than the ones that are farther away.
How would this scenario influence the first body?
1. the body which was still before starts moving in the direction of the water currents (rotation)
2. it starts spinning (in opposite direction to that of the ‘inducer’)
3. and it gets dragged towards the second body. Why?
Look at the force vectors in the picture below to understand why that happens.
Now replace water tank with Ether universe. Imagine our Earth spinning in that Ether ocean. Now we can explain why Earth attracts objects i.e. gravity.
Go to Demystifying Electromagnetism
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• cheesecookies On April 1, 2014 at 11:01 pm
I love relativity and astrophysics and I am hoping to major in it in university. Your blog really made me realise the implications beyond what I had learnt.
• SomeGuyFromNJ On March 19, 2017 at 2:55 am
Maybe this invisible water that’s acting on us to give us Gravity is “dark matter” and it’s just spinning around us right now as we type
• Mikhail Rakovsky On August 8, 2019 at 6:25 pm
It is result of rotating Dark Matter. All the planets position get determined by their density. Distance from Sun get reduced with increasing planets density.
Mercury 5.4 378.0
Venus 5.2 364.0
Earth 5.5 385.0
Mars 3.9 273.0
Jupiter 1.3 91.0
Saturn 0.7 49.0
Uranus 1.3 91.0
Neptune 1.6 112.0
The only not consistency is the orbit of Earth, because its density bigger than Venus, but if we calculate summary density the Earth-Moon system at the moon orbit the number of average density comes-out about 4.5, or less than Venus but bigger than Mars. Increasing density for Uranus and Neptune could be explained with in decreasing density of the dark matter (DM) to periphery of Sun system, suggesting “bagel” form high/low pressure DM and disturbances between them very similar picture to Jupiter’s atmosphere. The generally low DM pressure in Milky Way galaxy in rotating Sun system with combination of DM and VM get extremely low DM pressure in the center. DM pressure is growing-up to orbit of Saturn and stars to get reduced to the periphery.
• Aaron Do On July 3, 2014 at 12:21 pm
The ideas you have in your website are really interesting. The problem is you need to back up your theory with measurement.
• drgsrinivas On July 4, 2014 at 9:03 pm
The problem here is not lack of experimental proof or backup by ‘measurements’. The whole point is that observations and ‘measurements’ need to be interpreted logically to make sense out of them. When people don’t bother about logic, any ‘measurement’ can be used to back up any stupid statement.
The observation: Apple falls to the ground
Relativists’ explanation: because space is curved.
(Even then, why should the apple fall ‘down’? Why doesn’t it fly ‘away’? In other words, what force makes the apple to move from the less curved space to the more curved space? I am sure relativists resort to circular logic here: They might say ‘that is because of gravity’!!!)
Ether theory: due to ‘whirlpool effect’ as the Earth spins through the ocean of Ether.
The observation: A photon appears to pass through both slits and interfere with itself
Quantumists’ explanation: A photon particle travels through all paths simultaneously and hence is able to pass through both the slits.
Ether explanation: when a photon is fired inside the Ether Ocean, it creates a wave just like how a water particle fired inside a pool of water results in a water wave. And it is this wave which spreads and travels in all directions simultaneously. So it is not the particle itself which travels via both the slits, but it is the particle energy which does so in the form of ‘daughter waves’.
Similarly better logical explanations exist for cosmic ray muons reaching the Earth in large numbers, slowing of GPS clock, neutral pion decay, aberration of star light etc etc.
So all the observations and ‘measurements’ claimed by modern physicists as proof of their weird theories actually prove that our physicists are mad because all of them can be explained logically and without resorting to the stupid preachings of their relativity and quantum religions.
• Aaron Do On July 24, 2014 at 12:27 pm
So you’re saying that we have plenty of measurements, they just need to be interpreted properly…
Regarding the “whirlpool” theory, I have some difficulty visualizing how it would work in 3 dimensions. If you have a spherical object spinning in a tank of water, then it is only spinning on one axis. So would there be any attraction to the “poles” of the object, and how would it compare to the “equator” (pardon the terminology)?
• drgsrinivas On July 24, 2014 at 11:30 pm
You are actually on the right track. If you look at our solar system or the numerous galaxies, they are more or less disc shaped and not spherical. According to the ‘whirlpool’ model, the gravitational influence exerted by a celestial body is greatest in the ‘equatorial plane’ and it decreases towards the poles.
But how do we explain the observation that the weight of an object is more towards the Polar Regions than at the equator? A body’s weight is decided upon by two forces. One is the gravitational force which pulls the body inwards i.e. towards the Earth. And the other force is the centrifugal force which pushes the body away from the earth. It is the sum of these two forces which probably decide the actual weight of a body. As we move towards the poles, not only the gravitational attraction becomes weaker, but is also the centrifugal repulsion force. I think probably there is more reduction in the centrifugal repulsion force than in the gravitational attraction force as one moves from the equator towards the poles.
The phenomenon of gravity can be easily conceptualised by understanding how a centrifuge works- “Centrifuge model of gravity”. Of course it is ultimately Bernoulli’s effect that underlies both.
• Aaron Do On July 27, 2014 at 7:24 pm
Thanks for the reply!
I still have two doubts though. The first is that at the precise position of one of the poles, you would expect both the centrifugal and the “whirlpool” force to equal to zero. i.e. no gravity. I think that kind of effect would be well documented. Unless the earth’s precession has some effect too (I would expect it to be very small…).
My second doubt is that if you take a spinning sphere in a tank of water, and water is moving towards the “equator” then it would have to be moving away from the poles in order for the water to circulate. In your ether model, wouldn’t something similar occur?
• drgsrinivas On August 3, 2014 at 12:55 pm
Yes, the above described Bernoulli phenomenon can’t fully explain the gravitational ‘attraction’ near the polar regions. And thanks for your thought provoking question: I have stumbled upon a new concept that not only solves the gravity issue at the poles but also provides an insight into the phenomenon of magnetism.
Briefly, we know that the vast majority of the space inside the atoms is ’empty’. In other words, the vast majority of the space inside any ‘solid object’ (including our Earth) is empty. According to the Ether model of universe, all the space including this empty ‘internal mileau’ of all objects is pervaded by the Ether or photons. So our Earth may be considered as a highly ‘porous’ body suspended in the ether ocean and also filled with the ether fluid. Now as the body of our earth rotates, ether gets dragged in via the ‘poles’, flows outward at the ‘equator’.
This inward dragging of ether is what manifests as gravity near the poles and the differential spin of ether combined with Bernoulli effect explains the gravitational attraction near the equator. We can actually undertake a simple experiment to prove this: we just have to make a round porous body (made of say iron mesh) spin inside water and see how it affects near by smaller objects.
And I believe it is the flow pattern of ether in and out of the Earth which manifests as the magnetic lines/ field of Earth.
• Aaron Do On July 28, 2014 at 7:40 pm
I may have been a bit hasty with my second doubt. So I guess the water itself is only moving in a circle around the sphere, but not towards the sphere, and nearby objects are moved by the water towards the sphere? I think I might try this out in my kitchen sink just to verify… 😀
• curtweinstein2 On July 31, 2014 at 7:59 am
1) Is the “spin” real? OK, I can “buy” the ether, no problem.
2) If the Earth didn’t spin wouldn’t it yet create the same “amount” of gravity?
3) Oh, OK, I know why I am confused. I think “gravity” is the ether, and “gravity” doesn’t spin with the Earth, according to our experiences with Foucault’s Pendulum and also according to Dr. Petr Beckmann.
• drgsrinivas On August 6, 2014 at 9:50 am
If Earth didn’t spin, there wouldn’t be any gravity here. Let me correct you, Ether is not gravity, it is the differential spin of Ether which manifests as gravity.
• Spindizzy On September 7, 2018 at 9:17 am
I know this is a late reply, but I’m somewhat confused.
If an object does not spin, there is no gravity? Yes?
What about things like comets which don’t have spin, and yet appear to have gravity? I mean, we’ve had the Rosetta mission, which landed a probe on a comet. If there wasn’t any gravity, it would just bounce off, yes?
• drgsrinivas On September 9, 2018 at 5:42 pm
I don’t know whether comets spin or not. I haven’t researched enough. But objects in space can probably attract other objects without the ‘external’ spin. For example, if all or majority of the atoms that make up an object orient and spin in the same direction, that object could theoretically create ether winds and attract objects in its vicinity. That is, there could be internal spin even if the object isn’t spinning externally… and may be this is what happens with magnets.
And merely making some object land on some other object in space neither proves or nor disproves gravity.
And, I am very skeptical of the scientists’ distant observations and their interpretations. They are not intelligent enough to observe things and make interpretations even in our immediate neighborhood (for e.g. waves in a pond), why talk about their capacity to study things in the faraway space!
• Darcy Donelle (@Darcy_Donelle) On October 17, 2014 at 6:45 pm
The Ether? The first strong evidence against the ether emerged in the late 19th century via the Michelson–Morley experiment…
It seems like you’re having difficulty grasping the fundamental properties of nature. Your invoking ether as an explanation for gravity is no more convincing than accepting gravity as a fundamental property of nature. All you have done is introduced a new fundamental property of nature, i.e. that the earth spins through a whirlpool. Where does Earth get its energy to spin around its axis and orbit the Sun? If this energy comes from the whirlpool, then where does the whirlpool obtain its energy from?
“We are to admit no more causes of natural things, than such as are both true and sufficient to explain their appearances.” – Isaac Newton.
• drgsrinivas On October 18, 2014 at 10:38 am
If you want to religiously believe in what your mentors preached you, I have no objection. But don’t swear your beliefs as evidence. I have explained how your mentors have misinterpreted Michelson’s experiment and exposed their stupid reasoning here – https://debunkingrelativity.com/ether-wind-and-ether-drag/
I strongly suggest that you don’t read that if you are a weak hearted individual because that would tear apart your religious theories and your heart may not tolerate that. Rather keep chanting that Ether has been disproved, so that you remain healthy and your religion survives!
So where does earth get the energy to spin? What about posing the question to your religion? Also let me ask your religion another question, where does the matter that makes the Earth come from? I have never claimed that Ether model would answer all the questions down to the most fundamental level. https://debunkingrelativity.com/2014/03/29/the-divine-stuff-explains-all/
• Aerophos On October 17, 2014 at 8:36 pm
IMPORTANT: Good theory, I like your thinking, BUT: have you proven that objects in space that DONT spin have ZERO gravity? I strongly suspect that you will find at least ONE object in our solar system that doesnt have much of a spin or any spin but it still has gravity. Also, what about magnets? If magnets can attract or repel even when NOT spinning, then WHY cant larger objects like the earth not have the same quality? If magnets can do it, in other words, if magnets can exert a force that we can call “micro gravity” towards other magnets, then why cant a planet have a similar quality? Your theory could be correct, but as long as you are not saying that there doesnt exist any other attracting and/or repelling forces when a planet doesnt spin. As long as you’re not implying that, then I’m happy. In my personal opinion, I think that non-spinning large objects in space still have gravity. Therefore, we need to try and find a different explanation for the existence of gravity and how it works. We need to try and find an explanation independent of the “ether” theory. Have you ever considered that gravity could be related to magnetism in some way?
• drgsrinivas On October 17, 2014 at 9:19 pm
I feel that gravitational attraction and magnetism are fundamentally one and the same and can be explained by the Ether model. I will have to explore more on this issue. I have explained my thoughts briefly in the following reply-
• JJ On July 8, 2017 at 4:15 am
You are both correct in my opinion, they are definitely part of the same “force.” To me, time, space, gravity, EM, are all a singular phenomenon. And one can and do impact the other.
• Galacar On October 17, 2014 at 11:33 pm
To Darcy Donelle
Don’t believe what you have been spoonfed!
I know this is the original fairy tale!
But so much is wrong with it1
People later have tried to replicate it, to no avail!
You see, ; science’ is about propaganda, not real information and truth!
Once you see that, a lot becomes clear.
It really is a disguised religion.
Start unlearning what you have learned ( read: being programmed with),’
and you can start thinking rational and logical again.
• J Jagannath On May 1, 2015 at 7:39 am
I second that.
• Saiz On June 5, 2015 at 1:23 pm
I like micro explanations. You explain pressure by micro impacts of the particules againts the walls. But why the move of the particle changes from ‘randomly’ in all directions to ‘preferentially’, and why gases and liquids do this accordingly with Bernouilli law? If the particles continue moving randomly and rightward the collisions frequency would be the same, and so the pressure?
Thank you.
NB. I’m absolutly in agreement with you concerning the stupid relativity theory, even more after having seen the movie “Interstellar”
• Amitabh On July 24, 2015 at 11:07 pm
I like your explanation of floating objects. In a large vessel filled with water just leave some random free floating objects..they mimic a galaxy. The objects have different energies and exert wave patterns in the water pool…all the objects somehow keep a steady balance and flow as they chart their own orbit. They never collide. The container MUST BE ALIVE and so should be the environment. I mean a terracotta vessel is alive and water in plastic is inert ( ignoring the static) This is our cosmos. Of course the fluid contains or surrounds the planets, so is not planar in the sense of the experiment explained.
Now why the apple falls…is because the seed inside the apple wants to germinate. Are physicists blind to this simple phenomena of life meeting life ? So smoke goes to air and gross matter ones back to earth…and water evaporates to be with the clouds and reach the ocean…Is this poetry or science. .?
• Trevin On August 12, 2016 at 6:50 pm
Things do not happen only because life wants to meet life. That is like saying that dinosaurs just grew wings so that they could fly and not die when they jump off of trees. There are scientific explanations for these things (even though I do not really believe that dinosaurs grew wings).
• pk surendran On January 25, 2016 at 7:56 pm
All this boils down to one thing: Our seers short cut (learning by intuition) was the only way we could see the total from the fringes of fragments….
• aether On February 12, 2016 at 10:08 pm
I’m having trouble visualizing how ether spin alone can explain gravity on a sphere (earth). Can you elaborate?
• daniel3710 On February 14, 2016 at 5:42 pm
I believe what he said was that the spin of the earth within the ether medium causes a lower pressure area around the earth and this attracts nearby bodies due to the fact that any object in a pressurized environment will move towards areas of less pressure to create balance.
• aether On February 14, 2016 at 7:54 pm
Thanks for your explanation, but I’m still having trouble visualizing what happens especially at the poles.
I actually filled up my sink with water and used an electric hand held mixer to test the theory. The single mixing piece/extension (oblong, not spherical) was stuffed with steel wool. I used floating markers on the surface of the water. Admittedly crude experiment. I might try it in the bathtub or a bigger clear plastic container (round). It would help if I had colored suspended particles in the water.
Prior to accidentally finding this site, I had never considered Bernoulli effect was the cause of gravity. I’m not dismissing the idea, just having trouble visualizing it. Action at a distance just doesn’t cut it in my (simple?) mind. I always pictured an ether medium and universal pressure. Pressure differentials at points of mass.
• drgsrinivas On February 14, 2016 at 10:39 pm
aether, thanks for your interest.
I have explained that in the following post.
The gravitational attraction near the poles can be explained by the centripetal flow of ether towards the poles of the spinning body.
And thank you very much daniel3710, I truly appreciate your input. And this is what I really need. It is becoming rather difficult for me to address each and every query posted by the readers immediately. For most questions posed by the readers, explanations already exist at one place or the other on this blog. Little more elaboration is what is often required. (I know that animations would really help understand many concepts presented on this blog but I am neither a techie nor do I have time for that now. So please bear with me).
Having said that, It is because of the questions posed by the readers that I become enlightened everyday and able to solve the mysteries of this universe and creation. I owe a lot to all those people.
• Aether On February 15, 2016 at 3:05 am
I haven’t read everything on your thought provoking site, but I had read that piece. Still not sure, but that doesn’t mean you are wrong. I’ll play around with my kitchen mixer some more when I get the chance.
Also, at the equator, the rotational velocity of the earth is approximately 1037 mph. At the moon’s equator, the rotational velocity is about 10 mph. According to some sources, gravity on the moon is ~ 17% of gravity on earth – this doesn’t seem to square with spin differentials that are 100 times different. At least in my mind.
Then again, the moon travels farther than the earth on the trip around the sun and maybe this has an influence. And maybe the published density and gravity of the moon are wrong.
In any event, I can’t buy into action at a distance, constant SOL, time dilation…….all nonsense in my opinion. There are no paradoxes in nature imo. The Emperor’s New Clothes is the perfect analogy.
• drgsrinivas On February 16, 2016 at 10:56 pm
aether, I am unable to deduce the exact relation between the rotational velocity and the force of gravitational attraction in mathematical terms. That is beyond my mathematical brain. But I can tell you one thing: The rotational velocity of the fluid particles decrease as we go farther from the spinning object. And it is the velocity gradient between two adjacent layers of the fluid which determines the gravitational force at any locality in the space. Of course, the higher the the rotational velocity of a celestial body, the greater the velocity gradient that develops between the successive layers of ether, but I don’t think this relation is a linear one.
That probably explains why earth’s gravity is only 6 times more than that of moon despite its much faster rotational velocity. And of course, there probably exist many other factors- I would also propose ether density in addition to the ones you have mentioned.
• aether On February 17, 2016 at 1:42 am
Yes, this was readily observable in my sink experiments using fine black pepper particles. Unclear what happened at the poles of the steel wool or inside the steel wool for that matter.
<i.>but I don’t think this relation is a linear one.
I don’t either. Inversely proportionate to the ^2 of the distance? Do we really know the gravity on the moon? Our space gadgets typically have rocket boosters.
Anyway, I’m sure all this can all be explained by Einstein’s GR theory (at this point I’m surprised it’s still just a theory and not a law). No joke, GR can be used to explain everything.
Seemingly, everything proves Einstein was right. Behold: One year to collect the data and a whopping 5 years to massage and distort the data to maintain the religion of relativity!
What a spectacle! Theater of the absurd. A never ending Tamasha!
Liked by 1 person
• aether On February 18, 2016 at 3:40 am
Paper on Bernoulli effect and ether:
Click to access Jom%202014-Lin.pdf
Out of my league.
• bimbomechanic On February 21, 2016 at 5:03 pm
Has anyone considered gravity as density?
Objects denser then air sink, similar to how objects operate in water.
• drgsrinivas On February 21, 2016 at 8:58 pm
Denser/heavier objects sink in water because of gravity. If there was no gravity/ external force acting upon them, objects, whether heavier or lighter remain where ever they are left i.e. neither they float nor do they sink. So density per se can’t explain gravity.
Having said that, differences in Ether density can influence the strength and extent of the gravitational field generated by a spinning body.
• aether On February 23, 2016 at 12:28 am
A few thoughts.
In my comment above, I mentioned that the spin of the moon is very small relative to the spin of the earth. While true, the rotational velocities of both the earth and moon are tiny compared to their orbital velocities ~ 67,000mph. So one can think of the moon as orbiting the sun along with the earth in a vortex streamline. Of course, the (relative) spins of the earth and moon still play a role in the system
As expected, the farther from the sun the slower the orbital velocities of the planets:
Scientists talk about the barycenter of our solar system, but that might be theoretically calculated rather than actually observed. And even if it is observed, perhaps it could be explained in terms of a swirling vortex.
In a hurricane, the higher wind speeds and lower pressures are nearer the eye (center).
• Stephen On February 24, 2016 at 9:55 pm
All – though as men we are equal, I will not pretend to have equal understanding or education. I am just a man who loves to learn and am fascinated with understanding the universe (small task right – ha ha )…but I learn just for the joy of it. I am still absorbing these concepts. What lead me to this page was my inability to grasp the concept of space-time, which I believe is a component of Einstein’s theories. To me, time is simply a concept to quantify change. If nothing changed (all was frozen) in essence, time would stop, but with each passing second, my body changes, cells die and are born, the clouds move, the earth spins. The concept of time allows us to reference the changes around us. Is this a legitimate way of looking at this? I relate this to this discussion topic, be cause my limited understanding of relativity is that it is used to explain gravity. I do not understand how you can make time a principle of gravity other than to reference the effect it has on things. This becomes important because if space-time is a flawed concept, doesn’t the rest of his theory start to unravel? I am basing my comments on my limited understanding, so please forgive me if I come across as un educated (because I am). I also struggle with the concept that relativity applies to everything but light (in other words, the speed of light is constant, regardless of the travel speed of the observer.) Even if light has special properties, this still doesn’t make sense to me. I am not sure there is a question buried in my rambling, but any comment or response is welcomed.
• Galacar On February 25, 2016 at 12:27 pm
You sound more rational then an ‘educated’ men does.
You have been spared the deep brainwashing of our ‘scientific’ culture!
(being in a ‘culture’ is really telling us something!!)
Belessings to you.
Now, be less humble, and KNOW you can work things out.
Throw all that stuff away that tells you, that because you are ‘uneducated’
you can’t and ‘scientists’ can do better.
You have the very very deep advantage of NOT being brainwashed.
So, let your genius run.
As for your grasping or ‘grokking’ space-time.
Congrats! There is nothing to grasp!
Liked by 1 person
• Stephen On February 25, 2016 at 8:03 pm
Thank you very much for your good words. I continued trying to study Einstein’s theories after I made this original post yesterday. I will have to admit, your strong opinion seems extremely valid. To think that mass increases/decreases due to change in velocity (which in my mind, means you gain atoms or the atoms change), or that time slows down or that distance shrinks.. as you go faster…etc., makes absolutely no sense at all. At least not to me. At best, the function of the concept of relativity is simply to provide a reference point. (you cant know what light is if you don’t know what dark is). It is a concept of perception and not physical science. No wonder it was so hard to understand, I should worry if it did make sense to me. What is interesting is how many people who teach it, teach it as if it is undisputed fact. I am finding that happens a lot with people teaching science (and history for that matter). Often times things are just theories and not thoroughly validated in any way. In psychology you learn that beliefs are individual and have nothing to do with the truth, but our perceptions and our acceptance of what we are told or exposed to…as well as our faith in the source of information. I am quite sure when Einstein developed his ‘fuzzy’ math it made perfect sense. He was trying to solve un answered questions. He had great faith in math and assumed if he could make an equation work, then it MUST be true. I can understand, he is just human as am I. I think that as people, we feel foolish to challenge anyone who is accepted as ‘great’ or is highly esteemed. We also are typically willing to follow them blindly. I think it is part of our nature, but thankfully there are always those occasional minds that challenge the accepted and launch us forward in new ways of thinking and new ideas. Its actually quite brave in many ways. I’ll tell you something that happened to me as well. Up through my 20s I had no belief in Auras. In fact, I watched a special on 60 minutes about a girl who could see auras and even tell if someone was sick. My thought was either this is a hoax, or the little girl was gifted with rare psychic ability (which I also didn’t really know if I could believe in). Then one day, many years later, I realized I could see them!. First, they were always clear and I had always assumed I was seeing clear light reflecting off of people. When I realized I was seeing this in dim lit areas, and the entire shape of the person, it occurred to me that couldn’t be light reflecting. Then once I recognized it as an aura, I eventually could see colors too. I don’t see them all the time, and they have no ‘psychic’ meaning to me, but this has opened my mind. I realized just because I didn’t believe in something, or if I couldn’t see something, didn’t mean it wasn’t true. It also taught me to be open to change my views if there is new reasonable information to consider. This being said, I still hold tight to the idea of critical thinking and using logic. I even accept the possibility that perhaps I have something wrong with my eyes or the part of the brain that deals with vision. I do not believe this to be true at this time, because of several reasons, but I also realize I cannot prove or validate what I am seeing. It is simply something I experience. I suppose I am just talking to talk now. I do want to thank you for voicing your conviction about the errors of relativity. It has really helped me.
Liked by 1 person
• Galacar On February 28, 2016 at 1:01 am
Actually, You are starting to get in touch wich your
multi dimensional YOU!
Nearly all of this (mainstream) world is here to keep us in
a little box, being little me.That is the PURPOSE of mainstream media,
science, politics, whatever.
(what do you do if you look up to someone? That is right! You are looking down on YOU! Makes sense?)
free your mind!
That will scare some people up the ladder! lol
I hope not to sound too arrogant, but this is my deep, deep conviction.
See if this resonates with you.
My two cents,
• aether On March 2, 2016 at 1:39 am
Regarding infinity:
Derived Planck Units
Who can say for sure?
• John Davis On June 25, 2016 at 6:17 am
Interesting article on gravitational anomalies during eclipses. Known as the Allais Effect, Pendulums swing faster during an eclipse.
Interestingly Allais went on to deduct this from the findings
“Maurice Allais states that the eclipse effect is related to a gravitational anomaly, that is inexplicable in the framework of the currently admitted theory of gravitation, without giving any explanation of his own. Allais’s explanation for another anomaly (the lunisolar periodicity in variations of the azimuth of a pendulum) is that space evinces certain anisotropic characteristics, which he ascribes to motion through an aether which is partially entrained by planetary bodies. He has presented this hypothesis in his 1997 book L’Anisotropie de l’espace. This explanation has not gained significant traction amongst mainstream scientists.”
Liked by 2 people
• Galacar On June 25, 2016 at 1:06 pm
@John Davis,
You wrote:
Well. problem is that the ‘modern physics’ is full with gravitational anomalies.
Because there is no gravity at all.
Gravity by itself is a myth, and anomalies come easy with myths. 😉
I am not saying things don’t fall etc. Of course they do.
But gravity m which is non-existing, has nothing to do with it.
btw isn’t it interesting to see that if people have a word for something,
like ‘gravity’ they think they understand it I find that fascinating.
My two cents.
• John Davis On June 26, 2016 at 9:09 pm
I think he is on the right track though. His study of pendulum swing during eclipses alludes to an ether or emission based source of gravity. As another paper puts it – We jump and fall back down because we are entrenched in the ether – which is constantly being pulled towards the earth. If the sun is a provider of this emission then it would make sense that an eclipse would disrupt its flow – in the same way an island disrupts tidal swell as felt by the mainland shore. It also might help us better understand the dual tide phenomenon for which the current language seems a bit illusory.
Liked by 1 person
• Trevin On August 12, 2016 at 4:45 am
If your ether theory is right, why is it that wind does not increase dramatically with altitude? If your theory was correct, would not the photons at higher elevations push the atmosphere at a rate slower than the speed the atmosphere goes on the ground? Would not these air particles cause wind, since they would be moving over a shorter distance in the same time period as the earth is rotating?
In addition to that, here is a website that presents the Concave Earth Theory, which I do not necessarily agree with: http://www.wildheretic.com/ . This theory is different from the heliocentric model, with the entire known universe being inside a concave earth. You should probably check it out, since you are theorizing things that have to do with astronomy.
• Trevin On August 12, 2016 at 6:31 pm
There is an experiment that seemingly proves that the heavens move above the earth without the earth moving. This experiment was done by George Airy with a water filled telescope. You can find out about this experiment at exhibit D in the following link: http://www.wildheretic.com/heliocentric-theory-is-wrong-pt1/ . Can your particular heliocentric model explain this experiment just as well as the immovable earth model can? If so, how?
• John Davis On August 16, 2016 at 10:03 pm
I’m open to both models – helio or geo. Airy’s experiment seems to be a simplified version of the Michelson Morley. The supposed conclusion – Either there is no Ether or Earth is not moving. However if you read through this site you’ll see that Dr. G. has whirlpooling ether models which can support both a spinning earth & ether.
• Lance Nelson On September 24, 2016 at 3:24 am
I enjoyed your description of pressure and how it changes with flow speed. However, I’m not following one of the main points to your argument. You said,
Let’s say the flow(or pipe) is oriented horizontally. I think you are saying that as the gas particles are accelerated horizontally, the vertical components of their velocity vectors(perpendicular to flow) will decrease, thus making their change in momentum in that direction diminish? Indeed, if that were the case, I can see how the pressure would decrease. However, how can a horizontal acceleration affect the other components of the velocity?
One more thing: your explanation relies on thermal motion being the main source of pressure. However, in many instances, it is the weight of the particles pushing down on their lower-lying neighbors that is the main source of the pressure. Can you explain in these terms.
• drgsrinivas On September 28, 2016 at 6:05 pm
Lance Nelson, thanks for your comments.
The simplest answer for your question is that when horizontally moving gas particles flow into the pipe, they displace the random particles. So the vertical pressure drops in the pipe as the gas flows horizontally. And of course, accumulation of random particles causes an increase in vertical pressure at the leading end of the stream. This increase in vertical pressure is what causes the ‘buckle’ or swelling of the water pipe at the leading end of flow.
I wouldn’t say it is ‘thermal motion’. It is ultimately Energy that causes motion of particles. We experience that Energy or motion of particles as heat in some situations. The scenario of ‘weight of particles pushing down’ only occurs in a gravitational field. There, the particles get downward acceleration because of the gravitational force and hence are able to exert downward pressure. It is again motion ultimately.
• carlchristianbarfield1st On August 23, 2017 at 10:25 am
Internal motion, in the electric field, results in much of the mass, as photons spinning exerts momentum on the surrounding space.
• John Foster On September 25, 2016 at 12:58 am
You had me excited by your revelations until you included rotation of a body having an increased effect on gravity. If anything rotation only decreases the effect of gravity due to it being a centrifugal force. Pressure is definitely the key to gravity, but maybe we need to start thinking on a smaller particle level, ones that may cause a pressure effect, but also pass through the object, and accelerate after leaving, slowly back to a maximum velocity.
If two bodies have an equal amount of pressure exerted upon them, and the particles causing the pressure can pass through the objects at a slower velocity, then slowly regain their original momentum, then that would cause the effect of attraction.
==> O =><= O<==
Also pressure on a single mass would definitely explain gravity. We need to think of gravity as a Pushing force rather than a Pulling force. Pressure caused by particles passing through us, slowing down as they do, and then slowly regaining their original velocity as they exit would explain gravity and attraction.
One other correction I would make is c=maximum speed of light. it is not constant because space is not a vacuum, but it is a maximum and hence works with the best equation Einstein came up with.
As for quantum entanglement, well this is easily explained by pressure throuout the universe. Infact any object can exist anywhere in the universe at any one time, if we apply the correct pressure on an object at the correct particle level. It is like a domino effect.
|| || // //
|| || // //
Time is not a dimension, it is not variable, it progresses at a universal rate. People need to start thinking outside the universal box, and observe things that way. Relativity only works because we are inside the box!, Einstein gave us equations for being in the box, and that was awesome. To progress his work we need to, erm, well think outside the box…
One thing I do have a problem with is time dilation. I cannot explain it, I do not know why it exists. I look forward to reading your theories on this subject
• John Foster On September 26, 2016 at 1:06 pm
Please forgive my post, I had a few drinks too many and got very excited by your ideas, but I had only read a small amount before I rushed out my drunken comments.
The words relativity and space time makes my head want to explode in anger every time I hear them. Reading just this page alone was like a breath of fresh air.
As for what I was trying to say in all that garbled post….
I have been trying to understand the cause of gravity my entire life, and einstein’s theories are just a load of rubbish to me. It is not that I don’t understand what he means, I just think it’s incorrect. I wasn’t trying to correct your theories and ideas, I was trying to correct his.
E = M x the MAXMUM speed of light squared
That equation was what I may have referred to as awesome, and as for progressing his work, it would be from the time when he worked out the relationship between energy and mass.
I like to try and have ideas to explain things that have apparently been proved to exist, such as quantum entanglement and time dilation. By time dilation I mean the difference in the 2 clocks in the experiment. I don’t believe in it myself, but I personally could not explain why the clocks differed. As I have only so far read a small amount of your work, I was trying to say I was very much looking forward to reading what you have to say about it.
If I have learned anything from this, it is to not try and explain my ideas when I am drunk 😀
• drgsrinivas On September 26, 2016 at 8:49 pm
John Foster, thanks for your interest and input. I can understand the reason for the confusion. I have explained about the centrifugal force and other forces created in the vicinity of a spinning body here- https://debunkingrelativity.com/2015/11/06/demystifying-electromagnetism/
When a body spins in a pool of water, it is true that centrifugal force pushes the water particles away (and generates ripples that spread outward). But all the suspended denser objects get dragged towards the spinning body. That’s what happens in a centrifuge also.
Your centrifugal force is what generates the so called gravitational waves that spread outward from a spinning celestial body in the Ether ocean.
• Daniel25 On February 8, 2017 at 12:24 pm
drgsrinivas, i love all you reports im slowing getting through them all, i have a question for you, do you believe we are on a spinning ball or do you believe we are living on a flat earth? alot of you debunking fits in line with a flat earth, tesla knew the earth wasnt moving, id love to hear your thoughts? thanks
• Joe Deglman On February 17, 2017 at 6:59 pm
After seeing this portrayal of ether flow it seems to explain the General Theory well, what will happen to light as it passes a star or the Sun. It is almost as if Einstein knew what ether was and its effects, and did his best to cover it up! Or maybe just a dunder head?
• JJ On July 8, 2017 at 4:08 am
Excellent work DrG!
I saw your comments about weight at the Polar regions and equator, as well as the centrifugal and centripetal forces.
Have you studied Nikolai Kozyrev? One of my now deceased friends who worked In “Aerospace” projects most his life told me he is the basis of many of their technologies. He successfully managed to diminish weight and gain weight EXPERIMENTALLY , through the use of spin/torsion, and proved existence of Aether (using other words). It also indicates a connection between electromagnetism, time, gravity, and “Space.”
There are also deeper implications, such as that to biology! But some other time.
He is all but ignored by modern science
I also recommend friend of Tesla, Walter Russel. While less of a scientist, I think conceptually, he is spot on. He discusses the centrifugal and centripetal (pressure) forces, and how they cycle in the world. As well as the illusion of the speed of light, and how it doesn’t actually travel the way we think it does.
Thanks for great work.
• JJ On July 8, 2017 at 4:33 am
Also, my friend told me that they had a better explanation (using Kozyrev’s theories) as to why “time was local,” and so called differences in its passage, instead of relativities time dilation. It has to do with Ether (vacuum), and how suns, galaxies, and planets spin…and their “counter spin.” One appears to create a sort of centripetal pressure toward poles (and is invisible) and the other, which we see, is the centrifugal spin at the equator.
Like “patches” of dark energy around stellar bodies, absorbed and released, at specific ratios that differ depending on the area. Perhaps the sun, was the local “clock” or conductor of the ratios of time/space. Perhaps there is indeed, a “divine order” materialistic scientists are utterly ignorant of.
In a sense, Einstein was right that time is not absolute. But all he did was see the effect, not the cause. He said space time magically bends in presence of mass (thus gravity), when actually, it may be more that the Ether (space-time) is what dynamically compresses “space” or responsible for the formation of planets, while centrifugal spin assists in expanding space (like whirpools moving outwardly).. The equilibrium between the two as the stable structures we observe as matter.
He also told me so called “free energy” technologies which draw from Aether/vacuum, have an effect on this compression or expansion of space, thus the acceleration or slowing down of time. And that abusing such technologies, actually have an impact on acceleration of time, and thus, are in no way “free.” That you could also impact space, by working only in space domain, and vice versa.
One person asked, where does the “Aether” get its energy from. I think out of all the QM physicists, David Bohm was closer. He called the vacuum a plenum, and looked at reality as enfolded in layers, implicate and explicate Orders, and in terms of Wholes instead of “individual balls” we call particles. We may not know exactly what exists beyond Aether/Vacuum, like a veil between levels of manifestation- BUT we do know that energy does exist, even QM knows this through the spontaneous manifestation of “light” which they call “virtual photons.”
One analogy is that one is the “absorbtion” of ether, or the porous planet being held together by the etheric center seeking pressure (which is correlated to youth or negative entropy) and the other, the centrifugal, as a “release” of the etheric pressure (and thus also, radiation and some magnetic phenomenon and entropy).
Maybe, As a body gets older, it bulges more at equator perhaps and also begins to lose its density. Almost as if youth is the winding of Aether (Gravity and “time)” and older age is the unwinding of Aether (radiation, and expansion). The winding like an increase in Aetheric pressure (which in turn, correlates with density, like octaves of matter or pressure), and the unwinding like a release of the pressure.
Most of this my own thoughts of course. But thanks for stimulating!
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• michael ngan On March 13, 2018 at 9:57 am
Hey Bro.
I love your explanation on Gravity. It is the most lucid and logical, I have encountered yet, beside Euler’s explanation, who used very similar logic.
However, you seems stopped suddenly, when you reached the end. I have the audacity, to added a little bit more explanation.
“Now replace water tank, with Ether universe. Imagine a body suspended in still Ether. Imagine Earth nearby and allow it to spin.
The ether closest to the Sun, spins the fastest, so the lateral pressure, on the surface of the Sun is lowest, and further away, there is a stronger lateral pressure.
So there is a pressure vector or gradient, that is directed from outer space, toward the surface of the earth and Sun.
And the resultant etheric pressure vector, is isotropic, or is exerted equally and perpendicularly, in all direction, toward the surface of the rotating earth and Sun.
And this pressure vector, depends only, on the distance from the surface of earth and Sun.
The closer one is toward the surface, the stronger one will experience this etheric pressure vector force called Gravity.
That explains Gravity.”
Liked by 1 person
• A_Concerned_Student On June 18, 2018 at 10:59 pm
Hello Dr.G,
I am a senior physics student who has stumbled upon your blog while looking into a question regarding fluid flow and Bernoulli’s principle. Initially, I must say, I was very impressed with your thoughts and understanding of fluid flow and Bernoulli’s principle. While you are technically incorrect as to pressure being a vector, (it has no direction), changes in pressure from one region to another can very much behave like a vector would and for this reason you could very easily treat pressure as a vector in SOME basic fluid flow problems. This aside, I found your discussion of Bernoulli’s principle rather well written.
Now, onto the main reason for my posting. Your final statement as to pressure’s role in gravity and the comments following left me rather concerned. I felt that someone with a grounding in physics should comment on this post for anyone who stumbles across your blog as I have. I feel this because someone with this understanding of physics is precisely the type of person who is best suited to debate on the topic, which is why I felt compelled to do so.
I will start with what determines a good scientific theory which is twofold: the ability to make predictions, and the ability to be falsified (proven wrong), not whether or not people like the theory or find it intuitive. Your theory has both these qualities and as such is a good scientific theory, however I fear that experiment decides that your theory is (at least in its current form) incorrect. Before I get into your theory, I would first like to defend the relativity and quantum mechanics (hereafter referred to as QM), which are the mainstream (admittedly not intuitive) theories for the two reasons stated above, predictions and the ability to be proven wrong.
Initially, the VAST majority of the scientific community rejected Einstein’s relativity and believed it utter nonsense as you seem to believe. However, Einstein’s theory was easily falsified by experiment so they conducted tests to show him that he was wrong, but instead his theory correctly predicted what would happen. Undeterred, the scientific community continued to devise more and more complicated and outrageous tests of relativity, each time convinced it would prove relativity wrong but each time it was right. To this day, no test or experiment has found fault with relativity, (specifically GR), and as scientists attempted to hold onto their classical theory of the universe, they ultimately led to relativity becoming the most rigorously tested scientific theory of all time! Similarly for QM, it was an even more ridiculous and unintuitive theory. Scientists had their hearts set on a classical theory of the universe, but yet again, each experiment and test failed to disprove QM. QM made absurd predictions that NOBODY thought could possibly happen, but time after time when scientists tested them scientists were baffled when the predictions were correct. Despite the vast majority of the scientific community believing relativity and QM absurd and ridiculous, and doing their best to find a case where they failed to coincide with experiment, they never did. It is for these reasons that relativity and QM are the mainstream theories of the last century, not because people liked them, quite the opposite was true, but because they made predictions which continually agreed with experiment (even predicting the existence of undiscovered particles in some cases) and because they were easily falsifiable but never falsified. So you may disagree with them because they are absurd, I myself find them absurd as does nearly everyone, but they are extremely useful as a scientific theory and have never been proven wrong so much so that people who hated them ultimately accepted them as the next step forward in physics. Nowhere do I say they are correct, as no theory can ever be proven correct, only incorrect, but for over a century they haven’t been despite the ridiculous claims they make (which coincide with experiment). This wraps up (for now at least), my defense of the mainstream “religions” as you frequently call them.
As for your theory, initially I thought it a brilliant re-imagining of gravity and rather intuitive. However, after careful consideration I feel that it is most likely if not absolutely wrong. I considered it deeply for a few days before posting this. The main issue I can not get past is gravity at the poles. If differential spin alone were the cause of gravity then none would exist at the poles instead you would be pulled horizontally towards the axis of rotation (pole) when near the poles which would have definitely been documented by now. So if that’s not what is happening let’s consider your adjustment with the “porous Earth” theory where aether is sucked in through the poles and ejected equatorially (or vice-versa follows the same argument in reverse). If this inflow of aether is strong enough to produce the (nearly) identical gravitational attraction at the poles as at the equator then the ejection at the equator must be of a comparable strength. The disk at the equator where aether would be ejected would have far smaller area than the likely cone-like shape of inflow at the poles. This would lead to a much stronger ejection than inflow (which we established causes the gravity at the poles) so the aether would cause a strong repulsion at the equator which should lead to a much lessened if not completely nullified or reverse gravitational force at the equator. I find it extremely unlikely that every planet/star we have ever observed has the precise inflow/outflow ratio that the gravitational attraction is attractive everywhere on the object and nearly identical to the equator. Comparatively, the mainstream gravity theory correctly accounts for why this is much simpler using just the shape of the planets. Also the planet Venus spins in the same direction as the sun which is opposite to what your theory predicts. According to your theory, Venus should then drift away from the sun or begin spinning the “correct” way. Since this hasn’t happened in it’s ~4.5 billion year lifespan I believe this to be further evidence that your theory is wrong.
These are just my considerations and I hope that this helps anyone who reads this site not become convinced by these ideas as I (a senior physics major) almost was as they are quite convincing and well-formed. Any comments are welcome and I sincerely do hope for a rebuttal.
• drgsrinivas On August 24, 2018 at 11:31 am
Dear student
Please keep aside your religious assumptions and also what your science prophets think they have observed. Just make a ball spin inside a pond and you would see it ‘attracting’ objects in its vicinity, you would see that ‘attraction’ force both at the equator and also at the poles. Try to explain to yourself how and why it happens, then you would understand about gravity. You wouldn’t get that even if I explain it to you. Spoon feeding doesn’t always help!
I have talked about the retrograde spin of venus here – https://sciencevstruth.com/2014/06/25/explaining-the-retrograde-spin-of-venus/
• drgsrinivas On June 21, 2018 at 2:30 pm
Hello A_Concerned_Student
Welcome to the blog.
Your comments went into spam folder. I don’t know why, may be because they are long.
Thank you for your well thought out criticism. I will surely clarify your queries when I get some time.
Basically it is not that easy for physics majors to come to terms with truth despite one’s genuine interest and whole hearted attempts. The reason is that, by the time one becomes a physics major, one is taken too far away from truth.
Can I ask you to clarify why you think pressure is not a vector, in simple layman’s terms. I know it is not a vector according to your books. But I am sure you realize that we don’t blindly go by those books here. Thank you!
• A_Concerned_Student On June 21, 2018 at 9:23 pm
Hello Dr.G,
I do appreciate you getting back to me and am glad to hear that it was a genuine mistake that my comments were sent to spam. The reason I say that pressure is not a vector can be seen clearly in a very simple example.
Imagine we have a rectangular box with some air in it. Now imagine that by some method (how doesn’t really matter at this point) that all the air particles ended up bunched together in small rectangular section of the middle of the box. I would attach a picture if I could but the picture would look a lot like your figure showing why the pressure would be reduced when the air is flowing horizontally through the box, except with sides on the box and no flow with no particles near the sides of the box. Clearly, since all the particles are concentrated near the central strip of the box there is a high pressure in this area and low pressure near either side of the box.
Now the question becomes, which way do the particles in the center of the box go? Anyone who has seen this site would say that the particles move from high to low pressure but there is two different directions that they could go to get to low pressure. Since a vector has both a strength and a direction this simple example shows why pressure cannot be a vector. Since in the center of this box the pressure vector would need to point in opposite directions at the same time to denote which ways the particles could go. On top of this example there is also the issue of how long (strong) the vector would be. At the center of the box there would be a high pressure and there would be a slightly lower pressure slightly off center. Therefore we could draw a fairly small vector to show this, but we could also draw a larger vector going from the center to the sides of the box where there are no air particles and a very low pressure. Since vectors have a definite length and direction, and in this (and many more) example(s) we cannot with certainty say which direction the vector is in nor how long the vector is, we must conclude that pressure is not a vector. Changes in pressures between regions however result in a relative strength and give a sense of which directions the air particles will move, which can act very much like a vector does.
• King On June 21, 2018 at 11:26 pm
Pple keep bringing the issue of testing the theories of relativity etc when it is clear that drgsrinivas has a THEORETICAL dispute rather than an empirical one! This clearly show that these mainstream advocates are not very smart! If I may put it succintly, Drg is questioning (putting it in my own words):
1.)Do the aleged tests ACTUALY test the petinent theory? Drg correctly note that in all Theories, we say if A is true, B is true. We have tested and confirmed B. However, claiming that thus A is true is a nonsequitur logic. In Drg’s words, ‘no experiment proves a theory straight away’
2.)Is Occum’s Razor use properly? It make no sense to say that predictions prove a theory since there are myriads of theories that make similar predictions! This is essentialy the lesson here. It is irrelevant that QM predict interferance patterns by supposing that a single particle somehow pass through two slits. We can also predict it by supposing that particles create ripples as they move!
Liked by 1 person
• A_Concerned_Student On June 22, 2018 at 11:44 am
Your first point is true entirely. A theory can NEVER be proven to be true, only proven false. If as in your point we call the theory “A” and something the theory predicts “B” then showing that B is true does not mean A is. However if B was shown to be false then A must be false. The purpose of bringing up tests of relativity is simply for this reason. Relativity makes a lot of predictions (B’s, C’s, etc.) all of which if they were not found empirically would mean that relativity is not right. Since all of relativity’s predictions have been observed empirically (or at least not shown false) then we have not disproved it and added more evidence that it may be correct or at the very least a good approximation of what is actually happening.
In your second point you admit Occam’s Razor. The Razor is a sometimes useful tool to remind scientists not to go crazy and make up ridiculous theories when there are simpler ways to go about it. There are indeed tons of theories that predict the same outcome of any one experiment, but fail in predicting all the experiments that have been done. The reason QM is as complicated as it is comes about from the Razor (albeit secondhand). Scientists started with simple explanations to explain why nature does what it does and correctly explained some experiments but not all. So in order to explain more of what they saw, they made their theory more complex. This continued throughout history, throwing away theories that were proven wrong or adjusting them until a new experiment came along that proved them wrong until we reached QM. At some point, no doubt, an experiment will come along showing it is wrong too and it will be adjusted or thrown away completely for a theory which better fits what experiments show. This theory however will more than likely be as complicated, if not more so than QM is currently in order to explain more than QM does.
• King On June 21, 2018 at 11:47 pm
Anyway let me still debunk the claims conserning ‘tests’. The ‘student’ wants us to beleive that the modern mainstream phyc theories were significantly disliked and only embraced after they passed lots of tests. By this, he insinuates that mainstream scientists aren’t ordinary humans who are ameanable to confirmational bias!
Let me debunk the nonsense. In 1919, Einstein became an overnight celebrity! If science was done the way it is advertised, this should not have happened. We know that whenever an aleged observation goes against mainstream theories, the observer stands a chance of losing his job (Alton Harp). Furthermore, an observation during an eclipse requires independent verification. If mainstream phyc wasn’t biased, then why hasn’t Maurice Allais similarly hit headlines when he claimed to observe anomaly during an eclipse?? If they hated GR like they do, ‘theories that agree with Allais effect’, then Einstein and Eddington could have gone just like Allais: into obscurity for 50 years.
• A_Concerned_Student On June 22, 2018 at 12:05 pm
I now see that you have commented several times on my post and thank you for taking the time. I am indeed a student so the quotations are unnecessary, if I were simply looking to sound more credible I would claim to be a professor or researcher of some kind instead of just a student. As for your comment, some mainstream scientists are definitely subject to confirmation bias. Scientists often choose to research the theories they are interested in or like, but this also leads many to be defensive of the theories they choose. As such, when new theories come along that threaten their own, they go out of their way to try and destroy the new theory, This was very much the case with relativity and QM. Scientists very much loved their neat and tidy classical theory of the world and when these new non-classical theories came along threatening that they were far from happy about it.
As for Einstein’s “overnight success”. As far as I know it is true that by 1919 Einstein was a well-known figure in the scientific community, however he originally published his theory of special relativity in 1905! He then went on to publish general relativity in 1915 leaving 14 years for tests of special relativity to take place and 4 more for GR before he was this well-known. So I feel it is safe to say that Einstein was far from an “overnight success”.
For this last little bit please note that I know very little of Allais’ work on the eclipse anomaly that is named after him. From what I am aware the reason Allais has faded into relative obscurity and is not taught in any form throughout schooling is due to the lack of reproducible effects. Many teams of researchers have attempted to observe the Allais effect with varying success. Some have found large anomalies, some small, and still others have found absolutely no observable anomaly. Since we cannot say that a phenomenon is occurring with certainty if we can’t reliably have it happen in a controlled experimental setting, it remains a “claimed” effect. This is likely why Allais did not become “very” famous (he did win a Nobel prize in economics), because his anomaly could possibly be due to experimental error or some other phenomena interfering with the experiment. Please note that I am not saying that it isn’t a real effect, just that it could be caused by something other than eclipses and that is why we can’t make it happen every time we try.
• King On June 22, 2018 at 12:15 am
In early 1930s, it was observed that the way galaxies spin cannot be explained by GR! This was barely 15 years after the concoction of GR. If, like the ‘student’ want us to beleive, scientists hated GR and wanted to disprove it, this should have been an exellent opportunity!! Instead, they invented an invisible kinda ‘dark matter’ as an ad hoc!!
In 1920s and 30s, we also know that mainstream physicists e.g. Paul Dirac were thoroughly working on relativity. If this theory was hated by majority of scientists, we know what should have happened: working on it would endanger careers!! We know that Diracs Theory made incorect predictions with regard to g-factor, etc. If mainstream physicist were looking for ways to falsify QM and relativity, again this was an oportunity! Instead, they invented Quantum Field Theory.
Again, QFT hits a wall by yielding infinities. This should have meant that QM is incompartible with SR. But again, they invented blatant ad-ocs such as renormalizations!
• A_Concerned_Student On June 22, 2018 at 12:26 pm
As I put in my reply to your first post the process of obtaining newer and better theories relies on modifying already successful ones. The spins of galaxies showed that GR wasn’t working in its current form, this is true. But it worked very nicely if we assumed there was more matter in the galaxies then we could see by ~70%, then all the predictions worked really well. Thus they began looking for whether this “dark” (can’t be seen) matter existed and what it could be. Therefore it shows that GR wasn’t wrong but rather predicted the existence of an entirely new form of matter that we couldn’t see! Now I know that you’re going to say that we haven’t “proved” it exists and I say how can we? If the supposed matter doesn’t “react” (layman’s terms) with light but only through gravity, and we see it interacting with gravity causing this anomalous spin in galaxies, then we’ve seen all we can to say that it’s there! Yes this does feel very ad hoc and sketchy but it makes very accurate predictions which allows us to better predict and understand what happens in the universe around us, and ultimately that is the purpose of science.
Dirac’s relativistic theory is in fact QFT, he never made a “relativity” of his own but rather pioneered combining QM with relativity to make QFT. Since both theories are so successful in describing things separately the next logical step is to combine them and that’s what he did. Since they were founded separately the math used in each wasn’t initially easy to reconcile. e.g. Consider Leibniz and Newton both making calculus from two entirely different methods and the completely unrelated notation they are combined under nowadays. Renormalizations are a completely mathematical thing which mathematicians do frequently and is perfectly valid, and applying this to physics does make it more abstract but it works very well to make QM and SR work under a single notation. Infinities are special in the sense that getting them doesn’t necessarily mean what you are doing is wrong but rather the way you are going about it is. Math becomes very abstract once you bring calculus into the picture and in many problems the slightest error in calculation or wrong approach to a problem can yield infinities. It doesn’t mean the problem is wrong or the math is wrong, it just means that you are doing something wrong using the math in that problem. Such was the case for QFT, when combining SR and QM we were doing the math wrong. By introducing renormalizations however, the infinities went away and SR and Qm combined nicely and gave good consistent predictions.
• King On June 22, 2018 at 12:32 am
But compare this with what mainstream truely dislikes. Consider the observation made by Alton Harp. If mainstream could be ‘forced’ by observation against what they like, then observation that quasars shows no time dillation should have forced them to rethink of Arps work. They don’t so they are realy not doing science!!
Consider the so called ‘axis of evil’. Why are they still seem to doubt?? Compare this to the ‘discovery’ of higgs. Why didn’t it waite for ‘independent confirmation’ like cold fussion? You see, when they see what they want, next year, you hear of nobel prizes. But some observations must waite for confirmations, then wait, then wait,…
Consider dark matter. Why are they still searching for it? Why didn’t they dismiss it based on a single experiment that failed to detect it, like they did, luminiferous aether??
• A_Concerned_Student On June 22, 2018 at 12:54 pm
As for this final comment of yours, I am afraid that I do not know who Alton Harp is nor what he has done. If you would be so kind as to enlighten me then I could give my thoughts. As for the other suggestions I will try and each fairly quickly but I want to explain each as best as I can as well.
The observation that quasars show no time dilation is a mystery. There are a myriad of reasons why this could be and many possible explanations have been put forth. Until we have more experiments to show whether any of these are correct and if quasars truly have no time dilation then there is no reason to throw out a theory which works so well in so many cases. Much like how we still use classical mechanics when considering how a ball flies instead of the full GR calculation, we didn’t throw out CM but simply realized that we need something more accurate in certain cases.
The “axis of evil” is a political term coined by George W. Bush regarding governments that allegedly supported terrorism at the time and I feel that this has no place in the current topic as it is not science related nor do I feel that I know enough about the topic to properly discuss it.
The discovery of the Higgs did not wait for independent confirmation because it isn’t reasonable to ask, and because of the strong confidence of the result. To discover the Higgs, 22 countries had to come together to establish the world’s largest particle accelerator and spend roughly 40 years to find it. It is unlikely that any one person or even large team of universities could ever imagine doing this themselves and as such it is unlikely that anyone could ever independently confirm it. The main reason it wasn’t independently confirmed though was because of it’s 5 sigma confidence. 5 sigma means that they are 99.99994% confident they saw the Higgs. That means that only 1 in 3.5 million times would they be wrong. Since they are a team of 22 countries of the world’s top researchers working together and they’re that sure they saw it, then independent confirmation doesn’t add any significant amount more confidence. So it wasn’t “required”. As for cold fusion, anytime anyone says they’ve done it they’ve quickly corrected themselves and said it was a mistake. This combined with the fact that so many countries are working on it independently means that if one of them can do it, so could another likely. As such, independent confirmation is reasonably possible and so we require it to confirm cold fusion.
Finally, the aether. While much of the scientific community does not believe in it, some are still believers (a professor of mine is actually). It is worth saying that a lot of experiments have found evidence that there is no aether, the Michelson-Morley was just the nail in the coffin. At that it only showed that there was no significant movement in the aether. All our theories on light actually don’t require an aether to completely describe everything we see it do and as such we appeal to Occam’s Razor. If we don’t need it, then why add it? There are many theories which require one and assume it exists, however the mainstream theories don’t need it so they say it doesn’t exist. It may very well exist, and if it had a noticeable effect on anything we’ve observed so far then it would be apart of our current theories. Maybe one day it will be useful in describing why something happens, then it will be brought back. Maybe not.
• King On June 23, 2018 at 3:25 pm
A concerned student,
Thanks for those good replies. You have addressed some things very well, but there is still several things that I find them unsatisfying.
My main point is not to say ‘Allais is true’. It is to say that:
1.)An anouncement of a 1954 gravity anomaly at then would just be same as an anouncement of a 1919 gravity anomally in that both require independent confirmation.
2.)The 1919 anomally made Einstein an overnight GLOBAL celebrity (not a celebrity in a German scientific community) but the 1954 didn’t create any sensation.
3.)Therefore it is evident that scientists were especting to confirm Einstein, contrary to what you said, that they were eager to falsify it.
• A_Concerned_Student On June 23, 2018 at 9:05 pm
I see your point now regarding the 1919 eclipse. I think the main difference in the two cases comes from the nature of the two examples. Namely, that the 1919 eclipse aimed to see which of two theories is better (Newton or Einstein), and that the 1954 eclipse anomaly was not expected but rather Allais coming out and saying he saw something weird. In the case of the 1919 eclipse, researchers went in looking to see if the results agreed with Newton or Einstein and as such the media was aware of their intentions and the impact that the experiment had for the direction of physics. As such, when Eddington et al. released their findings that November the media was probably looking forward to it, as once again it is a big story. It is also worth noting that Eddington did have multiple teams in different locations around the Earth to observe the eclipse, not just a single guy with a telescope which lends to the credibility of his result. In the case of Allais in 1954 however, there was no build-up before the observation to receive the medias attention. He came out afterwards saying he saw something that was weird.
Now I ask you King, if you were a journalist looking to publish a story that'll sell newspapers and you had to choose between the two stories, which would you choose? The one of testing which theory is right, the well-known Newtonian theory or the up-and-coming relativity of Einstein and the impact this has on science? Or rather would you choose the story of an economist who made an observation that pendulums behave weirdly during an eclipse? I posit that it is the media who ultimately made Einstein a celebrity and Allais an unknown name. The desire for the media to put out sensationalist stories that are revolutionary leads them to put forth more stories like EInstein's before independent confirmation and more accurate analysis can be performed. So I believe that it is not science who is at fault for the Einstein/Allais case but rather how the media decides to report science. But that is just my opinion, it could very well be wrong.
I also do want to say that Eddington was a strong supporter of relativity. Much of the dislike that I mentioned in my original post towards relativity was towards special relativity when it first came out in 1905 and by 1919 many scientists had already come around to the theory. As such it is possible that Eddington was biased, and there are many articles discussing such possibilities elsewhere. I’m afraid that I don’t know much as to the specifics of this example that I can not defend nor refute Eddington’s results and as such I will not further comment on whether or not he was biased. I have said all I can about it already and would like to leave it there.
• King On June 23, 2018 at 3:57 pm
you said that MM experiment was the last amongst many experiments that failed to detect aether. However, it is exactly the opposit! Starlight aberation, Young’s slit experiment and Fizeau Experiment had earlier on shown compelling evidences for aether! Latter, Sagnac’s experiment, Dayton’s experiment etc showed evidence of aether. Also if there is a problem with detecting variation in speed of light is a problem due to Lorentz’s Transforms, electrons too show wave phenomena and their speed is lesser than c so they don’t suffer the same problem as light. Or is the speed of electrons also invariant?
You claim that Allais, effect is sometimes reproducable but sometimes not. But this is same in all experiments that involves small effects. That is why Hafele took 4 clocks on board. Sometimes the effect is reproduced, sometimes not, even in SR!
• A_Concerned_Student On June 23, 2018 at 9:16 pm
As for the first few experiments you mention, they did in fact give strong MOTIVATION for looking for an aether-dragging theory. However, any such theory that followed ran into its own issues, either contradicting its own assumptions or being disproved by experiment. As such, these theories were continually adjusted until they were eventually given up when relativity explained these experiments with less complex assumptions and a higher degree of accuracy. The Young’s double slit actually has no relevance to relativity and is instead handled by QM so I will disregard this one despite how an aether theory may play into it.
As for your question of the invariance of electron speed. This is a gross misunderstanding of the concept as very clearly electron speeds are not always the same. Electrons take very different speeds all the time and we see this daily. Therefore your comment regarding this is irrelevant to the topic at hand. Also the purpose of taking 4 clocks on board was for accuracy. Unlike as you claim that the effect is sometimes reproduced, in Hafele’s experiment all 4 clocks showed the effect, albeit not to the same exact extent each time. Thus the effect was reproduced each time, but not the exact same in each clock (minor differences).
• King On June 23, 2018 at 4:25 pm
‘student’ said that ‘renormalization’ is a problem not with the theory but with how we do calculations in it. True, but without the calculations the theory cannot predict anything! If there is a problem in how we predict (through calculation), then there is a problem in the aleged ‘confirmations’.
This is the case!! Through ‘renormalization’, a mathphysicist ‘exchanges the hats and pulls out the rabbit’!! If we are testing a theory which allgedly claim if A is true, then B is true, we gotta first be sure that indead B follows logically from A. We are not allow to say that ‘if there is a Santa, then there will be an eclipse today’. Such is the problem! the ‘non regularized theory’ is theory X. The ‘regularized theory’ is theory Y!
We are told that the lagrangian, which was deduced using tenets of relativity (eg lorentz invariance) actualy was infinity , ie meaningless!! It is replaced by a ‘regularized’ one, which isn’t relativistic!
• King On June 23, 2018 at 4:51 pm
‘A concerned student’
As a quick example to illustrate the problem with ‘regularization’, consider how they ‘predict’ the Casimir effect. Now, if there is energy in ’empty space’, QM demands that it be quantized. Relativity, on the other hand, demands that there is no smallest wavelength! So we have to sum infinite frequencies. So it isn’t a problem with ‘how we calculate’. It is the theories that demand so!!
If we introduce the ‘smallest cuttoff wavelength’, that theory won’t be relativistic in that I can ask: ‘relative to whon is that length’. Like fools, they go ahead anyway and introduce such concepts as ‘planck’s length’.
The quantum mechanistic equation (as taught by Planck) should add energies as hf+2hf+3hf+…but this is infinity! So they ‘regularize’ by multiplying each term by a continuous function. But then in what sence is energy still quantized? Or rather, why should we insist that the theory is using a constant such as h at all?
• A_Concerned_Student On June 23, 2018 at 10:03 pm
I admit I know little regarding regularization aside from that it is the process of adding additional information to assist in answering a ill-posed question. However your description of why the theories are the issue is again a misunderstanding. Let me show this by emphasizing that QM does not require that energy is quantized. Energy is only quantized when it is bound to something such as an electron in a bound orbit or a photon with a specific frequency. As such any energy is allowed in empty space so your issue with how relativity and QM relate in this example is non-existent. I also would like to point out that planck length is the theorized length at which we would need a working theory of quantum gravity to make predictions, as such it has nothing to do with relativity in this sense. The smallest cut-off wavelength would be relative to whomever is applying the cut-off as it is they that are looking into its effects. Applying a cut-off of any kind is used to see what effects what you are looking at has in this range, as such it must be relative to you as you are imposing the constraint of the cut-off. Ultimately regularization is not a problem in reconciling the two theories, but you have been lead to conclude it is by a misunderstanding of what QM and relativity actually entail.
• King On June 23, 2018 at 5:11 pm
But the problem with lagrangian is even worse. I hereby clarify.
The lagrangian defines the ‘laws’ in the theory. So we arive at the lagrangian using the tenets such as ‘conservation of energy’ (gauge invariance), Lorentz invariance etc.
The snake oil peddler of modern phyc goes ahead and introduces a gauge invariant and a lorentz invariant lagrangian and calls it, for instance, QED. So we have, he says, a theory which adheres to relativity etc.
So far so good. But now lets ‘predict’. Lo! He now says the parameters appearing in the lagrangian were infinity! (an infinit lagrangian can be any theory, ‘turned to infinity’, from Newt to santa). He introduces a ‘renormalized langrangian’ which doesn’t give a shit about the former invariances! He now says this is the definition of the theory!! He predict using this new theory and claims that this confirms relativity and QM!!
• A_Concerned_Student On June 23, 2018 at 10:13 pm
This comment you have made here is a real mess but I will try and address it the best I can. Firstly, a Lagrangian does not define any laws in a theory but merely is used to determine what a system will do in a given problem and how it will evolve over time. This Lagrangian may come from arguments of gauge or Lorentz invariance or may be derived by other means (conservation of energy, momentum, etc.). The renormalized Lagrangian will be introduced if the theory that it is being applied to is non-linear as that is where renormalization generally comes into play. Renormalization, I emphasize, is a tool. If we have issue with certain problems yielding difficulties, we can sometimes use renormalization to make sense of it. Nowhere, does this renormalized Lagrangian define any theory but once again just determines how a system will evolve over time, and a renormalization of a Lagrangian will still maintain any invariances the original had but it is rather the constraint of the cut-off that is chosen that will break these invariances since you are limiting what behaviors you are interested in.
• King On June 24, 2018 at 10:08 am
A Concerned Student
You said that QM doesn’t demand that energy be always quantized. This is true, but you did not understand my argument cause it was too breif. In this particular case of Casimir effect, where we are dealing with STANDING WAVES in a cavity, it does require that we quantize energy. My argument does not claim that in QM, energy is ALWAYS quantized. So in some way, you did a straw man’s attack.
So Let me explain the issue more. When we are constracting a quantum field theory, i.e ‘doing second quantization’, we consider standing waves (i.e quantum harmonic oscillators). In a classic theory, the AMPLITUDE of the energy of the standing waves varies continuously. All we do is drop this ‘continuous applitudes’ and replace them with creation and anihillation operators. These operators creats / anihilates energy in quanta, i.e in interger multiples of hf. So in QFT, the relevant case in Casimir, energy absolutely need to be quantized!
• King On June 24, 2018 at 10:28 am
A concerned student
Let me even explain it further so you may see that you are mistaken! When forming a quantum field theory, like I said, we replace the classic, continuos amplitude with a quantum, ‘creation and anihilation operators’ which adds/ removes energy in quanta. Relativity, on the other hand, demands that we add waves of all frequencies!
If, like you say, QM simply doesn’t demand that we quantize energy, then you should easily see that the problem with infinities would never arise at all! All we will have to do is choose the amplitudes of the standing waves such that they diminish continuosly with increasing frequency, as a INNITIAL/ BOUNDARY CONDITION of the theory. In other words, classic theory allows us to simply pick an appropriate, convergent, fourier series of the solutions. That is why infinity issues don’t arise in CM. Ergo, the demand for quantization of energies is the cause for infinities issues. Solving it by denying the former is contradictory!
So a ariving at hamiltoni
• A_Concerned_Student On June 25, 2018 at 9:29 am
I see now what you are going after in your argument. Yes in the case of the Casimir effect specifically, the amplitudes of the standing waves must be quantized, and for my confusion I apologize. But also in the specific case of the Casimir effect, while the sum of these amplitudes does tend to infinity which seems problematic as it leads to an infinite vacuum energy between the plates, it is the difference in energies which manifests as the Casimir effect. Thus the infinites inside and outside the plates cancel out in the theory yielding the effect. An unsatisfying statement without the actual mathematics to back it up surely, but ultimately it is what is accepted and the best I can do in this setting.
• King On June 24, 2018 at 11:04 am
You take reg/renorm issue lightly cause you din’t understand my point! It isn’t even so much about quantization of energy. The main point in my argument would still apply even if energy needed not be quantized in the issue (but it does). Here it goes:
1.)If we are predicting (making claim B )from theory A, the statement B must be a purely logical consiquence of statement A. In other words, if claim A (the theory and boundary/innitial conditions) is a mathematical statement, we must arive at a claim B ONLY by a purely mathematical proceedure that doesn’t add in stuffs from nowhere into the equations.
2.)Regularization does add in stuffs to the equations in the mid way (which weren’t included in the postulates of the theory). For example ‘fractional dimensions’. ‘fictitious heavy mass’, in the case of casimir, we introduce an exponential fact which isn’t a solution of any equation.
2.)Therefore a prediction B, arrived at by regularization is NOT a logical consequence of theory A.
• A_Concerned_Student On June 25, 2018 at 9:40 am
As for this comment, I agree. Regularization is adding in already known information obtained by other means that were not supposed in the initial theory/problem and as such any predictions made by a theory which has been regularized may not be logically sound. I say that it may not because this depends on the specific cases. Much of the time regularization is a tool to solve a problem that is not well stated or to prevent overfitting data. As such it is applied after a theory has already made it’s predictions and as such will not effect the logical status of any causal relationship between the theory and its predictions. Sometimes it may however effect the logical status, if the information added is not correct or changes the initial theory in some way such as supposing fractional dimensions changes the initial assumption of GR that spacetime is 4-dimensional (an example I just made-up). If this does occur then yes a prediction is not a logical consequence of the theory and if you can find an example of this then by all means tear it apart, I’m sure it does happen as some researchers can be quite sloppy with making assumptions and later breaking them unknowingly.
• King On June 24, 2018 at 11:31 am
The ‘student’ says that a lagrangian doesn’t define any laws of the theory but merely say how the system evolve with time!!
This is amusing! He is confusing LAGRANGIAN with HAMILTONIAN! (But the student isn’t that bad anyway, he can learn.;))
Lagrangian is the laws of a theory expressed in different mathematical formalism. Specifically we define the Laws by optimizing the lagrangian, or by the principle of least action. Einstein-Hilbert Action IS Einstein’s Field equation! GR fails as a quantum field because this Lagrangian isn’t renormalizable. In other words mere failure of the lagrangian of GR means failure of GR, because GR is COMPLETLY defined by its lagrangian! How can you fail to know this???
• A_Concerned_Student On June 25, 2018 at 9:53 am
I have not confused the Lagrangian with the Hamiltonian. In fact they both describe the time evolution of a system and are directly related by the fact that they are Legendre transforms of one another. While the Hamiltonian is a more direct way of obtaining the time evolution, the Lagrangian is equally as capable and provides more insight into the symmetries of the system at hand. Also please note that the Lagrangian is actually used along with the principle of least action and it is the action that is optimized to obtain the time evolution of the system. Once again, the Lagrangian does not define any laws of the system, but is merely used to obtain the time evolution of the system and gain some insight into the symmetries involved. The Lagrangian is central to GR in that it is used to obtain time evolutions of systems as in any other mechanics it is used in, not to define the mechanics. This is not a misconception by myself as you say but rather one by yourself. You seem to have confused the theories themselves with a recurring central calculation tool in those theories.
• King On June 24, 2018 at 12:10 pm
So now that you understand that lagrangian IS the theory, let me now hope that the student will now get concerned with the trouble with mainstream! If you are a true scientist, you must be critical to everyone, not just to those ‘outside mainstream’. In true search for knowledge, there is no provision that we pat the back of mainstream and critisize everyone else! If anyone can go wrong, why not mainstream??
The ‘student’ say ‘the invariances of the original lagrangian still apply’ halo! The original lagrangian is now infinity! What sense does it make to say that an infinite lagrangian is invariant? An infinit number is anything!
The lagrangian defines a theory, and in QFT, the original, invariant lagrangian is declared ‘unphysical’, bare one. So of course if the lorentz invariant lagrangian is ‘unphysical’ the Lorentz invariant theory flashes down the toilet, hence relativity goes the way of ‘aether’,! What is physical is the EFFECTIVE lagrangian which can be more reasonably seen as to describe AETHER!!
• A_Concerned_Student On June 25, 2018 at 10:14 am
I agree wholeheartedly that science must be critical of everyone. Please do not mistake my defense of the more mainstream theories as bias, I am simply trying to give reasons for why they do as they do and show that these are valid reasons. Nowhere do I say that mainstream is correct and everyone else is wrong. I originally came to this comments section because I found flaws with Dr.G’s theory and wanted to address them. Had this been a GR topic and I found issue with it I would have done the same. Criticism of everyone is a cornerstone of progress in science and as such should be applied to the mainstream and non-mainstream alike.
In regards to your issue with an infinite Lagrangian. The key takeaway here is that, while the Lagrangian may have units of energy, it has NO PHYSICAL INTERPRETATION. There is no object/quantity in the universe which is a Lagrangian, like there is one that is energy or momentum. It cannot be observed as it is not a physical object/quantity. As such there is no issue with it being infinite as it is only observable which need by finite. The Lagrangian in QFT depends on the erngy scale of the theory and can however be used to determine the masses of particles, couplings, etc. As such it is once again a tool. The effective Lagrangian also has no physical significance of any kind. It is just a specific type of renormalized Lagrangian which also depends on the cutoff as well as the energy scale like the bare Lagrangian. Once again, neither the bare nor the effective Lagrangian hold any physical significance at all. They are tools to determine parameters of the system in QFT.
• King On June 24, 2018 at 12:35 pm
The ‘student’ says that the Planck’s Length will be relative to whomever is applying the cuttoff.
This is funny! The Planck’s length is given by lp=(hG/c^3)^(1/2). The right hand side are UNIVERSAL CONSTANTS, not relativistic values! Neither h, G nor c depends on frame of rifference because they define various ‘laws of physics’ which, according to relativity, must be the same in all frames. Ergo expressing a ‘length’ this way means that this length doesn’t depend on ‘whomever’. It is a CONSTANT!
The student don’t quickly get this cause he dismises the relevance of electron waves in aether issue! Its relevant because if there are other waves travelling less than c, then we might as well use measurements of this speeds to determin an appropriate aether rather than use light waves.
Relativist knows this and he opts to solve this by demanding that electrons move in all manner of velocities (hence all manner of debroglie wavelengths)
• A_Concerned_Student On June 25, 2018 at 10:20 am
Correct again King, Planck’s length is defined in terms of universal constants. Since length is relative however, not everyone will measure the same length if they are in different reference frames. I may measure 2 meters and you may only measure 1. Thus, while Planck’s length is defined in terms of universal constants, I may measure Planck’s length in my frame and you could measure the same distance as 2 Planck’s lengths. Thus frame does matter. Not because the constants change in different reference frames (the number will always be the same when you calculate it), but instead because length is relative and two people may measure the same object as having different lengths.
• King On June 24, 2018 at 1:11 pm
Now let me show the ‘student’ how ‘relativistic length’ variance with frames is realized in a quantum field theory. It is simply that a certain wavelength varies depending on who is measuring. It is the case that whenever a wavelength ‘contract’ relativistically, a wavelength that was longer by the same factor will be ‘seen’ to replace the contracted one and so the ‘vacuum’ remain as it were. If this wasn’t the case, the contracted wavelength would indicate who is moving. I.e one would measure ‘ABSOLUTE MOTION’ through the vacuum by measuring debronglie wavelength of the wave he is moving throug, reintroducing aether!
But you should be able to see that if the set of harmonic waves in vacuum is FINIT (such as when we introduc a cuttoff), then ’empty space’ woun’t look symmetrical to all observers. It is this demand that waves should be infinite that yields infinities in QFT. So renormalization is NOT ‘just a math proceedure’. It involves introduction of unjustifiable tenets!!
• A_Concerned_Student On June 25, 2018 at 10:28 am
Having briefly considered this comment I tentatively agree. Your argument seems to be sound for why the number of harmonics must be infinite as otherwise it seems as though you could use this to determine who is moving. However this isn’t an issue in QFT. Summing an infinite number of wave amplitudes does not necessarily yield an infinite value as long as the amplitudes decay sufficiently (as you pointed out in an earlier comment). Furthermore, you’re argument for why there must be infinite harmonics was purely logical, it never used renormalization and still achieved infinite harmonics (which still isn’t an issue). So I find it difficult to see how you claim renormalization introduces unjustifiable tenets (I believe infinite harmonics was this tenet) when you derived this tenet solely with logic and never used renormalization.
• King On June 24, 2018 at 1:27 pm
The ‘student’ says that the reason the 1919 experiment created a sensation than the 1954 one was because the former was an attempt to check what theory was correct, which was known prior that experiment. So he blames JOURNALISTS, not SCIENTISTS. But SCIENTIFIC JOURNALISM is done by scientists.
But it doesn’t change my point, which was that scientists were especting to falsify Newt and in Allais case, they weren’t especting to falsify anything. The mere fact that it was popular prio to 1919 mean that scientists had already entertained the idea of replacing Newt. Of course they were not ‘just eager to falsify GR’ like ‘student’ claimed. If they were, they would not entertained invention of alternatives from within mainstream. Gr was just like SUSY now or String theory, i.e a mainstream theory. Non mainstream theories are aproached in a dismisive attitude. They can, for instance us only ONE experiment like in MM!
• A_Concerned_Student On June 25, 2018 at 10:43 am
I’m sorry if it came across as though I blame journalism for Allais’ dismissal. I do believe they have a strong role in what science gets to much of society but it is not ultimately their fault. Scientific journalism is far from done by scientists however. Buzzfeed for example is full of articles sensationalizing recent science, as are many other newspapers and news sites around the world and I guarantee you that these are not all (or even mostly) written by scientists.
Back to the main topic though, and here I say that they weren’t expecting to falsify Newton. They knew that only one of either Newton or Einstein (or neither) could be correct and it just so happened that Newton turned out to be shown wrong. Even afterward many scientists continued to dismiss Einstein and looked for alternative classical theories to relativity (and some continue to this day). Many modified Newtonian theories exist nowadays which build off Newton and try to maintain the classical view of nature but these are not the mainstream view obviously. There is nothing wrong with being a mainstream or non-mainstream theory in general, provided that you can experimentally determine things. If experiment shows that you are wrong, then you’re wrong. If it shows that you’re right, you can keep being a theory and go onto the next batch of tests. If you’re non-mainstream there may or may not be a dismissive attitude towards your theory, but this doesn’t matter if you’re right! If so then no experiment will show that you’re wrong and you will continue to gain popularity as you continually be right! But if you’re wrong then the theorists working on the theory need to either adjust it or possibly give it up. While the popularity of your theory may determine how many perceive the theory, ultimately it is how effective the theory is that determines who is right. Not whether it’s liked initially.
• King On June 24, 2018 at 1:41 pm
The ‘student’ claim that all the clocks reproduced the effect. But some pple say some didn’t! And this highlight some problems with ‘student’. He was neither there with Hafele nor was he there with Allais. In both cases, he relies on CLAIMS. However, he choses to say that the mainstream claim is always correct. But mainstream’s integrity is part of what we are questioning here. Ergo, a blind apeal to this authority is moot.
Allais and others claimed that the effect is reproducable. The mainstream say Hafele effect is reproducable. If we are true scientists, we should suspend the issue until we device our on experiment and arrive at our own conclusion irespective of concensus! that is what true science is. There is no provision that a body as ‘mainstream’ must accept it. Infact there is no ‘mainstream’ at all in the definition of ‘science’ We use our own eyes!
• A_Concerned_Student On June 25, 2018 at 10:49 am
King you are correct once again! I rely on claims in my argument because of your reliance on claims. You claimed that Hafele’s clocks weren’t all the same and you weren’t there. Now you appeal to the claims of other people who weren’t there that they weren’t the same. So, since you may appeal to claims, so do I. I appealed to the claim made by the experimenter (who WAS there) and those with him who verified that he was not lying that they were the same. Now ultimately this is still an appeal to claims, and isn’t conclusive, but since we aren’t capable of meeting up together and performing the experiment (as if I or you did it alone the other would say that you could just be claiming it did one or the other) this is the best either of us can do. And if that’s not good enough then the debate is moot itself. Sure we must not blindly accept what anyone says, but in this debate we have to appeal to something, and ultimately the appeal to someone who was there seems like stronger evidence than the claims of others who weren’t. But that is up to personal opinion…
• King On June 24, 2018 at 2:23 pm
The ‘student’ claims that all earlier ‘aether drag’ theories faced problems. No! MM was the only experiment that paused serious problems with aether cause all the other experiments were reasonably consistent with a partialy draged either. So my point stand: scientists USED ONLY ONE experiment to dismiss the whole aether theory, not many experiments like ‘student’ claims.
The ‘student’ dismisses the relevance of Young’s Slit Experiment in providing evidence for aether!!! Of course he assumes SR in this, which makes the silly claim that waves can propagate without a medium. However, we are questioning SR here, we are not assuming it. That argument is like saying that Lorentz could explain Lorentz transforms without SR, therefore SR is irrelevant to aether issue.
• A_Concerned_Student On June 25, 2018 at 11:10 am
MM was an experiment which posed issues for partial aether dragging yes but many of the theories involving aether dragging (partial or full) faced issues with self-consistency. They would make assumptions and then break them later to solve some other problem. Now, not all of them may have been like this, but it did happen with some for sure. Complete dragging had many experiments which showed issue with the theory (stellar aberration, Fizeau experiment, etc.), but for partial aether dragging there was also the Trouton-Noble experiment as well as partial aether dragging’s issue of aether and matter having different relative velocities for different colors of light. So it was not just one experiment but rather several issues.
I dismissed the double slit experiment as evidence for the aether yes, but not because of SR. I disregarded it specifically because it doesn’t have anything to do with SR, which we were debating at the time, but rather is handled by QM. I felt it was off-topic so i disregarded it. Nowhere did I claim that SR was the reason for doing so and so I apologize if it came across that way. Also SR does not claim that waves can propagate without a medium, in fact classical EM claims this (and gives compelling evidence as to how) which came about before relativity was proposed. As for if the double slit is EVIDENCE for the aether, no it is not. Theories exist where the double slit can be explained by an “aether” yes but that is not evidence for one. I can explain gravity in terms of chicken eggs but that does not mean that is evidence for chicken eggs causing gravity! Please do pardon my exaggeration as it is used to prove a point, not to belittle your side of the argument. While an aether COULD explain the double slit, that does not make the double slit evidence for an aether. That seems like some circular logic to me personally. I actually am quite a fan of Bohm’s pilot wave theory which could be compatible with a medium similar to an aether but that does not provide evidence for one sadly.
Also I would like to add that if my comments come across at all offensive or dismissive that this is not the intention. It is very hard to convey tone over writing and I do think that you, King, are a quite well informed individual on the topics we are discussing. As such I thank you for taking the time to write these posts and hope you continue.
• A_Concerned_Student On June 25, 2018 at 11:24 am
Also, I feel it necessary to clarify my intentions of my very first comment where I briefly defended the mainstream theories. I feel this deserved its own reply to the original as it appears to have been interpreted as though I feel that the mainstream theories are always right and should be blindly followed, which is far from the case. My intention behind defending relativity and QM were simply to provide some sense of background as to why they are the currently accepted theories. If there were serious flaws with them that couldn’t be fixed, or even if a better alternative were available, they wouldn’t have become the mainstream theories. Had a better theory come along at some point which better agreed with experiment and made better predictions, they would have been replaced since the ultimate goal of science is to describe nature as accurately as we can. The scientific community wouldn’t deny their ultimate goal simply because they are accustomed to the current theories.
If other theories were better, they'd be the mainstream ones.
That was my intention. That was the intended takeaway from my first defense. They may not be perfect, and they may be replaced one day by a better theory but they are the best we have at the moment. That is all I wanted to add.
• King On June 25, 2018 at 6:29 pm
A concerned student
Thanks for those good replies. You seem to be now understanding much of what I say. We may not agree but that isn’t an issue, as long as we can UNDERSTAND each other.:) However, there are still few issue you haven’t gotten!
Lagrangian does define the laws of physics!! This should be the biggest lesson you will learn from me.:) if you will ever want to contribute to mathematical phyc, all you will ever need is brainstorm an appropriate lagrangian. Please mark my words! If you want to unify both GR and QFT, just write the correct lagrangian.
You said that minimizing the action gives the dynamics of the system!! This is not true! If you minimize the lagrangian for an oscillating spring what you get is the equation:
The left hand side is Newt’s law while the right is Hook’s law. So minimizing the lagrangian gives not the DYNAMICS but the LAWS governing the dynamics!!
• King On June 25, 2018 at 6:50 pm
Likewise if you minimize an appropriate action (integral of lagrangian with time), you get Schrödinger equation (i.e the ‘law’ governing quantum mechanics). If you minimize EM action, you get Maxwell’s equations. If you minimize Einstein-Hilbert Action, you get Einstein’s Field equations. So the lagrangian does not give how a system evolves it spits the so called laws that governs the evolution. As you may know, to get the exact dynamics, we need additional information apart from the laws termed innitial and/or boundary conditions.
Since upon minimizing the action, we obtain the laws, we just cannot throw in any lagrangian (including infinite ones) claiming that ‘lagrangian isn’t observable after all’. If the laws of phyc must be Lorentz invariant, so must their lagrangians for the lagrangians are just a different mathematical way of stating those laws.
• A_Concerned_Student On June 25, 2018 at 10:36 pm
I now see the issue you are having concerning Lagrangians. You argue that the Lagrangian defines the laws of the system and in some sense I can see why you’d believe that. The Lagrangian is determined not by divine intervention or by being creative and picking one that works nicely as you seem to believe, but rather it has a rigorous mathematical definition. It is the difference between potential and kinetic energies of a system which defines a Lagrangian, from there minimizing the action results in the EQUATION OF MOTION of the system. An example of which is the spring equation you gave previously, and this equation of motion determines what state the system will be in at a given time. As such it completely describes the time evolution of a system and requires no inputs of the laws of a system. In a sense minimizing the action does result in “spitting out” some “laws” of the system such as conservation of momentum, but I do not believe these to be true laws (an example of which being Newton’s laws) so much as they are quantities which don’t change in a particular example. They are not always conserved in every system and are as such not a true law, I believe. These conservation laws do define what the system will do as they are a part of the time evolution of a system. In the example of the mass on a spring, while momentum may be conserved when the spring in oscillating back and forth, if you were to make the spring spin around it’s anchor point then momentum would not be conserved. As such this conservation “law” is not always true even for the same system. Thus the equation you stated earlier is only true in a specific case and as such I do not call it a law but rather just refer to it as an equation of motion or a time evolution.
• King On June 25, 2018 at 7:18 pm
It is more serious, my friend, it is more serious!! Now, we know one thing QM, via Heisenberg’s UP demands that there is a minimum amount of energy termed zero point energy given by E=hf/2. So if there is energy in ‘vacuum’, it must adhere to this.
We also know from math that if we have in infinit series, s=a0+a1+a2+,…,+an, for the series to converge, an must tend to zero as n tends to infinity. In other words if we are summing infinite series of energies then there mustn’t be a limit as to how small the energy can get! So it is quite easy to seen that the Heisenberg’s UP has a potential of yielding unsurmountable problems if it gives minimum energy that is greater than zero and yet we must sum infinite of them!
Now f ranges over all the harmonics so that total zero point energy must be E=1/2(hf+2hf+3hf+,…), which is infinity!! But solving this is, not just a matter of subtracting infinity from infinity. That is a meaningless math. And what, in the first place, is ‘negative energy’?
• King On June 25, 2018 at 7:42 pm
So since infinity-infinity=bullshit, they need to subtract two huge amount of energies, but not infinit amounts. The only way they have is to modify the series by multiplying each term by a factor that that makes the energies diminish smoothly as frequency increases and call this ‘regularization’. But this self defeats the Heisenberg’s UP!! Such a way of adding energies smoothly so that there isn’t any limmit as to how small they can get is a CLASSIC theory, not a quantum theory!! so indead ‘regularization’ is an exchange of huts and then pulling out the rabbit!
But I am sure that drgsrinivas is smilling if he get this math. There is a very simply way out if we think that there is a medium in vacuum and the waves are ACTUAL WAVES rather than these mathematical non entities of QM. Aether provides a natural cuttoff at the size of the individual aetheric molecules, which drgsrinivas calls them ‘photons’.
• King On June 25, 2018 at 8:23 pm
The ‘student’ says that ‘infinities isn’t an issue as long as the harmonics sufficiently diminishes’. Like I have shown, it doesn’t diminish in QM because the lowest energy is never allowed to get to zero but to hf or something like that. The main idea of QFT was to remove the amplitude of a classic harmonic waves and to replace it with a creation/anihillation operator which adds/remove energy in quanta. In math, though, for an infinit series to converge (not to yield infinities), there should be no limit as to how small an amplitude can get. So QFT must have problems with infinities when we demand that we add harmonics of all frequencies as required by SR!
Yes, l can deduce this infinities logically. ‘Renormalization’ does not introduce infinities. It attempts to get rid of them. But this proceedure introduces unjustifiable tenets during regularization in that they modify the equation that had been earlier on shown to be Lorentz invariant etc.
• A_Concerned_Student On June 25, 2018 at 10:36 pm
In response to your remaining issue with vacuum energy, I see where you are misinterpreting my argument. The key issue is that it is not the energies of the harmonics that we are summing, but the amplitudes of the harmonics. The amplitude of the standing waves represents how much of a contribution that energy level has in the total energy. While the energies do start at hf/2 and continue upwards towards infinity, the amplitudes start at some fixed number (say around 50% of the total) and decrease rather quickly. While the vacuum energy level may have a strong contribution (the 50% again), the second energy level of 3hf/2 may only have a contribution of 30% and the remaining energy levels will continue to have less prevalence in the total. Thus adding the amplitudes does not tend to infinity but rather a fixed number. So while there may be infinite energy levels, the amount of energy present may not necessarily be infinite. My wording previously may have been confusing and have lead to this misconception.
• King On June 25, 2018 at 8:37 pm
My original question was ‘relative to whom is planck’s length’. I did not ask about lengths in general. Since we have a length that is defined using UNIVERSAL CONSTANTS, such a length can’t be frame dependent, violating SR! I know the try to say ‘laws of phyc break down at planck’s length’, or ‘there is no length smaller than lp’. But these are just silly ad-ocs that could be easily solved by thinking in terms of aether in which cuttoff frequency isn’t an issue!
• A_Concerned_Student On June 25, 2018 at 10:43 pm
Once again, the number that you calculate that is the Planck length is not frame dependent. It’s a fixed number. So we do not have a length which is defined using universal constants, we have a number which is defined with them and can be APPLIED to a length. However, since length IS frame dependent, while I may measure a piece of wood as being a Planck length long, you may measure the wood as being 62 Planck lengths long. Thus, while the number is not frame dependent, what me measure objects as being is frame dependent, that’s where the relativity of the Planck length comes in. And once again, it’s not that there isn’t a smaller length than the Planck length, there is, it’s just that we need a working theory of quantum gravity to make any predictions about what happens at these lengths.
• King On June 25, 2018 at 8:55 pm
The student says that just because aether can be used to explain the experiment doesn’t mean the experiment constitute an evidence for aether. But if we use this logic, we can similarly dismiss all the so called evidences for all the mainstream! When we say hafele clock ticking is evidence for SR, we simply mean that the phenomenon can be explained by SR, nothing more!
The reason why we must regard young’s experiment as to have been evidence for aether is that the aether advocates at then (including young) had predicted so. If we reason having in mind modern view, this will be like saying that if in future, someone finds another explanation for hafele effect, then the effect is never an evidence for SR. What student is saying is that just cause someone can concoct another so called explanation for double slit experiment then the effect isn’t evidence for aether! However, same logic doesnt apply when someone offers an alternative explanation to mainstream theoris
• A_Concerned_Student On June 25, 2018 at 10:47 pm
I am sorry for the misconception here. I was mistaken in saying that this does not constitute evidence for the aether. I got wound up in my head at the time of posting and made a poor statement. If the aether theory can explain the double slit experiment, then fine, that does lend credence to the aether theory. I was mistaken and apologize.
• King On June 26, 2018 at 12:55 pm
A concerned student,
When quantizing Dirac Field to form a quantum field theory we do so just like in quantizing Electromagnetic field. In the latter, the square modulus of the amplitude of the waves is directly propotional to energy. It is exactly the same in the former! In Casimir effect, we are mainly summing quantized, EM waves. In solving QED equations, we are also summing the electron waves. So the situation is perfectly similar in both cases: the point is that if we are summing infinite waves of frequencies nf, then we are automatically also summing up infinite amounts of energies of the form nhf, for the hamiltonians of the systems is given as the square moduluses of the amplitudes of the waves.
Infact, in QED, what brings about the infinity issue is the hamiltonian, which is expressed as:
where y is electron wave equation. It is the summing of y that brings infinities. So it isn’t correct to say we are summing amplitudes but not energies!
• A_Concerned_Student On June 26, 2018 at 8:15 pm
While I see your point, technically it is the amplitudes being summed and not energies. The energies are the end result of the summing but they are not being summed, so I stand by this statement. And once again, summing these does yield infinities, this has been said many times, but it is considered a feature of the math not the physics. This is one of the main arguments for renormalization as while the energy calculated by summing these are infinite as the math states, only differences in energy can be physically observed.
• King On June 26, 2018 at 1:28 pm
A concerned student,
You say that the equation: md^2x/dt^2=-kx is an equation of motion of the system??? It can’t be because the equation doesn’t specify anything. According to theory, it is as true to say md^x/dt^2=-kx when the spring is at rest as it is to say so when it is swinging in its full force!
The equation of motion is given by x=Acos(wt)+Bsin(wt). i.e the equation of motion is given by the SOLUTION of the above differential equation. The differential eq itself is the LAW of the spring! Without knowing A and B, we have no clue of how the system moves, so we can’t an equation like that ‘an equation of motion’. Differential eqs never describe system’s dynamics. They describe their constrains (laws). It is exactly same when we are deriving Einstein’s field eqs, Schrödinger eqs etc. It is these differential eqs that Einstein demands them to be Lorentz invariant in the first postulate of SR.
• A_Concerned_Student On June 26, 2018 at 8:23 pm
The differential equation is often referred to as an equation of motion as in many cases they cannot be exactly solved analytically and must be done so numerically. The differential equation contains just as much (if not more in cases) information on how the system evolves over time as the solution to it. Just as you said it is true to say the differential equation holds at rest as at full stretch which is ultimately the purpose of an equation of motion. It must be true and describe a system at any point in time, which the differential equation does.
Again I am forced to reiterate that this does not constitute the laws of the a system. The equation of motion is simply how we model the system and attempt to describe what it will do at each point in time. The laws of the system are what we use to find an equation of motion such as conservation of energy or mass. Also I must point out that nowhere in the 2 postulates of SR does Lorentz invariance come about. It is a feature of SR, but it is derived as a consequence of these two postulates.
• King On June 26, 2018 at 2:18 pm
You ‘student’ is still not answering my question well. Ok, let me put it this way:
You say that you can measure a length and find that it is equal to Planck’s length (lp). Then when another person measures it, he finds that it equals 62lp. Then you say that to describe things at lenghts less than lp, we require new phyc. But then what is less than lp for you is far more than lp for that perso who sees your lp as 62 lp. So the exact same events that you are describing it with new phyc at lengths less than lp will be described by ordinary phyc for the person seeing your lp as 62lp! This contradicts the first postulate of SR which asserts that all laws of phyc are same as seen in all inertial frames of riferances!
Thing is that ‘cuttoffs’ (planck’s length included), like I showed you earlier, aren’t compartible with SR cause thus ‘vacuum’ fails to appear symmetrical for all observers! However, it is fully compattible with an aether theory!!
• A_Concerned_Student On June 26, 2018 at 8:32 pm
A very good point! However, not a contradiction to what I claimed. The Planck length (lp) is the supposed length at which our theories of physics break down. Therefore, if I see something as less than the lp and you as more and we describe it differently as you point out, this contradiction is due to our theories breaking down at the lp as predicted! The laws of physics no longer appear the same to everyone because they have broken down at the lp!
Now, I have been refuting your attacks on SR, QFT and so on for a while now. Now you claim an aether theory is compatible with “vacuum asymmetries” (not quite sure what you’re getting at there exactly) so I ask you to explain why you think it is a better theory in general. Why you feel that the aether is the correct next step forward in science instead of relativity and quantum. I already poked a hole in Dr.G’s aether theory of gravity and would like to see if your own holds up to modern theories or if you are simply claiming it is compatible without evidence (hopefully not).
• King On June 27, 2018 at 1:56 pm
The argument that md^2x/dt^2=-kx is a law is pretty straight foward
F=md^2x/dt^2 (Newt’s) is a law that says how an ‘mass’ accelerates.
F=-kx is a law (Hook’s) that says how a spring is streatched.
So md^2x/dt^2=-kx is just a law that says how a mass attatched to a spring moves.
It is not an equation of motion because it just says how the ‘mass’ ACCELERATES. From simple calculus, we know that a differential eq does not contain as much info as its solution. This is because the solutions have ARBITRARY CONSTANTS. In phyc, we obtain the info pertaining to the arbitrary constants by OBSERVING the innitial &/ boundary conditions. Therefore the solutions have more information than the differential equation.
You can understand it this way: the differential eq is like saying ‘I live in America’. The solution is like saying ‘I live in Los Angeles’. The latter inform someone more pertaining to where you live!
• A_Concerned_Student On June 27, 2018 at 9:23 pm
If you would like to refer to this equation as a law by all means. All I am saying is that I prefer not to as that seems confusing as the actual laws would be conservation of momentum and energy and so on. However, this differential equation (DE) directly implies the solution with its arbitrary constants. Unlike your example of living in LA and America, this is more akin to saying “I live in LA” which implies “I live in America”. The DE always leads to the same solution, (ie living in LA implies you live in America) but the solution does not imply it came from that DE (living in America does not mean you live in LA). The only reason a solution would have more information than the DE would be after imposing the initial/boundary conditions. Therefore it is these conditions that add the information, it is not a feature of the solution to the DE itself. As such it is perfectly acceptable to refer to the DE as an equation of motion as it always leads to the same arbitrary solution which can then have the initial/boundary conditions applied to it. The benefit to leaving it as the DE is that additional terms can be more easily added to account for new phenomena such as additional forces or friction which will lead to a new family of solutions.
• King On June 27, 2018 at 2:54 pm
A concerned student
So you now admitt that summing the infinite number of harmonics yield infinities, good! But you say it is considered a feature of math and not phyc! What sense does this make?? If I sum 1+2+3…, then I obtain infinity as a feature of math. But if I sum hf+2hf+3hf…, I get infinity as a feature of the theory that says what hf means! ‘energy’, ‘planck’s constants’, ‘frequencies’, ‘amplitudes’ etc are features of PHYC not MATH. Math deal solely with pure numbers!
• A_Concerned_Student On June 27, 2018 at 9:27 pm
I find this comment contradictory. You say that summing numbers leads to infinities is a feature of the math, but summing hf+2hf+… is a feature of the physics? In the latter case we are just repeating the same sum as before but instead we are using 1(energy unit)+2(energy units)+… The sum is still 1+2+3+.. but we have specified that these are units of energy now, thus it is still the math that results in the infinities, not the units of the quantities we are summing. Thus by your own admission, the math leads to the infinities.
• King On June 27, 2018 at 3:33 pm
A concerned student
What you are aluding to is a simple thermodynamic fact: if you are inside an ocean that is at thermodynamic equilibrium, then you can’t observe the total energy of the ocean FROM WITHIN THE OCEAN. You can only observe the difference between the total energy at two points in the ocean for thus the energy gradient creats entropic force. Same thing might be happening in, say Casimir Effect.
The point that I am stating is that ‘energy difference’ doesn’t make sense if we are subtracting infinity-infinity. This is a basic math fact that the theorists agree on. To make a sensible subtraction, they are forced to subtract a finit number from another finit number. But doing this involves throwing away the theories that had predicted infinity-infinity, i.e QM and SR! This is what I am trying to show you!
It even isn’t true to say that total energy isn’t observable. GR states that energy/mass warp spacetime. If energy in Vacuum=infinity, then spacetime should be warped into nonexistent dot!
• A_Concerned_Student On June 27, 2018 at 9:33 pm
Yes that is exactly what I was referring to. However, since in this case we are referring to the universe instead of an ocean, and since we can never leave the universe (as it encompasses everything in existence), then we can only observe the differences in energy anywhere and not the total. And yes, infinity minus infinity does not make sense mathematically. It is only by renormalization that this issue is avoided in QFT. I do not say that infinity minus infinity doesn’t occur in QFT, only that after renormalization that it does not. And total energy is not observable in all cases with the EXCEPTION of gravity (GR). Since gravity is not encompassed by any quantum theory and is purely classical at this point we can not apply this same condition to it. As such, gravity is the exception to this statement that total energy isn’t observable. This may change in the future or it may not.
• King On June 27, 2018 at 4:18 pm
The student seems to have gotten my point.:) But he isn’t repeating well! lp is tiny so it isn’t a major problem to SR, isn’t it? We need only a slight modification of QFT, isn’t it?
But their is a catch!! The relativist doesn’t get a cegar still! In SR, length is RELATIVE and what appears as lp for one observer should appear like a galaxy to another! So if laws of phyc breaks down at lp for observer P, then there are some observers such as observer Q who will be able to tell that he is moving at v without riferance to any frame by simply observing the point, lq, wherein the laws of phyc begines to break, in his own frame!! I.e v=c{1-(lp/lq)^2}^(1/2). This is cause lp is given by UNIVERSAL CONSTANTS. This contradicts the first postulate of SR which states that one should not tell that he is moving from measurements done solely within his frame!! Here, we can regard frame P as a PREFERED FRAME.
• A_Concerned_Student On June 27, 2018 at 9:37 pm
I believe I already responded to this point in my last batch of comments. SR gets contradicted in this case regarding the lp because that’s what the lp is! It is the point where our current theories break down. So even though someone can discern who is moving and who is at rest when near the lp this is not an issue that wasn’t already predicted. We know our theories don’t work at these small scales, so when they don’t why is that an issue? We believe we need a working theory of quantum gravity to reconcile what happens at these scales, as I pointed out previously.
• A_Concerned_Student On June 27, 2018 at 9:47 pm
Also, you never responded to my question as to why you feel a theory incorporating an aether should be the main theory of physics nowadays. While you may have forgotten to I find this unlikely as you haven’t failed to respond to anything I’ve posted yet. So I hope that you are simply preparing a well thought out response as to why you feel this is the better of the many theories out there.
• Galacar On June 27, 2018 at 10:28 pm
“The Planck length (lp)”
Hmmm. maybe the minimal size of the ‘pixel’ in our computer generated ‘reality’
we are living in?
Interesting times, to say the least!
• King On June 28, 2018 at 11:53 am
This equation implies a set of equations such as:
These set of eqs can be called the symmetries, or degrees of freedoms of the system. Eqs 1,2,3…describes not how a specific spring is moving. Rather, it describes infinite ways in which the spring can move. So the DE does not describe how the system moves. It merely describes how the moving system is CONSTRAINED, i.e it describes the FREQUENCY (w) of the system but not the AMPLITUDE of the system. You can see that wt is commot to all the eqs 1,2,3,…So wt is akin to ‘America’ while A,B,C,…is akin to ‘Los angeles’, ‘Chicago’ etc
In mathphyc, any eq that describes the CONSTRAINTS of the system rather than the actual state of the system is always called a ‘law’. This is important cause exactly such eqs is what SR was aluding to in first the postulate.
• A_Concerned_Student On June 29, 2018 at 9:31 am
If feel it is a stretch to call x=Acos(wt) a symmetry as it describes an oscillatory motion in time, but I see the point you are trying to get at and can accept that the DE can be thought of as a constraint. Thus the DE leads to a family of solution with the same properties and constraints but with different specific constants such as amplitude. That seems reasonable enough to me.
• King On June 28, 2018 at 1:01 pm
Difference between 1+2+3 and hf+2hf+3hf
Math does not specify what it is that we are summing. Math only say ‘infinite NUMBER’. When we say ‘infinit ENERGY’, we are nolonger talking about math. We are now talking about phyc. Therefore hf+2hf+3hf,… is not a math statement. It is a (meaningless) phyc claim. Performing an infinite sum of a physical quantity and then claiming ‘it is a feature of math but not phyc’ makes no sense! ‘energy’ is not a feature of math. It is a feature of phyc. If we could shove anything to math, then all the phyc eqs will be feature of math but not phyc, for in all cases, we perform addition, multiplications etc. When we as -ihdy/dt=(h^2/2m)d^2y/dx^2+vy,(Schrödinger eq) we are also adding units of energy. So why are we not saying Schrödinger eq is also a feature of math but not phyc? Why does it become a ‘feature of math’ when our eqs make no sense?
• A_Concerned_Student On June 29, 2018 at 9:24 am
It is not that the equations make no sense (as you say). What I claim is that if we were to call hf the number 1 as in 1 unit of energy, then summing these energies would yield the same infinity that we get by summing hf, 2hf, etc. Thus I say that it is a feature of math because it is the sum 1+2+3+… that yields the infinities, not the fact that it is energies that we are summing. Had it been pure math with no units of energy, we would have still arrived at infinity.
• King On June 28, 2018 at 1:49 pm
So you agree that infinity-infinity make no math sense. But this is how the infinity is gotten rid of:
1.)We perform an infinit sum of harmonics and note that it sums to infinity.
2.)we introduce another infinity (renormalization) so we may attempt to get a finit number by subtracting infinity-infinity
3.)But since infinity-infinity make no sense, we regularize the sum, ie we introduce another number equation, which is finit-finit, which now make sense.
It is particularly the third process that I have troubles with. I insist that QFT predict infinity-infinity. This other finit-finit eq is a different ‘hat’ from which our canning ‘pseudo-magician’ is ‘pulling out the rabbit’ having realized that the infinity-infinity ‘hat’ has no ‘rabbit’! The second proceedure, on the other hand, is a very troublesome ad oc!
• A_Concerned_Student On June 29, 2018 at 9:41 am
Your issue with regularization and renormalization is not new. Many top researchers in QFT had similar issues with it, most notably DIrac! As they can sometimes violate the underlying physics in the theory, or adds in additional information that is not known such as un-physical particles this issue is not new. This is a current issue even today as a search for a realistic regularization is always underway. But, it is here that I restate what I had said at the very beginning. That the purpose of any theory is to make predictions of what is going to happen. And it is for this reason that renormalization and regularization are used. They allow researchers and theorists to make predictions about interactions in QFT and see if this fits with experiment, As such it is a useful tool in QFT for that reason alone. While it may be counter-intuitive to the purpose of the theory at times, it is a strong tool in making predictions of the theory. Your skepticism is well placed in this area however.
• King On June 28, 2018 at 3:24 pm
Planck’s Length
Why is it an issue when the laws of phyc break down at these small lengths? Simple! In SR, ‘small’ is ambiguos. But let me see if this illustration help. Consider how they explain the ‘length contraction’ in the case of muons crossing the earth’s atmosphere. The muon manages to reach the earth cause in its frame, the atmosphere is very thin! So by simply moving sufficiently fast,(call it velocity v0), an ‘observer’ can find the entire earth’s atmosphere to be less that lp, so he needs new phyc to describe the events we see here! This contradicts the first postuale of special relativity as applied to phyc working at the length equal to the thickness of our atmosphere! The laws of phyc, in the length scale of our atmosphere, is nolonger same as seen in all inertial frames!!
• A_Concerned_Student On June 29, 2018 at 9:46 am
Once again King, your contradiction falls under the same argument I made above. If the muon sees the atmosphere as less than the Planck length, then we expect a contradiction since our theories do not work at these scales as that is what the Planck length defines. For any further comments where you make another example of a contradiction of SR using the Planck length, I feel it necessary to just leave a “RA” (“Refer Above”) as we are just repeating the same two comments over and over.
• King On June 28, 2018 at 3:54 pm
Another thing is that you seem to unwittingly think that the the first postulate of SR is a ‘law of physics’ which itself should break down at lp!! This postulate isn’t a law of phyc. It musn’t break down even if laws of phyc break down cause it is statement about the laws of phyc but is itself not a law of phyc!
As you said, the aether hypothesis was rejected by application of Occum’s Razor (OR). Even according to Occum himself, OR was never meant to be used to determine which of the two theories is correct (relativists us OR almost for this purpose!). The theory rejected via OR should be constantly reexermined upon new discoveries.
To put it breifly, aether theories don’t have the first postulate but can predict everything SR can! That laws of phyc appear same is only INCIDENTAL and as such, this ‘rule’ can break without ‘breaking the aether theory’.
• A_Concerned_Student On June 29, 2018 at 10:02 am
I do not believe that the first postulate of SR should break down at the Planck length. I believe that it is the math that SR uses that breaks down at these scales. We cannot calculate what happens at these scales using the current SR theory not because the postulate breaks down but because all of SR’s predictive power comes from the math it is built around and is currently not equipped to handle scaled where quantum effects of gravity are believed to become important.
Again, I do not say that the aether theory is wrong. I had said that the aether was not required in our theories of the time and by application of OR we should not have it. While it may exist, we have no reason to use it currently, so we won’t. That was the reasoning I presented.
The true issue I have with this last comment though is your claim that an aether theory can predict everything that SR can, and I aim to challenge this. I will start with a seemingly simple prediction that relativity makes. How does an aether theory, for instance, deal with the case of the muons produced in our atmosphere reaching the surface as you brought up earlier? More generally, how would an aether theory handle the (well documented and tested) observation that particles half-lives are longer (by the amounts predicted by relativity) when moving at higher speeds than when stationary?
• drgsrinivas On July 7, 2018 at 6:30 pm
I can see at least one person on this planet who can speak and communicate the truth to the physicists in their own language. Hats off King! 🙏
As physicists have abandoned commonsense, however much I speak, it probably makes little sense to them. I think people like King could rescue at least some physicists who are on the verge of the ‘science black hole’. But once fallen into that black hole, I don’t think anyone could rescue them! 😀
Dear student, you haven’t made any holes in my ether model. I am sorry but the conclusions you draw out of your box experiment are rather silly.
To start from the basics, pressure is force acting upon unit surface area. So pressure is nothing but force. Next, the pressure exerted by any fluid/gas is the result of the motion of the individual particles. Because of the random motion, particles move in multiple directions. Particles that move in the upward direction would exert upward pressure and those that are moving downward would exert downward pressure. So a body of fluid can exert pressure (i.e. force) in multiple directions. What is the big problem here?
If my right hand applies force towards the north and left hand applies force towards the south, does that stop force being a vector because I am applying force in two directions. The point is that a system can exert force in multiple directions. Similarly a system can exert pressure in multiple directions.
Your vector length argument is even more silly. You are basically confusing between pressure and pressure gradient. While pressure is measured at one point (surface), pressure gradient indicates the pressure difference between two point surfaces. When you keep the pressure sensor near the centre, it would measure more pressure because more particles hit the sensor and as you move farther from the centre, it would record less and less pressure because of fewer particle impacts. With a pressure sensor, we can define and map out the pressures at every point in every direction. Obviously the pressure gradient would vary between different points.
Further more, you seem to argue that because you can’t specify a definite length (magnitude), pressure can’t be a vector. Do you think inability to define a magnitude makes something a scalar? Do you think scalars don’t have definite magnitude?
I could actually rewrite the whole of my above gravity model in terms of force vectors and without uttering the word ‘pressure’! But I doubt if that opens the eyes of the science believers.
Dear student, I will address your other concerns in a little while.
• A_Concerned_Student On July 8, 2018 at 10:29 pm
I think the issues we are having here are merely a misunderstanding. I do not believe the inability to define a magnitude makes something a scalar. I used that example to show that if we assume pressure is a vector (the assumption), which has a definite magnitude and a direction by definition, that we conclude in the example that there is no definite magnitude thereby achieving a contradiction. Since we assumed something and resulted in a conclusion, then the assumption must be false, and pressure must not be a vector. The argument for pressure being a scalar of definite strength does not yield these conclusions. This is all I was trying to show.
As for your other arguments toward pressure, the main issue I see is with a main assumption of the kinetic theory of gasses where pressure is generally defined as force per unit area. The theory assumes that there are sufficiently high densities (and number of gas particles) in any region such that the particles have very frequent collisions with one another. This results in any one particle not moving very far between collisions. As such in any small region chosen the pressure will act in all directions equally (assuming random motion) leaving no specific direction of preference for the pressure and thus is treated as a scalar at any given position. Yes the individual particles of gas would cause forces, but these forces are counteracted by other particles moving in opposite directions (as the motions are assumed random) thus creating no net accelerations only a scalar pressure on the body. I reiterate that pressure differences (gradients) cause accelerations, not pressures themselves as a vector form of pressure would suggest.
Assuming that pressure were a vector, and since pressure is force per unit area, all pressures should result in accelerations in their respective directions. This is not the case however. A very simple example comes as looking at a diver at the bottom of the ocean. Anyone would agree that the diver is under immense pressure and the diver themselves would definitely notice the pressure on them. However, the diver does not begin accelerating towards the ocean floor, or the surface, nor left,right,forwards, or backwards. Since the conclusion is inescapable that if pressure were a vector that it would cause accelerations, and since we do not see immense pressures causing accelerations, it must not be true that pressures are vectors.
• drgsrinivas On August 24, 2018 at 3:52 pm
So pity Richard. I didn’t expect this from a science major.
It is true that when a force (or pressure) acts upon a body, the body would accelerate in the direction of the force. But what when an equal force (pressure) acts upon the same body in the opposite direction. The body would obviously remain stationary. It is the sum of all the forces and the resultant net force vector which decides whether a body moves and in what direction. The fact that a body remains stationary and doesn’t get accelerated despite lot of forces acting upon it doesn’t mean that force isn’t a vector.
I don’t get why people have so much difficulty in viewing pressure as a vector when the definition itself says pressure is force. Thanks to our science education! But any way, you can forget about that part because I can present the ether model of gravity without uttering the word ‘pressure’.
As explained in the above post, just make a ball spin inside a pond. You would see nearby suspended objects getting dragged towards the ball. That should help you grasp the ether model of gravity. I will leave it to your imagination to explain why and how that happens! No spoon feeding any more.
• King On July 10, 2018 at 6:48 pm
So you now agree that if a moving particle sees our atmosphere as to be less than lp, then there is a contradiction with SR in that we are describing a scenario merely differing by inertial frames we are perceiving it in with different phyc!
But that was my point when I introduced this issue and you, at first, denied that it has anything to do with SR, claiming that lp is only a length scale wherein we need a quantum theory of gravity!! It is only after I took lots of my time to show you that it contradicts SR, did you now switch gears and say ‘it is espected cause OUR THEORIES break down at lp’. In other words, you now include SR in the ‘our theories’ basket which don’t work at lp!
Unfortunately for your, SR is not ‘our theories’ in this case. SR is a theory that says how ‘our theories’ should look from different frames and it should not break at any length as length, according to it, is frame dependent!
• A_Concerned_Student On July 11, 2018 at 5:04 pm
From the outset I said that SR does not work at these small scales and that we would need a quantum theory of gravity to describe what would happen. This has been my argument the entire time, and nowhere did I divert from it. But SR is one of our theories and as such should break down at the lp. This is because it is built upon classical mechanics and does not consider any quantum effects. As such once we get to the scales at which quantum mechanics (or quantum gravity) should become important that SR does not work. This is because SR does not account for any of these effects or any new phenomena that exists at these scales that we are unaware of.
• King On July 10, 2018 at 7:10 pm
You answered yourself this, and I have no time to explain the details. You said aether theory is rejected in favour of SR via OCCAM’S RAZOR. This is the way we us OR:
If two/more theories can EQUALY explain all the phenomena, then the simplest one of them should be chosen.
So if you agree that aether theory was rejected because of OR, you should be agreeing that it too can explain what SR can. Otherwise, you are tacitly joining your fellow SR beleivers who have made the fallacy of thinking of OR as a criteria of telling which one of the theories correctly describes nature!
• richard johnston On July 11, 2018 at 5:44 pm
A clever redirect King, bu we have been going back and forth for weeks with me on the defensive but now that I pose you a question you suddenly don’t have time? Sounds like a weak attempt to avoid the question. But I will let it go as I have found a semi-satisfactory answer elsewhere. Basically, the aether theories response to explaining this (and other) phenomena like it, is to assume length contraction and time dilation! A weak solution I understand, to assume an entirely new phenomena to fix your theory instead of having the phenomena be derived by your theory from it’s postulates. But this happens all the time in science so I will not dwell on it nor hold it against an aether theory. With these additional assumptions (which are admittedly ad hoc), the aether theory could make any prediction SR does. But I wonder (and honestly do not know) how an aether theory accounts for gravitational effects as handled by GR? Again, that I do not know.
• King On July 13, 2018 at 9:56 am
So you conclude that aether theories just ‘assumes’ that there is a length contraction etc instead of deriving it from postulates. You conclude this even though you never know what kinda aether theory I have in my mind!
The postulates of an aether theory involves atomic models that attempts to RATIONALY explain the forces of nature. (like in Drg case, the hypothesis that atoms spin so as to cause gravity via Bernaulis Effects. But I am not saying it is this bernauli’s gravity that I am gonna us). It turns out that when we make a model that explains electromagnetism well, the relativistic effects follows directly from how the atoms must behave so they may cause the observed EM effects!! So it is far from being ‘just an ad oc’! It is deduced from the atomic structure of aether, meant to explain EM, not an ad oc to explain length contraction itself, as such, it is a powerfull explanation!
• richard johnston On July 14, 2018 at 3:14 am
My apologies King, i was referring to one of the most acconplished specific aether theories that I know of (namely Lorentz’s) where he did in fact just throw in length contraction to make it work. I admit not all theories will do this and again apologize for my vagueness.
• King On July 13, 2018 at 10:47 am
No you did not say that
SR does not work at length scale less than lp where we need quantum gravity… You said, and I quote:
‘planck’s length is the theorised length at which we would need a working theory of quntum gravity to make predictions as such, it has nothing to do with SR in this sense’
Please go back and remind yourself!.;)
By ‘nothing to do with SR’, It seems to mean planck’s length says nothing in regard to validity/invalidity of SR, wherelse ‘lp is length scale at which SR breaks down’ is certainly a consideration of lp such that lp has ‘something to do with SR’. This ‘something’ is SR breaking down at lp.
Furthermore, reading your original quote, one can’t tell wether, according to you, this new quantum gravity is compartible with SR or not. Lastly, what then was the point of your objections to my arguments about regularization yet they are just other ways of saying ‘we are dropping the original QFT’ (which must work at some scales)
• richard johnston On July 14, 2018 at 3:17 am
King, again I made a error in clarity by saying the planck length had nothing to do with SR. I did intend it as you stated afterwards that SR breaks down at these lengths and therefore does have to do with it. This new quantum gravity would be compatible with SR, as it would reduce to SR in the limiting case of negligable gravity and large scales.
• King On July 13, 2018 at 11:30 am
Guys, now before I embark on the course I would have taken if our ‘student’ was straight, let me show how ‘lost’ our student is.
1.)There is a length scale such as lp below which QFT must break down.
2.)The solutions of QFT must be summed over all wavelengths to ensure that vacuum state obeys SR (taking the continum limit)
You can see that point 2 again implies that we must sum the solutions of QFT even at length scales where it is aleged to break down there (continuum limit must include wavelengths less than lp)!!
To avoid this absurdity, we ‘regularize’ our theory. This mean we drop our original theory and replace it with another one that takes into consideration the ‘fact’ that QFT shouldn’t work below lp. In the case of Casimir effect, like I said, we multiply the terms of the sum by an exponential factor such that it begines to rapidly dampens the waves of wavelengths below some length scale. So reg is not ‘just math’. It is more accurately, AD OC!
• King On July 13, 2018 at 12:25 pm
So you can see that we are taking the continum limit to ensure that our theory is compartible with SR (mathematically we say ‘to make our theory Lorentz invariant). We are NOT taking the continuum limit (CL) to ensure that our theory describes phenomena happening below lp. Ergo, when we fail to take the CL (because of lp), our theory is simply not Lorentz Invariant (as seen in all length scales). It is not just that ‘our theory breaks down at lp’, and I will further elaborate below.
When we say that ‘our theory breaks down at a certain length scale’ we mean that only the experiments done at those length scales will reveal the short commings of our theory. However, in the case of SR, if it breaks down below lp, we can tell that SR is wrong by performing an experiment even at the length scale of our atmosphere! In a different wording, for a theory to be compartible with SR, it is the Lorentz invariant theory that must break down at length scales, not the Lorentz invariance itself.
• King On July 13, 2018 at 12:53 pm
Now let me illustrate in the most absurd way. The ‘student’, agrees that a speeding particle can observe the thickness of our atmosphere as to be less than lp. Good! But here is a coffee time. To such a particle, we can check that the predictions of SR fails by the measurements of a muon-like particle in the length scale of our atmosphere! So if SR breaks down in some lengths scale, it breaks down in all of them!
Specifically, we can check that we are in a prefered riferance by considering a particle moving at v. If the phyc describing it breaks down at some length d, and it is the same d in all directions, then we are in a prifered frame. Any other frame moving relative to this frame must have it’s d lesser in the direction it is moving through!! This breaks Lorentz’s Symmetry for a particle whizzing across length d, and not only lp!! (Notice that we cannot explain this failure by supposing that the particle sees d as lp as this assumes that the theory, which we are trying to test, is already correct!).
• King On July 13, 2018 at 1:48 pm
Lastly, even if SR did infact break down ONLY for a particle whizzing across lp, we still cannot pretend that SR is still correct because ‘lp is very small’. This mainstream reasoning is like saying that since we can explain many biological phenomena by assuming that microscopic organisms don’t exist (since being soo tiny as to need microscops to observe, their effects are often negligable), we can still entertain a theory that deny existence of microbs even if we have discovered them, provided we say ‘our theory is only correct at certain length scales’.
The silliness of the above reasoning becomes apparent when we consider the jarring claims like that the living things surreptitiously pop out from non living things! The verdict is that even if lp is ‘small’ from the point of view of human, it isn’t ‘small’ for nature to hid its secrets so that it eludes us into forming stupid theories that seem correct as seen from limmited scales just like the theory of ‘living things poping out from mud’ seems correct.
• richard johnston On July 14, 2018 at 3:30 am
King, I see your point regarding the CL, and again agree. By assuming a cut-off we are not using our original theory but rather an approximation of it. This approximation is iseful in getting an intuition or estimate of what will happen in an experiment and will likely violate the principles we built the model from but again is useful. However, in the case of QFT it os important to point out that QFT is a CONTINUOUS theory and we use discrete scales to make the calculations easier, then once we take the CL we actually regaib the original complete theory. Im not sure I follow your argument for how we can determine a preferred frame in the case of a muon in our atmosphere. Could you please expand on that idea so that I may attempt to understand your point?
• richard johnston On July 14, 2018 at 2:57 pm
This argument is one of the very core reasons as to why regularization is used in modern theoretical physics nowadays. The argument being that we can’t observe what happens at really small lengths (below lp) or really high energies so therefore we choose to cut out frequencies in these ranges as we believe there to be “new physics” which are needed to account for them. This way we can test how accurate our theory is IN THE SCALE that they are intended to be used in, while acknowledging where they no longer apply and since we can’t observe these regions yet, this leaves room for advancement once we can and can actually get some experimental evidence to support any new theory. In this way, regularization is similar to acknowledging that we don’t know what happens at these scales and can’t find out yet, but we can on other scales so we will focus on what we can predict/test until such a time that we can test the other scales. I understand why you wouldn’t like that reasoning but it is very practical.
• King On July 16, 2018 at 1:25 pm
Yes, in math phyc, they use ‘compatibility’ that way. ‘Theory X is compatible with theory Y if theory X reproduces theory Y when we take some limit’. But this isn’t what most pple understand by ‘compatibility’. Let me illustrate it using relativistic ‘length contraction’. We write:
When we put v=0, we find that x=x’, which is the ‘Newtonian phyc’. So we see that in this sense, SR is compatible with Newt’s phyc! But you should note that we should rather put it this way:
1.)Newtonian phyc claim that time is absolute.
2.)SR claims that time is relative
3.)Ergo these theories are not compatible!
This is very different from saying:
1.)Newt phyc claims that time dillation=1
2.)SR claims that time dillation=B
3.)Ergo these theories are compatible with each other in that B=1 when v=0
In the second statements, these are not what the theories claim it happens. These are what the theories claim IS THE EXTEND OF WHAT HAPPENS.
• King On July 16, 2018 at 2:09 pm
In other words Newt and Ein are not disagreeing on the EXTENT in which a phenomenon happens. They are disagreeing on the issue of whether a phenomenon happens at all! In other words, many of us understand the phyc world as to be amenable to a black and white type description that cannot be properly expressed mathematicaly. Sometimes we can drop the issue of MAGNITUDES alltogether and talk of whether or not a stick contracted (as a yes or no issue).
We understand the consept ‘inexistent’ as to be completely different from the consept of ‘small’. It is equaly daunting to claim that a particle of planck’s length size poped into existence as it is to say that a whole galaxy did, even if the former ‘creation ex-nihilo’ is ‘compatible’ with ex-nihilo-nihil’ in that both the event of ‘creation’ and ‘no creation’ aren’t observable in our scale.
• King On July 16, 2018 at 2:34 pm
Having said the above, I hope that someone will understand my following statement: the fact that two theories are ‘compatible’ with each other, in that what they predict aren’t much contradictory at some scale, doesn’t mean that the picture of the universe that these theories depict are compatible. A good example is SR being compatible with Galilien Transform whe v is near zero. It is, however, the pic of the universe, not the MAGNITUDE of the predicted effects that many pple are concerned when they question these theories that makes ‘weird’ claims. It is not the t=t’B that makes pple shout aaaah! When they hear of SR. it is the claim that ‘time’ can ‘dillate’ at all. Ergo, the compatibility with t=t’ at v=0 is irelevant to the issue.
So if a ‘counterintuitive’ theory at a smaller scale can be ‘compatible’ with an ‘intuitive’ theory at larger scale, what stops another ‘intuitive’ theory being compatible with the ‘counterintuitive’ one at even smaller scales?
• King On July 16, 2018 at 3:04 pm
If nothing, then how can we swear that we have verified that our world is ‘weird’ because a theory that is said to break down at smaller scales says so and has been confirmed at a bigger scale? For instance, if we know not what happens at lengths less than lp, might there be particles of sizes less than lp that incesantly and randomly bombards a quantum particle, making its position and momentum unpredictable? This will be an example of a commonsensical theory that correctly describes the length scales less than lp but which is ‘compatible’ with the sensless one! It is just like the common sensical theory of germs at the scale of microscopes is compatible with the non-sensical theory of ‘life popping out from mud’ in that germs aren’t observable in our scale.
So other than insisting that ‘our world is weird’, isn’t it more honest to say we just don’t know? Then add that math phyc can’t help us unlock the true secrets of our universe for they demand impossibly precise measurements?
• richard johnston On July 17, 2018 at 8:21 pm
Some really good points King! All of what you say is completely valid and I agree wholeheartedly. Nothing stops the weirdness of QM from being compatible with an intuitive theory at smaller scales but as of yet we have no intuitive theory which can do this. The lack of observations at these scales also leads modern science to say exactly that, “we don’t know”. But at least for the scales we observe our universe appears “weird”, but this could be reconciled or continually become weirder as we look smaller. Compatibility is extremely important as you say, as it “reconciles” theories with opposing viewpoints. As in the Newton/Einstein example they disagree on whether time dilates. Einstein says yes, Newton says no. But seeing as all of Newtons theories were developed using low speed experiments with less sensitive equipmemt, they are unable to meaaure the extremely small changes that relativity predicts. So in the case where the velocity is much much less than the speed of light where these effects are negligably small, Einstein’s relativity reproduces Newtonian mechanics but when the speed is comparable to the soeed of light, Newton does not reproduce relativity. Thus while they disagree, one can reproduce the other and we can attribute the correction to the fact that Newtonian experiments were insensitive to these small effects.
The problem with basing a theory around intuitive reasoning (as members of this site are often to do) is precisely this reasoning. Our intuition is built around our everyday scales (Newtonian) and are thus insensitive to smaller effects outside our perception. So while intuition is good at predicting what should happen on normal scales, it is not applicable to other scales as it has no experience with them and thus no bearing. I believe it was Feynman who said “The universe is under no obligation to make sense to you” which I feel is very fitting. The universe has no need to appeal to our intuition and could be very unintuitive and does appear to be so as we see it currently.
• King On July 20, 2018 at 1:55 pm
good! I see we realy don’t have much querell, if you agree that a theory that ‘isn’t compartible with common sense’ at a bigger scale can be ‘compatible with common sense’ at and even smaller scale. However, some relativists don’t maintain the scientific stand that if we find experiments ‘compatible with commonsese’ we should change our world view. They seem to think that ‘proving’ theory at one scale mean we have proven the kinda universe it depict.
That ‘the pple here rely on intuition’ is the standard, ilogical argument offered by relativists/quantum mechanists whenever they see anyone challenging their theories. This is ilogical and unscientific because ‘intuition’ is subjective. We can’t get into pple’s minds to verify what it is that they are ‘relying’ on. You only know your own intuition. And what does this gotta do with phyc? It seems to concern psychology/ epistemology etc.
• King On July 20, 2018 at 2:22 pm
I don’t agree that pple question modern phyc theories because they have never seen such a thing in day to day life. Like some die hard skeptic of SR has said, he can experience things like ‘a speeding vehicle apearing to receed at the same velocity no matter how he is chasing it’ by merely taking a glass of beer or puffing some weed!
The bottomline is that it is never the universe nor any experiment that is ‘counterintuitive’ or that ‘doesn’t make sense’. It is solely the EXPLANATIONS they offer that are! ‘Intuition’ does not affirm nor deny any observation for we have no A PRIORI idea of how the world must APPEAR. There is no a priori reason as to why a speeding muon must decay the same way as an otherwise similar, stationary muon for these are two things under different conditions. Ergo intuition frames no ‘prediction’ whatsoever in this! Ergo, saying that muon decay confirms that universe is ‘counterintuitive’ is silly!
• richard johnston On July 21, 2018 at 10:46 am
My reason for commenting that our intuition is a reason that many people have issue with theories like SR and QM is because they make predictions that we don’t see in our everyday lives. I make this appeal not to defend my own theory, but to try and explain why some people might reject these theories. For instance, time dilation is a common thing that people take issue with in SR and very often their first argument involves how they have never seen experienced time slowing down/speeding up. But then again, they’ve also never gone at a speed even close to the speed of light, so how could they experience any noticeable time dilation as SR predicts? I don’t say that SR is right or wrong, only that some people reject it due to this fact. And I agree that it is unscientific as there is no way to quantify why someone fully rejects something, but that does not mean that it isn’t a good reason that some people may reject it. As for whether the universe or some experiment can be counter-intuitive, they can be! Counter-intuitive literally means counter to what intuition predicts. Therefore, if my intuition predicts one thing and an experiment shows something else, then that result is counter-intuitive. Intuition definitely does not affirm nor deny any observation, but it is one of the first things people use when trying to predict what will happen and that is why I mention it.
• King On July 20, 2018 at 2:55 pm
It is LOGICS, not INTUITION that is the problem with modern phyc THEORIES (not the experiments they correctly predict). The easiest to deal with is SR. The following statements are logics
1.)if A=B, and B=C, then A=C
2.)if A is larger than B, and C is larger than A, then C cannot be as larger than A as it is larger than B.
It is the second, simple logic that the second postulate of SR violet. It goest this way:
1.)speed of light is larger than speed of B. speed of B is larger than the speed of A. So light speed can’t be as larger than B as it is larger than A.
The crucial point is that as we try to make a math model of speed, like in any quantity, this model must capture the notion of MAGNITUDE well. If measuring speed by rods and clocks and taking ratios leads to violation of the above logic, then such definition of ‘speed’ simply doesn’t capture ‘speed’ as a MAGNITUDE like any other magnitude.
• richard johnston On July 21, 2018 at 11:16 am
This is a very good logical statement, however, it is not really applicable in the case we are talking about. Basically, any theory involving vectors can not include this logical statement as it does not fit with the idea of magnitude. The reason it does not apply is because this statement only relates scalar numbers, not their magnitudes. Consider the scalar case where 2.) holds. Let A=3, B=10, and C=20. Clearly C>A, C>B, and B>A then C-BA, C>B and B>A, but in this case we actually have B<A! Therefore, this assumption can not be used in ANY vector theory, not just SR. So the fact the SR violates this logical statement isn’t an issue, THE ENTIRETY OF PHYSICS VIOLATES IT! It is worth noting that if you take the magnitudes of scalar numbers (absolute value) then this logical statement is still violated, so your argument for this statement being a measure of how well magnitude is defined is flawed. Really what this statement aims to show is that the order of numbers is well-defined.
• richard johnston On July 24, 2018 at 2:50 am
So a portion of my comment above got erased for some reason and it makes no sense to read now so I will rewrite it here (with some edits) beginning from “Let A=3,…”)
Let A=3, B=10, and C=20. If we write your logical statement 2.) using math then it becomes: If C>A, C>B, and B>A then C-B<C-A. In fact, we can prove this to be true even without using the assumptions that C>A and C>B but the C is important with respect to relativity so we will keep it. So using the numbers for A,B,C above we can see that it is true that C>A (20>3), C>B (20>10) and B>A (10>3) which then implies that C-B<C-A (20-10<20-3 or 10<17) and it is clearly true. To show this again let’s let A=-5, B=2 and C=5. C-B=3 and C-A=10 and thus C-B<C-A so it even works when negative numbers are involved!
Now, here’s where this logic doesn’t work. We have shown (really just seen but it can be very easily proved) it to hold for both positive and negative numbers but does it work for magnitudes? For those of you who are unfamiliar, when we talk of magnitudes with respect to scalar numbers or vectors we don’t care care about the direction of the vector or the sign of the scalar, only how “long” it is. So for a number like 3, the magnitude is just 3, but for -3 the magnitude is also 3! So we drop the sign for scalar numbers and that’s the magnitude. For vectors we drop any sign as well and just use Pythagorean theorem to find the vector’s length and that’s the magnitude. So let’s try our second example again but with magnitudes.
Let A=-5, B=2 and C=5. Now, the magnitude of -5 is 5, magnitude of 2 is 2, and magnitude of 5 is 5. So really we now have A=5, B=2 and C=5, therefore C-B=3 and C-A=0 and we see that C-B>C-A which is not what we want! In fact, by taking the magnitudes of our numbers we actually violate our assumption that B>A! We assumed that B>A but actually had B<A when we use magnitudes! So this shows that this logical statement doesn’t hold for magnitudes, only regular numbers (sign and all).
Now since any theory involving vectors uses the concept of magnitude inherently, this means that this logical statement can’t be used in any such theory! So it isn’t just SR that violates 2.), in fact ALL OF PHYSICS VIOLATES IT!!! From the above it is clear that this logical statement does not actually show that magnitude is well-defined, since the introduction of magnitudes breaks it after all. Really what this logic aims to show is that the order of numbers is well-defined.
• A_Concerned_Student On July 21, 2018 at 11:51 am
Side Note: An equivalent way that you could formulate logical statement 2.) would be to say: If A is to the left of B on a number line, and B is to the left of C on a number line, then A cannot be closer to C on the number line than B is.
From this formulation of 2.) it is clear that this statement does not apply to physics in general. This is because (one of many reasons) physics denotes many quantities as vectors which are not confined to the number line and whose magnitudes are strictly positive numbers.
• King On July 20, 2018 at 3:31 pm
My point gotta be easy to grasp. We defined speed as v=x/t and not as t/x because we wanted our math value for v to capture the ‘magnitude of v’ well, i.e the faster object must be assigned larger numbers and vice versa. So obviously, we defined v using a commonsensical consideration that larger speeds in the univers should be mapped into larger numbers in the paper. Ergo, speed cannot possibly defy common sense! If ‘t’ did not ‘tick’, then v=infinity and we realized that this defied the commonsensical notion of ‘magnitude for speed’. So no one was stupid enough to make a clock which does not tick (so at least an extreem example illustrates that a priori understanding of speed as a 1 dimensional magitude defines the correct clock and never the vice-versa)!
Next, we saw that an eratically ticking clock can’t capture the notion of ‘speed’ well. Were it not so, ‘accurate clocks’ could have been invented in stonnage! All these we did using COMMON SENSE!
• King On July 20, 2018 at 3:44 pm
So then how should we describe a claim that measurents of a speed has shown that ‘universe doesn’t make sense’? Alright, you have guessed it! If I may be very polite, it is stupid!!
It is never ‘universe’ that dictates that v=x/t nor does it asks us to swear by ‘accurate’ atomic clocks in Paris as to measure ‘passage of time’. These are human concoctions and requirements, not ‘the universe’. Only the nincompoops like Feynman can foolishly follow human made concepts like ‘speeds =x/t’ and say that that is ‘the universe’.
• King On July 20, 2018 at 4:17 pm
Lastly, let me stress again that the universe is NOT ‘counterintuitive’.
1.)a muon ariving at the earth surfice from the sky is NOT counterintuitive. It is just a muon!
2.)particles forming inteference patterns on a screen is NOT counterintuitive. They are just a parten of dots on a screen.
3.)a flown clock ticking slowlier than a stationary one is NOT counterintuitive (my clock also tick slowlier when I threw it into the dust pin)
Not a single experiment ever apear ‘counterintuitive’ when we just take a look at it. Rather it is when a human tries to force these observations into a simple, math description (which can be voted for in a science club) does he wind up with counterintuitive theories. However, if we shut up and just experience the universe, there is nothing counterintuitive. It is just what it is!
• richard johnston On July 21, 2018 at 11:25 am
Everything else here that you have said is very much correct, if a little misworded. Measurements of any kind do not show that the universe doesn’t make sense, only that the universe does not make sense to us. This is more of a statement of our own lack of knowledge than any attribution of weirdness to the universe. One has to consider that whenever we say something is “weird” or “stupid” or anything else that this is in reference to ourselves. We say “weird” relative to what we believe is normal. We like to think particles having a specific position is normal, so QM’s prediction of no absolute position of particles is “weird” relative to what we think is normal. If in your last post you had replaced the word “counter-intuitive” with “unnatural” then I would agree wholeheartedly! Everything in nature does exactly what it’s supposed to do and always has done, it is our intuition and our understanding which makes things counter-intuitive. We expect one thing to happen but see another, so we call it “weird” and counter-intuitive (as is the correct word to call it). But it is far from unnatural. As you said, it is just what it is!
• richard johnston On July 24, 2018 at 2:27 am
Galacar, I do not say that intuition isn’t important, it certainly is important in getting a quick idea of what we think should happen in any scenario. The key thing here is that it gives an idea of what WE THINK SHOULD happen. However, it is not great at giving exact details of what will happen and can very often be incorrect, but it is still useful. What I am trying to say is, while intuition is important, it should not be the basis for explaining everything that happens in the universe. More importantly, we should not disregard a theory or experimental finding simply because it does not agree with our intuition.
The links that you pointed me to however don’t seem to disprove my statements at all. In fact, I dare say that they aren’t even related to my statements on intuition. The first link leads to an article stating that intuitive people have made some good predictions relating to bipolar disorder and could be useful in other medical research. Sounds good to me, but it doesn’t say anything about basing a theory on intuition only that intuition can be useful in some medical settings. Thereby, it is not related to my claims, and it definitely does not disprove them in any shape or form. The second link leads to your own comment elsewhere detailing the story of some psychics who claim to have observed atoms at what they saw. Once again, how does this relate to basing theories on intuition? It says nothing on this topic and can therefore not disprove my claims.
While your contributions to the discussion are appreciated, I do ask that you be careful with not saying that you have disproved things when you have not. Disprove means to show that a statement or theory is false, and the evidence you provided did not do that. It could be seen as evidence not to believe my statement, but it does not show my statement to be false.
Best Regards!
• King On July 24, 2018 at 3:08 pm
Like you now stated it clearly in ‘side note’, you replaced my argument with another totaly different one and then attacked your own straw man!
There is a difference between saying that A lies to the left of B in the numberline and saying that B is LARGER than A, or at least in how I used the word ‘larger than’. When I said ‘larger than’, I mean ‘its magnitude is larger than’.
Since my A, B, and C already rifered to MAGNITUDES, non of A, B, and C rifers to negative numbers. So I was not rifering to a case where A,B and C are positions along a number line so that taking their magnitudes can result in swapping which of them is larger. I am talking about numbers that represent magnitudes already. So the logic absolutely apply to magnitudes as well when stated this way:
If B is larger than A in MAGNITUDE, and C is larger than B in MAGNITUDE then C-A is larger than C-B in magnitude.
• King On July 24, 2018 at 3:30 pm
Next, we see that this logic must apply in the case of the magnitudes of speeds. We begine with speed A being zero, i.e ‘stationary’ frame. Then speed B=v (speed of the ‘moving’ frame) is greater than A. finaly, c is both greater than A and greater than B as well!
So clearly, v is NOT negative to c as both are moving in the same direction! So the case when a ‘negative number’ swaps its sign is inexistent here. So it is a STRAW MAN invented by richard!
Apart from that, my main argument is that it is LOGIC, not INTUITION that we have a problem with SR. Even if my argument wasn’t valid, the more correct objection would be ‘you are relying on inapplicable PREMISES ‘ and not ‘you are relying on INTUITION’. But that is still not right! I am using a different DEFINITION of speed that captures the notion I have better than taking ratios of what rods and clocks indicates!
• King On July 24, 2018 at 3:47 pm
So clearly, SR violates the basic logic that applies to all magnitudes! It advocates that speed of light is greater than the speed of frame A (stationary frame), speed of frame B (frame moving at v) is greater than the speed of frame A, speed of light is greater than the speed of B, but the speed of light is as greater than B as it is greater than A!!
What I have done to expose the absurdity we are aluding to is to jettison the issue off all irelevancies (observers, clocks, rulers, switch back and fort between riferance frames etc). These are fake ‘cloths’ covered to SR. take them off and we see its nakedness!
Clocks and rulers offer us their TESTIMONIES with regard to what they ‘think’ is the speed of snail. But how this magnitude relates to others, like in all magnitudes is a matter of LOGIC, not MEASUREMENTS.
• King On July 24, 2018 at 4:19 pm
‘Intuition predicts…’
Whose intuition does that? The ‘intuition’ in the funny farm?
Intuition of sane pple is open to WHATEVER the universe might reveal. Then after that, it merely try to MAKE SENSE of it (of which I noted this is the intuition of Drgsrinivas.;)! Intuition is just like logic. They don’t predict anything. We DON’T say whether or not a muon produced above the atmosphere will reach the earth surface. We say, IF it reaches the surface, WHY does it do that, IF it fails, WHY doesn’t it do that. Sane intuition works AFTER, not BEFORE observation.
How stupid it is to say that we insist muon never reach the earth surface!!! Einstein’s nincompoops put this in our mouths so when they find muons, this confirms their religious stand that ‘universe is cuonterintuitive’. Every sane person knows that atmosphere isn’t empty. To expect ‘moving’ to be same as ‘not moving’ is insane!
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• King On July 24, 2018 at 4:38 pm
Lets consider MM experiment, for instance. While relativists would like you to beleive that all ‘intuitive’ pple didn’t expect null results, this is far from truth! There were DEBATES, and this is the reason why the experiment had to be done.
The classic physicist, WHALTER RITZ, for instance, beleived that the results should be NULL even though he was fully aware of Fizeau results, starlight aberation, etc and his inability to explain them all! However, he did not buy into SR either!
His intuition was good. When we don’t understand a phenomena, it is just that, we DON’T understand! Relativists teaches us that we should raise up our hands and declare ‘the universe is counterintuitive’ and close up any more attempts to understand. In other words, they tell us just what RELIGIONS have been telling us for 10000 years: our god does in ways that no man can understand!
• King On July 24, 2018 at 6:36 pm
Now, having talked of of those ‘logic’ vs ‘intuition’ of ‘speeds’, lets now come to a crucial issue: does logic or intuition tells us that when we MEASURE the speed of light, we must always find it to vary based on riferance frame cause both logic and intuition dictates that it must vary? It is realy neadless to say ‘naughta’ to reasonable pple! However, relativist requires pages to understand this!!
Common sense tell use that if I use a pendulum that swing once a day to define ‘second’, this pendulum clock can indicate that a snail is moving at the speed of light! So if a clock slows down, what is actualy moving slowlier can seem to move very fast as measure by that clock! Logic/commonsense then say this doesn’t mean that the object is ACTUALY moving fast.
In other words common sense does not deny that ANY of what SR predicts can actualy be observed. It merely deny SRs idea that we take these measured values LITERALY and ‘too seriously’. Unsurpringly then, physicists ( Lorentz) had predicted before SR
• King On July 24, 2018 at 7:01 pm
But my notion of ‘speed’ jetisones the clocks, riferance frames, observers and the rulers all together! If I remove these off the mouth of a relativist, he loses his major loop hole: that the issue is very complex! He can’t keep jumping, ‘it is clock’, ‘it is time’, ‘it is observer’, ‘it is non-euclidean’, ‘it is frame of riferance’..hopelessly confusing themselves and other idiots over a very simple issue!
Clocks are 100% useless when we want to say merely what is ‘FASTER THAN’, ‘MOVING AS FAST AS’ or ‘MOVING SLOWLIER THAN’. So why a relativist bring a confusing clock to such an issue? We don’t need a clock, for instance, to tell that in order for a cheetah to catch a gazel, it must move FASTER THAN a gazelle. Ergo we have an example of a good/scientificaly acurate notion of speed without riferance to a clock! However, a clock can indicate that a cheetah, which has got up a gazelle was moving slowlier than the gazelle! This mean we through the clock to museums! They have zero relevance to ‘universe’!
• Satyam On July 27, 2018 at 4:21 pm
When “counter-intuitive” was first proclaimed by physicists they meant that what they said was true and that what the “common” people thought was not true. The physicists then knew nothing about intuition and they still don’t know anything about it as you show us. You call yourself a student but this is no excuse because in order to become a “real”physicist you have to comply with the rules of your teachers. Now you are saying that things are counter-intuitive because you don’t understand them, so first the laymen were ignorant and now you say you and your teachers are ignorant. In both cases however there is no scientific knowledge of what intuition is. But the physicists still insist that they know what is going on even when they admit they know nothing and this is the real problem, they just pretend to know. This is what fake science does, it covers all bases and makes a mess of everything.
• drgsrinivas On August 24, 2018 at 5:04 pm
Relativity and Quantum prophets are always obsessive of making distinction between intuition, commonsense and rational thinking as that helps them to sell their absurd theories. First the science pastors brainwash their students and make them believe that commonsense and intuition only come in the way of knowing the Science. The faithful students so give up their commonsense and intuition in their eagerness to learn ‘Science’. The science prophets then sell their absurd theories to the faithful students. The students having given up their commonsense, readily accept all the absurd things and keep reciting them. Those students who thoroughly recite those absurd theories become science prophets and the preaching and ‘religious conversion’ continues in the name of education.
But let me tell you that it is ultimately Logic that underlies all of those processes (including maths). The only difference is that while someone goes by intuition or commonsense, one’s mind does the logical deduction process subconsciously, taking into account of the ‘data in hand’ or the immediately available information (something like RAM). In the so called rational thinking or critical thinking, one does the same logical deduction process rather consciously. In the conscious process, one will not just go by the ‘data in hand’ but tries to look for and take into account of the deeper and broader issues before arriving at a conclusion. So one may conclude that conscious rational thinking is superior to commonsense or intuition. While there can no argument about that, there are some very important points to bear in mind.
The depth and breadth of information processing done subconsciously by some individual could be greater than all that involved in the conscious rational analysis by some other individual. That is, great minds could arrive at the truth just by intuition while the ignorant people may not reach that despite their best conscious thinking and performing costly experiments and consuming all the funds in the state.
So intuition doesn’t mean some totally random, haphazard or irrational thinking. Rather it is a subconscious logical deduction process. Intelligent minds could know the truth instinctively while the less intelligent minds may not know the truth even after years of scientific research, experimentation and grounding. In fact, less intelligent minds who lack commonsense would only get mislead by the experimental data.
• CDUB On July 7, 2019 at 2:27 pm
This might qualify as a good idea if you could provide a more quantitative approach.
• Hector Estepan On June 28, 2020 at 5:01 pm
Interesting and delightful thoughts. New to your site and just found out about it. Here is my take on it. The major problem is mathematics, and specifically Euclid’s point definition “that which has no part”. Every mathematical equation uses Euclid’s point definition. The only conclusion is the universe is made out of nothing! It is of some concern because we are surrounded by real things!
To get around this problem why not start with a real Mathematical point such as a sphere of radius r and having a constant speed, zero mass, and zero charge. I guess it can be called a photon, but that is another story. The movement of such a point outlines a continuous mathematical function, and if the speed is constant, the point creates a uniform continuous mathematical universe. This is the stage.
We can then look at the mathematical function’s domain minus to positive infinity, thus making this mathematical point incapable of being created or destroyed, therefore if the photon is to be the mathematical point, the photon concept has to change. Good luck with that!
Because the point is real, it has to have the three dimensional Cartesian vector
in the x,y, z directions. In fact, all real objects have to have these three orthogonal vectors, and if it does not have these three vectors, then the object does not exist.
A volume in motion can be designated as V/t, but since all volumes have the vector then the velocity of the volume in motion is the vector/t, hence if the volume is moving in the positive x direction the velocity vector is just the x-component/t. Now here comes an interesting part. The x component in this sphere varies from zero to 2r, and each x component has to move with the constant speed v, which equals x/t. Concentrate on that relationship v=x/t, or x=vt. what does it say for a moving sphere? it clearly states that within, and on that moving sphere, time varies as x varies! It means that every time measurement we make on a moving object reflects nothing about the time in that moving object. Hence an object in motion having different lengths in the direction of motion, has different times in the direction of motion! So much for the constancy of time, but more importantly our external measurements tell us nothing about what is going on inside the moving object. Whatever interpretation we make is bound to be wrong. So much for experimental evidence.
There is some doubt about the Earth’s spin at the equator and at the poles.
The Earth spins with a speed at the equator, and the speed at the poles is zero. We utilize this speed gradient to hurl objects into space, but there is something wrong with this logic, because the Earth’s equator is not breaking off from the Earth’s poles, therefore the spin speed at the poles is equal to the spin speed at the equator, and yes the circumference at the equator is 2(pi)(diameter) and that at the poles is zero, but time at the equator is t and at the poles is zero! Once again it is time that is varying with the varying radius. Think about what this is doing to our concept of space time. Think about what this is doing to our concept of gravity.
There is obviously more if anyone is interested
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a01fee5102a3302d | July 4, 2019
The Quantum Zeno Effect: how to pause time on a quantum computer
Zeno of Elea lived around 2500 years ago and established a set of philosophical problems still taught by philosophers. Despite no surviving writings by Zeno, we can learn about the concept known as Zeno’s arrow paradox from Aristotle’s writings. The paradox has inspired a particularly intriguing concept in quantum mechanics: the quantum zeno effect. By just observing a particular system you can stop it from moving. Using IBM’s quantum composer I’ll show you how to try this experiment yourself on a real quantum computer.
Aristotle – Physics VI
Imagine recording an arrow soaring through the air with a high speed camera. When we play the clip we see that it follows a smooth trajectory towards its target. However, at each infintesimal point in time, or every frame, the arrow is stationary. This apparent paradox highlights the concept of discrete and continuous motion and is resolved by realising that the instantaneous velocity of a moving arrow can never be truly zero.
Quantum Zeno Effect
The quantum Zeno effect, on the other hand, demonstrates something highly suprising. Here we replace the arrow with a quantum arrow that can only be either in the bow or in the bullseye, or in a superposition of the two. By creating a force acting on the bow, to push the arrow towards the bullseye, we go from the state \ket{\text{bow}} to \ket{\text{bullseye}}.
The Quantum Zeno Effect's bow and arrow
Mathematically we can represent this as the state of the system evolving under a Hamiltonian of the form
\[H=\frac{1}{T}(\ket{\text{bow}}\bra{\text{bullseye}} + \ket{\text{bullseye}}\bra{\text{bow}})\]
where we have defined a time period T which dictates the speed of the evolution. This equation is solved using the Schrödinger equation. The state of the system at later times is
\[\ket{\psi(t)}=\cos(\frac{2\pi t}{T})\ket{\text{bow}}+\sin(\frac{2\pi t}{T})\ket{\text{bullseye}}\]
Something strange happens when we measure the state of a quantum system. It collapses into only one of the two states. When this happens the time evolution starts again, from this new state. If we keep measuring the system we can actually stop it from evolving in time!
Building the quantum circuit
The IBM Q allows us to explore this concept on a real quantum device. We will create a moving system which we will allow to evolve in time, like our arrow through the air.
We will then measure the state of our qubit repeatedly. By doing so the qubit will not be able to change state.
To start we need to decide where we will begin and where we want to end. To keep things simple lets just start our qubit in the state 0 and launch it towards the state 1. Coincidently, we can imagine our qubit as an arrow pointing to the surface of a sphere. If the arrow is pointing upwards, the qubit is in the state 0 and if it’s pointing down then it’s at 1.
The IBM Q automatically initialises our qubit in the 0 state. To test this we can try out a test circuit which simply measures our target (middle) qubit. The output tells us that all the qubits were measured to be in state 0.
Hello World: the qubits start in the state 00000
How do we let our state evolve in time? Luckily the IBM Q is a universal quantum computer, meaning we could, in theory, run any classical algorithm we like. The operation we choose is a rotation. We can tell the quantum computer to rotate our state around the bloch sphere by some angle. We have five qubits so we pick to rotate the qubit a fifth of the way around the bloch sphere.
Quantum Zeno Effect modelled on the Bloch sphere
The qubit is rotated by 180 degrees and is measured to be 1
After doing this five times the result is not suprising, we end up with a 1 every time: the arrow has made it from the bow to the bullseye!
Adding Measurement
But what happens if we place a detector in the way of each of the rotations? This is equivelant to creating a CNOT gate which has the effect of switching the state of the \ket{0} of a target qubit. Using the principle of deffered and implicit measurment (which states that if we leave some quantum wires untouched we can assume they are measured), we create the effect of observing the state of our middle qubit after each rotation.
We add CNOT gates to measure the state of each qubit after each rotation.
Running the quantum algorithm
First lets try running the alogirthm on IBMs circuit simulator to check we get the result we expect. The number of times each result was given is presented in a histogram.
Quantum Zeno Effect: results of simulation
Results of the simulation
We see that our middle qubit remains in its \ket 0 state 68% of the time! To see why, consider the evolution after each gate rotation. The state of the qubit will be \ket\psi(t_1)=\cos(\pi/10)\ket{0}+\sin(\pi/10)\ket{1}. Using our measurement gadget we see that the qubit will be projected either into the state \ket 0 with probability \cos^2(\pi/10) or into \ket 1 with probability \sin^2(\pi/10). Repeating this process a further 5 times we see that the chance of remaining in state \ket 0 is \cos^{10}(\pi/10)=0.605. By accounting for more complex transition including jumping into state \ket 1 and then jumping back to \ket 0, we recover the expected probability.
We can now send this circuit to the real quantum computer and we see that we have succesfully stopped the time evolution of our qubit about 50% of the time!
The Quantum Zeno Effect on IBM Q
Results from IBM Q on device ibmq_16_melbourne
So the result appears not to be perfect – the result isn’t exaclty what is expected from the theory. This is due to two reasons. First, the IBM Q only has limited connectivity. This means the circuit needs to be broken down and remodelled before it can run on the quantum computer. This makes the circuit more complicated and it takes longer to run. The qubits are also vunerable to noise. Extremely subtle effects can have the effect of shifting the states of the qubits and destroying their coherence.
The result of the quantum zeno effect demonstrates the suprising and seemingly paradoxical world of quantum mechanics. The measurment problem is largely unresolved and is explored in great detail in Adam Becker’s most recent book ‘What is real?’. Exaclty why measurement has the effect of collapsing the wavefunction, or splitting the unvierse in two as the many world theorists argue, is unknown.
Not only does quantum computing have the power to unlock many of the key mysteries of chemisty, technology and health, but I hope it can also help us to answer deep fundamental questions about the nature of our universe.
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e0c970408b2f2797 | Title & Abstract
Name of the speaker: Prof. Howard Wiseman, Griffith University, Australia
Why experimental metaphysics needs a quantum computer
Abstract: Experimental metaphysics is the study of how empirical results can reveal indisputable facts about the fundamental nature of the world, independent of any theory. It is a field born from Bell’s 1964 theorem, and the experiments it inspired, proving the world cannot be both local and deterministic. However, there is an implicit assumption in Bell’s theorem, that the observed result of any measurement is absolute (it has some value which is not ‘relative to its observer’). This assumption may be called into question when the observer becomes a quantum system (the “Wigner’s Friend” scenario), which has recently been the subject of renewed interest. Here, building on work by Brukner, we derive a theorem, in experimental metaphysics, for this scenario [1]. It is similar to Bell’s 1964 theorem but dispenses with the assumption of determinism. The remaining assumptions, which we collectively call "local friendliness", yield a strictly larger polytope of bipartite correlations than those in Bell's theorem (local determinism), but quantum mechanics still allows correlations outside the local friendliness polytope. We illustrate this in an experiment in which the friend system is a single photonic qubit [1]. I argue that a truly convincing experiment could be realised if that system were a sufficiently advanced artificial intelligence software running on a very large quantum computer, so that it could be regarded genuinely as a friend. I will briefly discuss the implications of this far-future scenario for various interpretations and modifications of quantum theory.
[1] Kok-Wei Bong, Aníbal Utreras-Alarcón, Farzad Ghafari, Yeong-Cherng Liang, Nora Tischler, Eric G. Cavalcanti, Geoff J. Pryde and Howard M. Wiseman, “A strong no-go theorem on the Wigner’s friend paradox", Nature Physics (2020).
Name of the speaker: Prof. Fabrizio Piacentini, INRiM, Italy
Title: Weak-interaction-based measurements: a new tool for quantum technologies
Abstract: Measurements can be considered one of the pillars of physics, especially in Quantum Mechanics, because of features without classical counterparts like the wave function collapse in "sharp" (projective) measurements.
In the last decades, among the most interesting measurement paradigms discussed and tested in the quantum physics community, one can find weak measurements. i.e. measurements characterized by an interaction weak enough to avoid the wave function collapse, representing an excellent tool for both fundamental research and quantum technologies.
Furthermore, in recent years new measurement paradigms have been proposed as a further evolution of weak measurements, e.g. protective measurements, able to obtain information on the expectation value of an observable even measuring a single particle. A second example is given by genetic quantum measurements, showing analogies with the typical evolution-inspired mechanisms of genetic algorithms and yielding uncertainties even below the quantum Cramér-Rao bound, while a third one is represented by robust weak measurements, able to reliably extract a weak value (even an anomalous one) with just a single click of the detector, without the usual average on multiple detection events.
In this talk, after a general introduction, I will present some of the latest results related to the experimental implementation of weak-interaction-based measurement protocols in different scenarios, highlighting their new, disruptive features and advantages related to both the quantum foundations and quantum technologies frameworks.
Name of the Speaker: Prof. G S Agarwal*, Texas A & M University
Title: Two-Photon Processes in Entangled Fields
Abstract: Two photon processes like nonlinear absorption and Raman scattering are known to provide wealth of information on systems of interest. These are typically studied by using coherent laser fields. The efficiency of such processes depends on the interference among different path ways from initial state to final state. The use of entangled light and more generally quantum light can be used to control these path ways and to enhance the efficiency of the two photon and more generally multiphoton and nonlinear coherent processes like up-conversion, leading to new directions of research in nonlinear spectroscopy. I would describe the fundamentals and recent progress in the study of two photon processes in Entangled fields.
Name of the Speaker: Prof. Lev vaidman, Tel Aviv University, Israel
Title: Experimental demonstrations of exotic quantum measurements
Abstract: I will report first demonstrations of various types of quantum measurements. Nonlocal measurement - measurement of a property of a composite quantum system with spatially separated parts. Protective measurement - measuring expectation value of an observable with a single click. Robust weak measurement - measuring weak value of an observable with a single click. Modified interaction-free measurement: measurement that tells us that the place is empty without any particle passing through it.
Name of the Speaker: Prof. Franco Nori, (RIKEN, Japan, and University of Michigan, Ann Arbor, USA)
Title: A few examples of Machine Learning and Artificial Neural Networks applied to Quantum Physics
Abstract: Machine learning provides effective methods for identifying topological features [1]. We show that unsupervised manifold learning can successfully retrieve topological quantum phase transitions [1]. We have also developed [2] machine learning-inspired quantum state tomography based on neural-network representations of quantum states. We also consider conditional generative adversarial networks (CGANs) to QST [3]. We demonstrate [4] that artificial neural networks can simulate first-principles calculations of extended materials.
[1] Y. Che, C. Gneiting, T. Liu, F. Nori, Topological Quantum Phase Transitions Retrieved from Manifold Learning, Phys. Rev. B 102, 134213 (2020).
[2] A. Melkani, C. Gneiting, F. Nori, Eigenstate extraction with neural-network tomography, Phys. Rev. A 102, 022412 (2020).
[3] S. Ahmed, C.S. Munoz, F. Nori, A.F. Kockum, Quantum State Tomography with Conditional Generative Adversarial Networks, (2020). [arXiv]
[4] N. Yoshioka, W. Mizukami, F. Nori, Neural-Network Quantum States for the Electronic Structure of Real Solids, Communications Physics, 4, 106 (2021).
[5] K. Bartkiewicz, et al., Experimental kernel-based quantum machine learning in finite feature space, Sci. Rep. 10, 12356 (2020).
PDF files of these publications can be found here: https://dml.riken.jp/pub/ai_meets_qp/
All of our work is accessible here: https://dml.riken.jp/pub/
*Our work is supported in part by NTT Research, JST, JSPS, ARO, AFOSR, AOARD, and FQXi.
Name of the Speaker: Prof. Michael Hall, Australian National University
Title: How to cheat at quantum cryptography: the roles of free will, causality and retrocausality
Abstract: The promise of strong quantum cryptography relies on assumptions of locality and free choice. If these assumptions hold, and if the measurement correlations between the coding devices satisfy a so-called Bell inequality, then it is impossible for an external eavesdropper to have knowledge of the code generated by the devices. I will discuss how the free choice assumption may be violated in practice, while maintaining locality, and some implications thereof. The minimal information resources that allow perfect eavesdropping is determined to be a mere 0.08 bits of causal correlation, between the physical source and the "random number generators" (possibly humans) that determine the measurement settings. Devices can be built that exploit this possibility, leading to a buyer-beware warning for off-the-shelf quantum cryptography apparatus. Further, and somewhat surprisingly, only ~0.04 bits are required if retrocausal correlations are permitted. While we cannot build devices that exploit the latter possibility, its greater efficiency provides an interesting "Occam's razor" argument for retrocausality in nature.
Further information:
[1] https://journals.aps.org/pra/abstract/10.1103/PhysRevA.102.052228
[2] https://physics.anu.edu.au/news_events/?NewsID=213
Name of the Speaker: Prof. Valerio Scarani, National University of Singapore
Title: optimal discrimination of optical modes
Abstract: This talk shall start with the classic Helstrom discrimination of two states. I shall then explain how discrimination among more states can be approached using semi-definite programs (SDP). With these tools, I shall introduce the notion of discrimination of optical modes. I shall describe how the SDP approach can be generalised to deal with this case, and show a couple of interesting examples [from: I.W. Primaatmaja, A. Ho, VS, Phys. Rev. A 103, 052410 (2021); https://arxiv.org/abs/2012.11104].
Name of the speaker: Prof. Alexandre MATZKIN, CNRS, France
Title: Wigner-Friend scenarios: from the Measurement problem to the consistency of Quantum Mechanics
Abstract: The measurement problem still hovers over the foundations of quantum theory. While in most situations we do not need to bother (at least for practical purposes) about these conceptual difficulties, in some instances the role of the Observer and the nature of the quantum state become prominent. This is the case for Wigner-Friend scenarios, which involve observers that measure other observers measuring a quantum system. Here different ways of understanding the measurement axioms lead to different predictions for the outcomes of (what are at least for now) thought-experiments. In this talk, I will discuss these issues, starting with the original Wigner's Friend setup and looking at other scenarios proposed more recently. I will argue that standard quantum mechanics, given eg in the celebrated Feynman Lectures textbook has no problems in dealing with such scenarios, though this involves underlying assumptions that are by no means obvious and might in the future prove to be incorrect.
Name of the Speaker: Prof. Francesco Buscemi, Nagoya University, Japan
Title: Prediction, retrodiction, and the Second Law of Thermodynamics
Abstract: In this talk I will present some recent work clarifying the role that prediction and retrodiction (and, more generally, Bayesian inference) play in the logical foundations of the Second Law of Thermodynamics and various fluctuation theorems for classical and quantum systems. The exposition will be very much pedagogical, assuming only a little background in elementary probability theory, classical thermodynamics, and quantum (information) theory. Based upon https://arxiv.org/abs/2003.08548 and https://arxiv.org/abs/2009.02849
Name of the speaker: Dr. Amit Rai, Jawaharlal Nehru University, New Delhi
Title: Non-classical light in a J_x photonic lattice
Abstract: We report the study of non-classical light in a photonic lattice having a parabolic coupling distribution, also known as a Jx photonic lattice. We focus on a two-photon Fock state, a two-photon N00N state, a single-mode squeezed state and a coherent state as inputs to the lattice. We investigate the possibility of a perfect transfer of the mean photon number as well as the quantum state from one waveguide mode to another. We study photon–photon correlation for the two-photon N00N state. For the single-mode squeezed state we perform a detailed study of the evolution of the squeezing factor and entanglement between the waveguide modes. Our findings suggest a perfect transfer of the average photon number in all cases and a perfect transfer of the quantum state in the cases of the two-photon Fock state and the two-photon N00N state only, but not in the cases of the squeezed and coherent states. Our results should have applications in the physical implementation of photonic continuous-variable quantum-information processing.
Name of the speaker: Dr. Suddhasatta Mahapatra, IIT Bombay
Title: Quantum Computing with Electron Spins in Silicon
Abstract: The spin states of electrons represent a promising two-level-system for realization of a scalable quantum computing architecture. As the naturally abundant isotope of Silicon (28Si) has zero nuclear-spin , (enriched) Si serves as an ideal solid-state environment to host electron spin qubits, ensuring long coherence and relaxation times. Moreover, the physical implementation of the spin quantum computing architecture relies on the mature CMOS process technology, enabling large scale integration of dense arrays of spin qubits. In the past couple of years, tremendous advances have been made in this field, with demonstration of high-fidelity control, manipulation, and measurement of spin qubits, as well as methods to enable qubit coupling over long-distances. In this talk, starting from the fundamental concepts of spin quantum computing, I will present a brief overview of the prospects and challenges towards development of a scalable architecture.
Name of the speaker: Dr. Stefanos Kourtis, Universite De Sherbrooke
Title: Classical and quantum computations as tensor networks
Abstract: Tensor networks are multilinear-algebra data structures that are finding application in diverse fields of science, from quantum many-body physics to artificial intelligence. I will introduce tensor networks and illustrate how they can be used to represent classical and quantum computations. I will then motivate tensor network algorithms that perform or simulate computations in practice and demonstrate their performance on benchmarks of current interest, such as model counting and quantum circuit simulation. I will close with an outline of ongoing work and an outlook on future directions.
Name of the Speaker: Dr. Aikaterini Mandilara, Nazarbayev University, Kazakhstan
Title: Methods for characterizing multipartite entanglement in pure and mixed states
Abstract: The lecture is going to be divided into 3 parts. In the first part, the problem of characterizing multipartite entanglement is going to be exposed and then the algebraic method of nilpotent operators [1] is going to be presented as a general solution to this problem. The presentation is going to be based on simple, instructive examples. In the second part, the problem of identifying entanglement in mixed multipartite states is going to be first analyzed in a geometric way. On the same setting I will provide a geometric understanding of the best separable approximation, a unique representation providing a clear picture of the entanglement content of a state. Then I will explain the steps of an efficient algorithm [2] for achieving the best separable approximation and present examples which concern open quantum systems and bound entangled states. Finally, in the third part, I will talk about entanglement within continuous quantum variables. I will revise well-known results about entanglement of Gaussian states and then present a method for detecting entanglement on non-Gaussian mixed states [3]. The latter is based on a generalized uncertainty relation that takes into account the non-Gaussianity of a state.
[1] Quantum entanglement via nilpotent polynomial}s, A. Mandilara,V. M. Akulin, A. V. Smilga and L. Viola, Phys. Rev. A 74, 022331 (2006).
[2] Essentially Entangled Component of Multipartite Mixed Quantum States, its Properties and an Efficient Algorithm for its Extraction}, V. M. Akulin, G. A. Kabatyanski and A. Mandilara, Phys. Rev. A 92, 042322 (2015).
[3] Detection of non-Gaussian entangled states with an improved continuous-variable separability criterion}, A. Hertz, E. Karpov, A. Mandilara, N. J. Cerf, Phys. Rev. A 93, 032330 (2016).
Name of the Speaker: Prof. Yueh-Nan Chen, National Cheng Kung University, Taiwan
Title: Benchmarking quantum state transfer in the Cloud
Abstract: Quantum state transfer (QST) provides a method to send arbitrary quantum states from one system to another. Such a concept is crucial for transmitting quantum information into the quantum memory, quantum processor, and quantum network. In this talk, I will first introduce the concept of EPR steering. I will then describe the temporal analogue of EPR steering, i.e. temporal quantum steering. For practical applications, I will show that the temporal steerability is preserved when the perfect QST process is successful. Otherwise, it decreases under imperfect QST processes. We then apply the temporal steerability measurement technique to benchmark quantum devices including the IBM quantum experience and QuTech quantum inspire under QST tasks. The experimental results show that the temporal steerability decreases as the circuit depth increases. Moreover, we show that the no-signaling in time condition could be violated because of the intrinsic non-Markovian effect of the devices.
Professor Yueh-Nan Chen received the B.S. and M.S. degrees in Dep. of Electrophysics from National Chiao Tung University, Hsinchu, Taiwan, in 1996 and 1998, respectively. In 2001, he received the Ph.D. degree in Dep. of Electrophysics from National Chiao Tung University. He is a Professor in the Department of Physics at National Cheng-Kung University (NCKU). He is now also the director of Center for Quantum Frontiers of Research & Technology (QFort) at NCKU. His research interests include quantum transport, quantum optics, and quantum information.
Name of the speaker: Prof. Emanuele Dalla Torre, Bar-Ilan University, israel
Title: Quantum simulations with quantum computers on the cloud: Floquet and topology
Abstract: Quantum computing holds the promise to solve specific computational problems much faster than any known classical algorithm. Current quantum computers are, however, too small and too noisy to perform useful calculations. In this talk I will follow a different route and show how to use these systems to study fundamental physical questions, and specifically those related to the dynamics of many-body quantum systems. I will focus on two specific applications of quantum computers on the cloud: the demonstration of the topological property of spin models [1] and the realization of a quantum system with long range interactions [2]. These works raise new basic questions concerning the effects of classical noise on quantum states of matter, and provide a useful benchmark for actual quantum computers.
1. Daniel Azses, Rafael Haenel, Yehuda Naveh, Robert Raussendorf, Eran Sela, Emanuele G. Dalla Torre, Identification of symmetry-protected topological states on noisy quantum computers, Physical Review Letters 125, 120502 (2020)
2. Mor M. Roses, Haggai Landa, Emanuele G. Dalla Torre, Simulating long-range hopping with periodically-driven superconducting qubits, https://arxiv.org/abs/2102.09590
Name of the Speaker: Prof. Salvatore Savasta, University of Messina, Italy
Title: Ultrastrong coupling between light and matter
Abstract: Ultrastrong coupling between light and matter has, in the past decade, transitioned
from a theoretical idea to an experimental reality. In this new regime of quantum light–matter
interaction, beyond weak and strong coupling, the coupling strength
is comparable to the transition frequencies in the system.
Here we review the theory of quantum systems with ultrastrong coupling, discussing entangled ground states with virtual excitations, and new avenues for nonlinear optics. We also overview a subset of the multitude of experimental setups, including superconducting circuits, organic molecules, semiconductor polaritons, and optomechanical systems, that have now achieved ultrastrong coupling. I also discuss recent achievements of the so-called deep strong coupling regime, where the coupling strength becomes larger than the transition frequencies of the system. I conclude by discussing the potential applications enabled by these
Name of the speaker: Prof. Ashwani K. Tiwari, Indian Institute of Science Education and Research Kolkata, Mohanpur 741246
Title: Quantum Dynamics, Wavepacket and Coherent Control
Abstract: Time-dependent Schrödinger equation (TDSE) is the most fundamental equation in the quantum mechanics. My talk will focus on the different techniques to propagate wavepacket using the TDSE. Some examples of coherent control of chemical reactions using wavepacket dynamics will also be discussed.
1. N. Balakrishnan, C. Kalyanaraman, and N. Sathyamurthy, Physics Reports 280, 79 (1997). 2. A. K. Tiwari and N. E. Henriksen J. Chem. Phys., 144, 014306 (2016).
3. A. K. Tiwari and N. E. Henriksen J. Chem. Phys., 141, 204301 (2014).
4. A. K. Tiwari, D. Dey, and N. E. Henriksen Phys. Rev. A, 89, 023417 (2014).
Name of the speaker: Dr. Said Sakhi, American University of Sharjah, USA
Title: Theoretical foundation of Josephson junction dynamics
Abstract: The Josephson junction effect (JJE) is one of the remarkable manifestations of quantum effects in condensed matter physics. It offers the potential to control and to manipulate the macroscopic wave function of a condensate, and it provides a wide variety of stimulating applications in quantum technologies. In this pedagogical talk, after introducing the essential physics of superconductivity, I discuss the theoretical foundation of Josephson dynamics and I highlight the peculiar features of Josephson phenomena using Ginzburg–Landau (GL) theory. This material should facilitate the description of solid state realization of a quantum computer that makes use of superconducting qubits based on Josephson junctions.
Name of the speaker: Prof. Marcin Pawłowski, University of Gdansk, Poland
Title: Information Theoretic Principles of Quantum Mechanics
Abstract: The violation of Bell Inequalities is arguably the strangest property of the quantum theory. It forces us to abandon intuitive principles of either locality or realism and leaves us with a lot of questions: If these principles should not be taken for granted, which should? Can the quantum theory be derived from operational principles in a way similar to relativity? Are some principles better than the others?
In this talk I will try to give the partial answers to these questions. Since deriving all the predictions of quantum theory seems like a monumental task, I will focus on something simpler but still far from trivial – deciding if a given probability distribution could be generated in a quantum experiment of a certain structure. This is what the studies of information theoretic principles aim to achieve.
I will present a few most well known principles and discuss motivation behind each of them, their predictive strengths and drawbacks. I’ll conclude with the latest results in the field.
Name of the speaker: Prof. Dieter Suter, TU Dortmund, Germany,
Title: Quantum information processing with hybrid quantum registers based on individual electronic and nuclear spins
Abstract: The "Digital Revolution" that transformed our lives and our economy is based on the ubiquity of information-processing devices whose processing power increased exponentially for many decades, following Moore's law. As this trend is approaching fundamental physical limits, new directions are explored for even more powerful computational devices based on quantum mechanical systems. Such devices can solve problems that will remain out of reach for conventional (super-)computers. This talk will provide an introduction into a specific physical platform for quantum information processing, which uses individual electronic and nuclear spins in defect centers in diamond and SiC. The combination of different types of qubits allows one to take advantage of the favourable properties of each type but also poses some challenges. We describe the relevant properties of these centers and show how the different degrees of freedom can be controlled effectively and efficiently.
Name of the speaker: Prof. Lorenzo Maccone, University of Pavia, Italy
Title: The four postulates of quantum mechanics are three
Abstract: The tensor product postulate of quantum mechanics states that the Hilbert space of a composite system is the tensor product of the components' Hilbert spaces. All current formalizations of quantum mechanics that do not contain this postulate contain some equivalent postulate or assumption (sometimes hidden). Here we give a natural definition of composite system as a set containing the component systems and show how one can logically derive the tensor product rule from the state postulate and from the measurement postulate. In other words, our paper reduces by one the number of postulates necessary to quantum mechanics.
Name of the speaker: Prof. Vinod Menon, The City College of New York, USA
Title: Light based Hamiltonian simulators
Speaker: Prof. S. Lakshmi Bala, Department of Physics, IIT Madras
Title of talk: What can we learn about the state of light from optical tomograms?
Abstract: Extraction of information from patterns is an important tool in a variety of diverse areas ranging from medical science where images carry details of the scanned object, to linguistics where patterns in the structure of sentences facilitate natural language processing, and provide information on the thoughts behind a string of words. In all these disciplines, much is inferred from the images or patterns, and further detailed
investigations are often proved to be merely corroborative of the lessons learnt from them. This aspect is true of optics as well. The optical tomograms or patterns obtained directly as histograms from the experiment can be used to extract information on nonclassical effects such as squeezing properties, quantum entanglement and revivals of the state of light when it propagates through a nonlinear optical medium. In this talk, I will present some aspects
of optical tomograms, and the message that lies buried in them.
Speaker: Dr. B. Sharmila, Department of Physics, IIT Madras
Title: Tomographic entanglement indicators from an NMR experiment and from the IBM quantum computing platform
Abstract: In this talk, we demonstrate the advantages of the tomographic entanglement indicators in the context of spin and hybrid quantum systems. We use data from an NMR experiment and compare the results with those obtained from performing both experiment and simulation using the IBM quantum computing program. First, the tomographic entanglement indicators from the NMR experiment are shown to agree well with standard entanglement measures calculated from the corresponding density matrices. Further, these indicators compare well with those obtained from the experimental execution and simulation of equivalent circuits corresponding to the NMR experimental set-up, using the IBM quantum computing platform. This exercise is also extended to the case of a hybrid quantum system described by the double Jaynes-Cumming model.
Name of the speaker: Dr. Sai Vinjanampathy, IIT Bombay
Title: Introduction to Variational Quantum Algorithms.
Abstract: Since quantum evolution is difficult to simulate on a classical computer, there is a recent push to use quantum computers to directly solve problems such as estimating the ground states of molecules. I will discuss the basic idea of "variational quantum algorithms" in the context of NISQ devices. I will take the students through "zoom blackboard" lectures that discuss (a) Setting up an ansatz for a quantum state, (b) setting up a representation for Hamiltonians and unitaries, (c) measuring various scalar quantities of interest and (d) some issues with optimization of functionals on NISQ computers that are the topic of discussion in the literature.
Name of the speaker: Dr.Kavita Dorai, IISER Mohali
Title: Quantum Information Processing Using Nuclear Spins as Qubits and Qudits
Abstract: Nuclear magnetic resonance (NMR) quantum computers were one of the first and the most successful quantum technologies to be used as a testbed for quantum information processing. This talk will begin with an overview of using nuclear spins as qubits and qudits for quantum information processing and will detail some of the early successes in the field, including experimental implementations of quantum algorithms and quantum simulation.
The later part of the talk will focus on some of the current challenges that this quantum technology is facing.
Name of the speaker: Dr. Ashok Kumar, IIST Trivandrum
Title: Spatial quantum correlation properties of bright twin beams of light
Abstract: Spatial quantum correlations promise to enhance the sensitivity of quantum imaging and quantum sensing, along with applications in quantum information processing. We will discuss spatial quantum correlation properties of bright twin beams of light generated with a four-wave mixing process in hot rubidium vapour cell. We will start with a general motivation of how the spatial correlation properties led to the famous Einstein-Podolsky-Rosen (EPR) paradox, and then introduce the experimental scheme that we have implemented to realize the EPR paradox with a macroscopically large number of photons. We use an electron-multiplying charge-coupled device camera to record images of the bright twin beams in the near and far field regimes to achieve an apparent violation of the uncertainty principle by more than an order of magnitude, which remains statistically significant even in the limit of a small number of images. We will also present some of our results on how the spatial distribution of cross-correlations of spatial noises of the twin beams can be engineered.
Name of the speaker: Prof. Anirban Pathak, Jaypee Inst. Inf. Tech. Noida, India
Title: The notion of security in the quantum world: advantages, expectations and challenges
Abstract: We will briefly introduce the notion of unconditional security in the context of quantum communication and multiparty computation. Expected advantages of the quantum and semi-quantum schemes over their classical counterparts will be discussed in detail with particular focus on the technological limitations. Specifically, it will be shown that the device imperfections can be exploited to perform quantum hacking.
Name of the speaker: Dr. Swarnamala Sirsi, Yuvaraja’s College
Title: Joint measurability of qubit, qutrit operators
Name of the speaker: Prof. Urbasi Sinha, Raman Research Institute, Bangalore
Title: Exotic world in the foundations of quantum mechanics: Precision experiments and beyond
Abstract: In this talk, I will present an overview on some of the work that is being pursued at the Quantum Information and Computing lab at Raman Research Institute, Bangalore particularly in the domain of experimental photonic quantum science and technologies. A photon being the fundamental unit of light presents itself as an ideal test bed for precision testing of the foundations of quantum theory ranging from phenomena related to interference, superposition, entanglement as well as quantum measurements. Through several experiments over the last decade, we have been probing and utilizing these fundamental phenomena on the one hand towards better understanding of the principles of quantum mechanics and on the other hand towards applying such knowledge in applications ranging from quantum metrology, quantum information processing as well as secure quantum communications. I will attempt to discuss some of this exciting journey in this talk, especially discussing our exciting results in higher dimensional quantum information, new findings related to the Hong-Ou-Mandel effect, newly devised method for precise quantum state estimation which we call quantum state interferography as well as our experiments towards long distance secure quantum communications as a part of our project on quantum experiments using satellite technology.
Name of the speaker: Dr. Debasis Sarkar, University of Culcutta
Title: Local Distinguishability and Indistinguishability of Quantum States- A Brief Introduction
Abstract: Distinguishability of quantum states has an immense importance in quantum information processing. For perfect discrimination, the set of states must be mutually orthogonal. However, for composite systems the situation is quite different. In such cases, it would be preferred to restrict the set of allowable operations to be local in nature (i.e., LOCC). It is really hard to distinguish locally a set of quantum states (entangled or not) shared between a number of parties situated at distant places. Rather, it shows many counter-intuitive results in quantum information theory. It is found that some orthogonal product states are locally indistinguishable. In contrast, there are orthogonal entangled states that are locally distinguishable. The study in this direction was motivated by the discovery of ‘quantum non-locality without entanglement’, which establishes a very strange phenomenon that there are sets of orthogonal product states, not LOCC distinguishable. Quantum nonlocality without entanglement in a multipartite quantum system is still incompletely studied except for some special class of completely orthogonal product bases(COPB) and some unextendible product bases(UPBs). In this lecture, we will try review some of the important results regarding the above issues and lastly we will show our recent results on tripartite systems, a direction to probe multipartite quantum systems.
Name of the speaker: Dr. Ashoka S. Vudaygiri, University of Hyderabad, India
Title & Abstract:
1. Quantum Key Distribution - I will discuss the basics of the QKD
2. Quantum Computation in cold atoms using spin - spin interaction.
Name of the speaker: Prof. Halliwell, Jonathan J, Imperial College London, Singapore
Title: Aspects of Leggett-Garg Tests of Macrorealism
Name of the Speaker: Prof. Sonjoy Majumder, Head, Centre for Theoretical Studies, IIT-Kharagpur
Title: Quantum control of spin-oscillation dynamics of ion-atom mixture using external field
Abstract: The study of trapped ions immersed in ultra-cold atomic gas is one of the most challenging coveted for quantum simulations and quantum computations. Spin dynamics of ions or atoms of the system under external fields can be exploited for quantum gates and circuits. Here we will discuss the light-control mechanism of the spin-exchange process among the ions and atoms under a magnetic field. The spin-oscillation property can predict different properties of the atomic gas once the inter-atomic interaction is considered.
1) M. Tomaza, et al., Rev. Mod. Phys., 91, 035001 (2019)
2) A Bhowmik, N N Dutta, and S Majumder, Phy. Rev. A, 102. 063116 (2020)
Name of the speaker: Dr. Anand Kumar Jha, IIT Kanpur, India
Title: Partial coherence: Applications in quantum state measurement, imaging and communication
Abstract: Fields with quantum correlations are resources to several quantum-information applications as they could be exploited for performing tasks that would otherwise be impossible. One of the major challenges faced in the implementation of several quantum-information protocols is the efficient measurement of quantum states and quantum correlations, especially the high-dimensional quantum states. In this talk, I will present how partial coherence properties could be utilized for efficient measurement of high-dimensional quantum states and correlations. I’ll also present some of our work on the applications of partially coherent light fields for imaging and communication.
Name of the Speaker: Dr. Bhaskar Kanseri, Experimental Quantum Interferometry and Polarization (EQUIP), Department of Physics, Indian Institute of Technology Delhi, Hauz Khas, New Delhi-110016, India , Email:[email protected]
Title: Quantum state engineering using hybrid variable resources for quantum informatics
Abstract: Hybrid variable quantum resources refer to the use of both continuous variable and discrete variable tools available in quantum physics. Both of these resources have been widely explored independently and have found several existing applications in quantum domain. However, since each of them has some benefits (shortcomings) over the other, their use in a complementary or joint manner may be quite advantageous. For instance, using them together, one can generate and characterize more exotic non-classical and non-Gaussian states of light [1]. This talk aims to familiarize students about this quantum optical toolkit and demonstrate the use of these hybrid resources for generation of single photon states and optical Schrödinger’s cat state. The role of cavities in pulsed domain would also be highlighted and we will see that one can employ synchronized pulse optical cavities for second harmonic generation [2], to approximate as an on-demand quantum source [3], for Fock state generation and realization of all cavity Schrödinger cat state for quantum information applications. We finally look forward to highlight some of our recent attempts using this quantum toolkit towards realizing methods for secure quantum communication in free space and optical fibres.
1. J. Etesse, M. Bouillard, B. Kanseri and R. Tualle-Brouri, Phys. Rev. Letts. 114, 193602 (2015)
2. B. Kanseri, M. Bouillard and R. Tualle-Brouri, Opt. Commun. 380, 148 (2016)
3. M. Bouillard, G. Boucher, J. F. Ortas, B. Kanseri, and R. Tualle-Brouri, Opt. Expr. 27, 3113 (2019)
Name of the speaker: Prof. Luiz Davidovich, Universidade Federal do Rio de Janeiro, Brazil
Title: Physics, information, and the new quantum technologies
Name of the Speaker: Prof. Krishna Thyagarajan , Bennett University, India
Title: The quantum nature of light and the Photon
Name of the Speaker: Dr. Debashis Saha,
Title - Introduction to self-testing
Abstract - To realize genuine quantum technology, the back-end user should be ensured that the quantum devices work as specified by the provider. Methods to certify that a quantum device operates in a nonclassical way are therefore needed. Among various certification methods, the most compelling one is self-testing (or blind-tomography). It exploits quantum nonlocal correlations and provides the complete characterization of quantum devices without any assumption on the internal features of the devices. This talk will be an introduction to self-testing.
Name of the speaker: Dr. Sankar De, Saha Institute of Nuclear Physics, HBNI, Kolkata, India.
Title: Electromagnetically Induced Transparency: Quantum memory and Atomic magnetometry
Abstract: The light-atom interaction is one of the key research areas in the present time due to its vast applications in the various fields along with its fundamental interest. Using quantum optical methods, one can control the properties of an atomic medium with lasers and therefore can create a new medium with distinctive characteristics. Atoms can be prepared in a coherent superposition of energy states under the interaction of two or more laser fields which are resonant with various atomic transitions. Among these, in a three-level atomic system, electromagnetically induced transparency (EIT) occurs when a strong control or pump laser induces a narrow spectral transparency window at a highly absorbing atomic resonance for a weak probe laser beam by creating coherence between the relevant atomic states. As a result, the properties of the atomic vapour are changed dramatically and in the vicinity of EIT, the medium becomes very dispersive. This leads to interesting phenomena culminating into observation of slow light and storage and precision atomic magnetometers. I shall briefly discuss these developments in my talk.
Name of the Speaker: Prof. Guruprasad Kar, Physics and Applied Mathematics Unit, Indian Statistical Institute
Title: Understanding Quantum Nonlocality
Abstract: Quantum mechanics has no contradiction with principle of relativity implying it is consistent with no-signaling condition. J.S. Bell proved that quantum mechanical correlations do not necessarily satisfy local realistic condition (put forward by Einstein) and quantum mechanics is nonlocal only in this sense. The issue of quantum nonlocality will be discussed by invoking some simple multipartite games.
Name of the speaker: Dr. Ashutosh Rai, Researcher, Slovak Academy of Sciences, Bratislava
Title: Communication Cost of Simulating Entanglement
Abstract: Say, two non communicating and space-like separated parties Alice and Bob share some entangled state, and perform on their respective parts some local measurement chosen randomly from their set of incompatible measurements. Then it is well known that non-classical correlations can result in the joint probability distribution of the outcomes of Alice and Bob. Such non-classical feature is witnessed by violation of some Bell-type inequality. Entanglement is a necessary condition for witnessing this non-classical effect. Then question of interest is to ask what is the minimal communication cost of simulating (non-classical) quantum correlations generated by some given states and measurements. In this talk, I plan to present on this topic and outline some open problems on the topic.
Name of the speaker: Prof. Sivakumar Srinivasan, Krea University
Title: Jaynes-Cummings-model and circuit QED.
Name of the speaker: Dr. Manabendra Nath Bera, IISER Mohali
Title: Quantum Heat Engines with Carnot Efficiency at Maximum Power
Abstract: Conventional heat engines, be these classical or quantum, with higher power yield lesser efficiency and vice versa and respect various power-efficiency trade-off relations. Here we show that these relations are not fundamental. We introduce quantum heat engines that deliver maximum power with Carnot efficiency in the one-shot finite-size regime. These engines are composed of working systems with a finite number of quantum particles and are restricted to one-shot measurements. The engines operate in a one-step cycle by letting the working system simultaneously interact with hot and cold baths via semi-local thermal operations. By allowing quantum entanglement between its constituents and, thereby, a coherent transfer of heat from hot to cold baths, the engine implements the fastest possible reversible state transformation in each cycle, resulting in maximum power and Carnot efficiency. We propose a physically realizable engine using quantum optical systems.
Name of the speaker: Anant V. Varma, IISER Kolkata
Title: Simulating non-Hermitian dynamics of a multi-spin quantum system and an emergent central spin model.
Abstract: In recent times there has been much discussion of non-Hermitian quantum systems in the context of many-body systems owing to the exotic manifestations like a violation of Lieb-Robinson bound (PRL 124,136802 (2020)), non-Hermitian skin effect (PRL 121, 086803 (2018)), suppression of defect production in Kibbel-Zurek mechanism (Nature Comm. 10, 2254 (2019)) and correspondence between (d+1) dimensional gapped Hermitian systems and d-dimensional point-gapped non-Hermitian systems (PRL 123, 206404 (2019)). Hence a possibility of simulating such a system that will facilitate the direct observation of such phenomena could be of great importance. It is known that the dynamics of a single spin-1/2 PT-symmetric system can be simulated by conveniently embedding it into a subspace of a larger Hilbert space with unitary dynamics. In the context of many-body physics, what would be the consequence of the complexity of such ideas of embedding non-Hermitian many-body systems in unknown. We show that such an embedding leads to non-trivial Hamiltonian which has complex interactions. We consider a simple example of N free PT-symmetric spin-1/2s to obtain the resulting many-body interacting Hamiltonian of N+1 spin-1/2s. We can visualize it as a strongly correlated central spin model with the additional spin-1/2 playing the role of central spin. We would show that due to the orthogonality catastrophe, even a vanishing small exchange field applied along the anisotropy axis of the central spin leads to a strong suppression of its decoherence arising from spin-flipping perturbations.
This talk would be based on the following paper: Anant V. Varma, and Sourin Das, Simulating non-Hermitian dynamics of a multi-spin quantum system and an emergent central spin model” arXiv:2012.13415 (communicated to Phys. Rev. B) .
Speaker: Dr. Ayan Khan, Bennett University
Title: Understanding some Exhotic Phases of Matter and Their Implication in Quantum Technology
Session: Quantum Chemistry & Applications to Condensed Matter Physics
Abstract: Ultra-cold atomic gases are considered as the ultimate testing ground for condensed matter physics theories as one can engineer the dynamics of these extremely cold atoms quite precisely and effectively. Of late, several new experimental evidences have emerged, inspiring us to relook at the conventional understanding of liquid and solid. In this lecture, we plan to introduce a couple of exciting phases which have been observed very recently. Among them, the quantum droplet is the liquid-like state which does not follow the conventional Van der Waals theory. The other entity, known as supersolid, is as unique as its name suggests. It is a spatially ordered material with superfluid properties. An introduction to these phases will enable us to comment on their implications in quantum technology.
Name of the speaker: Dr. P. D. Duraga Nandini, Pune University
Title: Factorization, coherence and asymmetry in the Heisenberg spin-1/2 XXZ chain with Dzyaloshinskii-Moriya interaction
Abstract: A certain class of quantum phase transitions(QPT) are associated with the intriguing property of the existence of a non-trivial 'factorizability' property, i.e., the quantum state becomes completely separable at certain parameter strengths which serve as precursors signalling the existence of a QPT associated with an entanglement transition(ET); there is a crossover from one type of entanglement to another across the factorizing field. The most notable example is that of the Heisenberg spin S= 1/2 chain where it was shown several years ago that a factorizable ground state emerges at a certain value of the external magnetic field. We address here the question of the effect of Dzyaloshinskii-Moriya interaction (DMI) on the factorization, coherence and asymmetry properties of the ground state. We compute using numerical DMRG, various bipartite entanglement and coherence estimators like the one-tangle, two-spin-concurrence and Wigner-Yanase skew information measure. We show that a longitudinal DMI destroys the factorizability property(which physically manifests in the existence of a non-zero chiral spin current) whilea transverse DMI preserves it. We relate the presence(absence) of factorizability to the presence(breaking) of the $U(1)$ rotation symmetry about the local magnetization axis at each lattice site. We show that although the longitudinal DMI destroys the factorization property, there is a 'pseudofactorizing' field at which the violation of the $U(1)$ symmetry is minimal. An entanglement crossover occurs across this field which is characterized by an enhanced but finite range of two-spin concurrence in its vicinity in contrast with the diverging range of the concurrence for the ET across the factorizing field. We discuss also the relation of the asymmetry to the 'frameness' or the abililty to specify a full reference frame for the many body state.
Pradeep Thakur and P. Durganandini, Phys.Rev. B 102, 064409 (2020); Pradeep Thakur and P. Durganandini, 2021
Name of the speaker: Prof. Binayak S. Choudhury, Department of Mathematics, IEST, Shibpur
Name of the speaker: Prof. Tabish Qureshi, Centre for Theoretical Physics, J.M.I., New Delhi.
Title: Wave-Particle Duality and the Quantum Eraser
Abstract: Wave and particle natures are two complementary aspects of quantum objects, and are believed to be mutually orthogonal. In interference experiments, which are the testbed of wave-particle duality, the information about which of the different possible ways, a particle followed, construes its particle nature. How good is the interference that the particle shows, construes its wave nature. If the "which-way" information about a particle exists, the interference is lost. An interesting concept is that if the potential "which-way" information, stored in a quantum device, is erased, the interference, which was lost, can come back. This phenomenon is called "quantum erasure." There has been a long standing debate on the meaning of quantum erasure, and whether the choice regarding erasing the which-way information can be made after the particle has already hit the screen. This would imply that the particle can be "retrocausally" forced to behave like a particle or a wave, much after it has hit the screen. These concepts will be explained and clarified in the talk.
Name of the Speaker: Prof. Aditi Sen De, Harish - Chandra Research Institute
Title: Quantum Communication (without security)
Abstract: The quantum theory of nature, formalized in the first few decades of the 20th century, contains elements that are fundamentally different from those required in the classical description of nature. Based on the laws of quantum mechanics, in recent years, several discoveries have been reported which can revolutionize the way we think about modern technologies. I will talk about such inventions in the field of communication and some of the recent results towards building communication network.
Name of the speaker: Prof. Chiranjib Mitra, IISER Kolkata
Title: Experimental quantification of entanglement in low dimensional spin systems
Abstract: We report the macroscopic entanglement properties of a low dimensional quantum spin system by investigating its magnetic properties at low temperatures and high magnetic fields. The temperature and magnetic field dependence of entanglement from the susceptibility and magnetization data is performed and comparison is made with corresponding theoretical estimates. Extraction of entanglement has been made possible through the macroscopic witness operator, magnetic susceptibility and heat capacity. The spin systems studied exhibit quantum phase transition (QPT) at low temperatures, when the magnetic field is swept through a critical value. We show explicitly, using tools used in quantum information processing (QIP), that quantum phase transition (QPT) can be captured experimentally using quantum complementary observables. Entanglement properties of the same quantum spin systems when investigated by heat capacity measurements also capture the QPT.
Name of the Speaker: Prof. Dipankar Home, Bose Institute, India
Title: Guaranteeing the Certainty of Randomness: Interface with No-signalling and Nonlocality
Abstract: Randomness is a fundamental feature of nature, and a key resource for myriad applications in diverse areas of physical and biological sciences, including, in particular, communication and cryptography. For such applications, certifying and quantifying Genuine Randomness (GR) is a crucial issue, which requires true unpredictability to be guaranteed in the presence of uncontrollable imperfections, and even if there is adversarial tampering of the random number generating device. The present talk will focus on this specific issue.
The talk will begin by pointing out the fundamental inadequacies of the currently available random number generating devices. Next, it will be explained how the argument based on solely the fundamental physical principle of No-signalling provides a way forward by enabling the use of nonlocal correlations embodied in quantum entanglement for certifying GR in the device-independent scenario. This will be illustrated with respect to both the Bell-CHSH inequality, as well as the Hardy and the Cabello-Liang relations.
In conclusion, the discussion will be confined to outlining the methods adopted for quantifying such certified GR in terms of the guaranteed minimum and maximum achievable bounds of GR, thereby giving a broad idea of how the key relevant results are obtained. For further details regarding this line of studies which have revealed a number of significant features of the relationship between the amounts of randomness, nonlocality and entanglement, one may look at the following mentioned works [1,2,3]:
1. S. Pironio, A. Acin, S. Massar, A. B. de la Giroday, D. N. Vaskevitch, P. Maunz et al., Random numbers certified by Bell’s theorem, Nature 464 (2010) 1021–1024.
2. A. Acin, S. Massar and S. Pironio, Randomness versus nonlocality and entanglement, Phys. Rev. Lett. 108 (2012) 100402.
3. S. Sasmal, S. Gangopadhyay, A. Rai, D. Home and U. Sinha, Genuine randomness vis-a-vis nonlocality: Hardy and Hardy type relations, arXiv:2011.12518 (2020).
Name of the speaker: Prof. Cyril BRANCIARD, French National Centre for Scientific Research
Title: Quantum indefinite causal relations
Abstract: Quantum theory allows for processes where events happen in some indefinite causal order. A new field has emerged in the last decade, that aims at investigating the kind of indefinite causal relations that can be found in the quantum world, and at looking for potential applications. I will give an overview of this new domain, and present some of the latest results in the area.
Name of the Speaker: Prof. Archan S. Majumdar, SNBCBS
Title : Single shot quantum correlations shared by multiple observers
Abstract: We explore the possibility of sharing of quantum correlations in single copies of two- and three-qubits by multiple parties. Various types of nonlocal quantum correlations, such as Bell-CHSH nonlocality, quantum steering, and entanglement detection are considered. We find the upper bound on the number of sequential observers who can share the above different kinds of correlations. This opens up the possibility of resource efficient quantum information processing involving multiple parties without having to create and preserve either multipartite entangled states, or many copies of bipartite entangled states.
Name of the speaker: Dr. Neetik Mukherjee, Department of Chemical Sciences, IISER Kolkata
Title: Confined quantum systems and Information entropy in Chemistry
Abstract: Since its inception, confined quantum systems has emerged as a subject of topical interest. In such stressed environment, the rearrangement of atomic orbitals leads to increase in coordination number. Atoms, molecules confined under cavities of varying size and shape, exhibit distinct fascinating changes in their physical and chemical properties from their free counterpart. Atoms under high pressure were first studied as early as 1937. Such a situation can be modeled by shifting the spatial boundary from infinity to a certain finite region. Depending upon the capacity of pressure one can simulate them by invoking two broad categories of confining potentials, impenetrable (hard) and penetrable (soft). A new virial-like theorem has been proposed for these systems. In recent years, appreciable attention was paid to investigate various information measures, namely, Fisher information (I), Shannon entropy (S), R enyi entropy (R), Tsallis entropy (T), Onicescu energy (E), and several complexities in a multitude of physical and chemical systems including central potentials. In confined condition S and E has been successfully used to uncover the effect of confinement on Compton profiles (CP). A deeply bound electron has a very flat and broad momentum distribution. As a consequence, its CP is also broad. This broadness of distribution can be quantified by S, E. Hence, these measures can act as descriptors about the bound effect on an electron within a quantum system. Current study will convincingly establish this interpretation.
1. Analysis of Compton profile through information theory in H-like atoms inside impenetrable sphere, Neetik Mukherjee and Amlan K. Roy, J Phys. B 53, 253002, (2020).
2. A quantum mechanical virial-like theorem for confined quantum systems Neetik Mukherjee and Amlan K. Roy, Phys. Rev. A 99, 022123 (2019).
3. Information-entropic measures in free and confined hydrogen atom, Neetik Mukherjee and Amlan K. Roy, Int. J. Quant. Chem. 118 e25596 (2018).
4. Quantum confinement in an asymmetric double-well potential through energy analysis and information entropic measure. Neetik Mukherjee and Amlan K. Roy, Ann. Phys. (Berlin) 528 412 (2016).
5. Information entropy as a measure of tunneling and quantum confinement in a symmetric double-well potential. Neetik Mukherjee , Arunesh Roy and Amlan K. Roy, Ann. Phys. (Berlin) 527 825 (2015).
Name of the Speaker: Dr. Ritabrata Sengupta, IISER Berhampur
Title: Quantum channels, PPT channels, and all that
Name of the Speaker: Dr. Arpita Maitra
Title: Quantum Supremacy and Its Implementation in Likelihood Theory with Quantum Coins and Computers
Abstract: In this talk, we discuss quantum supremacy, controversy with the term "supremacy" and its application in several fields like computation, communication, and security. Quantum supremacy comes in different forms; as processors/circuits, as entanglement, and as no cloning theorem. In the present talk, we show how entanglement provides an advantage in quantum likelihood theory to distinguish two density matrices. We also show how we implemented this theoretical result in IBM quantum computers. Our experimental results showed that, whereas the theory matches with the simulation, it differs significantly when we run the programme in actual quantum processors. Finally, we discuss the future research avenues in this direction.
Name of the Speaker: Prof. Supurna Sinha, RRI Bangalore
Title: Entropy and Geometry of Quantum States
Name of the Speaker: Dr. Utpal Roy, IIT Patna
Title: Bose-Einstein condensate: Quantum Simulation and Quantum Information
Name of the Speaker: Prof. Dr. Apoorva D Patel, IISC Bangalore, India
Title: Two Uses of the Density Matrix: Understanding Quantum Chaos and Quantum Machine Learning Kernel
Abstract: The density matrix generalises the concept of probability distribution to quantum theory. That offers a new perspective on many classical problems. (1) In classical physics, chaos is characterised as rapid divergence of evolution trajectories that are infinitesimally separated to begin with. This definition does not directly apply to the quantum case because the overlap of two quantum states is invariant under unitary evolution. A phase space evolution scenario can get around this problem and help us understand quantum chaos. (2) Classification of data is a basic problem in machine learning. In supervised learning, the algorithm first determines the variational parameters that best separate the data, using known training datapoints. It can then predict the class labels of a new unknown datapoint. The kernel represents the overlap of states associated with the datapoints, and it can be chosen as the Hilbert-Schmidt inner product for convenient quantum processing and state discrimination.
Name of the Speaker: Prof. Dr. MS Santhanam, IISER Pune, India
Title: Quantum entanglement and chaotic systems
Abstract: This talk will introduce how classical chaos affects quantum entanglement. It will cover some of the basic ideas and discuss some recent developments.
Name of the Speaker: Prof. Dr. Ozawa Masano, Chubu University, Japan
Title: Soundness and completeness of quantum root-mean-square errors
Abstract: Quantifying the error of a measurement is fundamental to experimental sciences. In the classical physics, the root-mean-square (rms) error has been used for the standard error measure. The rms of the noise-operator, called the noise operator based error measure, has been used in quantum physics as a quantum counterpart of the classical rms error. The noise operator based error measure satisfies the following requirements: (I) operational definability (to be definable by the POVM, the measured observable, and the state); (ii) correspondence principle (to coincide with the classical rms error if the POVM and the observable commute); and (iii) soundness (to vanish for accurate measurements). However, it fails to satisfy (iv) completeness (to vanish only for accurate measurements), if the POVM and the observable do not commute. We discuss how to modify the noise operator based error measure to satisfy all requirements (i)--(iv). We obtain an error measure that satisfies (i)--(iv), and moreover, it is shown that the new error measure maintains the previously derived universally valid uncertainty relations and their experimental confirmations without changing their forms and interpretations, in contrast to a prevailing view that a state-dependent formulation for measurement uncertainty relation is not tenable. This talk is based on [M. Ozawa, Soundness and completeness of quantum root-mean-square errors, npj Quantum Inf. 5, 1 (2019)].
Name of the Speaker: Prof. Dr. Prof. Marek Zukowski, University of Gdańsk, Poland
Title: Physics and Metaphysics of Wigner's Friends
Name of the Speaker: Prof. Sivakumar Srinivasan, KREA University
Title: Jaynes-Cummings model and qubits
Name of the Speaker: Debasish Parida & Uday Singla, IISER Kolkata. BITS Pilani
Title: Quantum Simulation of the Fermionic systems. (An introduction to Quantum Information in Quantum Chemistry.)
Name of the Speaker: Dr. Kumar Abhinav, Nakhon Sawan Studiorum for Advanced Studies - NAS, Mahidol University, Nakhon Sawan 60130, Thailand.
Title: PT-symmetric and Pseudo-Hermitian Systems: Hilbert space, scattering and other aspects
Abstract: PT-symmetric systems have been at the forefront for more than a couple of decades, enveloping real physical systems beyond their Hermitian counterparts. Belonging to the larger class of pseudo-Hermitian systems, they support both real and complex-conjugate eigenvalues characterized by certain parametric phases. In this talk, we look into the vector space and conservation laws for these systems and find novel results that should be observable. Further, recent studies about field-theoretic applications of such systems are mentioned.
Name of the Speaker: Dr. Rangeet Bhattacharyya, IISER Kolkata
Title: Quantum computation in open quantum systems: optimality of clock speeds
Abstract: We have recently demonstrated that quantum master equations could be extended to include the non-linear and dissipative terms from the drive acting on open quantum systems. We have also experimentally verified the theoretical predictions of drive-induced dissipation. In this talk, we show that such terms along with the system-bath interactions give rise to an optimal condition on qubit gate fidelity. We argue that the qubit gates have maximum fidelity only for a specific range of drive values; too weak or too strong a drive results in poor performance of a quantum circuit due to lower fidelity. We also demonstrate the universality of the results for the gate operations on single- and multiple-qubit gates.
Name of the Speaker: Dr. Sanjib Dey, IISER Mohali
Title: Resources of quantum information theories with PT-symmetry and cavity optomechanics
Abstract: Studies on nonclassicality, entanglement and decoherence of quantum systems are some of the key areas of research in quantum information science. Analysis and development of such features based on resource theoretical frameworks have unveiled a new avenue in the last decade. Quantum resource theory is perhaps the most revolutionary framework that quantum physics has ever experienced. It plays vigorous roles in unifying the quantification methods of a requisite quantum effect as wells as in identifying protocols that optimize its usefulness in a given application in areas ranging from quantum information to computation. Moreover, the resource theories have transmuted radical quantum phenomena like coherence, nonclassicality and entanglement from being just intriguing to being helpful in executing realistic tasks. Along with the rapid growth of various resource theories corresponding to standard quantum optical states, significant advancement has been expedited along the same direction for generalized quantum optical states. Generalized quantum optical framework strives to bring in several prosperous contemporary ideas including nonlinearity, PT-symmetric non-Hermitian theories, etc., to accomplish similar but elevated objectives of the standard quantum optics and information theories. In this talk, I will discuss our recent developments in the given context and their usefulness in the areas of quantum information theories. Certain remarkable features of quantum optomechanics within the field will also be discussed alongside. More specifically, I will come up with realistic and experimental ideas of cavity optomechanics to generate resourceful states and their utilization in different areas of quantum information theory.
Name of the Speaker: Dr. Shrobona Bagchi, Tel Aviv University Israel
Title : IID and problem specific samples of quantum states from Wishart Distributions
Abstract: Random samples of quantum states are an important resource for various tasks in quantum information science, and samples in accordance with a problem-specific distribution can be indispensable ingredients. Some algorithms generate random samples by a lottery that follows certain rules and yield samples from the set of distributions that the lottery can access. Other algorithms, which use random walks in the state space like the Monte Carlo, can be tailored to any distribution, at the price of autocorrelations in the sample and with restrictions to low-dimensional systems in practical implementations. We present a two-step algorithm for sampling from the quantum state space that overcomes some of these limitations. We first produce a CPU-cheap large proposal sample, of uncorrelated entries, by drawing from the family of complex Wishart distributions, and then reject or accept the entries in the proposal sample such that the accepted sample is strictly in accordance with the target distribution. We establish the explicit form of the induced Wishart distribution for quantum states. This enables us to generate a proposal sample that mimics the target distribution and, therefore, the efficiency of the algorithm, measured by the acceptance rate, can be many orders of magnitude larger than that for a uniform sample as the proposal. We demonstrate that this sampling algorithm is very efficient for one-qubit and two-qubit states, and reasonably efficient for three-qubit states, while it suffers from the "curse of dimensionality" when sampling from structured distributions of four-qubit states.
Name of the Speaker: Dr. Stephan Sponar, TU Wien, Atominstitut - Institute of Atomic & Subatomic Physics
Title: Quantum measurements - Theory and Experiment
Abstract: The uncertainty principle is an important tenet and active field of research in quantum physics. Information-theoretic uncertainty relations, formulated using entropies, provide one approach to quantifying the extent to which two non-commuting observables can be jointly measured. Recent theo- retical analysis predicts that general quantum measurements (i.e. positive-operator valued measures) are necessary to saturate certain uncertainty relations and thereby overcome certain limitations of projec- tive measurements. Here, we experimentally test a tight information-theoretic measurement uncertainty relation with neutron spin-1/2 qubits.
Name of the Speaker: Prof. Dipankar Home, Bose Institute
Name of the Speaker: Dr. Subhadeep De, Inter-University Centre for Astronomy and Astrophysics (IUCAA)
Title: Optical Clocks: An Indispensable Tool for Quantum Metrology & quantum Enabled Technology
Abstract: The optical atomic clock measures the “highly forbidden” atomic transition frequencies (clock transitions) in the optical domain with unprecedented accuracies. Neutral atoms stored in an optical lattice and a single atomic ion confined in an electrodynamic trap are the two most favorable approaches for building optical clocks. Apart from their use for accurate time-keeping, they are one of the most useful tools to hunt answers for several open science questions. Additional to the optical clocks, long-distance transfer of the phase preserved optical photons allows intercomparison of the geographically distributed clocks, which allows hunting for new physics such as the constancy of the dimensionless fundamental constants, violation of the fundamental symmetries, geodetic measurements, and so on. At the upcoming Precision & Quantum Measurement laboratory (PQM-lab: https://pqmlab.iucaa.in), IUCAA we are building a ytterbium-ion optical clock, which will be used to pursue quantum metrology, precision measurements, and developing quantum technologies. In this lecture, I shall focus on some of these science goals that we intend to pursue and recent development in the lab.
Name of the Speaker: Prof. N D Chavda, Department of Applied Physics, Faculty of Technology & Engineering, The Maharaja Sayajirao University of Baroda
Title: Entanglement in Interacting Particle Systems
Abstract: In the present work, we study entanglement entropy in interacting particle systems modeled by embedded one- plus two-body random matrix ensembles for both fermion and boson systems. Also, participation ratio is studied and its correlations with entanglement entropy are analyzed. The results are consistent with those obtained using Bose-Hubbard model and spin models.
Name of the Speaker: Dr. K. G. Paulson, Institute of Physics (IOP)-Bhubaneswar, India
Title: Speed of evolution of open quantum systems
Abstract: Quantum speed limit time defines the bound on the minimum time required for a quantum system to evolve between two states. It finds wide applications in various research fields such as quantum information processing, quantum computing, and quantum thermodynamics. Investigation of bounds on the speed of evolution of the system in open quantum dynamics is of fundamental interest, as it reveals the nature of the interaction between a quantum system and a bath. The behaviour of quantum speed limit time for initial pure and mixed states is investigated, and its connections with various channel properties are established. Dynamics of quantum correlation (QC) under (non-) Markovian dynamics are well studied; we discuss the relationship between quantum speed limit time and dynamics of QC under CP-(in) divisible unital and non-unital channels.
Name of the Speaker: Sangita Majumdar, IISER Kolkata, India
Title: Energy and information analysis of spatially confined atoms through Density functional theory
Abstract: Atom trapped inside a cavity introduces fascinating changes in the observable properties. While an atom confined by rigid walls is a simple model to study electrons restricted to small regions, penetrable walls are more convenient to contrast the corresponding results with an experimental counterpart. Here we present few exploratory results obtained from a newly proposed DFT-based method in our laboratory to address such confinement in atoms. The radial Kohn-Sham (KS) equation is solved invoking a physically motivated non variational, work-function-based exchange potential, along with a simple parametrized local Wigner functional and a nonlinear, gradient- and Laplacian-dependent functional (LYP). GPS method is used to construct an optimized non-uniformly discretized spatial gridfor solving the KS equation. Preliminary results are presented for both ground and excited states of atoms and ions enclosed within impenetrable and penetrable cages. This includes external potential in the form of harmonic confinement, atom/ion embedded in a fullerene cage. The exchange-only results are practically of Hartree-Fock quality. The interplay between ordering and crossing of states as functions of cavity radius is analyzed by constructing a traditional correlation diagram. Study of such two-electron harmonium atom inside spherical cavity is pursued to conclude whether this crossing behaviour is unique to the one-electron coulomb potential or it can be observed in other single-particleconfining potential.
Name of the Speaker: Prof. Ray-Kuang Lee, National Tsing Hua University, Taiwan
Title: Simulating non-Hermitian quantum systems by dilations
Abstract: Despite the initial motivation to establish an alternative framework of quantum theory, we can also take PT-symmetric systems as effective descriptions of large Hermitian systems in some subspaces [1, 2]. By using the Naimark dilation theorem, one can always find some four-dimensional Hermitian Hamiltonians to effectively realize two-dimensional unbroken PT-symmetric systems [3]. Then, passive PT-symmetric couplers can thus be implemented with a refractive index of real values and asymmetric coupling coefficients. This opens up the possibility to implement general PT-symmetric systems with state-of-the-art asymmetric slab waveguides, dissimilar optical fibers, or cavities with chiral mirrors [4]. As for the broken PT-symmetry, we disclose the relations between PT-symmetric quantum theory and weak measurement theory by embedding a PT-symmetric (pseudo-Hermitian) system into a large Hermitian one [5]. However, with only a global Hermitian Hamiltonian, how do we know whether it is a dilation and is useful for simulation? To answer this question, we consider the problem of how to extract the internal nonlocality in the Hermitian dilation. We unveil that the internal nonlocality brings nontrivial correlations between the subsystems. By evaluating the correlations with local measurements in three different pictures, the resulting different expectations of the Bell operator reveal the distinction of the internal nonlocality, which provides the figure of merit to test the reliability of the simulation, as well as to verify a PT-symmetric (sub)system [6].
[1] Yi-Chan Lee, Min-Hsiu Hsieh, Steven T. Flemmia, and RKL, "Local PT symmetry violates the no-signaling principle," Phys. Rev. Lett. 112, 130404 (2014); Editors' Suggestion; Featured in Physics: Reflecting on an Alternative Quantum Theory.
[2] Ludmila Praxmeyer, Popo Yang, and RKL, "Phase-space representation of a non-Hermitian system with PT-symmetry," Phys. Rev. A 93, 042122 (2016).
[3] Minyi Huang, RKL, and Junde Wu, "Manifestation of Superposition and Coherence in PT-symmetry through the $\eta$-inner Product," J. Phys. A: Math. Theor. 51, 414004 (2018).
[4] Yi-Chan Lee, Jibing Liu, You-Lin Chuang, Min-Hsiu Hsieh, and RKL, "Passive PT-symmetric couplers without complex optical potentials," Phys. Rev. A 92, 053815 (2015).
[5] Minyi Huang, RKL, Lijian Zhang, Shao-Ming Fei, and Junde Wu, "Simulating broken PT-symmetric Hamiltonian systems by weak measurement," Phys. Rev. Lett. 123, 080404 (2019).
[6] Minyi Huang, RKL, and Junde Wu, "Extracting the internal non-locality from the dilated Hermiticity," Phys. Rev. A (in press, 2021); [arXiv: 2009.06121].
Name of the Speaker: Athira B S, IISER Kolkata, India
Title: Interferometric Weak measurement
Abstract: The weak value amplification concept, introduced by Aharonov, Albert, and Vaidman, has proven to be fundamentally important and extremely useful for numerous metrological applications. This quantum mechanical concept can be understood using the wave interference phenomena and can therefore be realized in classical optical settings also. The weak value amplification concept can be formulated within the realm of classical electromagnetic theory of light. In this regard, our recent experimental work on the realization of the weak value of polarization observable by introducing a weak coupling between the path degree of freedom of an interferometer and the polarization degree of freedom of light will be presented. There is an upper bound on the maximum achievable weak value amplification in the conventional linear response regime of weak measurements. On the conceptual ground, it is also equally important to understand the weak values through physically meaningful and experimentally accessible properties such as the system response function. In an attempt to address these issues, we have demonstrated a fundamental relationship between the weak value of an observable and complex zero of the response function of a system by employing weak measurement on spin Hall effect of a Gaussian light beam.
Name of the Speaker: Prof. Aranya B Bhattacherjee, Department of Physics, Birla Institute of Technology and Science, Pilani,Hyderabad Campus, India
Title: Quantum Entanglement in Hybrid Systems
Abstract: Entanglement is one of the important elements of quantum mechanics as it is responsible for correlations between observables. I will first introduce some basic facts about entanglement and then go on to discuss some fundamental optomechanics. I will then discuss some hybrid optomechanical schemes where continuous variable entanglement can be realized between any two chosen degrees
of freedom. Such entanglement can lead to quantum state transfer between light and mechanical oscillators. In fact, entangled optomechanical systems have
potential profitable application in realizing quantum communication networks, in which the mechanical modes play the vital role of local nodes where quantum information can be stored and retrieved while optical modes carry this information between the nodes. In between I will show some recent experimental results.
Name of the Speaker: Bhallamudi Vidya Praveen, IIT Madras, India
Title: Making use of relaxation of spins for spectroscopy and sensing
Abstract: Spins are being extensively pursued for quantum enhanced technologies. This follows long-standing work in magnetic /spin resonance where techniques for controlling and manipulating spins were extensively developed, leading to successful applications in the form of spectroscopic characterization tools for the lab and a medical diagnostic tool in the form of Magnetic Resonance Imaging. I will introduce these topics and then focus on how spin relaxation, which is generally considered a bane for quantum technologies, for performing sensitive spectroscopic measurements. In particular, I will look at the use of quantum defects in diamond for such applications.
Name of the Speaker: Prof. Sibasish Ghosh, IMSc, Chennai
Title: Universal schemes for detecting entanglement in two-mode Gaussian states: Stokes-like operator based approach
Abstract:-- Detection of entanglement in quantum states is one of the most important problems in quantum information processing. However, it is one of the most challenging tasks to find a universal scheme which is also desired to be optimal to detect entanglement for all states of a specific class – as always preferred by experimentalist. Although, the topic is well studied, at least in the case of lower dimensional compound systems (e.g., two-qubit systems), but in the case of continuous variable systems, this remains as an open problem. Even in the case of two-mode Gaussian states, the problem is not fully resolved. In our work, we have tried to address this issue. At first, a limited number of Hermitian operators is given to test the necessary and sufficient criterion on the covariance matrix of separable two-mode Gaussian states. Thereafter, we present an interferometric scheme to test the same separability criterion in which the measurements are being done via
Stokes-like operators. In such case, we consider only single-copy measurements on a two-mode Gaussian state at a time and the scheme amounts to the full state tomography. Although this latter approach is a linear optics based one, nevertheless it is not an economic scheme. Resource-wise a more economical scheme than the full state tomography can be obtained if we consider measurements on two copies of the two-mode Gaussian state at a time. However, optimality of the scheme is not yet known.
Name of the Speaker: Prof. Pijushkanti Ghosh, Visva Bharati University
Title: Pseudo-hermitian quantum systems: Construction, Solvability, Supersymmetry
Abstract: After reviewing the basic ideas on PT-symmetric systems, mathematical formulation of
pseudo-hermitian system will be presented at an elementary level. In general, finding the operators relevant for defining the modified norm in the Hilbert space of a given quantum system is a non-trivial task for both PT-symmetric as well as pseudo-hermitian systems. Pseudo-hermitian systems with a pre-determined metric in the Hilbert space may be constructed from the known quantum systems via Dyson mapping. The construction of a few examples will be presented, which include pseudo-hermitian
Jaynes-Cummings model, Dicke model, transverse Ising model in one dimension, quadratic form of bosonic operators. The construction of pseudo-hermitian supersymmetric system will be presented.
Name of the Speaker: Dr. Ananya Ghatak, University of Amsterdam, Netherlands
Title: Observation of novel bulk-edge correspondence in non-Hermitian metamaterials
Abstract: Topological edge modes are excitations that are localized at the materials’ edges and yet are characterized by a topological invariant defined in the bulk. Recently, the advent of non-Hermitian topological systems—wherein energy is not conserved—has sparked considerable theoretical advances. In particular, novel topological phases that can only exist in non-Hermitian systems have been introduced. We will discuss an experimentally observed novel form of bulk-edge correspondence for non-Hermitian topological phases. It shows, a change in the bulk non-Hermitian topological invariant corresponds to a
change of localization of the topological edge mode. With quantum-to- classical analogy we create a mechanical metamaterial with non-reciprocal
interactions, in which our predicted bulk-edge correspondence has been observed experimentally and witnesses its robustness. Such novel topological
features in non-Hermitian systems boost metamaterials by opening new avenues to manipulate waves in unprecedented fashions.
Name of the Speaker: Dr. Rama Gupta, DAV College, India
Title: Insights of information Entropy in the Nonlinear World
Name of the Speaker: Prof. Gautam Vemuri, Indiana University Purdue University Indianapolis, USA
Title: Delay-coupled semiconductor lasers as a platform for PT-symmetry
Name of the Speaker: Prof. Bimalendu Deb, Indian Association for the Cultivation of Science (IACS)
Title: Exploring quantum information by atom-atom and atom-ion cold collisions
Name of the Speaker: Dr. Manas Kulkarni,
Title: Localisation, Quantum State Transfer and emergent PT symmetry in non-Hermitian systems
Abstract: In the first part of the talk [1], we will discuss localization in cavity-QED arrays. We show that a careful engineering of drive, dissipation and Hamiltonian results in achieving indefinitely sustained self-trapping. We show that the intricate interplay between drive, dissipation, and light-matter interaction results in requiring an optimal window of drive strengths in order to achieve such non-trivial steady states. In the second part of the talk [2], we will discuss optimal protocols for efficient photon transfer in a cavity-QED network. This is executed through a stimulated Raman adiabatic passage scheme where time-varying inductive or capacitive couplings (with carefully chosen sweep rate) play a key role. In the third part of the talk [3], we will discuss emergent PT symmetry in a double-quantum-dot circuit-QED set-up. Starting from a fully Hermitian microscopic description, we show that a non-Hermitian Hamiltonian emerges in a double quantum dot circuit-QED set-up, which can be controllably tuned to the PT symmetric point. Our results pave the way for an on-chip realization of a potentially scalable non-Hermitian system with a gain medium in quantum regime, as well as its potential applications for quantum technology.
[1] A. Dey, M. Kulkarni, Phys. Rev. A 101, 043801 (2020)
[2] A. Dey, M. Kulkarni, Phys. Rev. Research 2, 042004, Rapid Communications (2020)
[3] A. Purkayastha, M. Kulkarni, Y. N. Joglekar, Phys. Rev. Research 2, 043075 (2020)
Name of the Speaker: Dr. Aradhya Shukla,
Title: PT-symmetry and Supersymmetry: Broken and Unbroken Phases
Name of the Speaker: Prof. Usha Devi A R
Title: Continuous Measurements on Open Quantum Systems: Quantum Diffusion
Abstract: Continuous probing of a quantum system through non-demolition measurements on the environment result in stochastic master equations. Depending on the nature of measurements one obtains quantum diffusion or jump type stochastic differential equations (also called Belavkin-Schrodinger equations). Study of quantum stochastic equations governing open system dynamics has gained importance in measurement and feedback based quantum control, which paves way to efficient parameter estimation. I discuss quantum stochastic differential equation for diffusion of a system and outline its implications in parameter estimation.
Name of the Speaker: Prof. Jayendra Nath Bandyopadhyay, BITS Pilani, India
Title: Floquet Quantum Systems
Abstract: Periodically driven systems are ubiquitous in nature. These kinds of systems are studied theoretically using the Floquet theorem, hence these systems are also called Floquet systems. In this talk, I shall discuss the basics of Floqurt theory and systems and its application in designing quantum materials that may be useful in developing hardware for quantum computers.
Name of the Speaker: Prof. V. Ravishankar
Title: Q in QIQT
Abstract: What is meant by quantum in quantum information? This question offers a number of perspectives, both in formal and applied domains, glimpses of which I shall provide in this talk.
Name of the Speaker: Dr. Mamta Balodi, IISc Bangalore, India
Title: Categories, Functors and Natural transformations
Abstract: Almost half a century back, Eilenberg and Mac Lane gave us a revolutionary way of looking at Mathematical structures which are known as categories. This view point not only unifies. Mathematics but also breaks the barriers between different disciplines like Computer Programming, Quantum Information, Logic and Linguistics. In this talk, I will be introducing the basic notions of this discipline which are categories, functors and natural transformations by the way of examples.
Name of the Speaker: Dr. Kazi Rajibul Islam, Institute for Quantum Computing (IQC) and Dept of Physics and Astronomy, University of Waterloo, Canada, Affiliate Scientist, TRIUMF and Perimeter Institute for Theoretical Physics
Title: Programmable Quantum Simulations with Laser-cooled Trapped Ions
Abstract: Trapped ions are among the most advanced technology platforms for quantum information processing. When laser-cooled close to absolute zero temperature, atomic ions form a Coulomb crystal with micron-scale spacings in a radio-frequency ion trap. Qubit or spin-1/2 levels, encoded in hyperfine energy states of each ion, can be initialized, manipulated, and detected optically with high precision. Laser fields can also couple the qubit states of arbitrary pairs of ions through (virtual) excitation of collective phonon modes, creating programmable quantum logic operations and spin Hamiltonians. In this talk, I will focus on programmable trapped-ion quantum spin simulators and explain how techniques from holographic optical engineering to machine learning can be combined to harness the power of these simulators. I will also describe the development of QuantumION, an open-access, multi-user quantum computing facility for the academic community.
Name of the Speaker: Mr. Sooryansh Asthana, IIT Delhi, India
Title: Quantum communication using SU(2)–invariant 2 × N level separable states and its classical optical analogue
Abstract: Bloch vector contains the information encoded in a qubit. Employing this fact, we propose protocols for remote transfer of information in a qubit to a remote qudit using SU(2)– invariant 2 × N–level discordant (but separable) states as a quantum channel. These states have been identified as separable equivalents of the two-qubit entangled Werner states in [Bharath & Ravishankar, Phys. Rev. A 89, 062110]. We have also proposed a protocol for swapping of quantum discord from 2 × N–level states to N × N–level states. Employing these protocols, we believe that quantum information processing can be performed using highly mixed separable higher dimensional states. We show that the classical optical version of information transfer protocol can be employed for transferring information from the path to the orbital angular momentum degree of freedom of classical light
Name of the Speaker: Prof.Jacob Biamonte, Skolkovo Institute of Science and Technology Moscow, Russia
Title: Results on variational quantum circuits to minimise effective Hamiltonians
Abstract: Modern quantum processors enable the execution of short quantum circuits. These quantum circuits can be iteratively tuned to minimise an objective function and solve problem instances. This is known as variational quantum computation: local measurements are repeated and modified to determine the expected value of an effective Hamiltonian. Whereas solving practical problems appears to remain out of reach, many questions of theoretical interest surround the variational model. I will explain some recent limitations found in collaboration, including reachability deficits in QAOA (i.e. increasing problem density — the ratio of constraints to variables — induces under parameterisation at fixed circuit depth), parameter saturations in QAOA (that layer-wise training plateaus) and the existence of abrupt trainability transitions (that a critical number of layers exists where any fewer layers results in no training for certain objective functions). I will also explain some more forward looking findings, including the concentration of parameters in QAOA (showing a problem instance independence of optimised circuit parameters) and my proof that the variational model is, in theory, a universal model of quantum computation.
Name of the Speaker: Prof. Vyacheslav P. Spiridonov, Laboratory of Theoretical Physics, JINR, Dubna & Laboratory of Mirror Symmetry, NRU HSE, Moscow, Russia
Title: Solvable Potentials in Quantum Mechanics from Symmetry Reductions and Coherent States
Abstract: Solvable models of nonrelativistic quantum mechanics provide beautiful idealizations of reality. There is no completely regular way of generating them. We briefly describe the factorization method providing one of the options through reductions of an infinite chain of symmetry transformations. Self-similar potentials obtained in this way are described by very complicated nonlinear special functions including ordinary and q-deformed Painleve transcendents. They are related to polynomial quantum algebras, the q-harmonic oscillator algebra being the simplest case. Applications of these potentials to coherent states, solitons, Ising chains, Coulomb gases are shortly presented. In particular, we describe superpositions of coherent states of the harmonic oscillator associated with the parity and Fourier transformations.
Name of the Speaker: Dr. Himadri Shekhar Dhar, IIT Bombay, India
Title: Light matter interaction in quantum technology
Abstract: In this talk, we look at the role of light-matter interaction in modern day quantum technology. We introduce the topic with a quick prelude to the origin of quantum optics and the field of cavity quantum electrodynamics, before touching upon the era of “artificial atoms” and “hybrid quantum systems.” However, the main focus of the talk will be to look at how one can work with simple theoretical models to study these systems and how by building light-matter interaction some important results can emerge that are useful from the perspective of quantum technology, such as generation of nonclassical light and quantum storage.
Name of the Speaker: Nayana Das, ISI, Kolkata
Title: Two Efficient Measurement-Device-Independent Quantum Dialogue Protocols
Abstract: Quantum dialogue is a process of two-way secure and simultaneous communication using a single channel. Recently, a Measurement Device Independent Quantum Dialogue (MDI-QD) protocol has been proposed (Quantum Information Processing 16.12 (2017): 305). To make the protocol secure against information leakage, the authors have discarded almost half of the qubits remaining after the error estimation phase. We propose two modified versions of the MDI-QD protocol such that the number of discarded qubits is reduced to almost one-fourth of the remaining qubits after the error estimation phase. We use almost half of their discarded qubits along with their used qubits to make our protocol more efficient in qubits count. We show that both of our protocols are secure under the same adversarial model given in MDI-QD protocol.
Ref: International Journal of Quantum Information 18.07 (2020): 2050038.
Biography: Nayana Das is a Ph.D. student at Applied Statistics Unit in Indian Statistical Institute, Kolkata. She received her B.Sc. degree in Mathematics and M.Sc. degree in Pure Mathematics from the University of Calcutta. Her research interest is Quantum Cryptography, Quantum Information Theory and Security.
Name of the Speaker: Pritam chattopadhyay, CSRU, Indian Statistical Institute, Kolkata, India
Title: Thermal Engine from Uncertainty Relation Standpoint
Abstract: The study of thermal devices in the quantum regime has gathered more attraction for research in recent times. Various systems like the quantum amplifier, magnetic refrigerators and engines, semiconductors, thermoelectric generators, and many others explore quantum laws. With the
advent of quantum technology, the exploration of quantum heat engines have gathered more attraction like the Otto engine [1], Stirling engine [2, 3] and so on. Quantum cycles in established heat engines are generally modeled with various quantum systems as working substances. In our
approach, we have considered a heat engine that is modeled with an infinite potential well as the working substance to determine the efficiency and work done with the information of the uncertainty relation of the quantum system [4]. Along with that, the upper and lower bounds on the efficiency of the heat engine are proposed through the uncertainty relation. This work was further extended in the relativistic regime [5] where we have encountered that uncertainty relation has a significant connection with the thermodynamic process.
[1] T. D. Kieu, Phys. Rev. Lett. 93 (2004) 140403.
[2] G. Thomas, D. Das, and S. Ghosh. Phys. Rev. E 100, 012123.
[3] Y. Yin, L. Chen, and F. Wu. The European Physical Journal Plus 132.1 (2017): 1-10.
[4] P. Chattopadhyay et al. Entropy 2021, 23(4), 439.
[5] P. Chattopadhyay, G. Paul, Sci Rep 9, 16967 (2019). |
c8c35ccb33f67a01 | Bad science Medicine Physics Quackery Skepticism/critical thinking
Luminas Pain Relief Patches: Where the words “quantum” and “energy” really mean “magic”
Orac discovers the Luminas Pain Relief Patch. He is amused at how how quacks confuse the words “quantum” and “energy” with magic.
Luminas Pain Relief Patches Luminas Pain Relief Patches: They cure everything through…energy (wait, no, magic).
Energy. Quacks keep using that word. I do not think it means what they think it means. Certainly Luminas doesn’t. Yes, I know that I use a lot of variations on that famous quote from The Princess Bride all the time, probably more frequently than I should and likely to the point of annoying some of my readers, but, damn, if it isn’t a nearly all-purpose phrase to use to riff on various quackery.
Also, if there’s one concept that quacks love to abuse, it’s energy. Whether it’s “energy healing” like reiki, where practitioners claim to be able to channel healing energy from the magical mystical “universal source” specifically into their patient to specifically heal whatever ails them, even if it’s from a distance or you’re a dog, or “healing touch,” where practitioners claim to be able to manipulate their patients’ “life energy” fields, again to healing effect, so much quackery is based on a misunderstanding of “energy” as basically magic. So it is with some spectacularly hilarious woo that I came across last week and, given that it’s Friday, decided to feature as a sort of Friday Dose of Woo Lite. It even abuses quantum theory because of course it does. So much quackery does.
So what are we talking about here? What is Luminas? To be honest, more than anything else, it reminds me of the silly “Body Vibesenergy stickers that Gwyneth Paltrow and Goop were selling last year (and probably still are) that claim to “rebalance the energy frequency in our bodies,” whatever that means. So let’s look at the claims.
Right on the front page of the Luminas website, you’ll find a video. It’s well-produced, as many such videos for quackery are, and it blathers on about how the product being advertised takes advantage of “revolutions in quantum physics,” as a lot of quackery does. Let’s see how this lovely patch supposedly works.
The basic claim is that the Luminas patch is charged with the “energetic signatures of natural remedies known for centuries to reduce inflammation.” These natural remedies include “Acetyl-L-Carnitine, Amino Acids, Arnica, Astaxanthin, B-Complex, Berberis Vulgaris, Bioperine, Boluoke, Boswellia, Bromelain, Chamomile, Chinchona, Chondroitin, Clove, Colostrum, CoQ10, Cordyceps, Curcumin, Flower Essences Frankincense, Ginger, Ginseng, Glucosamine, Glutathione, Guggulu, Hops Extract, K2, Lavender, Magnesium, Motherwort, MSM, Olive Leaf, Omega-3, Peony, Proteolytic Enzymes, Polyphenols, Rosemary Extract, Telomerase Activators, Turmeric, Vinpocetine, Vitamin D, White Willow Bark and over 200 more!”
Luminas Pain Relief Patches
Luminas Pain Relief Patches: here’s the excuse to show partially naked bodies.
Don’t believe me? Take a look at this video on this page! It starts out with an announcer opining about how “energy is all around us.” (Well, yes it is, but that doesn’t mean your nonsense product works.) The announcer then goes on about Luminas somehow infuses its patches with the energy from she substances above:
…energy that your body inherently knows how to absorb and use with absolutely no side effects.
What? Not even skin irritation from the patch or any of the adhesive used to stick the patch to your body? I find that hard to believe. I mean, even paper tape can cause irritation! Fear not, though! The announcer continues:
Through the use of quantum physics scientists and doctors now have the ability to store the energetic signatures of hundreds of pain- and inflammation-relieving remedies on a single patch. Once applied, your body induces the flow of energy from the patch, choosing which electrons it needs to reduce inflammation. Science, relieving pain, with the power of nature.
So. Many. Questions. How, for instance, do the Luminas “scientists” store these “energetic signatures” on a patch? (More on that later.) What, exactly, is an “energetic signature”? How does the body know which electrons it needs to reduce inflammation and pain? As a surgeon and scientists with a PhD in cellular physiology, I’d love to know the physiologic mechanism by which the body can distinguish one electron from another, given that there really is no known biological (or, come to think of it, no physical) mechanism for that to happen and if Luminas has discovered one its scientists should be nominated for the Nobel Prize.
Let’s get back to a key question, though: How on earth is all this energy goodness concentrated into a little patch roughly the size of a playing card? Physicists and chemists are going to guffaw at the answer, I promise you. First, the same page linked to above also notes that the “patches contain no active ingredients” because they “are charged with electrons captured from” the substances listed above. So is this some form of homeopathy? Of course not! Look at the video, which shows magical energy swirling off of the natural remedies and winding its way into the patch! There’s your energy, you unbeliever, you! How can you possibly question it?
But, hey, the makers of Luminas know that there are science geeks out there; so for the benefit of them included in the FAQ is an explanation of just how much natural product-infused electrony goodness you can expect in a single patch:
For the geeks and scientists among us: Each patch contains 5.2 x 10^19 molecular structures, each with 2 oxygen polar bonding areas capable of holding a targeted, host electron, creating a total possible charging capacity equal to 10.4 x 10^19 host electrons. After considering the average transmission field voltage of humans (200 micro volts) we can calculate the relative capacity, per square inch of patch, at 333 Pico Farads.
So basically, they’re saying that each patch contains around 86 micromoles of…whatever…and that that whatever can bind…electrons, I guess. Somewhere, far back in the recesses of my mind and buried in the mists of time from decades ago, my knowledge from my undergraduate chemistry degree and the additional advanced physics courses stirred—and then screamed! I can’t wait to see what actual physicists and chemists whose knowledge is in active use think of this. I apologize in advance if I cause them too much pain by showing them this. Not everyone’s neurons are as resistant as mine to apoptosis caused by waves of burning stupid. It is a resistance built up over 14 years of examining claims like those of Luminas.
Who, I wondered, developed this amazing product? In the first video, we discover that it is a woman named Sonia Broglin, who is the director of product development at Luminas. Naturally, she’s featured with a monitor in the background showing what look like infrared heat images of people. I actually laughed out loud as the video went on, because it shows her in very obviously posed and scripted interactions with patients with no shirts on and up to several of these patches all over their torso and arms. Me being me, I had to Google her, and guess what I found? Surprise! Surprise! She’s listed as a certified EnergyTouch® practitioner who graduated from the EnergyTouch® School of Advanced Healing. What, you might ask, is EnergyTouch®? This:
Energy Touch® is an off-the-body multidimensional healing process that allows the Energy Touch® Practitioner to access outer levels of the human energy field. It is based on the understanding that the human energy field is a dynamic system of powerful influences, in unique relationship to physical, emotional, and spiritual wellbeing. This system consists of the field (aura), chakras (energy centers) and the energy of the organs and systems of the body.
We readily accept the many ways that our body functions and is powered by energy. Our heart beats using energy pulses. Our brain and nervous system communicates with our entire body through complex energetic pathways. Our human energy field is constantly reacting in response to the physical and emotional and spiritual needs of our body.
EnergyTouch® is distinctive in the field of energy healing in that the work takes place in a more expanded energy field allowing the practitioner to work on a cellular level. Our work includes accessing an energetic hologram of the physical body, which is a unique and vital aspect of EnergyTouch® Healing. This energetic hologram acts as a matrix connecting the energies of the outer levels of the field precisely with the physical body on a cellular level.
EnergyTouch® practitioners are skillfully capable of moving fluently throughout the levels of the human energy field, to access and utilize outer level energies to clear blocks and restore function at the most basic cellular level.
It’s all starting to make sense now. That is some Grade-A+, seriously energy woo there, and I’m guessing Broglin cranked it up to 11 when developing the Luminas patches.
Next up is someone named Dr. Craig Davies, who is billed as “Pro Sports Doctor.” Yes, but a doctor of what? It didn’t take much Googling to figure out that Davies is not a physician. He is a chiropractor, because of course he is. He ha actually worked on the PGA tour, apparently adjusting the spines of professional golfers.
Then there’s Dr. Ara Suppiah. Unlike Davies, Dr. Suppiah appears to be more legit:
He is a practicing ER physician, Chief Wellness Officer for Emergency Physicians of Florida and an assistant professor at the University of Central Florida Medical School. He also is the personal physician for several top PGA Tour professionals, including Henrik Stenson, Justin Rose, Gary Woodland, Graeme McDowell, Ian Poulter, Steve Stricker, Hunter Mahan, Jimmy Walker, Vijay Singh, Graham DeLaet, and Kevin Chappell, as well as LPGA Tour players Anna Nordqvist and Julieta Granada.
However, his Twitter bio describes him as doing “functional sports medicine,” which suggests to me functional medicine, which is not exactly science-based. Basically, Dr. Suppiah looks like an ER doc turned sports medicine doc who was a bit into woo but has dived both feet first into the deep end of energy medicine pseudoscience by endorsing these Luminas patches. Seriously, a physician should really know better, but clearly Dr. Suppiah doesn’t. Either that, or the money was good.
Ditto Dr. Ashley Anderson, a nurse practitioner who also gives an endorsement. She’s affiliated with Athena Health and Wellness, a practice that mixes standard women’s health treatments with “integrative medicine” quackery like acupuncture, reflexology, traditional Chinese medicine, and the like.
Given the claims being made, you’d think that Luminas would have some…oh, you know…actual scientific evidence to support its patch. The video touts “astounding results” from Luminas’ patient trials, but what are those trials? Certainly they are not published anywhere that I could find in the peer-reviewed literature. Certainly I could find no registered clinical trials in What I did find on the Luminas website is a hilariously inept trial in which patients were imaged using thermography (which, by the way, is generally quackery when used by alternative medicine practitioners).
Luminas Pain Control Patches
8/Luminas4.jpg”> Luminas Pain Control Patches: Wait! Don’t you believe our patient studies that are totally not clinical trials? Come on! It’s science, man![/caption]So. Many. Questions. About. This. Trial. For instance,
So. Many. Questions. About. This. Trial. For instance, was there a randomized controlled trial of the Luminas patch versus an identical patch that wasn’t infused with the magic electrony goodness of the Luminas patch? (My guess: No.) I also know from my previous studies that thermography is very dependent on maintaining standardized conditions and a rigorously controlled room temperature, as well as on using rigorously standardized protocols. Did Luminas do that? It sure doesn’t look like it. It looks as though Broglin just did thermography on people, slapped a patch on them, and then repeated the thermography. Of course, such shoddy methodology guarantees a positive result, at least with patients whose patch is applied to an area covered by clothing. The temperature of that skin can start out warmer and then cool over time after the clothing is taken off, regardless of whether a patch is applied or not. Did Broglin do any no-patch control runs, to make sure to correct for this phenomenon? Color me a crotchety old skeptic, but my guess is: Almost assuredly not. No, scratch that. There’s no way on earth it even occurred to these quacks to run such a basic control. They can, of course, prove me wrong by sending me their detailed experimental protocol to read.
I suspect I will wait a long time. After all,
After nearly 14 years of regular blogging and 20 years of examining questionable claims, it never ceases to amaze me that products like Luminas patches are still sold. Basically, it’s a variety of quantum quackery in which “energy” is basically magic that can do anything, and quantum is an invocation of the high priests of quackery.
By Orac
To contact Orac: [email protected]
99 replies on “Luminas Pain Relief Patches: Where the words “quantum” and “energy” really mean “magic””
By my reckoning, 10.4 x 10^19 electrons is 16.7 coulombs, To store 16.7 coulombs on 333 picofrads you need to charge it to 50 billion volts. Leyden would be impressed. Now I’m not quite clear on the claim, since the description says electrons per patch and the capacitance is per square inch.
But they are special electrons, like the marshmallow bits in Lucky Charms, so they probably don’t abide by the usual rules.
With all that charge, opening the package ought to result in the patches flying out like one of those spring snake gags.
You can figure out the area of the patches from their measurements. The large patches are 2.75″ x 4.0″ and the medium patches are 1.5″ x 2.75″. Just sayin’. 🤔
But the description says so many molecular structures per patch, not per unit area, then in practically the same breath talks about capacitance per square inch, hence my confusion: “Each patch contains 5.2 x 10^19 molecular structures,” I might be induced to think the numbers are total fabrications. But surely not!
Anyway, it makes little difference if the voltage is 50 billion or 50 million.
What cracks me up is that someone knew enough physics to come up with those numbers but not enough to know why they are ridiculous.
If they put Elvis’s mojo, or even Mojo’s mojo, into an energy patch, I might buy it.
“If you don’t have Mojo Nixon, then your patch could use some fixin’!”
I don’t know where Skid Roper has gone, but Mojo seems to have hooked up with Jello Biafra on at least one occasion. The “Love Me, I’m a Liberal” is somewhat amusing.
“Capacitance.” Get it together, Random Capitalization People.
But SERIOUSLY, it has the energetic signatures of all of those herbs, spices and nutraceuticals!
-btw- more legit ( semi-legit?) pain patches and liquids contain cayenne, menthol or lidocaine:
early on, in my continuing leg injury adventure, I used a liquid form of lidocaine which seemed to be helping HOWEVER at one point, it felt as though my leg were on fire and washing it off didn’t help.
Eventually, it wore off and I felt better but swore off Demon Lidocaine.
Fortunately, I am better enough that I don’t try these products but I can see how people rely upon them when they have pain.
Perhaps this is the doctrine of signatures updated for modern times.
I don’t know how you’d get lidocaine to penetrate intact skin. Perhaps it will if it is dissolved in dimethyl sulfoxide (DMSO). I think iontophoresis will work with lido.
@ doug:
I looked at the meds: they are OTC – standard drugstore stuff and one is 4% lidocaine/ another product has that plus 1% menthol. It did help against muscle/ nerve pain HOWEVER I had a bad reaction so I don’t use it.
Cripes, the first time I had my submandibular cyst biopsied (they eventually resorted to the aspiration gun), all I got was the cold spray, which is absolute crap as an analgesic.
Yes, it has the “signature” of a bunch of placebos, which makes it…
…a more convenient placebo!
The only thing that would be even more convenient would be placebos you can download from an app. Oh wait a minute. Haven’t we read, in this very column, about some “energy medicine” quack offering their own extra-special photons in an app?
This one sells electrons, that one sells photons, the only thing that hasn’t been tried yet is to sell neutrinos. Someone needs to put up a “surprise!” website offering “health-enhancing neutrinos,” and while they’re at it, “selected quarks and gluons.”
“We take out the strange quarks and leave only the charmed quarks, so you can have a charmed life!
Hmm, if only my close friend & coworker who does websites, had this type of sense of humor, I’d love to try it.
When you click the “Buy” button, you get a message about quantum quackery and a caution to not waste money on dreck.
BTW, if we proliferated those kinds of “surprise!” websites, they’d screw up the signal-to-noise ratio for the quantum quacks and other quacks, so badly that the quacks might suffer a loss of business, purely by way of losing placement in search engines. Anyone here up for a bit of guerrilla media?
As a chemist, I am amused by people who think that 10^19 is a large number. Or that it’s in any way impressive or unusual.
As for the oxygen polar bonding areas capable of holding a targeted host electron, I should put that on an exam to see if anyone can figure out that it just seems to be a florid description of an anion. You could say that lye (sodium hydroxide, aka Draino) contains the same: wouldn’t that make a great skin patch!:)
I think that the FDA should require that, if anyone wants to use the word “quantum”, or even “energy”, about a product, they should first be able to define it. That would do the trick. Even Deepak himself couldn’t pass that test.
Chopra especially couldn’t pass the test. He has had physicists try to explain it to him while he sits there with a blank look on his face so he knows he doesn’t understand it. That’s why he prefers quantum woo – you don’t have to understand anything and can just make $h!+ up as you go along confident that no acolyte or fellow woomeister will pull you up on it even though their own version is contradictory.
Chopra can actually be pretty good on comparative religion, so it’s doubly tragic that he goes down the quantum BS road. If he stuck to religion & philosophy, and stayed the hell away from the science he knows not, he could do some good.
Part of the blame for this rests with the media for giving his nonsense attention. Same as with Nobel laureates who’ve gone down various BS roads, such as Shockley and quack racial theories, etc. Same as with Silicon Valley big-wigs, look up “transhumanism” and “Alcor” and so on.
If we tried to educate reporters, it would be a constant game of whack-a-mole, and there would always be those who resist all efforts so they can keep pursuing cheap clickbait. But perhaps we can reach senior editors and publishers, at least in the major media such as newspapers of record, radio/TV networks, and so on?
Scientists could offer their grad students incentives to do the outreach. Postal mail to publishers, that leads off with “I’m writing on behalf of Dr. So-and-So (well known scientist) at Such-and Such University (major university)…” could work, because it’s leveraging name recognition, and postal mail gets through where email doesn’t. These letters and the replies could also be published to scientists’ blogs.
Thoughts? Ideas?
Yo Garnetstar, I’ll take your “energy challenge.”
Canonically, energy is the capacity to do work.
Work is somewhat circularly defined as conversion of energy from one form to another.
Mundane examples: a generator converts kinetic energy to electrical energy; and a motor converts electrical energy to kinetic energy. The same device can be used both ways, thus we get regenerative braking in electric and hybrid automobiles.
OK, so (excess capitalization intended for effect):
Energy is the capacity to do work. The special Energy embodied in our products, does its work by multiplying the Subtle Forces of your Bio-Energetic Field…”
Uh-ohski, looks like we’ll have to require them to define “force” (e.g. a measurable influence on the motion or other behavior of an object), and “field” (an area of spacetime in which a given force has a measurable effect e.g. a gravitational field around a star).
This could actually get fun.
Canonically, energy is the capacity to do work.
Which is why it’s a poor definition. There was a good post on this at the old SB, maybe Chad Orzel. Definitely not Ethan. I can’t remember whether there was another physics Scibling.
Fields exist throughout the entire universe and there are particle fields as well as force fields and, of course, the Higgs field.
I’d love to know the physiologic mechanism by which the body can distinguish one electron from another, given that there really is no known biological mechanism for that to happen
It’s even better than that: electrons are particles that by definition cannot be distinguished from one another. Each and every electron is fully identical to any other electron in a very fundamental way. All electrons have the exact same mass, charge and spin, and quantum physics also dictates that it is not possible to track the trajectories of individual electrons.
Absolutely, because it would completely overturn the amassed knowledge in the field of quantum physics from the past hundred years.
It’s even better than that: electrons are particles
The deuce you say. (Yes, I understand why one can’t walk through walls absent making a big mess).
So much so that John Wheeler wrote to Albert Einstein saying that he has figured out why all those electrons are identical. By using Richard Feynman’s idea that positrons are sort of like electrons traveling backwards in time, he concluded that there is only one electron in the universe (you test this for yourself using Feynman diagrams and it is indeed makes sense). Of course this was only an interesting idea, and no one really believes this is true.
correction…John Wheeler communicated this to Richard Feynman, not Albert Einstein.
We need a Wooday moment of silence in honor of Queen Elizabeth’s personal physician, “an international leader in homeopathic and holistic medicine”, who was killed on Wednesday when his bicycle was hit by a truck. On National Cycle to Work Day.
No word on whether he was treated in the Homepathic ER.
On National Cycle to Work Day.
Back when I was working for a university press, a fine young, long-waisted lady, year after year, would implore me to ride a bike to work. And I always pointed out that my apartment was a five-minute walk from work. She wanted me to rent one anyway. I’m somewhat hostile to the attempts by cyclists to try to Borg pedestrians, especially given that they represent a greater hazard than do cars.
I agree. Cycling is for leisure. In my neck of the woods, there are numerous trails for walking, cycling, and horse riding which were originally rail lines between the city and the outlying farmlands.
I used to cycle to school in college. It was about a 15 minute ride, and parking was definitely easier. I would do it again if I lived close enough to work, or if I worked on a campus large enough to make biking an easier way to get around. It’s good exercise. But that should be a choice, never a demand from others.
I do everything by bike. But hey, I’m Dutch. I even take the bike for distances that are a 5 minute walk.
Don’t like long distance biking and hate other cyclists, who think traffic regulations are not ment for them.
On the other hand, I rather get run over by a bike, than by a car.
I sometimes hate pedestrans as well, especially when they walk on the cycleway, instead of on the footpath next to it and let their small dog run free, while the cycleway is slippery, because of snow an ice.
Gotta be honest here. When my wife and I were in Amsterdam, we both thought that the bicyclists they were some of the biggest jerks we’d ever seen. Try as we might to stay in the footpath and obey the traffic signs and lights, we still had multiple near misses in just four days in the city.
I used to cycle to school as well and survived two near death experiences. I would definitely not use it for commuting these days. Motorists hate us even those of us who are polite. But I love long distance cycling. In fact I’m in training for my 7th consecutive 210km Around The Bay cycling event held in Melbourne each year in early October. And I do all my training on the rail trail that extends 44km into the countryside from where Ilive. I pity those who have to train in the city.
I presume that those in Chicago who bicycle on the sidewalk (which is prohibited if one is over 14 years of age) and wear helmets are doing the latter in case they get clocked. The next time I hear “on your right/left,” I’m moving in that direction.
Problem in the UK too. Some [email protected] in a track suit talking on a mobile phone while riding on the pavement. Makes me want to kick his wheels in. Not only is it illegal (But not really enforced) but bloody dangerous too. I always ride on the road. Unless there is a specific cycle path. Don’t get me started on running red lights. Grrrrgh.
Same. But mostly out of self interest. I never ride on footpaths – because cycling on the road is faster. And I always obey traffic lights – because motorists don’t see cyclists and their cars hurt when they hit you. I also wear a helmet and not only because it is legally required. I have been hit on the head several times and was grateful my head was covered by a helmet.
But some pedestrians are a bit of a worry as well, especially on shared trails where I cycle. Dogs are rarely under the control of their owners. Either the leash is way too long or the dogs actually disconnected from their leashes. Having walked my dog in the past before arthritis put an end to that (the dog, not me.Yet!), I sympathise. My solution is to slow down to a speed at which I am able to come to complete stop before hitting the dog as it inevitably walks directly into my line of travel. I also make a point of exchanging some pleasantry with its owner, hoping, I think in vain, that they will take better care of their mutt next time.
When I pass a pedestrian from behind. I have learnt that the only unambiguous call is “rider” – in a strong voice and at the the right distance. You approach them from the centre of the trail and sway to whichever side they don’t move to, because their choice is totally unpredictable, even if they are walking well to one side of the trail. When I approach from in front, I keep to my left (I live in Australia where we drive on the left) and hope that the pedestrian will sensibly move to their left as well. This is usually the case but also not guaranteed. The occasional pedestrian already walking on the left side of the trail will inexplicably move to the right side of the trail despite putting themselves into my direct line of travel.
in a strong voice and at the the right distance
Just bear in mind that not everyone can hear. It’s too long a story for me to recount in my current state of exhaustion, but quite a while ago, I basically wound up with a partial lateral meniscectomy as a result of impacted cerumen. And random street violence coming from behind. It was about a year before I stopped cocking my fist if anybody approached me too quickly from behind.
@ Orac,
Amsterdam and cyclists, that’s some combination. I think a Dutch lawyer started a case against the city council, because they should do more against cyclists, who didn’t follow the law. Alas he lost his case. (Actually, finding a cyclist who follows the rules, is something like finding a needle in a haystack.)
I still remember seeing a friend of my mother, a very civilised lady, cycling against the traffic, something that annoys the hell out of me and make me want to scream.
I can’t say I never cycle on a foothpath, but only if there are no pedestrians, or just one or so and I limit my speed to walking-speed.
Yes, I am well aware that not everyone has good hearing. I see many octogenarians walking on these trails. So, I should add, that I never hold it against pedestrians when they seem to do silly things.
Just yesterday there was a schoolgirl about fifteen years of age using part of the trail to walk home from school who was walking on the left side of centre of the trail and moved to over to the right side when I warned her by calling out “rider” (from past experience, many pedestrians don’t hear you coming and are startled when you pass them, so often it is for their benefit) and then followed up with “passing on your right”.
I always show pedestrians the utmost respect because they are doing exactly what I do – enjoying exercising on a nature trail – just using a different method. This is also partly so that they we remain on friendly terms because you often meet the same pedestrians repeatedly. I never reprimand dog owners for the actions of their dogs even when it is because they are not controlling their dogs. I understand that they prefer not to have their dog on a leash however impracticable.
OK, fine. Now, how long does it take for all those fancy electrons to be released? A few picoseconds, I’d guess, if the capacitance is of that order of magnitude. Not much point using a patch, then. Better just to rub an amber rod with a black cat fed with turmeric at midnight and apply it to the base of the victim’s skull, but there’s probably not so much money in that.
True, I suppose. There’s always someone willing to pay for witchcraft. After that it’s all down to the marketing.
Actually, Rich, while I do have several pieces of Baltic amber and some curry powder, I think that getting the semi-feral black cats to stay still long enough will be somewhat of a bitch.
I am reminded, somehow, of Doctor Science’s statement that you can generate animal magnetism by rubbing Amber with a cloth. But it all depends on Amber’s mood.
Handcuff, rope and the whip. Hand that to Amber and she’ll take care of your pain 😀
Not need for pain patches…
Al who’s finding it hard to type on a keyboard while being handcuffed and roped out
I was thinking, more efficient to sell 330 picofarad capacitors, with instructions to tape one lead to your arm and leave the other lead pointing into the air to “receive Healing Energy from the Life-Force Field.”
The leads would have to be curled up into little spiral curleycues, so the pointy ends weren’t sticking out, otherwise potentially serious injuries could occur.
Our competitors’ quantum healing patches quickly wear out. And they have to be applied as soon as you remove them from the packaging, otherwise the electrons wear off, much as overcooked vegetables lose their nutrition. But our Life Force Capacitors never wear out: they keep delivering Energy from the Life-Force Field, for as long as you wear them! You won’t ever have to buy another one, unless you lose yours or want to give them away as gifts.
Imagine going to a healing-woo convention and seeing people running around with capacitors taped to their arms, with curlycue leads sticking up.
That would be worth all the effort.
Damn, I really want to try this, just for the sake of seeing the pictures of wooskis with capacitors taped to their arms.
When the game gets old, shut down the website, post an official-looking “FBI notice” on the home page, and start spreading conspiracy theories about “government suppression of alternative medicine.” Then track the conspiracy theories to scope out how they propagate. A couple of years later, publish a story about the whole thing.
The leads would have to be curled up into little spiral curleycues
Please, no string theory.
The good thing about this ‘therapy’ is that you can have acupuncture for free if the cats is not amused by these shenanigans
OMG, I was laughing hard! Thanks…
Does anybody else see the direct self-contradiction? In quantum mechanics, multiple electrons must enter into a wave function as an antisymmetric superposition because they are fermions, which gives what is called exchange symmetry. The Pauli exclusion principle is a direct consequence of this. What I mean is that quantum mechanics says that electrons are so indistinguishable from one another that the wave function containing them is a sum of all situations where they have all individually traded positions in the configuration –at risk of repeating myself, literally because you can’t tell them apart.
Really kind of amazing: invoke quantum mechanics in the first sentence and then immediately posit a situation in the exact next sentence that quantum mechanics, by its very nature, says can’t happen.
Perhaps each electron is quantum-entangled with another electron in a much more advanced civilization’s hospital light year’s away where alien physicians do the choosing for us?
Apart from what others above have noted about this statement: The map is not the territory. We can compute the energetic signatures of the relevant molecules. We can store the results for hundreds of such molecules on media the size of those patches. That does not mean we have actually exposed the patient’s body to any of those substances–it’s more like taping a solid-state memory device (such as a thumb drive) to the body. And I suspect it is just as theraputically effective.
In addition, I have an instinctive distrust of quantitative statements based on color scales where they do not show me values. I refer to the diagram in which they claim to show a reduction of inflammation in a matter of minutes. How do I know that they haven’t fiddled with the range of the color scale between the “before” and “after” pictures? How do I know it is not the result of somebody walking off the street (or removing his shirt) and then sitting in an air-conditioned room for a few minutes? The one thing I do have to work with here is the relative levels, and I see that in general the parts of the body that have relatively high values of what they are measuring in the before picture have relatively high values of that quantity in the after picture. If these patches actually did anything, I would expect the parts of the body that are marked as having had patches put on them would see a greater reduction, and I am not seeing that in the diagram.
This is really just an expendible-buy-some-more variant of the magic-infused silicone wrist bands that first appeared several years ago. The magic in the wrist bands was better because it could make its way to the target site all on its own.
Someone made a ton of money off those silly bracelets; they re-named the basketball arena in Sacramento for them. I haven’t seen any ads for the bracelets recently, are they out of style or am I not watching the right ads?
Orac, I was wondering if you could comment on a new book by a Dr. Tom Cowan (he has been favorably reviewed by “Dr.” Mercola, if that helps 😉 ). He has polluted our local public radio current affairs program a couple of times and I would like to find a SBM review of his work to forward to the local news team. Here is a link to his interview (I don’t know if the booked the composting lady as an ironic comment).
I’m guessing that’s not the same Tom Cowan who used to play in defence for Huddersfield Town…
Whenever Orac highlights one of these scams, I always wonder how successful they are, how many people actually buy these things. All we can know here is that somebody with a sizable chunk of cash to invest thought this would be a winning item and funded the rather splashy (and definitely professionally produced i.e. not cheap) promotional video/website/etc. I’m pretty sure some of the past Friday Woo -ish howlers – e.g. Bill Gray’s coherence apps, QUANTUMMan – are no more than the failed pipe dreams of would-be alt-med entrepreneurial titans. The websites are ghosts that never get updated, or the LLCs are shuttered, other business records show no activity, or the proprieters are still busy working their day jobs, or something… But maybe the fact these things keep appearing is evidence that some of them have worked, well enough to encourage other overly-ambitious woo impresarios to try?
In addition to the expense displayed in setting up Luminas, I also interpret this as a straight-up scam, not the work of ‘true believers’. It’s not just the totally nonsense invocation of quantum physics. The clinching howler for me is the list of EVERY popular supplement “and over 200 more!” Gee, that’s a lot of energy to stick in one little patch. They must be using the quantum magic to keep all the different vibrations from all those different substances from interfering with one another or combining in a way that really f***s you up. And, yeah, I’m sure there’s an exhaustive complicated manufacturing process involved in charging the patches with electrons from each of those substances – which conveniently leaves “no active ingredients” in the product.
I hope Orac follows up on Luminas at some point in the future where it might be evident whether or not they’ve found a viable market and are making any money. If they do well that would be another drop of depressing news in the giant ocean of gullibility, magical thinking, conspiracy theory, denialism that seem to be pandemic here, at what is most likely the sunset of homo sapiens sapiens…
What with all this “quantum” stuff, I am sure they have someone on board to make sure they get it right. Probably someone even more qualified than Sean Carroll and Lawrence Krauss combined.
Probably someone even more qualified than Sean Carroll and Lawrence Krauss combined
Please don’t give them the “multiverse.”
Defibrillator pads…now those have some real electron charge. Why doesn’t Luminas sell those?
As I understand it, the typical defibrillator tops out at something around 400 joules. I re-ran the pad numbers assuming this time that the total charge was distributed over the area of the big pad, so the voltage would be reduced to only 4.56 gigavolts. For the total pad capacitance of 3663 picofarads, that works out to 3.8 x 10^19 joules – about a hundred million full charges for a defibrillator, or about 10500 kilowatt hours. Seems to me like that would really put the flame in inflammation.
Perhaps agencies in charge of airport security should be alerted.
OT but Mike Adams will soon be ranting…
( NYT) It seems that Alex Jones may have deleted evidence he was ordered to save of Sandy Hook conspiracy mongering he broadcast: the parents of the murdered children are trying to sue his pants off**, threatening his lucrative supplement, survivalist business.
He’s been taken off facebook, you tube and pirate radio.
But he’s available on Mike’s new
** which may be just but profoundly unattractive.
We’ve known about Acetyl-L Carnitine for centuries? Who knew!
But the reference to K2 is really what caught my attention. I wonder what the DEA will think of that after what happened in New Haven the other day.
Ah, so that’s what K2 is! I’d just assumed they were referring to the mountain (it seemed as reasonable as everything else). has a frequently asked question section:
A. Do not cut the patch. If you cut the patch, the charge will be lost and the patch will no longer be effective.
@ Narad or Denice Walter,
Is there a simple and an cost effect way to determine the validity of such a statement without purchasing the luminas patch?.
Please advise.
You brought it up, Michael. Either you come up with a clever work around, or you buy one, cut it, and see for yourself.
The rest of us have better things to do.
@ Panacea,
Thanks for doing better things! The patch must have been cut at one point in the manufacturing process. It must be one hell of a trade secret wherein a second cut completely destroys it efficacy. I know of only one other product that fails completely after being cut and that’s a water balloon.
I’m sure they’ve worked out that they need to cut before charging. But then again…
For anyone with even a basic layman’s knowledge of Quantum Physics, the nonsense in this claim is obvious. However, there are much less obvious forms of quantum quackery that can, and do, fool people even with advanced knowledge of QM.
The following video was made by Quantum Gravity Research. This organisation employs physics PhDs to do research into QM so it uses real physics. The problem is that its founder clings to the old “consciousness causes collapse” version of the Copenhagen Interpretation.
The vast majority of present day physicists who do favour the Copenhagen interpretation have long ago jettisoned the “consciousness causing collapse” nonsense because the evidence against it is so overwhelming.
However some have a vested interest in this idea and, as the video reveals, they cherrypick the science that seems to support their interpretation and ignore the multitude of disconfirming evidence. And they actually lie about there being no deterministic interpretation of QM.
Despite the backing from many physics PhDs doing this research, the idea is BS. The organisation was setup by Klee Irwin who made a fortune selling fake medical remedies. But it takes more than a smattering of knowledge of QM to see through it all.
If nothing else, it is a testament to the adage “sex sells” – watch it to see what I mean 😉
Beg pardon? I have little interest in “interpretations,” but that’s likely because I’m weary of MWI babbling around the Intertubes. It’s not “shut up and calculate,” but yes, the measurement problem is a real thing even if the Schrödinger equation is nominally deterministic.
If you can take it, check this out.
That link is to the ideas behind the group that calls itself “Quantum Gravity Research”. Forget about anything written by this research group. It is not peer reviewed. It is based on the underlying idea that consciousness collapses the wave function, a totally discredited concept that used to be promoted as an outworking of the Copenhagen interpretation but has long since been excised from that interpretation by the vast majority of today’s physicists on the basis of experiments in QM. Although the research is conducted by phd graduates, the organisation is actually funded by an individual who made his fortune selling medical scams to an unwary credulous public and who effectively believes The Matrix was a film about science.
Nice piece here, though (h/t Peter Woit). My few remaining neurons are never going to get it together to grasp geometric Langlands or representation theory, unless I start with Charles Pierce. I like his brother better in any event.
Unfortunately this is way beyond my pay grade. I don’t have any formal training or qualifications in QM, just enough to have a reasonably well developed layman’s BS meter for quantum woo (I hope).
I don’t understand your lack of interest in interpretations of QM.
The trick is to separate the physics from the interpretation. The fun is to see how some people who are committed to one interpretation or another denigrate other interpretations but seem blind to the problems with the interpretations they favour. For example, some people who favour the Copenhagen interpretation and criticise the MWI seem to be unaware that the Copenhagen interpretation is similarly burdened with its “multiple paths” which amounts to “infinitive paths” all of which much be traversed. And MWI is more parsimonious with one less assumption. Not that I support MWI, only that I find the discussion interesting.
And the Pilot wave interpretation is attractive because it is both mundanely physical and determnistic but, on the other hand, it requires the existence of global hidden variables which is problematic from the point of view of the very real evidence for non-locality, and it is at least incomplete because can’t account for special and general relativity. But who knows what future discoveries may yield.
I will read your link though.
I don’t understand your lack of interest in interpretations of QM.
Decoherence is decoherence. The interesting question from my viewpoint is whether GR needs to be quantizted or QFT needs to be superseded to get further. This requires experiment primarily, which, absent serendipity, requires connection to theory (or the other way around). I have no objection to the philosophy of physics, but at some point it’s just navel-gazing or worse.
^^ Let me try it this way: Are the “many worlds” a well-ordered (transfinite) set? If yes, which SR would seem to require, what’s labeling them? If no, then you’re just back where you started, whether the question is nonlocality or the simple emergence of classical behavior of quantum systems.
Dear Luminas,
In addition to ferking up all that quantum-y stuff, it would be Berberis vulgaris, not Vulgaris.
Carl Linnaeus
The parents donated $50,000 to the children’s hospital where she was treated, so I think they know who did all the work.
Follow-up: Apparently the child was first diagnosed with a primary brain tumour which is almost uniformly fatal and she was given weeks to months to live. However, a follow-up scan showed cavitation or cyst formation which created doubt about the diagnosis and therefore the tumour was biopsied. This changed the diagnosis to Juvenile Xanthgranuloma, which is essentially an abnormal proliferation of a type of blood cell called a histiocyte. So this was actually not a primary brain tumour. This changed the prognosis because these tumours can be treated and cured with chemotherapy. Her treating specialists were obviously still concerned because the site of the tumour and therefore were guarded about her prognosis. However, this is no miracle as it is being portrayed in the media. She was cured by chemotherapy administered by paediatric medical specialists.
Hmmm … an oxygen with two “electron binding spots” [???] …maybe it’s … WATER!!
ps://”> Luminas Pain Relief Patches: Here’s the excuse to show partially naked bodies.
I don’t see the picture but at least I got the description 😀
Yeah I tried for that was well, but the link disappointingly led nowhere. 😉
I’ve had Lyme disease for the past 4 years and just started treatment for that 2 months ago. To my surprise the patches do take away the radiating stabbing pain I get when my body is at rest.
I have to wear a lot though. 8 patches on my back and 3 per hand. It says they work for up to 3 days but I wear all of them for about 5 days and have success.
I put on 5 patches before bed one time and I was wired beyond belief and couldn’t fall asleep for the life of me.
The patches are really affordable the way I use them and are providing a lot of relief while this long 9 month treatment plan unfolds
Isn’t it pretty obvious that Dr. Craig Davies is a Doctor of Pro Sports? There are very few university medical schools (maybe Palmer College?) offering that degree, so I expect his services to be rather expensive.
Comments are closed. |
b6c167d5bb58552d |
1. Whether we should treat it as a wave equation or a heat equation with imaginary time, or both?
2. If it is a wave equation, how do we express it in the form of a wave equation?
3. Is there any physical significance that Schrödinger equation has the same form of a heat equation with imaginary time? For example, what is diffusing?
• $\begingroup$ I don't understand what physics principle you're asking about. There are many linear equations. So what? There are many exponential decay equations. It merely gives us analogues. $\endgroup$
– Bill N
Sep 19 '16 at 17:15
• 2
$\begingroup$ Possible duplicate of How is the Schroedinger equation a wave equation? $\endgroup$ Sep 19 '16 at 17:35
• 1
$\begingroup$ Is there any reason why Schrödinger equation can be written in the form of a heat equation? Also, Schrödinger has first order derivative with respect to time (instead of second order), why is it a wave equation? $\endgroup$
– Kevin Kwok
Sep 19 '16 at 17:46
• 1
$\begingroup$ The part with the heat equation is closely related to physics.stackexchange.com/q/80131/50583 and physics.stackexchange.com/q/144832/50583, the part about it being a wave equation is a duplicate of the already linked question. Please ask a single, focused question per post, or at least questions so closely related they can't meaningfully be split up. $\endgroup$
– ACuriousMind
Sep 20 '16 at 13:25
1) Both: it is apparently a heat equation in imaginary time and it is a wave equation because its solutions are waves.
2) Nonstationary Schrodinger equation (let us assume free particle) $$ i\hbar\frac{\partial\psi}{\partial t}=-\frac{\hbar^2\nabla^2}{2m}\psi $$ is essentially complex: it can never be satisfied by a real function, only by a complex one.
Nevertheless, its solutions are waves because the complex $\psi$ means it is actually a system of two real equations of the first order in time. Assuming $\psi=u+iv$ we have: $$ \hbar\frac{\partial u}{\partial t}=-\frac{\hbar^2\nabla^2}{2m}v,\qquad \hbar\frac{\partial v}{\partial t}=\frac{\hbar^2\nabla^2}{2m}u. $$ Eliminating, say, $v$, we get: $$ \hbar^2\frac{\partial^2 u}{\partial t^2}=-\frac{\hbar^4\nabla^4}{4m^2}u. $$ In two dimensions, this equation has the same form as a wave equation for bending (flexural) waves on a thin rigid plate. It is also of the 2-nd order in time and 4-th order in coordinates. The analogy extends also to wave dispersions: the bending waves have a quadratic dispersion $\omega\sim q^2$ similarly to free particle obeying Schrodinger equation $E=p^2/2m$.
3) This analogy is widely used in the diffusion Monte-Carlo method, where the Schrodinger equation is solved in imaginary time. In this case, its solution is decaying instead of being oscillatory and, if we normalize it properly, it will converge to the ground state wave function:
What is diffusing here? Taking imaginary time $\tau=it$, we have the following imaginary time Schrodinger equation for a particle in a potential $V$: $$ \hbar\frac{\partial\psi}{\partial t}=\frac{\hbar^2\nabla^2}{2m}-V\psi. $$ The first term in right hand side is usual diffusion. The second is something like heat production or burning, and its "minus" sign means this heat production is more intense in the minima of $V$.
Thus, the picture of diffusion in imaginary time is the following: the first term ("diffusion") tries to delocalize $\psi$, while the second term tries to lure $\psi$ to the minima of the potential $V$. Their interplay is the same as that between kinetic and potential energies in quantum mechanics, and its result is a ground state wave function - exactly what is used in diffusion Monte-Carlo calculations.
• $\begingroup$ As a reference, the governing equation of the dynamics of a rigid plate can be found at en.wikipedia.org/wiki/… $\endgroup$
– Kevin Kwok
Sep 20 '16 at 7:36
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a785aef7ea840b03 | Skip to main content
A modular design of molecular qubits to implement universal quantum gates
The inability of conventional computers to solve certain problems efficiently, such as the simulation of quantum systems1,2, is one of the main driving forces for the implementation of quantum computing and quantum information processing (QIP)3,4, which exploit the laws of quantum mechanics. At the theoretical level, several algorithms have been shown to outperform classical computers in certain computational tasks such as factoring large numbers into primes5 and searching of unsorted directories6. A variety of quantum systems have shown an excellent performance as the basic units for quantum information (qubits)7,8,9,10,11. However, they are difficult to link controllably into useful arrays producing the two-qubit entangling quantum gates (QGs) crucial for any quantum algorithm. Among the most important entangling QGs are the controlled NOT-gate (CNOT) and the gate. The gate brings the two-qubit state to the superposition . The CNOT gate flips the state of the target qubit if, and only if, the control qubit is in the state; this implies that each qubit has to respond inequivalently to an external stimulus. All other quantum gates, however complex, can be constructed from these two-qubit gates and single-qubit gates.
Therefore the major challenge for the physical implementation of QIP is bringing together qubits in an organized, scalable and addressable way to make such QGs3,4. Here we demonstrate that supramolecular chemistry could have a major impact in addressing this challenge.
Molecular nanomagnets have been proposed as qubits12,13,14,15,16,17,18,19,20,21,22,23,24,25,26 and coherence times of individual molecular qubits have been studied and have improved24,25. Supramolecular chemistry27 allows us to bring together, with great control, complex molecules into arrays, and recently this has been exploited to tune two-qubit interactions built from molecular nanomagnets28. Hence supramolecular chemistry is a promising tool to build multi-qubit devices; if this could be made scalable this is a competitive route towards the realization of a quantum computer.
Demonstrations of CNOT gates based on molecular nanomagnets have been published18,19,23. The interaction between two dissimilar qubits produces a splitting in the low-lying two-qubit states that allows selective addressing of the transition by means of resonant EPR pulses, while keeping the other components of the wave-function frozen. In this way, the excitation of the target depends on the state of the control qubit and a CNOT gate is implemented. The drawback of this approach is the permanent direct coupling between the qubits, which makes these proposals hard to scale as with a permanent coupling the state of the qubits experiences an unwanted many-body spontaneous evolution in time (as in NMR QIP schemes29), whose harmful effects increase with the number of qubits. Conversely, a switchable indirect coupling between the two qubits would make the register much more easily scalable as when the switch is in the off state the unwanted spontaneous evolution is suppressed.
We have proposed using {Cr7Ni} heterometallic rings as qubits14. They are two-level systems (S=1/2 ground state) that have sufficiently long phase memory times to allow many gate operations before state degradation occurs25, and assemblies of {Cr7Ni} rings have been made which show a permanent coupling between the spins26. Here we report two-qubit assemblies which include switchable links, that allow us to propose a fully modular supramolecular design strategy27,28 towards quantum computation schemes based on either the CNOT or gate. The strategy uses supramolecular chemistry to tailor the individual components, spatial configuration and hence the properties of the resulting supramolecules. These supramolecules have been studied by electrochemistry, continuous-wave and pulsed EPR spectroscopy to understand their static and dynamic spin properties. These measured parameters have been used to perform detailed simulations of the performance of both the CNOT and gates including the effects of decoherence. The two gates we propose are based on either global or local control of the qubit–qubit interaction. The first proposal exploits uniform magnetic pulses to manipulate two inequivalent Cr7Ni qubits in an asymmetric supramolecule, which would implement the CNOT gate. The second exploits the local electric control of a redox-active linker, which might be addressed by a tip to reversibly switch on and off the qubit–qubit interaction, to implement the gate. The scalability of these approaches is discussed in the last part of the paper.
Syntheses and structural characterization
We have earlier reported the selective functionalization of [nPr2NH2][Cr7NiF8(O2CtBu)16] 1 (ref. 30) with iso-nicotinate (O2C-py) to obtain [nPr2NH2][Cr7NiF8(O2CtBu)15(O2C-py)] 2 (ref. 26) (Supplementary Fig. 1a). This synthetic method can be extended to produce [nPr2NH2][Cr7NiF8(O2CtBu)15(O2C-terpy)] 3 and [nPr2NH2][Cr7NiF8(O2CtBu)15(O2C-Ph-terpy)] 4 (Supplementary Fig. 1b,c), from the controlled reaction of 1 with 4-carboxy-2,2′:6′,2″-terpyridine (O2C-terpy) and 4′-(4-carboxyphenyl)-2,2′:6′,2″-terpyridine (O2C-Ph-terpy), respectively (full experimental details are given in the Supplementary Methods)31. 2, 3 and 4 are hereafter abbreviated as {Cr7Ni-O2C-py}, {Cr7Ni-O2C-terpy} and {Cr7Ni-O2C-Ph-terpy}, respectively. They consist of CrIII7NiII rings, containing an octagon of metal centres, with the inner rim bridged by fluoride ions and the outer rim by carboxylates: the functionalized O2C-py 2, O2C-terpy 3 and O2C-Ph-terpy 4 ligands sit on a Cr…Ni edge of the octagon. This provides us with three supramolecular qubits with two different denticities, which allow us to assemble different QGs by appropriate choice of the central node (Fig. 1).
Figure 1: Supramolecular design strategy for the construction of two-qubits assemblies.
(a) Synthesis of functionalized {Cr7Ni} rings (where x=0 and 1). (b) Linkage of {Cr7Ni} coordination cages to different central nodes.
Reaction of equimolar quantities of 2 and 3 with cobalt(II) thiocyanate in a mixture of Et2O/acetone leads to [{Cr7Ni-O2C-py}→Co(SCN)2←{Cr7Ni-O2C-terpy}] 5 in good yield, which has been characterized by X-ray single crystal diffraction (Fig. 2a; for all crystallographic information see Supplementary Data 1). Compound 5 contains two inequivalent Cr7Ni qubits coordinated to a central cobalt(II) ion, which has a six-coordinate CoN6 octahedral environment with a cis-arrangement of the two thiocyanate N atoms. The Co-N bond distances are typical of high spin cobalt(II) ions32, with the bonds to the thiocyanate ligands shorter (average 2.056(12) Å) than those to terpy or pyridine N-donors (2.137(6) to 2.224(4) Å). Hence the d7 CoII site (SCo=3/2) has a 4T1g ground term (using Oh symmetry labels)33, which leads to a well-isolated effective Seff=1/2 ground state at low temperature (see Characterization of assemblies via EPR spectroscopy below). The cis coordination geometry at the CoII node means the {Cr7Ni-O2C-terpy}- and {Cr7Ni-O2C-py}-based qubits are arranged in an almost orthogonal orientation. Therefore the two qubits in 5 are symmetry inequivalent, as required for implementing a CNOT gate (Fig. 2b).
Figure 2: Molecular structure of the asymmetric two-qubit assembly for proposed implementation of a CNOT gate.
(a) Molecular structure of [{Cr7Ni-O2C-py}→Co(SCN)2←{Cr7Ni-O2C-terpy}] 5. nPr2NH2+ cations are not shown (H atoms and tert-butyl groups are omitted for clarity). Colour code: Co, dark red; Cr, green; Ni, purple; Ru, brown; N, cyan; O, red; S, yellow; C, grey; F, pale green. (b) Schematic representation of the effect of the CNOT gate on a pair of qubits, initialized in the computational basis states and , respectively. The CNOT flips the target qubit if the control is set to .
Reaction of two equivalents of 2 with a preformed oxo-centred pivalate-bridged triangular cluster with terminal pyridine groups [RuIII2CoIIO(O2CtBu)6(py)3] 6, hereafter abbreviated as [RuIII2CoII], in acetone gives [{Cr7Ni-O2C-py}→[RuIII2CoIIO(tBuCO2)6(py)]←{Cr7Ni-O2C-py}] 7 (Fig. 3a), where two of the terminal pyridine ligands of 6 were replaced by the iso-nicotinate group of 2. Reaction of two equivalents of 3 or 4 and either cobalt(II) perchlorate or tetrafluoroborate gives [{Cr7Ni-O2C-terpy}→Co←{Cr7Ni-O2C-terpy}][X]2 [X=ClO4 8a or BF4 8b] and [{Cr7Ni-O2C-Ph-terpy}→Co←{Cr7Ni-O2C-Ph-terpy}][X]2 [X=ClO4 9a or BF4 9b] (Fig. 3b,c) (full experimental details are given in the Supplementary Methods). The architecture in 7, 8 and 9 contains two equivalent qubits, separated by a redox-switchable centre34,35, which makes it suitable for implementation of a iSWAP gate (Fig. 3d).
Figure 3: Molecular structures of the redox-active two-qubit assemblies for proposed implementation of an iSWAP gate.
(a) [{Cr7Ni-O2C-py}→[RuIII2CoIIO(tBuCO2)6(py)]←{Cr7Ni-O2C-py}] 7. (b) the cation of [{Cr7Ni-O2C-terpy}→Co←{Cr7Ni-O2C-terpy}][ClO4]2 8a. (c) the cation of [{Cr7Ni-O2C-Ph-terpy}→Co←{Cr7Ni-O2C-Ph-terpy}][ClO4]2 9a. nPr2NH2+ cations and ClO4 anions are not shown (H atoms and tert-butyl groups are omitted for clarity). Colour codes as Fig. 2. (d) Schematic representation of the effect of the iSWAP gate on a pair of qubits, initialized in the computational basis state . The gate brings to the equal-weight superposition . In the scheme proposed below, it operates as soon as the inter-qubit interaction is turned on (double arrow).
The crystal structure of 7 consists of two {Cr7Ni-O2C-py} rings linked through the iso-nicotinate groups to a [RuIII2CoIIO(tBuCO2)6(py)] triangle (Fig. 3a)34,36. The metal ions are statistically disordered over the three sites within the triangular M3O unit. They have a six-coordinate, octahedral MO5N environment formed by four carboxylate oxygen atoms from the bridging pivalate ligands in the equatorial plane, with the central oxide and one nitrogen atom from iso-nicotinate or pyridine groups occupying the axial positions. The three metal ions are positioned at the corners of an isosceles triangle with two distinct intermetallic distances of 3.296 and 3.315 Å. The central oxide lies within the M3 plane of the metal atoms, while the six pivalate bridging ligands lie above and below the M3O plane and the pyridine N-donors rest perpendicular to this plane.
The crystal structure of 8a and 9a are made up of cationic [{Cr7Ni-O2C-terpy}CoII{Cr7Ni-O2C-terpy}] and [{Cr7Ni-O2C-Ph-terpy}CoII{Cr7Ni-O2C-Ph-terpy}] units together with counterbalancing ClO4 anions (Fig. 3b,c). Both compounds have a central cobalt(II) ion with a six-coordinate CoN6 octahedral environment, with the six Co-N bonds in the range expected for a low spin cobalt(II) ion37. The average value of the two N-donors from the central pyridines of the terpy are shorter (1.923(6) 8a and 1.920(8) 9a Å) than the other Co-N contacts (2.118(7) 8a and 1.986(9) 9a Å). This feature leads to an axially compressed octahedral environment for the low-spin d7 CoII ions giving a SCo=1/2 ground state. Compounds 8a and 9a differ in the shortest Co…M(ring) contact (8.675(2) 8a and 10.979(5) 9a Å) and the staggered 8a or eclipsed 9a arrangement of rings in the assemblies.
The two Cr7Ni qubits in each of 79 are linked by a redox-switchable central node34,38. Cyclic voltammetry on 79, and on the central nodes in isolation, show a one-electron reversible oxidation in each case (measured in CH2Cl2, 0.1 M nBu4NPF6). For 6 and 7 the half-wave potential (E1/2)=−0.31 (6), −0.35 (7) V versus Fc+/Fc (Fc=ferrocene)39, and this oxidation is assigned to the [RuIII2CoII] to [RuIII,IV2CoII] couple34. For 8b, 9b and reference complexes [Co(HO2C-terpy)2][BF4]2 10 and [Co(HO2C-Ph-terpy)2][BF4]2 11, E1/2=−0.30 (8b), −0.27 (9b), −0.27 (10) and −0.24 (11) V, which are assigned to the CoII/III couple. The anodic to cathodic peak separation values are similar to those of ferrocene under the same conditions (see Electrochemistry section in Supplementary Methods, Supplementary Fig. 4 and Supplementary Table 3).
Characterization of assemblies via EPR spectroscopy
To study the interactions between the molecular components in the supramolecular structures 5, 7, 8b and 9b we performed magnetometry and multi-frequency continuous wave EPR studies. The measured magnetometry data are essentially the sum of the Cr7Ni qubits and the linking nodes (Supplementary Figs 5–9), and therefore are uninformative other than confirming that the interactions are very weak. EPR spectroscopy is much more sensitive to such weak interactions.
The EPR spectra of a powder of 5 at 5K (Fig. 4a and Supplementary Fig. 10) have a complex multiplet structure because the Seff=1/2 of the Co(II) node has a very anisotropic effective g-tensor and also gives rise to very anisotropic exchange tensors (J) with the {Cr7Ni} rings. This structure is best resolved at W-band (94 GHz; Fig. 4a). The spectra can be simulated40 with the effective three-spin Hamiltonian (1), with two different anisotropic exchange interactions (J12 and J23) between the Seff=½ of the Co(II) ion (S2) and the two S=½ of {Cr7Ni-O2C-terpy} (S1) and {Cr7Ni-O2C-py} (S3) rings, respectively (Fig. 4a).
Figure 4: EPR spectroscopy of QG assemblies.
(a) Experimental powder W-band (94 GHz) EPR spectrum of 5 at 5K (black) and simulation (red) using Hamiltonian (1) and parameters in Table 1. (b) Experimental K-band (24 GHz) EPR spectra of 7 frozen solution (black), and simulation (red) using Hamiltonian (2) and parameters given in the text: experimental spectra after oxidation of 7 to 7box with [FeCp2](PF6) (ca. 3 mM, 1:1) (green), and after reducing 7box with cobaltocene (3 mM, 1:1) (blue). (c,d) Experimental Q-band (34 GHz) EPR spectra of 8b and 9b, respectively, in frozen solution (black), and simulations (red) using Hamiltonian (2) and parameters given in the text; experimental spectra after oxidation to 8box and 9box with AgBF4 (ca. 3 mM, 1:1) (green), and after reducing using with cobaltocene (3 mM, 1:1) (blue). The sharp peak marked * is a trace radical impurity.
The multiplets centred at ca. 1, 1.5 and 3.5 T (Fig. 4a) mark the effective g values of the Co(II) ion as g=6.50, 4.25 and ca. 2, while the Cr7Ni rings have well defined g=1.78, 1.78 and 1.74 (the unique value being for the orientation perpendicular to the ring)30. The large difference (Δg) between the g=6.5 and 4.3 orientations of the Co(II) ion and those of the rings, and the weak exchange interactions is useful as it allows us to treat the problem as an ABX spin system. The multiplet structure of the g=6.5 and 4.3 features is then due to weak exchange with two different S=1/2, giving doublets-of-doublets from which we can read the Jy and Jz components of J12 and J23. The high field region is more complicated, due to the much smaller difference between the third Co(II) g value and those of the {Cr7Ni} rings. The remaining parameters (the final components of J12, J23 and g2) were obtained by simulation (Fig. 4a). The parameters in Table 1 are labelled to a common reference frame, that is, taking into account the orthogonal orientation of the two {Cr7Ni} rings.
Table 1 Spin-Hamiltonian parameters used for simulation of the W-band EPR spectra of 5.
7, 8b and 9b, contain redox-active linking nodes and their EPR spectra change with the oxidation state. Hence, EPR was performed on both the reduced and oxidized forms at 5K as frozen solutions, allowing cycling of the redox state (Fig. 4b–d). Spectra were measured on the as prepared samples, then solutions were warmed to room temperature and the oxidized forms 7ox, 8box and 9box generated in situ by addition of [FeCp2](PF6) or AgBF4 followed by freezing and measurement. To complete the cycle, the solutions were thawed again and 7, 8b and 9b regenerated by reduction with cobaltocene. We have also measured spectra of the isolated nodes in the paramagnetic Co(II) form, viz. complex 6, and [Co(HO2C-terpy)2](BF4)2 10 and [Co(HO2C-Ph-terpy)2](BF4)2 11 (Supplementary Figs 2 and 3). Like 5, 6 also contains a 4T high-spin Co(II) ion with an Seff=1/2 ground state, giving g=5.61, 4.05, 2.77 (Supplementary Fig. 11), while 10 and 11 have low-spin Co(II) hence S=1/2, giving g=2.047, 2.076, 2.195 (Supplementary Fig. 12) and g=2.022, 2.111 and 2.215 (Supplementary Fig. 13), respectively.
For the supramolecular structures 7, 8b and 9b, the exchange coupling is nicely resolved in K- and Q-band (24 and 34 GHz) EPR spectra (Fig. 4b–d). In each case there is weak coupling with respect to the difference in Zeeman energies 41. This, and the equivalence of the two Cr7Ni rings, gives AB2 spin systems. The {Cr7Ni} ring resonances (the ‘B’ spins) are split into doublets (Fig. 4b–d), giving a direct measure of J (the exchange splittings of the central node resonances are not resolved). The spectra can be simulated with the simple Hamiltonian (2) using an isotropic exchange interaction between the central nodes (S2) and the two {Cr7Ni} rings (S1 and S3), even for 7, which has an effective spin 1/2 centre at the node.
Simulations with the isotropic exchange interaction J as the only variable gives J=−0.026, −0.026 and −0.024 cm−1 for 7, 8b and 9b, respectively; the g values were fixed to those measured for individual components. The EPR spectra of 7ox, 8box and 9box are very simple, resembling isolated {Cr7Ni} rings, because the oxidized forms of the central nodes are diamagnetic. Reduction of 7ox, 8box and 9box back to 7, 8b and 9b regenerates the original spectrum.
Implementing interesting quantum algorithms requires qubits with long phase memory times, such that they can be manipulated many times without errors. Individual {Cr7Ni} heterometallic rings and simpler paramagnetic centres have shown long enough phase memory times (TM)21,22,24,25 to perform coherent electron spin manipulations before state degradation occurs. To check that this key property is preserved when they are incorporated in the supramolecular assemblies, we have performed pulsed EPR measurements at resonances corresponding to both the central nodes and the {Cr7Ni} rings (Supplementary Figs 14–26). For the latter we find similar values for all compounds, and in both oxidation states where relevant (TM=683, 749, 767, 790, 984, 750 and 1,031 ns for 5, 7, 7ox, 8b, 8box, 9b and 9box, respectively, measured at Q-band and 3K; Supplementary Table 4), demonstrating that the phase memory times are not strongly influenced by structural or magnetic differences in the linkage of the rings. In addition, values of TM measured at resonances fields corresponding to the central node (in the paramagnetic state) are also similar for all compounds (500–700 ns at Q-band and 3K; Supplementary Tables 5–12). Hence, the phase memory time of the central node does not represent a limitation (see also the Discussion below) for the applicability of our schemes for implementing a universal set of quantum gates. These results are used in the simulations below, and are extremely promising, as these TM times are sufficiently long for spin manipulation even when they are integrated in a supramolecular assembly.
CNOT gate with uniform magnetic pulses
In the following, we introduce two quantum computation schemes, based on either local or non-local control of the inter-qubit interaction employing the structures described and the parameters obtained by EPR spectroscopy. First, we show that compound 5 is suitable for a CNOT gate, using uniform magnetic pulses as the only manipulation tool42. As we are treating three interacting inequivalent doublets this produces 23 energy levels. We define the computational basis within the low-energy subspace where CoII is frozen into its Sz=−1/2 state, which corresponds to the four lowest levels shown in red in Fig. 5a. The four levels correspond to arrangements of the spins of the two qubits, S1 and S3, having the relative orientations |↓↓>, |↓↑>, |↑↓> and |↑↑>, respectively, which we label as |00>, |01>, |10> and |11> in Fig. 5a.
Figure 5: A CNOT gate based on the structure of 5.
(a) Field-dependence of the energy levels of 5 resulting from the Hamiltonian (1). The low-energy group of levels (red), where Co is frozen into its Sz=−1/2 state, defines the computational basis. The high-energy group of levels, where the Co spin is inverted, is exploited to perform two-qubit gates. (b) Simulation of the pulse sequence implementing CNOT as Ry(π/2)CZ Ry(−π/2), where Ry(α) is a rotation of the target qubit by an angle α around the y axis and CZ is a controlled-Z gate. We illustrate the gate by starting at time t=0 with a superposition state , which transforms under a CNOT gate (with the left qubit acting as control) into . The latter state is actually obtained by the pulse sequence implementing Ry(π/2)CZRy(−π/2), with a fidelity of 99.7%. The envelope of the pulses implementing the two Ry rotations and the CZ are outlined at the bottom. Note that performing the CZ gate (two central pulses) requires temporarily leaving the computational subspace. The intensity of the oscillating field at the pulse maximum is 50 G, and we assume a static field of 5 T directed along z.
In a field of a few Teslas the eigenstates are factorized, that is, the eigenfunctions of the rings and the CoII are not entangled. Hence it is possible to implement high-fidelity single-qubit rotations by EPR pulses resonant with low-energy gaps (see, for example, the shorter arrow in Fig. 5a). The combination of the inequivalent and anisotropic ring-Co exchange interaction and of the perpendicular arrangement of the two rings makes the two qubits significantly inequivalent. In particular, there is a sizeable gap (about 33 μeV in a field of 5 T) between the |00>→|01> and |00>→|10> transitions, which enables independent single-qubit rotations. We note that a distribution of the spin Hamiltonian parameters arising from a reasonable g or J strain (1%, see ref. 43) would yield a broadening of the energy levels with s.d. 5 μeV, thus keeping the two transitions distinguishable in realistic conditions, possibly with the help of narrow and/or composite pulses44,45.
A controlled phase-shift (Cφ) gate is obtained by a pulse resonant with the transition corresponding to the longer arrow in Fig. 5a where the cobalt(II) ion is temporarily in its Sz=+1/2 state (blue energy levels), followed by a repetition of the same pulse that would bring the state back with an additional phase φ. The value of φ is controlled by the phase difference between the first and the second pulse. For a pulse with field along the y direction, the implementation of this gate is fast, owing to the large value of gy for CoII ion. Moreover, the relatively large exchange interaction in equation (1) allows us to employ large oscillating fields (50 G), since the desired transition is spectroscopically well resolved from all the others. Consequently, the Cφ gate can be performed in only about 12 ns, with fidelities close to 99.99 % (see Computational Details section in Supplementary Methods).
The CNOT gate can then be obtained by the sequence of gates Ry(π/2) CZ Ry(−π/2), where Ry(α) is a rotation of the target Cr7Ni qubit by an angle α around the y axis (a single-qubit operation) and where CZ is the phase-shift gate described above with φ=π. We have numerically solved the time-dependent Schrödinger equation for the Hamiltonian (1) in presence of this pulse sequence. Results are reported in Fig. 5b and show that this two-qubit gate can be obtained with very high fidelity in only 30 ns. As noted above, the two qubits are significantly inequivalent, even if the g tensor of the rings is nearly isotropic. This makes this complex well suited for the quantum simulation of antisymmetric Hamiltonians (Supplementary Figs 28 and 29 for an illustrative example).
gate with local electric control
The redox properties of the central node in 7, 8 and 9 can be exploited to perform the universal gate on the {Cr7Ni} qubits, whose effect on the computational basis is given by:
and .
When the central node is in the diamagnetic state ([RuIII,IV2CoII] 7 or CoIII 89), the two qubits are decoupled and only single-qubit operations can be performed. Conversely, when the paramagnetic oxidation state, [RuIII2CoII] 7 or CoII 89, is present the system behaves as a trimer described by the Hamiltonian (2). Previously, Lehmann et al.16 have proposed that two spins S=1/2 connected by a redox-active unit can be exploited for the implementation of the gate by switching the redox unit with a scanning tunnelling microscope (STM) tip at an appropriate potential. In this way one electron can be added or removed from the redox unit very quickly. It was demonstrated that particular sets of parameters of the trimer Hamiltonian lead to a pure evolution of the two qubits after specific time intervals. However, high fidelity for the gate is guaranteed only for fixed ratios between the qubit–qubit exchange (Jqq) and the qubit-redox unit exchange (Jqr). For the measured parameters of 7, 8 and 9 (JqrJ and Jqq=0 in equation (2)) the fidelity of the gate would be low.
Here we propose a more flexible scheme for the gate based on our measured parameters, which works if the two qubits have the same Zeeman energy but one different to that of the switch in the ‘on’ state. This is the case here, where the g values of the qubits and switches are very different, g1z=g3z=1.74 for {Cr7Ni} rings and g2z=2.77 (7), 2.195 (8) and 2.215 (9) for the central node. This difference in the g values means that in magnetic fields of a few T, the ring-central node exchange J is small compared with the difference between the Zeeman energies of the ring and the central node. Hence, the spin state of the central node is nearly frozen in the Sz=−1/2 state and has only tiny virtual fluctuations that lead to an effective interaction between the two {Cr7Ni} qubits given by
where the field is along z, S1 and S3 refer to the spins of the first and second Cr7Ni qubits and
The feasibility of the scheme relies only on the equivalence of the qubits and on the hierarchy of the interactions (see Supplementary Methods for more details).
Therefore when the central node is paramagnetic, the state of the two qubits evolves according to (3), with negligible entanglement with the central node. For specific times, this evolution coincides with the gate apart from single-qubit rotations along z due to the second term in (3).
This perturbative picture is confirmed by the results of detailed calculations for compound 9 using the full Hamiltonian (2) (Fig. 6). Starting from the |10> logical state, we report in Fig. 6a the time evolution of the trimer wavefunction that would implement the gate. In a magnetic field of 3 T, after 4 ns the wavefunction has equal-weight contributions from |10> and |01>, which is the gate, while after 8 ns the states of the two spins are exchanged, that is, we have the |01> state. An extremely good fidelity F (larger than 0.99) for compound 9 is obtained for fields of the order of about 2.5 T or larger, after a suitable gating time tf of the order of a few nanoseconds (Fig. 6b). For such fields the perturbative picture of equation (3) holds very well, and tf is proportional to B (Fig. 6b), which is consistent with the form of the effective qubit–qubit coupling Γ1/B.
Figure 6: Simulation of the gate.
(a) For 9, taking |10>≡|1/2,−1/2>|−1/2>Co as the initial state and an applied field of 3 T, we calculate the time-dependence of the oscillation of the trimer wavefunction between |10> and |01>≡|−1/2,1/2> |−1/2>Co. Other components are negligible. (b) Calculated average fidelity for 9 as a function of the magnetic field B and of the gating time, that is, the time the Co switch is in the on state. The fidelity is defined by , where for a given starting logical state, is the final state after an ideal gate, whereas is the actual final state. The average has been made over four random starting states. For each value of the field, the optimal gating time tf is the one maximizing ca. 4 ns for 3 T as shown in a. The oscillations corresponding to the fringes in the picture are associated with fluctuations of the Co spin state. As long as the frequency of these fluctuations is much larger than 1/tf these fluctuations are negligible, that is, the perturbative description of equation (3) is valid.
Analogous or better fidelities can be obtained with the parameters derived for compounds 7 and 8. Indeed, larger values of |g1zg2z| (as in 7) increase the validity of the perturbative picture (3), even for smaller values of B. Hence, we can exploit the modular strategy and optimize the performance of the gates by targeted chemical manipulations.
Our initial simulations above did not include decoherence. To gain a deeper insight into the performance of the proposed quantum gates, we have performed further simulations that include the effect of both relaxation and pure dephasing, allowing for the finite and measured values of T1 and TM, respectively. For the values of T1 measured by pulsed EPR (>10 μs in all compounds, see the Supplementary Tables 4–12), the effect of relaxation is found to be completely negligible.
Pure dephasing is accounted for by numerically solving the Lindblad master equation for the system density matrix ρ (refs 46, 47):
Here the commutator describes the coherent evolution induced by the full system Hamiltonian H, while the second term describes pure dephasing mechanisms. The subscript k=1, 2 and 3 labels the spins of the qubits and of the switch, while and are spin 1/2 raising and lowering operators.
Using the values of TM of the Cr7Ni qubits measured by pulsed-EPR spectroscopy (between 700 and 1,000 ns, see above) we still find very high fidelities: 99.3% for the implementation of the CNOT gate on compound 5 and 99.6% for the gate on 9. As expected, the fidelities are only marginally affected by decoherence because gating times are much shorter than TM.
It is also worth discussing the role of the decoherence of the switch in the present schemes. As far as the CNOT gate scheme (compound 5) is concerned, the contribution of the central Co switch to decoherence is similar to that of the rings, because the switch is temporarily excited to the SZ=+1/2 state during the CNOT gates. Conversely, only virtual excitations of the switch are involved in the implementation of the gate exploiting the redox-active linker. Hence, the fidelity remains very high even for dephasing time of the switch of the order of the gating time (some ns). This difference is shown by the two colour maps reported in Fig. 7. They show the dependence of the fidelity of a (left) and of a CNOT gate (right) on the dephasing times TM of the rings and of the switch. It is evident that the fidelity of the is practically independent on whereas its dependence on is much more pronounced. Conversely, the implementation of the CNOT on compound 5 leads to a similar dependence of the fidelity on and .
Figure 7: Effect of decoherence on quantum gates.
(a) Fidelity of the as a function of the dephasing times of the rings and of the switch. Remarkably, the fidelity is nearly independent on the
The chemistry described above can be extended to make one-dimensional (1D) chains incorporating the two-Cr7Ni supramolecules, linked by either single cobalt sites or oxo-centred metal triangles. Such 1D chains have already been made involving single Cr7Ni units and [2]- and [3]-rotaxanes and the principles established, especially for 1D chains of rotaxanes48, should work well. The key steps are to include two functional groups per Cr7Ni ring, which has already been done for iso-nicotinate acid, and by functionalizing the termini of the central organic thread of rotaxanes. Inclusion of two different functionalities is more taxing but entirely feasible (see the detailed schemes in Supplementary Figs 30 and 31). Therefore the chemistry is potentially scalable.
While this challenging chemistry proceeds, it is possible to show that the schemes are theoretically scalable. The extension of a two-qubit QG to a multi-qubit register raises important issues concerning the propagation of errors. In such a register, we can identify two main sources of errors, whose effect increases with the number of qubits: pure dephasing and imperfect operation of the switch arising from a residual inter-qubit interaction still present in the ‘off’ state.
Errors induced by pure dephasing (decoherence) increase with the overall computational time. However, a finite chain of qubits with interposed switches would allow simultaneous (parallel) manipulation of non-overlapping parts of the register, which drastically reduces the computation time, and hence decoherence-induced errors, with respect to a serial implementation. For instance, a setup based on compound 5 can be manipulated in parallel for interesting classes of quantum simulation algorithms17. This requires inclusion of two distinct switches in the structure—SwitchA and SwitchB—with different responses to external stimuli, such that the 1D chain has as a repeat pattern -Qubit-SwitchA-Qubit-SwitchB-.
For compounds 7, 8 and 9, the scheme requires a local addressability of individual switches on the chain, which are separated by about 3 nm. For a first proof-of-principle experiment with a single assembly, an STM tip could be used to provide the best control. A parallel implementation of gates would be possible if different Co switches could be addressed individually at the same time. To achieve a scalable structure, a molecular chain of qubits might be layered onto a surface and addressed by means of a cross-bar architecture similar to that proposed in refs 16, 49. It is worth noting that Cr7Ni rings have already been deposited on surfaces, without significant modification of their magnetic properties50.
We now examine the scaling of the errors with the number of qubits in the register. As a first step, we consider the effect of pure dephasing on a set of non-interacting qubits subject to Lindblad (Markovian) dynamics. It can be shown (see ref. 51 and Supplementary Information) that the decoherence error ɛ=1−F2 on N qubits scales at most as , valid for tf<<TM (where tf is the optimal gating time). In the parallel quantum-simulator implementation considered in ref. 17, manipulating a chain of N qubits takes the same time as manipulating the shortest chain that contains the two distinct switches (qubit-SwitchA-qubit-SwitchB-qubit). Therefore tf is limited by the value it assumes for a chain of three qubits, while in a serial scheme tf increases linearly with N. By chemical engineering of Cr7Ni qubits, a TM of 15 μs has been obtained25; this should allow the implementation of around twenty 2-qubit gate operations on a chain consisting of 10 qubits, while keeping the error below 0.1.
Finally, we analyse the consequences of an imperfect operation of the switch and their effect on scalability. For an ideal gate the inter-qubit exchange interaction would be completely turned off when the central node is in its diamagnetic state. Double electron–electron resonance measurements (Supplementary Fig. 27) reveal a very weak, residual interaction between the qubits. The resulting oscillations are on a time scale (ca. 0.2 μs) much longer than our gating times (a few ns) and could be corrected by means of refocusing techniques29. In the case of the CNOT scheme (compound 5), the always ‘on’ exchange interaction between the rings and the Co ion yields a small residual qubit–qubit coupling (Supplementary Methods). As a consequence, an unwanted slow evolution of the qubits occurs on a timescale TUE of the order of a few hundreds of ns. Although this is longer than the CNOT gate time (about 30 ns), larger values of TUE would be needed for performing sequences of many gates. This can be obtained by modifying the molecule to decrease the Co-ring exchange interaction. For instance, TUE can be increased by a factor of 50 by reducing the Co-ring exchange interaction by a factor of three, which could be achieved chemically by adding an extra phenyl group between the ring and the central node. A similar reduction of the residual qubit–qubit coupling can be obtained by increasing the static magnetic field.
In summary, we have described two different schemes for universal quantum information processing, based on either local or global control of the qubit–qubit interaction. We have demonstrated that the flexibility of molecular {Cr7Ni} qubits makes them suitable for the implementation of each of these two schemes, if properly functionalized and linked by means of a supra-molecular design strategy. The two-qubit units can be controlled either magnetically or electrically, and implement either the CNOT or perfectly entangling gates. Our realistic simulations, based on experimental parameters and including decoherence, show that these gates can be performed with remarkably high fidelity. The modular strategy proposed here offers a degree of control in terms of the magnitude of the coupling between molecular spin qubits, the spatial orientation of the modules and the possibility to have a switchable interaction that represents a significant step forward with respect to previous achievements on the assembly of qubits.
Since future developments of quantum technology are unpredictable, we emphasize the importance of pursuing both these two parallel roads towards the actual realization of a quantum computer.
Additional information
Accession codes: The X-ray crystallographic coordinates for structures reported in this Article have been deposited at the Cambridge Crystallographic Data Centre (CCDC), under deposition number CCDC 1029608–1029613 and 1415380–1415383. These data can be obtained free of charge from The Cambridge Crystallographic Data Centre via
How to cite this article: Ferrando-Soria, J. et al. A modular design of molecular qubits to implement universal quantum gates. Nat. Commun. 7:11377 doi: 10.1038/ncomms11377 (2016).
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This work was supported by the EPSRC (UK), the European Commission (Marie Curie Intra-European Fellowship to J.F.-S. (622659) and A.F. (300402)). E.M.P. thanks the Panamanian agency SENACYT-IFARHU for funding. R.E.P.W. thanks the Royal Society for a Wolfson Merit Award. We also thank EPSRC (UK) for funding an X-ray diffractometer (grant number EP/K039547/1) and the National EPR Facility. We thank Diamond Light Source for access to synchrotron X-ray facilities. A.C., S.C. and P.S. acknowledge financial support from the FIRB Project No. RBFR12RPD1 of the Italian Ministry of Education and Research.
Author information
J.F.S., E.M.P. and A.F. synthesized the majority of the compounds discussed; S.A.M. synthesized compound 6. G.T. was involved in design of the synthetic strategies. E.M.P. and I.J.V.-Y. carried out the X-ray single crystal diffraction studies. E.M.P., J.F.-S. and F.T. performed the EPR spectroscopic studies, and J.F.-S. carried out the electrochemical studies reported here. E.J.L.M. helped interpret the EPR spectra. A.C., S.C. and P.S. designed the schemes for quantum gates and A.C. carried out their detailed simulations. R.E.P.W. oversaw the project and wrote the paper with J.F.S., S.C. and P.S., with significant input from all other authors.
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Correspondence to Richard E.P. Winpenny.
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Supplementary Figures 1-31, Supplementary Tables 1-12, Supplementary Methods and Supplementary References (PDF 3345 kb)
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Cif files for compounds 3 to 11 (ZIP 8485 kb)
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Ferrando-Soria, J., Moreno Pineda, E., Chiesa, A. et al. A modular design of molecular qubits to implement universal quantum gates. Nat Commun 7, 11377 (2016).
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200596ac78bd164a | Erwin Schrödinger
Erwin Schrodinger
Erwin Rudolf Josef Alexander Schrödinger, more commonly known as Erwin Schrödinger, was an Austrian physicist and theoretical biologist. One of the founders of quantum mechanics, he is known for the Schrödinger equation and his brilliant contributions to the wave theory of matter. He shared the Nobel Prize for Physics with Paul Dirac in 1933.
Early Life and Contributions:
Erwin Schrödinger was born in Vienna, Austria in 1887. He actually had a Bavarian family that had settled in Vienna long ago. Exceptionally talented and highly educated, he learned almost everything, including the history of Italian painting and most of the recent theories related to theoretical physics.
He became an artillery officer in World War I. He took several positions at Stuttgart, Breslau, and Zurich from 1920 onwards. Zurich proved to be the most productive period for Schrödinger. The tremendous discovery of the Schrodinger Wave Equation took place in 1926. It explained how the quantum state of a physical system changes in time.
Schrödinger went to Berlin in 1927 as the successor of Max Planck. Berlin used to be a center of scientific activity, but he was soon made to leave for Oxford, from where he went to Princeton, and then got back to Austria.
Later Life and Death:
When the Anschluss was over, Erwin Schrödinger made an escape to Italy and then made it to the Institute for Advanced Studies in Dublin. He worked there until his retirement in 1955. He continued to write several important papers. Schrödinger died of tuberculosis in 1961.
He was 73 years old. The Erwin Schrödinger International Institute for Mathematical Physics was in Vienna was named after him in 1993. |
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How to derive the Schrodinger equation for a system with position dependent effective mass? For example, I encountered this equation when I first studied semiconductor hetero-structures. All the books that I referred take the equation as the starting point without at least giving an insight into the form of the Schrodinger equation which reads as
$$\big[-\frac{\hbar^2}{2}\nabla \frac{1}{m^*}\nabla + U \big]\Psi ~=~ E \Psi. $$
I feel that it has something to do with conserving the probability current and should use the continuity equation, but I am not sure.
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Hi ballkikhaal - I edited out the part of your question asking about a book, because we limit the number of book recommendation questions on the site. See if anything in this question helps you. – David Z Oct 10 '12 at 16:23
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5 Answers 5
For derivation of the PDM Schrodinger equation see K. Young, Phys. Rev. B 39, 13434–13441 (1989) "Position-dependent effective mass for inhomogeneous semiconductors". Abstract.:A systematic approach is adopted to extract an effective low-energy Hamiltonian for crystals with a slowly varying inhomogeneity, resolving several controversies. It is shown that the effective mass $m_R$ is, in general, position dependent and enters the kinetic energy operator as $ -\nabla({m_R-1})\nabla/2$. The advantage of using a basis set that exactly diagonalizes the Hamiltonian in the homogeneous limit is emphasized.
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Link in answer(v2) behind paywall. @MKB: In the future please link to abstract pages rather than pdf files, e.g. prb.aps.org/abstract/PRB/v39/i18/p13434_1 – Qmechanic Nov 30 '12 at 22:54
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A Hamiltonian must be self-adjoint. The equation must also reduce to the familiar equation in the case of a constant mass. Now the form of the operator is already determined as the only simple self-adjoint generalization of the position-independent Schroedinger equation to the position dependent case.
If you specialize to 1 dimension, you get the Sturm-Liouville equation. At
you can find a discussion of its self-adjointness. Everything genaralizes to the PDE case.
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In addition to Claudius' and Ron Maimon's answers, I would like to make three comments:
1. Classically, the Hamiltonian function for the effective mass approximation reads $$\tag{1} H({\bf r}, {\bf p})~:=~\frac{{\bf p}^2}{2m^*({\bf r})}+V({\bf r}).$$
2. Quantum mechanically, when one quantizes the classical model (1), one should pick a self-consistent choice for the Hamiltonian operator $\hat{H}$. It is natural to replace the classical variable ${\bf r}$ and ${\bf p}$ in the Hamiltonian (1) with the operators $$\tag{2}\hat{\bf r}~=~{\bf r} \qquad\text{and}\qquad \hat{\bf p}~=~\frac{\hbar}{i}\nabla $$ (in the Schrodinger representation). But which operator ordering prescription should one choose? One natural choice, which (under appropriate boundary conditions) makes the Hamiltonian Hermitian, is $$\tag{3} \hat{H}~:=~\hat{\bf p}\cdot \frac{1}{2m^*(\hat{\bf r})}\hat{\bf p}+V(\hat{\bf r})~=~-\frac{\hbar^2}{2}\nabla\cdot \frac{1}{m^*({\bf r})}\nabla+V({\bf r}).$$
3. Finally, let us mention a somewhat related/generalized Hermitian Hamiltonian operator $$\tag{4} \hat{H}~=~-\frac{\hbar^2}{2}\Delta_g +V({\bf r}), $$ which may give another useful (anisotropic) effective mass model. Here $\Delta_g$ is the Laplace-Beltrami operator for a Riemannian $3\times 3$ metric $g_{ij}=g_{ij}({\bf r})$, which, roughly speaking, may be viewed as an (anisotropic) effective mass tensor.
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The derivation is straightforward if you consider the source of the effective mass is a slowly varying hopping parameter on a tight-binding (lattice particle) model. Here you have a particle on a square lattice with a probability amplitude to go left, right, up and down, forward, backwards. The main physical requirement is Hermiticity, which in 1d can be used (with a phase choice on the wavefunction) to turn the phase everywhere real.
Once you do this, there is a real amplitude at site n to hop one square to the right r(n) and an amplitude to hop one square to the left, which by hermiticity and reality, must be r(n-1)--- it must be the complex conjugate of the amplitude to hop right from position n-1. So the amplitude equation is
$$ i{dC\over dt} = r(n-1) C_{n-1} - (r(n-1)+r(n))C_n + r(n) C_{n+1} $$
This is, when r is slowly varying, equivalent to the continuum equation found by Taylor expanding and keeping only the most relevant terms:
$$ i {d\psi \over dt} = {1\over 2} {\partial \over \partial x} (r(x) {\partial\over \partial x} \psi(x)) $$
As Feynman noted but never published (Dyson published this comment posthumously, in a paper in American Journal of Physics titles something like "Feynman's derivation of the Maxwell equations from Schrodinger equation"), Dirac's phase trick doesn't work in higher dimensions, because you can't fix all the phases. Then the commutators have a magnetic field addition, and to make it consistent, the magnetic field has to end up obeying Maxwell's equation, since the phase rotation gives a U(1) symmetry. This is not a true derivation of Maxwell's physics from quantum mechanics, it is just a way of showing you need the extra assumption of CP invariance to make the hopping hamiltonian real (which is true).
Then with the extra assumption, you just get
$$ i {d\psi\over dt} = {1\over 2} \nabla \cdot (t(x) \nabla \psi) + V(x) \psi $$
Where I have added back the potential. This is the continuum limit of a tight binding model with spatially slowly varying hopping, or inverse effective mass.
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I would be very surprised if you managed to find a strict mathematical derivation of the Schrödinger equation anywhere – at least I have not encountered one until now. However, it might be worth pointing out that the ‘general’ time-dependent Schrödinger equation, which is often taken as an axiom of quantum mechanics, is usually
$$i \hbar \partial_t \Psi = \hat H \Psi \quad .$$
In the case of a stationary Hamiltonian (usually $U(x,t) \equiv U(x)$), this equation separates and you get the stationary Schrödinger equation, namely
$$ \hat H \Psi = E \Psi \quad ,$$
that is, an eigenvalue equation for the Hamiltonian.
Given this equation, it is then relatively simple to work out the form of the Hamiltonian (in your case, $-\frac{\hbar^2}{2} \nabla \frac{1}{m^\star} \nabla + U$) and plug it into the equation. The exact form of the Hamiltonian is usually guesswork based on observation and analogies to classical mechanics. In general, we have
$$ \hat H = \hat T + \hat U $$
where $\hat T$ and $\hat U$ denote the operators for kinetic and potential energy, correspondingly.
It is worth noting that you can derive the continuity equation (which is identical to probability conservation in this case) from the Schrödinger equation by adding the complex conjugate of the Schrödinger equation to itself.
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thank you for the answer but what i am asking is how to derive it within the regime of effective mass approximation and that too when the mass has a spatial profile... for example in the case of Al/GaAs high electron mobility transistor we have a position dependent mass. – baalkikhaal Oct 10 '12 at 16:55
Are you looking for a derivation of the Hamiltonian $\hat H$ or the Schrödinger equation? I am positive that neither probability conservation nor the continuity equation have anything to do with the earlier. – Claudius Oct 10 '12 at 16:57
I have come across this equation in Hamaguchi on page 347 <books.google.co.in/…; – baalkikhaal Oct 10 '12 at 18:13
Where do you think Schrodinger equations come from, if not by some sort of a derivation? – Ron Maimon Oct 10 '12 at 19:31
The answer is not by guesswork, it is from the tight-binding approximation with CP invariance to guarantee that the hopping parameter is real, and then Hermiticity guarantees the hopping is symmetric and equal to the given Hamiltonian. If the hopping is slowly locally varying, then you get the Hamiltonian they say. The Schrodinger equation which is axiomatic is not as specific as the Schrodinger equation in space, which has a specific ansatz for the kinetic term which can be justified from tight binding, as Feynman does in his lectures. – Ron Maimon Oct 10 '12 at 19:47
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dc73cdb721704376 | Viewpoint: Weyl electrons kiss
Leon Balents, Kavli Institute for Theoretical Physics, University of California, Santa Barbara, CA 93106, USA
Published May 2, 2011 | Physics 4, 36 (2011) | DOI: 10.1103/Physics.4.36
Topological semimetal and Fermi-arc surface states in the electronic structure of pyrochlore iridates
Xiangang Wan, Ari M. Turner, Ashvin Vishwanath, and Sergey Y. Savrasov
Published May 2, 2011 | PDF (free)
+Enlarge image Figure 1
Figure 1 Schematic image of the structure of the Weyl semimetal in momentum space. Two diabolical points are shown in red, within the bulk three-dimensional Brillouin zone. The excitations near each diabolical point behave like Weyl fermions. Each point is a source or sink of the flux (i.e., a monopole in momentum space) of the U(1) Berry connection, defined from the Bloch wave functions, as indicated by the blue arrows. The grey plane indicates the surface Brillouin zone, which is a projection of the bulk one. Wan et al. show that an odd number of surface Fermi arcs terminate at the projection of each diabolical point, as drawn here in yellow. In the iridium pyrochlores studied in the paper, a nonschematic picture would be significantly more complex, as there are 24 diabolical points rather than 2.
Topology, the mathematical description of the robustness of form, appears throughout physics, and provides strong constraints on many physical systems. It has long been known that it plays a key role in understanding the exotic phenomena of the quantum Hall effect. Recently, it has been found to generate robust and interesting bulk and surface phenomena in “ordinary” band insulators described by the old Bloch theory of solids. Such “topological insulators,” insulating in the bulk and metallic on the surface, occur in the presence of strong spin-orbit coupling in certain crystals, with unbroken time-reversal symmetry [1].
It is usually believed that such topological physics is obliterated in materials where magnetic ordering breaks time-reversal symmetry. This is by far the most common fate for transition-metal compounds that manage to be insulators—so called “Mott insulators,” which owe their lack of conduction to the strong Coulomb repulsion between electrons. In an article appearing in Physical Review B, Xiangang Wan from Nanjing University, China, and collaborators from the University of California and the Lawrence Berkeley National Laboratory, US, show that this is not necessarily the case, and describe a remarkable electronic structure with topological aspects that is unique to such (antiferro-)magnetic materials [2]. The state they describe is remarkable in possessing interesting low-energy electron states in the bulk and at the surface, linked by topology. In contrast, topological insulators, like quantum Hall states, possess low-energy electronic states only at the surface.
The theory of Wan et al., which uses the LDA+U numerical method, is a type of mean field theory. As such, the low-energy quasiparticle excitations are described simply by noninteracting electrons in a background electrostatic potential and, in the case of a magnetically ordered phase, by a spatially periodic exchange field. It is possible to follow the evolution of the electronic states as a function of the U parameter, which is used to model the strength of Coulomb correlations. They apply the technique to iridium pyrochlores, R2Ir2O7, where R is a rare earth element. These materials are known to exhibit metal-insulator transitions (see, e.g., Ref. [3]), indicating substantial correlations, and are characterized by strong spin-orbit coupling due to the heavy element Ir (iridium). In the intermediate range of U, which they suggest is relevant for these compounds, Wan et al. find an antiferromagnetic ground state with the band structure of a “zero-gap semimetal,” in which the conduction and valence bands “kiss” at a discrete number (24!) of momenta. The dispersion of the bands approaching each touching point is linear, reminiscent of massless Dirac fermions such as those observed in graphene.
This would be interesting in itself, but there are important differences from graphene. Because of the antiferromagnetism, time-reversal symmetry is broken, and as a consequence, despite the centrosymmetric nature of the crystals in question, the bands are nondegenerate. Thus two—and only two—states are degenerate at each touching point, unlike in graphene where there are four. In fact, the kissing bands found by Wan et al. are an example of accidental degeneracy in quantum mechanics, a subject discussed in the early days of quantum theory by von Neumann and Wigner (1929), and applied to band theory by Herring (1937). The phenomenon of level repulsion in quantum mechanics tends to prevent such band crossings. To force two levels to be degenerate, one must consider the 2×2 Hamiltonian matrix projected into this subspace: not only must the two diagonal elements be made equal, the two off-diagonal elements must be made to vanish. This requires three real parameters to be tuned to achieve degeneracy. Thus, without additional symmetry constraints, such accidental degeneracies are vanishingly improbable in one and two dimensions, but can occur as isolated points in momentum space in three dimensions (the three components of the momentum serving as tuning parameters). An accidental touching of this type is called a diabolical point. The 2×2 matrix Schrödinger equation in the vicinity of this point is mathematically similar to a two-component Dirac-like one, known as the Weyl equation. Thus the low-energy electrons in this state behave like Weyl fermions. A property of such a diabolical point is that it cannot be removed by any small perturbation, but may only disappear by annihilation with another diabolical point.
Actually such diabolical points were suggested previously in a very similar context by Murakami [4], who argued that such a semimetallic state would naturally arise as an intermediate between a topological insulator and a normal band insulator. That theory does not directly apply here, since Murukami assumed time-reversal symmetry, which is broken in Wan et al.’s calculations. However, inversion symmetry plays a similar role, and indeed the latter authors find that the Weyl semimetal is intermediate between an “ordinary” Mott insulator and an “axion insulator,” somewhat analogous to the topological insulator. The axion insulator has a quantized magnetoelectric response identical to that of a (strong) topological insulator, but lacks the protected surface states of the topological insulator.
What is really new and striking about the recent paper is the implications for surface states. Remarkably, they find that certain surfaces (e.g., <110> and <111> faces) have bound states at the Fermi energy, and that these states do not form the usual closed Fermi surfaces found in 2d or 3d metals. Instead, the states at the Fermi energy form open “arcs,” terminating at the projection of diabolical points onto the surface Brillouin zone (see Fig. 1). Fermi arcs have appeared before in physics in experimental studies of high-temperature cuprate superconductors. However, in that context they are mysterious and puzzling, because they would seem to be prohibited by topology. The Fermi surface is by definition the boundary between occupied and unoccupied states, and if it terminates then one could go “around” the termination point and smoothly change from one to another, which is impossible at zero temperature. At a surface, this paradox is avoided because the surface states may unbind into the bulk when going around the end of a Fermi arc.
This theoretical proposal provides plenty of motivation for future experiments. Observation of the Fermi arcs would be striking, and provide a useful metric to gauge the empirical ones seen in the cuprates. The bulk Weyl fermions are also interesting and would be exciting to try to observe in transport. Simple phase-space considerations suggest that the low-energy electrons should be remarkably resistant to scattering by impurities. Most of this theory could apply to many other materials—the only necessary conditions are significant spin-orbit coupling and antiferromagnetic order preserving inversion symmetry (and the latter is only essential for some of the physics). More generally, the work widens the range of unusual states of matter that have been proposed to arise in the regime of strong spin-orbit coupling and intermediate correlation. Despite the uncommon aspects described above, Wan et al.’s work is a mean-field theory, and yet more exotic possibilities have been suggested that are not describable in this way (e.g., Ref. [5]). Hopefully this is just the beginning of the theoretical and experimental exploration of this fascinating regime.
1. M. Z. Hasan and C. L. Kane, Rev. Mod. Phys. 82, 3045 (2010).
2. X. Wan, A. M. Turner, A. Vishwanath, and S. Y. Savrasov, Phys. Rev. B 83, 205101 (2011).
3. K. Matsuhira and et al, J. Phys. Soc. Jpn. 76, 043706 (2007).
4. S. Murakami, New J. Phys. 9,356 (2007).
5. D. A. Pesin and L. Balents, Nature Phys. 6, 376 (2010).
About the Author: Leon Balents
Leon Balents
Leon Balents is Professor of Physics in the Physics Department and a Permanent Member of the Kavli Institute for Theoretical Physics at the University of California, Santa Barbara. He is active in theoretical condensed matter physics, where his research interests include quantum magnetism, strongly correlated electrons, low-dimensional systems, and topological phenomena in solids. He received his Ph.D. in physics from Harvard University in 1994 and has been on the faculty at UCSB since 1999.
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aabc00ec0533b765 | From Wikipedia, the free encyclopedia
(Redirected from Electrons)
Jump to: navigation, search
For other uses, see Electron (disambiguation).
Hydrogen atom orbitals at different energy levels. The brighter areas are where you are most likely to find an electron at any given time.
Composition Elementary particle[1]
Statistics Fermionic
Generation First
Interactions Gravity, Electromagnetic, Weak
Symbol e, β
Antiparticle Positron (also called antielectron)
Theorized Richard Laming (1838–1851),[2]
G. Johnstone Stoney (1874) and others.[3][4]
Discovered J. J. Thomson (1897)[5]
Mass 9.10938291(40)×10−31 kg[6]
5.4857990946(22)×10−4 u[6]
[1822.8884845(14)]−1 u[note 1]
0.510998928(11) MeV/c2[6]
Electric charge −1 e[note 2]
−1.602176565(35)×10−19 C[6]
−4.80320451(10)×10−10 esu
Magnetic moment −1.00115965218076(27) μB[6]
Spin 12
The electron is a subatomic particle, symbol e or β, with a negative elementary electric charge.[7] Electrons belong to the first generation of the lepton particle family,[8] and are generally thought to be elementary particles because they have no known components or substructure.[1] The electron has a mass that is approximately 1/1836 that of the proton.[9] Quantum mechanical properties of the electron include an intrinsic angular momentum (spin) of a half-integer value in units of ħ, which means that it is a fermion. Being fermions, no two electrons can occupy the same quantum state, in accordance with the Pauli exclusion principle.[8] Like all matter, electrons have properties of both particles and waves, and so can collide with other particles and can be diffracted like light. The wave properties of electrons are easier to observe with experiments than those of other particles like neutrons and protons because electrons have a lower mass and hence a higher De Broglie wavelength for typical energies.
Many physical phenomena involve electrons in an essential role, such as electricity, magnetism, and thermal conductivity, and they also participate in gravitational, electromagnetic and weak interactions.[10] An electron in space generates an electric field surrounding it. An electron moving relative to an observer generates a magnetic field. External magnetic fields deflect an electron. Electrons radiate or absorb energy in the form of photons when accelerated. Laboratory instruments are capable of containing and observing individual electrons as well as electron plasma using electromagnetic fields, whereas dedicated telescopes can detect electron plasma in outer space. Electrons have many applications, including electronics, welding, cathode ray tubes, electron microscopes, radiation therapy, lasers, gaseous ionization detectors and particle accelerators.
Interactions involving electrons and other subatomic particles are of interest in fields such as chemistry and nuclear physics. The Coulomb force interaction between positive protons inside atomic nuclei and negative electrons composes atoms. Ionization or changes in the proportions of particles changes the binding energy of the system. The exchange or sharing of the electrons between two or more atoms is the main cause of chemical bonding.[11] British natural philosopher Richard Laming first hypothesized the concept of an indivisible quantity of electric charge to explain the chemical properties of atoms in 1838;[3] Irish physicist George Johnstone Stoney named this charge 'electron' in 1891, and J. J. Thomson and his team of British physicists identified it as a particle in 1897.[5][12][13] Electrons can also participate in nuclear reactions, such as nucleosynthesis in stars, where they are known as beta particles. Electrons may be created through beta decay of radioactive isotopes and in high-energy collisions, for instance when cosmic rays enter the atmosphere. The antiparticle of the electron is called the positron; it is identical to the electron except that it carries electrical and other charges of the opposite sign. When an electron collides with a positron, both particles may be totally annihilated, producing gamma ray photons.
The ancient Greeks noticed that amber attracted small objects when rubbed with fur. Along with lightning, this phenomenon is one of humanity's earliest recorded experiences with electricity. [14] In his 1600 treatise De Magnete, the English scientist William Gilbert coined the New Latin term electricus, to refer to this property of attracting small objects after being rubbed. [15] Both electric and electricity are derived from the Latin ēlectrum (also the root of the alloy of the same name), which came from the Greek word for amber, ἤλεκτρον (ēlektron).
In the early 1700s, Francis Hauksbee and French chemist Charles François de Fay independently discovered what they believed were two kinds of frictional electricity—one generated from rubbing glass, the other from rubbing resin. From this, Du Fay theorized that electricity consists of two electrical fluids, vitreous and resinous, that are separated by friction, and that neutralize each other when combined.[16] A decade later Benjamin Franklin proposed that electricity was not from different types of electrical fluid, but the same electrical fluid under different pressures. He gave them the modern charge nomenclature of positive and negative respectively.[17] Franklin thought of the charge carrier as being positive, but he did not correctly identify which situation was a surplus of the charge carrier, and which situation was a deficit.[18]
In 1891 Stoney coined the term electron to describe these elementary charges, writing later in 1894: "... an estimate was made of the actual amount of this most remarkable fundamental unit of electricity, for which I have since ventured to suggest the name electron".[20] The word electron is a combination of the words electr(ic) and (i)on.[21] The suffix -on which is now used to designate other subatomic particles, such as a proton or neutron, is in turn derived from electron.[22][23]
A round glass vacuum tube with a glowing circular beam inside
A beam of electrons deflected in a circle by a magnetic field[24]
The German physicist Johann Wilhelm Hittorf studied electrical conductivity in rarefied gases: in 1869, he discovered a glow emitted from the cathode that increased in size with decrease in gas pressure. In 1876, the German physicist Eugen Goldstein showed that the rays from this glow cast a shadow, and he dubbed the rays cathode rays.[25] During the 1870s, the English chemist and physicist Sir William Crookes developed the first cathode ray tube to have a high vacuum inside.[26] He then showed that the luminescence rays appearing within the tube carried energy and moved from the cathode to the anode. Furthermore, by applying a magnetic field, he was able to deflect the rays, thereby demonstrating that the beam behaved as though it were negatively charged.[27][28] In 1879, he proposed that these properties could be explained by what he termed 'radiant matter'. He suggested that this was a fourth state of matter, consisting of negatively charged molecules that were being projected with high velocity from the cathode.[29]
In 1892 Hendrik Lorentz suggested that the mass of these particles (electrons) could be a consequence of their electric charge.[31]
In 1896, the British physicist J. J. Thomson, with his colleagues John S. Townsend and H. A. Wilson,[12] performed experiments indicating that cathode rays really were unique particles, rather than waves, atoms or molecules as was believed earlier.[5] Thomson made good estimates of both the charge e and the mass m, finding that cathode ray particles, which he called "corpuscles," had perhaps one thousandth of the mass of the least massive ion known: hydrogen.[5][13] He showed that their charge to mass ratio, e/m, was independent of cathode material. He further showed that the negatively charged particles produced by radioactive materials, by heated materials and by illuminated materials were universal.[5][32] The name electron was again proposed for these particles by the Irish physicist George F. Fitzgerald, and the name has since gained universal acceptance.[27]
Robert Millikan
While studying naturally fluorescing minerals in 1896, the French physicist Henri Becquerel discovered that they emitted radiation without any exposure to an external energy source. These radioactive materials became the subject of much interest by scientists, including the New Zealand physicist Ernest Rutherford who discovered they emitted particles. He designated these particles alpha and beta, on the basis of their ability to penetrate matter.[33] In 1900, Becquerel showed that the beta rays emitted by radium could be deflected by an electric field, and that their mass-to-charge ratio was the same as for cathode rays.[34] This evidence strengthened the view that electrons existed as components of atoms.[35][36]
Around the beginning of the twentieth century, it was found that under certain conditions a fast-moving charged particle caused a condensation of supersaturated water vapor along its path. In 1911, Charles Wilson used this principle to devise his cloud chamber so he could photograph the tracks of charged particles, such as fast-moving electrons.[39]
Atomic theory[edit]
By 1914, experiments by physicists Ernest Rutherford, Henry Moseley, James Franck and Gustav Hertz had largely established the structure of an atom as a dense nucleus of positive charge surrounded by lower-mass electrons.[40] In 1913, Danish physicist Niels Bohr postulated that electrons resided in quantized energy states, with the energy determined by the angular momentum of the electron's orbits about the nucleus. The electrons could move between these states, or orbits, by the emission or absorption of photons at specific frequencies. By means of these quantized orbits, he accurately explained the spectral lines of the hydrogen atom.[41] However, Bohr's model failed to account for the relative intensities of the spectral lines and it was unsuccessful in explaining the spectra of more complex atoms.[40]
Chemical bonds between atoms were explained by Gilbert Newton Lewis, who in 1916 proposed that a covalent bond between two atoms is maintained by a pair of electrons shared between them.[42] Later, in 1927, Walter Heitler and Fritz London gave the full explanation of the electron-pair formation and chemical bonding in terms of quantum mechanics.[43] In 1919, the American chemist Irving Langmuir elaborated on the Lewis' static model of the atom and suggested that all electrons were distributed in successive "concentric (nearly) spherical shells, all of equal thickness".[44] The shells were, in turn, divided by him in a number of cells each containing one pair of electrons. With this model Langmuir was able to qualitatively explain the chemical properties of all elements in the periodic table,[43] which were known to largely repeat themselves according to the periodic law.[45]
Quantum mechanics[edit]
In his 1924 dissertation Recherches sur la théorie des quanta (Research on Quantum Theory), French physicist Louis de Broglie hypothesized that all matter possesses a de Broglie wave similar to light.[49] That is, under the appropriate conditions, electrons and other matter would show properties of either particles or waves. The corpuscular properties of a particle are demonstrated when it is shown to have a localized position in space along its trajectory at any given moment.[50] Wave-like nature is observed, for example, when a beam of light is passed through parallel slits and creates interference patterns. In 1927, the interference effect was found in a beam of electrons by English physicist George Paget Thomson with a thin metal film and by American physicists Clinton Davisson and Lester Germer using a crystal of nickel.[51]
A symmetrical blue cloud that decreases in intensity from the center outward
De Broglie's prediction of a wave nature for electrons led Erwin Schrödinger to postulate a wave equation for electrons moving under the influence of the nucleus in the atom. In 1926, this equation, the Schrödinger equation, successfully described how electron waves propagated.[52] Rather than yielding a solution that determined the location of an electron over time, this wave equation also could be used to predict the probability of finding an electron near a position, especially a position near where the electron was bound in space, for which the electron wave equations did not change in time. This approach led to a second formulation of quantum mechanics (the first being by Heisenberg in 1925), and solutions of Schrödinger's equation, like Heisenberg's, provided derivations of the energy states of an electron in a hydrogen atom that were equivalent to those that had been derived first by Bohr in 1913, and that were known to reproduce the hydrogen spectrum.[53] Once spin and the interaction between multiple electrons were considered, quantum mechanics later made it possible to predict the configuration of electrons in atoms with higher atomic numbers than hydrogen.[54]
In 1928, building on Wolfgang Pauli's work, Paul Dirac produced a model of the electron – the Dirac equation, consistent with relativity theory, by applying relativistic and symmetry considerations to the hamiltonian formulation of the quantum mechanics of the electro-magnetic field.[55] To resolve some problems within his relativistic equation, in 1930 Dirac developed a model of the vacuum as an infinite sea of particles having negative energy, which was dubbed the Dirac sea. This led him to predict the existence of a positron, the antimatter counterpart of the electron.[56] This particle was discovered in 1932 by Carl Anderson, who proposed calling standard electrons negatrons, and using electron as a generic term to describe both the positively and negatively charged variants.
In 1947 Willis Lamb, working in collaboration with graduate student Robert Retherford, found that certain quantum states of hydrogen atom, which should have the same energy, were shifted in relation to each other, the difference being the Lamb shift. About the same time, Polykarp Kusch, working with Henry M. Foley, discovered the magnetic moment of the electron is slightly larger than predicted by Dirac's theory. This small difference was later called anomalous magnetic dipole moment of the electron. This difference was later explained by the theory of quantum electrodynamics, developed by Sin-Itiro Tomonaga, Julian Schwinger and Richard Feynman in the late 1940s.[57]
Particle accelerators[edit]
With a beam energy of 1.5 GeV, the first high-energy particle collider was ADONE, which began operations in 1968.[60] This device accelerated electrons and positrons in opposite directions, effectively doubling the energy of their collision when compared to striking a static target with an electron.[61] The Large Electron–Positron Collider (LEP) at CERN, which was operational from 1989 to 2000, achieved collision energies of 209 GeV and made important measurements for the Standard Model of particle physics.[62][63]
Confinement of individual electrons[edit]
Fundamental properties[edit]
The invariant mass of an electron is approximately 9.109×10−31 kilograms,[67] or 5.489×10−4 atomic mass units. On the basis of Einstein's principle of mass–energy equivalence, this mass corresponds to a rest energy of 0.511 MeV. The ratio between the mass of a proton and that of an electron is about 1836.[9][68] Astronomical measurements show that the proton-to-electron mass ratio has held the same value for at least half the age of the universe, as is predicted by the Standard Model.[69]
Electrons have an electric charge of −1.602×10−19 coulomb,[67] which is used as a standard unit of charge for subatomic particles, and is also called the elementary charge. This elementary charge has a relative standard uncertainty of 2.2×10−8.[67] Within the limits of experimental accuracy, the electron charge is identical to the charge of a proton, but with the opposite sign.[70] As the symbol e is used for the elementary charge, the electron is commonly symbolized by e, where the minus sign indicates the negative charge. The positron is symbolized by e+ because it has the same properties as the electron but with a positive rather than negative charge.[66][67]
The electron has an intrinsic angular momentum or spin of 12.[67] This property is usually stated by referring to the electron as a spin-12 particle.[66] For such particles the spin magnitude is 32 ħ.[note 3] while the result of the measurement of a projection of the spin on any axis can only be ±ħ2. In addition to spin, the electron has an intrinsic magnetic moment along its spin axis.[67] It is approximately equal to one Bohr magneton,[71][note 4] which is a physical constant equal to 9.27400915(23)×10−24 joules per tesla.[67] The orientation of the spin with respect to the momentum of the electron defines the property of elementary particles known as helicity.[72]
The electron has no known substructure.[1][73] and it is assumed to be a point particle with a point charge and no spatial extent.[8] In classical physics, the angular momentum and magnetic moment of an object depend upon its physical dimensions. Hence, the concept of a dimensionless electron possessing these properties might seem paradoxical and inconsistent to experimental observations in Penning traps which point to finite non-zero radius of the electron. A possible explanation of this paradoxical situation is given below in the "Virtual particles" subsection by taking into consideration the Foldy-Wouthuysen transformation. The issue of the radius of the electron is a challenging problem of the modern theoretical physics. The admission of the hypothesis of a finite radius of the electron is incompatible to the premises of the theory of relativity. On the other hand, a point-like electron (zero radius) generates serious mathematical difficulties due to the self-energy of the electron tending to infinity.[74] These aspects have been analyzed in detail by Dmitri Ivanenko and Arseny Sokolov.
Observation of a single electron in a Penning trap shows the upper limit of the particle's radius is 10−22 meters.[75] There is a physical constant called the "classical electron radius", with the much larger value of 2.8179×10−15 m, greater than the radius of the proton. However, the terminology comes from a simplistic calculation that ignores the effects of quantum mechanics; in reality, the so-called classical electron radius has little to do with the true fundamental structure of the electron.[76][note 5]
Quantum properties[edit]
Virtual particles[edit]
Main article: Virtual particle
The interaction with virtual particles also explains the small (about 0.1%) deviation of the intrinsic magnetic moment of the electron from the Bohr magneton (the anomalous magnetic moment).[71][87] The extraordinarily precise agreement of this predicted difference with the experimentally determined value is viewed as one of the great achievements of quantum electrodynamics.[88]
The apparent paradox (mentioned above in the properties subsection) of a point particle electron having intrinsic angular momentum and magnetic moment can be explained by the formation of virtual photons in the electric field generated by the electron. These photons cause the electron to shift about in a jittery fashion (known as zitterbewegung),[89] which results in a net circular motion with precession. This motion produces both the spin and the magnetic moment of the electron.[8][90] In atoms, this creation of virtual photons explains the Lamb shift observed in spectral lines.[83]
An electron generates an electric field that exerts an attractive force on a particle with a positive charge, such as the proton, and a repulsive force on a particle with a negative charge. The strength of this force is determined by Coulomb's inverse square law.[91] When an electron is in motion, it generates a magnetic field.[80]:140 The Ampère-Maxwell law relates the magnetic field to the mass motion of electrons (the current) with respect to an observer. This property of induction supplies the magnetic field that drives an electric motor.[92] The electromagnetic field of an arbitrary moving charged particle is expressed by the Liénard–Wiechert potentials, which are valid even when the particle's speed is close to that of light (relativistic).
A graph with arcs showing the motion of charged particles
When an electron is moving through a magnetic field, it is subject to the Lorentz force that acts perpendicularly to the plane defined by the magnetic field and the electron velocity. This centripetal force causes the electron to follow a helical trajectory through the field at a radius called the gyroradius. The acceleration from this curving motion induces the electron to radiate energy in the form of synchrotron radiation.[80]:160[93][note 6] The energy emission in turn causes a recoil of the electron, known as the Abraham–Lorentz–Dirac Force, which creates a friction that slows the electron. This force is caused by a back-reaction of the electron's own field upon itself.[94]
Photons mediate electromagnetic interactions between particles in quantum electrodynamics. An isolated electron at a constant velocity cannot emit or absorb a real photon; doing so would violate conservation of energy and momentum. Instead, virtual photons can transfer momentum between two charged particles. This exchange of virtual photons, for example, generates the Coulomb force.[95] Energy emission can occur when a moving electron is deflected by a charged particle, such as a proton. The acceleration of the electron results in the emission of Bremsstrahlung radiation.[96]
An inelastic collision between a photon (light) and a solitary (free) electron is called Compton scattering. This collision results in a transfer of momentum and energy between the particles, which modifies the wavelength of the photon by an amount called the Compton shift.[note 7] The maximum magnitude of this wavelength shift is h/mec, which is known as the Compton wavelength.[97] For an electron, it has a value of 2.43×10−12 m.[67] When the wavelength of the light is long (for instance, the wavelength of the visible light is 0.4–0.7 μm) the wavelength shift becomes negligible. Such interaction between the light and free electrons is called Thomson scattering or Linear Thomson scattering.[98]
When electrons and positrons collide, they annihilate each other, giving rise to two or more gamma ray photons. If the electron and positron have negligible momentum, a positronium atom can form before annihilation results in two or three gamma ray photons totalling 1.022 MeV.[99][100] On the other hand, high-energy photons may transform into an electron and a positron by a process called pair production, but only in the presence of a nearby charged particle, such as a nucleus.[101][102]
Atoms and molecules[edit]
Main article: Atom
Probability densities for the first few hydrogen atom orbitals, seen in cross-section. The energy level of a bound electron determines the orbital it occupies, and the color reflects the probability of finding the electron at a given position.
Electrons can transfer between different orbitals by the emission or absorption of photons with an energy that matches the difference in potential.[104] Other methods of orbital transfer include collisions with particles, such as electrons, and the Auger effect.[105] To escape the atom, the energy of the electron must be increased above its binding energy to the atom. This occurs, for example, with the photoelectric effect, where an incident photon exceeding the atom's ionization energy is absorbed by the electron.[106]
The chemical bond between atoms occurs as a result of electromagnetic interactions, as described by the laws of quantum mechanics.[108] The strongest bonds are formed by the sharing or transfer of electrons between atoms, allowing the formation of molecules.[11] Within a molecule, electrons move under the influence of several nuclei, and occupy molecular orbitals; much as they can occupy atomic orbitals in isolated atoms.[109] A fundamental factor in these molecular structures is the existence of electron pairs. These are electrons with opposed spins, allowing them to occupy the same molecular orbital without violating the Pauli exclusion principle (much like in atoms). Different molecular orbitals have different spatial distribution of the electron density. For instance, in bonded pairs (i.e. in the pairs that actually bind atoms together) electrons can be found with the maximal probability in a relatively small volume between the nuclei. On the contrary, in non-bonded pairs electrons are distributed in a large volume around nuclei.[110]
Four bolts of lightning strike the ground
A lightning discharge consists primarily of a flow of electrons.[111] The electric potential needed for lightning may be generated by a triboelectric effect.[112][113]
When cooled below a point called the critical temperature, materials can undergo a phase transition in which they lose all resistivity to electrical current, in a process known as superconductivity. In BCS theory, this behavior is modeled by pairs of electrons entering a quantum state known as a Bose–Einstein condensate. These Cooper pairs have their motion coupled to nearby matter via lattice vibrations called phonons, thereby avoiding the collisions with atoms that normally create electrical resistance.[122] (Cooper pairs have a radius of roughly 100 nm, so they can overlap each other.)[123] However, the mechanism by which higher temperature superconductors operate remains uncertain.
Electrons inside conducting solids, which are quasi-particles themselves, when tightly confined at temperatures close to absolute zero, behave as though they had split into three other quasiparticles: spinons, Orbitons and holons.[124][125] The former carries spin and magnetic moment, the next carries its orbital location while the latter electrical charge.
Motion and energy[edit]
The plot starts at zero and curves sharply upward toward the right
where me is the mass of electron. For example, the Stanford linear accelerator can accelerate an electron to roughly 51 GeV.[127] Since an electron behaves as a wave, at a given velocity it has a characteristic de Broglie wavelength. This is given by λe = h/p where h is the Planck constant and p is the momentum.[49] For the 51 GeV electron above, the wavelength is about 2.4×10−17 m, small enough to explore structures well below the size of an atomic nucleus.[128]
γ + γe+ + e
For reasons that remain uncertain, during the process of leptogenesis there was an excess in the number of electrons over positrons.[131] Hence, about one electron in every billion survived the annihilation process. This excess matched the excess of protons over antiprotons, in a condition known as baryon asymmetry, resulting in a net charge of zero for the universe.[132][133] The surviving protons and neutrons began to participate in reactions with each other—in the process known as nucleosynthesis, forming isotopes of hydrogen and helium, with trace amounts of lithium. This process peaked after about five minutes.[134] Any leftover neutrons underwent negative beta decay with a half-life of about a thousand seconds, releasing a proton and electron in the process,
np + e + ν
For about the next 300000400000 years, the excess electrons remained too energetic to bind with atomic nuclei.[135] What followed is a period known as recombination, when neutral atoms were formed and the expanding universe became transparent to radiation.[136]
Roughly one million years after the big bang, the first generation of stars began to form.[136] Within a star, stellar nucleosynthesis results in the production of positrons from the fusion of atomic nuclei. These antimatter particles immediately annihilate with electrons, releasing gamma rays. The net result is a steady reduction in the number of electrons, and a matching increase in the number of neutrons. However, the process of stellar evolution can result in the synthesis of radioactive isotopes. Selected isotopes can subsequently undergo negative beta decay, emitting an electron and antineutrino from the nucleus.[137] An example is the cobalt-60 (60Co) isotope, which decays to form nickel-60 (60Ni).[138]
A branching tree representing the particle production
At the end of its lifetime, a star with more than about 20 solar masses can undergo gravitational collapse to form a black hole.[139] According to classical physics, these massive stellar objects exert a gravitational attraction that is strong enough to prevent anything, even electromagnetic radiation, from escaping past the Schwarzschild radius. However, quantum mechanical effects are believed to potentially allow the emission of Hawking radiation at this distance. Electrons (and positrons) are thought to be created at the event horizon of these stellar remnants.
Cosmic rays are particles traveling through space with high energies. Energy events as high as 3.0×1020 eV have been recorded.[142] When these particles collide with nucleons in the Earth's atmosphere, a shower of particles is generated, including pions.[143] More than half of the cosmic radiation observed from the Earth's surface consists of muons. The particle called a muon is a lepton produced in the upper atmosphere by the decay of a pion.
πμ + ν
A muon, in turn, can decay to form an electron or positron.[144]
μe + ν
+ ν
Aurorae are mostly caused by energetic electrons precipitating into the atmosphere.[145]
The frequency of a photon is proportional to its energy. As a bound electron transitions between different energy levels of an atom, it absorbs or emits photons at characteristic frequencies. For instance, when atoms are irradiated by a source with a broad spectrum, distinct absorption lines appear in the spectrum of transmitted radiation. Each element or molecule displays a characteristic set of spectral lines, such as the hydrogen spectral series. Spectroscopic measurements of the strength and width of these lines allow the composition and physical properties of a substance to be determined.[147][148]
In laboratory conditions, the interactions of individual electrons can be observed by means of particle detectors, which allow measurement of specific properties such as energy, spin and charge.[106] The development of the Paul trap and Penning trap allows charged particles to be contained within a small region for long durations. This enables precise measurements of the particle properties. For example, in one instance a Penning trap was used to contain a single electron for a period of 10 months.[149] The magnetic moment of the electron was measured to a precision of eleven digits, which, in 1980, was a greater accuracy than for any other physical constant.[150]
Plasma applications[edit]
Particle beams[edit]
Electron beams are used in welding.[155] They allow energy densities up to 107 W·cm−2 across a narrow focus diameter of 0.1–1.3 mm and usually require no filler material. This welding technique must be performed in a vacuum to prevent the electrons from interacting with the gas before reaching their target, and it can be used to join conductive materials that would otherwise be considered unsuitable for welding.[156][157]
Electron-beam lithography (EBL) is a method of etching semiconductors at resolutions smaller than a micrometer.[158] This technique is limited by high costs, slow performance, the need to operate the beam in the vacuum and the tendency of the electrons to scatter in solids. The last problem limits the resolution to about 10 nm. For this reason, EBL is primarily used for the production of small numbers of specialized integrated circuits.[159]
Electron beam processing is used to irradiate materials in order to change their physical properties or sterilize medical and food products.[160] Electron beams fluidise or quasi-melt glasses without significant increase of temperature on intensive irradiation: e.g. intensive electron radiation causes a many orders of magnitude decrease of viscosity and stepwise decrease of its activation energy.[161]
Linear particle accelerators generate electron beams for treatment of superficial tumors in radiation therapy. Electron therapy can treat such skin lesions as basal-cell carcinomas because an electron beam only penetrates to a limited depth before being absorbed, typically up to 5 cm for electron energies in the range 5–20 MeV. An electron beam can be used to supplement the treatment of areas that have been irradiated by X-rays.[162][163]
Particle accelerators use electric fields to propel electrons and their antiparticles to high energies. These particles emit synchrotron radiation as they pass through magnetic fields. The dependency of the intensity of this radiation upon spin polarizes the electron beam—a process known as the Sokolov–Ternov effect.[note 8] Polarized electron beams can be useful for various experiments. Synchrotron radiation can also cool the electron beams to reduce the momentum spread of the particles. Electron and positron beams are collided upon the particles' accelerating to the required energies; particle detectors observe the resulting energy emissions, which particle physics studies .[164]
Low-energy electron diffraction (LEED) is a method of bombarding a crystalline material with a collimated beam of electrons and then observing the resulting diffraction patterns to determine the structure of the material. The required energy of the electrons is typically in the range 20–200 eV.[165] The reflection high-energy electron diffraction (RHEED) technique uses the reflection of a beam of electrons fired at various low angles to characterize the surface of crystalline materials. The beam energy is typically in the range 8–20 keV and the angle of incidence is 1–4°.[166][167]
The electron microscope directs a focused beam of electrons at a specimen. Some electrons change their properties, such as movement direction, angle, and relative phase and energy as the beam interacts with the material. Microscopists can record these changes in the electron beam to produce atomically resolved images of the material.[168] In blue light, conventional optical microscopes have a diffraction-limited resolution of about 200 nm.[169] By comparison, electron microscopes are limited by the de Broglie wavelength of the electron. This wavelength, for example, is equal to 0.0037 nm for electrons accelerated across a 100,000-volt potential.[170] The Transmission Electron Aberration-Corrected Microscope is capable of sub-0.05 nm resolution, which is more than enough to resolve individual atoms.[171] This capability makes the electron microscope a useful laboratory instrument for high resolution imaging. However, electron microscopes are expensive instruments that are costly to maintain.
Two main types of electron microscopes exist: transmission and scanning. Transmission electron microscopes function like overhead projectors, with a beam of electrons passing through a slice of material then being projected by lenses on a photographic slide or a charge-coupled device. Scanning electron microscopes rasteri a finely focused electron beam, as in a TV set, across the studied sample to produce the image. Magnifications range from 100× to 1,000,000× or higher for both microscope types. The scanning tunneling microscope uses quantum tunneling of electrons from a sharp metal tip into the studied material and can produce atomically resolved images of its surface.[172][173][174]
Other applications[edit]
In the free-electron laser (FEL), a relativistic electron beam passes through a pair of undulators that contain arrays of dipole magnets whose fields point in alternating directions. The electrons emit synchrotron radiation that coherently interacts with the same electrons to strongly amplify the radiation field at the resonance frequency. FEL can emit a coherent high-brilliance electromagnetic radiation with a wide range of frequencies, from microwaves to soft X-rays. These devices may find manufacturing, communication and various medical applications, such as soft tissue surgery.[175]
Electrons are important in cathode ray tubes, which have been extensively used as display devices in laboratory instruments, computer monitors and television sets.[176] In a photomultiplier tube, every photon striking the photocathode initiates an avalanche of electrons that produces a detectable current pulse.[177] Vacuum tubes use the flow of electrons to manipulate electrical signals, and they played a critical role in the development of electronics technology. However, they have been largely supplanted by solid-state devices such as the transistor.[178]
See also[edit]
for quantum number s = 12.
4. ^ Bohr magneton:
6. ^ a b c d e P.J. Mohr, B.N. Taylor, and D.B. Newell (2011), "The 2010 CODATA Recommended Values of the Fundamental Physical Constants" (Web Version 6.0). This database was developed by J. Baker, M. Douma, and S. Kotochigova. Available: http://physics.nist.gov/constants [Thursday, 02-Jun-2011 21:00:12 EDT]. National Institute of Standards and Technology, Gaithersburg, MD 20899.
7. ^ "JERRY COFF". Retrieved 10 September 2010.
25. ^ Dahl (1997:55–58).
28. ^ Dahl (1997:64–78).
30. ^ Dahl (1997:99).
34. ^ Becquerel, H. (1900). "Déviation du Rayonnement du Radium dans un Champ Électrique". Comptes rendus de l'Académie des sciences (in French) 130: 809–815.
35. ^ Buchwald and Warwick (2001:90–91).
47. ^ Uhlenbeck, G.E.; Goudsmith, S. (1925). "Ersetzung der Hypothese vom unmechanischen Zwang durch eine Forderung bezüglich des inneren Verhaltens jedes einzelnen Elektrons". Die Naturwissenschaften (in German) 13 (47): 953. Bibcode:1925NW.....13..953E. doi:10.1007/BF01558878.
48. ^ Pauli, W. (1923). "Über die Gesetzmäßigkeiten des anomalen Zeemaneffektes". Zeitschrift für Physik (in German) 16 (1): 155–164. Bibcode:1923ZPhy...16..155P. doi:10.1007/BF01327386.
52. ^ Schrödinger, E. (1926). "Quantisierung als Eigenwertproblem". Annalen der Physik (in German) 385 (13): 437–490. Bibcode:1926AnP...385..437S. doi:10.1002/andp.19263851302.
64. ^ Prati, E.; De Michielis, M.; Belli, M.; Cocco, S.; Fanciulli, M.; Kotekar-Patil, D.; Ruoff, M.; Kern, D. P.; Wharam, D. A.; Verduijn, J.; Tettamanzi, G. C.; Rogge, S.; Roche, B.; Wacquez, R.; Jehl, X.; Vinet, M.; Sanquer, M. (2012). "Few electron limit of n-type metal oxide semiconductor single electron transistors". Nanotechnology 23 (21): 215204. doi:10.1088/0957-4484/23/21/215204. PMID 22552118. edit
74. ^ Eduard Shpolsky, Atomic physics (Atomnaia fizika),second edition, 1951
78. ^ J. Beringer et al. (Particle Data Group) (2012). "Review of Particle Physics: [electron properties]". Physical Review D 86 (1): 010001. Bibcode:2012PhRvD..86a0001B. doi:10.1103/PhysRevD.86.010001.
80. ^ a b c d e Munowitz, M. (2005). Knowing, The Nature of Physical Law. Oxford University Press. ISBN 0-19-516737-6.
86. ^ Murayama, H. (March 10–17, 2006). "Supersymmetry Breaking Made Easy, Viable and Generic". Proceedings of the XLIInd Rencontres de Moriond on Electroweak Interactions and Unified Theories (La Thuile, Italy). arXiv:0709.3041. —lists a 9% mass difference for an electron that is the size of the Planck distance.
103. ^ Quigg, C. (June 4–30, 2000). "The Electroweak Theory". TASI 2000: Flavor Physics for the Millennium (Boulder, Colorado): 80. arXiv:hep-ph/0204104.
133. ^ Sather, E. (Spring–Summer 1996). "The Mystery of Matter Asymmetry". Beam Line. University of Stanford. Retrieved 2008-11-01.
142. ^ Halzen, F.; Hooper, D. (2002). "High-energy neutrino astronomy: the cosmic ray connection". Reports on Progress in Physics 66 (7): 1025–1078. arXiv:astro-ph/0204527. Bibcode:2002astro.ph..4527H. doi:10.1088/0034-4885/65/7/201.
151. ^ Mauritsson, J. "Electron filmed for the first time ever". Lund University. Archived from the original on March 25, 2009. Retrieved 2008-09-17.
158. ^ Ozdemir, F.S. (June 25–27, 1979). "Electron beam lithography". Proceedings of the 16th Conference on Design automation (San Diego, CA, USA: IEEE Press): 383–391. Retrieved 2008-10-16.
160. ^ Jongen, Y.; Herer, A. (May 2–5, 1996). "Electron Beam Scanning in Industrial Applications". APS/AAPT Joint Meeting (American Physical Society). Bibcode:1996APS..MAY.H9902J.
161. ^ Mobus G. et al. (2010). Journal of Nuclear Materials, v. 396, 264–271, doi:10.1016/j.jnucmat.2009.11.020
163. ^ Gazda, M.J.; Coia, L.R. (June 1, 2007). "Principles of Radiation Therapy". Retrieved 2013-10-31.
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d99b7eeeffc6235f | Support Options
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Nanoelectronic Modeling Lecture 09: Open 1D Systems - Reflection at and Transmission over 1 Step
By Gerhard Klimeck1, Dragica Vasileska2, Samarth Agarwal3
1. Purdue University 2. Electrical and Computer Engineering, Arizona State University, Tempe, AZ 3. Electrical and Computer Engineering, Purdue University, West Lafayette, IN
Published on
One of the most elemental quantum mechanical transport problems is the solution of the time independent Schrödinger equation in a one-dimensional system where one of the two half spaces has a higher potential energy than the other. The analytical solution is readily obtained using a scattering matrix approach where wavefunction amplitude and slope are matched at the interface between the two half-spaces. Of particular interest are the wave/particle injection from the lower potential energy half-space. In a classical system a particle will suffer from complete reflection at the half-space border if its kinetic energy is not larger than the potential energy difference at the barrier. The classical particle will be completely transmitted if its kinetic energy exceeds the potential barrier difference. A quantum mechanical particle or wave however exhibits a few interesting features: 1) it can penetrate into the potential barrier when ints kinetic energy is lower than the potential step energy, and 2) transmission over the barrier is not complete and energy dependent. Incomplete transmission implies a reflection probability for the wave even though its kinteci energy exceeds the potential barrier difference. This simple example shos the extended nature of wavefunctions and the non-local effects of local potential variations in its simplest form.
Cite this work
Researchers should cite this work as follows:
• Gerhard Klimeck; Dragica Vasileska; Samarth Agarwal (2010), "Nanoelectronic Modeling Lecture 09: Open 1D Systems - Reflection at and Transmission over 1 Step,"
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Università di Pisa, Pisa, Italy
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f8f740ff09fd832e | An example to illustrate how indistinguishable particles can behave as if they are distinguishable
Imagine two electrons bound inside two hydrogen atoms that are far apart. The Pauli exclusion principle says that the two electrons cannot be in the same quantum state because electrons are indistinguishable particles. But the exclusion principle doesn't seem at all relevant when we discuss the electron in a hydrogen atom, i.e. we don't usually worry about any other electrons in the Universe: it is as if the electrons are distinguishable. Our intuition says they behave as if they are distinguishable if they are bound in different atoms but as we shall see this is a slippery road to follow. The complete system of two protons and two electrons is made up of indistinguishable particles so it isn't really clear what it means to talk about two different atoms. For example, imagine bringing the atoms closer together - at some point there aren't two atoms anymore.
You might say that if the atoms are far apart, the two electrons are obviously in very different quantum states. But this is not as obvious as it looks. Imagine putting electron number 1 in atom number 1 and electron number 2 in atom number 2. Well after waiting a while it doesn't anymore make sense to say that "electron number 1 is still in atom number 1". It might be in atom number 2 now because the only way to truly confine particles is to make sure their wavefunction is always zero outside the region you want to confine them in and this is never attainable. We therefore really should treat our two electrons as being indistinguishable from each other, i.e. we are to think of two electrons in the potential of two protons. Let us simplify the situation a little bit by neglecting the interaction between the two electrons - this won't be a bad approximation if the protons are far apart and the electrons are below the ionization energy of 13.6 eV, and in any case it doesn't really affect our argument. The allowed energies of one of the electrons must therefore be approximately equal to the energy levels of an electron in the potential of a single proton (provided the energy is less than the ionization energy). Now the problem is clear - how can both electrons be in (e.g.) the ground state at the same time? Crucially, we are not allowed to appeal to the fact that the electrons are localized on one proton or the other to get round this problem. In the language of quantum mechanics, the energy eigenstates for each electron are not localized on one proton or the other. The initial wavefunction for one electron might be peaked in the region of one proton but after waiting for long enough the wavefunction will evolve to a wavefunction which is not localized at all. In short, the quantum state is completely specified by giving just the electron energies and then it is a puzzle why two electrons can have the same energy (we're also ignoring things like electron spin here but again that is a detail which doesn't affect the main line of the argument). A little thought and you may be able to convince yourself that the only way out of the problem is for there to be two energy levels whose energy difference is too small for us to have ever measured in an experiment. The example presented below is designed to illustrate that this is indeed what is going on.
We'll consider a simplified model. Our system will be two non-interacting particles moving in one dimension. The particles move in the potential illustrated in the figure below: it is an infinite square well with a finite potential barrier in the middle. One can think of putting one particle in the left-hand region and the other in the right-hand region with energies below the barrier height, V. The particles are, for a time, confined however there is always a non-zero tunnelling probability and we cannot therefore say with certainty that the particles remain on one side or the other of the potential barrier. Our goal is to determine the energy eigenvalues and eigenfunctions for the single particle states. If the potential barrier is sufficiently large, then the energy eigenstates will be approximately equal to those of a single particle in an infinite potential well. However, as argued above, we'll encounter the interesting result that there are in fact two slightly non-degenerate energy levels for each energy level of the infinite potential well.
Let V be the height of the potential barrier and 2*delta be its width. We'll work in units where the Schrödinger equation for the energy eigenstates reads
-(diff(diff(psi(x), x), x))+V(x)*psi(x) = E*psi(x)
We'll also choose our units so that L = 2(i.e. -1 < x and x < 1). Solving the Schrödinger equation in each of the three regions (I, II and III) gives:
proc (x) options operator, arrow; sin(alpha(E[i])*x)+A(E[i])*cos(alpha(E[i])*x) end proc
proc (x) options operator, arrow; B[i](E[i])*(exp(beta(E[i])*x)+n[i]*exp(-beta(E[i])*x)) end proc
proc (x) options operator, arrow; n[i]*psi[I](-x) end proc
where alpha = sqrt(E), beta = sqrt(V-E). The index ilabels whether we are considering those eigenstates which are even or odd about the centre of the potential (which I choose to be at x = 0). (You should be able to convince yourself that the eigenstates must be either even or odd functions as a result of symmetry.) Consequently, n[even] = 1, n[odd] = -1. Note that I have made life a little bit easier by not working with normalized wavefunctions; to get the normalization correct one just has to re-scale by an overall factor but we won't need to bother doing that here because we'll not be computing any probabilities. The boundary conditions are that the wavefunction must vanish for x < -1and 1 < xand that the wavefunction and its derivative must be continuous at x = `&+-`(delta). Implementing these conditions allows us to fix A and B[i]:
proc (E) options operator, arrow; tan(alpha(E)) end proc
proc (E) options operator, arrow; (A(E)*cos(alpha(E)*delta)-sin(alpha(E)*delta))/(exp(-beta(E)*delta)+n[i]*exp(beta(E)*delta)) end proc
and the boundary conditions also lead to the following transcendental equation in the energy E.
proc (i) options operator, arrow; beta(E)*(-sin(alpha(E)*delta)+A(E)*cos(alpha(E)*delta))/(alpha(E)*(cos(alpha(E)*delta)+A(E)*sin(alpha(E)*delta)))+(exp(beta(E)*delta)+n[i]*exp(-beta(E)*delta))/(exp(b...
In order to do some numerics, let us make a choice for the parameters defining the potential:
Before solving, we'll first consider the energy eigenvalues corresponding to an infinite square well of width which is the result we expect in the case that Vtends to infinity. The mth energy eigenstate of the infinte square well is
There are two corresponding energy eigenstates in our case. The energy of the even eigenstate is
and the odd eigenvalue is
Note that the two energy eigenvalues are (a) not very far from the infinite square well result and (b) slightly different from each other. The corresponding energy eigenfunctions are plotted below.
The important point is that there are two almost degenerate energy eigenstates for every one energy eigenstate in the corresponding infinite square well. If we had made the potential barrier higher, then the ground state energy would approach that of the infinite square well and the splitting between the energy levels would be even smaller. For example if
Note that I had to go to 50 significant digits to detect the splitting between the two energies (the difference is real, not my numerical error!) You might like to think what determines the size of the tiny splitting between the energy levels. (Hint: look at the eigenstates in the vicinity of x = 0.)
After working through this exercise you might like to think how things work for an infinite well divided into 3 regions using two finite potential barriers. You might be tempted to think that there are now 4 energy levels for each energy level of the infinite square well corresponding to energy eigenstates that are variously odd/even about the centre of the two potential barriers. However this must be wrong. If it were correct then one could put 4 identical fermions all into the ground state and the Pauli principle would then be violated. You should convince yourself that there are in fact only 3 linearly independent eigenfunctions and hence only 3 energy levels for each energy level of the infinite square well. Now the Pauli principle can still hold: 3 fermions can go into each of the 3 "ground state" levels but the 4th fermion must go into a higher level.
Time evolution
Let's now consider what happens subsequently if we start with the particle on one side of the potential. For the initial wavefunction let us take
which corresponds to a particle located on the left hand side:
This is not an exact energy eigenstate, it is a superposition of two nearly degenerate states and as such it will evolve slowly with time. The below animation shows the subsequent time evolution of the probability density abs(Phi)^2 which evolves according to
proc (x, t) options operator, arrow; psi[even](x)*exp(I*E[even]*t)+psi[odd](x)*exp(I*E[odd]*t) end proc |
8decd3bf9f67729f | Built on Facts
Generally speaking, if you play a movie backwards everything that happens is still physically possible. If I throw you a baseball and you catch it, reversing the video is just the equally plausible situation of you throwing the baseball followed by my catching it. If entropy is changing in the video – e.g., breaking an egg – the time-reversed video will not be especially likely. But it doesn’t break the laws of physics.
This is called time-reversal symmetry (sometimes T-symmetry for short). If you’re looking at planets orbiting in uniform circular motion about the sun, you know from your physics class that they have to satisfy a particular relationship between their velocity and the force F of gravity that the sun exerts on the planet:
The gravitational force doesn’t make any reference to time – if you have two masses, they attract. On the other hand, time reversal changes v to -v, since the planet is now moving in the opposite direction*. But the equation involves the square of v, and so the minus sign vanishes and the law retains the same form despite the time reversal.
This T-symmetry holds very generally in physics. both in Newtonian mechanics and Maxwellian electrodynamics. It came as a bit of a shock therefore when it was discovered that particle physics (the weak interaction, specifically) is not time-reversal invariant. It does hold to a more general CPT-symmetry where the laws are invariant if you flip the direction of time, flip left and right, and change matter into antimatter. Exactly why this is so is not so well understood.
Which is a long way of introducing an interesting APS article about a violation of T-symmetry in electromagnetism. This certainly raised my eyebrows when I read the headline, but as it turns out the symmetry is indeed broken in the device itself under the influence of a magnetic field. But magnetic fields do reverse direction under time reversal (because they’re generated by moving currents), and so the symmetry is preserved if you take the magnetic field into account. Still, it’s a pretty snazzy paper – both the summary and the paper itself are freely available online. Give ‘em a look!
*This is sort of handwaving since v in this notation is usually defined as a strictly positive magnitude in the first place. If this bothers you, define the quantity v^2 = v.v and it’s a little more clear.
1. #1 Eric Rodriguez
October 15, 2010
Where are these studies being found? is it accurate information? Are there proffesional scientist reviewing these studies?
2. #2 Chris Crawford
October 15, 2010
Make a movie of an electron’s probability cloud of positions immediately after being measured by collision with a photon. The probability cloud expands with time; that’s what we expect. Now play the movie backwards. The probability cloud contracts – time irreversibility.
3. #3 Russell
October 15, 2010
The puzzling thing is that the Schrödinger equation is time symmetric, too. What is violating it?
4. #4 Raskolnikov
October 15, 2010
There is nothing strange about that. You just started in a very special initial state, one which is unlikely to be realized and that’s why you feel the time reversed motion seems strange. But it doesn’t violate any fundamental law.
5. #5 Carl Brannen
October 15, 2010
Glad to see you blogging again. The symmetry of time reversal violation is equivalent to CP violation under the (fairly reasonable but possibly a bit dated due to anti-neutrino mass measurement) assumption that CPT is a perfect symmetry. My latest paper shows that CP violation in the CKM mixing matrix (i.e. weak interactions) can be attributed to Berry / Pancharatnam / quantum phase. The paper had a strange rejection at Foundations of Physics. Both reviewers said it should be published, but one suggested it was too mathematical and more suited to Journal of Mathematical Physics. The FoP editor agreed, so I’m getting it ready for JMP.
6. #6 Chris Crawford
October 15, 2010
I’m not sure whether comments #3 and #4 refer to my own comment or to the main post, but I’ll assume that they are. If so, then here are my responses:
#3: Yes, Schrodinger’s equation is time symmetric. The Uncertainty Principle (when applied over a period of time) isn’t.
#4: What’s so special about measuring a particle with another particle? Do that with a zillion particles and you’ve got a gas in which the behavior is macroscopically irreversible. You can argue that, classically, the behavior of the gas is microscopically reversible, but if you use QM instead of classical mechanics, all those particle collisions are really just “measurements”, and are not reversible.
7. #7 Uncle Al
October 17, 2010
Consider the difference between pseudovectors (axial vectors, e.g., magnetic field H) and chiral systems. Reflecting a current-carrying solenoid trivially reverses the direction of current flow and coil helicity, but field direction does not reverse if the mirror plane is parallel to the field axis. Normal to the field axis the field reverses, solenoid helicity reverses… but what of current flow? Reversing two of three axes in a chiral system does not reverse its chirality. One reflection or all three axes’ reflections reverse chirality.
It’s complicated and subtle. Footnotes are a rich source of wonder.
1) Entropy is a weak arrow of time for it is only statistical.
2) Angular momentum is a strong arrow of time. Feynman’s sprinkler only spins in one direction, blowing water out but not aspirating it.
3) Chirality is coupled to moments of inertia. Chirality is also a strong arrow of time – and it don’t need no stinkin’ magnetic field,
Nature 463 210 (2010)
“Chiral spin liquids are a hypothetical class of spin liquids in which the time-reversal symmetry is macroscopically broken in the absence of an applied magnetic field or any magnetic dipole long-range order. ”
Phys. Rev. D 71 057501 (2005)
“The so-called time-reversal odd distribution functions are known to be nonvanishing in QCD due to the presence of the link operator in the definition of these quantities. I show that T-odd distributions can be nonvanishing also in chiral models”
time-reversal symmetry of chiral systems
Phys Rev Lett. 91(24) 247404 (2003)
Broken time reversal of light interaction with planar chiral nanostructures.
4) Gravitation does not get off the hook. Though physics cannot imagine a universe non-identical to its mirror image (perfect theory derived from fundamental symemtries, then furiously ad hoc patched with inserted symmetry breakings), this universe does it everywhere. Physics denies emergent obervables can be fundamental. Nobody has examined gravitation for chiral divergence. Physics knows no structural chemistry. Cowards – two opposite geometric parity atomic mass distribution Eotvos experiments,
Theory predicts what observation tells it to predict.
Somebody should look. The worst it can do is succeed.
8. #8 wingcodavid
February 17, 2011
Where can I access more information on Time Reversal?
Time slipping and the Dimension of Timelessness? |
5becf444662f5a5f | L^2 in spherical coordinates.
by koroljov
Tags: coordinates, spherical
koroljov is offline
Sep29-06, 11:21 PM
P: 26
I am trying to calculate L^2 in spherical coordinates. L^2 is the square of L, the angular momentum operator. I know L in spherical coordinates. This L in spherical coordinates has only 2 components : one in the direction of the theta unit vector and one in the direction of the phi unit vector.
I get the correct result for L^2 by substituting cartesian values for the theta and phi unit vectors in L, and then squaring and adding the components.
I do not get the correct result by simply squaring and adding the theta and phi components of L directly.Why not? Surely if this were a classical vector whose components are scalars rather than operators, I could find its norm squared in both ways, isn't it?
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masudr is offline
Sep30-06, 03:26 AM
P: 932
Before I can answer your question, I'll need to see how you've got L in spherical co-ordinates.
koroljov is offline
Sep30-06, 04:50 AM
P: 26
L = - i * h * (r x nabla) = - i * h * ( u_phi * d/theta - u_theta/sin(theta) * d/dphi )
where h=hbar, nabla=grad operator, u_phi and u_theta=phi and theta unit vectors, x = vector product.
Substituting cartesian values
u_theta = (cos(theta)*cos(phi), cos(theta)*sin(phi), -sin(theta))
u_phi = (-sin(phi), cos(phi), 0)
and squaring and adding the components gives the desired result for L^2:
L^2 = -h^2 * (1/sin(theta)^2 * d^2/dphi^2 + 1/sin(theta) * d/dtheta (sin(theta) * d/dtheta))
Simply squaring and adding the components of L does not seem to give this result.
masudr is offline
Sep30-06, 07:19 AM
P: 932
L^2 in spherical coordinates.
Oh I see.
As far as I'm aware, the reasoning behind this is not entirely obvious. In classical Hamiltonian mechanics, the physics of a system with N degrees of freedom can be formulated in terms of 2N variables. Traditionally, these are the position and conjugate momentum in the various dimensions. However there are a class of variables called canonical variables. Any of these variables can be used used to do Hamiltonian mechanics. In going to spherical co-ordinates, you are suggesting using [itex]r, \theta, \phi[/itex] and the associated derivatives (for the momenta).
The reason all this is important is that the prescription for going from classical to quantum mechanics is to promote the Poisson brackets to commutators, and the functions on position and momenta to functions of the associated operators. I think it boils down to which derivative operators we need for the momentum operators to be canonical variables (and I'm guessing that the extra [itex]\sin(\theta)\mbox{'s}[/itex] appear because of that).
koroljov is offline
Sep30-06, 11:22 PM
P: 26
Thanks for your response.
I can't say I completely understand it though. In Quantum mechanics, as it is being taught to me, we never used classical Hamiltonian mechanics (except for the hamiltonian in the Schrödinger equation). Rather, we converted from classical mechanics to quantum mechanice by replacing the momentum with -i * h * nabla and the kinetic energy by i * h * d/dt.
markr is offline
Oct1-06, 04:51 PM
P: 1
I haven't done the proof, but the issue may be that the derivatives of the basis vectors are not zero.
For example d/d phi u_theta = cos theta u_phi.
koroljov is offline
Oct2-06, 05:47 AM
P: 26
Yes, I see, you are correct. How dumb of me to not see that. Purely out of curiosity: Is it even possible to do this in spherical coordinates directly? I mean, using the correct derivatives when "squaring" the components will give me another vector operator, but the end result (L^2) is an operator whose result is a scalar (rather than a vactor).
masudr is offline
Oct2-06, 06:49 AM
P: 932
NB. throughout, I use the following transformation
Hmm. I'm sure that's not enough to explain it. The Lagrangian in spherical polars is given by:
This gives the momenta as:
[tex]p_r = \partial L / \partial \dot{r}} = m\dot{r}[/tex]
[tex]p_\theta=\partial L / \partial \dot{\theta}} = mr^2\dot{\theta}[/tex]
[tex]p_\phi = \partial L / \partial \dot{\phi}} = mr^2 \sin^2{\theta}\dot{\phi}[/tex]
The reason this is important is because:
[\hat{q}_i,\hat{p}_j] = i\hbar\{q_i,p_j\}=i\hbar\delta_{ij}
where it is understood that q, p are the variables the problem is formulated in, and [itex]\hat{q}, \hat{p}[/itex] are the associated position and momentum operators, and the curly brackets are Poisson brackets.
What this means is that if we are to do our problem in a new set of variables, we must find what the momentum corresponds to, and then replace those with the operators [itex]-i\hbar\partial / \partial q_i[/itex]. So:
[tex]\hat{p}_r = -i\hbar\partial / \partial r[/tex]
[tex]\hat{p}_\theta = -i\hbar\partial / \partial \theta[/tex]
[tex]\hat{p}_\phi = -i\hbar\partial / \partial \phi[/tex]
In spherical polars, the cartesian components of angular momentum are given by:
[tex]L_x=-p_\theta \sin{\phi}\cos{\phi}\cos{\theta}-\frac{p_\phi}{\sin{\theta}}[/tex]
[tex]L_y=-p_\theta \sin{\phi}\cos{\phi}\sin{\theta}-\frac{p_\phi\cos{\theta}}{\sin^2{\theta}}[/tex]
[tex]L_z=p_\theta \sin^2{\theta}\end{multiline*}[/tex]
where the [itex]p_\theta, p_\phi[/itex] are the canonical momenta of the [itex]\theta, \phi[/itex] variables. This was obtained by changing the cartesian components of L from cartesian variables (i.e. [itex]L_x = yp_z - zp_y=my\dot{z}-mz\dot{y}[/itex] to spherical polars).
Now by doing our quantisation (i.e. replacing classical variables with their corresponding operators, whose commutators correspond to the classical Poisson bracket)
[tex]\hat{L}_x = -i\hbar(\sin{\phi}\cos{\phi}\cos{\theta}\frac{\partial}{\partial \theta}-\frac{1}{\sin{\theta}}\frac{\partial}{\partial \phi})[/tex]
[tex]\hat{L}_y=-i\hbar(\sin{\phi}\cos{\phi}\sin{\theta}\frac{\partial}{\partial \theta}-\frac{\cos{\theta}}{\sin^2{\theta}}\frac{\partial}{\partial \phi})[/tex]
[tex]\hat{L}_z=-i\hbar \sin^2{\theta}\frac{\partial}{\partial \theta}[/tex]
All that remains is to square these operators up (remembering that they apply to functions on the right hand side; this ensures that the product rule/Leibniz rule is applied accordingly) and add them up to see what [itex]\hat{L}^2[/itex] looks like in spherical polars.
I'm not 100% if I'm on the right tracks here, but as far as I know, I haven't made any mistakes. If I had the inclination/time to square those operators and sum them, I might have found out...
EDIT: lots of edits to get the [itex]\LaTeX[/itex] right.
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e29fd0ba9a87fe3a | Take the 2-minute tour ×
Apologies if this is a little vague. It might not have a good answer.
Given the interpretation of $|\psi(x)|^2$ as a probability distribution it's unsurprising that a wave function that is concentrated around a point $x$ should behave at least a little like a classical particle at the point $x$.
Is there a similarly intuitive explanation for why a plane wave function $\exp(ik\cdot x)$ behaves somewhat like a classical particle with momentum $\hbar k$? I'm not looking for the standard explanation in terms of eigenstates of the momentum operator, but something that can be used pedagogically for people whose linear algebra isn't sophisticated enough for that.
For example it's not hard to see that a plane wave in n-dimensions has a direction associated with it but it's not intuitively obvious to me that a higher frequency wave should have a higher momentum (unless I reason via the Schrödinger equation which I don't want to do). It's also not surprising that a plane EM wave carries momentum, after all it can interact with charged matter via the Lorentz force and transfer momentum to it, but wave functions don't have such a straightforwardly interpretable interaction.
So how can we make it unsurprising that a plane wave function has a definite momentum?
share|improve this question
Does the concept of group velocity of a wavepacket help? – Michael Brown Apr 11 '13 at 15:20
@MichaelBrown Having just yesterday worked through the basics of anomalous dispersion in media, seeing examples of group velocities that do all kinds of weird things, I think the answer has to be no :-) – Dan Piponi Apr 11 '13 at 15:48
1 Answer 1
The statement "the plane-wave wavefunction $\psi(x)=\exp(i\mathbf{k}\cdot\mathbf{x})$ has definite momentum $\mathbf{p}=\hbar\mathbf{k}$" is a slightly more elaborate restatement of de Broglie's rule, which essentially states "matter particles with momentum $p$ are associated with waves whose wavelength is $\lambda=\hbar/p$".
The only things you need to add to get the full statement is the directionality (i.e. saying the wavefronts are orthogonal to $\mathbf{p}$, which is easy), and the distinction between $e^{i\mathbf{k}\cdot\mathbf{x}}$, $e^{-i\mathbf{k}\cdot\mathbf{x}}$, $\cos(\mathbf{k}\cdot\mathbf{x})$, and so on. The latter is a bit trickier but the choice of $e^{i\mathbf{k}\cdot\mathbf{x}}$ can be justified, to within a sign convention, as the only plane-wave function which gives, by itself, an orientation for $\mathbf{k}$. The sign is simply that, a convention. (Since it's the same convention as $[x,p]=i\hbar$ vs $[p,x]=i\hbar$, it's not going to go away. So just say either convention is fine and we just choose the former.)
The important thing is that the weirdness of the plane-wave $\leftrightarrow$ definite-momentum identification is exactly the same weirdness of de Broglie's $\lambda=\hbar/p$, or that behind the commutator $[x,p]=i\hbar$. How could you possibly justify it? However you try, your argument will have a hole somewhere. It's a fundamental, distinguishing building blocks of (the weirdness of) quantum mechanics.
share|improve this answer
No classical argument can ever give rise to the physical constant $\hbar$ and so I agree there will always be some hole if we try to give classical intuition for quantum mechanics. Nonetheless I think there is still something useful you can do. For example if you rotate a wavefunction then you rotate $k$ in the obvious way. Similarly if you rotate a classical system, you rotate $p$ in exactly the same way. Find enough similarities like this and you go some way to smoothing over the hole. – Dan Piponi Apr 11 '13 at 15:52
Plane waves are weird in classical physics too. By this I mean they do not exist in nature. As a result you can draw conclusions that are formally correct but misleading. For example, in free space, finite EM beams can have nonzero longitudinal component. I.e., the E and B vectors are not purely transverse. – user27777 Aug 16 '13 at 20:07
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36dae96494237dca | Chapter 30
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History shows that scientific “truth” changes over time. The uncertainty is the reason why continued testing of our ideas is so important in science.
Science is the study of the natural world using the five senses. Because people use their senses every day, people have always done some sort of science. However, good science requires a systematic approach. While ancient Greek science did rely upon some empirical evidence, it was heavily dominated by deductive reasoning. Science as we know it began in the 17th century. The father of the scientific method is Sir Francis Bacon (1561–1626), who clearly defined the scientific method in his Novum Organum (1620). Bacon also introduced inductive reasoning, which is the foundation of the scientific method.
The first step in the scientific method is to define clearly a problem or question about how some aspect of the natural world operates. Some preliminary investigation of the problem can lead one to form a hypothesis. A hypothesis is an educated guess about an underlying principle that will explain the phenomenon that we are trying to explain. A good hypothesis can be tested. That is, a hypothesis ought to make predictions about certain observable phenomena, and we can devise an experiment or observation to test those predictions. If we conduct the experiment or observation and find that the predictions match the results, then we say that we have confirmed our hypothesis, and we have some confidence that our hypothesis is correct. On the other hand, if our predictions are not borne out, then we say that our hypothesis is disproved, and we can either alter our hypothesis or develop a new one and repeat the process of testing. After repeated testing with positive results, we say that the hypothesis is confirmed, and we have confidence that our hypothesis is correct.
Properly applied inductive reasoning does not necessarily lead to a true conclusion.
Notice that we did not “prove” the hypothesis, but that we merely confirmed it. This is a big difference between deductive and inductive reasoning. If we have a true premise, then properly applied deductive reasoning will lead to a true conclusion. However, properly applied inductive reasoning does not necessarily lead to a true conclusion. How can this be? Our hypothesis may be one of several different hypotheses that produce the same experimental or observational results. It is very easy to assume that our hypothesis, when confirmed, is the end of the matter. However, our hypothesis may make other predictions that future, different tests may not confirm. If this happens, then we must further modify or abandon our hypothesis to explain the new data. The history of science is filled with examples of this process, and we ought to expect that this will continue.
This puts the scientist in a peculiar position. While we can definitely disprove a number of propositions, we can never be entirely sure that what we believe to be true is indeed true. Thus, science is a very changing thing. History shows that scientific “truth” changes over time. The uncertainty is the reason why continued testing of our ideas is so important in science. Once we test a hypothesis many times, we gain enough confidence that it is correct, and we eventually begin to call our hypothesis a theory. So a theory is a grown-up, well-developed hypothesis.
At one time, scientists conferred the title of law to well-established theories. This use of the word “law” probably stemmed from the idea that God had imposed some order (law) onto the universe, and our description of how the world operates is a statement of this fact. However, with a less Christian understanding of the world, scientists have departed from using the word law. Scientists continue to refer to older ideas, such as Newton’s law of gravity or laws of motion as law, but no one has termed any new ideas in science as law for a very long time.
Isaac Newton
Isaac Newton (1643–1727)
In 1687, Sir Isaac Newton (1643–1727) published his Principia, which detailed work that he had done about two decades earlier. In the Principia, Newton presented his law of gravity and laws of motion, which are the foundation of the branch of physics known as mechanics. Because he required a mathematical framework to present his ideas, Newton invented calculus. His great breakthrough was to hypothesize that the force that held us to the earth was the same force that kept the moon orbiting around the earth each month. From knowledge of the moon’s distance from the earth and orbital period, Newton used his laws of motion to conclude that the moon is accelerated toward the earth 1/3600 of the measured acceleration of gravity at the surface of the earth. The fact that we on the earth’s surface are 60 times closer to the earth’s center than the moon allowed Newton to devise his inverse square law for gravity (602 = 3,600).
This unity of gravity on the earth and the force between the earth and moon was a good hypothesis, but could Newton test it? Yes. When Newton applied his laws of gravity and motion to the then-known planets orbiting the sun (Mercury, Venus, Earth, Mars, Jupiter, and Saturn), he was able to predict several things:
1. The planets orbit the sun in elliptical orbits with the sun at one focus of the ellipses.
2. The line between the sun and a planet sweeps out equal areas in equal intervals of time.
3. The square of a planet’s orbital period is proportional to the third power of the planet’s mean distance from the sun.
Johannes Kepler
Johannes Kepler (1571–1630)
These three statements are known as Kepler’s three laws of planetary motion, because the German mathematician Johannes Kepler (1571–1630) had found them in a slightly different form several decades before Newton. Kepler empirically found his three laws by studying data on planetary motions taken by the Danish astronomer Tycho Brahe (1546–1601) over a period of 20 years in the latter part of the 16th century. Kepler arrived at his result by laborious trial and error for over two decades, but he had no explanation of why the planets behaved the way that they did. Newton easily showed (or predicted) that the planets must follow Kepler’s law as a consequence of his law of gravity.
Many other predictions of Newton’s new physics followed. Besides Earth, Jupiter and Saturn had satellites that obeyed Newton’s formulation of Kepler’s three laws. Newton’s good friend who privately funded the publication of the Principia, Sir Edmond Halley (1656–1742), applied Newton’s work to the observed motions of comets. He found that comets also followed the laws, but that their orbits were much more elliptical and inclined than the orbits of planets. In his study, Halley noticed that one comet that he observed had an orbit identical to one seen about 75 years before and that both comets had a 75-year orbital period. Of course, when the comet returned once again, Halley was long dead, but this comet bears his name.
In 1704, Newton first published his other seminal work in physics, Optics. In this book, he presented his theory of the wave nature of light. Together, his Principia and Optics laid the foundation of physics as we know it. Over the next two centuries, scientists applied Newtonian physics to all sorts of situations, and in each case the predictions of the theory were borne out by experiment and observation. For instance, William Herschel stumbled upon the planet Uranus in 1781, and its orbit followed Kepler’s three laws as well. However, by 1840, astronomers found that there were slight discrepancies between the predicted and observed motion of Uranus. Two mathematicians independently hypothesized that there was an additional planet beyond Uranus whose gravity was tugging on Uranus. This led to the discovery of Neptune in 1846. These successes gave scientists a tremendous confidence in Newtonian physics, and thus Newtonian physics is one of the most well-established theories in history. However, by the end of the 19th century, experimental results began to conflict with Newtonian physics.
Quantum Mechanics
Near the end of the 19th century, physicists turned their attention to how hot objects radiate, with one practical application being the improvement of efficiency of the filament of the recently invented light bulb. Noting that at low temperatures good absorbers and emitters of radiation appear black, they dubbed a perfect absorber and emitter of radiation a black body. Physicists experimentally determined that a black body of a certain temperature emitted the greatest amount of energy at a certain frequency and that the amount of energy that it radiated diminished toward zero at higher and lower frequencies. Attempts to explain this behavior with classical, or Newtonian, physics worked very well at most frequencies but failed miserably at higher frequencies. In fact, at very high frequencies, classical physics required that the energy emitted increase toward infinity.
Max Planck
Max Planck (1858–1947)
In 1901, the German physicist Max Planck (1858–1947) proposed a solution. He suggested that the energy radiated from a black body was not exactly in waves as Newton had shown, but was instead carried away by tiny particles (later called photons). The energy of each photon was proportional to its frequency. This was a radical departure from classical physics, but this new theory did exactly explain the spectra of black bodies.
In 1905, the German-born physicist Albert Einstein (1879–1955) used Planck’s theory to explain the photoelectric effect. What is the photoelectric effect? A few years earlier, physicists had discovered that when light shone on a metal to which an electric potential was applied, electrons were emitted. Attempts to explain the details of this phenomenon with classical physics had failed, but Einstein’s application of Planck’s theory explained it very well.
Other problems with classical physics had mounted. Physicists found that excited gas in a discharge tube emitted energy at certain discrete wavelengths or frequencies. The exact wavelengths of emission depended upon the composition of the gas, with hydrogen gas having the simplest spectrum. Several physicists investigated the problem, with the Swedish scientist Johannes Rydberg (1854–1919) offering the most general description of the hydrogen spectrum in 1888. However, Ryberg did not offer a physical explanation. Indeed, there was no classical physics explanation for the spectral behavior of hydrogen gas until 1913, when the Danish physicist Niels Bohr (1885–1962) published his model of the hydrogen atom that did explain hydrogen’s spectrum.
In the Bohr model, the electron orbits the proton only at certain discrete distances from the proton, whereas in classical physics the electron can orbit at any distance from the proton. In classical physics the electron must continually emit radiation as it orbits, but in Bohr’s model the electron emits energy only when it leaps from one possible orbit to another. Bohr’s explanation of the hydrogen atom worked so well that scientists assumed that it must work for other atoms as well. The hydrogen atom is very simple, because it consists of only two particles, a proton and an electron. Other atoms have increasing numbers of particles (more electrons orbiting the nucleus, which contains more protons as well as neutrons) which makes their solutions much more difficult, but the Bohr model worked for them as well. The Bohr model is essentially the model that most of us learned in school.
While Bohr’s model was obviously successful, it seemed to pull some new principles out of the air, and those principles contradicted principles of classical physics. Physicists began to search for a set of underlying unifying principles to explain the model and other aspects of the emerging new physics. We will omit the details, but by the mid-1920s, those new principles were in place. The basis of this new physics is that in very small systems, as within atoms, energy can exist in only certain small, discrete amounts with gaps between adjacent values. This is radically different from classical physics, where energy can assume any value. We say that energy is quantized because it can have only certain discrete values, or quanta. The mathematical theory that explains the energies of small systems is called quantum mechanics.
Quantum mechanics is a very successful theory. Since its introduction in the 1920s, physicists have used it to correctly predict the behavior and characteristics of elementary particles, nuclei of atoms, atoms, and molecules. Many facets of modern electronics are best understood in terms of quantum mechanics. Physicists have developed many details and applications of the theory, and they have built other theories upon it.
Quantum mechanics is a very successful theory, yet a few people do not accept it. Why? There are several reasons. One reason for rejection is that the postulates of quantum mechanics just do not feel right. They violate our everyday understanding of how the physical world works. However, the problem is that very small particles, such as electrons, do not behave the same way that everyday objects do. We invented quantum mechanics to explain small things such as electrons because our everyday understanding of the world fails to explain them. The peculiarities of quantum mechanics disappear as we apply quantum mechanics to larger systems. As we increase the size and scope of small systems, we find that the oddities of quantum mechanics tend to smear out and assume properties more like our common-sense perceptions. That is, the peculiarities of quantum mechanics disappear in larger, macroscopic systems.
Another problem that people have with quantum mechanics is certain interpretations applied to quantum mechanics. For instance, one of the important postulates of quantum mechanics is the Schrödinger wave equation. When we apply the Schrödinger equation to a particle such as an electron, we get a mathematical wave as a description of the particle. What does this wave mean? Early on, physicists realized that the wave represented a probability distribution. Where the wave had a large value, the probability was large of finding the particle in that location, but where the wave had low value, there was little probability of finding the particle there. This is strange. Newtonian physics had led to determinism—the absolute knowledge of where a particle was at a particular time from the forces and other information involved. Yet, the probability function does accurately predict the behavior of small particles such as electrons. Even Albert Einstein, whose early work led to much of quantum mechanics, never liked this probability. He once famously remarked, “God does not play dice with the universe.” Erwin Schrödinger (1887–1961), who had formulated his famous Schrödinger equation stated in 1926, “If we are going to stick to this ****** quantum-jumping, then I regret that I ever had anything to do with quantum theory.”
Note that with the probability distribution we cannot know precisely where a particle is located. A statement of this is the Heisenberg Uncertainty Principle (named for Werner Heisenberg, 1901–1976). We explain this by acknowledging that particles such as electrons have a wave nature as well as a particle nature. For that matter, we also believe that waves (such as light and sound) also have a particle nature. This wave-particle duality is a bit strange to us, because we do not sense it in everyday experience, but it is borne out by numerous experimental results.
For instance, let us consider a double slit experiment. If we send a wave toward an obstruction with two slits in it, the wave will pass through both slits and produce a distinctive interference pattern behind the slits. This is because the wave passes through both slits. If we send a large number of electrons toward a similar apparatus, the electrons will also produce an interference pattern behind the slits, suggesting that the electrons (or their wave functions) went through both slits. However, if we send one electron at a time toward the slits and look for the emergence of each electron behind the slits, we will find that each electron will emerge through one slit or the other, but not both. How can this be? Indeed, this is perplexing. The most common resolution is the Copenhagen interpretation, named for the city where it was developed. This interpretation posits that an individual electron does not go through either slit, but instead exists in some sort of meta-stable state between the two states until we observe (detect) the electrons. At the point of observation, the electron’s wave equation collapses, allowing the electron to assume one state or the other. Now, this is weird, but most alternate explanations are even weirder, so you might understand why some people may have a problem with quantum mechanics.
Classical physics introduced determinism, quantum mechanics introduced indeterminism.
Is there a way out of this dilemma? Yes. Why do we need an interpretation to quantum mechanics? No one demanded any such interpretation of Newtonian physics. No one asked, “What does it mean?” There is no meaning, other than the fact that Newtonian physics does a good job of describing what we see in the macroscopic world. The same ought to be true for quantum mechanics. It does a good job of describing the microscopic world. Whereas classical physics introduced determinism, quantum mechanics introduced indeterminism. This indeterminism is fundamental in the sense that uncertainty in outcome will still exist even if we have all knowledge of the relevant input parameters. Newtonian determinism fit well with the concept of God’s sovereignty, but the fundamental uncertainty of quantum mechanics appears to rob God of that attribute. However, this assumes that quantum mechanics is a complete theory, that is, that quantum mechanics is an ultimate theory. There are limits to the applications of quantum mechanics, such as the fact that there is no theory of quantum gravity. If the history of science is any teacher, we can expect that quantum mechanics will one day be replaced by some other theory. This other theory probably will include quantum mechanics as a special case of the better theory. That theory may clear up the uncertainty question.
As an aside, we perhaps ought to mention that the determinism derived from Newtonian physics also produces a conclusion unpalatable to many Christians. If determinism is true, then all future events are predetermined from the initial conditions of the universe. Just as the Copenhagen interpretation of quantum mechanics led to even God not being able to know the outcome of an experiment, many people applying determinism concluded that God was unable to alter the outcome of an experiment. That is, God was bound by the physics that rules the universe. This quickly led to deism. Most, if not all, people today who reject quantum mechanics refuse to accept this extreme interpretation of Newtonian physics. They ought to recognize that just as determinism is a perversion of Newtonian physics, the Copenhagen interpretation is a perversion of quantum mechanics.
The important point is that just as classical mechanics does a good job in describing the macroscopic world, quantum mechanics does a good job in describing the microscopic world. We ought not expect any more of a theory. Consequently, most physicists who believe the biblical account of creation have no problem with quantum mechanics.
There are two theories of relativity, the special and general theories. We will briefly describe the special theory of relativity first. Even before Newton, Galileo (1564–1642) had conducted experiments with moving bodies. He realized that if we move toward or away from a moving object, the relative speed that we measure for that object depends upon that object’s motion and our motion. This Galilean relativity is a part of Newtonian mechanics. The same behavior is true for the speed of waves. For instance, if we ride in a boat moving through water with waves, the speed of the waves that we measure will depend upon our motion and on the motion of the waves. In 1881, Albert A. Michelson (1852–1931) conducted a famous experiment that he refined and repeated in 1887 with Edward W. Morley (1838–1923). In this experiment, they measured the speed of light parallel and perpendicular to our annual motion around the sun. Much to their surprise, they found that the speed of light was the same regardless of the direction they measured it. This null result baffled physicists, for if taken at face value, it suggested that the earth did not orbit the sun, while there is other evidence that the earth does indeed orbit the sun.
In 1905, Albert Einstein took the invariance of the speed of light as a postulate and worked out its consequences. He made three predictions concerning an object as its speed approaches the speed of light:
1. The length of the object as it passes will appear to shorten toward zero.
2. The object’s mass will increase without bound.
3. The passage of time as measured by the object will approach zero.
These behaviors are strange and do not conform to what we might expect from everyday experience, but keep in mind that in everyday experience we do not encounter objects moving at any speed close to that of light.
Eventually, these predictions were confirmed in experiments. For instance, particle accelerators accelerate small particles to very high speeds. We can measure the masses of the particles as we accelerate them, and their masses increase in the manner predicted by the theory. In other experiments, very fast-moving, short-lived particles exist longer than they do when moving very slowly. The rate of time dilation is consistent with the predictions of the theory. Length contraction is a little more difficult to directly test, but we have tested it as well.
Relativity Confirmed
Relativity Confirmed
Einstein’s theory of special relativity applies to particles moving at a constant rate but does not address their acceleration. Einstein addressed that problem with his general theory in 1916, but he also treated the acceleration due to gravity. In general relativity, space and time are physical things that have a structure in some ways similar to a fabric. Einstein treated time as a fourth dimension in addition to the normal three dimensions of space. We sometimes call this four-dimensional entity space-time or simply space. The presence of a large amount of matter or energy (Einstein previously had shown their equivalence) alters space. Mathematically, the alteration of space is like a curvature, so we say that matter or energy bends space. The curvature of space telegraphs the presence of matter and energy to other matter and energy in space, and this more deeply answered a question about gravity. Newton had hypothesized that gravity operated through empty space, but his theory could not explain at all how the information about an object’s mass and distance was transmitted through space. In general relativity, an object must move through a straight line in space-time, but the curvature of space-time induced by nearby mass causes that straight-line motion to appear to us as acceleration.
Einstein’s new theory made several predictions. The first opportunity to test the theory happened during a total solar eclipse in 1919. During the eclipse, astronomers were able to photograph stars around the edge of the sun. The light from those stars had to pass very close to the sun to get to the earth. As the stars’ light passed near the sun, the sun attracted the light via the curvature of space-time. This caused the stars to appear farther from the sun than they would have otherwise. Newtonian gravity also predicts a deflection of starlight toward the sun, but the deflection is less than with general relativity. The observed amount of deflection was consistent with the predictions of general relativity. Astronomers have repeated the experiment many times since 1919 with ever-improving accuracy.
For many years, radio astronomers have measured with great precision the locations of distant-point radio sources as the sun passed by, and those results beautifully agree with the predictions. Another early confirmation was the explanation of a small anomaly in the orbit of the planet Mercury that Newtonian gravity could not explain. Many other experiments of various types have repeatedly confirmed general relativity. Some experiments today even allow us to test for slight variations of Einstein’s theory.
We can apply general relativity to the universe as a whole. Indeed, when we do this, we discover that it predicts that the universe is either expanding or contracting; it is a matter of observation to determine which the universe actually is doing. In 1928, Edwin Hubble (1889–1953) showed that the universe is expanding. Most people today think that the expansion began with the big bang, the supposed sudden appearance of the universe 13.7 billion years ago. However, there are many other possibilities. For instance, the creation physicist Russell Humphreys proposed his white hole cosmology, assuming that general relativity is the correct theory of gravity (see his book Starlight and Time1). It is interesting to note that universal expansion is consistent with certain Old Testament passages (e.g., Psalm 104:2) that mention the stretching of the heavens.
Seeing that there is so much evidence to support Einstein’s theory of general relativity, why do some creationists oppose the theory? There are at least three reasons. One reason is that, as with quantum mechanics, modern relativity theory appears to violate certain common-sense views of the way that the world works. For instance, in everyday experience, we don’t see mass change and time appear to slow. Indeed, general relativity forces us to abandon the concept of simultaneity of time. Simultaneity means that time progresses at the same rate for all observers, regardless of where they are. As we previously stated, in special relativity, time slows with greater speed. However, with general relativity, the rate at which time passes depends not only upon speed but also on one’s location in a gravitational field. The deeper one is in a gravitational field, the slower that time passes. For example, a clock at sea level will record the passage of time more slowly than a clock at mile-high Denver. Admittedly, this is weird. However, the discrepancy between the clocks at these two locations is so miniscule as to not appear on most clocks, save the most accurate atomic clocks. This sort of thing has been measured several times, and the discrepancies between the clocks involved always are the same as those predicted by theory. Thus, while our perception is that time flows uniformly everywhere, the reality is that the passage of time does depend upon one’s location, but the differences are so small in the situations encountered on the earth that we cannot perceive them. That is, the predictions of general relativity on earth are consistent with our ability to perceive time. However, there are conditions beyond the earth that the loss of simultaneity would be very obvious if we could experience them.
A second reason why some creationists oppose modern relativity theory is the misappropriation of modern relativity theory to support moral relativism. Unfortunately, modern relativity theory arose at precisely the time that moral relativism became popular. Moral relativists proclaim that “all things are equal,” and they were very eager to snatch some of the triumph of relativity theory to support their cause. There are at least two problems with this misappropriation. First, it does not follow that a principle that works in the natural world automatically operates in the world of morality. The physical world is material, but the world of morality is immaterial. Second, the moral relativists either did not understand relativity or they intentionally misused it. Despite the common misconception, modern relativity theory does not tell us that everything is relative. There are absolutes in modern theory of relativity. The speed of light is a constant. While the passage of time may vary, general relativity provides an absolute way in which to compare the passage of time in two reference frames. The modern theory of relativity in no way supports moral relativism.
The third reason why some creationists reject modern relativity theory is that they think that general relativity inevitably leads to the big-bang model. However, the big-bang model is just one possible origin scenario for the universe; there are many other possibilities. We have already mentioned Russ Humphreys’s white hole cosmology, and there are other possible recent creation models based upon general relativity. True—if general relativity is not correct, then the big-bang model would be in trouble. However, if general relativity is correct, then the shortcut attempt to undermine the big-bang model will doom us from ever finding the correct cosmology.
String Theory
With the establishment of quantum mechanics in the 1920s, the development of the science of particle physics soon followed. At first, only a few particles were known: the electron, proton, and neutron. These particles all had mass and were thought at the time to be the fundamental building blocks of matter. Quantum mechanics introduced the concept that material particles could be described by waves, and conversely that waves could be described by particles. That led to the concept of particles that had no mass, such as photons, the particles that make up light. Eventually, physicists saw the need for other particles, such as neutrinos and antiparticles. Evidence for these odd particles soon followed. Experimental results suggested the existence of other particles, such as the meson, muon, and tau particles, as well as their antiparticles. Many of these new particles were very short-lived, but they were particles nevertheless.
Physicists began to see patterns in the growing zoo of particles. They could group particles according to certain properties. For instance, elementary particles possess angular momentum, a property normally associated with spinning objects, so physicists say that elementary particles have “spin.” Imagining elementary particles as small spinning spheres is useful, but modern theories view this as a bit naive. Spin comes in a quantum amount. Some particles have whole integer values of quantum spin. That is, they have integer multiples (0, ±1, ±2, etc.) of the basic unit of spin. Physicists call these particles Bosons. Other particles have half integer (±1/2, ±3/2, etc.) amounts of spin, and are known as fermions. Bosons and fermions have very different properties. Physicists also noticed that elementary particles tended to have certain mathematical relationships between one another. Physicists eventually began to use group theory, a concept from abstract algebra, to classify and study elementary particles.
By the 1960s, physicists began to suspect that many elementary particles, such as protons and neutrons, were not so elementary after all, but consisted of even more elementary particles. Physicists called these more elementary particles quarks, after an enigmatic word in a James Joyce poem. According to the theory, there are six types of quarks. Many particles, such as protons and neutrons, consist of the combination of two quarks. The different combinations of quarks lead to different particles. Some of those combinations of quarks ought to produce particles that no one had yet seen, so these combinations amounted to predictions of new particles. Particles physicists were able to create these particles in experiments in particle accelerators, so the successful search for those predicted particles was confirmation of the underlying theory. Therefore, quark theory now is well established.
In recent years, particle physicists have in similar fashion developed string theory. Physicists have noticed that certain patterns among elementary particles can be explained easily if particles behave as tiny vibrating strings. These strings would require the existence of at least six additional dimensions of space. We already know that the universe has three normal spatial dimensions as well as the dimension of time, so these six extra dimensions bring the total number of dimensions to ten. The reason why we do not normally see the other six dimensions is that they are tightly curled up and hidden within the tiny particles themselves. At extremely high energies, the extra dimensions ought to manifest themselves. Therefore, particle physicists can predict what kind of behavior strings ought to exhibit when they accelerate particles to extremely high energies. The problem is that current particle accelerators are not nearly powerful enough to produce these effects. As theoretical physicists refine their theories and we build new, powerful particle accelerators, physicists expect that one day we can test whether string theory is true, but for now there is no experimental evidence for string theory.
The Size of Strings
The Size of Strings
We realize the illustration used deuterium, a rare isotope of hydrogen, to help convey the point.
Currently, most physicists think that string theory is a very promising idea. Assuming that string theory is true, there still remains the question as to which particular version of string theory is the correct one. You see, string theory is not a single theory but instead is a broad outline of a number of possible theories. Once we confirm string theory, we can constrain which version properly describes our world. If true, string theory could lead to new technologies. Furthermore, a proper view of elementary particles is important in many cosmological models, such as the big bang. This is because in the big-bang model, the early universe was hot enough to reveal the effects of string theory.
Modern physics is a product of the 20th century and relies upon twin pillars: quantum mechanics and general relativity. Both theories have tremendous experimental support. Christians ought not to view these theories with such great suspicion. True, some people have perverted or hijacked these theories to support some nonbiblical principles, but some wicked people have even perverted Scripture to support nonbiblical things. We ought to recognize that modern physics is a very robust, powerful theory that explains much. At the same time, the theory is very incomplete in some respects. In time, we ought to expect that some new theories will come along that will better explain the world than these theories do. However, we know that God’s Word does not change.
String theory has emerged in the 21st century as the next great idea in physics. Time will tell if string theory will live up to our expectations. What ought to be the reaction of Christians to this? We must be vigilant to investigate the amount of nonbiblical influences that may have crept into modern thinking, particularly in the interpretation of string theory (as with modern physics). However, we must be careful not to throw out the baby with the bath water. That is, can we reject the anti-Christian thinking that many have brought to the discussion? The answer is certainly yes. As with the question of origins, we must strive to interpret these things on our terms, guided by the Bible. Do the new theories adequately describe the world? Can we see the hand of the Creator in our new physics? Can we find meaning in our studies that brings glory to God? If we can answer yes to each of these questions, then these new theories ought not to be a problem for the Christian.
The New Answers Book 2
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9b2e04ec444480d8 | photoI am a research fellow working on quantum-enhanced machine learning and applications of high-performance learning algorithms in quantum physics. Trained as a mathematician and computer scientist, I received my PhD from the National University of Singapore. Currently I work in the Quantum Information Theory group in ICFO-The Institute of Photonic Sciences and I am the Chief of Quantum Machine Learning in the Creative Destruction Lab in the University of Toronto. Previously I worked for the University of Borås and did longer research stints at several institutions, including the Indian Institute of Science, Barcelona Supercomputing Center, Tsinghua University, the Centre for Quantum Technologies in the National University of Singapore, and the Quantum Information Group in the University of Tokyo. I serve in an advisory role for various startups, and I am a member of the NUS Overseas Colleges Alumni.
Current Interests
Quantum-enhanced machine learning Quantum-enhanced machine learning: Current and near-future quantum technologies have a potential of improving learning algorithms. Of particular interest are algorithms that have a high computational complexity or that require sampling. The latter type includes many probabilistic graphical models in which not only the training phase, but also the inference phase has been infeasible at scale, prompting a need for quantum-enhanced sampling. This in turn will enable deep architectures for probabilistic models, as well as scalable implementations of statistical relational learning, both of which go beyond the black-box model of neural networks and shift the focus towards explainable artificial intelligence. While speedup is the primary consideration, we also investigate the fundamental limits of statistical learning theory in the framework of quantum physics.
Moment matrix Quantum many-body systems, optimization, and machine learning: Identifying the ground state of a many-particle system whose interactions are described by a Hamiltonian is an important problem in quantum physics. During the last decade, different relaxations of the previous Hamiltonian minimization problem have been proposed. These algorithms include the lower levels of a general hierarchy of semidefinite programming (SDP) relaxations for non-commutative polynomial optimization, which provide a lower bound on the ground-state energy, complementing the upper bounds that are obtainable using variational methods. The latest developments step away from optimization, and introduce machine learning as an ansatz for ground-state energy problems and for the study of quantum phase transitions. In fact, strong links between quantum many-body physics (tensor networks in particular) and deep learning are being established. We are developing a set of theoretical and numerical tools to pursue these synergies. Sponsored by the ERC grant QITBOX, by the Spanish Supercomputing Network (FI-2013-1-0008 and FI-2013-3-0004) and by the Swedish National Infrastructure for Computing (SNIC 2014/2-7 and 2015/1-162) and a hardware donation by Nvidia Corporation.
Past Projects
t-s-a Trotter-Suzuki Approximation (2012, 2015-2017): The Trotter-Suzuki decomposition leads to an efficient algorithm for solving the time-dependent Schrödinger equation and the Gross-Pitaevskii equation. Using existing highly optimized CPU and GPU kernels, we developed a distributed version of the algorithm that runs efficiently on a cluster. Our implementation also improves single node performance, and is able to use multiple GPUs within a node. The scaling is close to linear using the CPU kernels, whereas the efficiency of GPU kernels improve with larger matrices. We also introduced a hybrid kernel that simultaneously uses multicore CPUs and GPUs in a distributed system. The distributed extension was carried out while visiting the the Barcelona Supercomputing Centre funded by HPC-EUROPA2. Generalizing the capabilities of kernels was carried out by Luca Calderaro sponsored by the Erasmus+ programme. Computational resources were granted by the Spanish Supercomputing Network (FI-2015-2-0023 and FI-2016-3-0042), the High Performance Computing Center North (SNIC 2015/1-162 and SNIC 2016/1-320), and a hardware grant by Nvidia.
pericles Pericles (2013-2017): Promoting and Enhancing Reuse of Information throughout the Content Lifecycle taking account of Evolving Semantics (Pericles) is an integrated project in which academic and industrial partners have come together to investigate the challenge of preserving complex digital information in dynamically evolving environments, to ensure that it remains accessible and useful for future generations. We address contextuality and scalability within the project. Contextuality refers to probabilistic framework that considers the broader and narrower context of the data within a quantum-like formulation, whereas scalability allows executing algorithms on massive data sets using heterogeneous accelerator architectures. Funded by European Commission Seventh Framework Programme (FP7-601138).
chip-sl ChiP-SL (2013-2014): Big data asks for scalable algorithms, but scalability is just one aspect of the problem. Many applications also require the speedy processing of large volumes of data. Examples include supporting financial decision making, advanced services in digital libraries, mining medical data from magnetic resonance imaging, and also analyzing social media graphs. The velocity of machine learning is often boosted by deploying GPUs or distributed algorithms, but rarely both. We are developing high-performance supervised and unsupervised statistical learning algorithms that are accelerated on GPU clusters. Since the cost of a GPU cluster is high and the deployment is far from being trivial, the project Cloud for High-Performance Statistical Learning (ChiP-SL) enables the verification, rapid dissemination, and quick adaptation of the algorithms being developed. Funded by Amazon Web Services.
squalar SQUALAR (2011): High-performance computational resources and distributed systems are crucial for the success of real-world language technology applications. The novel paradigm of general-purpose computing on graphics processors offers a feasible and economical alternative: it has already become a common phenomenon in scientific computation, with many algorithms adapted to the new paradigm. However, applications in language technology do not readily adapt to this approach. Recent advances show the applicability of quantum metaphors in language representation, and many algorithms in quantum mechanics have already been adapted to GPU computing. Scalable Quantum Approaches in Language Representation (SQUALAR) aimed to match quantum-inspired algorithms with heterogeneous computing to develop new formalisms of information representation for natural language processing. Co-funded by Amazon Web Services.
shaman SHAMAN (2010-2011) was an integrated project on large-scale digital preservation. As part of the preservation framework, advanced services aid the discovery of archived digital objects. These services are based on machine learning and data processing, which in turn asks for scalable distributed computing models. Given the requirements for reliability, the project took a middleware approach based on MapReduce to perform computationally demanding tasks. Since memory organizations which are involved in digital preservation potentially lack the necessary infrastructure, a high-performance cloud computing component was also developed. Funded by Framework Programme 7. |
2c346243d38f0f51 | Saturday, December 31, 2005 ... Français/Deutsch/Español/Česky/Japanese/Related posts from blogosphere
Testing E=mc2 for centuries
Chad Orzel seems to disagree with my comments about the interplay between the theory and experiment in physics. That's too bad because I am convinced that a person who has at least a rudimentary knowledge about the meaning, the purpose, and the inner workings of physics should not find anything controversial in my text at all.
Orzel's text is titled "Why I could never be a string theorist" but it could also be named "Why I could never be a theorist or something else that requires to use the brain for extended periods of time". Note that the apparently controversial theory won't be string theory; it will be special relativity. The critics who can't swallow string theory always have other, much older and well-established theories that they can't swallow either.
The previous text about the theory vs. experiment relations
Recall that I was explaining a trivial fact that in science in general and physics in particular, we can predict the results of zillions of experiments without actually doing them. It's because we know general enough theories - that have been found by combining the results of the experiments in the past with a great deal of theoretical reasoning - and we know their range of validity and their accuracy. And a doable experiment of a particular kind usually fits into a class of experiments whose results are trivially known and included in these theories. This is what we mean by saying that these are the correct theories for a given class of phenomena. An experiment with a generic design is extremely unlikely to be able to push the boundaries of our knowledge.
When we want to find completely new effects in various fields, we must be either pretty smart (and lucky) or we must have very powerful apparata. For example, in high-energy physics, it's necessary that we either construct accelerators that accelerate particles to high energies above 100 GeV or so - this is why we call the field high-energy physics - or we must look for some very weak new forces, for example modifications of gravity at submillimeter experiments, or new, very weakly interacting particles. (Or some new subtle observations in our telescopes.)
If someone found a different, cheaper way to reveal new physics, that would be incredible; but it would be completely foolish to expect new physics to be discovered in a generic cheap experiment.
Random experiments don't teach us anything
It's all but guaranteed that if we construct a new low-energy experiment with the same particles that have been observed in thousands of other experiments and described by shockingly successful theories, we are extremely unlikely to learn anything new. This is wasting of taxpayers' money especially if the experiments are very expensive.
In the particular case of the recent "E=mc^2 tests", the accuracy was "10^{-7}" while we know experimentally that the relativistic relations are accurate with the accuracy "10^{-10}", see Alan Kostelecky's website for more concrete details. We just know that we can't observe new physics by this experiment.
Good vs. less good experiments
In other fields of experimental physics, there are other rules - but it is still true that one must design a smart enough experiment to be able to see something new or to be able to measure various things (or confirm the known physical laws) with a better accuracy than the previous physicists. There are good experimentalists and less-good experimentalists (and interesting and not-so-interesting experiments) which is the basic hidden observation of mine that apparently drives Ms. or Mr. Orzel up the wall.
Once again: What I am saying here is not just a theorist's attitude. Of course that it is also the attitude of all good experimentalists. It is very important for an experimentalist to choose the right doable experiments where something interesting and/or new may be discovered (or invented) with a nonzero probabilitity. There is still a very large difference between the experiments that reveal interesting results or inspire new ideas and experiments that no one else finds interesting.
Every good experimentalist would subscribe to the main thesis that experiments may be more or less useful, believe me. Then there are experimentalists without adjectives who want to be worshipped just for being experimentalists and who disagree with my comments; you may guess what is the reason.
Of course that one may design hundreds of experiments that are just stamp-collecting - or solving a homework problem for your experimental course. I am extremely far from thinking that this is the case everywhere outside high-energy physics. There have been hundreds of absolutely fabulous experiments done in all branches of physics and dozens of such experiments are performed every week. But there have also been thousands of rather useless experiments done in all these fields. Too bad if Ms. or Mr. Orzel finds it irritating - but it is definitely not true that all experiments are created equal.
Interpreting the results
Another issue is that if something unexpected occured in the experiment that was "testing E=mc^2", the interpretation would have to be completely different than the statement that "E=mc^2" has been falsified. It is a crackpot idea to imagine that one invents something - or does an experiment with an iron nucleus or a bowl of soup - that will show that Einstein was stupid and his very basic principles and insights are completely wrong.
Hypothetical deviations from the Lorentz invariance are described by terms in our effective theories. Every good experimentalist first tries to figure out which of them she really measures. Neither of these potential deviations deserves the name "modification of the mass-energy relation" because even the Lorentz-breaking theories respect the fact that since 1905, we know that there only exists one conserved quantity to talk about - mass/energy - that can have various forms. We will never return to the previous situation in which the mass and energy were thought to be independent. It's just not possible. We know that one can transform energy into particles and vice versa. We can never unlearn this insight.
New physics vs. crackpots' battles against Einstein
Einstein was not so stupid and the principles of his theories have been well-tested. (The two parts of the previous sentence are not equivalent but they are positively correlated.) To go beyond Einstein means to know where is the room for any improvement, clarification, or deformation of his theories and for new physics, and the room is simply not in the space of ideas that "E=mc^2 is wrong" or "relativity is flawed". A good experimentalist must know something about the theory, to avoid testing his own laymen's preconceptions about physics that have nothing to do with the currently open questions in physics.
Whether or not an experimental physicist likes it or not, we know certain facts about the possible and impossible extensions and variations of the current theories - and a new law that "E=mc^2" will be suddenly violated by one part in ten million in a specific experiment with a nucleus is simply not the kind of modification that can be done with the physical laws as we know them. Anyone who has learned the current status of physics knows that this is not how serious 21st century physics looks like. The current science is not about disproving some dogmatic interpretations of Bohr's complementarity principle either.
Chad Orzel is not the only one who completely misunderstands these basic facts. Hektor Bim writes:
• Yeah, this post from Lubos blew me away, and I’ve been trained as a theorist.
Well, it does not look like a too well-trained one.
• As long as we are still doing physics (and not mathematics), experiment rules.
Experiments may rule, but there are still reasonable (and even exciting) experiments and useless (and even stupid) experiments. If someone thinks that the "leading role" of the experiments means that the experimentalists' often incoherent ideas about physics are gonna replace the existing theories of physics and that every experiment will be applauded even if it is silly, is profoundly confused. Weak ideas will remain weak ideas regardless of the "leading role" of the experiments.
• What also blew me away is that Lubos said that “There is just no way how we could design a theory in which the results will be different.” This is frankly incredible. There are an infinite number of ways that we could design the theory to take into account that the results would be different.
Once again, there are no ways how to design a scientific theory that agrees with the other known experiments but that would predict a different result of this particular experiment. If you have a theory that agrees with the experiments in the accelerators but gives completely new physics for the iron nucleus, you may try to publish it - but don't be surprised if you're described as a cook.
Of course that crackpots always see millions - and the most spectacular among them infinitely many ;-) - ways to construct their theories. The more ignorant they are about the workings of Nature, the more ways to construct the theories of the real world they see. The most sane ones only think that it is easy to construct a quantum theory of gravity using the first idea that comes to your mind; the least sane ones work on their perpetuum mobile machines.
I only mentioned those whose irrationality may be found on the real axis. If we also included the cardinal numbers as a possible value of irrationality, a discussion of postmodern lit crits would be necessary.
Scientific theories vs. crackpots' fantasies
Of course someone could construct a "theory" in which relativity including "E=mc^2" is broken whenever the iron nuclei are observed in the state of Massachusetts - much like we can construct a "theory" in which the law of gravity is revoked whenever Jesus Christ is walking on the ocean. But these are not scientific theories. They're unjustifiable stupidities.
The interaction between the theory and experiments goes in both ways
It is extremely important for an experimental physicist to have a general education as well as feedback from the theorists to choose the right (and nontrivial) things to measure and to know what to expect. It is exactly as important as it is for a theorist to know the results of the relevant experiments.
Another anonymous poster writes:
• What Lumo seems to argue is that somehow we can figure out world just by thinking about it. This is an extremely arrogant and short-sighted point of view, IMPO – and is precisely what got early 20th century philosophers in trouble.
What I argue is that it is completely necessary for us to be thinking about the world when we construct our explanations of the real world as well as whenever we design our experiments. And thinking itself is responsible at least for one half of the big breakthroughs in the history of science. For example, Einstein had deduced both special relativity as well as general relativity more or less by pure thought, using only very general and rudimentary features of Nature known partially from the experiments - but much more deeply and reliably from the previous theories themselves. (We will discuss Einstein below.)
Thinking is what the life of a theoretical physicist is mostly about - and this fact holds not only for theoretical physicists but also other professions including many seemingly non-theoretical ones. If an undereducated person finds this fact about the real world "arrogant", it is his personal psychological problem that does not change the fact that thinking and logical consistency are among the values that matter most whenever physical theories of the real world are deduced and constructed.
The anonymous poster continues:
• By the same reasoning the orbits of the planets must be circular – which is what early “philosophers” argued at some point.
Circular orbits were an extremely useful approximation to start to develop astrophysics. We have gone through many other approximations and improvements, and we have also learned how to figure out which approximations may be modified and which cannot. Cutting-edge physics today studies neither circular orbits nor the questions whether "E=mc^2" is wrong; it studies very different questions because we know the answers to the questions I mentioned.
Pure thought in the past and present
A wise physicist in 2005 respects the early scientists and philosophers for what they have done in the cultural context that was less scientifically clear than the present era, but she clearly realizes their limitations and knows much more than those early philosophers. On the other hand, a bad and arrogant scientist in 2005 humiliates the heroes of the ancient science although he is much more dumb than they were, and he is asking much more stupid questions and promoting a much more rationally unjustifiable criticism of science in general than the comparably naive early philosophers could have dreamed about.
Of course that in principle, one can get extremely far by pure thought, if the thought is logically coherent and based on the right principles, and many great people in the history of science indeed had gotten very far. These are the guys whom we try to follow, and the fact that there have been people who got nowhere by thinking cannot change the general strategy either.
• Anthropic principle completely destroys whatever is left of the “elegance” argument, which is why it’s entertaining to see what will happen next.
I know that some anti-scientific activists would like to destroy not only the "elegance" of science but the whole science - and join forces with the anthropic principle or anything else if necessary - but that does not yet mean that their struggle has any chance to succeed or that we should dedicate them more than this single paragraph.
Another anonymous user writes:
• As far as what Lubos meant, only he can answer that. But it would be obviously foolish to claim relativity could have been deduced without experimental input, and Lubos, whatever else he might be, is no fool.
History of relativity as a victory of pure thought
If interpreted properly, it would not be foolish; it is a historical fact. For example, I recommend you The Elegant Universe by Brian Greene, Chapter 2, for a basic description of the situation. Einstein only needed a very elementary input from the experiments - namely the invariance of physical laws under uniform motion; and the constancy of speed of light - which naturally follows from Maxwell's equations and Einstein was sure that the constancy was right long before the experiments showed that the aether wind did not exist.
It is known pretty well that the Michelson-Morley experiments played a rather small role for Einstein, and for some time, it was even disputed whether Einstein knew these experiments at all back in 1905. (Yes, he did.) Some historians argue that the patented ideas about the train synchronization themselves played a more crucial role. I don't believe this either - but the small influence of the aether wind experiments on Einstein's thinking seems to be a consensus of the historians of science.
Einstein had deeply theoretical reasons to be convinced about both of these two assumptions. Symmetry such as the Galilean/Lorentz symmetry or "the unity of physical explanations" are not just about some irrelevant or subjective concepts of "beauty". They are criteria that a good physicist knows how to use when he or she looks for better theories. The observation that the world is based on more concise and unified principles than what the crackpots and laymen would generally expect is an experimentally verified fact.
These two observations are called the postulates of special relativity, and the whole structure of special relativity with all of its far-reaching consequences such as the equivalence of matter and energy follows logically. Needless to say, all of these effects have always been confirmed - with accuracy that currently exceeds the accuracy available to the experimentalists of Einstein's era by very many orders of magnitude. Special relativity is a genuine and true constraint on any theory describing non-gravitational phenomena in our Universe, and it is a strong constraint, indeed.
Importance of relativity
Whoever thinks that it is not too important and a new experiment with a low-energy nucleus may easily show that these principles are wrong, which essentially allows us to ignore special relativity, and that everything goes after all, is a crackpot.
General relativity: even purer thought
In a similar way, the whole structure of general relativity was derived by the same Einstein purely by knowing the previous special theory of relativity plus Newton's approximate law of gravity, including the equivalence of the inertial and gravitational mass; the latter laws were 250 years old. There was essentially no room for experiments. The first experiments came years after GR was finished, and they always confirmed Einstein's predictions.
The known precession of Mercury's perihelion is an exception; this prediction of GR was known before Einstein, but Einstein only calculated the precession after he had completed his GR, and henceforth, the precession could not directly influence his construction of GR. He was much more influenced and impressed by Ernst Mach, an Austrian philosopher. I don't intend to promote Mach - but my point definitely is to show that the contemporary experiments played a very small role when both theories of relativity were being developed.
There were also some experiments that argued that they rejected the theory, and Einstein knew that these experiments had to be wrong because "God was subtle but not malicious". Of course that Einstein was right and the experiments were wrong. (Similar stories happened to many great theoretical physicists; an experiment of renowned experimentalists that claimed to have falsified Feynman-Gell-Mann's theory of V-A interactions was another example - and Feynman knew right away when he was reading the paper that the experimentalists were just being silly.) Our certainty today that special relativity (or the V-A nature of the weak interactions) is correct in the "simply doable" experiments is much higher than our confidence in any single particular experimentalist. You may be sad or irritated, but that's about everything that you can do against this fact.
Other theories needed more experiments
It would be much harder to get that far without experiments in quantum mechanics and particle physics, among many other branches of physics and science, but whoever questions the fact that there are extremely important insights and principles that have been found - and/or could be found or can be found - by "pure thought" (or that were correctly predicted long before they were observed), is simply missing some basic knowledge about science.
Although I happily admit that we could not have gotten that far without many skillful (and lucky) experimentalists and their experiments, there have been many other examples beyond relativity in which important theories and frameworks were developed by pure mathematical thinking whose details were independent of experiments. The list includes, among hundreds of other examples,
• Dirac's equation. Dirac had to reconcile the first-order Schrödinger equation with special relativity. As a by-product, he also predicted something completely unknown to the experimentalists, namely antiparticles. Every successful prediction may be counted as an example of theoretical work that was not driven by experiments.
• Feynman's diagrams and path integral. No one ever really observed "diagrams" or "multiple trajectories simultaneously contributing to an experiment". Feynman appreciated Dirac's theoretical argument that the classical concept of the action (and the Lagrangian) should play a role in quantum mechanics, too, and he logically deduced that it must play role because of his sum over trajectories. The whole Feynman diagram calculus for QED (generalizable to all other QFTs) followed by pure thought. Today we often say that an experiment "observes" a Feynman diagram but you should not forget about the huge amount of pure thought that was necessary for such a sentence to make any sense.
• Supersymmetry and string theory. I won't provoke the readers with a description.
Lorentz violations are not too interesting and they probably don't exist
• If he is claiming that Lorentz invariance must be exact at all scales, then I agree that he’s being ridiculous. But I think it is reasonable to claim that this experiment was not really testing Lorentz invariance at a level where it has not been tested before.
What I am saying is that it is a misguided approach to science to think that the next big goal of physics is to find deviations from the Lorentz invariance. We won't find any deviations. Most likely, there aren't any. The hypotheses about them are not too interesting. They are not justified. They don't solve any puzzles. Even if we find the deviations and write down the corresponding corrections to our actions, we will probably not be able to deduce any deep idea from these effects. Since 1905 (or maybe the 17th century), we know that the Lorentz symmetry is as fundamental, important and natural as the rotational symmetry.
The Lorentz violation is just one of many hypothetical phenomenological possibilities that can in principle be observed, but that will probably never be observed. I find it entertaining that those folks criticize me for underestimating the value of the experiments when I declare that the Lorentz symmetry is a fundamental property of the Universe that holds whenever the space is sufficiently flat. Why is it entertaining? Because my statement is supported by millions of accurate experiments while their speculation is supported by 0.0001 of a sh*t. It looks like someone is counting negative experiments as evidence that more such experiments are needed.
The only reason why the Lorentz symmetry irritates so many more people than the rotational symmetry is that these people misunderstand 20th century physics. From a more enlightened perspective, the search for the Lorentz breaking is equally (un)justified as a search for the violation of the rotational symmetry. The latter has virtually no support because people find the rotational symmetry "natural" - but this difference between rotations and boosts is completely irrational as we have known since 1905.
Parameterizing Lorentz violation
In the context of gravity, the deviations from the Lorentz symmetry that can exist can be described as spontaneous symmetry breaking, and they always include considering the effect of gravity as in general relativity and/or the presence of matter in the background. In the non-gravitational context, these violations may be described by various effective Lorentz-breaking terms, and all of their coefficients are known to be zero with a high and ever growing degree of accuracy. Look at the papers by Glashow and Coleman, among others.
Undoing science?
The idea that we should "undo" the Lorentz invariance, "undo" the energy-mass equivalence, or anything like that is simply an idea to return physics 100 years into the past. It is crackpotism - and a physics counterpart of creationism. The experiments that could have been interesting in 1905 are usually no longer so interesting in 2005 because many questions have been settled and many formerly "natural" and "plausible" modifications are no longer "natural" or "plausible". The previous sentence comparing 1905 and 2005 would be obvious to everyone if it were about computer science - but in the case of physics, it is not obvious to many people simply because physics is harder to understand for the general public.
But believe me, even physics has evolved since 1905, and we are solving different questions. The most interesting developments as of 2005 (for readers outside the Americas: 2006) are focusing on significantly different issues, and whoever describes low-energy experiments designed to find "10^{-7}" deviations from "E=mc^2" as one of the hottest questions in 2005 is either a liar or an ignorant. It is very fine if someone is doing technologically cute experiments; but their meaning and importance should not be misinterpreted.
Internet gender gap
First, an off-topic answer. Celal asks me about the leap seconds - why has not the Earth already stopped to rotate if there are so many leap seconds. The answer is that we are now indeed inserting a leap second in most of the years - which means that one year is longer by roughly 1 second than it was back in 1820 when the second was defined accurately enough. More precisely, what I want to say is that one solar day is now longer by roughly 1/365 of a second than it was in the 19th century; what matters is of course that the noon stays at 12 pm.
Although the process of slowing down the Earth's rotation has some irregularities, you can see that you need roughly 200 years to increase the number of the required leap seconds per year by one. In order to halve the angular velocity, you need to increase the number of leap seconds roughly by 30 million (the number of seconds per year), which means that you need 30 million times 200 years which is about 6 billion years. Indeed, at time scales comparable to the lifetime of the solar system, the length of the day may change by as much as 100 percent.
100 percent is a bit of exaggeration because a part of the recent slowing is due to natural periodic fluctuations and aperiodic noise, not a trend. However, coral reefs indeed seem to suggest that there were about 400 days per year 0.4 billion years ago. Don't forget that the slowing down is exponential, I think, and therefore the angular velocity will never quite drop to zero (which has almost happened to our Moon).
BBC informs that
While the same percentage of men and women use the internet, they use it in very different ways & they search for very different things. Women focus on maintaining human contacts by e-mail etc. while men look for new technologies and ways to do new things in novel ways.
• "This moment in internet history will be gone in a blink," said Deborah Fallows, senior research fellow at Pew who wrote the report.
I just can't believe that someone who is doing similar research is simultaneously able to share such feminist misconceptions. The Internet has been around for ten years and there has never been any political or legal pressure for the men and women to do different things - the kind of pressures in the past that is often used to justify similar hypotheses about the social origin of various effects.
Friday, December 30, 2005 ... Français/Deutsch/Español/Česky/Japanese/Related posts from blogosphere
Next target of terrorists: Indian string theorists
A newspaper in Bombay informs that
The terror attack at the Indian Institute of Science campus in Bangalore on Wednesday that killed a retired IIT professor has sent shockwaves through the Indian blogosphere.
Blogger and researcher, Kate, wondered if Tata Institute of Fundamental Research [the prominent Indian center of string theory] would be the next target.
Rashmi Bansal expressed sadness at scientists becoming the latest terror victims. “I mean, sure, there would be some routine security checks at the gate, but who seriously believes that a bunch of scientists gathered to discuss string theory or particle physics could be of interest to the Lashkar-e-Toiba?” she wrote in her blog, Youth Curry (
Ms. Bansal may change her mind if she analyzed some posters here - to see at least a "demo" how the anger against the values of modern science can look like. More generally, I emphasize that my warning is absolutely serious. It is not a joke, and I've erased a misleading anonymous comment that suggested that.
Finally, I think that whoever thinks that a scientist cannot become a victim of terrorists is plain stupid. The islamic extremists fight against the whole modern civilization, and the string theorists in India and elsewhere - much like the information technologies experts - are textbook examples of the infiltration of the modern civilization and, indeed, the influence of the Western values - or at least something that was associated with the Western values at least for 500 years.
Everyone who observes the situation and who is able to think must know that Bangalore has been on the terrorists' hit list for quite a while.
If the person who signed as "Indian physicist" does not realize that and if he or she were hoping that the terrorists would treat him or her as a friend (probably because they have the same opinions about George W. Bush?), I recommend him or her to change the field because the hopes were completely absurd.
I give my deepest condolences to the victim's family but I am not gonna dedicate special sorrow to the victim, Prof. Puri, just because he was a retired professor. There are many other innocent people being killed by the terrorists and I am equally sad for all of them. The death of the innocent people associated with "our" society is of course the main reason why I support the war on terror - or at least its general principles. The attack against the conference is bad, but for me it is no surprise. And the casualties of 9/11 were 3,000 times higher which should still have a certain impact on the scale of our reactions.
Third string revolution predicted for physics
CapitalistImperialistPig has predicted
for 2006, started by someone who is quite unexpected. It would be even better if the revolution appeared in the first paper of the year.
Sidney Coleman Open Source Project
Update: See the arXiv version of Sidney Coleman's QFT notes (click)
Jason Douglas Brown has been thinking about a project to transcribe the QFT notes of a great teacher into a usable open source book. I am going to use the notes in my course QFT I in Fall 2006; see the Course-notes directory.
We are talking about 500 pages and about 10 people who would share the job. If you want to tell Jason that it is a bad or good idea, or join his team, send an e-mail to
• jdbrown371 at
Bayesian probability I
See also a positive article about Bayesian inference...
Two days ago, we had interesting discussions about "physical" situations where even the probabilities are unknown.
Reliable quantitative values of probabilities can only be measured by the same experiment repeated many times. The measured probability is then "n/N" where "n" counts the "successful measurements" among all experiments of a certain kind whose total number is "N". This approach defines the "frequentist probability", and whenever we know the correct physical laws, we may also predict these probabilities. If you know the "mechanism" of any system in nature - which includes well-defined and calculable probabilities for all well-defined questions - you can always treat the system rationally.
Unknown probabilities
It is much more difficult when you are making bets about some events whose exact probabilities are unknown. Even in these cases, we often like to say a number that expresses our beliefs quantitatively. Such a notion of probability is called Bayesian probability and it does not really belong to exact sciences.
Thursday, December 29, 2005 ... Français/Deutsch/Español/Česky/Japanese/Related posts from blogosphere
All stem cell lines were fabricated
Wednesday, December 28, 2005 ... Français/Deutsch/Español/Česky/Japanese/Related posts from blogosphere
Comment about the new colors: I believe that the new colors are not a registered trademark of Paul Ginsparg. Moreover, mine are better.
Just a short comment about this creation of Jimbo Wales et al. I am impressed how unexpectedly efficient Wikipedia is. Virtually all of its entries can be edited by anyone in the world, even without any kind of registration. When you realize that there are billions of not-so-smart people and hundreds of millions of active idiots living on this blue planet - and many of them have an internet access - it is remarkable that Wikipedia's quality matches that of Britannica.
But this kind of hypertext source of knowledge is exactly what the web was originally invented for.
Moreover I am sure that Wikipedia covers many fields much more thoroughly than Britannica - and theoretical physics may be just another example. Start with list of string theory topics, 2000+ of my contributions, or any other starting point you like. Try to look for the Landau pole, topological string theory, heterotic string, or thousands of other articles that volunteers helped to create and improve. Are you unsatisfied with some of these pages? You can always edit them.
Tuesday, December 27, 2005 ... Français/Deutsch/Español/Česky/Japanese/Related posts from blogosphere
Hubble: cosmic string verdict by February
Let me remind you that the Hubble pictures of the cosmic-string-lensing CSL-1 candidate, taken by Craig Hogan et al., should be available by February 2006. Ohio's Free Times interviews Tanmay Vachaspati who has studied cosmic strings for 20 years. (Via Rich Murray.)
Monday, December 26, 2005 ... Français/Deutsch/Español/Česky/Japanese/Related posts from blogosphere
Evolution and the genome
Stem cell fraud
Back to the positive story: the genetic evidence for evolution.
New tools make questions solvable
The death of hidden variables
Sun's chemistry and spectroscopy
The last we-will-never-know people
Speed of evolution
Reply to Pat Buchanan
Saturday, December 24, 2005 ... Français/Deutsch/Español/Česky/Japanese/Related posts from blogosphere
Merry Christmas
Background sound (press ESC to stop): Jakub Jan Ryba's "Czech Christmas Mass" (Hey master, get up quickly); a 41:39 MP3 recording here
Merry Christmas! This special season is also a great opportunity for Matias Zaldarriaga and Nima Arkani-Hamed to sing for all the victims of the anthropic principle who try to live in the bad universes (audio - sorry, the true artists have not been recorded yet):
Friday, December 23, 2005 ... Français/Deutsch/Español/Česky/Japanese/Related posts from blogosphere
Thursday, December 22, 2005 ... Français/Deutsch/Español/Česky/Japanese/Related posts from blogosphere
TeX for PowerPoint: TeX4PPT
Aurora is a new commercial LaTeX system for MS Office
Some readers may have installed TeXpoint as an add-in to their PowerPoint. Let me now mention that
is probably superior and everyone who uses TeX as well as PowerPoint should install this piece of free software. In this framework, you may create a new "text box" using the drawing toolbar. Inside the text box, you may write some $tex$. When you're finished, you right-click and choose TeXify. It will convert the text box into a nice piece of LaTeX. One internal advantage over TeXpoint is that it is directly the DVI that is being converted to Microsoft's own fonts. (TeXpoint was also generating a postscript as well as an image.) This means, among other things, that the text respects the background.
The father of Bott periodicity died
Via David G.
Raoul Bott - a Harvard mathematician who was fighting against cancer in San Diego and who discovered, among other things, the Bott periodicity theorem in the late 1950s - died the night of December 19-20, 2005.
His mother and aunts spoke Hungarian. However, his Czech stepfather did not, and therefore the principal language at home was German. At the high school, on the other hand, he had to speak Slovak. His nanny was English which helped young Bott to learn authentic English. To summarize this paragraph: one should not be surprised that Bott hated foreign languages.
Blog of WWW inventor
The person who invented the World Wide Web has started to write
No, it is not a blog of Al Gore - Al Gore has only invented the Al Gore rhythms. The new blog belongs to Tim Berners-Lee who made his invention while at CERN, and currently lives here in Boston.
Figure 1: The first web server in the world (1990)
MIT talk: a theory of nothing
Today, John McGreevy gave an entertaining MIT seminar mainly about the theory of nothing, a concept we will try to define later. The talk described both the work about the topology change induced by closed string tachyon condensation as well as the more recently investigated role that the tachyons may play for a better understanding of the Big Bang singularity. Because we have discussed both of these related projects on this blog, let's try to look at everything from a slightly complementary perspective.
Defining nothing
First of all, what is nothing? John's Nothing is a new regime of quantum gravity where the metric tensor - or its vev - equals zero. This turns out to be a well-defined configuration in three-dimensional gravity described as Chern-Simons theory. It is also the ultimate "paradise" studied in canonical gravity and loop quantum gravity.
Does "nothing" exist and is there anything to study about it? I remain somewhat sceptical. If the metric is equal to zero in a box, it just means that the proper lengths inside the box are zero, too. In other words, they are subPlanckian. The research of "nothing" therefore seems to me as nothing else from the research of the subPlanckian distances. This form of "nothing" is included in every piece of space you can think of, as long as you study it at extremely short distances. And we should not forget that the subPlanckian distances, in some operational sense, do not exist. I guess that John would disagree and he would argue that nothing is an "independent element" of existence; a phase in a phase diagram. I have some problems with this picture.
Tachyons create nothing
Wednesday, December 21, 2005 ... Français/Deutsch/Español/Česky/Japanese/Related posts from blogosphere
MIT talk: Susanne Reffert
Yesterday we went to MIT to see the talk by Susanne Reffert who will be finishing her PhD under Dieter Lüst and who will probably continue her investigation of string theory in Amsterdam, turning down offers from the KITP and CERN. And it was a very nice talk. First of all, she uses Keynote, an Apple-based alternative for the PowerPoint which reconciles TeX and animations into a consistent whole.
Moduli stabilization of F-theory flux vacua again
There have been too many points in the talk to describe all of them here. They studied, among other things, all possible orientifolded and simultaneously orbifolded toroidal (T^6) vacua of type IIB string theory, their resolution, description in terms of toric geometry, flops, and especially the stabilization of the moduli. One of the unexpected insights was that one can't stabilize the Kähler moduli and the dilaton after the uplift to the de Sitter space if there are no complex structure moduli to start with; rigid stabilized anti de Sitter vacua may be found but can't be promoted to the positive cosmological constant case. Some possibilities are eliminated, some possibilities survive, if you require all moduli to be stabilized.
Recall that the complex structure moduli and the dilaton superfield are normally stabilized by the Gukov-Vafa-Witten superpotential - the integral of the holomorphic 3-form wedged with a proper combination of the 3-form field strengths - while the Kähler moduli are stabilized by forces that are not necessarily supernatural but they are non-perturbative which is pretty similar. The latter nonperturbative processes used to stabilize the Kähler moduli include either D3-brane instantons or gaugino condensation in D7-branes.
At this level, one obtains supersymmetric AdS4 vacua. Semirealistic dS4 vacua may be obtained by adding anti-D3-branes, but Susanne et al. do not deal with these issues.
43rd known Mersenne prime: M30402457
One of the GIMPS computers that try to find the largest prime integers of the form
• 2^p - 1
i.e. the Mersenne primes has announced a new prime which will be the 43rd known Mersenne prime. The discovery submitted on 12/16 comes 10 months after the previous Mersenne prime. It seems that the lucky winner is a member of one of the large teams. Most likely, the number still has less than 10 million digits - assuming that 9,152,052 is less than 10 million - and the winner therefore won't win one half of the $100,000 award.
The Reference Frame is the only blog in the world that also informs you that the winner is Curtis Cooper and his new greatest exponent is p = 30,402,457. (Steven Boone became a co-discoverer; note added on Saturday.) You can try to search for this number on the whole internet and you won't find anything; nevertheless, on Saturday, it will be announced as the official new greatest prime integer after the verification process is finished around 1 am Eastern time. If you believe in your humble correspondent's miraculous intuition, you may want to make bets against your friends. ;-)
Tuesday, December 20, 2005 ... Français/Deutsch/Español/Česky/Japanese/Related posts from blogosphere
Temperatures' autocorrelation
Imagine that the Church would start to control the whole society once again. A new minister of science and propaganda would be introduced to his office. His name would not quite be Benedict but rather Benestad. How would they use scientific language to argue that the Bible in general and Genesis in particular literally describes the creation? They would argue that Genesis predicts water, grass, animals, the Sun, the Earth, and several other entities, and the prediction is physically sound. If anyone tried to focus on a possible discrepancy or a detail, Benestad would say that the heretics were pitching statistics against solid science.
The choice of the name "Benestad" will be explained later.
Do you think that the previous sentences are merely a fairy-tale? You may be wrong. First, we need to look at one scientific topic.
Monday, December 19, 2005 ... Français/Deutsch/Español/Česky/Japanese/Related posts from blogosphere
Cosmological constant seesaw
One of the reasons why I have little understanding for the Rube Goldberg landscape machines is that their main goal is to explain just one number, namely the cosmological constant, which could eventually have a simple rational explanation. Let me show you two explanations leading to the same estimate. Recall that the observed cosmological constant is of order
This is almost exactly the same seesaw game with the scales like the neutrino seesaw game. In the case of the neutrinos, we assume the right-handed SU(5)-neutral neutrinos to acquire the GUT scale masses - which is almost the same thing as the Planck scale above - and the unnaturally small value of the observed neutrino masses comes from the smaller eigenvalue(s) of the matrix ((mGUT, mEW), (mEW,0)).
Blogs against decoherence
If you're interested in a blog whose main enemy is decoherence - because they want to construct a quantum computer - see
Everything new you need to know about the realization of quantum bits.
LHC on schedule
2005, the international year of physics, has so far been a flawless year for the LHC. 1000 out of 1232 magnets are already at CERN; 200 magnets have already been installed. See
Update, September 2008: the protons start to orbit in the LHC on September 10th, 9:00 am, see the webcast. But the collisions will only start in October 2008, before a winter break. In 2009, everything will be operating fully. Click the "lhc" category in the list below to get dozens of articles about the Large Hadron Collider.
Distasteful Universe and Rube Goldberg machines
A famous colleague of ours from Stanford has become very popular among the Intelligent Design bloggers. Why is it so? Because he is the unexpected prophet that suddenly revived Intelligent Design - an alternative framework for biology that almost started to disappear. How could he have done so? Well, he offered everyone two options.
• Either you accept the paradigm shifting answer to Brian Greene's "Elegant Universe" - namely the answer that the Universe is not elegant but, instead, it is very ugly, unpredictable, unnatural, and resembling the Rube Goldberg machines (and you buy the book that says so)
• Or you accept Intelligent Design.
You may guess which of these two bad options would be picked by your humble correspondent and which of them would be chosen by most Americans. What does it mean? A rather clear victory for Intelligent Design.
The creationist and nuclear physicist David Heddle writes something that makes some sense to me:
• His book should be subtitled String Theory and the Possible Illusion of Intelligent Design. He has done nothing whatsoever to disprove fine-tuning. Nothing. He has only countered it with a religious speculation in scientific language, a God of the Landscape. Snatching victory from the jaws of defeat, he tells us that we should embrace the String Theory landscape, not in spite of its ugliness, but rather because of it. Physics should change its paradigm and sing praises to inelegance. Out with Occam’s razor, in with Rube Goldberg.
This statement is also celebrated by Jonathan Witt, another fan of ID. Tom Magnuson, one more creationist, assures everyone that if the people are given the choice to choose between two theories with the same predictive power - and one of them includes God - be sure that they will pick the religious one. And he may be right. Well, not everyone will make the same choice. Leon Brooks won't ever accept metaphysics and Evolutionblog simply applaudes our famous Stanford colleague for disliking supernatural agents. But millions of people with the same emotions as William Dembski will make a different choice and it is rather hard to find rational arguments that their decision is wrong because this is a religious matter that can't be resolved scientifically at this point. Discussions about the issue took place at Cosmic Variance and Not Even Wrong.
Intelligent design in physics
Several clarifications must be added. Just like the apparent complexity of living forms supports the concept of Intelligent Design in biology (when I saw the beautiful fish today in the New England Aquarium, I had some understanding for the creationists' feelings), the apparent fine-tuning supports a similar idea in physics. A person like me who expects the parameters of the low-energy effective field theory to emerge from a deeper theory - which is not a religious speculation but a straightforward extrapolation of the developments of the 20th century physics - indeed does believe in some sort of "intelligent design". But of course its "intelligence" has nothing to do with human intelligence or the intelligence of God; it is intelligence of the underlying laws extending quantum field theory.
Opposite or equivalent?
The anthropic people and the Intelligent Design people agree with each other that their pictures of the real world are exactly opposite to one another. In my opinion, this viewpoint about their "contradiction" already means a victory for Intelligent Design and irrational thinking in general. The scientific opinion about this question - whether the two approaches are different - is of course diametrically different. According to a scientific kind of thinking, there is no material difference between
• the theory that God has skillfully engineered our world, or has carefully chosen the place for His creation among very many possibilities
• and the theory that there are uncontrollably many possibilities and "ours" is where we live simply because most of the other possibilities don't admit life like ours
From a physics perspective, these things are simply equivalent. Both of them imply that the parameters "explained" by either of these two theories are really unexplainable. They are beyond our thinking abilities and it does not matter whether we use the word "God" to describe our ignorance about the actual justification of the parameters.
Both of these two approaches may possibly be improved when we reduce the set of possibilities to make some predictions after all. For example, we can find which vacuum is the correct one. Once we do so, the questions whether some "God" is responsible for having chosen the right vacuum, or whether no "God" is necessary, becomes an unphysical question (or metaphysical question, if you prefer an euphemism). Again, the only way how this question may become physical is that we actually understand some rational selection mechanism - such as the Hartle-Hawking wavefunction paradigm - that will lead to a given conclusion. Or if we observe either God or the other Universes; these two possibilities look comparably unlikely to me.
Without these observations and/or nontrivial quantitative predictions, God and the multiverse are just two different psychological frameworks. In this sense, the creationists are completely correct if they say that the multiverse is so far just another, "naturalistic" religion.
As they like to say, the two pillars of the religion of "naturalism" - Freud and Marx - are dead. And Darwin is not feeling too well, they add - the only thing I disagree with. ;-) Marx and Freud are completely dead, indeed.
Friday, December 16, 2005 ... Français/Deutsch/Español/Česky/Japanese/Related posts from blogosphere
Intelligent Design: answers to William Dembski
William Dembski is one of the most active intellectual promoters of Intelligent Design. He also has a blog
in which he tries to collect and create various arguments and pseudoarguments to support his agenda. Just like a certain one-dimensional blog where every piece of news is projected onto the one-dimensional axis "may it hurt string theory?" - and if the projection is positive, the news is published - Uncommon descent evaluates articles and sentences according to their ability to hurt mainstream biology and to support Intelligent Design.
While I am among those who find all one-dimensional blogs and especially most of their readers kind of uninspiring, let me admit that in my opinion, neither of the two Gentlemen mentioned above seems to be a complete moron and many of their questions may deserve our time.
Dembski vs. Gross and Susskind
Because of the description of the blog above, it should not be surprising that Dembski celebrates and promotes both Susskind's anthropic comments indicating that many physicists have accepted opinions remotely analogous to Intelligent Design - as well as Gross's statement that we don't know what we're talking about.
Incidentally, when Dembski quotes David Gross, he says "remember that string theory is taught in physics courses". That's a misleading remark. String theory is only taught in courses on string theory, and with the exception of Barton Zwiebach's award-winning MIT undergraduate course, all such courses are graduate courses. What the advocates of Intelligent Design classes at schools want is definitely much more than the current exposure of the basic school and high school students to string theory.
Although Dembski and some of his readers may find these quotations of the famous physicists relevant, they are not. Maybe, we don't know what we're talking about when we study quantum Planckian cosmology, but we know what we're talking about whenever we discuss particle physics below 100 GeV, the history of our Universe after the first three minutes, and millions of other situations.
What Dembski wants to modify about our picture of the Universe are not some esoteric details about the workings of the Universe at the Planck scale or the mechanisms of vacuum selection. He wants to revert our knowledge about very low energy processes in physics and biology. That makes all his comparisons of biology with uncertainty in quantum gravity irrelevant.
Scientists may be confused about cutting-edge physics but that's very different from being confused about the insights in biology that have been more or less settled in the 19th century. Some scientists may think that a coincidence whose probability was 10^{-350} had to happen before our Universe was created or "chosen", but they don't need probabilities of order 10^{-10^{100}}
OK, the answers
Finally, let me answer 5 questions from Dembski's most recent blog article about microbiology:
• (1) Why does biology hand us technical devices that human design engineers drool over?
It is because the natural length scale of human beings is 1 meter. This is the size of humans as Nature created them. This is the length scale at which humans are very good in designing things. I claim that the human engineers are better than Mother Nature in creating virtually any object whose structure is governed by the length scale of one meter. The engineers are also better at longer distance scales - and the trip to the Moon is an example. Engineers had to develop some technology before the humans could directly affect matter at shorter distance scales than the size of our hands. We are getting better and we may get better than Mother Nature in a majority of nanotechnologies in the near future. William Dembski shows a remarkable short-sightedness if he justifies his opinion by saying that Nature is superior over technology - because it is all but guaranteed that technology will be taking a lead and the strength of Dembski's position will therefore definitely decrease with time.
At any rate, even the successes of engineers themselves reflect the miraculous powers of Mother Nature because engineers were created by Her, too. I am afraid that this fact is not appreciated by many advocates of Intelligent Design and many other people.
• (2) Why don’t we ever see natural selection or any other unintelligent evolutionary mechanisms produce such systems?
Of course that we do. When microprocessors are produced, for example, there is a heavy competition between different companies that produce the chips. Although Intel is planning to introduce their 65 nanometer technology in 2006, AMD may be ahead because of other reasons. This competition is nothing else than the natural selection acting at a different level, with different, "non-biological" mechanisms of reproduction, and such a competition causes the chips to evolve in an analogous way like in the case of animals. (If you want to see which factors drive the decisions about the "survival of the fittest" in the case of chipmakers, open the fast comments.)
Competition also works in the case of ideas, computer programs, ideologies, cultures, "memes", and other things. Indeed, we observe similar mechanisms in many contexts. The detailed technical implementation of the reproduction, mutation, and the rules that determine the survival of the fittest depend on the situation. Some of the paradigms are however universal.
• (3) Why don’t we have any plausible detailed step-by-step models for how such evolutionary mechanisms could produce such systems?
In some cases we do - and some of these models are really impressive - but if we don't, it reflects several facts. The first fact is that the scientists have not been given a Holy Scripture that would describe every detail how the Universe and species were created. They must determine it themselves, using the limited data that is available today, and the answers to such questions are neither unique nor canonical. The evolution of many things could have occured in many different ways. There are many possibilities what things could have evolved and even more possibilities how they could have evolved.
The fact that Microsoft bought Q-DOS at one moment is a part of the history of operating systems, but this fact was not really necessary for the actual evolution of MS Windows that followed afterwards. In the same way, the species were evolved after many events that occured within billions of years - but almost neither of them was absolutely necessary for the currently seen species to be evolved. Because the available datasets about the history of the Earth are limited - which is an inevitable consequence of various laws of Nature - it is simply impossible to reconstruct the unique history in many cases. However, it is possible in many other cases and people are getting better.
• (4) Why in the world should we think that such mechanisms provide the right answer?
Because of many reasons. First of all, we actually observe the biological mechanisms and related mechanisms - not only in biology. They take place in the world around us. We can observe evolution "in real time". We observe mutations, we observe natural selection, we observe technological progress driven by competition, we observe all types of processes that are needed for evolution to work. Their existence is often a fact that can't really be denied.
Also, we observe many universal features of the organisms, especially the DNA molecules, proteins, and many other omnipresent entities. Sometimes we even observe detailed properties of the organisms that are predicted by evolution. Moreover, the processes mentioned above seem to be sufficient to describe the evolution of life, at least in its broad patterns. Occam's razor dictates us that we should not invent new things - and miracles - unless they become necessary. Moreover, evolution of life from simple forms seems to be necessary. We know that the Universe has been around for 13.7 billion years and the Earth was created about 5 billion years ago. We know that this can happen. We observe the evolution of more complex forms in the case of chips and in other cases, too.
According to the known physical laws and the picture of cosmology, the Earth was created without any life on it. Science must always prefer the explanations that use a minimal amount of miracles, a minimal set of arbitrary assumptions and parameters, and where the final state looks like the most likely consequence of the assumptions. This feature of science was important in most of the scientific and technological developments and we are just applying the same successful concepts to our reasoning about everything in the world, including the origin of species.
In this sense, I agree with William Dembski when he says that science rejects the creation by an unaccessible and unanalyzable Creator a priori. Rejecting explanations based on miracles that can be neither analyzed nor falsified is indeed a defining feature of science, and if William Dembski finds it too materialistic, that's too bad but this is how science has worked since the first moment when the totalitarian power of the Church over science was eliminated.
• (5) And why shouldn’t we think that there is real intelligent engineering involved here, way beyond anything we are capable of?
Because of the very same reasons as in (4). Assuming the existence of pre-existing intelligent engineering is an unnatural and highly unlikely assumption with an extremely small explanatory power. One of the fascinating properties of science as well as the real world is that simple beginnings may evolve into impressive outcomes, and modest assumptions are sufficient for us to derive great and accurate conclusions. The idea that there was a fascinating intelligent engineer - and the result of thousands or billions of years of his or her work is an intellectually weak creationist blog - looks like the same development backwards: weak conclusions derived from very strong and unlikely assumptions; poor future evolved from a magnificent past. Such a situation is simply just the opposite of what we are looking for in science - and not only in science - which is why we consider the opinion hiding in the "question" number (5) to be an unscientific preconception. (The last word of the previous sentence has been softened.)
We don't learn anything by assuming that everything has to be the way it is because of the intent of a perfect pre-engineer. We used to believe such things before the humans became capable to live with some degree of confidence and before science was born. Today, the world is very different. For billions of years, it was up to the "lower layers" of Nature to engineer progress. For millions of years, monkeys and humans were mostly passive players in this magnificent game.
More recently, however, humans started to contribute to the progress themselves. Nature has found a new way how to make the progress more efficient and faster - through the humans themselves. Many details are very new but many basic principles underlying these developments remain unchanged. Science and technology is an important part of this exciting story. They can only solve their tasks if they are done properly. Rejecting sloppy thinking and unjustified preconceptions is needed to achieve these goals.
Incidentally, Inquisition and censorship works 100% on "Uncommon Descent". Whoever will be able to post a link on Dembski's blog pointing to this article will be a winner of a small competition. ;-)
Technical note: there are some problems with the Haloscan "fast comments", so please be patient. Right-clicking the window offers you to go "Back" which you may find useful.
String theory is phrase #7
The non-profit organization
located in San Diego, CA, has released its top word list for 2005 (news). The top words are led by "refugee" and "tsunami". Names are led by "God", "tsunami", "Katrina", and "John Paul II". Included are also musical terms and youthspeak.
The top seven phrases are the following:
• out of the mainstream
• bird flu
• politically correct
• North/South divide
• purple thumb
• climate change and global warming
• string theory
You see that almost all of the words and things that The Reference Frame dislikes are above string theory. The defeat of string theory by the global warming is particularly embarassing. ;-) But the 7th place is not so bad after all.
Concerning political correctness, it is just not the phrase itself that was successful. Many new political correct words were successful, too. For example, the word "failure" was replaced by "deferred success" in Great Britain. On the other hand, the politically incorrect word "refugee" - that many people wanted to replace with "evacuee" - was a winner, too.
Incidentally, Jim Simons, after having discovered Chern-Simons theory and earned billions of dollars from his hedge fun(d), wants to investigate autism.
Roy Spencer has a nice essay on sustainability in TCS daily. The only sustainable thing is change, he says. He also argues that if the consumption of oil or production of carbon dioxide were unsustainable, a slower rate of the same processes would be unsustainable, too.
Sustainability becomes irrelevant because of technological advances in almost all cases. Spencer chooses Michael Crichton's favorite example - the unsustainable amount of horseshit in New York City 100 years ago when there were 175,000 horses in the city. Its growth looked like a looming disaster but it was stopped because of cars that suddenly appeared.
Also, he notices that the employees of a British Centre for Ecology and Hydrology - that had to be abolished - were informed that the center was unsustainable which is a very entertaining explanation for these people who fought for sustainability in their concerned scientific work. Also, Spencer gives economical explanations to various social phenomena. For example, the amount of possible catastrophic links between our acts and natural events as well as the number of types of our activities that will be claimed to be "unsustainable" in the scientific literature is proportional to the amount of money we pay to this sector of science.
It looks like we can run out of oil soon because the companies have no interest to look for more oil than what is needed right now - it is expensive to look for oil. That makes it almost certain that we will find much more oil than we know today.
Pure heterotic MSSM
As announced in October here, Braun, He, Ovrut, and Pantev have finally found an exact MSSM constructed from heterotic string theory on a specific Calabi-Yau.
The model has the Standard Model group plus the U(1)B-L, three generations of quarks and leptons including the right-handed neutrino, and exactly one pair of Higgs doublets which is the right matter content to obtain gauge coupling unification.
By choosing a better gauge bundle - with some novel tricks involving the ideal sheaves - they got rid of the second Higgs doublet. While they use the same Calabi-Yau space with h11=h12=3 i.e. with 6 complex geometric moduli, they now only have 13 (instead of 19) complex bundle moduli.
The probability that this model describes reality is roughly 10450 times bigger than the probability for a generic flux vacuum, for example the vacua that Prof. Susskind uses in his anthropic interview in New Scientist. ;-)
Thursday, December 15, 2005 ... Français/Deutsch/Español/Česky/Japanese/Related posts from blogosphere
Something between 2 and 3 billion visitors
This is how you can make quarter a million sound like a lot. ;-)
There is a counter on the right side. If you happen to see the number 250,000, you may write your name as a comment here. The prize for the round visitor includes 3 articles that he or she can post here.
The number 250,000 counts unique visitors - in the sense that every day, one IP address can only increase the number by one. The total number of hits is close to 1 million.
The Reference Frame does not plan any further celebrations. ;-)
Update: Robert Helling and Matt B. both claim to have grabbed 250,000, and I still have not decided who is right. Matt B. has sent me a screenshot so his case is pretty strong. It is academically possible that the number 250,000 was shown to two people - because by reloading, one can see the current "score" without adding a hit.
Lisa's public lecture
I just returned from a public lecture of Lisa Randall - who promoted science of extra dimensions and her book Warped Passages - and it was a very nice and impressive experience. Not surprisingly, the room was crowded - as crowded as it was during a lecture of Steve Pinker I attended some time ago. As far as I can say today, she is a very good speaker. There was nothing in her talk that I would object to and nothing that should have been said completely differently.
As you can guess, I was partially feeling as a co-coach whose athlete has already learned everything she should have learned. ;-)
Nima Arkani-Hamed introduced Lisa in a very professional and entertaining way. Randall used a PowerPoint presentation, showed two minutes of a cartoon edition of Abbott's Flatland, explained what are different ways to include and hide extra dimensions (with a focus on warped geometry), how they are related to some of the problems of particle physics such as the hierarchy problem, how do they fit into the framework of string theory and what string theory is, and what are the methods with which we're possibly gonna observe them. After the talk, she answered many questions from the audience in a completely meaningful way.
Wednesday, December 14, 2005 ... Français/Deutsch/Español/Česky/Japanese/Related posts from blogosphere
Coldest December
My time for writing on the blog continues to be limited, so let me just offer you a short provocation. The scientists may have been right after all, the global cooling is coming. ;-) This December will almost surely become one of the coldest American Decembers since the 19th century. Daily record lows have been breached in New York State (10 degrees F below the previous record), the Midwest (Illinois), Utah, Texas (classes canceled), Oklahoma, Colorado, Kansas, Pennsylvania (previous record was 1958), and elsewhere. More snow and cold is forecast. Natural gas is propelled to record.
You may say that it is just the U.S. However, severe cold wave grips North India, too, with at least 21 casualties. The capital sees the coldest day in 6 years. The same thing applies to China and the Communist Party of China helps poor to survive bitter winter. You may complain that I only talk about countries that host one half of the world's population. You're right: the global temperature continues to be stable, around 2.7 Kelvins. ;-)
We are doing fine in Massachusetts, the temperature is -10 Celsius degrees with windchill at -18 Celsius degrees. Tonight, it will be around 6 Fahrenheit. Don't forget your sweaters and gloves.
The consensus scientists have may found a sign error in their calculations. The carbon dioxide causes global cooling. This occassional sign flip is called the climate change.
Tuesday, December 13, 2005 ... Français/Deutsch/Español/Česky/Japanese/Related posts from blogosphere
Shut up and calculate
I would not promote overly technical lecture notes, especially not about things covered in many books. But the interpretation of quantum mechanics in general and decoherence in particular - a subject that belongs both to physics as well as advanced philosophy - is usually not given a sufficient amount of space in the textbooks, and some people may be interested in Lecture23.pdf.
Monday, December 12, 2005 ... Français/Deutsch/Español/Česky/Japanese/Related posts from blogosphere
Riemann's hypothesis
I just received a lot of interesting snail mail. The first one is from Prof. Winterberg, one of the discoverers of cold fusion. He argues against the extra dimensions, using a picture of naked fat people (actually, some of them are M2-branes) and a German letter he received from his adviser, Werner Heisenberg. Very interesting but I apologize to Prof. Winterberg - too busy to do something with his nice mail and the attached paper.
A publisher wants to sell the 1912 manuscript of Einstein about special relativity. Another publisher offers books about the Manhattan project and Feynman's impressive thesis.
One of the reasons I am busy now is Riemann's hypothesis. Would you believe that a proof may possibly follow from string theory? I am afraid I can't tell you details right now. It's not the first time when I am excited about a possible proof like that. After some time, I always realize how stupid I am and how other people have tried very similar things. The first time I was attracted to Riemann's hypothesis, roughly 12 years ago, I re-discovered a relation between zeta(s) and zeta(1-s). That was too elementary an insight that was far from a proof but at least it started to be clear why the hypothesis "should be" true. The time I need to figure out that these ideas are either wrong or old and standard is increasing with every new attempt - and the attempts become increasingly similar to other attempts of mathematicians who try various methods. Will the time diverge this time? :-)
Sunday, December 11, 2005 ... Français/Deutsch/Español/Česky/Japanese/Related posts from blogosphere
52.3 percent growth
What is a reasonable size of the GDP growth? 10 percent like in China? 4 percent like in the U.S.? Around 1 percent like in many European countries?
What if I tell you that a particular country had the GDP growth of 52.3 percent in 2004? Moreover, it is a country that is usually described as such a failure that the president of another country who more or less caused all these developments, including the number 52.3, should be hated or maybe even impeached according to hundreds of thousands of activists?
Don't you think that something is crazy about this whole situation? The country has not only a terrific growth potential but also a big potential to become an extremely civilized territory, just like it was thousands of years ago when Europe was their barbarian borderland.
Whether or not these things will happen depends on the acts of many people. Especially the people in that country itself. And also the people from other places in the world, especially America. Who do you think is a better human? Someone who tries to support positive developments in the world, including the country above, or someone who dreams about a failure in that country that would confirm his or her misconceptions that the president is a bad president?
I, for one, think that the members of the second group are immoral bastards. Moreover, it is pretty clear that most of them will spend the eternity at the dumping ground of history, unlike the president who will be written down as an important U.S. president in the future history textbooks.
All those critics who still retain at least a flavor of some moral values: please stop your sabotage as soon as possible. Even if you achieve what you want - a failure - it will be clear to everyone that the failure is not Bush's fault but your fault. |
8e93f9e674994cb8 | Techniques of Molecular Modelling: Quantum Mechanical Methods
Molecular orbital methods date back to the origins of quantum mechanics and the Schrödinger equation.
1. The first practical chemical solution to this equation was formulated by Erich Hückel in 1931 and dealt exclusively with planar conjugated systems.
2. This molecular orbital theory was thereafter extended by Pauling, Hartree and Fock, Mulliken and many others,
3. and first effectively implemented in the 1960s by Hoffmann and Lipscomb to all valence electron systems (EHT),
4. by Pople in the 1970s who defined a family of semi-empirical methods
5. which were first effectively paramterised by Dewar (MINDO/3, MNDO, AM1), Stewart PM3 and 6)
6. In parallel, effective implementations of ab initio and density functional methods emerged in the 1980s (Gaussian, GAMESS, Turbomole, Orca, etc)
In essence, QM methods differ from the MM methods in providing explicit solutions for the electronic component of a molecule rather than just the nuclear terms.
These methods all adopt the Born-Oppenheimer approximation. Stated approximately, we assume that the motion and (point) position of the nuclei is described by classical equations, whereas the electrons is described by non-classical wavefunctions. This means the nuclear-nuclear energy contributions are classical mechanical (electrostatic point charge), but the nuclear-electron and electron-electron terms are quantum mechanical and the total energy is the sum of these nuclear and electronic components. The starting point for solving the electronic part is the Schrödinger equation:
H.Ψ = E.Ψ
A Quick Summary of the Principle Quantum Mechanical Methods used today
Methods which integrate Hamiltonian (H) and Basis set (Ψ)
• AM1 (Austin Model 1, based on a parametrised Hartree-Fock model), a Semiempirical method parametrised for ~42 elements. The method uses only valence shell atomic orbitals, representing them with Slater functions (see below), whilst the inner shells (e.g. 1s for C) and other terms in the Hartree-Fock equations are modelled with parametric functions (often referred to as the NDDO set of approximations) in a manner similar to molecular mechanics.
• PM3, a reparametrisation of AM1, introduced in 1989 and a bit like the curate's egg (much better for e.g. H-bonding). Parameters for 42 elements in all combinations.
• PM6, a reparametrisation of PM3 introduced in 2007 and about twice as accurate. Includes parameters for 69 elements (but not necessarily all possible combinations) DOI: 10.1007/s00894-007-0233-4. In 2009, the method was extended with explicit terms for dispersion energies (van der Waals) and delocalized hydrogen bonds, PM6-DH2+.
• RM1 is the latest (2008) method in this stable, offering high accuracy parameters for all combinations of the 10 most common elements (H, C, N, O, P, S, F, Cl, Br, I).
• Oniom: A method first introduced in 1996 which partitions a molecule into layers, each treated using a different approximation. The most common is a two layer model, the outer layer treated using molecular mechanics (or even a simpler unified field model), the inner one treated using MO theories. Three layer models apply one of mechanics, one of semi-empirical and one of ab initio MO theories (DOI: 10.1021/jp962071j)
Η: List of Ab initio/Density functional Hamiltonians Ψ: List of Basis sets
• RHF, or (Spin) Restricted Hartree Fock. Applied to singlet electronic states (all electrons paired) and used for relatively rapid prototyping of a calculation. No consideration for electron correlation.
• UHF, or (Spin) Unrestricted Hartree Fock, for use with open shell systems (doublets, triplets, etc).
• ROHF. Restricted openshell Hartree Fock, for open shell systems. More limited than UHF for some systems (ie 2nd derivatives, etc) but free of spin contamination.
• B3LYP (UB3LYP): A popular density functional method (not strictly an ab initio one), offering good reliability and short range electron correlation treatment at the penalty of about half the speed of a RHF calculation. There are many other functionals that have been proposed, but none tested so thoroughly as this one.
• wB97XD (Density functional theory with empirical long-range correlation correction; DOI: 10.1039/b615319b). These dispersion corrections have been incorporated into new generations of double-hybrid methods, such as BML and B2-PLYP/B2GP-PLYP which address many deficiencies of B3LYP (DOI: 10.1021/jp710179r) and which occupy the so-called fifth rung of Jacob's ladder of approximations (DOI: 10.1063/1.1390175)
• MP2 (MP4), Moller-Plesset methods based on perturbation theory, an older method than the density functionals for incorporating electron correlation. Perhaps 10-100 times slower than HF, but in some cases more reliable, especially for so-called "long range correlation effects". New double-hybrid DFT methods rely on MP2 for the correlation part of the DFT functional.
• CISD; Configuration Interaction based on a Hartree-fock reference state, for systems where multiple electronic configurations and electron correlation are important, e.g. excited states etc.
• CASSCF(n,n); a keyword for invoking multi-configurational SCF where the Hartree-Fock reference state is simply not good enough (i.e. transition metal series, excited states, etc).
• CCSD(T), CCSDTQ; the so-called coupled-cluster approach to the correlation problem. Very very expensive, and can only be properly applied to about 10-12 (non-hydrogen) atoms.
• STO-3G. A minimal (single-ζ) basis set. Use for very rapid prototyping of a calculation, or initial geometry optimisation from a poor starting guess. The 3 means using three Gaussian-type functions sharing a common exponent (ζ) to represent a shell (modelled on the Slater-type-function or STO, this being adapted from the Schrödinger solutions for a hydrogen atom).
• 3-21G*. a small double-ζ basis set due to Pople, and available for most of the periodic table. The 3 is used for the core electrons (1s for eg carbon), the "2" and "1" are used for the valence electrons, and are represented by respectively two and one Gaussian functions, each with its own ζ exponent. The "*" means polarization (d) functions on e.g. S, P, Cl etc. Used for relatively rapid prototyping of a calculation.
• SDD, an alternative basis set for the entire period table using effective core potentials (Pseudopotentials) to reduce the number of basis functions (for the core electrons) and to include relativistic effects (for heavy elements).
• 6-31+(d,p). A reasonably high quality double-ζ basis set with anionic diffuse (+) and "polarisation functions" on the atoms ("p" orbital for Hydrogen, "d and f" orbital for carbon, "f and g" orbital for transition metals, etc).
• cc-pVTZ. A triple-ζ correlation-consistent basis set with valence polarization functions, now regarded as appropriate for reasonably high quality on small to medium sized molecules.
• aug-cc-pV5Z and aug-cc-pV5Z-pp. An augmented and polarized "pentuple ζ", correlation-consistent basis set for best quality calculations; available significantly for the first transition series. The higher elements are treated using pseudopotentials (pp). Also included in this type of basis is the aug-pcx (x=1-4) Kohn-Sham consistent basis for use on the fifth rung of Jacob's ladder.
• Gen: A general input for basis sets. Get them from here and mix-n-match your basis sets for the problem at hand.
Methods which extrapolate Hamiltonian and Basis set
• G2/G3 (Gaussian-3): Effectively an extrapolation of both the basis set, and the correlation treatment, the combination of which heads off towards exact solutions of the Schrödinger equation. Hugely expensive in computer time.
• W1-W4 Theories, currently considered the most accurate solutions to the Schrödinger equations available. This level of theory is now routinely correcting "faulty" experimental data! See DOI: 10.1063/1.2348881, 10.1021/jp071690x, 10.1021/jp900056w and 10.1063/1.3489113
How the various methods scale with size: Let us define N as ~ the number of atoms (slightly more accurately, it represents the number of basis functions associated with all the atoms in the molecule under consideration, and so it also goes up with the quality of the basis set)
One also has to consider how the 1st and 2nd derivatives of the energy with respect to the nuclear coordinates behave. The most recent implementations of density functional theory have low scaling 1st and 2nd analytical derivatives, which now routinely enable good geometry optimisations using BFGS-like algorithms.
Case Study 6: Directing effects in the Electrophilic substitution of Aromatic rings
There are two ways of approaching this problem. The simplest is to look at either the properties of the reactant, or the properties of the initial product, the Wheland intermediate. We will look at the first of these initially.
1. The Properties of the Reactant: A substituted Benzene.
Henry Armstrong, working at (the precursor to) Imperial College around 1887, was one of the first chemists to categorise substituents (R, or Radicles as he calls them) on a benzene ring in terms of the effect they had towards electrophilic substitution reactions (of X) of the ring. With the electron yet to be discovered, he attributed the two observed modes of behaviour (i.e. o/p in Table I vs m in Table II of 10.1039/CT8875100258) as being due to an electrically polarizable entity he called an "affinity", of which he suggested a benzene ring had six. He also suggested these affinities acted at a distance over the whole ring, but which differed in their directionality (or Resultant as he puts it, the modern term for which is Vector) according to the nature of R (a difference we nowadays categorize as being the electron donating or withdrawing properties of R). Armstrong's description of the properties of his affinity (See DOI: 10.1039/CT8875100258 where he writes "the introduction of a radical/substituent [onto the benzene ring] doubtless involves an altered distribution of the affinity, much as the distribution of the electric charge in a body is altered by bringing it near to another body". In 10.1039/PL8900600095, p102, he also clearly describes what we now know as a Wheland intermediate) may well constitute the first glimmering of the wave-like properties of the electron!
Erich Hückel some 44 years later was able to derive from Schrödinger's wave equation a simple expression which predicted how these "affinities", now named electrons of course, could be polarized. The resulting (π) molecular orbital functions (eigenvectors) can be used to illustrate graphically how both the directing effects (i.e. the probability of finding electrons in any particular position, i.e. the MOs) and the activating/deactivating effects (i.e. the probability of interacting with the electrons, i.e. the orbital energies) operate.
To maximise the visual effect, we are going to use two substituents R that were not in fact in Armstrong's original list; CH2+ for an electron withdrawing group and CH2- for an electron donating group. Its worth considering just for a moment why we are not using more conventional neutral groups (i.e. NO2 and NH2 respectively) for this illustration. Being neutral, the latter groups can only polarize the molecule by separating charge to produce a dipolar ionic species. Such ionic charge separation always takes a fair bit of energy to achieve compared to neutral covalent bonding, and actually requires proper solvation to treat quantitatively. It is much easier if the R group is already ionic, because then moving the charge from one part of the molecule to another takes little energy and no change in ionicity, and hence eliminates the need for any solvation treatment.
o/p Directing m-Directing
HOMO, R=CH2- HOMO, R=CH2+ HOMO-1, R=CH2+
The Highest occupied π canonical Molecular orbital (HOMO) for R=CH2- corresponds to the electron-pair least bound to the molecule and is hence the most available to react with an Electrophile. It has an energy of -1.7eV, which shows how relatively weakly it is bound (most π electrons have energies around -10ev or even less). This weak binding is also associated with it being a good nucleophile. Note how this wavefunction has density on the ortho and the para positions of the ring, but none at all on the meta position. This of course matches the resonance bond formulation familiar to all organic chemists, and hence the recognition of o/p direction and activation for such a substituent.
Contrast this with R=CH2+. The HOMO in this case has an energy of -15.0eV, very much more tightly bound, with hence with vastly reduced nucleophilic properties. It has a node in the para position. The next highest orbital is almost degenerate with the HOMO (-15.6 eV) and has a node in the ortho positions. Because these two orbitals are essentially the same in energy, they have to be considered together; the only position where both of them do have density is in the meta position. This substituent therefore is meta directing, but strongly deactivating (due to the very negative orbital energies).
If we now assume that the reactant has similar properties to the transition state for this reaction (the Hammond principle, in part) we can infer that the transition state will inherit the same properties, and that the electrophilic reactivity of such substituted benzene derivatives can be derived purely from the properties of the reactant (if its an early transition state)
2. The Properties of a Heterocycle: Pyridine.
What about a real system? The differing behaviour of Pyridine and its N-Oxide towards electrophilic substitution is well known. In particular, these two, apparently almost identical species, have quite different properties when nitrated. The former nitrates (albeit slowly) in the 3-position, whilst the latter nitrates (rather faster) in the 4-position. Does the HOMO for each reflect this? The answer is that the corresponding orbitals look very similar to the ones shown above! Inded, the size of the orbital at the 4-position even looks larger than that at the 2-position.
HOMO, Pyridine, m-directing HOMO, Pyridine N-Oxide, o/p directing
3. Properties of the Wheland/Armstrong Intermediate.
An prequel to looking at reactivity of aromatics is to consider the relative energy of the possible Wheland intermediates as being better indicators of the outcome of the reaction than simply the wavefunction of the reactant. How can one calculate the relative energies? The mechanics method, since in general the force field constants do not cancel out for the various pararmeters, is going to give a very unreliable relative energy. It will also not reflect on the polarization of the π-electrons as illustrated above. Only proper treatment of the electrons via quantum mechanics can treat this sort of problem. The energies shown below have been obtained from the PM6 semi-empirical method (which calculates the energies and optimizes the geometries of the protonated Wheland intermediate in just a few seconds). The outcome shows clearly the differentiation between Pyridine and its N-oxide, and also reproduces the preference for 4- rather than 2-substitution in the latter (But see DOI: 10.1039/P29750000277 where an argument is presented for 4-nitration resulting not by reaction of the free N-oxide but the protonated species).
Substrate, X ortho (Relative to para) meta (Relative to para) para
Pyridine, X=N -3.9 (234.6) -15.1 (223.4) 0.0 (238.5)
Pyridine N-Oxide, X=NO 4.8 (208.9) 25.6 (231.2) 0.0 (205.6)
X=NS 3.3 (232.4) 23.5 (251.1) 0.0 (227.6)
Phosphabenzene X=P 0.1 (210.7) 0.0 (210.6) 0.0 (210.6)
Phosphabenzene P-oxide, X=PO 1.7 (145.4) 45.1 (188.8) 0.0 (143.7)
X=PS 2.2 (170.7) 45.1 (213.6) 0.0 (168.5)
Arsabenzene X=As 3.6 (229.6) 19.6 (245.6) 0.0 (226.0)
Iridiabenzene X=Ir(PH3)3 3.4 39.4 0.0
What about other substituents? An almost entirely unstudied problem is the electrophilic substitution of phosphabenzene. It's a great surprise to find that in modelling terms at least, it behaves quite differently to pyridine! Using this approach, one can even predict reactivity for something as unconventional as metallabenzenes (for a review of the chemistry of metallabenzenes, s ee 10.1039/b517928a). For examples of such metallabenzenes with or
. For details of the bonding in the Ru system, see this blog post.
Case Study 2 Revisited: Where ARE the electrons in the Pirkle Reagent?
As a visualisation problem, viewing the Crystal structure of the Pirkle reagent provided a rationalisation of how the intermolecular interactions might proceed via a novel form of hydrogen bonding involving an entire face of one aromatic ring. We saw how Molecular mechanics is not parameterised to deal with such an unusual interaction, and so might not handle it properly. Clearly, the location of the electrons is critical, since the H+ binds to regions of high electron density. There are three aromatic rings to choose from in this molecule, and moreover, because of the chiral centre present, two faces to each ring, ie six possibilities in all. Can calculating the wavefunction of this molecule cast light on this?
Part 1: The π-electrons. One solution of the Schrödinger equation provides a canonical set of energy levels for electron (pairs) in the molecule, that having the highest energy being referred to as the HOMO (Highest occupied molecular orbital). As we saw before, the HOMO corresponds to the electron (pair) which is least strongly bound to the molecule, and hence can be regarded as describing where the most basic (= proton seeking) or nucleophilic (= nucleus seeking) electrons in the molecule are. So a good first start to understand where a hydrogen bond might occur is to plot the HOMO for the Pirkle reagent. Unfortunately, the HOMO is a π type orbital and shows almost no discrimination for binding to a proton!
Part 2: The integrated σ- and π-electrons. In fact the effect we are seeking is actually transmitted via the σ framework (originating in the inductive effect of the C-F bonds) and not the π system. The next level of approximation is to recognise that the basicity of a molecule derives from not just a single MO, but a suitably weighted sum of the contributions of all the electrons, and particularly of the σ bonds. This function is called the molecular electrostatic potential or MEP. This is a workfunction, and represents the amount of energy needed to remove a bare proton from any specified position close to a molecule out to an infinite distance away. If this energy is positive, the proton was clearly bound to the molecule, if the energy is negative, it was clearly repelled by the molecule. The function needs to be computed for all points in space around the molecule, and it is conventional to calculate an "iso-value surface", or actually two iso-surfaces, one representing a positive value of the MEP, the other a negative value. This is then rendered using computer graphics. In this display the negative potential which in effect attracts a proton (i.e. give rise to a positive work function) is rendered in purple.
Notice how the two faces of the anthracene ring are quite different, an effect induced by the electron withdrawing effect of the C-CF3 group which withdraws electrons anti-periplanar to the bond, i.e. on the face opposite to it. But in turn the lone pair of the oxygen atom re-focuses density on to one ring of this opposite face, and in fact the hydrogen bond forms at almost exactly the centre of this electron density. Thus this hydrogen bond is created by a complex stereo-electronic effect caused by the shaping of the electron density by one withdrawing group and one lone pair.
Part 3: The electron Topology (AIM or Atoms in Molecules). What happens when one places two of these molecules into the close proximity found in the crystal structure? The π-MOs are of little help, and the electrostatic potentials of the two components, as used above, interfere to the point of not really revealing anything. Yet another way of analyzing where the electrons are is needed. This is found in a technique known as Quantum Chemical Topology. It is the study of the topology of electron density in molecules, and uses the language of dynamical systems (critical points, manifolds, gradient vector fields etc, first introduced by the mathematician Henri Poincare) to identify key points in the electron densities.
There are precisely four types of points, differing in the characteristics of the curvature of the electron density ρ(r) in the region of the critical point. This curvature is obtained from the second derivative of the electron density with respect to cartesian space (the density Hessian). The eigenvalues of this 3 by 3 matrix can have exactly four conditions, which are summarised by the signature of this matrix, being the sum of the signs of these eigenvalues:
1. NCP: A critical point (all three eigenvalues -ve) which identifies where an attractor (i.e. a nucleus) is in a molecule
2. BCP: A critical point (two -ve, one +ve) which identifies bonds in a molecule. Normally found between a pair of attractors, but can also involve three attractors (a 3-centre bond).
3. RCP: A critical point (one -ve, two +ve) which identifies rings in a molecule
4. CCP: A critical point (all three eigenvalues -+ve) which identifies cages in molecules
5. The total number of these various points must satisfy a topological relationship known as the Poincaré-Hopf condition, which states that num(NCP-BCP+RCP-CCP) = 1.
6. One property is known as the Laplacian of the electron density, ∇2ρ(r) (being the sum of the diagonal elements of the density Hessian) at a BCP is used to characterise the type of bond in that region
The most relevant of the critical points is the second type; a BCP. It tells us in effect if a bond exists between any pair of nuclei. What it does not tell is is what the bond order of that bond is; it could easily for example be less than 1. That is a chemical rather than a topological interpretation.
What happens when such an analysis is conducted on the dimer of the Pirkle reagent? In the resulting analysis (left), the light blue points are the positions of some of the critical bond points identified in the electron density topology (there are many more than shown here). These reveal some fascinating insights into how the Pirkle reagent binds with itself.
The QTAIM Critical point analysis
1. There is indeed a bond critical point between the H of the OH group, and the π-face of one ring. It shows it binding to approximately three atoms of that ring, in exactly the manner first revealed by visualisation, and then by the MEP plot above. The electron density at this point (0.013 - 0.015 au) corresponds very approximately to an interaction energy of about 2.5 - 3.0 kcal/mol.
2. A set of four bond critical points occur between four pairs of atoms in the π-π- stacked anthracene rings, with densities of around 0.004 - 0.005 au. This reveals that such stacking is indeed (weakly) attractive and of significance (totalling around 3 kcal/mol).
3. A further pair of bond points is shown between the oxygen lone pair and the adjacent ring C-H bond; the density (0.018) indicates each to be worth about 3 kcal/mol.
4. Thus the OH group actually participates in two concurrent types of hydrogen bonding, one to a π-face via the H, and one to a C-H bond via the lone pair. This last interaction was not spotted in this molecule until this topological analysis was done! It also re-inforces the concept that these interactions are co-operative rather than independent.
The NCI Method: Surface based on (reduced) density gradient ∇ρ(r); surface colour based on ∇2ρ(r)
• red =strongly repulsive
• yellow = weakly repulsive
• green = weakly attractive
• blue= strongly attractive.
• For more explanation, see this summary.
The very recent NCI (non covalent interactions) method takes this analysis one step further by developing the concept of surfaces of interaction rather than just critical points. These surfaces are derived from the density gradient ∇ρ(r) (the deviation from a homogenous electron distribution), coloured by the value of the λ2 eigenvalue of the Laplacian of the density ∇2ρ(r). The NCI method is showing great promise for explaing e.g. stereoselectivity in chemical reactions.
Part 4: Energies again. Our analysis of the Pirkle reagent concludes here with a revisitation of the binding energy. We saw previously how this could be estimated using Molecular Mechanics. This had the advantage of including the so called dispersion, or long range correlation effects, but the disadvantage that there was no ready access to the entropy of the process. A very new procedure which incorporates both advantages is the dispersion-corrected density functional method, one very recent implementation of which is ωB97XD (DOI: 10.1039/b810189b), and this gives the following values:
ΔH dimerisation -25.1; ΔS -49.2, TΔS = -14.7; ΔG dimerisation -10.4 kcal/mol.
Case Study 1 Revisited: How do orbitals interact within a molecule?
We (think we) know where covalent σ-bonds are in molecules, they lie along the straight line connecting the nuclei of two atoms. We can idealize lone pairs on atoms such as oxygen, and in case study 1, all we needed was an (X-ray) structure to spot a connection between the (antiperiplanar) orientation of (certain) such idealized lone pairs and the C-O bonds in the molecule. But are the lone pairs really where we have idealized them? And in dealing the orientation of the lone pair with a σ-bond, we become aware that we are actually dealing with a concept derived from the Klopman-Salem equation, which uses not so much the two-electron σ-bond, as a vacant orbital occupying the (assumed) same orientation known as the σ*-bond. This was the basis of the earlier lecture course on stereo-electronics and also conformational analysis. It is time to take some of these concepts and show how they can be properly quantified. Let us proceed as follows:
1. We will solve the Schrödinger equation for a molecule (to any required degree of accuracy). This gives us functions (the canonical molecular orbitals) which tell us the electron density probability over the molecule as a whole.
2. We now need to convert this (global) function to some sort of local property, since the concept of a bond is local, and not descriptive of the molecule as a whole. One such procedure is known as the NBO (Natural bond orbitals). A more formal description of this type of orbital is: Natural Bond Orbitals (NBOs) are localized few-center orbitals ("few" meaning typically 1 or 2, but occasionally more) that describe the Lewis-like molecular bonding pattern of electron pairs in optimally compact form. This can be precisely expressed in mathematical form (but we shall not do so here!); it is sufficient to say that the electron density probability predicted by summing all the occupied canonical molecular orbitals and alternatively all the local NBOs comes out the same! (i.e. there is more than one way of constructing a wavefunction for a molecule that predicts its overall electron density distribution, which in fact is the only properly measurable property).
3. Just as with canonical MOs, where we had a (doubly) occupied set and a virtual (unoccupied) set, so the NBOs turn out to geometrically occupy the bonds (BD), the lower energy non-valence cores (CR), and the valence lone pairs (LP) of the molecule, with the unoccupied orbitals being represented by BD* and RY* (don't worry about this last one, it stands for Rydberg).
4. In the NBO formalism, one can calculate an interaction energy between any doubly occupied BD or LP NBO, and any unoccupied BD* NBO. This can be represented in very simple terms by the diagram:
5. This is exactly the form of the Klopman-Salem diagram you may well recollect from 2nd year conformational analysis lectures. But we have now made progress. By using the NBO expressions, we can obtain an exact value for the value of the quantity E(2).
6. When this is done for the two dioxepins discussed in case study 1, the following results (at the B3LYP/6-31G(d) level) are obtained (with the keyword pop(nbo)) for the stable (green) and unstable (pink) forms;
7. The only two large E(2) terms (both ~15.5 kcal/mol) are between an oxygen LP and a C-O BD* involving the adjacent oxygen. They correspond exactly to the two antiperiplanar interactions we first noticed by visualisation, but now we can precisely quantify these interactions.
8. For the unstable isomer, only one large E(2) term is found (13.4); the other (7.5) is now significantly reduced because the orientation is no longer exactly antiperiplanar. The molecule is less stabilised and the two C-O bonds are no longer equal in length.
NBO analysis is very useful in a variety of situations, and is increasingly found in the literature. For example, it can be used to rationalise the gauche preference of 1,2-difluoroethane, and many others. Such E(2) NBO terms are thought to indicate useful chemistry down to energies of ~3 kcal/mol, and ones as high as 130 kcal/mol have been found (~15 is however typical of the anomeric effect). See this blog for another example where the unexpected was highlighted by such techniques.
Case Study 4 Revisited: The Helical structure of DNA
I left this case study dangling, by posing the question: how was it determined that the helix is right handed? Watson and Crick actually answered this by stating (very briefly with no details) that a left handed helix could only be constructed by violating the permissible van der Waals contacts. The vdW contacts are actually very well modelled using molecular mechanics (the 4th term in the force field described here) and so a reasonable model could have been constructed using this technique to test this assertion. It suffers one slight deficiency. Because it uses force constants to define the terms, these are assumed to be the same everywhere throughout the molecule. Thus the rungs of the DNA ladder in the middle are treated the same as at the end of the helix. Is this justified? It is only very recently that vdW effects have been incorporated into quantum mechanical models in a manner which can be applied to large molecules (ωB97XD, see above). So we will kill two birds with one stone by jumping straight to this technique. We will also overcome another defect of mechanics, and that is the general lack of implementation for incorporating entropic effects on the energies. We will pose the new question: what are the relative free energies of and handed DNA helices (which, by the way are diastereomers, not enantiomers!). It turns out the answer depends on the nature of the base pairs.
1. If they are exclusively C-G, as in d(CGCG)2, ΔG298 favours the left handed Z-helix by ~12 kcal/mol.
2. If they are A-T, as in d(ATAT)2, the right handed B form is favoured by ~4.3 kcal/mol. As it happens, this is exactly what is known for small (tetramers) of DNA!
3. Dispersion terms alone (van der Waals) make Z-d(CGCG)2 5.1 kcal/mol LESS stable than B-d(CGCG)2,
4. Dispersion terms alone (van der Waals) make Z-d(ATAT)2 12.5 kcal/mol LESS stable than B-d(ATAT)2.
System ΔΔG
1. Furanose oxygen within 2.85Å of a guanine ring
2. Furanose OC-H hydrogen within 2.48Å of a second furanose oxygen
3. Facilitated by C-Hσ/C-Oσ* anti-periplanar acidification (E2 5.8, magenta bonds)
4. Anomeric interaction between iguanine and the ribose; NLp/C-Oσ*; E2 11.6 (violet bond).
5. Cytosine-furanose anomeric (E2 6.8, indigo bond).
6. Gauche-like conformation of the ethane-1,2-diol fragment (gold bond).
We conclude that Watson and Crick's model building, based on van der Waals analysis, did indeed favour the B-DNA helical form, but that there are other aspects that can overcome this for CG-rich DNA strands. See for example this blog for more details.
Bsck to scales|| Back to Visualisation| Back to Mechanics|On to Transition states| On to Advanced| |
bf96bd494d09ac86 | Stanford Encyclopedia of Philosophy
Bohmian Mechanics
First published Fri Oct 26, 2001; substantive revision Tue Jul 3, 2012
Bohmian mechanics inherits and makes explicit the nonlocality implicit in the notion, common to just about all formulations and interpretations of quantum theory, of a wave function on the configuration space of a many-particle system. It accounts for all of the phenomena governed by nonrelativistic quantum mechanics, from spectral lines and scattering theory to superconductivity, the quantum Hall effect and quantum computing. In particular, the usual measurement postulates of quantum theory, including collapse of the wave function and probabilities given by the absolute square of probability amplitudes, emerge from an analysis of the two equations of motion: Schrödinger's equation and the guiding equation. No invocation of a special, and somewhat obscure, status for observation is required.
1. The Completeness of the Quantum Mechanical Description
Conceptual difficulties have plagued quantum mechanics since its inception, despite its extraordinary predictive successes. The basic problem, plainly put, is this: It is not at all clear what quantum mechanics is about. What, in fact, does quantum mechanics describe?
It might seem, since it is widely agreed that any quantum mechanical system is completely described by its wave function, that quantum mechanics is fundamentally about the behavior of wave functions. Quite naturally, no physicist wanted this to be true more than did Erwin Schrödinger, the father of the wave function. Nonetheless, Schrödinger ultimately found this impossible to believe. His difficulty had little to do with the novelty of the wave function (Schrödinger 1935): “That it is an abstract, unintuitive mathematical construct is a scruple that almost always surfaces against new aids to thought and that carries no great message.” Rather, it was that the “blurring” that the spread out character of the wave function suggests “affects macroscopically tangible and visible things, for which the term ‘blurring’ seems simply wrong.”
For example, in the same paper Schrödinger noted that it may happen in radioactive decay that
the emerging particle is described … as a spherical wave … that impinges continuously on a surrounding luminescent screen over its full expanse. The screen however does not show a more or less constant uniform surface glow, but rather lights up at one instant at one spot ….
And he observed that one can easily arrange, for example by including a cat in the system, “quite ridiculous cases” with
the ψ-function of the entire system having in it the living and the dead cat (pardon the expression) mixed or smeared out in equal parts.
It is thus because of the “measurement problem,” of macroscopic superpositions, that Schrödinger found it difficult to regard the wave function as “representing reality.” But then what does? With evident disapproval, Schrödinger observes that
the reigning doctrine rescues itself or us by having recourse to epistemology. We are told that no distinction is to be made between the state of a natural object and what I know about it, or perhaps better, what I can know about it if I go to some trouble. Actually — so they say — there is intrinsically only awareness, observation, measurement.
Many physicists pay lip service to the Copenhagen interpretation — that quantum mechanics is fundamentally about observation or results of measurement. But it is becoming increasingly difficult to find any who, when pressed, will defend this interpretation. It seems clear that quantum mechanics is fundamentally about atoms and electrons, quarks and strings, not those particular macroscopic regularities associated with what we call measurements of the properties of these things. But if these entities are not somehow identified with the wave function itself — and if talk of them is not merely shorthand for elaborate statements about measurements — then where are they to be found in the quantum description?
There is, perhaps, a very simple reason why it is so difficult to discern in the quantum description the objects we believe quantum mechanics ought to describe. Perhaps the quantum mechanical description is not the whole story, a possibility most prominently associated with Albert Einstein. (For a general discussion of Einstein's scientific philosophy, and in particular of his approach to the conflicting positions of realism and positivism, see the entry on Einstein's philosophy of science.)
In 1935 Einstein, Boris Podolsky and Nathan Rosen defended this possibility in their famous EPR paper (Einstein et al. 1935). They concluded with this observation:
The argument that the EPR paper advances to support this conclusion invokes quantum correlations and an assumption of locality. (See the entries on the Einstein-Podolsky-Rosen argument in quantum theory and on quantum entanglement and information.)
Later, on the basis of more or less the same considerations as those of Schrödinger quoted above, Einstein again concluded that the wave function does not provide a complete description of individual systems, an idea he called “this most nearly obvious interpretation” (Einstein 1949, p. 672). In relation to a theory incorporating a more complete description, Einstein remarked that “the statistical quantum theory would … take an approximately analogous position to the statistical mechanics within the framework of classical mechanics.” We note here, and show below, that Bohmian mechanics exactly fits this description.
2. The Impossibility of Hidden Variables … or the Inevitability of Nonlocality?
John von Neumann, one of the greatest mathematicians of the twentieth century, claimed that he had proven that Einstein's dream of a deterministic completion or reinterpretation of quantum theory was mathematically impossible. He concluded that (von Neumann 1932, p. 325 of the English translation)
It is therefore not, as is often assumed, a question of a re-interpretation of quantum mechanics — the present system of quantum mechanics would have to be objectively false, in order that another description of the elementary processes than the statistical one be possible.
Physicists and philosophers of science almost universally accepted von Neumann's claim. For example, Max Born, who formulated the statistical interpretation of the wave function, assured us that (Born 1949, p. 109)
No concealed parameters can be introduced with the help of which the indeterministic description could be transformed into a deterministic one. Hence if a future theory should be deterministic, it cannot be a modification of the present one but must be essentially different.
Bohmian mechanics is a counterexample to the claims of von Neumann. Thus von Neumann's argument must be wrong. In fact, according to John Bell (Mermin 1993, p. 805), von Neumann's assumptions (about the relationships among the values of quantum observables that must be satisfied in a hidden-variables theory) are so unreasonable that the “the proof of von Neumann is not merely false but foolish!” Nonetheless, some physicists continue to rely on von Neumann's proof.
Recently, however, physicists more commonly cite the Kochen-Specker Theorem and, more frequently, Bell's inequality in support of the contention that a deterministic completion of quantum theory is impossible. We still find, a quarter of a century after the rediscovery of Bohmian mechanics in 1952, statements such as these (Wigner 1976):
The proof he [von Neumann] published …, though it was made much more convincing later on by Kochen and Specker, still uses assumptions which, in my opinion, can quite reasonably be questioned. … In my opinion, the most convincing argument against the theory of hidden variables was presented by J. S. Bell (1964).
Now there are many more statements of a similar character that we could cite. This quotation is significant because Wigner was one of the leading physicists of his generation. Unlike most of his contemporaries, moreover, he was also profoundly concerned about the conceptual foundations of quantum mechanics and wrote on the subject with great clarity and insight.
There was, however, one physicist who wrote on this subject with even greater clarity and insight than Wigner himself: the very J. S. Bell whom Wigner praises for demonstrating the impossibility of a deterministic completion of quantum theory such as Bohmian mechanics. Here's how Bell himself reacted to Bohm's discovery (Bell 1987, p. 160):
Wigner to the contrary notwithstanding, Bell did not establish the impossibility of a deterministic reformulation of quantum theory, nor did he ever claim to have done so. On the contrary, until his untimely death in 1990, Bell was the prime proponent, and for much of this period almost the sole proponent, of the very theory, Bohmian mechanics, that he supposedly demolished.
Bohmian mechanics is of course as much a counterexample to the Kochen-Specker argument for the impossibility of hidden variables as it is to the one of von Neumann. It is obviously a counterexample to any such argument. However reasonable the assumptions of such an argument, some of them must fail for Bohmian mechanics.
Wigner was quite right to suggest that the assumptions of Kochen and Specker are more convincing than those of von Neumann. They appear, in fact, to be quite reasonable indeed. However, they are not. The impression that they are arises from a pervasive error, an uncritical realism about operators, that we discuss below in the sections on quantum observables, spin, and contextuality.
John Bell replaced the “arbitrary axioms” (Bell 1987, page 11) of Kochen-Specker and others by an assumption of locality, of no action-at-a-distance. It would be hard to argue against the reasonableness of such an assumption, even if one were so bold as to doubt its inevitability. Bell showed that any hidden-variables formulation of quantum mechanics must be nonlocal, as, indeed, Bohmian mechanics is. But he showed much much more. (For more detail on Bell's locality assumption, see Bell's theorem in Scholarpedia.)
In a celebrated paper he published in 1964, Bell showed that quantum theory itself is irreducibly nonlocal. (More precisely, Bell's analysis applies to any single-world version of quantum theory, i.e., any version for which measurements have outcomes that, while they may be random, are nonetheless unambiguous and definite, in contrast to the situation with Everett's many-worlds version of quantum theory.) This fact about quantum mechanics, based as it is on a short and mathematically simple analysis, could have been recognized soon after the discovery of quantum theory in the 1920's. That this did not happen is no doubt due in part to the obscurity of orthodox quantum theory and to the ambiguity of its commitments. It was, in fact, his examination of Bohmian mechanics that led Bell to his nonlocality analysis. In the course of investigating Bohmian mechanics, he observed that (Bell 1987, p. 11):
Bohm of course was well aware of these features of his scheme, and has given them much attention. However, it must be stressed that, to the present writer's knowledge, there is no proof that any hidden variable account of quantum mechanics must have this extraordinary character. It would therefore be interesting, perhaps, to pursue some further “impossibility proofs,” replacing the arbitrary axioms objected to above by some condition of locality, or of separability of distant systems.
In a footnote, Bell added that “Since the completion of this paper such a proof has been found.” He published it in his 1964 paper, “On the Einstein-Podolsky-Rosen Paradox.” In this paper he derives Bell's inequality, the basis of his conclusion of quantum nonlocality. (See the entry on Bell's Theorem. For a discussion of how nonlocality emerges in Bohmian mechanics, see Section 13.)
It is worth stressing that Bell's analysis indeed shows that any (single-world) account of quantum phenomena must be nonlocal, not just any hidden variables account. Bell showed that the predictions of standard quantum theory itself imply nonlocality. Thus if these predictions govern nature, then nature is nonlocal. [That nature is so governed, even in the crucial EPR-correlation experiments, has by now been established by a great many experiments, the most conclusive of which is perhaps that of Aspect (Aspect et al., 1982).]
Bell, too, stressed this point (by determinism Bell here means hidden variables):
The “problem” and “difficulty” to which Bell refers above is the conflict between the predictions of quantum theory and what can be inferred, call it C, from an assumption of locality in Bohm's version of the EPR argument, a conflict established by Bell's inequality. C happens to concern the existence of a certain kind of hidden variables, what might be called local hidden variables, but this fact is of little substantive importance. What is important is not so much the identity of C as the fact that C is incompatible with the predictions of quantum theory. The identity of C is, however, of great historical significance: it is responsible for the misconception that Bell proved that hidden variables are impossible, a belief that physicists until recently almost universally shared, as well as for the view, even now almost universally held, that what Bell's result does is to rule out local hidden variables, a view that is misleading.
Here again is Bell, expressing the logic of his two-part demonstration of quantum nonlocality, the first part of which is Bohm's version of the EPR argument:
As with just about everything else in the foundations of quantum mechanics, there remains considerable controversy about what exactly Bell's analyis demonstrates. Nonetheless, the opinion of Bell himself about what he showed is perfectly clear.
3. History
The pilot-wave approach to quantum theory was initiated by Einstein, even before the discovery of quantum mechanics itself. Einstein hoped that interference phenomena involving particle-like photons could be explained if the motion of the photons was somehow guided by the electromagnetic field — which would thus play the role of what he called a Führungsfeld or guiding field (see Wigner 1976, p. 262 and Bacciagaluppi and Valentini 2009, Ch. 9). While the notion of the electromagnetic field as guiding field turned out to be rather problematical, Max Born explored the possibility that the wave function could play this role, of guiding field or pilot wave, for a system of electrons in his early paper founding quantum scattering theory (Born 1926). Heisenberg was profoundly unsympathetic.
Not long after Schrödinger's discovery of wave mechanics in 1926, i.e., of Schrödinger's equation, Louis de Broglie in effect discovered Bohmian mechanics: In 1927, de Broglie found an equation of particle motion equivalent to the guiding equation for a scalar wave function (de Broglie 1928, p. 119), and he explained at the 1927 Solvay Congress how this motion could account for quantum interference phenomena. However, despite what is suggested by Bacciagaluppi and Valentini (2009), de Broglie responded very poorly to an objection of Wolfgang Pauli (Pauli 1928) concerning inelastic scattering, no doubt making a rather bad impression on the illustrious audience at the congress.
Born and de Broglie very quickly abandoned the pilot-wave approach and became enthusiastic supporters of the rapidly developing consensus in favor of the Copenhagen interpretation. David Bohm (Bohm 1952) rediscovered de Broglie's pilot-wave theory in 1952. He was the first person to genuinely understand its significance and implications. John Bell became its principal proponent during the sixties, seventies and eighties.
4. The Defining Equations of Bohmian Mechanics
In Bohmian mechanics the wave function, obeying Schrödinger's equation, does not provide a complete description or representation of a quantum system. Rather, it governs the motion of the fundamental variables, the positions of the particles: In the Bohmian mechanical version of nonrelativistic quantum theory, quantum mechanics is fundamentally about the behavior of particles; the particles are described by their positions, and Bohmian mechanics prescribes how these change with time. In this sense, for Bohmian mechanics the particles are primary, or primitive, while the wave function is secondary, or derivative.
Warning: It is the positions of the particles in Bohmian mechanics that are its “hidden variables,” an unfortunate bit of terminology. As Bell (1987, page 201) writes, referring to Bohmian mechanics and similar theories,
Absurdly, such theories are known as ‘hidden variable’ theories. Absurdly, for there it is not in the wavefunction that one finds an image of the visible world, and the results of experiments, but in the complementary ‘hidden’(!) variables. Of course the extra variables are not confined to the visible ‘macroscopic’ scale. For no sharp definition of such a scale could be made. The ‘microscopic’ aspect of the complementary variables is indeed hidden from us. But to admit things not visible to the gross creatures that we are is, in my opinion, to show a decent humility, and not just a lamentable addiction to metaphysics. In any case, the most hidden of all variables, in the pilot wave picture, is the wavefunction, which manifests itself to us only by its influence on the complementary variables.
Bohmian mechanics is the minimal completion of Schrödinger's equation, for a nonrelativistic system of particles, to a theory describing a genuine motion of particles. For Bohmian mechanics the state of a system of N particles is described by its wave function ψ = ψ(q1,…,qN) = ψ(q), a complex (or spinor-valued) function on the space of possible configurations q of the system, together with its actual configuration Q defined by the actual positions Q1,…,QN of its particles. (The word ‘spinor’ refers to a suitable array of complex numbers in place of a single one. Spinor-valued wave functions are used in quantum mechanics to describe electrons and other quantum particles that ‘have spin’.) The theory is then defined by two evolution equations: Schrödinger's equation
iℏ(∂ψ/∂t) = Hψ
for ψ(t), where H is the nonrelativistic (Schrödinger) Hamiltonian, containing the masses of the particles and a potential energy term, and (writing Im[z] for the imaginary part b of a complex number z = a +ib) a first-order evolution equation,
The Guiding Equation:
dQk/dt = (ℏ/mk) Im [ψ*∂kψ/ ψ*ψ] (Q1,…,QN)
for Q(t), the simplest first-order evolution equation for the positions of the particles that is compatible with the Galilean (and time-reversal) covariance of the Schrödinger evolution (Dürr et al. 1992, pp. 852–854). Here ℏ is Planck's constant divided by 2π, mk is the mass of the k-th particle, and ∂k = (∂/∂xk,∂/∂yk,∂/∂zk) is the gradient with respect to the generic coordinates qk = (xk,yk,zk) of the k-th particle. If ψ is spinor-valued, the two products involving ψ in the equation should be understood as scalar products (involving sums of products of spinor components). When external magnetic fields are present, the gradient should be understood as the covariant derivative, involving the vector potential. (Since the denominator on the right hand side of the guiding equation vanishes at the nodes of ψ, global existence and uniqueness for the Bohmian dynamics is a nontrivial matter. It is proven in Berndl, Dürr, et al. 1995 and in Teufel and Tumulka 2005.)
For an N-particle system these two equations (together with the detailed specification of the Hamiltonian, including all interactions contributing to the potential energy) completely define Bohmian mechanics. This deterministic theory of particles in motion accounts for all the phenomena of nonrelativistic quantum mechanics, from interference effects to spectral lines (Bohm 1952, pp. 175–178) to spin (Bell 1964, p. 10). It does so in an entirely ordinary manner, as we explain in the following sections.
For a scalar wave function, describing particles without spin, the form of the guiding equation above is a little more complicated than necessary, since the complex conjugate of the wave function, which appears in the numerator and the denominator, cancels out. If one looks for an evolution equation for the configuration compatible with the space-time symmetries of Schrödinger's equation, one almost immediately arrives at the guiding equation in this simpler form as the simplest possibility.
However, the form above has two advantages: First, it makes sense for particles with spin — and, in fact, Bohmian mechanics without further ado accounts for all the apparently paradoxical quantum phenomena associated with spin. Secondly, and this is crucial to the fact that Bohmian mechanics is empirically equivalent to orthodox quantum theory, the right hand side of the guiding equation is J/ρ, the ratio of the quantum probability current to the quantum probability density. This shows that it should require no imagination whatsoever to guess the guiding equation from Schrödinger's equation, provided one is looking for one, since the classical formula for current is density times velocity. Moreover, it follows from the quantum continuity equation ∂ρ/∂t + div J = 0, an immediate consequence of Schrödinger's equation, that if at some time (say the initial time) the configuration Q of our system is random, with distribution given by |ψ|2 = ψ*ψ, this will always be true (provided the system does not interact with its environment).
This demonstrates that it is wrong to claim that the predictions of quantum theory are incompatible with the existence of hidden variables, with an underlying deterministic model in which quantum randomness arises from averaging over ignorance. Bohmian mechanics provides us with just such a model: For any quantum experiment we merely take as the relevant Bohmian system the combined system, including the system upon which the experiment is performed as well as all the measuring instruments and other devices used to perform the experiment (together with all other systems with which these have significant interaction over the course of the experiment). We then obtain the “hidden variables” model by regarding the initial configuration of this big system as random in the usual quantum mechanical way, with distribution given by |ψ|2. The guiding equation for the big system then transforms the initial configuration into the final configuration at the conclusion of the experiment. It then follows that this final configuration of the big system, including in particular the orientation of instrument pointers, will also be distributed in the quantum mechanical way. Thus our deterministic Bohmian model yields the usual quantum predictions for the results of the experiment.
As the preceding paragraph suggests, and as we discuss in more detail later, Bohmian mechanics does not need any “measurement postulates” or axioms governing the behavior of other “observables”. Any such axioms would be at best redundant and could be inconsistent.
Besides the guiding equation, there are other velocity formulas with nice properties, including Galilean symmetry, and yielding theories that are empirically equivalent to orthodox quantum theory — and to Bohmian mechanics (Deotto and Ghirardi, 1998). The Bohmian choice is arguably the simplest. Moreover, Wiseman (2007) has shown that it is the Bohmian velocity formula, given by the guiding equation, that, according to orthodox quantum theory, would be found in a “weak measurement” of the velocity of a particle. And, somewhat paradoxically, it can be shown (Dürr et al., 2009) that according to Bohmian mechanics such a measurement is indeed a genuine measurement of the particle's velocity — despite the existence of empirically equivalent velocity formulas! Similarly, weak measurements could be used to measure trajectories. In fact, quite recently Kocsis et al. (2011) have used weak measurements to reconstruct the trajectories for single photons “as they undergo two-slit interference,” finding “those predicted in the Bohm-de Broglie interpretation of quantum mechanics.”
5. The Quantum Potential
Bohmian mechanics as presented here is a first-order theory, in which it is the velocity, the rate of change of position, that is fundamental. It is this quantity, given by the guiding equation, that the theory specifies directly and simply. The second-order (Newtonian) concepts of acceleration and force, work and energy do not play any fundamental role. Bohm, however, did not regard his theory in this way. He regarded it, fundamentally, as a second-order theory, describing particles moving under the influence of forces, among which, however, is a force stemming from a “quantum potential.”
In his 1952 hidden-variables paper (Bohm 1952), Bohm arrived at his theory by writing the wave function in the polar form ψ = Rexp(iS/ℏ), where S and R are real, with R nonnegative, and rewriting Schrödinger's equation in terms of these new variables to obtain a pair of coupled evolution equations: the continuity equation for ρ = R2 and a modified Hamilton-Jacobi equation for S. This differs from the usual classical Hamilton-Jacobi equation only by the appearance of an extra term, the quantum potential
U = −∑k(ℏ2/2mk) (∂k2 R / R ),
alongside the classical potential energy term.
Bohm then used the modified Hamilton-Jacobi equation to define particle trajectories just as one does for the classical Hamilton-Jacobi equation, that is, by identifying ∂kS with mkvk, i.e., by setting
dQk/dt = ∂kS / mk.
This is equivalent to the guiding equation for particles without spin. [In this form the (pre-Schrödinger equation) de Broglie relation p = ℏk, as well as by the eikonal equation of classical optics, already suggest the guiding equation.] The resulting motion is precisely what would be obtained classically if the particles were acted upon by the force generated by the quantum potential, in addition to the usual forces.
The quantum potential formulation of the de Broglie-Bohm theory is still fairly widely used. For example, the monographs by Bohm and Hiley and by Holland present the theory in this way. And regardless of whether or not we regard the quantum potential as fundamental, it can in fact be quite useful. In order to see most clearly that Newtonian mechanics should be expected to emerge from Bohmian mechanics in the classical limit, it is convenient to transform the theory into Bohm's Hamilton-Jacobi form. Then the (size of the) quantum potential provides a measure of the deviation of Bohmian mechanics from its classical approximation. Moreover, the quantum potential is also useful for developing approximation schemes for solutions to Schrödinger's equation (Nerukh and Frederick 2000).
However, Bohm's rewriting of Schrödinger's equation in terms of variables that seem interpretable in classical terms is not without a cost. The most obvious is an increase in complexity: Schrödinger's equation is rather simple, and it is linear, whereas the modified Hamilton-Jacobi equation is somewhat complicated, and highly nonlinear. Moreover the latter, since it involves R, requires the continuity equation for its closure. The quantum potential itself is neither simple nor natural. Even to Bohm it seemed “rather strange and arbitrary” (Bohm 1980, p. 80). And it is not very satisfying to think of the quantum revolution as amounting to the insight that nature is classical after all, except that there is in nature what appears to be a rather ad hoc additional force term, the one arising from the quantum potential. The artificiality that the quantum potential suggests is the price one pays for casting a highly nonclassical theory into a classical mold.
Moreover, the connection between classical mechanics and Bohmian mechanics that the quantum potential suggests is rather misleading. Bohmian mechanics is not simply classical mechanics with an additional force term. In Bohmian mechanics the velocities are not independent of positions, as they are classically, but are constrained by the guiding equation. (In classical Hamilton-Jacobi theory we also have this equation for the velocity, but there the Hamilton-Jacobi function S can be entirely eliminated and the description in terms of S simplified and reduced to a finite-dimensional description, with basic variables the positions and the (unconstrained) momenta of all the particles, given by Hamilton's or Newton's equations.)
Arguably, the most serious flaw in the quantum potential formulation of Bohmian mechanics is that it gives a completely false impression of the lengths to which we must go in order to convert orthodox quantum theory into something more rational. The quantum potential suggests, as has often been stated, that transforming Schrödinger's equation into a theory that can account in “realistic” terms for quantum phenomena, many of which are dramatically nonlocal, requires adding to the theory a complicated quantum potential of a grossly nonlocal character. It should be clear that this view is inappropriate. After all, the quantum potential need not even be mentioned in the formulation of Bohmian mechanics, and it in any case merely reflects the wave function, which Bohmian mechanics shares with orthodox quantum theory.
6. The Two-Slit Experiment
According to Richard Feynman, the two-slit experiment for electrons is (Feynman et al. 1963, p. 37–2) “a phenomenon which is impossible, absolutely impossible, to explain in any classical way, and which has in it the heart of quantum mechanics. In reality it contains the only mystery.” This experiment (Feynman 1967, p. 130) “has been designed to contain all of the mystery of quantum mechanics, to put you up against the paradoxes and mysteries and peculiarities of nature one hundred per cent.” As to the question (Feynman 1967, p. 145), “How does it really work? What machinery is actually producing this thing? Nobody knows any machinery. Nobody can give you a deeper explanation of this phenomenon than I have given; that is, a description of it.”
But Bohmian mechanics is just such a deeper explanation. It resolves in a rather straightforward manner the dilemma of the appearance of both particle and wave properties in one and the same phenomenon: Bohmian mechanics is a theory of motion describing a particle (or particles) guided by a wave. Here we have a family of Bohmian trajectories for the two-slit experiment.
figure 1
Figure 1: An ensemble of trajectories for the two-slit experiment, uniform in the slits.
(Adapted by Gernot Bauer from Philippidis et al. 1979.)
While each trajectory passes through only one slit, the wave passes through both; the interference profile that therefore develops in the wave generates a similar pattern in the trajectories guided by the wave.
Compare Feynman's presentation with Bell's (Bell 1987, p. 191):
Is it not clear from the smallness of the scintillation on the screen that we have to do with a particle? And is it not clear, from the diffraction and interference patterns, that the motion of the particle is directed by a wave? De Broglie showed in detail how the motion of a particle, passing through just one of two holes in screen, could be influenced by waves propagating through both holes. And so influenced that the particle does not go where the waves cancel out, but is attracted to where they cooperate. This idea seems to me so natural and simple, to resolve the wave-particle dilemma in such a clear and ordinary way, that it is a great mystery to me that it was so generally ignored.
Perhaps the most puzzling aspect of the two-slit experiment is the following: If, by any means whatsoever, it is possible to determine the slit through which the particle passes, the interference pattern will be destroyed. This dramatic effect of observation is, in fact, a simple consequence of Bohmian mechanics. To see this, one must consider the meaning of determining the slit through which the particle passes. This must involve interaction with another system that the Bohmian mechanical analysis must include.
The destruction of interference is related, naturally enough, to the Bohmian mechanical analysis of quantum measurement (Bohm 1952). It occurs via the mechanism that in Bohmian mechanics leads to the “collapse of the wave function.”
7. The Measurement Problem
The measurement problem is most the commonly cited of the conceptual difficulties that plague quantum mechanics. (It amounts, more or less, to the paradox of Schrödinger's cat.) Indeed, for many physicists the measurement problem is not merely one conceptual difficulty of quantum mechanics; it is the conceptual difficulty.
The problem is as follows. Suppose that the wave function of any individual system provides a complete description of that system. When we analyze the process of measurement in quantum mechanical terms, we find that the after-measurement wave function for system and apparatus that arises from Schrödinger's equation for the composite system typically involves a superposition over terms corresponding to what we would like to regard as the various possible results of the measurement — e.g., different pointer orientations. In this description of the after-measurement situation it is difficult to discern the actual result of the measurement — e.g., some specific pointer orientation. But the whole point of quantum theory, and the reason we should believe in it, is that it is supposed to provide a compelling, or at least an efficient, account of our observations, that is, of the outcomes of measurements. In short, the measurement problem is this: Quantum theory implies that measurements typically fail to have outcomes of the sort the theory was created to explain.
In contrast, if we, like Einstein, regard the description provided by the wave function as incomplete, the measurement problem vanishes: There is no measurement problem with a theory or interpretation in which, as in Bohmian mechanics, the description of the after-measurement situation includes, in addition to the wave function, at least the values of the variables that register the result. In Bohmian mechanics pointers always point.
Often, the measurement problem is expressed a little differently. Textbook quantum theory provides two rules for the evolution of the wave function of a quantum system: A deterministic dynamics given by Schrödinger's equation when the system is not being “measured” or observed, and a random collapse of the wave function to an eigenstate of the “measured observable” when it is. However, the objection continues, textbook quantum theory does not explain how to reconcile these two apparently incompatible rules.
That this formulation of the measurement problem and the preceding one are more or less equivalent should be reasonably clear: If a wave function provides a complete description of the after-measurement situation, the outcome of the measurement must correspond to a wave function that describes the actual result, that is, a “collapsed” wave function. Hence the collapse rule. But it is difficult to take seriously the idea that different laws than those governing all other interactions should govern those interactions between system and apparatus that we happen to call measurements. Hence the apparent incompatibility of the two rules.
The second formulation of the measurement problem, though basically equivalent to the first, raises an important question: Can Bohmian mechanics itself reconcile these two dynamical rules? How does Bohmian mechanics justify the use of the “collapsed” wave function instead of the original one? This question was answered in Bohm's first papers on Bohmian mechanics (Bohm 1952, Part I, Section 7, and Part II, Section 2). What would nowadays be called effects of decoherence, which interaction with the environment (air molecules, cosmic rays, internal microscopic degrees of freedom, etc.) produces, make difficult the development of significant overlap between the component of the after-measurement wave function corresponding to the actual result of the measurement and the other components of the after-measurement wave function. (This overlap refers to the configuration space of the very large system that includes all systems with which the original system and apparatus come into interaction.) But without such overlap that component all by itself generates to a high degree of accuracy the future evolution of the configuration of the system and apparatus. The replacement is thus justified as a practical matter. (See also Dürr et al. 1992, Section 5.)
Many proponents of orthodox quantum theory believe that decoherence somehow resolves the measurement problem itself. It is not easy to understand this belief. In the first formulation of the measurement problem, nothing prevents us from including in the apparatus all sources of decoherence. But then decoherence can no longer be in any way relevant to the argument. Be that as it may, Bohm (Bohm 1952) gave one of the best descriptions of the mechanisms of decoherence, though he did not use the word itself. He recognized its importance several decades before it became fashionable. (See also the encyclopedia entry on The Role of Decoherence in Quantum Mechanics.)
8. The Collapse of the Wave Function
In the previous section we indicated that collapse of the wave function can be regarded in Bohmian mechanics as a pragmatic affair. However, there is a sense in which the collapse of the wave function in Bohmian mechanics is more than a matter of convenience. If we focus on the appropriate notion of the wave function, not of the composite of system and apparatus — which strictly speaking remains a superposition if the composite is treated as closed during the measurement process — but of the system itself, we find that for Bohmian mechanics this does indeed collapse, precisely as the quantum formalism says. The key element here is the notion of the conditional wave function of a subsystem of a larger system, which we describe briefly in this section and that Dürr et al. 1992, Section 5, discuss in some detail, together with the related notion of the effective wave function.
For the evolution of the wave function, Bohmian mechanics is formulated in terms of Schrödinger's equation alone. Nonetheless the textbook collapse rule is a consequence of the Bohmian dynamics. To appreciate this one should first note that, since observation implies interaction, a system under observation cannot be a closed system but rather must be a subsystem of a larger closed system, which we may take to be the entire universe, or any smaller more or less closed system that contains the system to be observed, the subsystem. The configuration Q of this larger system naturally splits into X, the configuration of the subsystem, and Y, the configuration of the environment of the subsystem.
Suppose the larger system has wave function Ψ = Ψ(q) = Ψ(x, y). According to Bohmian mechanics, the larger system is then completely described by Ψ, evolving according to Schrödinger's equation, together with X and Y. The question then arises — and it is a critical question — as to what should be meant by the wave function of the subsystem.
There is a rather obvious answer for this, a natural function of x that suitably incorporates the objective structure at hand, namely the conditional wave function
ψ(x) = Ψ(x, Y)
obtained by plugging the actual configuration of the environment into the wave function of the larger system. (This definition is appropriate only for scalar wave functions; for particles with spin the situation would be a little more complicated.) It then follows immediately that the configuration of the subsystem obeys the guiding equation with the conditional wave function on its right-hand side.
Moreover, taking into account the way that the conditional wave function depends upon time t
ψt(x) = Ψt(x, Yt)
via the time dependence of Y as well as that of Ψ, it is not difficult to see (Dürr et al. 1992) the following two things about the evolution of the conditional wave: First, that it obeys Schrödinger's equation for the subsystem when that system is suitably decoupled from its environment. Part of what is meant by this decoupling is that Ψ has a special form, what might be called an effective product form (similar to but more general than the superposition produced in an “ideal quantum measurement”), in which case the conditional wave function of the subsystem is also called its effective wave function. Second, using the quantum equilibrium hypothesis, that it randomly collapses according to the usual quantum mechanical rules under precisely those conditions on the interaction between the subsystem and its environment that define an ideal quantum measurement.
It is perhaps worth noting that orthodox quantum theory lacks the resources that make it possible to define the conditional wave function, namely, the actual configuration Y of the environment. Indeed, from an orthodox point of view what should be meant by the wave function of a subsystem is entirely obscure.
9. Quantum Randomness
According to the quantum formalism, for a system with wave function ψ the probability density for finding its configuration to be q is |ψ(q)|2. To the extent that the results of measurement are registered configurationally, at least potentially, it follows that the predictions of Bohmian mechanics for the results of measurement must agree with those of orthodox quantum theory (assuming the same Schrödinger equation for both) provided that it is somehow true for Bohmian mechanics that configurations are random, with distribution given by the quantum equilibrium distribution |ψ(q)|2. Now the status and justification of this quantum equilibrium hypothesis is a rather delicate matter, one that has been explored in considerable detail (Dürr et al. 1992). Here are a few relevant points.
It is nowadays a rather familiar fact that dynamical systems quite generally give rise to behavior of a statistical character, with the statistics given by the (or a) stationary probability distribution for the dynamics. So it is with Bohmian mechanics, except that for the Bohmian system stationarity is not quite the right concept. Rather it is the notion of equivariance that is relevant. A probability distribution on configuration space ρψ, depending upon the wave function ψ, is equivariant if
ψ)t = ρψt
where the dependence on t on the right arises from Schrödinger's equation and on the left from the evolution on probability distributions arising from the flow that the guiding equation induces. Thus equivariance expresses the mutual compatibility, relative to ρψ, of the Schrödinger evolution of the wave function and the Bohmian motion of the configuration. It is an immediate consequence of the guiding equation and the quantum continuity equation that ρψ = |ψ(q)|2 is equivariant. (It can be shown in fact that this is more or less the only equivariant possibility that is suitably local (Goldstein and Struyve 2007).)
In trying to understand the status in Bohmian mechanics of the quantum equilibrium distribution, it is perhaps helpful to think of
quantum equilibrium, ρ = |ψ|2
as roughly analogous to (classical)
thermodynamic equilibrium, ρ = exp(-H/kT) /Z,
the probability distribution of the phase-space point of a system in equilibrium at temperature T. (Z is a normalization constant called the partition function and k is Boltzmann's constant.) This analogy has several facets: In both cases the probability distributions are naturally associated with their respective dynamical systems. In particular, these distributions are stationary or, what amounts to the same thing within the framework of Bohmian mechanics, equivariant. In both cases it seems natural to try to justify these equilibrium distributions by means of mixing-type, convergence-to-equilibrium arguments (Bohm 1953, Valentini and Westman 2005). It has been argued, however, that in both cases the ultimate justification for these probability distributions must be in terms of statistical patterns that ensembles of actual subsystems within a typical individual universe exhibit (Bell 1987, page 129, Dürr et al. 1992). In both cases the status of, and justification for, equilibrium distributions is still controversial. It is also perhaps worth noting that the typicality-grounded account of quantum randomness in Bohmian mechanics is extremely similar to Everett's account (Everett III 1957) of quantum randomness for “many worlds,” despite the huge metaphysical differences that exist between these two versions of quantum theory. It can be shown (Dürr et al. 1992) that probabilities for positions given by the quantum equilibrium distribution emerge naturally from an analysis of “equilibrium” for the deterministic dynamical system that Bohmian mechanics defines, much as the Maxwellian velocity distribution emerges from an analysis of classical thermodynamic equilibrium. (For more on the thermodynamic side of the analogy see Goldstein 2001.) Thus with Bohmian mechanics the statistical description in quantum theory indeed takes, as Einstein anticipated, “an approximately analogous position to the statistical mechanics within the framework of classical mechanics.”
10. Quantum Observables
Orthodox quantum theory supplies us with probabilities not merely for positions but for a huge class of quantum observables. It might thus appear that it is a much richer theory than Bohmian mechanics, which seems exclusively concerned with positions. Appearances are, however, misleading. In this regard, as with so much else in the foundations of quantum mechanics, Bell made the crucial observation (Bell 1987, p. 166):
[I]n physics the only observations we must consider are position observations, if only the positions of instrument pointers. It is a great merit of the de Broglie-Bohm picture to force us to consider this fact. If you make axioms, rather than definitions and theorems, about the “measurement” of anything else, then you commit redundancy and risk inconsistency.
Consider classical mechanics first. The observables are functions on phase space, functions of the positions and momenta of the particles. The axioms governing the behavior of the basic observables — Newton's equations for the positions or Hamilton's for positions and momenta — define the theory. What would be the point of making additional axioms, for other observables? After all, the behavior of the basic observables entirely determines the behavior of any observable. For example, for classical mechanics, the principle of the conservation of energy is a theorem, not an axiom.
The situation might seem to differ in quantum mechanics, as usually construed. Here there is no small set of basic observables having the property that all other observables are functions of them. Moreover, no observables at all are taken seriously as describing objective properties, as actually having values whether or not they are or have been measured. Rather, all talk of observables in quantum mechanics is supposed to be understood as talk about the measurement of the observables.
But if this is so, the situation with regard to other observables in quantum mechanics is not really that different from that in classical mechanics. Whatever quantum mechanics means by the measurement of (the values of) observables — that, we are urged to believe, don't actually have values — must at least refer to some experiment involving interaction between the “measured” system and a “measuring” apparatus leading to a recognizable result, as given potentially by, say, a pointer orientation. But then if some axioms suffice for the behavior of pointer orientations (at least when they are observed), rules about the measurement of other observables must be theorems, following from those axioms, not additional axioms.
It should be clear from the discussion towards the end of Section 4 and at the beginning of Section 9 that, assuming the quantum equilibrium hypothesis, any analysis of the measurement of a quantum observable for orthodox quantum theory — whatever it is taken to mean and however the corresponding experiment is performed — provides ipso facto at least as adequate an account for Bohmian mechanics. The only part of orthodox quantum theory relevant to the analysis is the Schrödinger evolution, and it shares this with Bohmian mechanics. The main difference between them is that orthodox quantum theory encounters the measurement problem before it reaches a satisfactory conclusion while Bohmian mechanics does not. This difference stems of course from what Bohmian mechanics adds to orthodox quantum theory: actual configurations.
The rest of this section will discuss the significance of quantum observables for Bohmian mechanics. (It follows from what has been said in the three preceding paragraphs that what we conclude here about quantum observables for Bohmian mechanics holds for orthodox quantum theory as well.)
Bohmian mechanics yields a natural association between experiments and so-called generalized observables, given by positive-operator-valued measures (Davies 1976), or POVM's, O(dz), on the value spaces for the results of the experiments (Berndl, Daumer, et al. 1995). This association is such that the probability distribution of the result Z of an experiment, when performed upon a system with wave function ψ, is given by <ψ | O(dz)ψ> (where < | > is the usual inner product between quantum state vectors).
Moreover, this conclusion is follows immediately from the very meaning of an experiment from a Bohmian perspective: a coupling of system to apparatus leading to a result Z that is a function of the final configuration of the total system, e.g., the orientation of a pointer. Analyzed in Bohmian mechanical terms, the experiment defines a map from the initial wave function of the system to the distribution of the result. It follows directly from the structure of Bohmian mechanics, and from the fact that the quantum equilibrium distribution is quadratic in the wave function, that this map is bilinear (or, more precisely, sesquilinear, in that its dependence on one factor of the wave function is antilinear, involving complex conjugation, rather than linear). Such a map is equivalent to a POVM.
The simplest example of a POVM is a standard quantum observable, corresponding to a self-adjoint operator A on the Hilbert space of quantum states (i.e., wave functions). For Bohmian mechanics, more or less every “measurement-like” experiment is associated with this special kind of POVM. The familiar quantum measurement axiom that the distribution of the result of the “measurement of the observable A” is given by the spectral measure for A relative to the wave function (in the very simplest cases just the absolute squares of the so-called probability amplitudes) is thus obtained.
For various reasons, after the discovery of quantum mechanics it quickly became almost universal to speak of an experiment associated with an operator A in the manner just sketched as a measurement of the observable A — as if the operator somehow corresponded to a property of the system that the experiment in some sense measures. It has been argued that this assumption, which has been called naive realism about operators, has been a source of considerable confusion about the meaning and implications of quantum theory (Daumer et al., 1997).
11. Spin
The case of spin illustrates nicely both the way Bohmian mechanics treats non-configurational quantum observables, and some of the difficulties that the naive realism about operators mentioned above causes.
Spin is the canonical quantum observable that has no classical counterpart, reputedly impossible to grasp in a nonquantum way. The difficulty is not quite that spin is quantized in the sense that its allowable values form a discrete set (for a spin-1/2 particle, ±ℏ/2). Energy too may be quantized in this sense. Nor is it precisely that the components of spin in the different directions fail to commute — and so cannot be simultaneously discussed, measured, imagined, or whatever it is that we are advised not to do with noncommuting observables. Rather the problem is that there is no ordinary (nonquantum) quantity which, like the spin observable, is a 3-vector and which also is such that its components in all possible directions belong to the same discrete set. The problem, in other words, is that the usual vector relationships among the various components of the spin vector are incompatible with the quantization conditions on the values of these components.
For a particle of spin-1 the problem is even more severe. The components of spin in different directions aren't simultaneously measurable. Thus, the impossible vector relationships for the spin components of a quantum particle are not observable. Bell (1966), and, independently, Simon Kochen and Ernst Specker (Kochen and Specker 1967) showed that for a spin-1 particle the squares of the spin components in the various directions satisfy, according to quantum theory, a collection of relationships, each individually observable, that taken together are impossible: the relationships are incompatible with the idea that measurements of these observables merely reveal their preexisting values rather than creating them, as quantum theory urges us to believe. Many physicists and philosophers of physics continue to regard the Kochen-Specker Theorem as precluding the possibility of hidden variables.
We thus might naturally wonder how Bohmian mechanics copes with spin. But we have already answered this question. Bohmian mechanics makes sense for particles with spin, i.e., for particles whose wave functions are spinor-valued. When such particles are suitably directed toward Stern-Gerlach magnets, they emerge moving in more or less a discrete set of directions — 2 possible directions for a spin-1/2 particle, having 2 spin components, 3 for spin-1 with 3 spin components, and so on. This occurs because the Stern-Gerlach magnets are so designed and oriented that a wave packet (a localized wave function with reasonably well defined velocity) directed towards the magnet will, by virtue of the Schrödinger evolution, separate into distinct packets — corresponding to the spin components of the wave function and moving in the discrete set of directions. The particle itself, depending upon its initial position, ends up in one of the packets moving in one of the directions.
The probability distribution for the result of such a Stern-Gerlach experiment can be conveniently expressed in terms of the quantum mechanical spin operators — for a spin-1/2 particle given by certain 2 by 2 matrices called the Pauli spin matrices — in the manner alluded to above. From a Bohmian perspective there is no hint of paradox in any of this — unless we assume that the spin operators correspond to genuine properties of the particles.
12. Contextuality
The Kochen-Specker Theorem, the earlier theorem of Gleason (Gleason 1957 and Bell 1966), and other no-hidden-variables results, including Bell's inequality (Bell 1964), show that any hidden-variables formulation of quantum mechanics must be contextual. It must violate the noncontextuality assumption “that measurement of an observable must yield the same value independently of what other measurements may be made simultaneously” (Bell 1987, p. 9). To many physicists and philosophers of science contextuality seems too great a price to pay for the rather modest benefits — largely psychological, so they would say — that hidden variables provide.
Even many Bohmians suggest that contextuality departs significantly from classical principles. For example, Bohm and Hiley (1993) write that “The context dependence of results of measurements is a further indication of how our interpretation does not imply a simple return to the basic principles of classical physics.”
However, to understand contextuality in Bohmian mechanics almost nothing needs to be explained. Consider an operator A that commutes with operators B and C (which however don't commute with each other). What is often called the “result for A” in an experiment for “measuring A together with B” usually disagrees with the “result for A” in an experiment for “measuring A together with C.” This is because these experiments differ and different experiments usually have different results. The misleading reference to measurement, which suggests that a pre-existing value of A is being revealed, makes contextuality seem more than it is.
Seen properly, contextuality amounts to little more than the rather unremarkable observation that results of experiments should depend upon how they are performed, even when the experiments are associated with the same operator in the manner alluded to above. David Albert (Albert 1992, p. 153) has given a particularly simple and striking example of this dependence for Stern-Gerlach experiments “measuring” the z-component of spin. Reversing the polarity in a magnet for “measuring” the z-component of spin while keeping the same geometry yields another magnet for “measuring” the z-component of spin. The use of one or the other of these two magnets will often lead to opposite conclusions about the “value of the z-component of spin” prior to the “measurement” (for the same initial value of the position of the particle).
As Bell insists (Bell 1987, p. 166):
A final moral concerns terminology. Why did such serious people take so seriously axioms which now seem so arbitrary? I suspect that they were misled by the pernicious misuse of the word ‘measurement’ in contemporary theory. This word very strongly suggests the ascertaining of some preexisting property of some thing, any instrument involved playing a purely passive role. Quantum experiments are just not like that, as we learned especially from Bohr. The results have to be regarded as the joint product of ‘system’ and ‘apparatus,’ the complete experimental set-up. But the misuse of the word ‘measurement’ makes it easy to forget this and then to expect that the ‘results of measurements’ should obey some simple logic in which the apparatus is not mentioned. The resulting difficulties soon show that any such logic is not ordinary logic. It is my impression that the whole vast subject of ‘Quantum Logic’ has arisen in this way from the misuse of a word. I am convinced that the word ‘measurement’ has now been so abused that the field would be significantly advanced by banning its use altogether, in favour for example of the word ‘experiment.’
13. Nonlocality
Bohmian mechanics is manifestly nonlocal. The velocity, as expressed in the guiding equation, of any particle of a many-particle system will typically depend upon the positions of the other, possibly distant, particles whenever the wave function of the system is entangled, i.e., not a product of single-particle wave functions. This is true, for example, for the EPR-Bohm wave function, describing a pair of spin-1/2 particles in the singlet state, that Bell and many others analyzed. Thus Bohmian mechanics makes explicit the most dramatic feature of quantum theory: quantum nonlocality, as discussed in Section 2.
It should be emphasized that the nonlocality of Bohmian mechanics derives solely from the nonlocality, discussed in Section 2, built into the structure of standard quantum theory. This nonlocality originates from a wave function on configuration space, an abstraction which, roughly speaking, combines — or binds — distant particles into a single irreducible reality. As Bell (1987, p. 115) has stressed,
Thus the nonlocal velocity relation in the guiding equation is but one aspect of the nonlocality of Bohmian mechanics. There is also the nonlocality, or nonseparability, implicit in the wave function itself, which is present even without the structure — actual configurations — that Bohmian mechanics adds to orthodox quantum theory. As Bell has shown, using the connection between the wave function and the predictions of quantum theory about experimental results, this nonlocality cannot easily be eliminated (see Section 2).
The nonlocality of Bohmian mechanics can be appreciated perhaps most efficiently, in all its aspects, by focusing on the conditional wave function. Suppose, for example, that in an EPR-Bohm experiment particle 1 passes through its Stern-Gerlach magnet before particle 2 arrives at its magnet. Then the orientation of the Stern-Gerlach magnet for particle 1 will significantly affect the conditional wave function of particle 2: If the Stern-Gerlach magnet for particle 1 is oriented so as to “measure the z-component of spin,” then after particle 1 has passed through its magnet the conditional wave function of particle 2 will be an eigenvector (or eigenstate) of the z-component of spin (in fact, belonging to the eigenvalue that is the negative of the one “measured” for particle 1), and the same thing is true for any other component of spin. You can dictate the kind of spin eigenstate produced for particle 2 by appropriately choosing the orientation of an arbitrarily distant magnet. As to the future behavior of particle 2, in particular how its magnet affects it, this of course depends very much on the character of its conditional wave function and hence the choice of orientation of the distant magnet strongly influences it.
This nonlocal effect upon the conditional wave function of particle 2 follows from combining the standard analysis of the evolution of the wave function in the EPR-Bohm experiment with the definition of the conditional wave function. (For simplicity, we ignore permutation symmetry.) Before reaching any magnets the EPR-Bohm wave function is a sum of two terms, corresponding to nonvanishing values for two of the four possible joint spin components for the two particles. Each term is a product of an eigenstate for a component of spin in a given direction for particle 1 with the opposite eigenstate (i.e., belonging to the eigenvalue that is the negative of the eigenvalue for particle 1) for the component of spin in the same direction for particle 2. Moreover, by virtue of its symmetry under rotations, the EPR-Bohm wave function has the property that any component of spin, i.e., any direction, can be used in this decomposition. (This property is very interesting.)
Decomposing the EPR-Bohm wave function using the component of spin in the direction associated with the magnet for particle 1, the evolution of the wave function as particle 1 passes its magnet is easily grasped: The evolution of the sum is determined (using the linearity of Schrödinger's equation) by that of its individual terms, and the evolution of each term by that of each of its factors. The evolution of the particle-1 factor leads to a displacement along the magnetic axis in the direction determined by the (sign of the) spin component (i.e., the eigenvalue), as described in the fourth paragraph of Section 11. Once this displacement has occurred (and is large enough) the conditional wave function for particle 2 will correspond to the term in the sum selected by the actual position of particle 1. In particular, it will be an eigenstate of the component of spin “measured by” the magnet for particle 1.
The nonlocality of Bohmian mechanics has a remarkable feature: it is screened by quantum equilibrium. It is a consequence of the quantum equilibrium hypothesis that the nonlocal effects in Bohmian mechanics don't yield observable consequences that can be controlled — we can't use them to send instantaneous messages. This follows from the fact that, given the quantum equilibrium hypothesis, the observable consequences of Bohmian mechanics are the same as those of orthodox quantum theory, for which instantaneous communication based on quantum nonlocality is impossible (see Eberhard 1978). Valentini (1991) emphasizes the importance of quantum equilibrium for obscuring the nonlocality of Bohmian mechanics. (Valentini (2010a) has also suggested the possibility of searching for and exploiting quantum non-equilibrium. However, in contrast with thermodynamic non-equilibrium, we have at present no idea what quantum non-equilibrium, should it exist, would look like, despite claims and arguments to the contrary.)
14. Lorentz Invariance
Like nonrelativistic quantum theory, of which it is a version, Bohmian mechanics and special relativity, a central principle of physics, are not compatible: Bohmian mechanics is not Lorentz invariant. Nor can it easily be modified to accommodate Lorentz invariance. Configurations, defined by the simultaneous positions of all particles, play too crucial a role in its formulation, with the guiding equation defining an evolution on configuration space. (Lorentz invariant extensions of Bohmian mechanics for a single particle, described by the Dirac equation (Bohm and Hiley 1993, Dürr et al. 1999) or the Klein-Gordon equation (Berndl et al. 1996, Nikolic 2005), can easily be achieved, though for a Klein-Gordon particle there are some interesting subtleties, corresponding to what might seem to be a particle traveling backwards in time.)
This difficulty with Lorentz invariance and the nonlocality in Bohmian mechanics are closely related. Since quantum theory itself, by virtue merely of the character of its predictions concerning EPR-Bohm correlations, is irreducibly nonlocal (see Section 2), one might expect considerable difficulty with the Lorentz invariance of orthodox quantum theory as well with Bohmian mechanics. For example, the collapse rule of textbook quantum theory blatantly violates Lorentz invariance. As a matter of fact, the intrinsic nonlocality of quantum theory presents formidable difficulties for the development of any (many-particle) Lorentz invariant formulation that avoids the vagueness of orthodox quantum theory (see Maudlin 1994).
Bell made a somewhat surprising evaluation of the importance of the problem of Lorentz invariance. In an interview with the philosopher Renée Weber, not long before he died, he referred to the paradoxes of quantum mechanics and observed that “Those paradoxes are simply disposed of by the 1952 theory of Bohm, leaving as the question, the question of Lorentz invariance. So one of my missions in life is to get people to see that if they want to talk about the problems of quantum mechanics — the real problems of quantum mechanics — they must be talking about Lorentz invariance.”
The most common view on this issue is that a detailed description of microscopic quantum processes, such as would be provided by a putative extension of Bohmian mechanics to the relativistic domain, must violate Lorentz invariance. In this view Lorentz invariance is an emergent symmetry obeyed by our observations — for Bohmian mechanics a statistical consequence of quantum equilibrium that governs the results of quantum experiments. This is the opinion of Bohm and Hiley (1993), of Holland (1993), and of Valentini (1997).
However — unlike nonlocality — violating Lorentz invariance is not inevitable. It should be possible, it seems, to construct a fully Lorentz invariant theory that provides a detailed description of microscopic quantum processes. One way to do this is by using an additional Lorentz invariant dynamical structure, for example a suitable time-like 4-vector field, that permits the definition of a foliation of space-time into space-like hypersurfaces providing a Lorentz invariant notion of “evolving configuration” and along which nonlocal effects are transmitted. See Dürr et al. 1999 for a toy model. Another possibility is that a fully Lorentz invariant account of quantum nonlocality can be achieved without the invocation of additional structure, exploiting only what is already at hand, for example, the wave function of the universe (see Dürr et al. 1999) or light-cone structure. For a step in this direction, see Goldstein and Tumulka 2003's model in which they reconcile relativity and nonlocality through the interplay of opposite arrows of time.
Be that as it may, Lorentz invariant nonlocality remains somewhat enigmatic. The issues are extremely subtle. For example, Bell (1987, page 155) rightly would find “disturbing … the impossibility of ‘messages’ faster than light, which follows from ordinary relativistic quantum mechanics in so far as it is unambiguous and adequate for procedures we [emphasis added] can actually perform. The exact elucidation of concepts like ‘message’ and ‘we’, would be a formidable challenge.” While quantum equilibrium and the absolute uncertainty that it entails (Dürr et al. 1992) may be of some help here, the situation remains puzzling.
15. Objections and Responses
Bohmian mechanics has never been widely accepted in the mainstream of the physics community. Since it is not part of the standard physics curriculum, many physicists—probably the majority—are simply unfamiliar with the theory and how it works. Sometimes the theory is rejected without explicit discussion of reasons for rejection. One also finds objections that are based on simple misunderstandings; among these are claims that some no-go theorem, such as von Neumann's theorem, the Kochen-Specker theorem, or Bell's theorem, shows that the theory cannot work. Such objections will not be dealt with here, as the reply to them will be obvious to those who understand the theory. In what follows only objections that are not based on elementary misunderstandings will be discussed.
A common objection is that Bohmian mechanics is too complicated or inelegant. To evaluate this objection one must compare the axioms of Bohmian mechanics with those of standard quantum mechanics. To Schrödinger's equation, Bohmian mechanics adds the guiding equation; standard quantum mechanics instead requires postulates about experimental outcomes that can only be formulated in terms of a distinction between a quantum system and the experimental apparatus. And, as noted by Hilary Putnam (2005),
In Putnam ([1965]), I rejected Bohm's interpretation for several reasons which no longer seem good to me. Even today, if you look at the Wikipedia encyclopaedia on the Web, you will find it said that Bohm's theory is mathematically inelegant. Happily, I did not give that reason in Putnam ([1965]), but in any case it is not true. The formula for the velocity field is extremely simple: you have the probability current in the theory anyway, and you take the velocity vector to be proportional to the current. There is nothing particularly inelegant about that; if anything, it is remarkably elegant!
One frequent objection is that Bohmian mechanics, since it makes precisely the same predictions as standard quantum mechanics (insofar as the predictions of standard quantum mechanics are unambiguous), is not a distinct theory but merely a reformulation of standard quantum theory. In this vein, Heisenberg wrote,
Bohm's interpretation cannot be refuted by experiment, and this is true of all the counter-proposals in the first group. From the fundamentally “positivistic” (it would perhaps be better to say “purely physical”) standpoint, we are thus concerned not with counter-proposals to the Copenhagen interpretation, but with its exact repetition in a different language (Heisenberg 1955, p. 18).
More recently, Sir Anthony Leggett has echoed this charge. Referring to the measurement problem, he says (Leggett 2005) that Bohmian mechanics provides “little more than verbal window dressing of the basic paradox.” And in connection with the double-slit experiment, he writes,
No experimental consequences are drawn from [the assumption of definite particle trajectories] other than the standard predictions of the QM formalism, so whether one regards it as a substantive resolution of the apparent paradox or as little more than a reformulation of it is no doubt a matter of personal taste (the present author inclines towards the latter point of view) (Leggett 2002, p. R419).
Now Bohmian mechanics and standard quantum mechanics provide clearly different descriptions of what is happening on the microscopic quantum level. So it is only with a purely instrumental attitude towards scientific theories that Bohmian mechanics and standard quantum mechanics can possibly be regarded as different formulations of exactly the same theory. But even if they were, why would this be an objection to Bohmian mechanics? Even if they were, we should still ask which of the two formulations is superior. Those impressed by the “not-a-distinct-theory” objection presumably give considerable weight to the fact that standard quantum mechanics came first. Supporters of Bohmian mechanics give more weight to its greater simplicity and clarity.
The position of Leggett, however, is very difficult to understand. There should be no measurement problem for a physicist with a purely instrumentalist understanding of quantum mechanics. But for more than thirty years Leggett has forcefully argued that quantum mechanics indeed suffers from the measurement problem. For Leggett the problem is so serious that it has led him to suggest that quantum mechanics might fail on the macroscopic level. Thus Leggett is no instrumentalist, and it is hard to understand why he so cavalierly dismisses a theory like Bohmian mechanics that obviously doesn't suffer from the measurement problem, with which he has been so long concerned.
Sir Roger Penrose (2005, page 811) also seems to have doubts as to whether Bohmian mechanics indeed resolves the measurement problem. He writes that
it seems to me that some measure of scale is indeed needed, for defining when classical-like behaviour begins to take over from small-scale quantum activity. In common with the other quantum ontologies in which no measurable deviations from standard quantum mechanics is expected, the point of view (e) [Bohmian mechanics] does not possess such a scale measure, so I do not see that it can adequately address the paradox of Schrödinger's cat.
But contrary to what he writes, his real concern seems to be with the emergence of classical behavior, and not with the measurement problem per se. With regard to this, we note that the Bohmian evolution of particles, which is always governed by the wave function and is always fundamentally quantum, turns out to be approximately classical when the relevant de Broglie wave length, determined in part by the wave function, is much smaller than the scale on which the potential energy term in Schrödinger's equation varies (see Allori et al., 2002). Under normal circumstances this condition will be satisfied for the center of mass motion of a macroscopic object.
It is perhaps worth mentioning that despite the empirical equivalence between Bohmian mechanics and orthodox quantum theory, there are a variety of experiments and experimental issues that don't fit comfortably within the standard quantum formalism but are easily handled by Bohmian mechanics. Among these are dwell and tunneling times (Leavens 1996), escape times and escape positions (Daumer et al. 1997a), scattering theory (Dürr et al., 2000), and quantum chaos (Cushing 1994, Dürr et al., 1992a).
Another claim that has become popular in recent years is that Bohmian mechanics is an Everettian, or “many worlds,” interpretation in disguise (see entry on the many worlds interpretation of quantum mechanics for an overview of such interpretations). The idea is that Bohmians, like Everettians, must take the wave-function as physically real. Moreover, since Bohmian mechanics involves no wave-function collapse (for the wave function of the universe), all of the branches of the wave function, and not just the one that happens to be occupied by the actual particle configuration, persist. These branches are those that Everettians regard as representing parallel worlds. As David Deutsch expresses the charge,
the ‘unoccupied grooves’ must be physically real. Moreover they obey the same laws of physics as the ‘occupied groove’ that is supposed to be ‘the’ universe. But that is just another way of saying that they are universes too. … In short, pilot-wave theories are parallel-universes theories in a state of chronic denial (Deutsch 1996, p. 225).
See Brown and Wallace (2005) for an extended version of this argument. Not surprisingly, Bohmians do not agree that the branches of the wave function should be construed as representing worlds. For one Bohmian response, see Maudlin (2010). Other Bohmian responses have been given by Lewis (2007) and Valentini (2010b).
The claim of Deutsch, Brown, and Wallace is of a novel character that we should perhaps pause to examine. On the one hand, for anyone who, like Wallace, accepts the viability of a functionalist many-worlds understanding of quantum mechanics — and in particular accepts that it follows as a matter of functional and structural analysis that when the wave function develops suitable complex patterns these ipso facto describe what we should regard as worlds — the claim should be compelling. On the other hand, for those who reject the functional analysis and regard many worlds as ontologically inadequate (see Maudlin 2010), or who, like Vaidman (see the SEP entry on the many-worlds interpretation of quantum mechanics), accepts many worlds on non-functionalist grounds, the claim should seem empty. In other words, one has basically to have already accepted a strong version of many worlds and already rejected Bohm in order to feel the force of the claim.
Another interesting aspect of the claim is this: It seems that one could consider, at least as a logical possibility, a world consisting of particles moving according to some well-defined equations of motion, and in particular according to the equations of Bohmian mechanics. It seems entirely implausible that there should be a logical problem with doing so. We should be extremely sceptical of any argument, like the claim of Deutsch, Brown, and Wallace, that suggests that there is. Thus what, in defense of many worlds, Deutsch, Brown, and Wallace present as an objection to Bohmian mechanics should perhaps be regarded instead as an objection to many worlds itself.
There is one striking feature of Bohmian mechanics that is often presented as an objection: in Bohmian mechanics the wave function acts upon the positions of the particles but, evolving as it does autonomously via Schrödinger's equation, it is not acted upon by the particles. This is regarded by some Bohmians, not as an objectionable feature of the theory, but as an important clue about the meaning of the quantum-mechanical wave function. Dürr et al. 1997 and Goldstein and Teufel 2001 discuss this point and suggest that from a deeper perspective than afforded by standard Bohmian mechanics or quantum theory, the wave function should be regarded as nomological, as an object for conveniently expressing the law of motion somewhat analogous to the Hamiltonian in classical mechanics, and that a time-dependent Schrödinger-type equation, from this deeper (cosmological) perspective, is merely phenomenological.
Bohmian mechanics does not account for phenomena such as particle creation and annihilation characteristic of quantum field theory. This is not an objection to Bohmian mechanics but merely a recognition that quantum field theory explains a great deal more than does nonrelativistic quantum mechanics, whether in orthodox or Bohmian form. It does, however, underline the need to find an adequate, if not compelling, Bohmian version of quantum field theory, and of gauge theories in particular. Some rather tentative steps in this direction can be found in Bohm and Hiley 1993, Holland 1993, Bell 1987 (p. 173), and in some of the articles in Cushing et al. 1996. A crucial issue is whether a quantum field theory is fundamentally about fields or particles — or something else entirely. While the most common choice is fields (see Struyve 2010 for an assessment of a variety of possibilities), Bell's is particles. His proposal is in fact the basis of a canonical extension of Bohmian mechanics to general quantum field theories, and these “Bell-type quantum field theories” (Dürr et al., 2004 and 2005) describe a stochastic evolution of particles that involves particle creation and annihilation. (For a general discussion of this issue, and of the point and value of Bohmian mechanics, see the exchange of letters between Goldstein and Weinberg by following the link provided in the Other Internet Resources section below.)
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physics: holism and nonseparability | quantum mechanics | quantum mechanics: Copenhagen interpretation of | quantum mechanics: Kochen-Specker theorem | quantum mechanics: many-worlds interpretation of | quantum mechanics: modal interpretations of | quantum mechanics: the role of decoherence in | quantum theory: measurement in | quantum theory: quantum entanglement and information | quantum theory: quantum gravity | quantum theory: quantum logic and probability theory | quantum theory: the Einstein-Podolsky-Rosen argument in | Uncertainty Principle
I am grateful to Joanne Gowa, Paul Oppenheimer, and the subject editors, Guido Bacciagaluppi and Wayne Myrvold, for very careful readings and many valuable suggestions. |
c8c35ccb33f67a01 | Bad science Medicine Physics Quackery Skepticism/critical thinking
Luminas Pain Relief Patches: Where the words “quantum” and “energy” really mean “magic”
Orac discovers the Luminas Pain Relief Patch. He is amused at how how quacks confuse the words “quantum” and “energy” with magic.
Luminas Pain Relief Patches Luminas Pain Relief Patches: They cure everything through…energy (wait, no, magic).
Energy. Quacks keep using that word. I do not think it means what they think it means. Certainly Luminas doesn’t. Yes, I know that I use a lot of variations on that famous quote from The Princess Bride all the time, probably more frequently than I should and likely to the point of annoying some of my readers, but, damn, if it isn’t a nearly all-purpose phrase to use to riff on various quackery.
Also, if there’s one concept that quacks love to abuse, it’s energy. Whether it’s “energy healing” like reiki, where practitioners claim to be able to channel healing energy from the magical mystical “universal source” specifically into their patient to specifically heal whatever ails them, even if it’s from a distance or you’re a dog, or “healing touch,” where practitioners claim to be able to manipulate their patients’ “life energy” fields, again to healing effect, so much quackery is based on a misunderstanding of “energy” as basically magic. So it is with some spectacularly hilarious woo that I came across last week and, given that it’s Friday, decided to feature as a sort of Friday Dose of Woo Lite. It even abuses quantum theory because of course it does. So much quackery does.
So what are we talking about here? What is Luminas? To be honest, more than anything else, it reminds me of the silly “Body Vibesenergy stickers that Gwyneth Paltrow and Goop were selling last year (and probably still are) that claim to “rebalance the energy frequency in our bodies,” whatever that means. So let’s look at the claims.
Right on the front page of the Luminas website, you’ll find a video. It’s well-produced, as many such videos for quackery are, and it blathers on about how the product being advertised takes advantage of “revolutions in quantum physics,” as a lot of quackery does. Let’s see how this lovely patch supposedly works.
The basic claim is that the Luminas patch is charged with the “energetic signatures of natural remedies known for centuries to reduce inflammation.” These natural remedies include “Acetyl-L-Carnitine, Amino Acids, Arnica, Astaxanthin, B-Complex, Berberis Vulgaris, Bioperine, Boluoke, Boswellia, Bromelain, Chamomile, Chinchona, Chondroitin, Clove, Colostrum, CoQ10, Cordyceps, Curcumin, Flower Essences Frankincense, Ginger, Ginseng, Glucosamine, Glutathione, Guggulu, Hops Extract, K2, Lavender, Magnesium, Motherwort, MSM, Olive Leaf, Omega-3, Peony, Proteolytic Enzymes, Polyphenols, Rosemary Extract, Telomerase Activators, Turmeric, Vinpocetine, Vitamin D, White Willow Bark and over 200 more!”
Luminas Pain Relief Patches
Luminas Pain Relief Patches: here’s the excuse to show partially naked bodies.
Don’t believe me? Take a look at this video on this page! It starts out with an announcer opining about how “energy is all around us.” (Well, yes it is, but that doesn’t mean your nonsense product works.) The announcer then goes on about Luminas somehow infuses its patches with the energy from she substances above:
…energy that your body inherently knows how to absorb and use with absolutely no side effects.
What? Not even skin irritation from the patch or any of the adhesive used to stick the patch to your body? I find that hard to believe. I mean, even paper tape can cause irritation! Fear not, though! The announcer continues:
Through the use of quantum physics scientists and doctors now have the ability to store the energetic signatures of hundreds of pain- and inflammation-relieving remedies on a single patch. Once applied, your body induces the flow of energy from the patch, choosing which electrons it needs to reduce inflammation. Science, relieving pain, with the power of nature.
So. Many. Questions. How, for instance, do the Luminas “scientists” store these “energetic signatures” on a patch? (More on that later.) What, exactly, is an “energetic signature”? How does the body know which electrons it needs to reduce inflammation and pain? As a surgeon and scientists with a PhD in cellular physiology, I’d love to know the physiologic mechanism by which the body can distinguish one electron from another, given that there really is no known biological (or, come to think of it, no physical) mechanism for that to happen and if Luminas has discovered one its scientists should be nominated for the Nobel Prize.
Let’s get back to a key question, though: How on earth is all this energy goodness concentrated into a little patch roughly the size of a playing card? Physicists and chemists are going to guffaw at the answer, I promise you. First, the same page linked to above also notes that the “patches contain no active ingredients” because they “are charged with electrons captured from” the substances listed above. So is this some form of homeopathy? Of course not! Look at the video, which shows magical energy swirling off of the natural remedies and winding its way into the patch! There’s your energy, you unbeliever, you! How can you possibly question it?
But, hey, the makers of Luminas know that there are science geeks out there; so for the benefit of them included in the FAQ is an explanation of just how much natural product-infused electrony goodness you can expect in a single patch:
For the geeks and scientists among us: Each patch contains 5.2 x 10^19 molecular structures, each with 2 oxygen polar bonding areas capable of holding a targeted, host electron, creating a total possible charging capacity equal to 10.4 x 10^19 host electrons. After considering the average transmission field voltage of humans (200 micro volts) we can calculate the relative capacity, per square inch of patch, at 333 Pico Farads.
So basically, they’re saying that each patch contains around 86 micromoles of…whatever…and that that whatever can bind…electrons, I guess. Somewhere, far back in the recesses of my mind and buried in the mists of time from decades ago, my knowledge from my undergraduate chemistry degree and the additional advanced physics courses stirred—and then screamed! I can’t wait to see what actual physicists and chemists whose knowledge is in active use think of this. I apologize in advance if I cause them too much pain by showing them this. Not everyone’s neurons are as resistant as mine to apoptosis caused by waves of burning stupid. It is a resistance built up over 14 years of examining claims like those of Luminas.
Who, I wondered, developed this amazing product? In the first video, we discover that it is a woman named Sonia Broglin, who is the director of product development at Luminas. Naturally, she’s featured with a monitor in the background showing what look like infrared heat images of people. I actually laughed out loud as the video went on, because it shows her in very obviously posed and scripted interactions with patients with no shirts on and up to several of these patches all over their torso and arms. Me being me, I had to Google her, and guess what I found? Surprise! Surprise! She’s listed as a certified EnergyTouch® practitioner who graduated from the EnergyTouch® School of Advanced Healing. What, you might ask, is EnergyTouch®? This:
Energy Touch® is an off-the-body multidimensional healing process that allows the Energy Touch® Practitioner to access outer levels of the human energy field. It is based on the understanding that the human energy field is a dynamic system of powerful influences, in unique relationship to physical, emotional, and spiritual wellbeing. This system consists of the field (aura), chakras (energy centers) and the energy of the organs and systems of the body.
We readily accept the many ways that our body functions and is powered by energy. Our heart beats using energy pulses. Our brain and nervous system communicates with our entire body through complex energetic pathways. Our human energy field is constantly reacting in response to the physical and emotional and spiritual needs of our body.
EnergyTouch® is distinctive in the field of energy healing in that the work takes place in a more expanded energy field allowing the practitioner to work on a cellular level. Our work includes accessing an energetic hologram of the physical body, which is a unique and vital aspect of EnergyTouch® Healing. This energetic hologram acts as a matrix connecting the energies of the outer levels of the field precisely with the physical body on a cellular level.
EnergyTouch® practitioners are skillfully capable of moving fluently throughout the levels of the human energy field, to access and utilize outer level energies to clear blocks and restore function at the most basic cellular level.
It’s all starting to make sense now. That is some Grade-A+, seriously energy woo there, and I’m guessing Broglin cranked it up to 11 when developing the Luminas patches.
Next up is someone named Dr. Craig Davies, who is billed as “Pro Sports Doctor.” Yes, but a doctor of what? It didn’t take much Googling to figure out that Davies is not a physician. He is a chiropractor, because of course he is. He ha actually worked on the PGA tour, apparently adjusting the spines of professional golfers.
Then there’s Dr. Ara Suppiah. Unlike Davies, Dr. Suppiah appears to be more legit:
He is a practicing ER physician, Chief Wellness Officer for Emergency Physicians of Florida and an assistant professor at the University of Central Florida Medical School. He also is the personal physician for several top PGA Tour professionals, including Henrik Stenson, Justin Rose, Gary Woodland, Graeme McDowell, Ian Poulter, Steve Stricker, Hunter Mahan, Jimmy Walker, Vijay Singh, Graham DeLaet, and Kevin Chappell, as well as LPGA Tour players Anna Nordqvist and Julieta Granada.
However, his Twitter bio describes him as doing “functional sports medicine,” which suggests to me functional medicine, which is not exactly science-based. Basically, Dr. Suppiah looks like an ER doc turned sports medicine doc who was a bit into woo but has dived both feet first into the deep end of energy medicine pseudoscience by endorsing these Luminas patches. Seriously, a physician should really know better, but clearly Dr. Suppiah doesn’t. Either that, or the money was good.
Ditto Dr. Ashley Anderson, a nurse practitioner who also gives an endorsement. She’s affiliated with Athena Health and Wellness, a practice that mixes standard women’s health treatments with “integrative medicine” quackery like acupuncture, reflexology, traditional Chinese medicine, and the like.
Given the claims being made, you’d think that Luminas would have some…oh, you know…actual scientific evidence to support its patch. The video touts “astounding results” from Luminas’ patient trials, but what are those trials? Certainly they are not published anywhere that I could find in the peer-reviewed literature. Certainly I could find no registered clinical trials in What I did find on the Luminas website is a hilariously inept trial in which patients were imaged using thermography (which, by the way, is generally quackery when used by alternative medicine practitioners).
Luminas Pain Control Patches
8/Luminas4.jpg”> Luminas Pain Control Patches: Wait! Don’t you believe our patient studies that are totally not clinical trials? Come on! It’s science, man![/caption]So. Many. Questions. About. This. Trial. For instance,
So. Many. Questions. About. This. Trial. For instance, was there a randomized controlled trial of the Luminas patch versus an identical patch that wasn’t infused with the magic electrony goodness of the Luminas patch? (My guess: No.) I also know from my previous studies that thermography is very dependent on maintaining standardized conditions and a rigorously controlled room temperature, as well as on using rigorously standardized protocols. Did Luminas do that? It sure doesn’t look like it. It looks as though Broglin just did thermography on people, slapped a patch on them, and then repeated the thermography. Of course, such shoddy methodology guarantees a positive result, at least with patients whose patch is applied to an area covered by clothing. The temperature of that skin can start out warmer and then cool over time after the clothing is taken off, regardless of whether a patch is applied or not. Did Broglin do any no-patch control runs, to make sure to correct for this phenomenon? Color me a crotchety old skeptic, but my guess is: Almost assuredly not. No, scratch that. There’s no way on earth it even occurred to these quacks to run such a basic control. They can, of course, prove me wrong by sending me their detailed experimental protocol to read.
I suspect I will wait a long time. After all,
After nearly 14 years of regular blogging and 20 years of examining questionable claims, it never ceases to amaze me that products like Luminas patches are still sold. Basically, it’s a variety of quantum quackery in which “energy” is basically magic that can do anything, and quantum is an invocation of the high priests of quackery.
By Orac
To contact Orac: [email protected]
99 replies on “Luminas Pain Relief Patches: Where the words “quantum” and “energy” really mean “magic””
By my reckoning, 10.4 x 10^19 electrons is 16.7 coulombs, To store 16.7 coulombs on 333 picofrads you need to charge it to 50 billion volts. Leyden would be impressed. Now I’m not quite clear on the claim, since the description says electrons per patch and the capacitance is per square inch.
But they are special electrons, like the marshmallow bits in Lucky Charms, so they probably don’t abide by the usual rules.
With all that charge, opening the package ought to result in the patches flying out like one of those spring snake gags.
You can figure out the area of the patches from their measurements. The large patches are 2.75″ x 4.0″ and the medium patches are 1.5″ x 2.75″. Just sayin’. 🤔
But the description says so many molecular structures per patch, not per unit area, then in practically the same breath talks about capacitance per square inch, hence my confusion: “Each patch contains 5.2 x 10^19 molecular structures,” I might be induced to think the numbers are total fabrications. But surely not!
Anyway, it makes little difference if the voltage is 50 billion or 50 million.
What cracks me up is that someone knew enough physics to come up with those numbers but not enough to know why they are ridiculous.
If they put Elvis’s mojo, or even Mojo’s mojo, into an energy patch, I might buy it.
“If you don’t have Mojo Nixon, then your patch could use some fixin’!”
I don’t know where Skid Roper has gone, but Mojo seems to have hooked up with Jello Biafra on at least one occasion. The “Love Me, I’m a Liberal” is somewhat amusing.
“Capacitance.” Get it together, Random Capitalization People.
But SERIOUSLY, it has the energetic signatures of all of those herbs, spices and nutraceuticals!
-btw- more legit ( semi-legit?) pain patches and liquids contain cayenne, menthol or lidocaine:
early on, in my continuing leg injury adventure, I used a liquid form of lidocaine which seemed to be helping HOWEVER at one point, it felt as though my leg were on fire and washing it off didn’t help.
Eventually, it wore off and I felt better but swore off Demon Lidocaine.
Fortunately, I am better enough that I don’t try these products but I can see how people rely upon them when they have pain.
Perhaps this is the doctrine of signatures updated for modern times.
I don’t know how you’d get lidocaine to penetrate intact skin. Perhaps it will if it is dissolved in dimethyl sulfoxide (DMSO). I think iontophoresis will work with lido.
@ doug:
I looked at the meds: they are OTC – standard drugstore stuff and one is 4% lidocaine/ another product has that plus 1% menthol. It did help against muscle/ nerve pain HOWEVER I had a bad reaction so I don’t use it.
Cripes, the first time I had my submandibular cyst biopsied (they eventually resorted to the aspiration gun), all I got was the cold spray, which is absolute crap as an analgesic.
Yes, it has the “signature” of a bunch of placebos, which makes it…
…a more convenient placebo!
The only thing that would be even more convenient would be placebos you can download from an app. Oh wait a minute. Haven’t we read, in this very column, about some “energy medicine” quack offering their own extra-special photons in an app?
This one sells electrons, that one sells photons, the only thing that hasn’t been tried yet is to sell neutrinos. Someone needs to put up a “surprise!” website offering “health-enhancing neutrinos,” and while they’re at it, “selected quarks and gluons.”
“We take out the strange quarks and leave only the charmed quarks, so you can have a charmed life!
Hmm, if only my close friend & coworker who does websites, had this type of sense of humor, I’d love to try it.
When you click the “Buy” button, you get a message about quantum quackery and a caution to not waste money on dreck.
BTW, if we proliferated those kinds of “surprise!” websites, they’d screw up the signal-to-noise ratio for the quantum quacks and other quacks, so badly that the quacks might suffer a loss of business, purely by way of losing placement in search engines. Anyone here up for a bit of guerrilla media?
As a chemist, I am amused by people who think that 10^19 is a large number. Or that it’s in any way impressive or unusual.
As for the oxygen polar bonding areas capable of holding a targeted host electron, I should put that on an exam to see if anyone can figure out that it just seems to be a florid description of an anion. You could say that lye (sodium hydroxide, aka Draino) contains the same: wouldn’t that make a great skin patch!:)
I think that the FDA should require that, if anyone wants to use the word “quantum”, or even “energy”, about a product, they should first be able to define it. That would do the trick. Even Deepak himself couldn’t pass that test.
Chopra especially couldn’t pass the test. He has had physicists try to explain it to him while he sits there with a blank look on his face so he knows he doesn’t understand it. That’s why he prefers quantum woo – you don’t have to understand anything and can just make $h!+ up as you go along confident that no acolyte or fellow woomeister will pull you up on it even though their own version is contradictory.
Chopra can actually be pretty good on comparative religion, so it’s doubly tragic that he goes down the quantum BS road. If he stuck to religion & philosophy, and stayed the hell away from the science he knows not, he could do some good.
Part of the blame for this rests with the media for giving his nonsense attention. Same as with Nobel laureates who’ve gone down various BS roads, such as Shockley and quack racial theories, etc. Same as with Silicon Valley big-wigs, look up “transhumanism” and “Alcor” and so on.
If we tried to educate reporters, it would be a constant game of whack-a-mole, and there would always be those who resist all efforts so they can keep pursuing cheap clickbait. But perhaps we can reach senior editors and publishers, at least in the major media such as newspapers of record, radio/TV networks, and so on?
Scientists could offer their grad students incentives to do the outreach. Postal mail to publishers, that leads off with “I’m writing on behalf of Dr. So-and-So (well known scientist) at Such-and Such University (major university)…” could work, because it’s leveraging name recognition, and postal mail gets through where email doesn’t. These letters and the replies could also be published to scientists’ blogs.
Thoughts? Ideas?
Yo Garnetstar, I’ll take your “energy challenge.”
Canonically, energy is the capacity to do work.
Work is somewhat circularly defined as conversion of energy from one form to another.
Mundane examples: a generator converts kinetic energy to electrical energy; and a motor converts electrical energy to kinetic energy. The same device can be used both ways, thus we get regenerative braking in electric and hybrid automobiles.
OK, so (excess capitalization intended for effect):
Energy is the capacity to do work. The special Energy embodied in our products, does its work by multiplying the Subtle Forces of your Bio-Energetic Field…”
Uh-ohski, looks like we’ll have to require them to define “force” (e.g. a measurable influence on the motion or other behavior of an object), and “field” (an area of spacetime in which a given force has a measurable effect e.g. a gravitational field around a star).
This could actually get fun.
Canonically, energy is the capacity to do work.
Which is why it’s a poor definition. There was a good post on this at the old SB, maybe Chad Orzel. Definitely not Ethan. I can’t remember whether there was another physics Scibling.
Fields exist throughout the entire universe and there are particle fields as well as force fields and, of course, the Higgs field.
I’d love to know the physiologic mechanism by which the body can distinguish one electron from another, given that there really is no known biological mechanism for that to happen
It’s even better than that: electrons are particles that by definition cannot be distinguished from one another. Each and every electron is fully identical to any other electron in a very fundamental way. All electrons have the exact same mass, charge and spin, and quantum physics also dictates that it is not possible to track the trajectories of individual electrons.
Absolutely, because it would completely overturn the amassed knowledge in the field of quantum physics from the past hundred years.
It’s even better than that: electrons are particles
The deuce you say. (Yes, I understand why one can’t walk through walls absent making a big mess).
So much so that John Wheeler wrote to Albert Einstein saying that he has figured out why all those electrons are identical. By using Richard Feynman’s idea that positrons are sort of like electrons traveling backwards in time, he concluded that there is only one electron in the universe (you test this for yourself using Feynman diagrams and it is indeed makes sense). Of course this was only an interesting idea, and no one really believes this is true.
correction…John Wheeler communicated this to Richard Feynman, not Albert Einstein.
We need a Wooday moment of silence in honor of Queen Elizabeth’s personal physician, “an international leader in homeopathic and holistic medicine”, who was killed on Wednesday when his bicycle was hit by a truck. On National Cycle to Work Day.
No word on whether he was treated in the Homepathic ER.
On National Cycle to Work Day.
Back when I was working for a university press, a fine young, long-waisted lady, year after year, would implore me to ride a bike to work. And I always pointed out that my apartment was a five-minute walk from work. She wanted me to rent one anyway. I’m somewhat hostile to the attempts by cyclists to try to Borg pedestrians, especially given that they represent a greater hazard than do cars.
I agree. Cycling is for leisure. In my neck of the woods, there are numerous trails for walking, cycling, and horse riding which were originally rail lines between the city and the outlying farmlands.
I used to cycle to school in college. It was about a 15 minute ride, and parking was definitely easier. I would do it again if I lived close enough to work, or if I worked on a campus large enough to make biking an easier way to get around. It’s good exercise. But that should be a choice, never a demand from others.
I do everything by bike. But hey, I’m Dutch. I even take the bike for distances that are a 5 minute walk.
Don’t like long distance biking and hate other cyclists, who think traffic regulations are not ment for them.
On the other hand, I rather get run over by a bike, than by a car.
I sometimes hate pedestrans as well, especially when they walk on the cycleway, instead of on the footpath next to it and let their small dog run free, while the cycleway is slippery, because of snow an ice.
Gotta be honest here. When my wife and I were in Amsterdam, we both thought that the bicyclists they were some of the biggest jerks we’d ever seen. Try as we might to stay in the footpath and obey the traffic signs and lights, we still had multiple near misses in just four days in the city.
I used to cycle to school as well and survived two near death experiences. I would definitely not use it for commuting these days. Motorists hate us even those of us who are polite. But I love long distance cycling. In fact I’m in training for my 7th consecutive 210km Around The Bay cycling event held in Melbourne each year in early October. And I do all my training on the rail trail that extends 44km into the countryside from where Ilive. I pity those who have to train in the city.
I presume that those in Chicago who bicycle on the sidewalk (which is prohibited if one is over 14 years of age) and wear helmets are doing the latter in case they get clocked. The next time I hear “on your right/left,” I’m moving in that direction.
Problem in the UK too. Some [email protected] in a track suit talking on a mobile phone while riding on the pavement. Makes me want to kick his wheels in. Not only is it illegal (But not really enforced) but bloody dangerous too. I always ride on the road. Unless there is a specific cycle path. Don’t get me started on running red lights. Grrrrgh.
Same. But mostly out of self interest. I never ride on footpaths – because cycling on the road is faster. And I always obey traffic lights – because motorists don’t see cyclists and their cars hurt when they hit you. I also wear a helmet and not only because it is legally required. I have been hit on the head several times and was grateful my head was covered by a helmet.
But some pedestrians are a bit of a worry as well, especially on shared trails where I cycle. Dogs are rarely under the control of their owners. Either the leash is way too long or the dogs actually disconnected from their leashes. Having walked my dog in the past before arthritis put an end to that (the dog, not me.Yet!), I sympathise. My solution is to slow down to a speed at which I am able to come to complete stop before hitting the dog as it inevitably walks directly into my line of travel. I also make a point of exchanging some pleasantry with its owner, hoping, I think in vain, that they will take better care of their mutt next time.
When I pass a pedestrian from behind. I have learnt that the only unambiguous call is “rider” – in a strong voice and at the the right distance. You approach them from the centre of the trail and sway to whichever side they don’t move to, because their choice is totally unpredictable, even if they are walking well to one side of the trail. When I approach from in front, I keep to my left (I live in Australia where we drive on the left) and hope that the pedestrian will sensibly move to their left as well. This is usually the case but also not guaranteed. The occasional pedestrian already walking on the left side of the trail will inexplicably move to the right side of the trail despite putting themselves into my direct line of travel.
in a strong voice and at the the right distance
Just bear in mind that not everyone can hear. It’s too long a story for me to recount in my current state of exhaustion, but quite a while ago, I basically wound up with a partial lateral meniscectomy as a result of impacted cerumen. And random street violence coming from behind. It was about a year before I stopped cocking my fist if anybody approached me too quickly from behind.
@ Orac,
Amsterdam and cyclists, that’s some combination. I think a Dutch lawyer started a case against the city council, because they should do more against cyclists, who didn’t follow the law. Alas he lost his case. (Actually, finding a cyclist who follows the rules, is something like finding a needle in a haystack.)
I still remember seeing a friend of my mother, a very civilised lady, cycling against the traffic, something that annoys the hell out of me and make me want to scream.
I can’t say I never cycle on a foothpath, but only if there are no pedestrians, or just one or so and I limit my speed to walking-speed.
Yes, I am well aware that not everyone has good hearing. I see many octogenarians walking on these trails. So, I should add, that I never hold it against pedestrians when they seem to do silly things.
Just yesterday there was a schoolgirl about fifteen years of age using part of the trail to walk home from school who was walking on the left side of centre of the trail and moved to over to the right side when I warned her by calling out “rider” (from past experience, many pedestrians don’t hear you coming and are startled when you pass them, so often it is for their benefit) and then followed up with “passing on your right”.
I always show pedestrians the utmost respect because they are doing exactly what I do – enjoying exercising on a nature trail – just using a different method. This is also partly so that they we remain on friendly terms because you often meet the same pedestrians repeatedly. I never reprimand dog owners for the actions of their dogs even when it is because they are not controlling their dogs. I understand that they prefer not to have their dog on a leash however impracticable.
OK, fine. Now, how long does it take for all those fancy electrons to be released? A few picoseconds, I’d guess, if the capacitance is of that order of magnitude. Not much point using a patch, then. Better just to rub an amber rod with a black cat fed with turmeric at midnight and apply it to the base of the victim’s skull, but there’s probably not so much money in that.
True, I suppose. There’s always someone willing to pay for witchcraft. After that it’s all down to the marketing.
Actually, Rich, while I do have several pieces of Baltic amber and some curry powder, I think that getting the semi-feral black cats to stay still long enough will be somewhat of a bitch.
I am reminded, somehow, of Doctor Science’s statement that you can generate animal magnetism by rubbing Amber with a cloth. But it all depends on Amber’s mood.
Handcuff, rope and the whip. Hand that to Amber and she’ll take care of your pain 😀
Not need for pain patches…
Al who’s finding it hard to type on a keyboard while being handcuffed and roped out
I was thinking, more efficient to sell 330 picofarad capacitors, with instructions to tape one lead to your arm and leave the other lead pointing into the air to “receive Healing Energy from the Life-Force Field.”
The leads would have to be curled up into little spiral curleycues, so the pointy ends weren’t sticking out, otherwise potentially serious injuries could occur.
Our competitors’ quantum healing patches quickly wear out. And they have to be applied as soon as you remove them from the packaging, otherwise the electrons wear off, much as overcooked vegetables lose their nutrition. But our Life Force Capacitors never wear out: they keep delivering Energy from the Life-Force Field, for as long as you wear them! You won’t ever have to buy another one, unless you lose yours or want to give them away as gifts.
Imagine going to a healing-woo convention and seeing people running around with capacitors taped to their arms, with curlycue leads sticking up.
That would be worth all the effort.
Damn, I really want to try this, just for the sake of seeing the pictures of wooskis with capacitors taped to their arms.
When the game gets old, shut down the website, post an official-looking “FBI notice” on the home page, and start spreading conspiracy theories about “government suppression of alternative medicine.” Then track the conspiracy theories to scope out how they propagate. A couple of years later, publish a story about the whole thing.
The leads would have to be curled up into little spiral curleycues
Please, no string theory.
The good thing about this ‘therapy’ is that you can have acupuncture for free if the cats is not amused by these shenanigans
OMG, I was laughing hard! Thanks…
Does anybody else see the direct self-contradiction? In quantum mechanics, multiple electrons must enter into a wave function as an antisymmetric superposition because they are fermions, which gives what is called exchange symmetry. The Pauli exclusion principle is a direct consequence of this. What I mean is that quantum mechanics says that electrons are so indistinguishable from one another that the wave function containing them is a sum of all situations where they have all individually traded positions in the configuration –at risk of repeating myself, literally because you can’t tell them apart.
Really kind of amazing: invoke quantum mechanics in the first sentence and then immediately posit a situation in the exact next sentence that quantum mechanics, by its very nature, says can’t happen.
Perhaps each electron is quantum-entangled with another electron in a much more advanced civilization’s hospital light year’s away where alien physicians do the choosing for us?
Apart from what others above have noted about this statement: The map is not the territory. We can compute the energetic signatures of the relevant molecules. We can store the results for hundreds of such molecules on media the size of those patches. That does not mean we have actually exposed the patient’s body to any of those substances–it’s more like taping a solid-state memory device (such as a thumb drive) to the body. And I suspect it is just as theraputically effective.
In addition, I have an instinctive distrust of quantitative statements based on color scales where they do not show me values. I refer to the diagram in which they claim to show a reduction of inflammation in a matter of minutes. How do I know that they haven’t fiddled with the range of the color scale between the “before” and “after” pictures? How do I know it is not the result of somebody walking off the street (or removing his shirt) and then sitting in an air-conditioned room for a few minutes? The one thing I do have to work with here is the relative levels, and I see that in general the parts of the body that have relatively high values of what they are measuring in the before picture have relatively high values of that quantity in the after picture. If these patches actually did anything, I would expect the parts of the body that are marked as having had patches put on them would see a greater reduction, and I am not seeing that in the diagram.
This is really just an expendible-buy-some-more variant of the magic-infused silicone wrist bands that first appeared several years ago. The magic in the wrist bands was better because it could make its way to the target site all on its own.
Someone made a ton of money off those silly bracelets; they re-named the basketball arena in Sacramento for them. I haven’t seen any ads for the bracelets recently, are they out of style or am I not watching the right ads?
Orac, I was wondering if you could comment on a new book by a Dr. Tom Cowan (he has been favorably reviewed by “Dr.” Mercola, if that helps 😉 ). He has polluted our local public radio current affairs program a couple of times and I would like to find a SBM review of his work to forward to the local news team. Here is a link to his interview (I don’t know if the booked the composting lady as an ironic comment).
I’m guessing that’s not the same Tom Cowan who used to play in defence for Huddersfield Town…
Whenever Orac highlights one of these scams, I always wonder how successful they are, how many people actually buy these things. All we can know here is that somebody with a sizable chunk of cash to invest thought this would be a winning item and funded the rather splashy (and definitely professionally produced i.e. not cheap) promotional video/website/etc. I’m pretty sure some of the past Friday Woo -ish howlers – e.g. Bill Gray’s coherence apps, QUANTUMMan – are no more than the failed pipe dreams of would-be alt-med entrepreneurial titans. The websites are ghosts that never get updated, or the LLCs are shuttered, other business records show no activity, or the proprieters are still busy working their day jobs, or something… But maybe the fact these things keep appearing is evidence that some of them have worked, well enough to encourage other overly-ambitious woo impresarios to try?
In addition to the expense displayed in setting up Luminas, I also interpret this as a straight-up scam, not the work of ‘true believers’. It’s not just the totally nonsense invocation of quantum physics. The clinching howler for me is the list of EVERY popular supplement “and over 200 more!” Gee, that’s a lot of energy to stick in one little patch. They must be using the quantum magic to keep all the different vibrations from all those different substances from interfering with one another or combining in a way that really f***s you up. And, yeah, I’m sure there’s an exhaustive complicated manufacturing process involved in charging the patches with electrons from each of those substances – which conveniently leaves “no active ingredients” in the product.
I hope Orac follows up on Luminas at some point in the future where it might be evident whether or not they’ve found a viable market and are making any money. If they do well that would be another drop of depressing news in the giant ocean of gullibility, magical thinking, conspiracy theory, denialism that seem to be pandemic here, at what is most likely the sunset of homo sapiens sapiens…
What with all this “quantum” stuff, I am sure they have someone on board to make sure they get it right. Probably someone even more qualified than Sean Carroll and Lawrence Krauss combined.
Probably someone even more qualified than Sean Carroll and Lawrence Krauss combined
Please don’t give them the “multiverse.”
Defibrillator pads…now those have some real electron charge. Why doesn’t Luminas sell those?
As I understand it, the typical defibrillator tops out at something around 400 joules. I re-ran the pad numbers assuming this time that the total charge was distributed over the area of the big pad, so the voltage would be reduced to only 4.56 gigavolts. For the total pad capacitance of 3663 picofarads, that works out to 3.8 x 10^19 joules – about a hundred million full charges for a defibrillator, or about 10500 kilowatt hours. Seems to me like that would really put the flame in inflammation.
Perhaps agencies in charge of airport security should be alerted.
OT but Mike Adams will soon be ranting…
( NYT) It seems that Alex Jones may have deleted evidence he was ordered to save of Sandy Hook conspiracy mongering he broadcast: the parents of the murdered children are trying to sue his pants off**, threatening his lucrative supplement, survivalist business.
He’s been taken off facebook, you tube and pirate radio.
But he’s available on Mike’s new
** which may be just but profoundly unattractive.
We’ve known about Acetyl-L Carnitine for centuries? Who knew!
But the reference to K2 is really what caught my attention. I wonder what the DEA will think of that after what happened in New Haven the other day.
Ah, so that’s what K2 is! I’d just assumed they were referring to the mountain (it seemed as reasonable as everything else). has a frequently asked question section:
A. Do not cut the patch. If you cut the patch, the charge will be lost and the patch will no longer be effective.
@ Narad or Denice Walter,
Is there a simple and an cost effect way to determine the validity of such a statement without purchasing the luminas patch?.
Please advise.
You brought it up, Michael. Either you come up with a clever work around, or you buy one, cut it, and see for yourself.
The rest of us have better things to do.
@ Panacea,
Thanks for doing better things! The patch must have been cut at one point in the manufacturing process. It must be one hell of a trade secret wherein a second cut completely destroys it efficacy. I know of only one other product that fails completely after being cut and that’s a water balloon.
I’m sure they’ve worked out that they need to cut before charging. But then again…
For anyone with even a basic layman’s knowledge of Quantum Physics, the nonsense in this claim is obvious. However, there are much less obvious forms of quantum quackery that can, and do, fool people even with advanced knowledge of QM.
The following video was made by Quantum Gravity Research. This organisation employs physics PhDs to do research into QM so it uses real physics. The problem is that its founder clings to the old “consciousness causes collapse” version of the Copenhagen Interpretation.
The vast majority of present day physicists who do favour the Copenhagen interpretation have long ago jettisoned the “consciousness causing collapse” nonsense because the evidence against it is so overwhelming.
However some have a vested interest in this idea and, as the video reveals, they cherrypick the science that seems to support their interpretation and ignore the multitude of disconfirming evidence. And they actually lie about there being no deterministic interpretation of QM.
Despite the backing from many physics PhDs doing this research, the idea is BS. The organisation was setup by Klee Irwin who made a fortune selling fake medical remedies. But it takes more than a smattering of knowledge of QM to see through it all.
If nothing else, it is a testament to the adage “sex sells” – watch it to see what I mean 😉
Beg pardon? I have little interest in “interpretations,” but that’s likely because I’m weary of MWI babbling around the Intertubes. It’s not “shut up and calculate,” but yes, the measurement problem is a real thing even if the Schrödinger equation is nominally deterministic.
If you can take it, check this out.
That link is to the ideas behind the group that calls itself “Quantum Gravity Research”. Forget about anything written by this research group. It is not peer reviewed. It is based on the underlying idea that consciousness collapses the wave function, a totally discredited concept that used to be promoted as an outworking of the Copenhagen interpretation but has long since been excised from that interpretation by the vast majority of today’s physicists on the basis of experiments in QM. Although the research is conducted by phd graduates, the organisation is actually funded by an individual who made his fortune selling medical scams to an unwary credulous public and who effectively believes The Matrix was a film about science.
Nice piece here, though (h/t Peter Woit). My few remaining neurons are never going to get it together to grasp geometric Langlands or representation theory, unless I start with Charles Pierce. I like his brother better in any event.
Unfortunately this is way beyond my pay grade. I don’t have any formal training or qualifications in QM, just enough to have a reasonably well developed layman’s BS meter for quantum woo (I hope).
I don’t understand your lack of interest in interpretations of QM.
The trick is to separate the physics from the interpretation. The fun is to see how some people who are committed to one interpretation or another denigrate other interpretations but seem blind to the problems with the interpretations they favour. For example, some people who favour the Copenhagen interpretation and criticise the MWI seem to be unaware that the Copenhagen interpretation is similarly burdened with its “multiple paths” which amounts to “infinitive paths” all of which much be traversed. And MWI is more parsimonious with one less assumption. Not that I support MWI, only that I find the discussion interesting.
And the Pilot wave interpretation is attractive because it is both mundanely physical and determnistic but, on the other hand, it requires the existence of global hidden variables which is problematic from the point of view of the very real evidence for non-locality, and it is at least incomplete because can’t account for special and general relativity. But who knows what future discoveries may yield.
I will read your link though.
I don’t understand your lack of interest in interpretations of QM.
Decoherence is decoherence. The interesting question from my viewpoint is whether GR needs to be quantizted or QFT needs to be superseded to get further. This requires experiment primarily, which, absent serendipity, requires connection to theory (or the other way around). I have no objection to the philosophy of physics, but at some point it’s just navel-gazing or worse.
^^ Let me try it this way: Are the “many worlds” a well-ordered (transfinite) set? If yes, which SR would seem to require, what’s labeling them? If no, then you’re just back where you started, whether the question is nonlocality or the simple emergence of classical behavior of quantum systems.
Dear Luminas,
In addition to ferking up all that quantum-y stuff, it would be Berberis vulgaris, not Vulgaris.
Carl Linnaeus
The parents donated $50,000 to the children’s hospital where she was treated, so I think they know who did all the work.
Follow-up: Apparently the child was first diagnosed with a primary brain tumour which is almost uniformly fatal and she was given weeks to months to live. However, a follow-up scan showed cavitation or cyst formation which created doubt about the diagnosis and therefore the tumour was biopsied. This changed the diagnosis to Juvenile Xanthgranuloma, which is essentially an abnormal proliferation of a type of blood cell called a histiocyte. So this was actually not a primary brain tumour. This changed the prognosis because these tumours can be treated and cured with chemotherapy. Her treating specialists were obviously still concerned because the site of the tumour and therefore were guarded about her prognosis. However, this is no miracle as it is being portrayed in the media. She was cured by chemotherapy administered by paediatric medical specialists.
Hmmm … an oxygen with two “electron binding spots” [???] …maybe it’s … WATER!!
ps://”> Luminas Pain Relief Patches: Here’s the excuse to show partially naked bodies.
I don’t see the picture but at least I got the description 😀
Yeah I tried for that was well, but the link disappointingly led nowhere. 😉
I’ve had Lyme disease for the past 4 years and just started treatment for that 2 months ago. To my surprise the patches do take away the radiating stabbing pain I get when my body is at rest.
I have to wear a lot though. 8 patches on my back and 3 per hand. It says they work for up to 3 days but I wear all of them for about 5 days and have success.
I put on 5 patches before bed one time and I was wired beyond belief and couldn’t fall asleep for the life of me.
The patches are really affordable the way I use them and are providing a lot of relief while this long 9 month treatment plan unfolds
Isn’t it pretty obvious that Dr. Craig Davies is a Doctor of Pro Sports? There are very few university medical schools (maybe Palmer College?) offering that degree, so I expect his services to be rather expensive.
Comments are closed. |
412fe8bb3e1fba4a | In the comments to a question I asked recently, there is a discussion between user1271772 and myself on positive operators.
I know that for a positive trace-preserving operator $\Lambda$ (e.g. the partial transpose) if acting on a mixed state $\rho$ then although $\Lambda(\rho)$ is a valid density matrix it mucks up the density matrix of the system it is entangled to - hence this is not a valid operator.
This and user1271772's comments, however, got me thinking. $\Lambda$ acting on a state which is not part of a larger system does indeed give a valid density matrix and there is no associated entangled system to muck it up.
My question is, therefore: Is such an operation allowed (i.e. the action of a positive map on a state which is not part of a larger system). If not, why not? And if so, is it true that any positive map can be extended to a completely positive map (perhaps nontrivially)?
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$\begingroup$ Regarding the last sentence of the question, it may be helpful to note that any linear map $\Lambda$ from square matrices to square matrices, irrespective of being positive or completely positive, is uniquely determined by its action on pure state density matrices (simply because the pure state density matrices span the space of all matrices). So, there is no way to "extend" such a map to make it completely positive without changing its action on pure states. $\endgroup$ May 15 '18 at 12:05
• $\begingroup$ Why would the partial transpose acting on a pure state give a valid density matrix? Or do you just mean "acting on a state which is not part of a larger system"? (The former doesn't seem to make sense - any map will be "more positive" on mixed states than on pure states. The latter is simply called a "positive map".) $\endgroup$ May 15 '18 at 16:41
• $\begingroup$ @NorbertSchuch I do mean "acting on a state which is not part of a larger system" - is this not one and the same as a pure state? $\endgroup$ May 15 '18 at 16:48
• $\begingroup$ @Quantumspaghettification No. (Well, it is a bit a matter of belief, but the way it is phrased it is highly misleading with regard to the usual language. I had to read it several times to guess what you mean. I would suggest to rephrase it accordingly. $\endgroup$ May 15 '18 at 18:00
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$\begingroup$ @Quantumspaghettification $\rho=|\psi\rangle\langle\psi|$: A pure state. Otherwise (i.e., the rank of $\rho$ is $>1$): mixed state. On either of them, the transpose yields a positive $\Lambda(\rho)$. Only if we apply $\Lambda\otimes I$ to a larger state (be it pure or mixed), we obtain a non-postive state. $\endgroup$ May 15 '18 at 20:22
Any map which is not Completely Positive, Trace Preserving (CPTP), is not possible as an "allowed operation" (a more-or-less complete account of how some system transforms) in quantum mechanics, regardless of what states it is meant to act upon.
The constraint of maps being CPTP comes from the physics itself. Physical transformations on closed systems are unitary, as a result of the Schrödinger equation. If we allow for the possibility to introduce auxiliary systems, or to ignore/lose auxiliary systems, we obtain a more general CPTP map, expressed in terms of a Stinespring dilation. Beyond this, we must consider maps which may occur only with a significant probability of failure (as with postselection). This is perhaps one way of describing an "extension" for non-CPTP maps to CPTP maps — engineering it so that it can be described as a provocative thing with some probability, and something uninteresting with possibly greater probability; or at least a mixture of a non-CPTP map with something else to yield a total evolution which is CPTP — but whether it is useful to do so in general is not clear to me.
On a higher level — while we may consider entanglement a strange phenomenon, and in some way special to quantum mechanics, the laws of quantum mechanics themselves make no distinctions between entangled states and product states. There is no sense in which quantum mechanics is delicate or sensitive to the mere presence of nonlocal correlations (which are correlations in things which we are concerned with), which would render impossible some transformation on entangled states merely because it might produce an embarrassing result. Either a process is impossible — and in particular not possible on product states — or it is possible, and any embarrassment about the outcome for entangled states is our own, on account of the difficulty in understanding what has happened. What is special about entanglement is the way it challenges our classically-motivated preconceptions, not how entangled states themselves evolve in time.
• $\begingroup$ What physics law requires that subsystems of the universe must evolve this way? If we only assume that the universe evolves according to the Schroedinger equation, can we prove that all subsystems must evolve in a CPTP way? I have never seen such a proof, and others agree: sciencedirect.com/science/article/pii/S0375960105005748. I asked the question here: quantumcomputing.stackexchange.com/questions/2073/…. $\endgroup$ May 17 '18 at 0:28
• $\begingroup$ After more reading, I have found a counter-example to your claim that dynamics must be CPTP. When the initial density matrix is given by Eq. 6 of sciencedirect.com/science/article/pii/S0375960105005748, and the Hamiltonian is given in that same paragraph, $e^{-iHt}\rho e^{iHt}$ leads to a "total" density matrix where the subsystem density matrix is not even positive. The key idea is that the system and its bath are entangled even at time $t=0$. I believe you have to assume no entanglement between system and bath at $t=0$ in order to force CPTP in Choi's way or Alicki's way. $\endgroup$ May 17 '18 at 0:56
• $\begingroup$ @user1261772: if you are not allowed to assume no entanglement between system and bath, then in what respect is it even meaningful to consider a map on the system alone? The pre-existing entanglement makes a nonsense of the idea that we're even trying to provide a "more-or-less complete account" of how the system evolves. And --- finally --- if the subsystem operator is not even positive, how on earth do we interpret the possibility of obtaining negative probabilities (or supernormalised probabilities) of some of the eigenstates? $\endgroup$ May 17 '18 at 7:53
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$\begingroup$ "his is perhaps one way of describing an "extension" for non-CPTP maps to CPTP maps — engineering it so that it can be described as a provocative thing with some probability, and something uninteresting with possibly greater probability" -- do you have any example for that? It seems to me that this would with some probability produce an output which is non-positive, which cannot be. $\endgroup$ May 17 '18 at 8:24
• $\begingroup$ @Neil: I never said you are not allowed to assume no entanglement between system and bath. The paper said that the arguments made for CPTP maps by Choi and Alicki both assumed no initial correlation, then gave an example of how an OQS that is initially correlated with its bath, can have non-positive evolution when the system+bath are evolved using $e^{-iHt}\rho e^{iHt}$ and then the bath is traced out. You say that the pre-entanglement idea is "nonsense", but if you search "initial correlations" you will find a huge body of literature on OQSs that are initially correlated with their baths. $\endgroup$ May 17 '18 at 8:24
The situation of non-completely positive maps (or more generally non-linear maps) is controversial partly due to the precise definition of how you should construct the map. But it is easy to come up with an example of something that would seem to be NCP or even not linear.
1. Non linear map.
Consider a preparation device that can create a qubit in an arbitrary state $\rho$ (this device has 3 dials). Now let this device be constructed so that it also prepares a second state $\rho$ in the environment. I.e, you think you prepared a one qubit state $\rho$ but actually you prepared a two qubit state $\rho\otimes\rho$. The second qubit is the environment (which you cannot access), so if you perform tomography on your qubit, everything seems ok.
No imagine that you also have the following black box - it has (as far as you can tell) one input and two outputs. In reality (unknown to you) it has two inputs and two outputs and it simply spits out both the system qubit and the environement qubit. As far as you can tell, this black box is a cloning machine, violating linearity.
1. NCP
Similar to the idea above, but the preparation device prepares $\rho\otimes\rho^T$ (clearly this could be done in the lab). The black box will now be a one rail box (one qubit input one qubit output as far as the user is concerned), which swaps the system and environement. To you, it seems like a trasposition map.
Note that both preparation devices are physical, but the way you construct the map might depend on how you use them. In the example above I assumed that a mixed state $\rho$ would only be constructed by using the three dials in the machine. In principle, I could try to construct a mixed state by flipping coins and preparing pure states with the right probability. Tomorgraphy would show that the processes are equivalent, but the environment would be different, and the map you would construct for the black boxes would be different.
No law of physics states that we must be able to evolve a sub-system of the universe on its own.
There would be no way to definitively test such a law.
The density matrix of the universe must have a trace of 1 and be positive semi-definite, by the mathematical definition of probabilities1. Any change in the universe must1 preserve this, for mathematical reasons and due to definitions. If $\rm{Tr}(\rho_{\rm{universe}})\lt1$, you just haven't included the whole universe in $\rho_{\rm{universe}}$. If it's more than 1, or if $\rho_{\rm{universe}}<0$, what you have is not actually a density matrix, by the definition of probability1.
So the map: $\rho_{\rm{universe}}(0)\rightarrow\rho_{\rm{universe}}(t)$ must1 be positive and trace-preserving.
For convenience, we like to model sub-regions of the universe, and introduce complete positivity for that. But one day an experiment might come along that we find impossible to explain2, perhaps because we have chosen to model the universe in a way that's not compatible with how the universe actually works.
If we assume gravity doesn't exist, and we can magically compute anything we want, we believe that evolving $\rho_{\rm{universe}}$ using the right positive trace-preserving map, then doing a partial trace over all parts of the universe not of concern, will give accurate predictions. Introducing the notion of modeling only a sub-system of $\rho_{\rm{universe}}$, using a CPT map, is also something we believe will work, but we might bet slightly less on this, because we've added the assumption that sub-systems evolve this way, not just the universe as a whole.
1: Even this is debatable because the relationship between a wavefunction or density matrix and probabilities comes from a postulate of quantum mechanics called the Born rule, which until fewer than 10 years ago was never tested at all, and still has only been confirmed to be true within an $\epsilon$, and for a particular system: If Born's rule isn't true, Eq. 6 of this would not be zero. To test if Born's rule is true for a particular system (in this case, photons coming from some particular source), you would have to do an infinite number of instances, of all 7 of these experiments, or come up with a different way to test Born's rule (and I don't know of any). In 2009 we published this saying that Born's rule was true (for this system) to within an $\epsilon$ that was smaller than the experimental uncertainty, so we only know Born's rule is true for this system, and to within a precision limited by the experiment.
2: This is actually already the case, but let's pretend that gravity does not exist and that quantum mechanics (QED+QFD+QCD) is correct, and we still find it impossible to explain something, despite having (somehow) magical computer power to compute anything we want instantly.
• $\begingroup$ You're bringing up field theories, and there the notion of traces is much more subtle. But it was unnecessary for the question. No need to say anything like $Tr \rho_{universe}$ $\endgroup$
– AHusain
May 16 '18 at 6:35
• $\begingroup$ @AHusain: The question was about trace-preserving maps, which involves the trace. The question was directed at me. Let me decide how I would like to answer the question. $\endgroup$ May 16 '18 at 19:16
• $\begingroup$ Just wanted to point out that finite and infinite dimensional Hilbert spaces have some substantial differences. States on different sorts of VonNeumann algebras. That is all. $\endgroup$
– AHusain
May 16 '18 at 20:52
• $\begingroup$ @AHusain: Okay. The Hilbert space of a single particle can be uncountably infinite dimensional too, so these substantial differences don't just occur for $\rho_{\rm{universe}}$. Anyway the point I was trying to make in my answer was that quantum mechanics (QED+QFD+QCD) requires that $\rho_{\rm{universe}}$ evolves in a way that preserves trace and positivity (assuming the Born's rule axiom to be true). Does this mean all subsystems of the universe need to evolve by a CPT map? I have never seen a proof of this. $\endgroup$ May 16 '18 at 23:58
• $\begingroup$ If you're going to downvote an answer that took a whole morning (maybe 3-4 hours?) to write and format, would it not be fair to explain what you didn't like about it? $\endgroup$ May 17 '18 at 8:43
Your Answer
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The liquid drop model
Main article: Semi-empirical mass formula
The liquid drop model is one of the first models of nuclear structure, proposed by Carl Friedrich von Weizsäcker in 1935.[1] It describes the nucleus as a semiclassical fluid made up of neutrons and protons, with an internal repulsive electrostatic force proportional to the number of protons. The quantum mechanical nature of these particles appears via the Pauli exclusion principle, which states that no two nucleons of the same kind can be at the same state. Thus the fluid is actually what is known as a Fermi liquid. In this model, the binding energy of a nucleus with protons and neutrons is given by
where is the total number of nucleons (Mass Number). The terms proportional to and represent the volume and surface energy of the liquid drop, the term proportional to represents the electrostatic energy, the term proportional to represents the Pauli exclusion principle and the last term is the pairing term, which lowers the energy for even numbers of protons or neutrons. The coefficients and the strength of the pairing term may be estimated theoretically, or fit to data. This simple model reproduces the main features of the binding energy of nuclei.
The assumption of nucleus as a drop of Fermi liquid is still widely used in the form of Finite Range Droplet Model (FRDM), due to the possible good reproduction of nuclear binding energy on the whole chart, with the necessary accuracy for predictions of unknown nuclei.[2]
The shell model
Main article: Nuclear shell model
The expression "shell model" is ambiguous in that it refers to two different eras in the state of the art. It was previously used to describe the existence of nucleon shells in the nucleus according to an approach closer to what is now called mean field theory. Nowadays, it refers to a formalism analogous to the configuration interaction formalism used in quantum chemistry. We shall introduce the latter here.
Introduction to the shell concept
Difference between experimental binding energies and the liquid drop model prediction as a function of neutron number for Z>7
Systematic measurements of the binding energy of atomic nuclei show systematic deviations with respect to those estimated from the liquid drop model. In particular, some nuclei having certain values for the number of protons and/or neutrons are bound more tightly together than predicted by the liquid drop model. These nuclei are called singly/doubly magic. This observation led scientists to assume the existence of a shell structure of nucleons (protons and neutrons) within the nucleus, like that of electrons within atoms.
Indeed, nucleons are quantum objects. Strictly speaking, one should not speak of energies of individual nucleons, because they are all correlated with each other. However, as an approximation one may envision an average nucleus, within which nucleons propagate individually. Owing to their quantum character, they may only occupy discrete energy levels. These levels are by no means uniformly distributed; some intervals of energy are crowded, and some are empty, generating a gap in possible energies. A shell is such a set of levels separated from the other ones by a wide empty gap.
The energy levels are found by solving the Schrödinger equation for a single nucleon moving in the average potential generated by all other nucleons. Each level may be occupied by a nucleon, or empty. Some levels accommodate several different quantum states with the same energy; they are said to be degenerate. This occurs in particular if the average nucleus has some symmetry.
The concept of shells allows one to understand why some nuclei are bound more tightly than others. This is because two nucleons of the same kind cannot be in the same state (Pauli exclusion principle). So the lowest-energy state of the nucleus is one where nucleons fill all energy levels from the bottom up to some level. A nucleus with full shells is exceptionally stable, as will be explained.
As with electrons in the electron shell model, protons in the outermost shell are relatively loosely bound to the nucleus if there are only few protons in that shell, because they are farthest from the center of the nucleus. Therefore, nuclei which have a full outer proton shell will be more tightly bound and have a higher binding energy than other nuclei with a similar total number of protons. This is also true for neutrons.
Furthermore, the energy needed to excite the nucleus (i.e. moving a nucleon to a higher, previously unoccupied level) is exceptionally high in such nuclei. Whenever this unoccupied level is the next after a full shell, the only way to excite the nucleus is to raise one nucleon across the gap, thus spending a large amount of energy. Otherwise, if the highest occupied energy level lies in a partly filled shell, much less energy is required to raise a nucleon to a higher state in the same shell.
Some evolution of the shell structure observed in stable nuclei is expected away from the valley of stability. For example, observations of unstable isotopes have shown shifting and even a reordering of the single particle levels of which the shell structure is composed.[3] This is sometimes observed as the creation of an island of inversion or in the reduction of excitation energy gaps above the traditional magic numbers.
Basic hypotheses
Some basic hypotheses are made in order to give a precise conceptual framework to the shell model:
Brief description of the formalism
The general process used in the shell model calculations is the following. First a Hamiltonian for the nucleus is defined. Usually, for computational practicality, only one- and two-body terms are taken into account in this definition. The interaction is an effective theory: it contains free parameters which have to be fitted with experimental data.
The next step consists in defining a basis of single-particle states, i.e. a set of wavefunctions describing all possible nucleon states. Most of the time, this basis is obtained via a Hartree–Fock computation. With this set of one-particle states, Slater determinants are built, that is, wavefunctions for Z proton variables or N neutron variables, which are antisymmetrized products of single-particle wavefunctions (antisymmetrized meaning that under exchange of variables for any pair of nucleons, the wavefunction only changes sign).
In principle, the number of quantum states available for a single nucleon at a finite energy is finite, say n. The number of nucleons in the nucleus must be smaller than the number of available states, otherwise the nucleus cannot hold all of its nucleons. There are thus several ways to choose Z (or N) states among the n possible. In combinatorial mathematics, the number of choices of Z objects among n is the binomial coefficient CZ
. If n is much larger than Z (or N), this increases roughly like nZ. Practically, this number becomes so large that every computation is impossible for A=N+Z larger than 8.
To obviate this difficulty, the space of possible single-particle states is divided into core and valence, by analogy with chemistry (see core electron and valence electron). The core is a set of single-particles which are assumed to be inactive, in the sense that they are the well bound lowest-energy states, and that there is no need to reexamine their situation. They do not appear in the Slater determinants, contrary to the states in the valence space, which is the space of all single-particle states not in the core, but possibly to be considered in the choice of the build of the (Z-) N-body wavefunction. The set of all possible Slater determinants in the valence space defines a basis for (Z-) N-body states.
The last step consists in computing the matrix of the Hamiltonian within this basis, and to diagonalize it. In spite of the reduction of the dimension of the basis owing to the fixation of the core, the matrices to be diagonalized reach easily dimensions of the order of 109, and demand specific diagonalization techniques.
The shell model calculations give in general an excellent fit with experimental data. They depend however strongly on two main factors:
Mean field theories
The independent-particle model (IPM)
The interaction between nucleons, which is a consequence of strong interactions and binds the nucleons within the nucleus, exhibits the peculiar behaviour of having a finite range: it vanishes when the distance between two nucleons becomes too large; it is attractive at medium range, and repulsive at very small range. This last property correlates with the Pauli exclusion principle according to which two fermions (nucleons are fermions) cannot be in the same quantum state. This results in a very large mean free path predicted for a nucleon within the nucleus.[4]
The main idea of the Independent Particle approach is that a nucleon moves inside a certain potential well (which keeps it bound to the nucleus) independently from the other nucleons. This amounts to replacing an N-body problem (N particles interacting) by N single-body problems. This essential simplification of the problem is the cornerstone of mean field theories. These are also widely used in atomic physics, where electrons move in a mean field due to the central nucleus and the electron cloud itself.
The independent particle model and mean field theories (we shall see that there exist several variants) have a great success in describing the properties of the nucleus starting from an effective interaction or an effective potential, thus are a basic part of atomic nucleus theory. One should also notice that they are modular enough, in that it is quite easy to extend the model to introduce effects such as nuclear pairing, or collective motions of the nucleon like rotation, or vibration, adding the corresponding energy terms in the formalism. This implies that in many representations, the mean field is only a starting point for a more complete description which introduces correlations reproducing properties like collective excitations and nucleon transfer.[5][6]
Nuclear potential and effective interaction
A large part of the practical difficulties met in mean field theories is the definition (or calculation) of the potential of the mean field itself. One can very roughly distinguish between two approaches:
In the case of the Hartree–Fock approaches, the trouble is not to find the mathematical function which describes best the nuclear potential, but that which describes best the nucleon–nucleon interaction. Indeed, in contrast with atomic physics where the interaction is known (it is the Coulomb interaction), the nucleon–nucleon interaction within the nucleus is not known analytically.
There are two main reasons for this fact. First, the strong interaction acts essentially among the quarks forming the nucleons. The nucleon–nucleon interaction in vacuum is a mere consequence of the quark–quark interaction. While the latter is well understood in the framework of the Standard Model at high energies, it is much more complicated in low energies due to color confinement and asymptotic freedom. Thus there is yet no fundamental theory allowing one to deduce the nucleon–nucleon interaction from the quark–quark interaction. Furthermore, even if this problem were solved, there would remain a large difference between the ideal (and conceptually simpler) case of two nucleons interacting in vacuum, and that of these nucleons interacting in the nuclear matter. To go further, it was necessary to invent the concept of effective interaction. The latter is basically a mathematical function with several arbitrary parameters, which are adjusted to agree with experimental data.
Most modern interaction are zero-range so they act only when the two nucleons are in contact, as introduced by Tony Skyrme.[7]
The self-consistent approaches of the Hartree–Fock type
In the Hartree–Fock approach of the n-body problem, the starting point is a Hamiltonian containing n kinetic energy terms, and potential terms. As mentioned before, one of the mean field theory hypotheses is that only the two-body interaction is to be taken into account. The potential term of the Hamiltonian represents all possible two-body interactions in the set of n fermions. It is the first hypothesis.
The second step consists in assuming that the wavefunction of the system can be written as a Slater determinant of one-particle spin-orbitals. This statement is the mathematical translation of the independent-particle model. This is the second hypothesis.
There remains now to determine the components of this Slater determinant, that is, the individual wavefunctions of the nucleons. To this end, it is assumed that the total wavefunction (the Slater determinant) is such that the energy is minimum. This is the third hypothesis.
Technically, it means that one must compute the mean value of the (known) two-body Hamiltonian on the (unknown) Slater determinant, and impose that its mathematical variation vanishes. This leads to a set of equations where the unknowns are the individual wavefunctions: the Hartree–Fock equations. Solving these equations gives the wavefunctions and individual energy levels of nucleons, and so the total energy of the nucleus and its wavefunction.
This short account of the Hartree–Fock method explains why it is called also the variational approach. At the beginning of the calculation, the total energy is a "function of the individual wavefunctions" (a so-called functional), and everything is then made in order to optimize the choice of these wavefunctions so that the functional has a minimum – hopefully absolute, and not only local. To be more precise, there should be mentioned that the energy is a functional of the density, defined as the sum of the individual squared wavefunctions. The Hartree–Fock method is also used in atomic physics and condensed matter physics as Density Functional Theory, DFT.
The process of solving the Hartree–Fock equations can only be iterative, since these are in fact a Schrödinger equation in which the potential depends on the density, that is, precisely on the wavefunctions to be determined. Practically, the algorithm is started with a set of individual grossly reasonable wavefunctions (in general the eigenfunctions of a harmonic oscillator). These allow to compute the density, and therefrom the Hartree–Fock potential. Once this done, the Schrödinger equation is solved anew, and so on. The calculation stops – convergence is reached – when the difference among wavefunctions, or energy levels, for two successive iterations is less than a fixed value. Then the mean field potential is completely determined, and the Hartree–Fock equations become standard Schrödinger equations. The corresponding Hamiltonian is then called the Hartree–Fock Hamiltonian.
The relativistic mean field approaches
Born first in the 1970s with the works of John Dirk Walecka on quantum hadrodynamics, the relativistic models of the nucleus were sharpened up towards the end of the 1980s by P. Ring and coworkers. The starting point of these approaches is the relativistic quantum field theory. In this context, the nucleon interactions occur via the exchange of virtual particles called mesons. The idea is, in a first step, to build a Lagrangian containing these interaction terms. Second, by an application of the least action principle, one gets a set of equations of motion. The real particles (here the nucleons) obey the Dirac equation, whilst the virtual ones (here the mesons) obey the Klein–Gordon equations.
In view of the non-perturbative nature of strong interaction, and also in view of the fact that the exact potential form of this interaction between groups of nucleons is relatively badly known, the use of such an approach in the case of atomic nuclei requires drastic approximations. The main simplification consists in replacing in the equations all field terms (which are operators in the mathematical sense) by their mean value (which are functions). In this way, one gets a system of coupled integro-differential equations, which can be solved numerically, if not analytically.
The interacting boson model
The interacting boson model (IBM) is a model in nuclear physics in which nucleons are represented as pairs, each of them acting as a boson particle, with integral spin of 0, 2 or 4. This makes calculations feasible for larger nuclei. There are several branches of this model - in one of them (IBM-1) one can group all types of nucleons in pairs, in others (for instance - IBM-2) one considers protons and neutrons in pairs separately.
Spontaneous breaking of symmetry in nuclear physics
One of the focal points of all physics is symmetry. The nucleon–nucleon interaction and all effective interactions used in practice have certain symmetries. They are invariant by translation (changing the frame of reference so that directions are not altered), by rotation (turning the frame of reference around some axis), or parity (changing the sense of axes) in the sense that the interaction does not change under any of these operations. Nevertheless, in the Hartree–Fock approach, solutions which are not invariant under such a symmetry can appear. One speaks then of spontaneous symmetry breaking.
Qualitatively, these spontaneous symmetry breakings can be explained in the following way: in the mean field theory, the nucleus is described as a set of independent particles. Most additional correlations among nucleons which do not enter the mean field are neglected. They can appear however by a breaking of the symmetry of the mean field Hamiltonian, which is only approximate. If the density used to start the iterations of the Hartree–Fock process breaks certain symmetries, the final Hartree–Fock Hamiltonian may break these symmetries, if it is advantageous to keep these broken from the point of view of the total energy.
It may also converge towards a symmetric solution. In any case, if the final solution breaks the symmetry, for example, the rotational symmetry, so that the nucleus appears not to be spherical, but elliptic, all configurations deduced from this deformed nucleus by a rotation are just as good solutions for the Hartree–Fock problem. The ground state of the nucleus is then degenerate.
A similar phenomenon happens with the nuclear pairing, which violates the conservation of the number of baryons (see below).
Extensions of the mean field theories
Nuclear pairing phenomenon
The most common extension to mean field theory is the nuclear pairing. Nuclei with an even number of nucleons are systematically more bound than those with an odd one. This implies that each nucleon binds with another one to form a pair, consequently the system cannot be described as independent particles subjected to a common mean field. When the nucleus has an even number of protons and neutrons, each one of them finds a partner. To excite such a system, one must at least use such an energy as to break a pair. Conversely, in the case of odd number of protons or neutrons, there exists an unpaired nucleon, which needs less energy to be excited.
This phenomenon is closely analogous to that of Type 1 superconductivity in solid state physics. The first theoretical description of nuclear pairing was proposed at the end of the 1950s by Aage Bohr, Ben Mottelson, and David Pines (which contributed to the reception of the Nobel Prize in Physics in 1975 by Bohr and Mottelson).[8] It was close to the BCS theory of Bardeen, Cooper and Schrieffer, which accounts for metal superconductivity. Theoretically, the pairing phenomenon as described by the BCS theory combines with the mean field theory: nucleons are both subject to the mean field potential and to the pairing interaction.
The Hartree–Fock–Bogolyubov (HFB) method is a more sophisticated approach, [9] enabling one to consider the pairing and mean field interactions consistently on equal footing. HFB is now the de facto standard in the mean field treatment of nuclear systems.
Symmetry restoration
Peculiarity of mean field methods is the calculation of nuclear property by explicit symmetry breaking. The calculation of the mean field with self-consistent methods (e.g. Hartree-Fock), breaks rotational symmetry, and the calculation of pairing property breaks particle-number.
Several techniques for symmetry restoration by projecting on good quantum numbers have been developed.[10]
Particle vibration coupling
Mean field methods (eventually considering symmetry restoration) are a good approximation for the ground state of the system, even postulating a system of independent particles. Higher-order corrections consider the fact that the particles interact together by the means of correlation. These correlations can be introduced taking into account the coupling of independent particle degrees of freedom, low-energy collective excitation of systems with even number of protons and neutrons.
In this way, excited states can be reproduced by the means of random phase approximation (RPA), also eventually consistently calculating corrections to the ground state (e.g. by the means of nuclear field theory[6]).
See also
Further reading
General audience
Introductory texts
Fundamental texts
2. ^ Moeller, P.; Myers, W. D.; Swiatecki, W. J.; Treiner, J. (3 Sep 1984). "Finite Range Droplet Model". Conference: 7. International Conference on Atomic Masses and Fundamental Constants (AMCO-7), Darmstadt-Seeheim, F.R. Germany. OSTI 6441187.
3. ^ Sorlin, O.; Porquet, M.-G. (2008). "Nuclear magic numbers: New features far from stability". Progress in Particle and Nuclear Physics. 61 (2): 602–673. arXiv:0805.2561. Bibcode:2008PrPNP..61..602S. doi:10.1016/j.ppnp.2008.05.001. S2CID 118524326.
4. ^ Brink, David; Broglia, Ricardo A. (2005). Nuclear Superfluidity. Cambridge University Press. ISBN 9781139443074.
5. ^ Ring, P.; Schuck, P. (1980). The nuclear many-body problem. Springer Verlag. ISBN 978-3-540-21206-5.
6. ^ a b Idini, A.; Potel, G.; Barranco, F.; Vigezzi, E.; Broglia, R. A. (2015). "Interweaving of elementary modes of excitation in superfluid nuclei through particle-vibration coupling: Quantitative account of the variety of nuclear structure observables". Physical Review C. 92 (3): 031304. arXiv:1504.05335. Bibcode:2015PhRvC..92c1304I. doi:10.1103/PhysRevC.92.031304. S2CID 56380507.
7. ^ Beiner, M.; Flocard, H.; Van Giai, Nguyen; Quentin, P. (1975). "Nuclear ground-state properties and self-consistent calculations with the skyrme interaction". Nuclear Physics A. 238: 29–69. Bibcode:1975NuPhA.238...29B. doi:10.1016/0375-9474(75)90338-3.
8. ^ Broglia, Ricardo A.; Zelevinsky, Vladimir (2013). Fifty Years of Nuclear BCS: Pairing in Finite Systems. World Scientific. doi:10.1142/8526. ISBN 978-981-4412-48-3.
9. ^ "Hartree-Fock-Bogoliubov Method".
10. ^ Bayman, B. F. (1960). "A derivation of the pairing-correlation method". Nucl. Phys. 15: 33–38. Bibcode:1960NucPh..15...33B. doi:10.1016/0029-5582(60)90279-0. |
aef8626cea1bbd10 | University of Cambridge > Talks.cam > Theoretical Chemistry Informal Seminars > Application of normal forms and TST to the reaction dynamics of quantum wave packets
If you have a question about this talk, please contact Dr. Judith B. Rommel.
Transition state theory (TST) is a powerful framework to describe reactions which are mediated by a transition state between reactants and products. Due to its formulation in phase space and its general assumptions, it has numerous applications in chemistry and physics. However, because there exists no such phase space in the Schrödinger theory, TST cannot be applied directly to spatially extended wave packets as they appear in several quantum mechanical systems.
In this talk, I will present a general method which allows for the application of TST to the dynamics of quantum wave packets in a variational framework. Within the latter, the original wave function is replaced by an appropriate trial wave function depending on a set of variational parameters, and the Schrödinger equation is approximately solved by applying a time-dependent variational principle. The latter defines a noncanonical Hamiltonian system for the variational parameters, in which common structures such as ground or transition states, dividing surfaces, reactants and products can be identified.
In order to construct a dividing surface which is free of local recrossings, a normal form expansion in variational space is performed. The latter’s generating function can be chosen in such a way that it extracts the normal form of the dynamical equations as well as canonical coordinates in a natural way. The resulting classical Hamiltonian then directly allows to apply TST to the quantum system. Applications of the method will be demonstrated for a model potential within the linear Schrödinger theory and for Bose-Einstein condensates as nonlinear Schrödinger systems.
This talk is part of the Theoretical Chemistry Informal Seminars series.
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bb0f4411f0361420 | Einstein's unsuccessful investigations
From Wikipedia, the free encyclopedia
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Albert Einstein conducted several unsuccessful investigations. These pertain to force, superconductivity, and other research.
Special relativity[edit]
In the special relativity paper, in 1905, Einstein noted that, given a specific definition of the word "force" (a definition which he later agreed was not advantageous), and if we choose to maintain (by convention) the equation mass x acceleration = force, then one arrives at as the expression for the transverse mass of a fast moving particle. This differs from the accepted expression today, because, as noted in the footnotes to Einstein's paper added in the 1913 reprint, "it is more to the point to define force in such a way that the laws of energy and momentum assume the simplest form", as was done, for example, by Max Planck in 1906, who gave the now familiar expression for the transverse mass.
As Miller points out, this is equivalent to the transverse mass predictions of both Einstein and Lorentz. Einstein had commented already in the 1905 paper that "With a different definition of force and acceleration, we should naturally obtain other expressions for the masses. This shows that in comparing different theories... we must proceed very cautiously."[1]
Einstein published (in 1922) a qualitative theory of superconductivity based on the vague idea of electrons shared in orbits. This paper predated modern quantum mechanics, and today is regarded as being incorrect. The current theory of low temperature superconductivity was only worked out in 1957, thirty years after the establishing of modern quantum mechanics. However, even today, superconductivity is not well understood, and alternative theories continue to be put forward, especially to account for high-temperature superconductors.[citation needed]
Black holes[edit]
Einstein denied several times that black holes could form.[citation needed] In 1939 he published a paper that argues that a star collapsing would spin faster and faster, spinning at the speed of light with infinite energy well before the point where it is about to collapse into a Schwarzchild singularity, or black hole.
The essential result of this investigation is a clear understanding as to why the "Schwarzschild singularities" do not exist in physical reality. Although the theory given here treats only clusters whose particles move along circular paths it does not seem to be subject to reasonable doubt that more general cases will have analogous results. The "Schwarzschild singularity" does not appear for the reason that matter cannot be concentrated arbitrarily. And this is due to the fact that otherwise the constituting particles would reach the velocity of light.[2]
Einstein's argument itself only shows that stable spinning objects have to spin faster and faster to stay stable before the point where they collapse. But it is well understood today (and was understood well by some even then) that collapse cannot happen through stationary states the way Einstein imagined. Nevertheless, the extent to which the models of black holes in classical general relativity correspond to physical reality remains unclear, and in particular the implications of the central singularity implicit in these models are still not understood.
Closely related to his rejection of black holes, Einstein believed that the exclusion of singularities might restrict the class of solutions of the field equations so as to force solutions compatible with quantum mechanics, but no such theory has ever been found.[citation needed]
Quantum mechanics[edit]
In the early days of quantum mechanics, Einstein tried to show that the uncertainty principle was not valid. By 1927 he had become convinced of its utility, but he always opposed it.[citation needed]
EPR paradox[edit]
In the EPR paper, Einstein argued that quantum mechanics cannot be a complete realistic and local representation of phenomena, given specific definitions of "realism", "locality", and "completeness". The modern consensus is that Einstein's concept of realism is too restrictive.[citation needed]
Cosmological term[edit]
Einstein himself considered the introduction of the cosmological term in his 1917 paper founding cosmology as a "blunder".[3] The theory of general relativity predicted an expanding or contracting universe, but Einstein wanted a universe which is an unchanging three-dimensional sphere, like the surface of a three-dimensional ball in four dimensions.
He wanted this for philosophical reasons, so as to incorporate Mach's principle in a reasonable way. He stabilised his solution by introducing a cosmological constant, and when the universe was shown to be expanding, he retracted the constant as a blunder. This is not really much of a blunder – the cosmological constant is necessary within general relativity as it is currently understood, and it is widely believed to have a nonzero value today.
Minkowski's work[edit]
Einstein did not immediately appreciate the value of Minkowski's four-dimensional formulation of special relativity, although within a few years he had adopted it within his theory of gravitation.[citation needed]
Heisenberg's work[edit]
Finding it too formal, Einstein believed that Heisenberg's matrix mechanics was incorrect. He changed his mind when Schrödinger and others demonstrated that the formulation in terms of the Schrödinger equation, based on wave–particle duality was equivalent to Heisenberg's matrices.[citation needed]
Unified field theory[edit]
Einstein spent many years pursuing a unified field theory, and published many papers on the subject, without success.
1. ^ Miller, Arthur I. (1981), Albert Einstein's special theory of relativity. Emergence (1905) and early interpretation (1905–1911), Reading: Addison–Wesley, pp. 325–331, ISBN 978-0-201-04679-3
2. ^ Einstein, Albert (October 1939). "On a Stationary System With Spherical Symmetry Consisting of Many Gravitating Masses". Annals of Mathematics. 40 (4): 922–936. doi:10.2307/1968902. JSTOR 1968902.
3. ^ Wright, Karen (30 September 2004). "The Master's Mistakes". Discover Magazine. Retrieved 15 October 2009. |
ecfa40e479ee8518 | 薛定谔方程 Schrödinger equation
The Schrödinger equation is a linear partial differential equation that governs the wave function of a quantum-mechanical system. It is a key result in quantum mechanics, and its discovery was a significant landmark in the development of the subject. The equation is named after Erwin Schrödinger, who postulated the equation in 1925, and published it in 1926, forming the basis for the work that resulted in his Nobel Prize in Physics in 1933.
Conceptually, the Schrödinger equation is the quantum counterpart of Newton’s second law in classical mechanics. Given a set of known initial conditions, Newton’s second law makes a mathematical prediction as to what path a given physical system will take over time. The Schrödinger equation gives the evolution over time of a wave function, the quantum-mechanical characterization of an isolated physical system. The equation can be derived from the fact that the time-evolution operator must be unitary, and must therefore be generated by the exponential of a self-adjoint operator, which is the quantum Hamiltonian.
The Schrödinger equation is not the only way to study quantum mechanical systems and make predictions. The other formulations of quantum mechanics include matrix mechanics, introduced by Werner Heisenberg, and the path integral formulation, developed chiefly by Richard Feynman. Paul Dirac incorporated matrix mechanics and the Schrödinger equation into a single formulation. When these approaches are compared, the use of the Schrödinger equation is sometimes called “wave mechanics”. |
186affd08e35a891 | Frontiers reaches 6.4 on Journal Impact Factors
Original Research ARTICLE
Front. Phys., 01 October 2013 |
Umbral Vade Mecum
Thomas L. Curtright1 and Cosmas K. Zachos2*
• 1Department of Physics, University of Miami, Coral Gables, FL, USA
• 2High Energy Physics Division, Argonne National Laboratory, Argonne, IL, USA
In recent years the umbral calculus has emerged from the shadows to provide an elegant correspondence framework that automatically gives systematic solutions of ubiquitous difference equations—discretized versions of the differential cornerstones appearing in most areas of physics and engineering—as maps of well-known continuous functions. This correspondence deftly sidesteps the use of more traditional methods to solve these difference equations. The umbral framework is discussed and illustrated here, with special attention given to umbral counterparts of the Airy, Kummer, and Whittaker equations, and to umbral maps of solitons for the Sine-Gordon, Korteweg–de Vries, and Toda systems.
1. Introduction
Robust theoretical arguments have established an anticipation of a fundamental minimum measurable length in Nature, of order LPlanckGN/c3=1.6162×1035m, the corresponding mass and time being MPlanckc/GN=2.1765×108kg and LPlanck/c=5.3911×1044s. The essence of such arguments is the following (in relativistic quantum geometrical units, wherein ħ, c, and MPlanck are all unity).
In a system or process characterized by energy E, no lengths smaller than L can be measured, where L is the larger of either the Schwarzschild horizon radius of the system (~ E) or, for energies smaller than the Planck mass, the Compton wavelength of the aggregate process (~1/E). Since the minimum of max(E, 1/E) lies at the Planck mass (E = 1), the smallest measurable distance is widely believed to be of order LPlanck. Thus, continuum laws in Nature are expected to be deformed, in principle, by modifications at that minimum length scale.
Remarkably, however, if a fundamental spacetime lattice of spacing a = O(LPLanck) is the structure that underlies conventional continuum physics, then it turns out that continuous symmetries, such as Galilei or Lorentz invariance, can actually survive unbroken under such a deformation into discreteness, in a non-local, umbral realization (10, 11, 18).
Umbral calculus, pioneered by Rota and associates in a combinatorial context (4, 16), specifies, in principle, how functions of discrete variables in infinite domains provide systematic “shadows” of their familiar continuum limit properties. By preserving Leibniz's chain rule, and by providing a discrete counterpart of the Heisenberg algebra, observables built from difference operators shadow the Lie algebras of the standard differential operators of continuum physics. [For a review relevant to physics, see (13).] Nevertheless, while the continuous symmetries and Lie algebras of umbrally deformed systems might remain identical to their continuum limit, the functions of observables themselves are modified, in general, and often drastically so.
Traditionally, the controlling continuum differential equations of physics are first discretized (2, 5, 18), and then those difference equations are solved to yield umbral deformations of the continuum solutions. But quite often, routine methods to solve such discrete equations become unwieldy, if not intractable. On the other hand, some technical difficulties may be bypassed by directly discretizing the continuum solutions. That is, through appropriate umbral deformation of the continuum solutions, the corresponding discrete difference equations may be automatically solved. However, as illustrated below for the simplest cases of oscillations and wave propagation, the resulting umbral modifications may present some subtleties when it comes to extracting the underlying physics.
In (21) the linearity of the umbral deformation functional was exploited, together with the fact that the umbral image of an exponential is also an exponential, albeit with interesting modifications, to discretize well-behaved functions occurring in solutions of physical differential equations through their Fourier expansion. This discrete shadowing of the Fourier representation functional should thus be of utility in inferring wave disturbance propagation in discrete spacetime lattices. We continue to pursue this idea here with some explicit examples. We do this in conjunction with the umbral deformation of power series, especially those for hypergeometric functions. We compare both Fourier and power series methods in some detail to gain further insight into the umbral framework.
Overall, we utilize essentially all aspects of the elegant umbral calculus to provide systematic solutions of discretized cornerstone differential equations that are ubiquitous in most areas of physics and engineering. We pay particular attention to the umbral counterparts of the Airy, Kummer, and Whittaker equations, and their solutions, and to the umbral maps of solitons for the Sine-Gordon, Korteweg–de Vries, and Toda systems.
2. Overview of the Umbral Correspondence
For simplicity, consider discrete time, t = 0, a, 2a, …, na, …. Without loss of generality, broadly following the summary review of (13), consider an umbral deformation defined by the forward difference discretization of ∂t,
and whence of the elementary oscillation Equation, (t) = −x(t), namely,
Now consider the solutions of this second-order difference equation. Of course, (2) can be easily solved directly by the textbook Fourier-component Ansatz x(t) ∝ rt, (2), to yield (1 ± ia)t/a. However, to illustrate instead the powerful systematics of umbral calculus (13, 18), we produce and study the solution in that framework.
The umbral framework considers associative chains of operators, generalizing ordinary continuum functions by ultimately acting on a translationally-invariant “vacuum” 1, after manipulations to move shift operators to the right and have them absorbed by that vacuum, which we indicate by T · 1 = 1. Using the standard Lagrange-Boole shift generator
Teat, so that Tf(t)1=f(t+a)T1=f(t+a),(3)
the umbral deformation is then
ttT1, (5)
so that [t]0 = 1, and, for n > 0, [0]n = 0. The [t]n are called “basic polynomials”1 for positive n (5, 13, 16), and they are eigenfunctions of tT−1Δ.
A linear combination of monomials (a power series representation of a function) will thus transform umbrally to the same linear combination of basic polynomials, with the same series coefficients, f(t) ↦ f(tT−1). All observables in the discretized world are thus such deformation maps of the continuum observables, and evaluation of their direct functional form is in order. Below, we will be concluding the correspondence by casually eliminating translation operators at the very end, first through operating on the vacuum and then leaving it implicit, so that F(t) ≡ f(tT−1) · 1.
The umbral deformation relies on the respective umbral entities obeying operator combinatorics identical to their continuum limit (a → 0), by virtue of obeying the same Heisenberg commutation relation (18),
Thus, e.g., by shift invariance, TΔT−1 = Δ,
[t,tn]=ntn1 [Δ,[t]nTn]=n[t]n1T1n,(8)
so that, ultimately, Δ[t]n = n[t]n − 1. For commutators of associative operators, the umbrally deformed Leibniz rule holds (10, 11),
ultimately to be dotted onto 1. Formally, the umbral deformation reflects (unitary) equivalences of the unitary irreducible representation of the Heisenberg-Weyl group, provided for by the Stone-von Neumann theorem. Here, these equivalences reflect the alternate consistent realizations of all continuum physics structures through systematic maps such as the one we have chosen. It is worth stressing that the representations of this algebraic relation on the real or complex number fields can only be infinite dimensional, that is, the lattices covered must be infinite.
Now note that, in this case the basic polynomials [t]n are just scaled falling factorials, for n ≥ 0, i.e., generalized Pochhammer symbols, which may be expressed in various ways:
[t]n(tT1)n·1=t(ta)(t(n1)a)=an(t/a)!(t/an)! =anΓ(ta+1)Γ(tan+1)=(a)nΓ(nta)Γ(ta) .(10)
Thus [−t]n = (−)n[t + a(n − 1)]n. Furthermore, [an]n = ann!; [t]m[tam]nm = [t]n for 0 ≤ mn; and for integers 0 ≤ m < n, [am]n = 0. Thus, Δm[t]n = [an]m[t]nm/am.
Negative umbral powers, by contrast, are the inverse of rising factorials, instead:
[ 1t ]n=(T1t)n1=1(t+a)(t+2a)(t+na) =an(t/a)!(t/a+n)! =anΓ(ta+1)Γ(ta+n+1)=(a)nΓ(tan)Γ(ta) .(11)
These correspond to the negative eigenvalues of tT−1Δ.
The standard umbral exponential is then natural to define as (6, 11, 16)2
E(λt,λa)eλ[t]eλtT11=n=0λnn![t]n =n=0(λa)n(t/an)=(1+λa)t/a,(12)
the compound interest formula, with the proper continuum limit (a → 0). N.B. There is always a 0 at λ = −1/a.
Evidently, since Δ · 1 = 0,
and, as already indicated, one could have solved this equation directly3 to produce the above Et, λa).
Serviceably, the umbral exponential E happens to be an ordinary exponential,
and it actually serves as the generating function of the umbral basic polynomials,
Conversely, then, this construction may be reversed, by first solving directly for the umbral eigenfunction of Δ, and effectively defining the umbral basic polynomials through the above parametric derivatives, in situations where these might be more involved, as in the next section.
As a consequence of linearity, the umbral deformation of a power series representation of a function is given formally by
This may not always be easy to evaluate, but, in fact, the same argument may be applied to linear combinations of exponentials, and hence the entire Fourier representation functional, to obtain
F(t)=dτ f(τ)dω2πeiωτ(1+iωa)t/a =(1+aτ)t/af(τ)|τ=0 .(17)
The rightmost equation follows by converting iω into ∂τ derivatives and integrating by parts away from the resulting delta function. Naturally, it identifies with Equation (16) by the (Fourier) identity f(∂x)g(x)|x = 0 = g(∂x)f(x)|x = 0. It is up to individual ingenuity to utilize the form best suited to the particular application at hand.
It is also straightforward to check that this umbral transform functional yields
tfΔF ,(18)
and to evaluate the umbral transform of the Dirac delta function, which amounts to a cardinal sine or sampling function,
or to evaluate umbral transforms of rational functions, such as
to obtain an incomplete Gamma function (1), and so on. Note how the last of these is distinctly, if subtly, different from the umbral transform of negative powers, as given in (11).
In practical applications, evaluation of umbral transforms of arbitrary functions of observables may be more direct, at the level of solutions, through this deforming functional, Equation (17). For example, one may evaluate in this way the umbral correspondents of trigonometric functions,
Sin[t]ei[t]ei[t]2i, Cos[t]ei[t]+ei[t]2,(21)
so that
ΔSin[t]=Cos[t], ΔCos[t]=Sin[t].(22)
As an illustration, consider phase-space rotations of the oscillator. The umbral deformation of phase-space rotations,
x˙=p, p˙=x ΔX(t)=P(t), ΔP(t)=X(t),(23)
readily yields, by directly deforming continuum solutions, the oscillatory solutions,
In view of (14), and also
the umbral sines and cosines in (24) are seen to amount to discrete phase-space spirals,
with a frequency decreased from the continuum value (i.e., 1) to
So the frequency has become, effectively, the inverse of the cardinal tangent function.4 Note that the umbrally conserved quantity is,
such that Δ=0, with the proper energy as the continuum limit.
3. Reduction from Second-Order Differences to Single Term Recursions
In this section and the following, to conform to prevalent conventions, the umbral variable will be denoted by x, instead of t. In this case there is a natural way to think of the umbral correspondence that draws on familiar quantum mechanics language (16): The discrete difference equations begin as operator statements, for operator xs and Ts, but are then reduced to equations involving classical-valued functions just by taking the matrix element 〈x|…|vac〉 where |vac〉 is translationally invariant. The overall x-independent non-zero constant 〈x|vac〉 is then ignored.
To be specific, consider Whittaker's equation (1) for μ = 1/2,
This umbrally maps to the operator statement
Considering either y(xT−1)·1 ≡ Y(x), or else 〈x|y(xT−1)|vac〉 = Y(x) 〈x|vac〉, this operator statement reduces to a classical difference equation,
Before using umbral mapping to convert continuous solutions of (29) into discrete solutions (14, 15) of (31), here we note a simplification of the latter equation upon choosing a = 2, which amounts to setting the scale of x. With this choice (31) collapses to a mere one-term recursion. Shifting xx − 2 this is
Despite being a first-order difference equation, however, the solutions of this equation still involve two independent “constants of summation” even for x restricted to only integer values, because the choice a = 2 has decoupled adjacent vertical strips of unit width on the complex x plane. To be explicit, for integer x > 0, forward iteration gives (2)
Y(2k+1)=2k(j=1kj2kj)Y(1) andY(2k+2)=2k(j=1kjkj)Y(2) for integer k0,(33)
with Y(1) and Y(2) the two independent constants that determine values of Y for all larger odd and even integer points, respectively.
Or, if generic x is contemplated, the Equation (32) has elementary solutions, for arbitrary complex constants C1 and C2, given by
Y(x)=2x/2Γ(x2κ)Γ(x2)C1+(2)x/2Γ(x2)Γ(1x2+κ) C2(34)
In the second expression, we have used Γ(z)Γ(1 − z) = π/sin πz. Note the C2 part of this elementary solution differs from the C1 part just through multiplication by a particular complex function with period 2. This is typical of solutions to difference equations since any such periodic factors are transparent to Δ, as mentioned in an earlier footnote (12).
As expected, even for generic x the constants C1 and C2 may be determined given Y(x) at two judiciously chosen points, not necessarily differing by an integer. For example, if 0 < κ < 1,
C1=Γ(1+κ)21+κY(2+2κ) , C2=πsinπκC112Γ(κ)Y(2).(36)
Moreover, poles and zeros of the solution are manifest either from the Γ functions in (34), or else from continued product representations such as (33). For the latter, either forward or backward iterations of the first-order difference Equation (32) may be used. Schematically,
or alternatively,
Although both terms in (34) have zeroes, the C1 term also has poles while the C2 term has none—it is an entire function of x—and it is complex for any nonzero choice of C2. Of course, since the Equation (32) is linear, real and imaginary parts may be taken as separate real solutions. All this is evident in the following plots for various selected integer κ.
2x/2Γ(12xκ)Γ(12x) for κ = 1, 2, and 3 in red, blue, and green.
2x/2cosπ(12x)Γ(12x)Γ(112x+κ) for κ = 1, 2, and 3 in red, blue, and green.
Collapse to a mere one-term recursion also occurs for an inverse-square potential,
For μa2 = 1, which amounts to setting the scale of the energy of the solution, the umbral version of this equation reduces to
Y(x)=12(1+κa2x(x+a))Y(x+a) =12(1+aκxaκa+x)Y(x+a).(40)
That is to say,
Y(x+a)=2(1+xa)xa(xa+1+14κ2)(xa+114κ2)Y(x) .(41)
Elementary solutions for generic x, for arbitrary complex constants C1 and C2, are given by
Again, the C2 part of this elementary solution differs from the C1 part just through multiplication by a particular complex function with period a. And again, poles and zeros of these and other solutions are manifest either from those of the Γ functions, or else from a continued product form, e.g.
It is not surprising that (29) and (39) share the privilege to become only first-order difference equations for specific choices of a, as in (32) and (41), because they are both special cases of Whittaker's differential equation, as discussed in the next section. No other linear second-order ODEs lead to umbral equations with this property.
4. Discretization Through Hypergeometric Recursion
In this section we discuss several examples using umbral transform methods to convert solutions of continuum differential equations directly into solutions of the corresponding discretized equations. We use both Fourier and power series umbral transforms.
As an explicit illustration of the umbral transform functional (17), inserting the Fourier representation of the Airy function (1) yields
AiryAi(x) UmAiryAi(x,a)Re(1π0+e13ik3(1+ika)xadk).(45)
This integral is expressed in terms of hypergeometric functions and evaluated numerically in Appendix A.
Likewise, gaussians also map to hypergeometric functions, as may be obtained by formal series manipulations:
ex2G(x,a) n=0()n[x]2nn! =n=01n!(1)na2nΓ(xa+1)Γ(xa2n+1)(46)
where the reflection and duplication formulas were used to write
While the series (47) actually has zero radius of convergence, it is Borel summable, and the resulting regularized hypergeometric function is well-defined. See Appendix B for some related numerics.
For another example drawn from the familiar repertoire of continuum physics, consider the confluent hypergeometric equation of Kummer (A&S 13.1.1):
x y+(βx)yαy=0,(50)
whose regular solution at x = 0, expressed in various dialects, is
with series and integral representations
1F1(α;β;x)=n=0Γ(α+n)Γ(α)Γ(β)Γ(β+n)xnn! =Γ(β)Γ(α)Γ(βα)01exssα1(1s)βα1ds =1+αβx+12α(α+1)β(β+1)x2+16α(α+1)(α+2)β(β+1)(β+2)x3+O(x4).(52)
The second, independent solution of (50), with branch point at x = 0, is given by Tricomi's confluent hypergeometric function (1), sometimes known as HypergeometricU:
U(α,β,x)=πsinπβ (M(α,β,x)Γ(1+αβ)Γ(β)x1βM(1+αβ,2β,x)Γ(α)Γ(2β)).(53)
Invoking the umbral calculus for x, either of these confluent hypergeometric functions can be mapped onto their umbral counterparts using
1F1(α;β;x) 2F1(α,xa;β;a),(54)
where 2F1 is the well-known Gauss hypergeometric function (1). This map from 1F1 to 2F1 follows from the basic monomial umbral map,
and from the series (52). When combined, these give the well-known series representation of 2F1.
Next, reconsider the one-dimensional Coulomb problem defined by Whittaker's equation for general μ (1):
Since κ and μ are both arbitrary, this also encompasses the inverse-square potential, (39). Exact solutions of this differential equation are
whittakerW(κ,μ,x)=xμ+1/2ex/2 (Γ(2μ)Γ(μκ+12) 1F1(μκ+12;2μ+1;x)+Γ(2μ)Γ(μκ+12)x2μ1F1(μκ+12;2μ+1;x)).(59)
Umbral versions of these solutions are complicated by the exponential and overall power factors in the classical relations between the 1F1's and the Whittaker functions, but this complication is manageable. (In part this is because in the umbral calculus there are no ordering ambiguities (20).)
To obtain the umbral version of the Whittaker functions, we begin by evaluating
e12xT11F1(α;β;xT1)·1=m=0n=0(12)mΓ(α+n)Γ(α)Γ(β+n)Γ(β)[ x ]m+nm!n! =(1a2)xa 2F1(α,xa;β;2aa2),(60)
where we have performed the sum over m first, to obtain
The sum over n then gives the Gauss hypergeometric function in (60).
Next, to deal with the umbral deformations of the Whittaker functions, we need to use the continuation of (10) and (11) to an arbitrary power of xT−1, namely,
This continuation leads to the following:
(xT1)γe12xT11F1(α;β;xT1)·1 =aγΓ(xa+1)Γ(xaγ+1)Tγe12xT11F1(α;β;xT1)·1 =aγΓ(xa+1)Γ(xaγ+1)e12(xγa)T11F1(α;β;(xγa)T1)·1.(63)
Thus we obtain the umbral map
xγe12x1F1(α;β;x)Γ(xa+1)Γ(xaγ+1) aγ(1a2)xaγ 2F1(α,γxa;β;2aa2).(64)
Finally then, specializing to the relevant α, β, and γ, we find the umbral Whittaker functions. In particular,
whittakerM(κ,μ,x)Γ(xa+1)Γ(xaμ+12)aμ+1/2(1a2)xaμ12 2F1(μ+12κ,μ+12xa;2μ+1;2aa2).(65)
This result for general a exhibits what is special about the choice a = 2, as exploited in the previous section. To realize that choice from (65) requires taking a limit a ↗ 2, hence it requires the asymptotic behavior of the Gauss hypergeometric function (1):
Now with sufficient care, a = 2 solutions can be coaxed from the umbral version of whittakerM in (65), and/or the corresponding umbral counterpart of whittakerW, upon taking lima↗2 and making use of (66). Moreover, in principle the umbral correspondents of both Whittaker functions could be used to obtain from this limit a solution with two arbitrary constants.
On the other hand, for a = 2, the umbral equation corresponding to (56) again reduces to a one-term recursion, namely,
For generic x, solutions for arbitrary complex constants C1 and C2 are then given by
=2x/2Γ(1+x2)Γ(x2κ)Γ(x2+12+μ)Γ(x2+12μ)(C1+1π2 C2sin(πx2)sinπ(x2κ)),(69)
which agrees with (34) when μ = 1/2, of course. As in that previous special case, the C2 part of (68) differs from the C1 part just through multiplication by a particular complex function with period 2 (12).
We graph some examples to show the differences between the Whittaker functions and their umbral counterparts, for a = 1.
whittakerM(κ, 1/2, x) for κ = 1, 2, and 3 in red, blue, and green.
Umbral whittakerM(κ, 1/2, x) for a = 1, and for κ = 1, 2, and 3 in red, blue, and green.
The examples above are specific illustrations of combinatorics that may be summarized in a few umbral hypergeometric mapping lemmata, the simplest being
Lemma 1:
pFq(α1,,αp;β1,,βq;x)p+1Fq(α1,,αp,x/a;β1,,βq;a) ,(70)
where the series representation of the generalized hypergeometric function pFq is5
A proof of (70) follows from formal manipulations of these series.
The umbral version of a more general class of functions is obtained by replacing xxT−1 in functions of xk for some fixed positive integer k. Thus, again for hypergeometric functions, we have
Lemma 2:
pFq(α1,,αp;β1,,βq;xk)p+kFq( α1,,αp,1k(xa),1k(1xa),, 1k(k1xa);β1,,βq;(ak)k ).(72)
And again, a proof follows from formal series expansions.
Multiplication by exponentials produces only minor modifications of these general results, as was discussed above in the context of Whittaker functions, namely,
Lemma 3:
eλxpFq(α1,,αp;β1,,βq;xk)(1+aλ)xap+kFq( α1,,αp,1k(xa),1k(1xa),, 1k(k1xa);β1,,βq;(ak1+aλ)k ).(73)
In addition, multiplication by an overall power of x gives
Lemma 4:
xγeλxpFq(α1,,αp;β1,,βq;xk)Γ(xa+1)aγ(1+aλ)xaγΓ(xaγ+1) p+kFq( α1,,αp,γxak,1+γxak,,k1+γxak; β1,,βq;(ak1+aλ)k ).(74)
5. Wave Propagation
Given the umbral features of discrete time and space equations discussed above, separately, it is natural to combine the two.
For example, the umbral version of simple plane waves in 1+1 spacetime would obey an equation of the type (6, 11, 12),
(Δx2Δt2) F=0 ,(75)
on a time-lattice with spacing a and a space-lattice with spacing b, not necessarily such that b = a in all spacetime regions. For generic frequency, wavenumber and velocity, the basic solutions are
For right-moving waves, say, these have phase velocity
v(ω,k)=ωk aarcsin(b)barcsin(a).(77)
Thus, the effective index of refraction in the discrete medium is (b arcsin(a))/(a arcsin(b)), i.e., modified from 1. Small inhomogeneities of a and b in the fabric of spacetime over large regions could therefore yield interesting effects.
Technically, a more challenging application of umbral methods involves nonlinear, solitonic phenomena (21), such as the one-soliton solution of the continuum Sine-Gordon equation,
(x2t2)f(x,t)=sin(f(x,t)) , fSG(x,t)=4arctan(mexvt1v2).(78)
The corresponding umbral deformation of the PDE itself would now also involve a deformed potential sin(f(xT−1x, tT−1t))·1. But rather than tackling this difficult nonlinear difference equation, one may instead use the umbral transform (17) to infer that fSG(x, t) maps to
The continuum Korteweg–de Vries soliton is likewise mapped:
Closed-form evaluations of these Fourier integrals are not available, but the physical effects of the discretization could be investigated numerically, and compared to the Lax pair integrability machinery of (13), or to the results on a variety of discrete KdVs in (17), or to other studies (8, 9).
However, a more accessible example of umbral effects on solitons may be found in the original Toda lattice model (19). For this model the spatial variable is already discrete, usually with spacing b = 1 so x = n is an integer, while the time t is continuous. The equations of motion in that case are
for integer n. Though x = n is discrete, nevertheless there are exact multi-soliton solutions valid for all continuous t, as is well-known.
Specific one-soliton Toda solutions are given for constant α, β, γ, and q0 by
provided that
So the soliton's velocity is just v=±2βsinh(β2).
While obtained only for discrete x = n, for plotting purposes q(n, t) may be interpolated for any x (see graph below). To carry out the complete umbral deformation of this system, it is then only necessary to discretize t in the equations of motion (81). Consider what effects this approach to discrete time has on the specified one-soliton solutions.
To that end, expand the exact solutions in (82) as series,
Upon umbralizing t, the one-soliton solutions then map as
and these are guaranteed to give solutions to the umbral operator equations of motion,
Δp(n,tT1)1a(T1)p(n,tT1) =(e(q(n+1,tT1)q(n,tT1))e(q(n,tT1)q(n1,tT1))),(88)
upon projecting onto a translationally invariant “vacuum” (i.e., Q(n, t) ≡ q(n, tT−1) · 1).
Now, for integer time steps, t/a = m, consider the series at hand:
S(m,c,z)=k=1zkk(ekβ1)(1+ck)m =ln(1z1zeβ)+j=1mcj(mj)R(j,z),(89)
where c = γa, z = −αe−βn, and where for j > 0,
R(j,z)=k=0(ekβ1)zkkj1 Φ(eβz,1j,0)Φ(z,1j,0).(90)
Fortunately, for positive integer t/a, we only need the Lerch transcendent function,
for those cases where the sums are expressible as elementary functions. For example,
The ln(…) term on the RHS of (89) then reproduces the specified classical one-soliton solutions at t = 0, while the remaining terms give umbral modifications for t ≠ 0.
Altogether then, we have
These umbral results are compared to some time-continuum soliton profiles for t/a = 0, 1, 2, 3, and 4 in the following Figure (with q0 = 0, α = 1 = β, and γ = 2 sinh(1/2) = 1.042).
Toda soliton profiles q interpolated for all x ∈ [−5, 5] at integer time slices superimposed with their time umbral maps Q (thicker curves) for a = 1.
Thus, the umbral-mapped solutions no longer evolve just by translating the profile shape. Rather, they develop oscillations about the classical fronts that dramatically increase with time, that evince not only dispersion but also generation of harmonics, and that, strictly speaking, disqualify use of the term soliton for their description. Be that as it may, this model is referred to in some studies as integrable (8, 9).
These umbral effects on wave propagation evoke scattering and diffraction by crystals. But here the “crystal” is spacetime itself. It is tempting to speculate based on this analogy. In particular, were a well-formed wave packet to pass through a localized region of crystalline spacetime, with sufficiently large lattice spacings, the packet could undergo dramatic deformations in shape, wavelength, and frequency—far greater than and very different from what would be expected just from the dispersion of a free packet propagating through continuous space and time.
6. Concluding Remarks
We have emphasized how the umbral calculus has visibly emerged to provide an elegant correspondence framework that automatically gives solutions of ubiquitous difference equations as maps of well-known continuous functions. This correspondence systematically sidesteps the use of more traditional methods to solve these difference equations.
We have used the umbral calculus framework to provide solutions to discretized versions of several differential equations that are widespread building-blocks in many if not all areas of physics and engineering, thereby avoiding the rather unwieldy frontal assaults often engaged to solve such discrete equations directly.
We have paid special attention to the Airy, Kummer, and Whittaker equations, and illustrated several basic principles that transform their continuum solutions to umbral versions through the use of hypergeometric function maps. The continuum limits thereof are then manifest.
Finally, we have applied the solution-mapping technique to single solitons of the Sine-Gordon, Korteweg–de Vries, and Toda systems, and we have noted how their umbral counterparts—particular solutions of corresponding discretized equations—evince dispersion and other non-solitonic behavior, in general. Such corrections to the continuum result may end up revealing discrete spacetime structure in astrophysical wave propagation settings.
We expect to witness several applications of the framework discussed and illustrated here.
Conflict of Interest Statement
This work was supported in part by NSF Award PHY-1214521; and in part, the submitted manuscript has been created by UChicago Argonne, LLC, Operator of Argonne National Laboratory. Argonne, a U.S. Department of Energy Office of Science laboratory, is operated under Contract No. DE-AC02-06CH11357. The U.S. Government retains for itself, and others acting on its behalf, a paid-up nonexclusive, irrevocable worldwide license in said article to reproduce, prepare derivative works, distribute copies to the public, and perform publicly and display publicly, by or on behalf of the Government. Thomas L. Curtright was also supported in part by a University of Miami Cooper Fellowship.
1. ^We stress that the notation [t]n is shorthand for the product t(ta)…(t − (n − 1)a). It is not just the nth power of [t] = t.
2. ^Again we stress that eλ[t] is a short-hand notation, and not just the usual exponential of λ[t] = λt.
3. ^N.B. There is an infinity of “non-umbral” extensions of the Et, λa) solution (12): Multiplying the umbral exponential by an arbitrary periodic function g(t + a) = g(t) will pass undetected through Δ, and thus will also yield an eigenfunction of Δ. Often, such extra solutions have either a vanishing continuum limit, or else an ill-defined one.
4. ^That is, for Θ ≡ arctan(a), the spacing of the zeros, period, etc, are scaled up by a factor of tanc(Θ)tan(Θ)Θ1. For complete periodicity on the time lattice, one further needs return to the origin in an integral number of N steps, thus a solution of N = 2πn/arctan a. Example: For a = 1, the solutions' radius spirals out as 2t/2, while ω = π/4, and the period is τ = 8.
5. ^Recall results from using the ratio test to determine the radius of convergence for the pFq1, …, αp1,…,βq;x) series:
If p < q + 1 then the ratio of coefficients tends to zero. This implies that the series converges for any finite value of x.
If p = q + 1 then the ratio of coefficients tends to one, hence the series converges for |x| < 1 and diverges for |x| > 1.
If p > q + 1 then the ratio of coefficients grows without bound. The series is then divergent or asymptotic, and is a symbolic shorthand for the solution to a differential equation.
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Appendix A: Umbral Airy Functions
Formally, these can be obtained by expressing the Airy functions in terms of hypergeometric functions and then umbral mapping the series. The continuum problem is given by
yxy=0, y(x)=C1AiryAi(x)+C2AiryBi(x),(94)
AiryAi(x)=132/3Γ(2/3) 0F1(;23;19x3)131/3Γ(1/3) 0F1(;43;19x3),(95)
AiryBi(x)=131/6Γ(2/3) 0F1(;23;19x3)+31/6zΓ(1/3) 0F1(;43;19x3).(96)
The yY umbral images of these, solving the umbral discrete difference equation (3, 12)
are then given by (72) for k = 3. In particular,
UmAiryAi(x,a)=132/3Γ(2/3) 3F1(13xa,13(1xa),13(2xa);23;3a3) 131/3Γ(1/3) 3F1(13xa,13(1xa),13(2xa);43;3a3).(98)
Since the number of “numerator parameters” in the hypergeometric function 3F1 exceeds the number of “denominator parameters” by 2, the series expansion is at best asymptotic. However, the series is Borel summable. In this respect, the situation is the same as for the umbral gaussian (see Appendix B).
Alternatively, as previously mentioned in the text, using the familiar integral representation of AiryAi(x), the umbral map devolves to that of an exponential. That is to say,
AiryAi(xT1)=12π+exp(13is3+isxT1)ds (99)
Just as AiryAi(x) is a real function for real x, UmAiryAi(x, a) is a real function for real x and a,
After some hand-crafting, the final result may be expressed in terms of just three 2F2 generalized hypergeometric functions. To wit,
where the hypergeometric functions 2F2 (a, b; c, d; z) appear in the expression as
H1(w,z)=Γ(13z)Γ(13+13z) 2F2(13z,13+13z;13,23;13w3),(103)
H2(w,z)=Γ(13+13z)Γ(23+13z) 2F2(13+13z,23+13z;23,43;13w3),(104)
H3(w,z)=Γ(23+13z)Γ(1+13z) 2F2(23+13z,1+13z;43,53;13w3),(105)
and where the coefficients in (102) are
C3(w,z)=336cos(12πz12πzsignum(w))636cos(16πz+12πzsignum(w)) +636cos(56πz+12πzsignum(w))+3×323sin(12πz12πzsignum(w)) 336cos(12πz12πzsignum(w))+2×323sin(16πz+12zπsignum(w)) +323sin(12πz+12πzsignum(w))+2×323sin(56πz+12πzsignum(w)).(109)
While the coefficient functions C0−3 are not pretty, they are comprised of elementary functions, and they are nonsingular functions of z. On the other hand, the hypergeometric functions do have singularities and discontinuities for negative z. However, the net result for UmAiryAi is reasonably well-behaved.
We plot UmAiryAi(x, a) for a=0, ±14, ±12, and ±1.
UmAiryAi(x, a) for a = ±1, ±1/2, and ±1/4 (red, blue, & green dashed/solid curves, resp.) compared to AiryAi(x) = UmAiryAi(x, 0) (black curve).
Appendix B: Umbral Gaussians
As discussed in the text, straightforward discretization of the series yields the umbral gaussian map:
(NB G(x, a) ≠ G(−x, a).) Now, it is clear that term by term the series (110) reduces back to the continuum gaussian as a → 0. Nonetheless, since the series is asymptotic and not convergent for |a| > 0, it is interesting to see how this limit is obtained from other representations of the hypergeometric function in (111), in particular from using readily available numerical routines to evaluate 2F0 for specific small values of a. Some examples are shown here.
G(x, 1/2n) vs. x ∈ [−3,2], for n = 1, 2, and 3, in red, blue, and green, respectively, compared to G(x, 0) = exp(−x2), in black.
Mathematicaő code is available online to produce similar graphs, for those interested. It is amusing that Mathematica manipulates the Borel regularized sum to render the 2F0 in question in terms of Tricomi's confluent hypergeometric function U, as discussed above in the context of Kummer's Equation, cf. (53). Thus G can also be expressed in terms of 1F1s. The relevant identities are:
Keywords: umbral correspondence, discretization, difference equations, umbral transform, hypergeometric functions
Citation: Curtright TL and Zachos CK (2013) Umbral Vade Mecum. Front. Physics 1:15. doi: 10.3389/fphy.2013.00015
Received: 27 June 2013; Paper pending published: 04 September 2013;
Accepted: 10 September 2013; Published online: 01 October 2013.
Edited by:
Manuel Asorey, Universidad de Zaragoza, Spain
Reviewed by:
Apostolos Vourdas, University of Bradford, UK
An Huang, Harvard University, USA
Mariano A. Del Olmo, Universidad de Valladolid, Spain
*Correspondence: Cosmas K. Zachos, High Energy Physics Division 362, Argonne National Laboratory, Argonne, IL 60439-4815, USA e-mail: |
34fd3042d31a077c | Physicists Want to Rebuild Quantum Theory From Scratch
Physicists Want to Rebuild Quantum Theory From Scratch
Ulises Farinas for Quanta Magazine
Physicists Want to Rebuild Quantum Theory From Scratch
Ulises Farinas for Quanta Magazine
Scientists have been using quantum theory for almost a century now, but embarrassingly they still don’t know what it means. An informal poll taken at a 2011 conference on Quantum Physics and the Nature of Reality showed that there’s still no consensus on what quantum theory says about reality—the participants remained deeply divided about how the theory should be interpreted.
Quanta Magazine
author photo
Some physicists just shrug and say we have to live with the fact that quantum mechanics is weird. So particles can be in two places at once, or communicate instantaneously over vast distances? Get over it. After all, the theory works fine. If you want to calculate what experiments will reveal about subatomic particles, atoms, molecules and light, then quantum mechanics succeeds brilliantly.
If these efforts succeed, it’s possible that all the apparent oddness and confusion of quantum mechanics will melt away, and we will finally grasp what the theory has been trying to tell us. “For me, the ultimate goal is to prove that quantum theory is the only theory where our imperfect experiences allow us to build an ideal picture of the world,” said Giulio Chiribella, a theoretical physicist at the University of Hong Kong.
There’s no guarantee of success—no assurance that quantum mechanics really does have something plain and simple at its heart, rather than the abstruse collection of mathematical concepts used today. But even if quantum reconstruction efforts don’t pan out, they might point the way to an equally tantalizing goal: getting beyond quantum mechanics itself to a still deeper theory. “I think it might help us move towards a theory of quantum gravity,” said Lucien Hardy, a theoretical physicist at the Perimeter Institute for Theoretical Physics in Waterloo, Canada.
The Flimsy Foundations of Quantum Mechanics
The basic premise of the quantum reconstruction game is summed up by the joke about the driver who, lost in rural Ireland, asks a passer-by how to get to Dublin. “I wouldn’t start from here,” comes the reply.
Where, in quantum mechanics, is “here”? The theory arose out of attempts to understand how atoms and molecules interact with light and other radiation, phenomena that classical physics couldn’t explain. Quantum theory was empirically motivated, and its rules were simply ones that seemed to fit what was observed. It uses mathematical formulas that, while tried and trusted, were essentially pulled out of a hat by the pioneers of the theory in the early 20th century.
Take Erwin Schrödinger’s equation for calculating the probabilistic properties of quantum particles. The particle is described by a “wave function” that encodes all we can know about it. It’s basically a wavelike mathematical expression, reflecting the well-known fact that quantum particles can sometimes seem to behave like waves. Want to know the probability that the particle will be observed in a particular place? Just calculate the square of the wave function (or, to be exact, a slightly more complicated mathematical term), and from that you can deduce how likely you are to detect the particle there. The probability of measuring some of its other observable properties can be found by, crudely speaking, applying a mathematical function called an operator to the wave function.
I think quantum theory as we know it will not stand. Alexei Grinbaum
But this so-called rule for calculating probabilities was really just an intuitive guess by the German physicist Max Born. So was Schrödinger’s equation itself. Neither was supported by rigorous derivation. Quantum mechanics seems largely built of arbitrary rules like this, some of them—such as the mathematical properties of operators that correspond to observable properties of the system—rather arcane. It’s a complex framework, but it’s also an ad hoc patchwork, lacking any obvious physical interpretation or justification.
Compare this with the ground rules, or axioms, of Einstein’s theory of special relativity, which was as revolutionary in its way as quantum mechanics. (Einstein launched them both, rather miraculously, in 1905.) Before Einstein, there was an untidy collection of equations to describe how light behaves from the point of view of a moving observer. Einstein dispelled the mathematical fog with two simple and intuitive principles: that the speed of light is constant, and that the laws of physics are the same for two observers moving at constant speed relative to one another. Grant these basic principles, and the rest of the theory follows. Not only are the axioms simple, but we can see at once what they mean in physical terms.
What are the analogous statements for quantum mechanics? The eminent physicist John Wheeler once asserted that if we really understood the central point of quantum theory, we would be able to state it in one simple sentence that anyone could understand. If such a statement exists, some quantum reconstructionists suspect that we’ll find it only by rebuilding quantum theory from scratch: by tearing up the work of Bohr, Heisenberg and Schrödinger and starting again.
Quantum Roulette
One of the first efforts at quantum reconstruction was made in 2001 by Hardy, then at the University of Oxford. He ignored everything that we typically associate with quantum mechanics, such as quantum jumps, wave-particle duality and uncertainty. Instead, Hardy focused on probability: specifically, the probabilities that relate the possible states of a system with the chance of observing each state in a measurement. Hardy found that these bare bones were enough to get all that familiar quantum stuff back again.
Hardy assumed that any system can be described by some list of properties and their possible values. For example, in the case of a tossed coin, the salient values might be whether it comes up heads or tails. Then he considered the possibilities for measuring those values definitively in a single observation. You might think any distinct state of any system can always be reliably distinguished (at least in principle) by a measurement or observation. And that’s true for objects in classical physics.
Lucien Hardy, a physicist at the Perimeter Institute, was one of the first to derive the rules of quantum mechanics from simple principles.
Gabriela Secara, Perimeter Institute for Theoretical Physics
In quantum mechanics, however, a particle can exist not just in distinct states, like the heads and tails of a coin, but in a so-called superposition—roughly speaking, a combination of those states. In other words, a quantum bit, or qubit, can be not just in the binary state of 0 or 1, but in a superposition of the two.
But if you make a measurement of that qubit, you’ll only ever get a result of 1 or 0. That is the mystery of quantum mechanics, often referred to as the collapse of the wave function: Measurements elicit only one of the possible outcomes. To put it another way, a quantum object commonly has more options for measurements encoded in the wave function than can be seen in practice.
Hardy’s rules governing possible states and their relationship to measurement outcomes acknowledged this property of quantum bits. In essence the rules were (probabilistic) ones about how systems can carry information and how they can be combined and interconverted.
Hardy then showed that the simplest possible theory to describe such systems is quantum mechanics, with all its characteristic phenomena such as wavelike interference and entanglement, in which the properties of different objects become interdependent. “Hardy’s 2001 paper was the ‘Yes, we can!’ moment of the reconstruction program,” Chiribella said. “It told us that in some way or another we can get to a reconstruction of quantum theory.”
More specifically, it implied that the core trait of quantum theory is that it is inherently probabilistic. “Quantum theory can be seen as a generalized probability theory, an abstract thing that can be studied detached from its application to physics,” Chiribella said. This approach doesn’t address any underlying physics at all, but just considers how outputs are related to inputs: what we can measure given how a state is prepared (a so-called operational perspective). “What the physical system is is not specified and plays no role in the results,” Chiribella said. These generalized probability theories are “pure syntax,” he added — they relate states and measurements, just as linguistic syntax relates categories of words, without regard to what the words mean. In other words, Chiribella explained, generalized probability theories “are the syntax of physical theories, once we strip them of the semantics.”
Shouldn’t this shock anyone who thinks of quantum theory as an expression of properties of nature? Adán Cabello
The general idea for all approaches in quantum reconstruction, then, is to start by listing the probabilities that a user of the theory assigns to each of the possible outcomes of all the measurements the user can perform on a system. That list is the “state of the system.” The only other ingredients are the ways in which states can be transformed into one another, and the probability of the outputs given certain inputs. This operational approach to reconstruction “doesn’t assume space-time or causality or anything, only a distinction between these two types of data,” said Alexei Grinbaum, a philosopher of physics at the CEA Saclay in France.
To distinguish quantum theory from a generalized probability theory, you need specific kinds of constraints on the probabilities and possible outcomes of measurement. But those constraints aren’t unique. So lots of possible theories of probability look quantum-like. How then do you pick out the right one?
“We can look for probabilistic theories that are similar to quantum theory but differ in specific aspects,” said Matthias Kleinmann, a theoretical physicist at the University of the Basque Country in Bilbao, Spain. If you can then find postulates that select quantum mechanics specifically, he explained, you can “drop or weaken some of them and work out mathematically what other theories appear as solutions.” Such exploration of what lies beyond quantum mechanics is not just academic doodling, for it’s possible—indeed, likely—that quantum mechanics is itself just an approximation of a deeper theory. That theory might emerge, as quantum theory did from classical physics, from violations in quantum theory that appear if we push it hard enough.
Bits and Pieces
Some researchers suspect that ultimately the axioms of a quantum reconstruction will be about information: what can and can’t be done with it. One such derivation of quantum theory based on axioms about information was proposed in 2010 by Chiribella, then working at the Perimeter Institute, and his collaborators Giacomo Mauro D’Ariano and Paolo Perinotti of the University of Pavia in Italy. “Loosely speaking,” explained Jacques Pienaar, a theoretical physicist at the University of Vienna, “their principles state that information should be localized in space and time, that systems should be able to encode information about each other, and that every process should in principle be reversible, so that information is conserved.” (In irreversible processes, by contrast, information is typically lost—just as it is when you erase a file on your hard drive.)
What’s more, said Pienaar, these axioms can all be explained using ordinary language. “They all pertain directly to the elements of human experience, namely, what real experimenters ought to be able to do with the systems in their laboratories,” he said. “And they all seem quite reasonable, so that it is easy to accept their truth.” Chiribella and his colleagues showed that a system governed by these rules shows all the familiar quantum behaviors, such as superposition and entanglement.
Giulio Chiribella, a physicist at the University of Hong Kong, reconstructed quantum theory from ideas in information theory.
Courtesy of CIFAR
One challenge is to decide what should be designated an axiom and what physicists should try to derive from the axioms. Take the quantum no-cloning rule, which is another of the principles that naturally arises from Chiribella’s reconstruction. One of the deep findings of modern quantum theory, this principle states that it is impossible to make a duplicate of an arbitrary, unknown quantum state.
It sounds like a technicality (albeit a highly inconvenient one for scientists and mathematicians seeking to design quantum computers). But in an effort in 2002 to derive quantum mechanics from rules about what is permitted with quantum information, Jeffrey Bub of the University of Maryland and his colleagues Rob Clifton of the University of Pittsburgh and Hans Halvorson of Princeton University made no-cloning one of three fundamental axioms. One of the others was a straightforward consequence of special relativity: You can’t transmit information between two objects more quickly than the speed of light by making a measurement on one of the objects. The third axiom was harder to state, but it also crops up as a constraint on quantum information technology. In essence, it limits how securely a bit of information can be exchanged without being tampered with: The rule is a prohibition on what is called “unconditionally secure bit commitment.”
These axioms seem to relate to the practicalities of managing quantum information. But if we consider them instead to be fundamental, and if we additionally assume that the algebra of quantum theory has a property called non-commutation, meaning that the order in which you do calculations matters (in contrast to the multiplication of two numbers, which can be done in any order), Clifton, Bub and Halvorson have shown that these rules too give rise to superposition, entanglement, uncertainty, nonlocality and so on: the core phenomena of quantum theory.
Another information-focused reconstruction was suggested in 2009 by Borivoje Dakić and Časlav Brukner, physicists at the University of Vienna. They proposed three “reasonable axioms” having to do with information capacity: that the most elementary component of all systems can carry no more than one bit of information, that the state of a composite system made up of subsystems is completely determined by measurements on its subsystems, and that you can convert any “pure” state to another and back again (like flipping a coin between heads and tails).
Dakić and Brukner showed that these assumptions lead inevitably to classical and quantum-style probability, and to no other kinds. What’s more, if you modify axiom three to say that states get converted continuously—little by little, rather than in one big jump—you get only quantum theory, not classical. (Yes, it really is that way round, contrary to what the “quantum jump” idea would have you expect—you can interconvert states of quantum spins by rotating their orientation smoothly, but you can’t gradually convert a classical heads to a tails.) “If we don’t have continuity, then we don’t have quantum theory,” Grinbaum said.
Christopher Fuchs, a physicist at the University of Massachusetts, Boston, argues that quantum theory describes rules for updating an observer’s personal beliefs.
Katherine Taylor for Quanta Magazine
A further approach in the spirit of quantum reconstruction is called quantum Bayesianism, or QBism. Devised by Carlton Caves, Christopher Fuchs and Rüdiger Schack in the early 2000s, it takes the provocative position that the mathematical machinery of quantum mechanics has nothing to do with the way the world really is; rather, it is just the appropriate framework that lets us develop expectations and beliefs about the outcomes of our interventions. It takes its cue from the Bayesian approach to classical probability developed in the 18th century, in which probabilities stem from personal beliefs rather than observed frequencies. In QBism, quantum probabilities calculated by the Born rule don’t tell us what we’ll measure, but only what we should rationally expect to measure.
In this view, the world isn’t bound by rules—or at least, not by quantum rules. Indeed, there may be no fundamental laws governing the way particles interact; instead, laws emerge at the scale of our observations. This possibility was considered by John Wheeler, who dubbed the scenario Law Without Law. It would mean that “quantum theory is merely a tool to make comprehensible a lawless slicing-up of nature,” said Adán Cabello, a physicist at the University of Seville. Can we derive quantum theory from these premises alone?
“At first sight, it seems impossible,” Cabello admitted—the ingredients seem far too thin, not to mention arbitrary and alien to the usual assumptions of science. “But what if we manage to do it?” he asked. “Shouldn’t this shock anyone who thinks of quantum theory as an expression of properties of nature?”
Making Space for Gravity
In Hardy’s view, quantum reconstructions have been almost too successful, in one sense: Various sets of axioms all give rise to the basic structure of quantum mechanics. “We have these different sets of axioms, but when you look at them, you can see the connections between them,” he said. “They all seem reasonably good and are in a formal sense equivalent because they all give you quantum theory.” And that’s not quite what he’d hoped for. “When I started on this, what I wanted to see was two or so obvious, compelling axioms that would give you quantum theory and which no one would argue with.”
So how do we choose between the options available? “My suspicion now is that there is still a deeper level to go to in understanding quantum theory,” Hardy said. And he hopes that this deeper level will point beyond quantum theory, to the elusive goal of a quantum theory of gravity. “That’s the next step,” he said. Several researchers working on reconstructions now hope that its axiomatic approach will help us see how to pose quantum theory in a way that forges a connection with the modern theory of gravitation—Einstein’s general relativity.
Perhaps when we finally get our hands on quantum gravity, the interpretation will suggest itself. Lucien Hardy
Look at the Schrödinger equation and you will find no clues about how to take that step. But quantum reconstructions with an “informational” flavor speak about how information-carrying systems can affect one another, a framework of causation that hints at a link to the space-time picture of general relativity. Causation imposes chronological ordering: An effect can’t precede its cause. But Hardy suspects that the axioms we need to build quantum theory will be ones that embrace a lack of definite causal structure—no unique time-ordering of events—which he says is what we should expect when quantum theory is combined with general relativity. “I’d like to see axioms that are as causally neutral as possible, because they’d be better candidates as axioms that come from quantum gravity,” he said.
Hardy first suggested that quantum-gravitational systems might show indefinite causal structure in 2007. And in fact only quantum mechanics can display that. While working on quantum reconstructions, Chiribella was inspired to propose an experiment to create causal superpositions of quantum systems, in which there is no definite series of cause-and-effect events. This experiment has now been carried out by Philip Walther’s lab at the University of Vienna—and it might incidentally point to a way of making quantum computing more efficient.
“I find this a striking illustration of the usefulness of the reconstruction approach,” Chiribella said. “Capturing quantum theory with axioms is not just an intellectual exercise. We want the axioms to do something useful for us—to help us reason about quantum theory, invent new communication protocols and new algorithms for quantum computers, and to be a guide for the formulation of new physics.”
But can quantum reconstructions also help us understand the “meaning” of quantum mechanics? Hardy doubts that these efforts can resolve arguments about interpretation—whether we need many worlds or just one, for example. After all, precisely because the reconstructionist program is inherently “operational,” meaning that it focuses on the “user experience”—probabilities about what we measure—it may never speak about the “underlying reality” that creates those probabilities.
“When I went into this approach, I hoped it would help to resolve these interpretational problems,” Hardy admitted. “But I would say it hasn’t.” Cabello agrees. “One can argue that previous reconstructions failed to make quantum theory less puzzling or to explain where quantum theory comes from,” he said. “All of them seem to miss the mark for an ultimate understanding of the theory.” But he remains optimistic: “I still think that the right approach will dissolve the problems and we will understand the theory.”
Maybe, Hardy said, these challenges stem from the fact that the more fundamental description of reality is rooted in that still undiscovered theory of quantum gravity. “Perhaps when we finally get our hands on quantum gravity, the interpretation will suggest itself,” he said. “Or it might be worse!”
Right now, quantum reconstruction has few adherents—which pleases Hardy, as it means that it’s still a relatively tranquil field. But if it makes serious inroads into quantum gravity, that will surely change. In the 2011 poll, about a quarter of the respondents felt that quantum reconstructions will lead to a new, deeper theory. A one-in-four chance certainly seems worth a shot.
Grinbaum thinks that the task of building the whole of quantum theory from scratch with a handful of axioms may ultimately be unsuccessful. “I’m now very pessimistic about complete reconstructions,” he said. But, he suggested, why not try to do it piece by piece instead—to just reconstruct particular aspects, such as nonlocality or causality? “Why would one try to reconstruct the entire edifice of quantum theory if we know that it’s made of different bricks?” he asked. “Reconstruct the bricks first. Maybe remove some and look at what kind of new theory may emerge.”
“I think quantum theory as we know it will not stand,” Grinbaum said. “Which of its feet of clay will break first is what reconstructions are trying to explore.” He thinks that, as this daunting task proceeds, some of the most vexing and vague issues in standard quantum theory—such as the process of measurement and the role of the observer—will disappear, and we’ll see that the real challenges are elsewhere. “What is needed is new mathematics that will render these notions scientific,” he said. Then, perhaps, we’ll understand what we’ve been arguing about for so long.
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60d39644db536750 | We connect to each other through particles. Calls and texts ride flecks of light, Web sites and photographs load on electrons. All communication is, essentially, physical. Information is recorded and broadcast on actual objects, even those we cannot see.
Physicists also connect to the world when they communicate with it. They dispatch glints of light toward particles or atoms, and wait for this light to report back. The light interacts with the bits of matter, and how this interaction changes the light reveals a property or two of the bits—although this interaction often changes the bits, too. The term of art for such a candid affair is a measurement.
Particles even connect to each other using other particles. The force of electromagnetism between two electrons is conveyed by particles of light, and quarks huddle inside a proton because they exchange gluons. Physics is, essentially, the study of interactions.
Information is always conveyed through interactions, whether between particles or ourselves. We are compositions of particles who communicate with each other, and we learn about our surroundings by interacting with them. The better we understand such interactions, the better we understand the world and ourselves.
Physicists already know that interactions are local. As with city politics, the influence of particles is confined to their immediate precincts. Yet interactions remain difficult to describe. Physicists have to treat particles as individuals and add complex terms to their solitary existence to model their intimacies with other particles. The resulting equations are usually impossible to solve. So physicists have to approximate even for single particles, which can interact with themselves as a boat rolls in its own wake. Although physicists are meticulous, it is a wonder they ever succeed. Still, their contentions are the most accurate theories we have.
One of the strangest of these implications refutes the material basis of communication as well as common sense. Some physicists believe that we may be able to communicate without transmitting particles. In 2013 a once amateur physicist named Hatim Salih even devised a protocol, alongside professionals, in which information is obtained from a place where particles never travel. Information can be disembodied. Communication may not be so physical after all.
For if we can process information without particles, we may build a computer that need not turn on, and we may be able to communicate with absolute secrecy. There would be nothing to intercept and nothing to hack. This possibility derives from the information contained inside wave functions—and from the way that the imaginary manifests as real. So before we can disembody communication, we must give body to the quantum theory.
Embodying Quantum Mechanics
The basic instrument of quantum mechanics, from which all its oddities are composed, is the wave function. Every possible state of a quantum object, every possible outcome of its measurement, is a solution to the Schrödinger equation. This simple equation resembles the one that describes moving waves—enough to have confused Erwin Schrödinger into naming its solutions wave functions—but quantum waves are abstract, not real. Unlike the solutions for ocean breakers or sound, wave functions always contain imaginary numbers.
To obtain real answers from this complex math, physicists multiply a wave function by a negative version of itself. The result is the probability of observing an object with the properties that the wave function details. Summing all the squares of all the solutions for any quantum object always totals 100 percent. The Schrödinger equation accounts for every possibility. It confounds, but it does not surprise.
When we solve the Schrödinger equation to predict the location of a particle, there are usually many possibilities—much as there would be in establishing the precise location of surf. Positions and trajectories are ill-defined in quantum mechanics because of the well-known duality of particles and waves. But measurements offer a certainty that wave functions cannot. When we observe the location of an electron, we know it for sure. Such knowledge, however, has a price. Once we know the position, we cannot know the speed. If we measure the speed, we forfeit all knowledge of the position. This gnostic trade-off is called the Heisenberg uncertainty principle. Many other observables, such as time and energy, are equally incompatible.
One notable quirk of this mathematics is that combining solutions to the Schrödinger equation for any particular object, according to their probabilities, is also a possible solution. This is called a superposition, although that is a misnomer. One solution is not placed atop another, but rather they are added together into a blend. And as with juicing, the flavor of the whole surpasses what was added in.
Quantum mechanics is counterintuitive, and superpositions are why. We have never experienced one in our daily lives, despite the shifting probabilities and blends of truth that we live with. So, to understand superpositions, let’s consider a thought experiment that can be made real. This example illustrates most of the oddities of quantum mechanics, and underlies the actual experiments undertaken by Pan and his colleagues.
A Trick of Light
Point a laser toward a piece of glass coated partially with aluminum, as in a one-way mirror. If the glass is at an angle of 45 degrees relative to the incoming light, half the beam continues through and the other half reflects away, perpendicular to the original beam. There is no road less traveled by—the choice of path, like quantum mechanics, is perfectly random.
Now, set a regular mirror in each of these paths and reunite the beams. The light acts as a wave so the beams interfere with each other where they meet, producing a pattern of ripples on a fluorescent screen that glows where it is struck (Figure 1). The interference pattern on the screen looks like someone took a comb to it—a result equivalent to the famous double-slit experiment. But our setup has a fancier name—a Mach–Zehnder interferometer.
Credit: Jen Christiansen
We can alter the pattern on the screen by inserting a pane of glass in one beam’s path. Glass slows the light, so the peaks and valleys of its waves no longer match those from the other beam. A certain thickness of glass slows one beam just enough so its peaks arrive with the valleys of the other. Different areas of the screen now turn dark, where the light in the two beams interfere with each other destructively. If we were to place a photon detector at such a spot, no light would register.
Physicists have learned how to produce single photons and how to detect them (even with their eyes), so they often conduct such experiments using particles rather than beams. When they direct photons one at a time toward a one-way mirror, otherwise known as a beam splitter, half continue blithely through and the other half reflect away—the same as before with a beam. Nothing changes for single particles. Although only one photon travels on either path at a time, an interference pattern still emerges on the screen. We can even alter the pattern by inserting a pane of glass. Photons still act like waves. But what does each photon now interfere with? The answer to that question is the essence of quantum mechanics.
A photon cannot split in half and interfere with itself—we always detect photons whole. The photons exist as superpositions, so perhaps they take both paths at once. To explain superpositions, writers often say that a particle exists in two places at a time. But this is wrong.
If we place detectors on both of a photon’s possible paths, one always clicks and the other does not. If we place a detector in one path, it clicks half the time. Yet when a detector registers the photons on either path, an interference pattern no longer appears on the screen. Even if we interact with photons but let them pass, just to know where they are, the pattern still disappears. The act of a measurement, the very acquisition of knowledge, alters the result. Once we observe a particle, it does not act like a wave. Light may lead a double life, but it only leads one life at a time.
Schrödinger believed that wave functions corresponded to real objects. Since 1926, most physicists have interpreted the wave function as an abstract parcel of knowledge, not an inhabitant of our world. There is a sense, however, in which the mathematics must be real.
Whenever a photon is described by a superposition of paths, they somehow interfere. If we ruin the superposition by distinguishing the paths, the interference always disappears. Whenever we find out which path a photon takes, the other path is no longer possible. Wave functions detail possibilities. So after one path becomes impossible, the wave function changes to reflect our knowledge of the world. Physicists say the wave function collapses. The quantum world collapses, too. Superpositions are more tenuous than any of our classical experiences.
Blowing Up
In 1993, Avshalom Elitzur and Vaidman pushed interferometers past the surreal and into the absurd, with a thought experiment that others would make real. Instead of a fluorescent screen, imagine a second beam-splitter where the paths reunite (Figure 2a). Now place a detector in line with each possible path after the splitter. The photons are equally likely to proceed to either detector. Alter one of the original paths again by adding a pane of glass, so there is destructive interference at one detector but not the other—a photon always registers in the second detector, but never in the first. We can actually observe this.
Now place an obstacle in one of the paths after the original split. Half the photons are absorbed and the other half travel the unimpeded path. These unimpeded photons should proceed as before, to the second detector. They do not. Half register in the first detector, which did not click when there were two paths (Figure 2b). The interference disappears because the other path is no longer possible. The photons definitely travel the path without the obstruction, but somehow they know what happens to the other path and change their behavior accordingly. In fact, a photon appearing in the forbidden detector—just once—is enough to intuit the presence of the obstruction.
Elitzur and Vaidman claimed their thought experiment was an example of the nonlocality of quantum mechanics. Two particles born together can exist in a superposition of complementary properties—and as the particles separate across the universe, we can measure the property of one and instantly know the other. This interdependence is called entanglement. Classical objects have distant influences—the moon orbits the Earth, magnets attract metals—but these influences are communicated through local interactions, traveling no faster than light. Particles separated by a universe, however, lose their superpositions immediately. The photons on our paths have no mass or charge, so they do not emit physical influences across space. Quanta are still local. Yet somehow the photon on the clear path knows about the obstacle in the other path instantaneously, without interacting with it at all. The photon acquired information from afar.
“It is common to think that unlike classical mechanics,” Elitzur and Vaidman explained, “quantum mechanics poses severe restrictions on the minimal disturbance of the system due to the measurement procedure.” This cannot be true. A path may be undisturbed, yet our observations will change. The mere presence of an obstacle on the other path acts like a measurement, conveying information to the photons and to us.
Dennis Gabor, who developed holograms, said that every observation requires a photon. But light does not have to strike an object to reveal it. We can see without looking. (This is neither a shell game nor ESP. Most photons in the real world have many more than two possible paths, and these usually cancel one another, leaving the straight, shortest path for light, which we observe. Most light acts as we classically believe.)
Elitzur and Vaidman, who were then working in Tel Aviv, plotted their idea more dramatically (Figure 2b) Instead of an inert obstacle in one path, they imagined a bomb set to explode when struck by a photon. If a photon travels that path, the bomb explodes and we know for certain that the photon was there. If a photon travels the clear path, we can still discern the presence of an obstacle—in this case, the bomb—without shining a light on it. The photons on the unimpeded path will register in the forbidden detector half the time, alerting us to the bomb’s presence. Elitzur and Vaidman called this an interaction-free measurement. Sir Roger Penrose, the noble mathematical physicist, called their insight a counterfactual. But the thought experiment is not counter to established fact. Evert du Marchie van Voorthuysen demonstrated interaction-free measurements using inexpensive instruments—and obstacles other than bombs—at a science expo in Groningen, in 1995. Afterward, physicists could not explain the demonstration any better than the spectators.
Credit: Jen Christiansen
Wave functions and superpositions describe actual phenomena with actual consequences. The mathematics is set; the interpretation is not. Some physicists believe, again, that wave functions are real objects, similar to a magnetic field. Others contend that wave functions describe ensembles, not single particles. Still others take the mathematics so seriously that they argue superpositions create many worlds, one for each possibility.
Most physicists insist that the math details only the many possibilities of our one world. But Elitzur and Vaidman converted themselves to the more radical idea. “This paradox can be avoided in the framework of the many-worlds interpretation,” they wrote. If there are many worlds, interaction-free measurements are easy to explain. The wave function does not collapse and every possibility still exists, somewhere—we discern an obstacle here because an explosion happened in another universe.
In the reckoning of Elitzur and Vaidman the probability of conveying information without interactions, in any universe, is at best 50 percent. But in 1994, two young men who had recently completed their PhDs in the Bay Area—Mark Kasevich and Paul Kwiat—met in the lab of Anton Zeilinger in Innsbruck. Kasevich told Kwiat, and a few other colleagues in Austria, how they might improve the odds. If an obstacle transmits information without interactions half the time, more obstacles should transmit information more frequently. Repeatedly splitting the paths in an interferometer and inserting obstacles in them is akin to repeating measurements and gaining knowledge from each one. Kwiat and his colleagues called this an interrogation.
In theory, physicists could extract perfect information without interactions if they used an infinite number of obstacles. In experiments, Kwiat and his colleagues routed one of the paths of a photon through an obstacle six times and increased the number of interaction-free measurements to 70 percent. During the 1970s, two physicists at The University of Texas at Austin, Baidyanath Misra and E. C. George Sudarshan, studied the weird capacity for repeated measurements to prolong quantum effects. They called it Zeno’s paradox for quantum mechanics. The Greek philosopher had argued that measuring the position of an arrow repeatedly, as it progresses half the distance to its mark, implies the arrow never lands. Half a distance always remains.
In the nearly 25 years since the introduction of counterfactuals, physicists have realized many applications that are less volatile than detecting bombs. In 1998, Kwiat and his collaborators at Los Alamos developed photographs of human hair inside an interferometer, on a path that light did not traverse. Two years later in England, two theorists, Graeme Mitchison and Richard Jozsa, described how to compute without interactions.
Quantum computers are hard to build, in part because measurements are heavy-handed. To know the outcome of an algorithm, we have to ruin the very superpositions on which such a computer runs. In 2006, Onur Hosten, Kwiat and other collaborators at the University of Illinois at Urbana–Champaign, appended a chain of quantum Zeno effects to counterfactuals and designed a quantum computer that could deliver information without running at all. “This is possible only in the realm of quantum computers,” they explained, “where the computer can exist in a quantum superposition of ‘running’ and ‘not running’ at the same time.”
When Vaidman read that theoretical computers need not be turned on to work, he thought Kwiat had bested him again, improving the efficiency of counterfactuals. But the idea is not as straightforward as discerning an obstacle on a path without light. As Vaidman says, Kwiat and his collaborators’ computer relies on “the absence of an object in a particular place, allegedly without [photons] being there.” But no information can come from nothing. After analyzing the experiment for several months, Vaidman explained that “the photon did not enter the interferometer, the photon never left the interferometer, but it was there.” The particle had to be where it could not, if information was derived from the absence of an object. Kwiat wrote that Vaidman’s interpretation is “nonsense.”
At the Electronics and Telecommunications Research Institute in South Korea, in 2009, Tae-Gon Noh took the next logical step. Instead of a fanciful computer that does not have to run, Noh applied counterfactuals “to a real-world communication task.” He developed a protocol for sending a key to unlock shared data. When a photon travels the unobstructed path in an interferometer, the information acquired about the other path—through which the photon could not have traveled—may be used to reveal the secret key. The crests and troughs of the light can be made to undulate up and down or side to side, and this binary property (called polarization) can be used to encode bits. Information can then be transmitted through the obstructed channel, which the receiver controls. The sender and receiver also share regular information, but if they follow a simple protocol, no one can eavesdrop or steal their key. There is nothing to intercept—the photons live and die, as Noh explained, inside the sender’s device. Even stranger than the lack of a signal, he said, is “the mere possibility that an eavesdropper can commit a crime is sufficient to detect the eavesdropper, even though the crime is not in fact carried out.” He compared counterfactuals to the preemptive arrests in the film Minority Report.
In 2011, Pan and a few other collaborators in Hefei realized Noh’s “engrossing” scheme in the real world, on a tabletop in their lab. They sent a secure key—at a rate of 51 bits per second—over a kilometer of fiber-optic cable, although not without significant errors. Pan and his group did not achieve the fidelity needed to convert their science into a technology, but they claimed, “we have given proof-in-principle demonstrations.” Some information really could travel without particles.
While living in England in 2009, a young man named Hatim Salih read Noh’s paper and asked himself, “Why didn’t I think of that?” He had a degree in electronics but had taught himself quantum physics after reading a few popular books by Roger Penrose and attending seminars in York . A year later Salih returned to his native Sudan, where he marketed solar panels, and a friend invited him to be a visiting researcher at the King Abdulaziz City for Science and Technology in Saudi Arabia. He did not have a PhD, but with a colleague there and two other theorists at Texas A&M University, he took “the logic of counterfactual communication to its natural conclusion.” As they explained, “using a chained version of the Zeno effect, information can be directly exchanged between Alice and Bob with no physical particles traveling between them, thus achieving direct counterfactual communication.” (Instead of labeling senders and receivers A and B, physicists call them Alice and Bob.)
First, Salih and his colleagues devised a protocol for communicating some information without particles. Split photons down two paths and reunite them at a second beam splitter, as before. Now do this again and again, adding one interferometer after another (Figure 3a). Alter the paths with special beam splitters, so the photons always proceed to the same detector at the end. The theoretical Bob, who controls obstacles in the series of paths, can use them to send information to Alice’s detectors. If he lets a photon through, the guaranteed click at the first detector is defined as a 0 in binary logic. If he blocks the paths after every split, a photon very likely appears in the second detector, a result defined as a 1. Thus Bob transmits information to Alice, even when he does not let some particles through.
In theory this set-up transmits 0s with certainty, but the counterfactual information—the 1s transmitted without particles—are less reliable. Photons from the unimpeded path occasionally pass to the other detector that registers 0s, even if there are hundreds of obstacles.
But Salih and his colleagues then claimed that they knew how to accomplish what no one had before: Make each bit counterfactual. It should be possible to transmit signals between a sender and receiver simply by blocking the paths that a photon should never take.
After the initial split for the photons in an interferometer, divide one of these two paths again. Now add one small interferometer after another on this path, placing obstacles that Bob controls in each (Figure 3b). Many small interferometers are thus nested inside a large one, and this can be done again and again. The obstacles on the interior paths act as repeated measurements, and the more interaction-free measurements there are, the more efficient the communication will be. The paths can even be made to interfere, so the particles that arrive at Alice’s detectors can never travel the paths that Bob obstructs. They are truly blocked. But the detectors will still register differently when he obstructs his paths or not. Bob sends information without interacting with any particles.
The protocol that Salih and his colleagues designed is difficult to imagine, even inside a lab. So they conceived another protocol using a similar interferometer, one developed by Albert Michelson to determine the existence of the aether during the 1880s (and used, more recently, to detect gravitational waves). In a Michelson interferometer, light is again divided onto two paths, but mirrors reflect the beams back to where they originally split. They interfere there. Experimenters can nest these interferometers and distinguish the light where it interferes by the two polarizations, which serve as the bits.
At the end of their paper, Salih and his colleagues declared, “we strongly challenge the long-standing assumption that information transfer requires physical particles to travel between sender and receiver.” In 2014, they even obtained a patent on direct communication without “physically real” entities. Salih then founded a company, called Qubet Research, to monetize the idea.
Credit: Jen Christiansen
Weak Measurements, Strong Opinions
Lev Vaidman is a prolific commenter. Twelve of his most recent 25 papers are replies to other physicists or criticisms of their work. He is sometimes impolitic enough to assert that someone’s paper should never have been published, but he also lists his rejected papers on his Web site. He comments so frequently, he says, because “open discussion and disagreements help to move physics forward.”
Physicists can agree on the mathematics and the results of experiments, but still dispute their interpretations. Perhaps surprisingly—but also rationally—Vaidman doubts that communication happens in the absence of particles, as Salih and others describe. Vaidman has complained that Salih et al.’s protocol “was based on a naive classical approach to the past of the photons.” He consented that a “process is counterfactual if it occurs without any real physical particle traveling between the two parties. But what is the meaning of this definition? For the quantum particle, there is no clear definition of ‘traveling’.” Vaidman’s argument is not just about language, but what we can say about the world.
Vaidman insists that particles have no past. And if they don’t, we cannot actually know if one was ever near an object and interacted with it. When we measure a particle to find out, the wave function that could have told us collapses. We do not learn history from particles, we force history upon them.
But, in 1988, Vaidman and two colleagues at the University of South Carolina imagined a new kind of measurement—one that was so weak that it did not collapse quantum states. A weak measurement cannot release the information we seek from photons, but coupled with many such measurements and one strong one, it might. In fact, a weak measurement followed by a strong one gives us more information than we have any right to know. Sending an electron through a slight magnetic field and then a strong perpendicular field, for instance, reveals two incompatible properties at the same time. Weak measurements disclose what Heisenberg had deemed uncertain.
Vaidman and his colleagues have converted their theory of weak measurements into a new version of quantum mechanics. They combine the information from a weak measurement and a strong one into a single wave function. The past is set by the weak measurement, and Vaidman then builds a superposition between the particle’s past and its future to know what happened in between. When Vaidman applies his theory to counterfactuals, the photon always appears where it should not—on the obstructed path. Few understand his approach, and many doubt it. The results are imaginary numbers that give negative probabilities, which should be impossible.
But in 2013, Ariel Danan and a few colleagues in Tel Aviv, including Vaidman, studied interaction-free measurements in actual weak experiments. They vibrated one of the mirrors on one of the paths inside an interferometer to locate the photons on this path. “The experiment is analogous to the following scenario,” they wrote. “If our radio plays Bach, we know that the photons come from a classical music station, but if we hear a traffic report, we know that the photons come from a local radio station.” What they heard was surprising. The photons flitted about, even on forbidden paths, guided by their wave functions.
Many physicists doubt that a photon that neither enters a path nor exits it can still somehow be there. Salih argues that Vaidman is using his own version of quantum mechanics, so he naturally believes that other interpretations are wrong. Salih even implied that Vaidman is telling photons what to say when other physicists interrogate them.
This past April, Pan and his colleagues wrote in their paper: “Although several publications are presently available regarding the theoretical aspects of [counterfactual communication], a faithful experimental demonstration, however, is missing.” It was time for an experiment on communication to speak. The group started planning their experiment to end the “heated debate” before Salih and others even formally published their idea.
An infinite number of interferometers was required for perfect communication, which Pan and his group acknowledged was impractical. So they simplified the protocol for Michelson interferometers and built four, with two smaller ones nested inside. They set their source of single photons, their beam splitters and their mirrors on a small table that was temperature-controlled and isolated against vibrations. The counterfactual communication would occur across 50 centimeters, inside a lab in Shanghai. Pan’s collaborators, Cao and Li, designed a number of possible images to send, and the group voted for a Chinese knot. As Cheng-Zhi Peng explained, “it is symmetric and beautiful.”
Jian-Wei Pan (seated at center) and colleagues in their lab in Shanghai. Credit: Bo Li
The group wrote software to run their experiment automatically, without any human interference. On May 31, 2013, they sat at a computer and waited through the night to see if the image loaded on a screen. They trusted their instruments, but they quietly hoped that nothing would appear. A negative result would imply that quantum mechanics is wrong. No one had ever observed that.
Over five hours, 10 kilobytes of information passed the 50 empty centimeters between the sender and receiver. Many of the bits had to be transmitted several times before they registered, and the computer was better at recognizing 1s than 0s. But a monochrome bitmap appeared through static, although the group had not transmitted any particles that they could discern. Once they saw the image, after sunrise, they disbanded to sleep before they celebrated. They posted a short article one year later but did not submit their paper for publication for more than three years. They were too busy building communication satellites, and they wanted some time to think about the result.
Pan and his colleagues are now working to transmit a picture in shades of gray, and they hope to send pure quantum information based on another protocol by Salih. To ensure that no photons pass through the transmission channel, they also plan to do a weak measurement to determine where the photon goes.
Although Pan is in the business of communication satellites, and counterfactuals pique banks and militaries, the group reported another potential application for their experiment: “imaging ancient arts where shining a light directly is not permitted.” Kwiat has implied that counterfactuals might not be useful for anything else. He wrote: “In order to achieve a high level of counterfactuality, one needs many cycles, and this greatly slows down the rate of communication.” Information moves slower without particles than with them.
Credit: Jen Christiansen; Source: “Direct counterfactual communication via quantum Zeno effect,” By Yuan Cao et al., PNAS, No. 19, May 9, 2017 (Chinese knot panels)
Pan and colleagues attribute the mystery of counterfactual communication to the wave/particle duality. Salih has another interpretation. “I believe this experiment has something to say in support of the reality of the quantum wave function: If physical particles did not transfer information then what did?” Imaginary wave functions may be the last preserve of the real.
Salih is now working on a proof of counterfactuals, using weak measurements, to outflank his critics. When I asked Vaidman what would convince him that no particles were ever transmitted, he replied, tautologically: “If an object was found in a counterfactual way, there should be zero trace near it.” Pan’s collaborators told me, perhaps jokingly: “Although our demonstration hasn’t solved the issue entirely, we do believe that our work shed some light on the discussion.”
Quantum mechanics has survived nearly 100 years, and the unorthodox theory remains fabulous. Experiments routinely verify its predictions, and the normative theories invented to reform it have failed. Physicists continue to uncover new ways to adapt its mysteries to information technology and realize its wonders in the world. They are still waiting for the theory to communicate its meaning to us, however—with or without particles. |
c31e87459a0d2f63 | 2. BASIC
2. Basic results that shows how QM arises, written in BASIC
Pick a line of size “l” throw a random number denoting a position (“p”) on the line and associate a line whose length (“li”) is also chosen randomly but cannot exceed the length “l”. Set a constraint with a particular relation between “l”, ”p”, and “li” such that if this relation holds ignore the outcome, otherwise register the position. Boom, the outcome is the solution for Schrödinger equation for a particle in an infinite potential well (probability wave sin^2) (Fig. 1). My line of thought was: if nature is made of math then the best place to start is with a line. And what could be happening on this line. Nothing much, really, a point and a piece of line over and over. For the next energy level divide the line using natural numbers (how convenient). Simple line and simple rules lead to this gigantic dance of reality. It has been a dream of many that we owe our existence to some kind of automata. Well, this is it, maybe.
The program is listed here It is written in liberty basic and you can download it here. And if that is not enough I simulated two particles where the second acts like a potential. The two particles interact according to some basic rule (see the program). When you make the potential narrow i.e. the energy high, the first particle gets constrained and its probability goes to zero at the potential (fig. 2). When the second particle has a wide base then it very much acts like a square well and the probability looks exponential inside it (fig. 3). And tunneling phenomenon is clear, continuity automatically satisfied. In other words Schrödinger equation popped out of some geometry with rules. Crazy enough? No? Here is some more: I can make the particles interact without their wave functions overlapping just by not restricting the size of the associated length “li”. That is spooky action at distance. Left as an exercise. |
cf89ca5cac3105ff | In many important practical situations, beam propagation through a nonlinear media can be accurately modeled by the following nonlinear Schrödinger equation (NLS)
i Qη + ΔQ + |Q|2 Q = 0 (NLS)
with a function Q0(ρ) as an incident radially symmetric pulse Q(ρ, η=0)= Q0(ρ) (here Qη = Q / η and the Laplacian Δ is given by Δ= ∂2/∂ρ2 + (1/ρ)(∂/∂ρ), η=z / Ldf is a normalized propagation distance, and ρ= (x2+y2)1/2 / R0 is the normalized distance from the center of the pulse in the transverse two-dimensional plane, Ldf is the diffraction length and R0 is the beam’s FW1/eM radius).
NLS equation is a paraxial approximation of the following nonlinear Helmholtz (NHL) equation [1,2]
Ezz(x,y,z,t) + ΔE + (k0)2(1+(2n2/n0)|E|2) E = 0 (NHL)
derived by assuming the following:
• Electric field is written in the following SVEA form, with Q(x,y,z,t) slowly varying along propagation direction z, 0 / 2πR0)2Qηη ≈ 0, which is satisfied when λ0 << R0:
E(x,y,z,t) = R0 k0 (2n2/n0)1/2 Q(x,y,z,t) exp[ -i0t– k0z) ]
• where back-propagation term Q*(x,y,z,t) is excluded
See also: Nonparaxial approximation, Slowly varying envelope approximation (SVEA).
[1] G. Fibich A and S. Tsynkov B, Numerical solution of the nonlinear Helmholtz equation using nonorthogonal expansions, J. Comp. Phys. 210, (2005), 183-224.
[2] G. Baruch, G. Fibich, and Semyon Tsynkov, Simulations of the nonlinear Helmholtz equation: arrest of beam collapse, nonparaxial solitons and counter-propagating beams, Opt. Express 16, 13323-13329 (2008).
SimphoSOFT logo SimphoSOFT® propagation mathematical model is based on paraxaial approximation.
• e-mail
• phone: +1 (973) 621-2340 |
8f29418bed712145 | What Is Urea Nitrate?
Urea nitrate is a fertilizer-based high explosive that has been used in improvised explosive devices in Afghanistan, Pakistan, Iraq, and various other terrorist acts elsewhere in the world, like the 1993 World Trade Center bombings. It has a destructive power similar to better-known ammonium nitrate explosives, with a velocity of detonation between 11,155 ft/s (3,400 m/s) and 15,420 ft/s (4,700 m/s).
Urea nitrate explosions may be initiated using a blasting cap.
Ball-and-stick models of the ions in urea nitrate.
Urea contains a carbonyl group. The more electronegative oxygen atom pulls electrons away from the carbon forming a greater electron density around the oxygen, giving the oxygen a partial negative charge and forming a polar bond. When nitric acid is presented, it ionizes. A hydrogen cation contributed by the acid is attracted to the oxygen and forms a covalent bond [electrophile H+]. The electronegative NO3− ion then is attracted to the positive hydrogen ion. This forms an ionic bond and hence the compound urea nitrate.
(NH2)2CO (aq) + HNO3 (aq) → (NH2)2COHNO3 (s)
The compound is favored by many amateur explosive enthusiasts as a principal explosive for use in larger charges. In this role it acts as a substitute for ammonium nitrate based explosives. This is due to the ease of acquiring the materials necessary to synthesize it, and its greater sensitivity to initiation compared to ammonium nitrate based explosives.
Spectrum = gas discharge tube filled with hydrogen H2. Used with 1,8kV, 18mA, 35kHz. ≈8" length.
Spectrum = gas discharge tube filled with hydrogen H2. Used with 1,8kV, 18mA, 35kHz. ≈8″ length.
Hydrogen is a chemical element with chemical symbol H and atomic number 1. With an atomic weight of 1.00794 u, hydrogen is the lightest element on the periodic table. Its monatomic form (H) is the most abundant chemical substance in the Universe, constituting roughly 75% of all baryonic mass. Non-remnant stars are mainly composed of hydrogen in the plasma state. The most common isotope of hydrogen, termed protium (name rarely used, symbol 1H), has one proton and no neutrons.
The universal emergence of atomic hydrogen first occurred during the recombination epoch. At standard temperature and pressure, hydrogen is a colorless, odorless, tasteless, non-toxic, nonmetallic, highly combustible diatomic gas with the molecular formula H2. Since hydrogen readily forms covalent compounds with most nonmetallic elements, most of the hydrogen on Earth exists in molecular forms such as water or organic compounds. Hydrogen plays a particularly important role in acid–base reactions because most acid-base reactions involve the exchange of protons between soluble molecules. In ionic compounds, hydrogen can take the form of a negative charge (i.e., anion) when it is known as a hydride, or as a positively charged (i.e., cation) species denoted by the symbol H+. The hydrogen cation is written as though composed of a bare proton, but in reality, hydrogen cations in ionic compounds are always more complex. As the only neutral atom for which the Schrödinger equation can be solved analytically, study of the energetics and bonding of the hydrogen atom has played a key role in the development of quantum mechanics.
H2 spectra using a 600lpm diffraction grating.
H2 spectra using a 600lpm diffraction grating.
Hydrogen gas was first artificially produced in the early 16th century by the reaction of acids on metals. In 1766–81, Henry Cavendish was the first to recognize that hydrogen gas was a discrete substance, and that it produces water when burned, the property for which it was later named: in Greek, hydrogen means “water-former”.
Industrial production is mainly from steam reforming natural gas, and less often from more energy-intensive methods such as the electrolysis of water. Most hydrogen is used near the site of its production site, the two largest uses being fossil fuel processing (e.g., hydrocracking) and ammonia production, mostly for the fertilizer market. Hydrogen is a concern in metallurgy as it can embrittle many metals, complicating the design of pipelines and storage tanks.
Hydrogen gas (dihydrogen or molecular hydrogen) is highly flammable and will burn in air at a very wide range of concentrations between 4% and 75% by volume. The enthalpy of combustion is −286 kJ/mol:
2 H2(g) + O2(g) → 2 H2O(l) + 572 kJ (286 kJ/mol)
Hydrogen gas forms explosive mixtures with air in concentrations from 4–74% and with chlorine at 5–95%. The explosive reactions may be triggered by spark, heat, or sunlight. The hydrogen autoignition temperature, the temperature of spontaneous ignition in air, is 500 °C (932 °F). Pure hydrogen-oxygen flames emit ultraviolet light and with high oxygen mix are nearly invisible to the naked eye, as illustrated by the faint plume of the Space Shuttle Main Engine, compared to the highly visible plume of a Space Shuttle Solid Rocket Booster, which uses an ammonium perchlorate composite. The detection of a burning hydrogen leak may require a flame detector; such leaks can be very dangerous. Hydrogen flames in other conditions are blue, resembling blue natural gas flames.
The destruction of the Hindenburg airship was a notorious example of hydrogen combustion and the cause is still debated. The visible orange flames in that incident were the result of a rich mixture of hydrogen to oxygen combined with carbon compounds from the airship skin.
H2 reacts with every oxidizing element. Hydrogen can react spontaneously and violently at room temperature with chlorine and fluorine to form the corresponding hydrogen halides, hydrogen chloride and hydrogen fluoride, which are also potentially dangerous acids. |
bdff822e96f2c7de | Quantum Gravity and String Theory
Quantized Space-Time and Internal Structure of Elementary Particles: a New Model
Authors: Hamid Reza Karimi
In this paper we present a model in which the time and length are considered quantized. We try to explain the internal structure of the elementary particles in a new way. In this model a super-dimension is defined to separate the beginning and the end of each time and length quanta from another time and length quanta. The beginning and the end of the dimension of the elementary particles are located in this super-dimension. This model can describe the basic concepts of inertial mass and internal energy of the Elementary particles in a better way. By applying this model, some basic calculations mentioned below, can be done in a new way: 1- The charge of elementary particles such as electrons and protons can be calculated theoretically. This quantity has been measured experimentally up to now. 2- By using the equation of the particle charge obtained in this model, the energy of the different layers of atoms such as hydrogen and helium is calculated. This approach is simpler than using Schrödinger equation. 3- Calculation of maximum speed of particles such as electrons and positrons in the accelerators is given.
Comments: 23 pages.
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[v1] 16 Nov 2009
Unique-IP document downloads: 1349 times
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4e5ae708e91e1b94 | Mathematical Physics
On an Entropic Universal Turing Machine Isomorphic to Physics (draft)
Authors: Alexandre Harvey-Tremblay
According to the second law of thermodynamics, a physical system will tend to increase its entropy over time. In this paper, I investigate a universal Turing machine (UTM) running multiple programs in parallel according to a scheduler. I found that if, over the course of the computation, the scheduler adjusts the work done on programs so as to maximize the entropy in the calculation of the halting probability Ω, the system will follow the laws of physics. Specifically, I show that the computation will obey algorithmic information theory (AIT) analogues to general relativity, entropic dark energy, the Schrödinger equation, a maximum computation speed analogous to the speed of light, the Lorentz's transformation, light cone, the Dirac equation for relativistic quantum mechanics, spins, polarization, etc. As the universe follows the second law of thermodynamics, these results would seem to suggest an affinity between an "entropic UTM" and the laws of physics.
Comments: 39 Pages.
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[v1] 2017-08-13 20:34:15
[v2] 2017-08-14 09:19:09
[v3] 2017-08-16 20:22:20
[v4] 2017-09-09 17:23:30
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2527776af89790a0 | Scielo RSS <![CDATA[Revista mexicana de física]]> vol. 62 num. 3 lang. es <![CDATA[SciELO Logo]]> <![CDATA[Colloids and composite materials Au/PVP and Ag/PVP generated by laser ablation in polymeric liquid environment]]> Abstract Pulsed laser ablation of Silver and Gold targets, immersed in a polymeric solution of Polyvinylpyrrolidone (PVP), is used to generate colloids and composite metal-polymer. Solutions of PVP in deionized water at different concentrations are employed. Two PVP number average molecular weights were considered, 10000 g/mol and 55000 g/mol. The high purity targets are irradiated between 20 min and40 min with the third harmonic (THG) ( λ = 335 nm) of a Nd:YAG laser operating at a rate of 10 Hz with pulses of 8 ns. Optical spectroscopy in UV and vis regions, Scanning Electron Microscopy (SEM), High Resolution Scanning Electron Microscopy (HR-SEM) and X-Ray are used to identify and determine the shape and size of the produced particles. Very stable sub-micrometric spherical particles for Au/PVP y Ag/PVP samples are obtained with diameters of 0.72 μ m y 0.40 μm, respectively. The preparation of colloids is performed in one step and no surfactant or dispersing agent is used in this process. <![CDATA[Density of modes maps for design of photonic crystal devices]]> Abstract In the paper, we present numerical results on characterization of transmitting properties of wideband filters based on linear and nonlinear photonic crystals confined with the waveguide. Novel characteristics of the PhC filters such as density of modes maps and transmission maps are computed, and their efficiency is analyzed. Presented characteristics can be used as an auxiliary optimization tools to reduce optical losses when designing high-efficient optical interconnects. <![CDATA[Gravitational collapse in brane-worlds revisited]]> Abstract This paper is dedicated to revisit the modifications caused by branes in the collapse of a stellar structure under the Snyder-Oppenheimer scheme. Due to the homogeneity and isotropy of the model, we choose study the case of a closed geometry described by k = 1, through the tool of dynamical systems. We revisit the different components of the star and its evolution during the stellar collapse, paying particular attention to the non-local effects and the quadratic terms of the energy momentum tensor that come from branes corrections. In the same vein we realize a phase portrait together with a stability analysis with the aim of obtain information about the attractors or saddle points of the dynamical system under different initial conditions in the density parameters, remarking the parameters that come from branes contributions. <![CDATA[Dissociation-ionization and ionization-dissociation by multiphoton absorption of acetaldehyde at 266 and 355 nm. Dissociation pathways]]> Abstract The experimental results from the interaction of a sample of acetaldehyde (CH3CHO) with laser radiation at intensities between 109 and 1011Wcm2 and wavelengths of 266 and 355 nm are reported. As a result of multiple photon absorption, cations from ionization-dissociation (I-D) or dissociation-ionization (D-I) processes, were detected using a reflectron time of flight mass spectrometer. The processes I-D is predominant at 355 nm and D-I is predominant at 266 nm. The formation of different ions is discussed. From analysis of the ratios between the ion currents [I( C H 3 C O +)+I( C O +)]/[ I( C H 3 +)+I( H C O +)], originated from the C-C bond or from the C-H bond breaking at different laser intensities, the predominant channels are determined. <![CDATA[Cervical cancer detection based on serum sample surface enhanced Raman spectroscopy]]> Abstract In the presence of nanoparticles, the Raman signal is enhanced to the levels sufficient to detect a single molecule, therefore spectroscopy Surface-Enhanced Raman Scattering (SERS) is currently recognized as a detection technique extremely sensitive with high levels of molecular specificity. This is the first report in the cervical cancer detection based on serum SERS. The serum samples were obtained from 14 patients who were clinically diagnosed with cancer and 14 healthy volunteer controls. The serum samples were mixed with colloidal silver nanoparticles of 40 nm in the same proportion, using sonication. About 10 spectra were collected of each serum sample using a Horiba Jobin-Yvon LabRAM Raman Spectrometer with a laser of 830 nm. The enhanced Raman bands allowed identifying biomolecules present at low concentration as amide I and III, carotene, glutathione, tryptophan, tyrosine and phenylalanine. Subsequently, the processed SERS spectra were analyzed using multivariate statistical analysis including principal component analysis and linear discriminant analysis (LDA). Preliminary results showed that SERS and PCA-LDA can be used to discriminate between cervical cancer and control samples with high sensitivity and specificity, forming an excellent support technique for current detection techniques.<hr/>Resumen En presencia de nanopartículas, la señal Raman se amplifica hasta niveles suficientes como para detectar una molécula individual, por tanto la espectroscopia Surface-Enhanced Raman Scattering (SERS) es actualmente reconocida como una técnica de detección extremadamente sensible con altos niveles de especificidad molecular. Este es el primer reporte en la detección del cáncer cervicouterino basado en la dispersión Raman de superficie amplificada de muestras de suero. Las muestras de suero fueron obtenidas de 14 pacientes quienes fueron clínicamente diagnosticadas con cáncer y 14 voluntarios saludables. Las muestras de suero y nanopartículas de plata de 40 nm en forma coloidal fueron mezcladas usando sonicación. Alrededor de 10 espectros SERS por cada paciente fueron recolectados usando un espectrómetro Raman LabRAM Horiba Jobin-Yvon con un láser de 830 nm. Las bandas Raman fuertemente amplificadas permitieron identificar biomoléculas presentes en bajas concentraciones como amidas I y III, carotenos, glutatión, triptófano, tirosina y fenilalanina. Posteriormente, los espectros SERS procesados fueron analizados utilizando análisis estadístico multivariado incluyendo análisis de componentes principales (PCA) y analisis de discriminante lineal (LDA). Resultados preliminares demostraron que SERS y PCA-LDA pueden ser usados para discriminar entre muestras control y cáncer cervicouterino con alta sensibilidad y especificidad, conformando una excelente técnica de apoyo para las actuales técnicas de detección. <![CDATA[GaN nanowires and nanotubes growth by chemical vapor deposition method at different NH<sub>3</sub> flow rate]]> Abstract GaN nanowires and nanotubes have been successfully synthesized via the simple chemical vapor deposition method. NH3 flow rate was found to be a crucial factor in the synthesis of different type of GaN which affects the shape and the diameter of generated GaN nanostructures. X-ray diffraction confirms that GaN nanowires grown on Si(111) substrate under 900o C and with NH3 flow rate of 50 sccm presents the preferred orientation growth in the (002) direction. It is beneficial to the growth of nanostructure through catalyst annealing. Transmission electron microscopy and scanning electron microscopy were used to measure the size and structures of the samples. <![CDATA[Impact of planarized gate electrode in bottom-gate thin-film transistors]]> Abstract In this work, the fabrication of bottom-gate TFTs with unplanarized and planarized gate electrode are reported, as well simulations of the impact of the gate planarization in the TFTs are presented. Previously in literature, a reduction of the contact resistance has been attributed to this planarized structure. In order to provide a physical explanation of this improvement, the electrical performance of ambipolar a-SiGe:H TFTs with planarized gate electrode by Spin-On Glass is compared with unplanarized ambipolar a-SiGe:H TFTs. Then, the properties in the main device interfaces are analyzed by physically-based simulations. The planarized TFTs have better characteristics such as field-effect mobility, on-current, threshold voltage and on/off-current ratio which are consequence of the improved contact resistance. <![CDATA[A nonextensive wavelet (<em>q,q´</em>)-entropy for 1/𝑓<sup>α</sup> signals]]> Abstract This paper proposes a nonextensive wavelet ( q , q ' )-entropy computed as a wavelet-domain generalization of the time-domain ( q , q ' ) entropy of Borges and obtains a closed-form expression of this measure for the class of 1 / f α signals. Theoretical wavelet ( q , q ' )-entropy planes are obtained for these signals and the effect of parameters q and q ' on the shape and behaviour of these wavelet entropies are discussed with sufficient detail. The relationship of this entropy with Shannon and Tsallis entropies is studied and some applications of the proposed two-parameter wavelet entropy for the analysis/estimation of 1 / f signals are outlined. <![CDATA[Reduction of the Salpeter equation for massless constituents]]> Abstract Different from the case of the limit, it is shown in this paper that in the ultrarelativistic limit, the L = j, j+1 wave components are large terms for both state with parity ȠP = (-1)j and state with parity ȠP = (-1)j+1. Moreover, it is found that the states with parity ȠP = (-1) j are degenerate with the states with parity ȠP = (-1) j +1 if the Lorentz structure of the interaction between the massless constituents is four-vector or time-component of four-vector. The scalar interaction violates this degeneracy. <![CDATA[Implications of the Ornstein-Uhlenbeck-like fractional differential equation in cosmology]]> Abstract In this paper we introduce a generalized fractional scale factor and a time-dependent Hubble parameter obeying an “Ornstein-Uhlenbeck-like fractional differential equation” which serves to describe the accelerated expansion of a non-singular universe with and without the presence of scalar fields. Some hidden cosmological features were captured and discussed consequently. <![CDATA[Remarks on the (1+1)-Matrix-Branes, qubit theory and non-compact Hopf maps]]> Abstract We discuss different aspects of a possible link between the (1+1)-matrix-brane system with qubit theory and non-compact Hopf maps. In these scenarios, the (2+2)-signature plays an important role. We argue that such links may shed some light on the (2+2)-dimensional sector of a (2+10)-dimensional target background. <![CDATA[Convergence of resonance expansions in quantum wave buildup]]> Abstract The convergence of stationary and dynamical resonance expansions that involve complex eigenenergies of the system is analyzed in the calculation of the electronic probability density along the internal region of a resonant structure. We show that an appropriate selection of the resonance contributions leads to results that are numerically indistinguishable from the exact Hermitian calculation. In particular, the role played by the anti-resonances in the convergence process is emphasized. An interesting scaling property of the Schrödinger equation, and the stationary resonance expansion, useful for the analysis of convergence of families of systems, is also demonstrated. The convergence of a dynamical resonance expansion based on a Moshinsky shutter setup, is explored in the full time domain. In particular, we explore the build process of the electronic probability density in the transient regime, analyzing the contributions of different resonant states in the earliest stages of the buildup process. We also analyze the asymptotic limit of very long times, converging in the latter case to the stationary solution provided by the exact Hermitian calculation. <![CDATA[Additive and multiplicative noises acting simultaneously on Ermakov-Ray-Reid systems]]> Abstract We investigate numerically the effect of additive and multiplicative noises on parametric oscillator systems of Ermakov-Ray-Reid type when both noises act simultaneously. We find that the main perturbation effects on the dynamical invariant of these systems are produced by the additive noise. Different from the separate action of the multiplicative noise when the dynamical invariant of these systems is robust, we also find a weak effect that can be attributed to the multiplicative noise.<hr/>Resumen Se investigan numéricamente los efectos de los ruidos aditivos y multiplicativos sobre los sistemas dinámicos de osciladores paramétricos de tipo Ermakov-Ray-Reid cuando los dos tipos de ruidos actúan de manera simultánea. La mayor parte de la perturbación del invariante proviene del ruido aditivo. A diferencia del caso cuando el ruido multiplicativo actúa por separado y el invariante dinámico presenta robustez, encontramos que en la acción simultánea de los dos ruidos hay también un efecto pequeño atribuible al ruido multiplicativo. <![CDATA[Performance of three violins made in Mexico evaluated through standard measurements from a legendary violin]]> Abstract A set of Mexican violins handmade using traditional woods (i.e. spruce for the soundboard, maple for the body, and ebony for the fingerboard) were studied. Standard mobility measurements of these instruments were obtained, and the sound of each violin was recorded when a professional musician played a Bruch Concerto opening excerpt. One of the violins showed a mobility with high harmonic content remarking strong components of high frequencies, resembling the response of old Italian violins, and particularly one made by Stradivari; its sound was the brightest of the set. Meanwhile, other violin exhibited an opposite performance, with weak components of high frequency and having the darkest sound. The performance of the third violin was located between the other two, both in mobility and in its sound. The sound recordings are available for download; so these discussions can be actively judged by the reader. These Mexican violins covered a considerably different range of performance, so there is no reason to consider that violins made in Mexico could have some kind of limitation; quite the opposite, the results of this work show that obtaining a desired dynamical behavior (quantified by mobility measurements) of a new violin is totally feasible. <![CDATA[CO<sub>2</sub> measurement system based on pyroelectric detector]]> Abstract CO2 concentration sensor based on the infrared (IR) absorption is presented, by using a pyroelectric detector the sample absorbed IR radiation is obtained and the CO2 concentration is calculated, system capabilities, sensibility, repeatibility and, time response of developed system are studied, results show that based on a photopyroelectric technique is possible to get an accurate CO2 measurement system. <![CDATA[Solution-processed transparent dielectric based on spin-on glass for electronic devices]]> Abstract In this work, the fabrication and characterization of MOS and MIM capacitors using a solution-processed dielectric based on spin-on glass (SOG) solution is presented. The SOG solution is diluted with deionized water (DI) in order to make easier the evaporation of the organic material presented in the SOG. The films are highly transparent in the visible range, which makes feasible they use in transparent electronics. The capacitance-voltage and current-voltage characteristics of the MOS and MIM capacitors demonstrate the application of the solution-processed SOG/DI film as dielectric for electronic devices. |
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click titles to read articles
Whose Physics Is It Anyway? Q&A with Chanda Prescod-Weinstein
Why physics and astronomy communities must take diversity issues seriously in order to do good science.
Why Time Might Not Be an Illusion
Einstein’s relativity pushes physicists towards a picture of the universe as a block, in which the past, present, and future all exist on the same footing; but maybe that shift in thinking has gone too far.
The Complexity Conundrum
Quantum Dream Time
Our Place in the Multiverse
April 21, 2018
Riding the Rogue Quantum Waves
Could the formation of giant sea swells help explain how the macroscopic world emerges from the quantum microworld?
by Steven Ashley
November 6, 2016
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Thomas Durt
École Centrale de Marseille
In February 1933 the U.S. Navy oiler USS Ramapo was making good time on its run across the South Pacific when an officer spied a monster directly astern on the horizon. A huge rogue wave—a solitary sea swell that is much larger and more powerful than the surrounding waves—rapidly overtook the ship. Later, the Ramapo’s crew, having somehow survived the freak encounter, triangulated the wave’s height at an astounding 34 meters (112 feet)—the tallest rogue wave ever recorded.
Now, a trio of physicists is taking inspiration from such rogue waves—and the model commonly used to describe how they grow to such immense heights—to see if they can help solve one of the biggest mysteries in physics. Supported by a research grant of over $50,000 from FQXi, Thomas Durt of the École Centrale de Marseille, in France, Ralph Willox at the University of Tokyo, in Japan, and Samuel Colin of the Brazilian Center for Physics Research, in Rio de Janeiro, are investigating an alternative to quantum theory which can explain how the definite everyday world we see around us emerges from the uncertain microscopic realm, where objects can be in multiple places at the same time.
In the decade before the Ramapo’s momentous meeting in the South Pacific, leading European theorists had begun laying the foundations of quantum theory. They were grappling with the notion that on small scales, particles can behave as waves, and waves as particles, depending on how they are measured. Stranger still was that a quantum particle-cum-wave has no location until it is observed; only when it is measured does it settle in one spot. In 1926, Austrian physicist Erwin Schrödinger encapsulated this uncertainty by describing quantum objects mathematically as "wavefunctions." Schrödinger’s equation enables physicists to predict the probability of finding the quantum object in a particular place, or indeed with other fixed properties, when they carry out their experiment to measure the object’s features.
According to standard quantum theory, the observer carrying out the experiment in some way causes the collapse of the quantum wave-function, forcing the quantum object to take on definite properties. But nobody can explain how or why that should happen. So Durt, Willox and Colin have turned to rogue ocean waves—which scientists today actually describe using a more complicated version of the Schrödinger equation—for an answer.
Soaking Energy
Although rogue waves have many causes, scientists believe they sometimes develop spontaneously from natural processes that occur amid a random background of smaller waves. Researchers hypothesize that an unusual wave type can form that somehow ’sucks’ energy from surrounding waves to grow to enormous heights. The version of the Schrödinger equation that is used to describe rogue wave formation is described as a "non-linear" equation because—unlike the linear Schrödinger equation that is commonly used in quantum theory—it allows for the possibility that the waves in the system interact with themselves, amplifying effects. One of the simplest models says that through such non-linear processes, a normal ocean wave ’soaks’ energy from the adjacent waves, reducing them to mere ripples as it rises in turn.
Sea Monster
Understanding rogue waves could help unravel a quantum mystery.
Credit: MIT News
Could a similar effect be happening in quantum systems, enabling one type of quantum wave—corresponding to the quantum system being in one place, rather than spread over multiple locations, say—to grow at the expense of others? If so, this could explain how the quantum wavefunction collapses, as this single wave dominates over the others. "We aim to explain this spontaneous localization of the wave based on a process similar to the formation of rogue waves, whose birth is best described by a non-linear wave equation that describes extreme event amplification arising from small perturbations," Durt explains (see Classical and Quantum Gravity 31 (2014)).
Some years back, the mainstream view would have been that this approach is stretching an analogy too far, because subatomic systems and ocean waves are simply too different in character to be treated with the same math. But that’s changing: "Three or four years ago, I would have told you ’no, you will not find rogue wave-like phenomena in quantum mechanics’," notes Majid Taki, a physicist at the Lille University of Science and Technology, in France, who is an expert on non-linear waves in macroscopic environments.
"That’s because at the time we believed that rogue waves come only from highly non-linear conditions," Taki continues. Now, however, new research on rogue waves shows that they can be built in nearly linear systems that have only a small degree of non-linearity, a situation that is much closer to the quantum case. "I think now is the moment to try to find such effects in near-linear systems," says Taki, who is so convinced by the similarities that he advised Durt to pursue this approach.
It means pushing existing
technology to extremes,
which is a good thing.
- Catalina Curceanu
Durt, Willox and Colin hope to develop new ways to test their model by carrying out experiments in an optical trap—a focused laser beam that generates small forces that physically hold tiny objects in empty space like ’optical tweezers.’ The plan is to drop a quantum object, such as a nano-sized sphere, in a gravity-free environment. Ideally, this test would be performed in space because, far from Earth’s gravity, it will be possible to see whether gravitational effects induced between the components of the nanosphere itself (the self-interaction required by the model) causes the object’s wavefunction to collapse (Physical Review A 93, 062102 (2016)).
"This is an ambitious proposal," says FQXi member Catalina Curceanu, a quantum physicist and expert on collapse models at the National Institute of Nuclear Physics in Frascati, Italy. "Such experiments are very difficult because of the extreme precision that’s required." Curceanu says. "It means pushing existing technology to extremes, which is a good thing."
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fae1bc4c5b26a02a | Interaction picture
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In quantum mechanics, the interaction picture (also known as the Dirac picture) is an intermediate representation between the Schrödinger picture and the Heisenberg picture. Whereas in the other two pictures either the state vector or the operators carry time dependence, in the interaction picture both carry part of the time dependence of observables.[1] The interaction picture is useful in dealing with the changes to the wave functions and observable due to the interactions. Most field theoretical calculations[2][3] use the interaction representation because they construct the solution to the many body Schrödinger equation as the solution to the free particle problem plus some unknown interaction part.
Equations that include operators acting at different times, which hold in the interaction picture, don't necessarily hold in the Schrödinger or the Heisenberg picture. This is because time-dependent unitary transformations relate operators in one picture to the analogous operators in the others.
Operators and state vectors in the interaction picture are related by a change of basis (unitary transformation) to those same operators and state vectors in the Schrödinger picture.
To switch into the interaction picture, we divide the Schrödinger picture Hamiltonian into two parts,
Any possible choice of parts will yield a valid interaction picture; but in order for the interaction picture to be useful in simplifying the analysis of a problem, the parts will typically be chosen so that H0,S is well understood and exactly solvable, while H1,S contains some harder-to-analyze perturbation to this system.
If the Hamiltonian has explicit time-dependence (for example, if the quantum system interacts with an applied external electric field that varies in time), it will usually be advantageous to include the explicitly time-dependent terms with H1,S, leaving H0,S time-independent. We proceed assuming that this is the case. If there is a context in which it makes sense to have H0,S be time-dependent, then one can proceed by replacing by the corresponding time-evolution operator in the definitions below.
State vectors
A state vector in the interaction picture is defined as[4]
where |ψS(t)〉is the state vector in the Schrödinger picture.
An operator in the interaction picture is defined as
Note that AS(t) will typically not depend on Template:Mvar, and can be rewritten as just AS. It only depends on Template:Mvar if the operator has "explicit time dependence", for example due to its dependence on an applied, external, time-varying electric field.
Hamiltonian operator
For the operator H0 itself, the interaction picture and Schrödinger picture coincide,
This is easily seen through the fact that operators commute with differentiable functions of themselves. This particular operator then can be called H0 without ambiguity.
For the perturbation Hamiltonian H1,I, however,
where the interaction picture perturbation Hamiltonian becomes a time-dependent Hamiltonian—unless [H1,S, H0,S] = 0 .
It is possible to obtain the interaction picture for a time-dependent Hamiltonian H0,S(t) as well, but the exponentials need to be replaced by the unitary propagator for the evolution generated by H0,S(t), or more explicitly with a time-ordered exponential integral.
Density matrix
The density matrix can be shown to transform to the interaction picture in the same way as any other operator. In particular, let ρI and ρS be the density matrix in the interaction picture and the Schrödinger picture, respectively. If there is probability pn to be in the physical state |ψn〉, then
Evolution Picture
of: Heisenberg Interaction Schrödinger
Ket state constant
Observable constant
Density matrix constant
Time-evolution equations in the interaction picture
Time-evolution of states
Transforming the Schrödinger equation into the interaction picture gives:
This equation is referred to as the SchwingerTomonaga equation.
Time-evolution of operators
If the operator AS is time independent (i.e., does not have "explicit time dependence"; see above), then the corresponding time evolution for AI(t) is given by
In the interaction picture the operators evolve in time like the operators in the Heisenberg picture with the Hamiltonian H' =H0.
Time-evolution of the density matrix
Transforming the Schwinger–Tomonaga equation into the language of the density matrix (or equivalently, transforming the von Neumann equation into the interaction picture) gives:
Use of interaction picture
The purpose of the interaction picture is to shunt all the time dependence due to H0 onto the operators, thus allowing them to evolve freely, and leaving only H1,I to control the time-evolution of the state vectors.
The interaction picture is convenient when considering the effect of a small interaction term, H1,S, being added to the Hamiltonian of a solved system, H0,S. By utilizing the interaction picture, one can use time-dependent perturbation theory to find the effect of H1,I, e.g., in the derivation of Fermi's golden rule, or the Dyson series, in quantum field theory: In 1947, Tomonaga and Schwinger appreciated that covariant perturbation theory could be formulated elegantly in the interaction picture, since field operators can evolve in time as free fields, even in the presence of interactions, now treated perturbatively in such a Dyson series.
1. Albert Messiah (1966). Quantum Mechanics, North Holland, John Wiley & Sons. ISBN 0486409244 ; J. J. Sakurai (1994). Modern Quantum Mechanics (Addison-Wesley) ISBN 9780201539295 .
2. J. W. Negele, H. Orland (1988), Quantum Many-particle Systems,
3. Piers Coleman, The evolving monogram on Many Body Physics
4. The Interaction Picture, lecture notes from New York University
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See also
es:Imagen de evolución temporal |
a5cf4f80afd16295 | Friday, February 27, 2015
Mitochondria: who mentioned God?
Oh, they used the G word. The Guardian put “playing God” in the headline of my article today on mitochondrial replacement, and now everyone on the comments thread starts ranting about God. I’m not sure God has had much to say in this debate so far, and it’s a shame to bring him in now. But for the sake of the record, I’ll just add here what I said about this phrase in my book Unnatural. I hope that some of the people talking about naturalness and about concepts of the soul in relation to embryos might be able to take a peek at that book too. So here’s the extract:
“Time and again, the warning sounded by the theocon agenda is that by intervening in procreation we are ‘playing God’. Paul Ramsey made artful play of this notion in his 1970 book Fabricated Man, saying that ‘Men ought not to play God before they learn to be men, and after they have learned to be men they will not play God.’ To the extent that ‘playing God’ is simply a modern synonym for the accusation of hubris, this charge against anthropoeia is clearly very ancient. Like evocations of Frankenstein, the phrase ‘playing God’ is now no more than lazy, clichéd – and secular – shorthand, a way of expressing the vague threat that ‘you’ll be sorry’. It is telling that this notion of the man-making man becoming a god was introduced into the Frankenstein story not by Mary Shelley but by Hollywood. For ‘playing God’ was never itself a serious accusation levelled at the anthropoetic technologists of old – one could tempt God, offend him, trespass on his territory, but it would have been heretical seriously to entertain the idea that a person could be a god. As theologian Ted Peters has pointed out,
“The phrase ‘playing God?’ has very little cognitive value when looked at from the perspective of a theologian. Its primary role is that of a warning, such as the word ‘stop’. In common parlance it has come to mean just that: stop.’”
And yet, Peters adds, ‘although the phrase ‘playing God’ is foreign to theologians and is not likely to appear in a theological glossary, some religious spokespersons employ the idea when referring to genetics.’ It has, in fact, an analogous cognitive role to the word ‘unnatural’: it is a moral judgement that draws strength from hidden reservoirs while relying on these to remain out of sight.”
OK, there you go. Now here’s the pre-edited article.
It was always going to be a controversial technique. Sure, conceiving babies this way could alleviate suffering, but as a Tory peer warned in the Lords debate, “without safeguards and serious study of safeguards, the new technique could imperil the dignity of the human race, threaten the welfare of children, and destroy the sanctity of family life.” Because it involved the destruction of embryos, the Catholic Church inevitably opposed it. Some scientists warned of the dangers of producing “abnormal babies”, there were comparisons with the thalidomide catastrophe and suggestions that the progeny would be infertile. Might this not be just the beginning of a slippery slope towards a “Frankenstein future” of designer babies?
I’m not talking about mitochondrial replacement and so-called “three person babies”, but about the early days of IVF in the 1970s and 80s, when governments dithered about how to deal with this new reproductive technology. Today, with more than five million people having been conceived by IVF, the term “test-tube baby” seems archaic if not a little perverse (not least because test tubes were never involved). What that debate about assisted conception led to was not the breakup of the family and the birth of babies with deformities, but the formation of the HFEA in the Human Fertilisation and Embryology Act of 1990, providing a clear regulatory framework in the UK for research involving human embryos.
It would be unscientific to argue that, because things turned out fine on that occasion, they will inevitably do so for mitochondrial replacement. No one can be wholly certain what the biological consequences of this technique will be, which is why the HFEA will grant licenses to use it only on the careful worded condition that they are deemed “not unsafe”. But the parallels in the tone of the debate then and now are a reminder of the deep-rooted fears that technological intervention in procreation seems to awaken.
Scientists supportive of such innovations often complain that the opponents are motivated by ignorance and prejudice. They are right to conclude that public engagement is important – in a poll on artificial insemination in 1969, the proportion of people who approved almost doubled when they were informed about the prospects for treating infertility rather than just being given a technical account. But they shouldn’t suppose that science will banish these misgivings. They resurface every time there is a significant advance in reproductive technology: with pre-implantation genetic diagnosis, with the ICSI variant of IVF and so on. They will undoubtedly do so again.
In all these cases, much of the opposition came from people with a strong religious faith. As one of the versions of mitochondrial replacement involves the destruction of embryos, it was bound to fall foul of Catholic doctrine. But rather little was made of that elsewhere, perhaps an acknowledgement that in terms of UK regulation that battle was lost some time ago. (In Italy and the US, say, it is a very different story.) The Archbishops’ Council of the Church of England, for example, stressed that it was worried about the safety and ethical aspects of the technique: the Bishop of Swindon and the C of E’s national adviser for medical ethics warned of “unknown interactions between the DNA in the mitochondria and the DNA in the nucleus [that] might potentially cause abnormality or be found to influence significant personal qualities or characteristics.” Safety is of course paramount in the decision, but the scientific assessments have naturally given it a great deal of attention already.
Lord Deben, who led opposition to the bill in the Lords, addressed this matter head on by denying that his Catholicism had anything to do with it. “I hope no one will say that I am putting this case for any reason other than the one that I put forward,” he said. We can take it on trust that this is what he believes, while finding it surprising that the clear and compelling responses to some of his concerns offered by scientific peers such as Matt Ridley and Robert Winston left him unmoved.
Can it really be coincidental, though, that the many of the peers speaking against the bill are known to have strong religious convictions? Certainly, there are secular voices opposing the technology too, in particular campaigners against genetic manipulations in general such as Marcy Darnovsky of the Center for Genetics and Society, who responded to the ongoing deliberations of the US Food and Drug Administration over mitochondrial transfer not only by flagging up alleged safety issues but also insisting that we consider babies conceived this way to be “genetically modified”, and warning of “mission creep” and “high-tech eugenics”. “How far will we go in our efforts to engineer humans?” she asked in the New York Times.
Parallels between these objections from religious and secular quarters suggest that they reflect a deeper and largely unarticulated sense of unease. We are unlikely to progress beyond the polarization into technological boosterism or conservative Luddites and theologians unless we can get to the core of the matter – which is evidently not scriptural, the Bible being somewhat silent about biotechnological ethics.
Bioethicist Leon Kass, who led the George W. Bush administration’s Council on Bioethics when in 2001 it blocked public funding of most stem-cell research, has argued that instinctive disquiet about some advances in assisted conception and human biotechnology is “the emotional expression of deep wisdom, beyond reason’s power fully to articulate it”: an idea he calls the wisdom of repugnance. “Shallow are the souls”, he says, “that have forgotten how to shudder.” I strongly suspect that, beneath many of the arguments about the safety and legality of mitochondrial replacement lies an instinctive repugnance that is beyond reason’s power to articulate.
The problem, of course, is that what one person recoils from, another sees as a valuable opportunity for human well-being. Yet what are these feelings really about?
Like many of our subconscious fears, they are revealed in the stories we tell. Disquiet at the artificial intervention in procreation goes back a long way: to the tales of Prometheus, of the medieval homunculus and golem, and then to Goethe’s Faust and Shelley’s Victor Frankenstein, E.T.A. Hoffmann’s automaton Olympia, the Hatcheries of Brave New World, modern stories of clones and Ex Machina’s Ava. On the surface these stories seem to interrogate humankind’s hubris in trying to do God’s work; so often they turn out on closer inspection to explore more intimate questions of, say, parenthood and identity. They do the universal job of myth, creating an “other” not as a cautionary warning but in order more safely to examine ourselves. So, for example, when we hear that a man raising a daughter cloned from his wife’s cells (not, I admit, an unproblematic scenario) will be irresistibly attracted to her, we are really hearing about our own horror of incestuous fantasies. Only in Hollywood does Frankenstein’s monster turn bad because he is tainted from the outset by his origins; for Shelley, it is a failure of parenting.
I don’t think it is reading too much into the “three-parent baby” label to see it as a reflection of the same anxieties. Many children already have three effective parents, or more - through step-parents, same-sex relationships, adoption and so forth. When applied to mitochondrial transfer, this term shows how strongly personhood has become equated now with genetics, and indicates to geneticists that they have some work to move the public on from the strictly deterministic view of genetics that the early rhetoric of the field unwittingly fostered.
We can feel justifiably proud that the UK has been the first country to grapple with the issues raised by this new technology. It has shown already that embracing reproductive technologies can be the exact opposite of a slippery slope: what IVF led to was not a Brave New World of designer babies, but a clear regulatory framework that is capable of being permissive and casuistic, not bound by outmoded principles. The UK is not alone in declining to prohibit the technique, but it is right to have made that decision actively.
It is also right that that decision canvassed a wide range of opinions. Some scientists have questioned why religious leaders should be granted any special status in pronouncing on ethics. But the most thoughtful of them often turn out to have a subtle and humane moral sensibility of the kind that faith should require. There is a well-developed strand of philosophical thought on the moral authority of nature, and theology is a part of it. But on questions like this, we have a responsibility to examine our own responses as honestly as we can.
Monday, February 23, 2015
Why dogs aren't enough in Many Worlds
I'm very glad some folks are finding this exchange on Many Worlds instructive. That was really all I wanted: to get a proper discussion of these issues going. The tone that Sean Carroll found “snide and aggressive” was intended as polemical: it’s just a rhetorical style, you know? What I certainly wanted to avoid (forgive me if I didn’t) was any name-calling or implications of stupidity, fraud, chicanery etc. (It doesn’t surprise me that some of the responses failed to do the same.) My experience has been that it is necessary to light a fire under the MWI in order to get a response at all. Indeed, even then it is proving very difficult to keep the feedback to the point and not get led astray by red herrings. For example, Sean made a big point of saying:
I’m genuinely unsure if this is supposed to be referring to me. Since I said in my article
“Certainly, to say that the world(s) surely can’t be that weird is no objection at all”
then I kind of assume it isn’t – so I’m not sure why he brings the point up. I even went to the trouble of trying explicitly to ward off attempts to dismiss my arguments that way:
“Many Worlders harp on about this complaint precisely because it is so easily dismissed.”
But what Sean said next seems to get (albeit obliquely) to the heart of the matter:
“Hilbert space is big, regardless of one’s personal feelings on the matter.”
Whatever these arguments are about, they are surely not about what Hilbert space looks like, since Hilbert space is a mathematical construct – that is simply true by definition, and there is no argument about it. The argument is about what ontological status we ascribe to the state vectors that appear in Hilbert space. I do see the MW reasoning here: the reality we currently experience corresponds to a state vector in Hilbert space, and so why do we have any grounds for denying reality to the other states into which it can evolve by smooth unitary transformation? The problem, of course, is that a single state in quantum mechanics can evolve into multiple states. Yet if we are going to exclude any of those from having objective reality, we surely must have some criterion for doing so. Absent that, we have the MWI. I do understand that reasoning.
So it seems that the arguments could be put like this: is it an additional axiom to say “All states in Hilbert space accessible from an initial one that describes our real world are also describing real worlds” – or is it not? To objectors, it is, and a very expensive one at that. To MWers, it is merely what we do for all theories. “Give us one good reason why it shouldn’t apply here”, they say.
It’s a fair point. One objection, which has nothing whatsoever to do with the vastness of Hilbert space, is to say, well, no one has seriously posited such a vast number of multiple and in some sense “parallel” (initially) worlds before, so it seems fair to ask you to work a bit harder, since don’t we in science say that extraordinary claims require extraordinary evidence?* Might we not ask you to work a bit harder in this particular case to establish the relationship between what the formalism says and what exists in physical reality? After all, whether or not we admit all accessible states in Hilbert space a physical reality, we seem to get identical observational consequences. So right now, the only way we can choose between them is philosophically. And we don’t usually regard philosophy as the final arbiter in science.
*For example, Sean emphasizes that the many worlds are a prediction, not a postulate of the theory. But most other theories (all others?) can tell us some specific things that they don’t predict too about what we will see happen. But I’m not clear if the MWI can rule out any particular thing actually coming to pass that is consistent with the laws of physics. For example, the Copenhagen interpretation (just to take an example) can exclude the “prediction” that human life came to an end following a nuclear conflict sparked by the Bay of Pigs incident. Correct me if I am wrong, but the MWI cannot rule out this “prediction”. It cannot rule out the “prediction” that Many Worlders were never bothered by this irritating science writer. Even if MWI does not exactly say “everything happens”, can it tell us there is anything in particular (consistent with the laws of physics) that does not?
So up to this point, I can appreciate both points of view. What makes me uncomfortable is that the MWers seem so determined to pretend that what they are telling us is actually not so remarkable after all. What’s so surprising, they ask, about the idea that you can instantly duplicate a consciousness, again and again and again? What is frustrating is the blithe insistence that we should believe this, I suspect the most extraordinary claim that science has ever made, on the basis simply of Occam’s (fallible) razor. This is not, do please note, at all the same as worrying about “too many worlds”.
Still, who cares about my discomfort, right? But I wanted to suggest that it’s not just a matter of whether we are prepared to accept this extraordinary possibility. We need to acknowledge that it is rather more complicated than coming to terms with a cute gaggle of sci-fi Doppelgängers. This is not about whether or not people are “all that different from atoms”. It is about whether what people say can be ascribed a coherent meaning. Those responses that have acknowledged this point at all have tended to say “Oh who cares about selfhood and agency? How absurd to expect the theory to deal with unplumbed mysteries like that!” To which I would say that interpretations of quantum theory that don’t have multiple physical worlds don’t even have to think about dealing with them. So perhaps even that Ocaam’s razor argument is more complicated than you think.
It’s been instructive to see that the MWI is something of a hydra: there are several versions, or at least several views on it. Some say that the “worlds” bit is itself a red herring, a bit of gratuitous sci-fi that we could do without. Others insist that the worlds must be actual: Sean says that people must be copied, and that only makes any kind of sense if the world is copied around them. Some say that invoking problems with personhood is irrelevant since Many Worlds would be true anyway even without people in it. (The inconvenience with this argument is that there are people in it.) Sean, interestingly, says that copying people is not only real but essential, “for deriving the Born rule” in MWI. This is a pointer to his fascinating paper on “self-locating uncertainty”. Here he and Charles Sebens points out that, in the MWI where branch states are rendered distinct and non-interacting by decoherence, the finite time required for an observer to register which branch she is on means that there is a tiny but inescapable interval during which she exists as two identical copies but doesn’t know which one she is. In this case, Carool and Sebens argue, the rational way to “apportion credence to the different possibilities” is to use the Born rule, which allows us to calculate from the wavefunction the likelihood of finding a particular result when we make a measurement. This, they say, is why probability seems to come into the situation at all, given that the MWI says that everything that can happen does happen with 100% probability.
This sounds completely bizarre: a rule of quantum physics works because of us? But I think I can see how it makes sense. The universe doesn’t care about the Born rule: it’s not forever calculating “probabilities”. Rather, the Born rule is only needed in our mathematical theory of quantum phenomena – and this argument offers an explanation of why it works when it is put there. Now, there is a bit of heavy pulling still to do in order to get from a “rational way to make predictions while we are caught in that brief instant after the universe has split but before we have been able to determine which branch we are in” and a component of the theory that we use routinely even while we are not agreed that this situation arises in the first place. I’m still not clear how that bit works. Neither is it fully clear to me how we are ever really in that limbo between the universe splitting and us knowing which branch we took, given that, in one view of the Many Worlds at least, the universe has split countless times again during that interval. Maybe the answer would be that all those subsequent split produce versions that are identical with respect to the initial “experiment”, unless they involve processes that interact with the “experiment” and so are part of it anyway. I don’t know.
I do think I can see the answer to my question to Sean (not meant flippantly) of whether it has to be humans who split in order to get the Born rule, and not merely dogs. The answer, I think, is that dogs won’t do because dogs don’t do quantum mechanics. What seems weird is that we’re then left with an aspect of quantum theory that, in this argument, is the way it is not because of some fundamental underlying physical reason so much as because we asked the question in the first place. It feels a bit like Einstein’s moon: was the Born rule true before we invented quantum theory? Or to put it another way, how is consciousness having this agency without appearing explicitly anywhere in the theory? I’m not advancing these as critiques, just saying it seems odd. I’m happy to believe that, within the MWI, the logic of this derivation of the Born rule is sound.
But doesn’t that mean that deriving the Born rule, a longstanding problem in QM, is evidence for the MWI? Sadly not. There are purported derivations within the other interpretations too. None is universally accepted.
The wider point is that, if this is Sean’s reason for insisting we include dividing people in MWI, then the questions about identity raised in my article stand. You know, perhaps they really are trivial? But no one seems to want to say why. This refusal to confront the apparent logical absurdities and contradictions of a theory which predicts that “everything” really happens is curious. It feels as though the MWers find something improper about it – as though this is not quite the respectable business for a physicist who should be contemplating rates of decoherence and the emergence of pointer states and so on. But if you insist on a theory like this, you’re stuck with all its implications – unless, that is, you have some means of “disappearing worlds” that scramble the ability to make meaningful statements about anything.
Saturday, February 21, 2015
Many Worlds: can we make a deal?
OK, picking up from my last post, I think I see a way whereby we can leave this. Advocates of the Many World Interpretation will agree that it does not pretend to say anything about humans and stuff, and that expecting it to do so is as absurd as expecting someone to write down and solve the Schrödinger equation for a football game. They will agree that all those popular (and sometimes technical) books and articles telling us about our alternative quantum selves and Many-Worlds morality and so forth, are just the wilder speculative fringes of the theory that struggle with problems of logical coherence. They agree that statements like DeWitt’s that “every quantum transition taking place on every star, in every galaxy, in every remote corner of the universe is splitting our local world on earth into myriads of copies” aren’t actually what the theory says at all. They acknowledge a bit more clearly that the Alices and Bobs in their papers are just representations of devices that can make an observation (yes, I know this is all they have ever been intended as anyway.) They agree that when they say “The world is described by a quantum state”, they are using “world” in quite a special sense that makes no particular claims about our place(s) or even our existence(s) in it*. They admit that if one tries to broaden this sense of “world”, some difficult conundrums arise. They admit that the mathematical and ontological status of these “worlds” are not the same thing, and that the difference is not resolved by saying that the “worlds” are “really” there in Hilbert space, waiting to be realized.
Then – then – I’m happy to say, sure, the Many Worlds Interpretation, which yes indeed we might better relabel the Everettian Interpretation (shall we begin now?), is a coherent way to think about quantum theory. Possibly even a default way, though I shall want to seek advice on that.
Is that a deal?
*I submit that most physicists and chemists, if they write down the Schrödinger equation for, say, a molecular orbital, are not thinking that they are actually writing down the equation for a “world” but with some bits omitted. One might respond “Well, they should, unless they are content to be “shut up and calculate” scientists”. But I would submit that they are just being good scientists in recognizing the boundaries of the system their equations describe and are not trying to make claims about things they don’t know about or understand.
Friday, February 20, 2015
The latest on the huge number of unobservable worlds
OK, I get the point. Sean Carroll really doesn’t care about problems of the ontology of personhood in the Many World Interpretation. I figured that, as a physicist, these would not be at the forefront of his mind, which is fair enough. But philosophically they are valid questions – which is why David Lewis thought a fair bit about them in his Model Realism theory. It seems to me that a supposedly scientific theory that walks up and says “Sorry, but you are not you – I can’t say what it is you are, but it’s not what you think you are” is obliged to take questions afterwards. I wrote my article in Aeon to try to get those questions, so determinedly overlooked in many expositions of Many Worlds (though clearly acknowledged, if not really addressed, by one of its thoughtful proponents Lev Vaidman) on the table.
But no. We’re not having that, apparently. Sean Carroll’s response doesn’t even mention them. Perhaps he feels as Chad Orzel does: “Who cares? All that stuff is just a collection of foggily defined emergent phenomena that arising from vast numbers of simple quantum systems. Absent a concrete definition, and most importantly a solid idea of how you would measure any of these things, any argument about theories of mind and selfhood and all that stuff is inescapably incoherent.” I’m sort of hoping that isn’t the case. I’m hoping that when Carroll writes of an experiment on a spin superposition being measured by Alice, “There's a version of Alice who saw up and a version who saw down”, he doesn’t really think we can treat Alice – I mean real-world Alices, not the placeholder for a measuring device – like a CCD camera. It’s the business of physics to simplify, but we know what Einstein said about that.
All he picks up on is the objection that I explicitly call minor in comparison: the matter of testing the MWI. His response baffles me:
"The MWI does not postulate a huge number of unobservable worlds, misleading name notwithstanding. (One reason many of us like to call it “Everettian Quantum Mechanics” instead of “Many-Worlds.”) Now, MWI certainly does predict the existence of a huge number of unobservable worlds. But it doesn’t postulate them. It derives them, from what it does postulate."
(I don’t quite get the discomfort with the “Many Worlds” label. It seems to me that is a reasonable name for a theory that “predicts the existence of a huge number of unobservable worlds.” Still, call it what you will.)
I’m missing something here. By and large, scientific theories make predictions, and then we do experiments to see if those predictions are right. MWI predicts “a huge number of worlds”, but apparently it is unreasonable to ask if we might examine that prediction in the laboratory.
But, Carroll says, “You don’t hold it against a theory if it makes some predictions that can’t be tested. Every theory does that. You don’t object to general relativity because you can’t be absolutely sure that Einstein’s equation was holding true at some particular event a billion light years away.” The latter is a non-sequitur: accepting a prediction that can’t be tested is not the same as accepting the possibility of exceptions. And you might reasonably say that there is a difference between accepting a theory even if you can’t get experimentally at what it implies in some obscure corner of parameter space and accepting a theory that “predicts a huge number of unobservable worlds”, some populated by other versions of you doing unobservable things. But OK, might we then have just one prediction that we can test please?
I was dissatisfied with Carroll’s earlier suggestion that you can test MWI just by finding a system that violates the Schrödinger equation or the principle of superposition, because, as I pointed out, it is not a unique interpretation of quantum theory in that regard. His response? “So what?” Alternatives to MWI, he says, have to add to its postulates (or change them), and so they too should predict something we can test. And some do. I understand that Carroll thinks the MWI is uniquely exempt from having to defend its interpretation in particular in the experimental arena, because its axioms are the minimal ones. The point I wanted to raise in my article, though, was that the wider implications of the MWI make it less minimal than its advocates claim. If a “minimal” physical theory predicted something that seemed nonsensical about how cells work, but a more complex theory with an experimentally unsupported postulate took away that problem, would we be right to assert that the minimal theory must be right until there was some evidence for that other postulate? Of course, there may be a good argument for why trashing any coherent notion of self and identity and agency is not a problem. I’d love to hear it. I’d rather it wasn’t just ignored.
“Those worlds happen automatically” – sure, I see that. They are a prediction – sure, I see that. But this point-blank refusal to think any more about them? I don’t get that. Perhaps if Many Worlders were to stop, just stop, trying to tell us anything about how those many unobservable worlds are peopled, to stop invoking copies of Alice as placeholders for quantum measurements, to stop talking about quantum brothers, to say simply that they don’t really have a clue what their interpretation can mean for our notions of identity, then I would rest easier. And so would many, many other physicists. That, I think, would make them a lot happier than being told they don’t understand quantum theory or that they are being silly.
I’m concerned that this sounds like a shot at Sean Carroll. I really don’t want that. Not only is he a lot smarter than me, but he writes so damned well on such intensely interesting stuff. I’m not saying that to flatter him away. I just wanted to get these things discussed.
Many Worlds - a longer view
Here is the pre-edited version of my article for Aeon on the Many Worlds Interpretation of quantum theory. I’m putting it here not because it is any better than the published version (Aeon’s editing was as excellent and improving as ever), but because it gives me a bit more room to go into some of the issues.
In my article I stood up for philosophy. But that doesn’t mean philosophers necessarily get it right either. In the ensuing discussion I have been directed to a talk by philosopher of science David Wallace. Here he criticizes the Copenhagen view that theories are there to make predictions, not to tell us how the world works. He gets a laugh from his audience for suggesting that, if this were so, scientists would have been forced to ask for funding for the LHC not because of what we’d learn from it but so that we could test the predictions made for it.
This is wrong on so many levels. Contrasting “finding out about the world” against “testing predictions of theories” is a totally false opposition. We obviously test predictions of theories to find out if they do a good job of helping us to explain and understand the world. The hope is that the theories, which are obviously idealizations, will get better and better at predicting the fine details of what we see around us, and thereby enable us to tell ever more complete and satisfying stories about why things are this way (and, of course, to allow us to do some useful stuff for “the relief of man’s estate). So there is a sense in which the justification for the LHC derided by Wallace is in fact completely the right one, although that would have been a very poor way of putting it. Almost no one in science (give or take the [very] odd Nobel laureate who capitalizes Truth like some religious crank) talks about “truth” – they recognize that our theories are simply meant to be good working descriptions of what we see, with predictive value. That makes them “true” not in some eternal Platonic sense but as ways of explaining the world that have more validity than the alternatives. No one considers Newtonian mechanics to be “untrue” because of general relativity. So in this regard, Wallace’s attack on the Copenhagen view is trivial. (I don’t doubt that he could put the case better – it’s just that he didn’t do so here.)
What I really object to is the idea, which Wallace repeats, that Many Worlds is simply “what the theory tells you”. To my mind, a theory tells you something if it predicts the corresponding states – say, the electrical current flowing through a circuit, or the reaction rate of an enzymatic process. Wallace asserts that quantum theory “predicts” a you seeing a live Schrödinger’s cat and a you seeing a dead one. I say, show me the equation where those “yous” appear (along with the universes they are in). The best the MWers can do is to say, well, let’s just denote those things as Ψ(live cat) and Ψ(dead cat), with Ψ representing the corresponding universes. Oh please.
Some objectors to my article have been keen to insist that the MWI really isn’t that bizarre: that the other “yous” don’t do peculiar things but are pretty much just like the you-you. I can see how some, indeed many, of them would be. But there is nothing to exclude those that are not, unless you do so by hand: “Oh, the mind doesn’t work that way, they are still rational beings.” What extraordinary confidence this shows in our ability to understand the rules governing human behaviour and consciousness in more parallel worlds than we can possibly imagine: as if the very laws of physics will make sure we behave properly. Collapsing the wavefunction seems a fairly minor sleight of hand (and moreover one we can actually continue to investigate) compared to that. The truth is that we no nothing about the full range of possibilities that the MWI insists on, and nor can we ever do so.
One of the comments underneath my article – and others will doubtless repeat this – makes the remark that Many Worlds is not really about “many universes branching off” at all. Well, I guess you could choose to believe Anonymous Pete instead of Brian Greene and Max Tegmark, if you wish. Or you could follow his link to Sean Carroll’s article, which is one of the examples I cite in my piece of why MWers simple evade the “self” issue altogether.
But you know, my real motivation for writing my article is not to try to bury the MWI (the day I start imagining I am capable of such things, intellectually or otherwise, is the day to put me out to grass), but to provoke its supporters into actually addressing these issues rather than blithely ignoring them while bleating about the (undoubted) problems with the alternatives. Who knows if it will work.
In 2011, participants at a conference on the placid shore of Lake Traunsee in Austria were polled on what the conference was about. You might imagine that this question would have been settled before the meeting was convened – but since the subject was quantum theory, it’s not surprising that there was still much uncertainty. The conference was called “Quantum Physics and the Nature of Reality”, and it grappled with what the theory actually means. The poll, completed by 33 of the participating physicists, mathematicians and philosophers, posed a range of unresolved questions, one of which was “What is your favourite interpretation of quantum mechanics?”
The mere question speaks volumes. Isn’t science supposed to be decided by experiment and observation, free from personal preferences? But experiments in quantum physics have been obstinately silent on what it means. All we can do is develop hunches, intuitions and, yes, favourite ideas.
Which interpretations did these experts favour? There were no fewer than 11 answers to choose from (as well as “other” and “none”). The most popular (42%) was the view put forward by Niels Bohr, Werner Heisenberg and their colleagues in the early days of quantum theory, now known as the Copenhagen Interpretation. In third place (18%) was the Many Worlds Interpretation (MWI).
You might not have heard of most of the alternatives, such as Quantum Bayesianism, Relational Quantum Mechanics, and Objective Collapse (which is not, as you might suppose, saying “what the hell”). Maybe you’ve not heard of the Copenhagen Interpretation either. But the MWI is the one with all the glamour and publicity. Why? Because it tells us that we have multiple selves, living other lives in other universes, quite possibly doing all the things that we dream of but will never achieve (or never dare). Who could resist that idea?
Yet you should. You should resist it not because it is unlikely to be true, or even because, since no one knows how to test it, the idea is not truly scientific at all. Those are valid criticisms, but the main reason you should resist it is that it is not a coherent idea, philosophically or logically. There could be no better contender for Wolfgang Pauli’s famous put-down: it is not even wrong.
Or to put it another way: the MWI is a triumph of canny marketing. That’s not some wicked ploy: no one stands to gain from its success. Rather, its adherents are like giddy lovers, blinded to the flaws beneath the superficial allure.
The measurement problem
To understand how this could happen, we need to see why, more than a hundred years after quantum theory was first conceived, experts are still gathering to debate what it means. Despite such apparently shaky foundations, it is extraordinarily successful. In fact you’d be hard pushed to find a more successful scientific theory. It can predict all kinds of phenomena with amazing precision, from the colours of grass and sky to the transparency of glass, the way enzymes work and how the sun shines.
This is because quantum mechanics, the mathematical formulation of the theory, is largely a technique: a set of procedures for calculating what properties substances have based on the positions and energies of their constituent subatomic particles. The calculations are hard, and for anything more complicated than a hydrogen atom it’s necessary to make simplifications and approximations. But we can do that very reliably. The vast majority of physicists, chemists and engineers who use quantum theory today don’t need to go to conferences on the “nature of reality” – they can do their job perfectly well if, in the famous words of physicist David Mermin, they “shut up and calculate”, and don’t think too hard about what the equations mean.
It’s true that the equations seem to insist on some strange things. They imply that very small entities like atoms and subatomic particles can be in several places at the same time. A single electron can seem to pass through two holes at once, interfering with its own motion as if it was a wave. What’s more, we can’t know everything about a particle at the same time: Heisenberg’s uncertainty principle forbids such perfect knowledge. And two particles can seem to affect one another instantly across immense tracts of space, in apparent (but not actual) violation of Einstein’s theory of special relativity.
But quantum scientists just accept such things. What really divides opinion is that quantum theory seems to do away with the notion, central to science from its beginnings, of an objective reality that we can study “from the outside”, as it were. Quantum mechanics insists that we can’t make a measurement without influencing what we measure. This isn’t a problem of acute sensitivity; it’s more fundamental than that. The most widespread form of quantum maths, devised by Erwin Schrodinger in the 1920s, describes a quantum entity using an abstract concept called a wavefunction. The wavefunction expresses all that can be known about the object. But a wavefunction doesn’t tell you what properties the object has; rather, it enumerates all the possible properties it could have, along with their relative probabilities.
Which of these possibilities is real? Is an electron here or there? Is Schrödinger’s cat alive or dead? We can find out by looking – but quantum mechanics seems to be telling us that the very act of looking forces the universe to make that decision, at random. Before we looked, there were only probabilities.
The Copenhagen Interpretation insists that that’s all there is to it. To ask what state a quantum entity is in before we looked is meaningless. That was what provoked Einstein to complain about God playing dice. He couldn’t abandon the belief that quantum objects, like larger ones we can see and touch, have well defined properties at all times, even if we don’t know what they are. We believe that a cricket ball is red even if we don’t look at it; surely electrons should be no different? This “measurement problem” is at the root of the arguments.
Avoiding the collapse
The way the problem is conventionally expressed is to say that measurement – which really means any interaction of a particle with another system that could be used to deduce its state – “collapses” the wavefunction, extracting a single outcome from the range of probabilities that the wavefunction encodes. But the quantum mechanics offers no prescription for how this collapse occurs; it has to be put in by hand. That’s highly unsatisfactory.
There are various ways of looking at this. A Copenhagenist view might be simply to accept that wavefunction collapse is an additional ingredient of the theory, which we don’t understand. Another view is to suppose that wavefunction collapse isn’t just a mathematical sleight-of-hand but an actual, physical process, a little like radioactive decay of an atom, which could in principle be observed if only we had an experimental technique fast and sensitive enough. That’s the Objective Collapse interpretation, and among its advocates is Roger Penrose, who suspects that the collapse process might involve gravity.
Proponents of the Many Worlds Interpretation are oddly reluctant to admit that their preferred view is simply another option. They often like to insist that There Is No Alternative – that the MWI is the only way of taking quantum theory seriously. It’s surprising, then, that in fact Many Worlders don’t even take their own view seriously enough.
That view was presented in the 1957 doctoral thesis of the American physicist Hugh Everett. He asked why, instead of fretting about the cumbersome nature of wavefunction collapse, we don’t just do away with it. What if this collapse is just an illusion, and all the possibilities announced in the wavefunction have a physical reality? Perhaps when we make a measurement we only see one of those realities, yet the others have a separate existence too.
An existence where? This is where the many worlds come in. Everett himself never used that term, but his proposal was championed in the 1970s by the physicist Bryce De Witt, who argued that the alternative outcomes of the experiment must exist in a parallel reality: another world. You measure the path of an electron, and in this world it seems to go this way, but in another world it went that way.
That requires a parallel, identical apparatus for the electron to traverse. More, it requires a parallel you to measure it. Once begun, this process of fabrication has no end: you have to build an entire parallel universe around that one electron, identical in all respects except where the electron went. You avoid the complication of wavefunction collapse, but at the expense of making another universe. The theory doesn’t exactly predict the other universe in the way that scientific theories usually make predictions. It’s just a deduction from the hypothesis that the other electron path is real too.
This picture really gets extravagant when you appreciate what a measurement is. In one view, any interaction between one quantum entity and another – a photon of light bouncing off an atom – can produce alternative outcomes, and so demands parallel universes. As DeWitt put it, “every quantum transition taking place on every star, in every galaxy, in every remote corner of the universe is splitting our local world on earth into myriads of copies”.
Recall that this profusion is deemed necessary only because we don’t yet understand wavefunction collapse. It’s a way of avoiding the mathematical ungainliness of that lacuna. “If you prefer a simple and purely mathematical theory, then you – like me – are stuck with the many-worlds interpretation,” claims MIT physicist Max Tegmark, one of the most prominent MWI popularizers. That would be easier to swallow if the “mathematical simplicity” were not so cheaply bought. The corollary of Everett’s proposal is that there is in fact just a single wavefunction for the entire universe. The “simple maths” comes from representing this universal wavefunction as a symbol Ψ: allegedly a complete description of everything that is or ever was, including the stuff we don’t yet understand. You might sense some issues being swept under the carpet here.
What about us?
But let’s stick with it. What are these parallel worlds like? This hinges on what exactly the “experiments” that produce or differentiate them are. So you’d think that the Many Worlders would take care to get that straight. But they’re oddly evasive, or maybe just relaxed, about it. Even one of the theory’s most thoughtful supporters, Russian-Israeli physicist Lev Vaidman, seems to dodge the issue in his entry on the MWI in the Stanford Encyclopedia of Philosophy:
“Quantum experiments take place everywhere and very often, not just in physics laboratories: even the irregular blinking of an old fluorescent bulb is a quantum experiment.”
Vaidman stresses that every world has to be formally accessible from the others: it has to be derived from one of the alternatives encoded in the wavefunction of one of the particles. You could say that the universes are in this sense all connected, like stations on the London Underground. So what does this exclude? Nobody knows, and there is no obvious way of finding out.
I put the question directly to Lev: what exactly counts as an experiment? An event qualifies, he replied “if it leads to more than one ‘story’”. He added: “If you toss a coin from your pocket, does it split the world? Say you see tails – is there parallel world with heads?” Well, that was certainly my question. But I was kind of hoping for an answer.
Most popularizers of the MWI are less reticent. In the “multiverse” of the Many Worlds view, says Tegmark, “all possible states exist at every instant”. One can argue about whether that’s the quite same as DeWitt’s version, but either way the result seems to accord with the popular view that everything that is physically possible is realized in one of the parallel universes.
The real problem, however, is that Many Worlders don’t seem keen to think about what this means. No, that’s too kind. They love to think about what it means – but only insofar as it lets them tell us wonderful, lurid and beguiling stories. The MWI seduces us by multiplying our selves beyond measure, giving us fantasy lives in which there is no obvious limit to what we can do. “The act of making a decision”, says Tegmark – a decision here counting as an experiment – “causes a person to split into multiple copies.”
That must be a pretty big deal, right? Not for theoretical physicist Sean Carroll of the California Institute of Technology, whose article “Why the Many-Worlds formulation of quantum mechanics is probably correct” on his popular blog Preposterous Universe makes no mention of these alter egos. Oh, they are there in the background all right – the “copies” of the human observer of a quantum event are casually mentioned in the midst of the 40-page paper by Carroll that his blog cites. But they are nothing compared with the relief of having to fret about wavefunction collapse. It’s as though the burning question about the existence of ghosts is whether they observe the normal laws of mechanics, rather than whether they would radically change our view of our own existence.
But if some Many Worlders are remarkably determined to avert their eyes, others delight in this multiplicity of self. They will contemplate it, however, only insofar as it lets them tell us wonderful, lurid and beguiling stories about fantasy lives in which there is no obvious limit to what we can do, because indeed in some world we’ve already done it.
Most MWI popularizers think they are blowing our minds with this stuff, whereas in fact they are flattering them. They delve into the implications for personhood just far enough to lull us with the uncanniness of the centuries-old Doppelgänger trope, and then flit off again. The result sounds transgressively exciting while familiar enough to be persuasive.
Identity crisis
In what sense are those other copies actually “us”? Brian Greene, another prominent MW advocate, tells us gleefully that “each copy is you.” In other words, you just need to broaden your mind beyond your parochial idea of what “you” means. Each of these individuals has its own consciousness, and so each believes he or she is “you” – but the real “you” is their sum total. Vaidman puts the issue more carefully: all the copies of himself are “Lev Vaidman”, but there’s only one that he can call “me”.
““I” is defined at a particular time by a complete (classical) description of the state of my body and of my brain”, he explains. “At the present moment there are many different “Levs” in different worlds, but it is meaningless to say that now there is another “I”.” Yet it is also scientifically and, I think, logically meaningless to say that there is an “I” at all in his definition, given that we must assume that any “I” is generating copies faster than the speed of thought. A “complete description” of the state of his body and brain never exists.
What’s more, this half-baked stitching together of quantum wavefunctions and the notion of mind leads to a reductio ad absurdum. It makes Lev Vaidman a terrible liar. He is actually a very decent fellow and I don’t want to impugn him, but by his own admission it seems virtually inevitable that “Lev Vaidman” has in other worlds denounced the MWI as a ridiculous fantasy, and has won a Nobel prize for showing, in the face of prevailing opinion, that it is false. (If these scenarios strike you as silly or frivolous, you’re getting the point.) “Lev Vaidman” is probably also a felon, for there is no prescription in the MWI for ruling out a world in which he has killed every physicist who believes in the MWI, or alternatively, every physicist who doesn’t. “OK, those Levs exist – but you should believe me, not them!” he might reply – except that this very belief denies the riposte any meaning.
The difficulties don’t end there. It is extraordinary how attached the MWI advocates are to themselves, as if all the Many Worlds simply have “copies” leading other lives. Vaidman’s neat categorization of “I” and “Lev” works because it sticks to the tidy conceit that the grown-up "I" is being split into ever more "copies" that do different things thereafter. (Not all MWI descriptions will call this copying of selves "splitting" - they say that the copies existed all along - but that doesn't alter the point.)
That isn't, however, what the MWI is really about – it's just a sci-fi scenario derived from it. As Tegmark explains, the MWI is really about all possible states existing at every instant. Some of these, it’s true, must contain essentially indistinguishable Maxes doing and seeing different things. Tegmark waxes lyrical about these: “I feel a strong kinship with parallel Maxes, even though I never get to meet them. They share my values, my feelings, my memories – they’re closer to me than brothers.”
He doesn't trouble his mind about the many, many more almost-Maxes, near-copies with perhaps a gene or two mutated – not to mention the not-much-like Maxes, and so on into a continuum of utterly different beings. Why not? Because you can't make neat ontological statements about them, or embrace them as brothers. They spoil the story, the rotters. They turn it into a story that doesn't make sense, that can't even be told. So they become the mad relatives in the attic. The conceit of “multiple selves” isn’t at all what the MWI, taken at face value, is proposing. On the contrary, it is dismantling the whole notion of selfhood – it is denying any real meaning of “you” at all.
Is that really so different from what we keep hearing from neuroscientists and psychologists – that our comforting notions of selfhood are all just an illusion concocted by the brain to allow us to function? I think it is. There is a gulf between a useful but fragile cognitive construct based on measurable sensory phenomena, and a claim to dissolve all personhood and autonomy because it makes the maths neater. In the Borgesian library of Many Worlds, it seems there can be no fact of the matter about what is or isn’t you, and what you did or didn’t do.
State of mind
Compared with these problems, the difficulty of testing the MWI experimentally (which would seem a requirement of it being truly scientific) is a small matter. ‘It’s trivial to falsify [MWI]’, boasts Carroll: ‘just do an experiment that violates the Schrödinger equation or the principle of superposition, which are the only things the theory assumes.’ But most other interpretations of quantum theory assume them (at least) too – so an experiment like that would rule them all out, and say nothing about the special status of the MWI. No, we’d quite like to see some evidence for those other universes that this particular interpretation uniquely predicts. That’s just what the hypothesis forbids, you say? What a nuisance.
Might this all simply be a habit of a certain sort of mind? The MWI has a striking parallel in analytic philosophy that goes by the name of modal realism. Ever since Gottfried Leibniz argued that the problem of good and evil can be resolved by postulating that ours is the best of all possible worlds, the notion of “possible worlds” has supplied philosophers with a scheme for debating the issue of the necessity or contingency of truths. The American philosopher David Lewis pushed this line of thought to its limits by asserting, in the position called model realism, that all worlds that are possible have a genuine physical existence, albeit isolated causally and spatiotemporally from ours. On what grounds? Largely on the basis that there is no logical reason to deny their existence, but also because accepting this leads to an economy of axioms: you don’t have to explain away their non-existence. Many philosophers regard this as legerdemain, but the similarities with the MWI of quantum theory are clear: the proposition stems not from any empirical motive but simply because it allegedly simplifies matters (after all, it takes only four words to say “everything possible is real”, right?). Tegmark’s so-called Ultimate Ensemble theory – a many-worlds picture not explicitly predicated on quantum principles but still including them – has been interpreted as a mathematical expression of modal realism, since it proposes that all mathematical entities that can be calculated in principle (that is, which are possible in the sense of being “computable”) must be real. Lewis’s modal realism does, however, at least have the virtue that he thought in some detail about the issues of personal identity it raises.
If I call these ideas fantasies, it is not to deride or dismiss them but to keep in view the fact that beneath their apparel of scientific equations or symbolic logic they are acts of imagination, of “just supposing”. Who can object to imagination? Not me. But when taken to the extreme, parallel universes become a kind of nihilism: if you believe everything then you believe nothing. The MWI allows – perhaps insists – not just on our having cosily familial ‘quantum brothers’ but on worlds where gods, magic and miracles exist and where science is inevitably (if rarely) violated by chance breakdowns of the usual statistical regularities of physics.
Certainly, to say that the world(s) surely can’t be that weird is no objection at all; Many Worlders harp on about this complaint precisely because it is so easily dismissed. MWI doesn’t, though, imply that things really are weirder than we thought; it denies us any way of saying anything, because it entails saying (and doing) everything else too, while at the same time removing the “we” who says it. This does not demand broad-mindedness, but rather a blind acceptance of ontological incoherence.
That its supporters refuse to engage in any depth with the questions the MWI poses about the ontology and autonomy of self is lamentable. But this is (speaking as an ex-physicist) very much a physicist’s blind spot: a failure to recognize, or perhaps to care, that problems arising at a level beyond that of the fundamental, abstract theory can be anything more than a minor inconvenience.
If the MWI were supported by some sound science, we would have to deal with it – and to do so with more seriousness than the merry invention of Doppelgängers to measure both quantum states of a photon. But it is not. It is grounded in a half-baked philosophical argument about a preference to simplify the axioms. Until Many Worlders can take seriously the philosophical implications of their vision, it’s not clear why their colleagues, or the rest of us, should demur from the judgement of the philosopher of science Robert Crease that the MWI is ‘one of the most implausible and unrealistic ideas in the history of science’ [see The Quantum Moment, 2014]. To pretend that the only conceptual challenge for a theory that allows everything conceivable to happen (or at best fails to provide any prescription for precluding the possibilities) is to accommodate Sliding Doors scenarios shows a puzzling lacuna in the formidable minds of its advocates. Perhaps they should stop trying to tell us that philosophy is dead.
Monday, February 16, 2015
General relativity's big year?
For the record, my op-ed in the International New York Times.
You might think that physicists would be satisfied by now. They have been testing Einstein’s theory of general relativity (GR), which explains what gravity is, ever since he first described it one hundred years ago this year. And not once has it been found wanting. But they are still investigating its predictions to the nth decimal place, and this centenary year should see some particularly stringent tests. Perhaps one will uncover the first tiny flaw in this awesome mathematical edifice.
Stranger still is that, although GR is celebrated and revered among physicists like no other theory in science, they would doubtless react with joy if it is proved to fail. That’s science: you produce a smart idea and then test it to breaking point. But this determination to expose flaws isn’t really about skepticism, far less wanton nihilism. Most physicists are already convinced that GR is not the final word on gravity. That’s because the theory, which is applied mostly at the scale of stars and galaxies, doesn’t mesh with quantum theory, the other cornerstone of modern physics, which describes the ultra-small world of atoms and subatomic particles. It’s suspected that underlying both theories is a theory of quantum gravity, from which GR and conventional quantum theory emerge as excellent approximations just as Isaac Newton’s theory of gravity, posed in the late seventeenth century, works fine except in some extreme situations.
The hope is, then, that if we can find some dark corner of the universe where GR fails, perhaps because the gravitational fields it describes are so enormously strong, we might glimpse what extra ingredient is needed – one that might point the way to a theory of quantum gravity.
General relativity was not just the last of Einstein’s truly magnificent ideas, but arguably the greatest of them. His annus mirabilis is usually cited as 1905, when, among other things, he kick-started quantum theory and came up with special relativity, describing the distortion of time and space caused by travelling close to the speed of light. General relativity offered a broader picture, embracing motion that changes speed, such as objects accelerating as they fall in a gravitational field. Einstein explained that gravity can be thought of as curvature induced in the very fabric of time and space by the presence of a mass. This too distorts time: clocks run slower in a strong gravitational field than they do in empty space. That’s one prediction that has now been thoroughly confirmed by the use of extremely accurate clocks on space satellites, and in fact GPS systems have to adjust their clocks to allow for it.
Einstein presented his theory of GR to the Prussian Academy of Sciences in 1915, although it wasn’t officially published until the following year. The theory also predicted that light rays will be bent by strong gravitational fields. In 1919 the British astronomer Arthur Eddington confirmed that idea by making careful observations of the positions of stars whose light passes close to the sun during a total solar eclipse. The discovery assured Einstein as an international celebrity. When he met Charlie Chaplin in 1931, Chaplin is said to have told Einstein that the crowds cheered them both because everyone understood him and no one understood Einstein.
General relativity predicts that some burnt-out stars will collapse under their own gravity. They might become incredibly dense objects called neutron stars only a few miles across, from which a teaspoon of matter would weigh around 10 billion tons. Or they might collapse without limit into a “singularity”: a black hole, from whose immense gravitational field not even light can escape, since the surrounding space is so bent that light just turns back on itself. Many neutron stars have now been seen by astronomers: some, called pulsars, rotate and send out beams of intense radio waves from their magnetic poles, lighthouse beams that flash on an off with precise regularity when seen from afar. Black holes can only be seen indirectly from the X-rays and other radiation emitted by the hot gas that surrounds and is sucked into them. But astrophysicists are certain that they exist.
While Newton’s theory of gravity is mostly good enough to describe the motions of the solar system, it is around very dense objects like pulsars and black holes that GR becomes indispensible. That’s also where it might be possible to test the limits of GR with astronomical investigations. Last year, astronomers at the National Radio Astronomy Observatory in Charlottesville, Virginia, discovered the first pulsar orbited by two other shrunken stars, called white dwarfs. This situation, with two bodies moving in the gravitational field of a third, should allow one of the central pillars of GR, called the strong equivalence principle, to be put to the test by making very detailed measurements of the effects of the white dwarfs on the pulsar’s metronome flashes as they circulate. The team hopes to carry out that study this year.
But the highest-profile test of GR is the search for gravitational waves. The theory predicts that some astrophysical processes involving very massive bodies, such as supernovae (exploding stars) or pulsars orbited by another star (binary pulsars), should excite ripples in space-time that radiate outwards as waves. The first binary pulsar was discovered in 1974, and we now know the two bodies are getting slowly closer at just the rate expected if they are losing energy by radiating gravitational waves.
The real goal, though, is to see such waves directly from the tiny distortions of space that they induce as they ripple past our planet. Gravitational-wave detectors use lasers bouncing off mirrors in two kilometres-long arms at right angles, like an L, to measure such minuscule contractions or stretches. Two of the several gravitational-wave detectors currently built – the American LIGO, with two observatories in Louisiana and Washington, and the European VIRGO in Italy – have just been upgraded to boost their sensitivity, and both will start searching in 2015. The European Space Agency is also launching a pilot mission for a space-based detector, called LISA Pathfinder, this September.
If we’re lucky, then, 2015 could be the year we confirm both the virtues and the limits of GR. But neither will do much to alter the esteem with which it is regarded. The Austrian-Swiss physicist Wolfgang Pauli called it “probably the most beautiful of all existing theories.” Many physicists (including Einstein himself) believed it not so much because of the experimental teats but because of what they perceived as its elegance and simplicity. Anyone working on quantum gravity knows that it is a very hard act to follow.
Holding Rome together
Here’s my latest Material Witness column for Nature Materials.
Calling it the world’s earliest shopping mall is perhaps a qualified accolade, but Trajan’s Market in Rome is certainly a remarkable structure. These vaulted arcades, built early in the second century AD and perhaps originally administrative offices, have withstood almost two millennia of moderate-scale earthquakes. They aren’t alone in that: the Pantheon, Hadrian’s Mausoleum and the Baths of Diocletian in Rome have all shown comparable longevity and resilience. What is their secret?
The structures use concrete made from the pyroclastic volcanic rock of the region: coarse rubble of tuff and brick bound with a mortar made from volcanic ash. It is this mortar that provides structural stability, but the properties that give it such durability have only now been examined. Jackson et al. [Proc. Natl Acad. Sci. USA 111, 18484 (2014) – here] have reproduced the mortar used by Roman builders and used microdiffraction and tomography to study how it acquires its remarkable cohesion.
The Roman mortar was the result of a century or more of experimentation. It used pozzolan, an aluminosilicate volcanic pumice found in the region of the town of Pozzuoli, near Naples, which, when mixed with slaked lime (calcium hydroxide) in the presence of moisture, recrystallizes into a hydrated cementitious material. Although named for its Roman use, pozzolan has a much longer history in building and remained in use until the introduction of Portland cements in the eighteenth century.
The production of the volcanic ash–lime cement was described by the Roman engineer Vitruvius in his book an architecture from the first century BC, and Jackson et al. followed his recipe to make modern analogues. They found that the tensile strength and fracture energy increased steadily over several months, and used electron microscopy and synchrotron X-ray diffraction to look at the fracture surfaces and the chemical nature of the cementitious phases. Among the poorly crystalline matrix are platey crystals of a calcium aluminosilicate phase called strätinglite, crystallized in situ, which seem to act rather like the steel or glass microfibres added to some cements today to toughen them by providing obstacles to crack propagation. Unlike them, however, strätlingite resists corrosion.
Since the cement industry is a major producer of carbon dioxide liberated during the production of Portland cement, there is considerable interest in finding environmentally friendly alternatives. Some of these have a binding matrix of similar composition to the Roman mortar, and so Jackson et al. suggest that an improved understanding of what makes it so durable could point to approaches worth adopting today – such as using chemical additives that promote the intergrowth of reinforcing platelets.
Of course, the Roman engineers knew of the superior properties of their mortar only by experience. A similar combination of astute empiricism and good fortune lies behind the medieval lime mortars that, because of their slow setting, have preserved some churches and other buildings in the face of structural shifting. They tempt us to celebrate the skills of ancient artisans, but we should also remember that what we see today is selective: time has already levelled the failures. |
e640d6c30b0d791d | Time's arrow and Boltzmann's entropy
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Joel L. Lebowitz (2008), Scholarpedia, 3(4):3448. doi:10.4249/scholarpedia.3448 revision #137152 [link to/cite this article]
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Curator: Joel L. Lebowitz
The arrow of time expresses the fact that in the world about us the past is distinctly different from the future. Milk spills but doesn't unspill; eggs splatter but do not unsplatter; waves break but do not unbreak; we always grow older, never younger. These processes all move in one direction in time - they are called "time-irreversible" and define the arrow of time. It is therefore very surprising that the relevant fundamental laws of nature make no such distinction between the past and the future. This in turn leads to a great puzzle - if the laws of nature permit all processes to be run backwards in time, why don't we observe them doing so? Why does a video of an egg splattering run backwards look ridiculous? Put another way: how can time-reversible motions of atoms and molecules, the microscopic components of material systems, give rise to the observed time-irreversible behavior of our everyday world? The resolution of this apparent paradox is due to Maxwell, Thomson and (particularly) Boltzmann. These ideas also explain most other arrows of time - in particular; why do we remember the past but not the future?
What is time
Time is arguably among the most primitive concepts we have—there can be no action or movement, no memory or thought, except in time. Of course this does not mean that we understand, whatever is meant by that loaded word "understand", what time is. As put by Saint Augustine.
In a book entitled Time's Arrow and Archimedes' Point the Australian philosopher Huw Price describes well the ``stock philosophical debates about time. These have not changed much since the time of Saint Augustine or even earlier.
"... Philosophers tend to be divided into two camps. On one side there are those who regard the passage of time as an objective feature of reality, and interpret the present moment as the marker or leading edge of this advance. Some members of this camp give the present ontological priority, as well, sharing Augustine's view that the past and the future are unreal. Others take the view that the past is real in a way that the future is not, so that the present consists in something like the coming into being of determinate reality. .... Philosophers in the opposing camp regard the present as a subjective notion, often claiming that now is dependent on one's viewpoint in much the same way that here is. Just as "here" means roughly "this place", so "now" means roughly "this time", and in either case what is picked out depends where the speaker stands. In this view there is no more an objective division of the world into the past, the present, and the future than there is an objective division of a region of space into here and there.
Often this is called the block universe view, the point being that it regards reality as a single entity of which time is an ingredient, rather than as a changeable entity set in time."
A very good description of the block universe point of view is given by Kurt Vonnegut in his novel Slaughterhouse-Five. The coexistence of past, present and future forms one of the themes of the book. The hero, Billy Pilgrim, speaks of the inhabitants of Tralfamadore a planet in a distant galaxy: "The Tralfamadorians can look at all different moments just the way we can look at a stretch of the Rocky Mountains, for instance. They can see how permanent all the moments are, and they can look at any moment that interests them. It is just an illusion we have here on earth that one moment follows another like beads on a string, and that once a moment is gone it is gone forever."
This view (with relativity properly taken into account) is certainly the one held by most physicists—at least when they think as physicists. It is well expressed in the often quoted passage from Einstein's letter of condolences upon the death of his youthful best friend Michele Besso: "Michele has left this strange world just before me. This is of no importance. For us convinced physicists the distinction between past, present and future is an illusion, although a persistent one."
There are however also more radical views about time among physicists. At a conference on the Physical Origins of Time Asymmetry which took place in Mazagon, Spain, in 1991, the physicist Julian Barbour conducted an informal poll about whether time is fundamental. Here is Barbour's account of that from his book The End of Time.
"During the Workshop, I conducted a very informal straw-poll, putting the following question to each of the 42 participants: Do you believe time is a truly basic concept that must appear in the foundations of any theory of the world, or is it an effective concept that can be derived from more primitive notions in the same way that a notion of temperature can be recovered in statistical mechanics?
The results were as follows: 20 said there was no time at a fundamental level, 12 declared themselves to be undecided or wished to abstain, and 10 believed time did exist at the most basic level. However, among the 12 in the undecided/abstain column, 5 were sympathetic to or inclined to the belief that time should not appear at the most basic level of theory."
Matter in space-time
In this article, the intuitive notion of space-time as a primitive undefined concept is taken as a working hypothesis. This space-time continuum is the arena in which matter, radiation and all kinds of other fields exist and change.
Many of these changes have a uni-directional order "in time", or display an arrow of time. One might therefore expect, as Feynman puts it, that there is some fundamental law which says, that "uxels only make wuxels and not vice versa." But we have not found such a law.... "so this manifest fact of our experience is not part of the fundamental laws of physics." The fundamental microscopic laws (with some, presumably irrelevant, exceptions) all turn out to be time symmetric. Newton's laws, the Schrödinger equation, the special and general theory of relativity, etc., make no distinction between the past and the future—they are "time-symmetric". As put by Brian Greene in his book "The Fabric of the Cosmos: Space, Time and the Structure of Reality", "no one has ever discovered any fundamental law which might be called the Law of the Spilled Milk or the Law of the Splattered Egg."
It is only secondary laws, which describe the behavior of macroscopic objects containing many, many atoms, such as the second law of thermodynamics, (discussed below), which explicitly contain this time asymmetry. The obvious question then is; how does one go from a time symmetric description of the dynamics of atoms to a time asymmetric description of the evolution of macroscopic systems made up of atoms.
In answering that question, one may mostly ignore relativity and quantum mechanics. These theories, while essential for understanding both the very large scale and the very small scale structure of the universe, have a "classical limit" which is adequate for a basic understanding of time's arrow. One may also for simplicity ignore waves, made up of photons, and any entities smaller than atoms and talk about these atoms as if they were point particles interacting with each other via some pair potential, and evolving according to Newtonian laws.
In the context of Newtonian theory, the "theory of everything" at the time of Thomson, Maxwell and Boltzmann, the problem can be formally presented as follows: The complete microscopic (or micro) state of a classical system of \(N\) particles, is represented by a point \(X\) in its phase space \(\Gamma\ ,\) \( X =(r_1, p_1, r_2, p_2, ..., r_N, p_N), r_i\) and \(p_i\) being three dimensional vectors representing the position and momentum (or velocity) of the \(i\)th particle. When the system is isolated, say in a box \(V\) with reflecting walls, its evolution is governed by Hamiltonian dynamics with some specified Hamiltonian \(H(X)\) which we will assume for simplicity to be an even function of the momenta: no magnetic fields. Given \(H(X)\ ,\) the microstate \(X(t_0)\ ,\) at time \(t_0\ ,\) determines the microstate \(X(t)\) at all future and past times \(t\) during which the system will be or was isolated. Let \(X(t_0)\) and \(X(t_0+\tau)\ ,\) with \(\tau\) positive, be two such microstates. Reversing (physically or mathematically) all velocities at time \(t_0+\tau\ ,\) we obtain a new microstate, \(RX\ .\) \[ RX = (r_1,-p_1, r_2,-p_2, ...,r_N,-p_N). \] If we now follow the evolution for another interval \(\tau\) we find that the new microstate at time \(t_0 + 2\tau\) is just \(RX(t_0)\ ,\) the microstate \(X(t_0)\) with all velocities reversed: Hence if there is an evolution, i.e. a trajectory \(X(t)\ ,\) in which some property of the system, specified by a function \(f(X(t))\ ,\) behaves in a certain way as \(t\) increases, then if \(f(X) = f(RX)\) there is also a trajectory in which the property evolves in the time reversed direction. So why is one type of evolution, the one consistent with an entropy increase in accord with the "second law" of thermodynamics, common and the other never seen?
An example of the entropy increasing evolution is the approach to a uniform temperature of systems initially kept isolated at different temperatures, as exemplified by putting a glass of hot tea and a glass of cold water into an insulated container. It is common experience that after a while the two glasses and their contents will come to the same temperature.
This is one of the "laws" of thermodynamics, a subject developed in the eighteenth and nineteenth century, purely on the basis of macroscopic observations—primarily the workings of steam engines—so central to the industrial revolution then taking place. Thermodynamics makes no reference to atoms and molecules, and its validity remains independent of their existence and nature—classical or quantum. The high point in the development of thermodynamics came in 1865 when Rudolf Clausius pronounced his famous two fundamental theorems: 1. The energy of the universe is constant. 2. The entropy of the universe tends to a maximum.
The "second law" says that there is a quantity called entropy associated with macroscopic systems which can only increase, never decrease, in an isolated system. In Clausius' poetic language, the paradigm of such an isolated system is the universe itself. But even leaving aside the universe as a whole and just considering our more modest example of two glasses of water in an insulated container, this is clearly a law which is asymmetric in time. Entropy increase is identified with heat flowing from hot to cold regions leading to a uniformization of the temperature. But, if we look at the microscopic dynamics of the atoms making up the systems then, as noted earlier, if the energy density or temperature inside a box \(V\) gets more uniform as time increases, then, since the energy density profile is the same for \(X\) and \(RX\ ,\) there is also an evolution in which the temperature gets more nonuniform.
There is thus clearly a difficulty in deriving or showing the compatibility of the second law with the microscopic dynamics. This is illustrated by the impossibility of time ordering of the snapshots in {Fig. 1} using solely the microscopic dynamical laws: the time symmetry of the microscopic dynamics implies that if (a, b, c, d) is a possible ordering so is (d, c, b, a).
Figure 1: A sequence of "snapshots", a, b, c, d taken at times \(t_a, t_b, t_c, t_d\ ,\) each representing a macroscopic state of a system, say a fluid with two "differently colored" atoms or a gas in which the shading indicates the local density. How would one order this sequence in time?.
The explanation of this apparent paradox, due to Thomson, Maxwell and Boltzmann, shows that not only is there no conflict between reversible microscopic laws and irreversible macroscopic behavior, but, as clearly pointed out by Boltzmann in his later writings, there are extremely strong reasons to expect the latter from the former. (Boltzmann's early writings on the subject are sometimes unclear, wrong, and even contradictory. His later writings, however, are generally very clear). These reasons involve several interrelated ingredients which together provide the required distinction between microscopic and macroscopic variables and explain the emergence of definite time asymmetric behavior in the evolution of the latter despite the total absence of such asymmetry in the dynamics of the former.
To describe the macroscopic state of a system of \(N\) atoms in a box \(V\ ,\) say \( N {}^>_\sim 10^{20}\ ,\) we make use of a much cruder description than that provided by the microstate \(X\ .\) We shall denote by \(M\) such a macroscopic description or macrostate. As an example we may take \(M\) to consist of the specification, to within a given accuracy, of the energy and number of particles in each half of the box \(V\ .\) A more refined macroscopic description would divide \(V\) into \(K\) cells, where \(K\) is large but still \(K << N\ ,\) and specify the number of particles, the momentum, and the amount of energy in each cell, again with some tolerance.
Clearly \(M\) is determined by \(X\) but there are many \(X\)'s (in fact a continuum) which correspond to the same \(M\ .\) Let \(\Gamma_M\) be the region in \(\Gamma\) consisting of all microstates \(X\) corresponding to a given macrostate \(M\) and denote by \(|\Gamma_M|=(N! h^{3N})^{-1} \int_{\Gamma_M}\prod_{i=1}^Ndr_{i}p_i\ ,\) its symmetrized \(6N\) dimensional Liouville volume in units of \(h^{3N}\ .\) At this point this is simply an arbitrary choice of units. It is however a very convenient one for dealing with the classical limit of quantum systems. ath.nyu.edu/faculty/varadhan/
Time evolution of macrostates: An example
Consider a situation in which a gas of \(N\) atoms with energy \(E\) (with some tolerance) is initially confined by a partition to the left half of the box \(V\ ,\) and suppose that this constraint is removed at time \(t_a\ ,\) see Fig. 1. The phase space volume available to the system for times \(t>t_a\) is then fantastically enlarged compared to what it was initially, roughly by a factor of \(2^N\ .\) If the system contains 1 mole of gas then the volume ratio of the unconstrained phase space region to the constrained one is far larger than the ratio of the volume of the known universe to the volume of one atom.
Let us now consider the macrostate of this gas as given by \(M=\left({N_L \over N} , {E_L \over E}\right)\ ,\) the fraction of particles and energy in the left half of \(V\) (within some small tolerance). The macrostate at time \(t_a, M=(1, 1)\ ,\) will be denoted by \(M_a\ .\) The phase-space region \(|\Gamma| = \Sigma_E\ ,\) available to the system for \(t> t_a\ ,\) i.e., the region in which \(H(X) \in (E, E + \delta E), \delta E << E\ ,\) will contain new macrostates, corresponding to various fractions of particles and energy in the left half of the box, with phase space volumes very large compared to the initial phase space volume available to the system. We can then expect (in the absence of any obstruction, such as a hidden conservation law) that as the phase point \(X\) evolves under the unconstrained dynamics and explores the newly available regions of phase space, it will with very high probability enter a succession of new macrostates \(M\) for which \(|\Gamma_{M}|\) is increasing. The set of all the phase points \(X_t\ ,\) which at time \(t_a\) were in \(\Gamma_{M_a}\ ,\) forms a region \(T_t \Gamma_{M_a}\) whose volume is, by Liouville's Theorem, equal to \(|\Gamma_{M_a}|\ .\) The shape of \(T_t\Gamma_{M_a}\) will however change with \(t\) and as \(t\) increases \(T_t\Gamma_{M_a}\) will increasingly be contained in regions \(\Gamma_M\) corresponding to macrostates with larger and larger phase space volumes \(|\Gamma_M|\ .\) This will continue until almost all the phase points initially in \(\Gamma_{M_a}\) are contained in \(\Gamma_{M_{eq}}\ ,\) with \(M_{eq}\) the system's unconstrained macroscopic equilibrium state. This is the state in which approximately half the particles and half the energy will be located in the left half of the box, \(M_{eq} = ({1\over 2}, {1 \over 2})\) i.e. \(N_L /N\) and \(E_L/ E\) will each be in an interval \(\left({1 \over 2} - \epsilon, {1 \over 2} + \epsilon\right)\ ,\) \(N^{-1/2} << \epsilon << 1\ .\)
\(M_{eq}\) is characterized, in fact defined, by the fact that it is the unique macrostate, among all the \(M_\alpha\ ,\) for which \(|\Gamma_{M_{eq}}| / |\Sigma_E| \simeq 1\ ,\) where \(|\Sigma_E|\) is the total phase space volume available under the energy constraint \(H(X) \in (E, E + \delta E)\ .\) (Here the symbol \(\simeq\) means equality when \(N \to \infty\ .\)) That there exists a macrostate containing almost all of the microstates in \(\Sigma_E\) is a consequence of the law of large numbers. The fact that \(N\) is enormously large for macroscopic systems is absolutely critical for the existence of thermodynamic equilibrium states for any reasonable definition of macrostates, in the above example e.g. for any \(\epsilon\ ,\) such that, \(N^{-1/2} << \epsilon << 1\ .\) Indeed thermodynamics does not apply (is even meaningless) for isolated systems containing just a few particles. Nanosystems are interesting and important intermediate cases: Note however that in many cases an \(N\) of about 1,000 will already behave like a macroscopic system: see related discussion about computer simulations below.
After reaching \(M_{eq}\) we will (mostly) see only small fluctuations in \(N_L(t) / N\) and \(E_L(t) / E\ ,\) about the value \({1 \over 2}\ :\) typical fluctuations in \(N_L\) and \(E_L\) being of the order of the square root of the number of particles involved. (Of course if the system remains isolated long enough we will occasionally also see a return to the initial macrostate—the expected time for such a Poincaré recurrence is however much longer than the age of the universe and so is of no practical relevance when discussing the approach to equilibrium of a macroscopic system.)
As already noted earlier, the scenario in which \(|\Gamma_{M(X(t))}|\) increase with time for the \(M_a\) shown in Fig.1 cannot be true for all microstates \(X\subset \Gamma_{M_a}\ .\) There will of necessity be \(X\)'s in \(\Gamma_{M_a}\) which will evolve for a certain amount of time into microstates \(X(t)\equiv X_t\) such that \(|\Gamma_{M(X_t)}|<|\Gamma_{M_a}|\ ,\) e.g. microstates \(X\in \Gamma_{M_a}\) which have all velocities directed away from the barrier which was lifted at \(t_a\ .\) What is true however is that the subset \(B\) of such "bad" initial states has a phase space volume which is very very small compared to that of \(\Gamma_{M_a}\ .\) This is what is meant by the statement that entropy increasing behavior is typical; a more extensive discussion of typicality is given later.
Boltzmann's entropy
The end result of the time evolution in the above example, that of the fraction of particles and energy becoming and remaining essentially equal in the two halves of the container when \(N\) is large enough (and `exactly equal' when \(N \to\infty\)), is of course what is predicted by the second law of thermodynamics.
It was Boltzmann's great insight to connect the second law with the above phase space volume considerations by making the observation that for a dilute gas \(\log |\Gamma_{M_{eq}}|\) is proportional, up to terms negligible in the size of the system, to the thermodynamic entropy of Clausius. Boltzmann then extended his insight about the relation between thermodynamic entropy and \(\log |\Gamma_{M_{eq}}|\) to all macroscopic systems; be they gas, liquid or solid. This provided for the first time a microscopic definition of the operationally measurable entropy of macroscopic systems in equilibrium.
Having made this connection Boltzmann then generalized it to define an entropy also for macroscopic systems not in equilibrium. That is, he associated with each microscopic state \(X\) of a macroscopic system a number \(S_B\) which depends only on \(M(X)\) given, up to multiplicative and additive constants (which can depend on \(N\)), by \[\tag{1} S_B(X) = S_B (M(X)) \]
with \[\tag{2} S_B(M) = k \log|\Gamma_{M}|, \]
This is the Boltzmann entropy of a classical system, Penrose (1970) N. B. This definition uses two equations to emphasize their logical independence which is important for the discussion of quantum systems.
Boltzmann then used phase space arguments, like those given above, to explain (in agreement with the ideas of Maxwell and Thomson) the observation, embodied in the second law of thermodynamics, that when a constraint is lifted, an isolated macroscopic system will evolve toward a state with greater entropy. In effect Boltzmann argued that due to the large differences in the sizes of \(\Gamma_M\ ,\) \(S_B(X_t) = k \log |\Gamma_{M(X_t)}|\) will typically increase in a way which explains and describes qualitatively the evolution towards equilibrium of macroscopic systems.
These very large differences in the values of \(|\Gamma_M|\) for different \(M\) come from the very large number of particles (or degrees of freedom) which contribute, in an (approximately) additive way, to the specification of macrostates. This is also what gives rise to typical or almost sure behavior. Typical, as used here, means that the set of microstates corresponding to a given macrostate \(M\) for which the evolution leads to a macroscopic increase (or non-decrease) in the Boltzmann entropy during some fixed macroscopic time period \(\tau\) occupies a subset of \(\Gamma_M\) whose Liouville volume is a fraction of \(|\Gamma_M|\) which goes very rapidly (exponentially) to one as the number of atoms in the system increases. The fraction of "bad" microstates, which lead to an entropy decrease, thus goes to zero as \(N\to \infty\ .\)
Typicality is what distinguishes macroscopic irreversibility from the weak approach to equilibrium of probability distributions (ensembles) of systems with good ergodic properties having only a few degrees of freedom, e.g. two hard spheres in a cubical box. While the former is manifested in a typical evolution of a single macroscopic system the latter does not correspond to any appearance of time asymmetry in the evolution of an individual system. Maxwell makes clear the importance of the separation between microscopic and macroscopic scales when he writes: "the second law is drawn from our experience of bodies consisting of an immense number of molecules. ... it is continually being violated, ..., in any sufficiently small group of molecules ... . As the number ... is increased ... the probability of a measurable variation ... may be regarded as practically an impossibility."
On the other hand, because of the exponential increase of the phase space volume with particle number, even a system with only a few hundred particles, such as is commonly used in molecular dynamics computer simulations, will, when started in a nonequilibrium `macrostate' \(M\ ,\) with `random' \(X \in \Gamma_M\ ,\) appear to behave like a macroscopic system. After all, the likelihood of hitting, in the course of say one thousand tries, something which has probability of order \(2^{-N}\) is, for all practical purposes, the same, whether \(N\) is a hundred or \(10^{23}\ .\) Of course the fluctuation in \(S_B\) both along the path towards equilibrium and in equilibrium will be larger when \(N\) is small, c.f. [2b]. This will be so even when integer arithmetic is used in the simulations so that the system behaves as a truly isolated one; when its velocities are reversed the system retraces its steps until it comes back to the initial state (with reversed velocities), after which it again proceeds (up to very long Poincare recurrence times) in the typical way.
We might take as a summary of such insights in the late part of the nineteenth century the statement by Gibbs and quoted by Boltzmann (in a German translation) on the cover of his book Lectures on Gas Theory II:
``In other words, the impossibility of an uncompensated decrease of entropy seems to be reduced to an improbability.
Initial conditions
Once we accept the statistical explanation of why macroscopic systems evolve in a manner that makes \(S_B\) increase with time, there remains the nagging problem (of which Boltzmann was well aware) of what we mean by "with time": since the microscopic dynamical laws are symmetric, the two directions of the time variable are a priori equivalent and thus must remain so a posteriori.
In terms of Fig. 1 this question may be put as follows: why can one use phase space arguments to predict the macrostate at time \(t\) of an isolated system whose macrostate at time \(t_b\) is \(M_b\ ,\) in the future, i.e. for \(t > t_b\ ,\) but not in the past, i.e. for \(t < t_b\ ?\) After all, if the macrostate \(M\) is invariant under velocity reversal of all the atoms, then the same argument should apply equally to \(t_b + \tau\) and \(t_b -\tau\ .\) A plausible answer to this question is to assume that the nonequilibrium macrostate \(M_b\) had its origin in an even more nonuniform macrostate \(M_a\ ,\) prepared by some experimentalist at some earlier time \(t_a < t_b\) (as is indeed the case in Figure 1) and that for states thus prepared we can apply our (approximately) equal a priori probability of microstates argument, i.e. we can assume its validity at time \(t_a\ .\) But what about events on the sun or in a supernova explosion where there are no experimentalists? And what, for that matter, is so special about the status of the experimentalist? Isn't he or she part of the physical universe?
Put differently, where ultimately do initial conditions, such as those assumed at \(t_a\ ,\) come from? In thinking about this we are led more or less inevitably to introduce cosmological considerations by postulating an initial "macrostate of the universe" having a very small Boltzmann entropy. To again quote Boltzmann: "That in nature the transition from a probable to an improbable state does not take place as often as the converse, can be explained by assuming a very improbable [small \(S_B\)] initial state of the entire universe surrounding us. This is a reasonable assumption to make, since it enables us to explain the facts of experience, and one should not expect to be able to deduce it from anything more fundamental". While this requires that the initial macrostate of the universe, call it \(M_0\ ,\) be very far from equilibrium with \(|\Gamma_{M_0}|<< |\Gamma_{M_{eq}}|\ ,\) it does not require that we choose a special microstate in \(\Gamma_{M_0}\ .\) As also noted by Boltzmann elsewhere "We do not have to assume a special type [read microstate] of initial condition in order to give a mechanical proof of the second law, if we are willing to accept a statistical viewpoint...if the initial state is chosen at random...entropy is almost certain to increase." This is a very important aspect of Boltzmann's insight: it is sufficient to assume that this microstate is typical of an initial macrostate \(M_0\) which is far from equilibrium.
This going back to the initial conditions, i.e. the existence of an early state of the universe (presumably close to the big bang) with a much lower value of \(S_B\) than the present universe, as an ingredient in the explanation of the observed time asymmetric behavior, bothers some scientists. A common question is: how does the mixing of the two colors after removing the partitions in Fig. 1 depend on the initial conditions of the universe? The answer is that once you accept that the microstate of the system in 1a is typical of its macrostate the future evolution of the macrostates of this isolated system will indeed look like those depicted in Fig 1. It is the existence of inks of different colors separated in different compartments by an experimentalist, indeed the very existence of the solar system, etc. which depends on the initial conditions. In a "typical" universe everything would be in equilibrium.
It is the initial state of the universe plus the dynamics which determines what is happening at present. Conversely, we can deduce information about the initial state from what we observe now. As put by Feynman, Feynman, et al. (1967) "It is necessary to add to the physical laws the hypothesis that in the past the universe was more ordered, in the technical sense, [i.e. low \(S_B\)] than it is today...to make an understanding of the irreversibility."
Figure 2: With a gas in a box, the maximum entropy state (thermal equilibrium) has the gas distributed uniformly; however, with a system of gravitating bodies, entropy can be increased from the uniform state by gravitational clumping leading eventually to a black hole. From Reference number 8
A very clear discussion of initial conditions is given by Roger Penrose in connection with the "big bang" cosmology, Penrose, (1990 and 2005). He takes for the initial macrostate of the universe the smooth energy density state prevalent soon after the big bang: an equilibrium state (at a very high temperature) except for the gravitational degrees of freedom which were totally out of equilibrium, as evidenced by the fact that the matter-energy density was spatially very uniform. That such a uniform density corresponds to a nonequilibrium state may seem at first surprising, but gravity, being purely attractive and long range, is unlike any of the other fundamental forces. When there is enough matter/energy around, it completely overcomes the tendency towards uniformization observed in ordinary objects at high energy densities or temperatures. Hence, in a universe dominated, like ours, by gravity, a uniform density corresponds to a state of very low entropy, or phase space volume, for a given total energy, see Fig. 2.
The local `order' or low entropy we see around us (and elsewhere)—from complex molecules to trees to the brains of experimentalists preparing macrostates—is perfectly consistent with (and possibly even a necessary consequence of, i.e. typical of) this initial macrostate of the universe. The value of \(S_B\) at the present time, \(t_p\ ,\) corresponding to \(S_B (M_{t_p})\) of our current clumpy macrostate describing a universe of planets, stars, galaxies, and black holes, is much much larger than \(S_B(M_0)\ ,\) the Boltzmann entropy of the "initial state", but still quite far away from \(S_B(M_{eq})\) its equilibrium value. The `natural' or `equilibrium' state of the universe, \(M_{eq}\ ,\) is, according to Roger Penrose, Penrose (1990 and 2005), one with all matter and energy collapsed into one big black hole. Penrose gives an estimate \(S_B(M_0) / S_B(M_{t_p}) / S_{eq}\sim 10^{88} / 10^{101} / 10^{123}\) in natural (Planck) units, see Fig. 3.
Figure 3: The creator locating the tiny region of phase-space—one part in \(10^{10^{123}}\)—needed to produce a \(10^{80}\)-baryon closed universe with a second law of thermodynamics in the form we know it. From Reference number 8. If the initial state was chosen randomly it would, with overwhelming probability, have led to a universe in a state with maximal entropy. In such a universe there would be no stars, planets, people or a second law.
It is this fact that we are still in a state of low entropy that permits the existence of relatively stable neural connections, of marks of ink on paper, which retain over relatively long periods of time shapes related to their formation. Such nonequilibrium states are required for memories- in fact for the existence of living beings and of the earth itself.
We have no such records of the future and the best we can do is use statistical reasoning which leaves much room for uncertainty. Equilibrium systems, in which the entropy has its maximal value, do not distinguish between past and future.
Penrose's consideration about the very far from equilibrium uniform density "initial state" of the universe is quite plausible, but it is obviously far from proven. In any case it is, as Feynman says, both necessary and sufficient to assume a far from equilibrium initial state of the universe, and this is in accord with all cosmological evidence. The "true" equilibrium state of the universe may also be different from what Penrose proposes. There are alternate scenarios in which the black holes evaporate and leave behind mostly empty space, c.f. Chen and Carroll.
The question as to why the universe started out in such a very unusual low entropy initial state worries Penrose quite a lot (since it is not explained by any current theory) but such a state is just accepted as a given by Boltzmann. Clearly, it would be nice to have a theory which would explain the "cosmological initial state", but such a theory is not available at present. The "anthropic principle" in which there are many universes and ours just happens to be right, or we would not be here, is too speculative for an encyclopedic article.
• R. P. Feynman, The Character of Physical Law, MIT Press, Cambridge, Mass. (1967), ch. 5.
• S. Goldstein and J. L. Lebowitz, On the Boltzmann Entropy of Nonequilibrium Systems, Physica D, 193, 53-66, (2004); {b)} P. Garrido, S. Goldstein and J. L. Lebowitz, The Boltzmann Entropy of Dense Fluids Not in Local Equilibrium, Phys. Rev. Lett. 92, 050602, (2003).
• J. L. Lebowitz, {a)} Macroscopic Laws and Microscopic Dynamics, Time's Arrow and Boltzmann's Entropy, Physica A 194, 1–97(1993);
• J.L. Lebowitz, {b}}Boltzmann's Entropy and Time's Arrow, Physics Today, 46, 32–38(1993); see also letters to the editor and response in "Physics Today", 47, 113-116 (1994);{c)} Microscopic Origins of Irreversible Macroscopic Behavior, Physica A, 263, 516–527, (1999);
• J.L. Lebowitz, {d)} A Century of Statistical Mechanics: A Selective Review of Two Central Issues, Reviews of Modern Physics, 71, 346–357, 1999; {e)} From Time-symmetric Microscopic Dynamics to Time-asymmetric Macroscopic Behavior: An Overview, to appear in European Mathematical Publishing House, ESI Lecture Notes in Mathematics and Physics.
• O. Penrose, Foundations of Statistical Mechanics, Pergamon, Elmsford, N.Y. (1970): reprinted by Dorer (2005).
• R. Penrose, The Emperor's New Mind, Oxford U.P., New York(1990), ch. 7: The Road to Reality, A. E. Knopf, New York(2005), ch. 27–29.
• S.M. Carroll and J. Chen, Spontaneous Inflation and the Origin of the Arrow of Time, arXiv:hep-th/0410270v1
Internal references
Recommended reading
• For a general history of the subject and references to the original literature see S.G. Brush, The Kind of Motion We Call Heat, Studies in Statistical Mechanics, vol. VI, E.W. Montroll and J.L. Lebowitz, eds. North-Holland, Amsterdam, (1976).
• For a historical discussion of Boltzmann and his ideas see articles by M. Klein, E. Broda, L. Flamn in The Boltzmann Equation, Theory and Application, E.G.D. Cohen and W. Thirring, eds., Springer-Verlag, 1973.
• For interesting biographies of Boltzmann, which also contain many quotes and references, see E. Broda, Ludwig Boltzmann, Man—Physicist—Philosopher, Ox Bow Press, Woodbridge, Conn (1983); C. Cercignani, Ludwig Boltzmann; The Man Who Treated Atoms, Oxford University Press (1998); D. Lindley, Boltzmann's Atom: The Great Debate that Launched a Revolution in Physics, Simon & Shuster (2001).
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ad0d51ba32083fa8 | All quantum operations must be unitary to allow reversibility, but what about measurement? Measurement can be represented as a matrix, and that matrix is applied to qubits, so that seems equivalent to the operation of a quantum gate. That's definitively not reversible. Are there any situations where non-unitary gates might be allowed?
Unitary operations are only a special case of quantum operations, which are linear, completely positive maps ("channels") that map density operators to density operators. This becomes obvious in the Kraus-representation of the channel, $$\Phi(\rho)=\sum_{i=1}^n K_i \rho K_i^\dagger,$$ where the so-called Kraus operators $K_i$ fulfill $\sum_{i=1}^n K_i^\dagger K_i\leq \mathbb{I}$ (notation). Often one considers only trace-preserving quantum operations, for which equality in the previous inequality holds. If additionally there is only one Kraus operator (so $n=1$), then we see that the quantum operation is unitary.
However, quantum gates are unitary, because they are implemented via the action of a Hamiltonian for a specific time, which gives a unitary time evolution according to the Schrödinger equation.
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$\begingroup$ +1 Everyone interested in quantum mechanics (not just quantum information) should know about quantum operations e.g. from Nielsen and Chuang. I think it is worth mentioning (since the Wikipedia page on Stinespring dilation is too technical) that every finite-dimensional quantum operation is mathematically equivalent to some unitary operation in a larger Hilbert space followed by a restriction to the subsystem (by the partial trace). $\endgroup$ – Ninnat Dangniam Mar 20 '18 at 5:31
Short Answer
Quantum operations do not need to be unitary. In fact, many quantum algorithms and protocols make use of non-unitarity.
Long Answer
Measurements are arguably the most obvious example of non-unitary transitions being a fundamental component of algorithms (in the sense that a "measurement" is equivalent to sampling from the probability distribution obtained after the decoherence operation $\sum_k c_k\lvert k\rangle\mapsto\sum_k |c_k|^2\lvert k\rangle\langle k\rvert$).
More generally, any quantum algorithm that involves probabilistic steps requires non-unitary operations. A notable example that comes to mind is HHL09's algorithm to solve linear systems of equations (see 0811.3171). A crucial step in this algorithm is the mapping $|\lambda_j\rangle\mapsto C\lambda_j^{-1}|\lambda_j\rangle$, where $|\lambda_j\rangle$ are eigenvectors of some operator. This mapping is necessarily probabilistic and therefore non-unitary.
Any algorithm or protocol that makes use of (classical) feed-forward is also making use of non-unitary operations. This is the whole of one-way quantum computation protocols (which, as the name suggests, require non-reversible operations).
The most notable schemes for optical quantum computation with single photons also require measurements and sometimes post-selection to entangle the states of different photons. For example, the KLM protocol produces probabilistic gates, which are therefore at least partly non-reversible. A nice review on the topic is quant-ph/0512071.
Less intuitive examples are provided by dissipation-induced quantum state engineering (e.g. 1402.0529 or srep10656). In these protocols, one uses an open map dissipative dynamic, and engineers the interaction of the state with the environment in such a way that that the long-time stationary state of the system is the desired one.
At risk of going off-topic from quantum computing and into physics, I'll answer what I think is a relevant subquestion of this topic, and use it to inform the discussion of unitary gates in quantum computing.
The question here is: Why do we want unitarity in quantum gates?
The less specific answer is as above, it gives us 'reversibility', or as physicists often talk about it, a type of symmetry for the system. I'm taking a course in quantum mechanics right now, and the way unitary gates cropped up in that course was motivated by the desire to have physical transformations $\hat{U}$: that act as symmetries. This imposed two conditions on the transformation $\hat{U}$:
1. The transformations should act linearly on the state (this is what gives us a matrix representation).
2. The transformations should preserve probability, or more specifically inner product. This means that if we define:
$$|\psi '\rangle = U |\psi\rangle, |\phi'\rangle = U |\phi\rangle$$
Preservation of inner product means that $\langle \phi | | \psi \rangle= \langle \phi' | | \psi'\rangle$. From this second specification, unitarity can be derived (for full details see Dr. van Raamsdonk's notes here).
So this answers the question of why operations that keep things "reversible" have to be unitary.
The question of why measurement itself is not unitary is more related to quantum computation. A measurement is a projection on to a basis; in essence, it must "answer" with one or more basis states as the state itself. It also leaves the state in a way that is consistent with the "answer" to the measurement, and not consistent with the underlying probabilities that the state began with. So the operation satisfies specification 1. of our transformation $U$, but definitively does not satisfy specification 2. Not all matrices are created equal!
To round things back to quantum computation, the fact that measurements are destructive and projective (ie. we can only reconstruct the superposition through repeated measurements of identical states, and every measurement only gives us a 0/1 answer), is part of what makes the separation between quantum computing and regular computing subtle (and part of why it's difficult to pin that down). One might assume quantum computing is more powerful because of the mere size of the Hilbert space, with all those state superpositions available to us. But our ability to extract that information is heavily limited.
As far as I understand it this shows that for information storage purposes, a qubit is only as good as a regular bit, and no better. But we can be clever in quantum computation with the way that information is traded around, because of the underlying linear-algebraic structure.
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$\begingroup$ I find the last paragraph a bit cryptic. What do you mean by "slippery" separation here? It is also non-obvious how the fact that measurements are destructive implies something about such separation. Could you clarify these points? $\endgroup$ – glS Mar 15 '18 at 20:05
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$\begingroup$ @glS, good point, that was worded poorly. Does this help? I don't think I'm saying anything particularly deep, simply that Hilbert space size alone isn't a priori what makes quantum computation powerful (and it doesn't give us any information storage advantages) $\endgroup$ – Emily Tyhurst Mar 15 '18 at 20:41
There are several misconceptions here, most of them originate from exposure to only the pure state formalism of quantum mechanics, so let's address them one by one:
1. All quantum operations must be unitary to allow reversibility, but what about measurement?
This is false. In general, the states of a quantum system are not just vectors in a Hilbert space $\mathcal{H}$ but density matrices $-$ unit-trace, positive semidefinite operators acting on the Hilbert space $\mathcal{H}$ i.e., $\rho: \mathcal{H} \rightarrow \mathcal{H}$, $Tr(\rho) = 1$, and $\rho \geq 0$ (Note that the pure state vectors are not vectors in the Hilbert space but rays in a complex projective space; for a qubit this amounts to the Hilbert space being $\mathbb{C}P^1$ and not $\mathbb{C}^2$). Density matrices are used to describe a statistical ensemble of quantum states.
The density matrix is called pure if $\rho^2 = \rho$ and mixed if $\rho^2 < \rho$. Once we are dealing with a pure state density matrix (that is, there's no statistical uncertainty involved), since $\rho^2 = \rho$, the density matrix is actually a projection operator and one can find a $|\psi\rangle \in \mathcal{H}$ such that $\rho = |\psi\rangle \langle\psi|$.
The most general quantum operation is a CP-map (completely positive map), i.e., $\Phi: L(\mathcal{H}) \rightarrow L(\mathcal{H})$ such that $$\Phi(\rho) = \sum_i K_i \rho K_i^\dagger; \sum_i K_i^\dagger K_i \leq \mathbb{I}$$ (if $\sum_i K_i^\dagger K_i = \mathbb{I}$ then these are called CPTP (completely positive and trace-preserving) map or a quantum channel) where the $\{K_i\}$ are called Kraus operators.
Now, coming to the OP's claim that all quantum operations are unitary to allow reversibility -- this is just not true. The unitarity of time evolution operator ($e^{-iHt/\hbar}$) in quantum mechanics (for closed system quantum evolution) is simply a consequence of the Schrödinger equation.
However, when we consider density matrices, the most general evolution is a CP-map (or CPTP for a closed system to preserve the trace and hence the probability).
1. Are there any situations where non-unitary gates might be allowed?
Yes. An important example that comes to mind is open quantum systems where Kraus operators (which are not unitary) are the "gates" with which the system evolves.
Note that if there is only a single Kraus operator then, $\sum_i K_i^\dagger K_i = \mathbb{I}$. But there's only one $i$, therefore, we have, $K^\dagger K = \mathbb{I}$ or, $K$ is unitary. So the system evolves as $\rho \rightarrow U \rho U^\dagger$ (which is the standard evolution that you may have seen before). However, in general, there are several Kraus operators and therefore the evolution is non-unitary.
Coming to the final point:
In standard quantum mechanics (with wavefunctions etc.), the system's evolution is composed of two parts $-$ a smooth unitary evolution under the system's Hamiltonian and then a sudden quantum jump when a measurement is made $-$ also known as wavefunction collapse. Wavefunction collapses are described as some projection operator say $|\phi\rangle \langle\phi|$ acting on the quantum state $|\psi\rangle$ and the $|\langle\phi|\psi\rangle|^2$ gives us the probability of finding the system in the state $|\phi\rangle$ after the measurement. Since the measurement operator is after all a projector (or as the OP suggests, a matrix), shouldn't it be linear and physically similar to the unitary evolution (also happening via a matrix). This is an interesting question and in my opinion, difficult to answer physically. However, I can shed some light on this mathematically.
If we are working in the modern formalism, then measurements are given by POVM elements; Hermitian positive semidefinite operators, $\{M_{i}\}$ on a Hilbert space $\mathcal{H}$ that sum to the identity operator (on the Hilbert space) $\sum _{{i=1}}^{n}M_{i}=\mathbb{I}$. Therefore, a measurement takes the form $$ \rho \rightarrow \frac{E_i \rho E_i^\dagger}{\text{Tr}(E_i \rho E_i^\dagger)}, \text{ where } M_i = E_i^\dagger E_i.$$
The $\text{Tr}(E_i \rho E_i^\dagger) =: p_i$ is the probability of the measurement outcome being $M_i$ and is used to renormalize the state to unit trace. Note that the numerator, $\rho \rightarrow E_i \rho E_i^\dagger$ is a linear operation, but the probabilistic dependence on $p_i$ is what brings in the non-linearity or irreversibility.
Edit 1: You might also be interested Stinespring dilation theorem which gives you an isomorphism between a CPTP map and a unitary operation on a larger Hilbert space followed by partial tracing the (tensored) Hilbert space (see 1, 2).
I'll add a small bit complementing the other answers, just about the idea of measurement.
Measurement is usually taken as a postulate of quantum mechanics. There's usually some preceding postulates about hilbert spaces, but following that
• Every measurable physical quantity $A$ is described by an operator $\hat{A}$ acting on a Hilbert space $\mathcal{H}$. This operator is called an observable, and it's eigenvalues are the possibly outcomes of a measurement.
• If a measurement is made of the observable $A$, in the state of the system $\psi$, and the outcome is $a_n$, then the state of the system immediately after measurement is $$\frac{\hat{P}_n|\psi\rangle}{\|\hat{P}_n|\psi\rangle\|},$$ where $\hat{P}_n$ is the projector onto the eigen-subspace of the eigenvalue $a_n$.
Normally the projection operators themselves should satisfy $\hat{P}^\dagger=\hat{P}$ and $\hat{P}^2=\hat{P}$, which means they themselves are observables by the above postulates, and their eigenvalues $1$ or $0$. Supposing we take one of the $\hat{P}_n$ above, we can interpret the $1,0$ eigenvalues as a binary yes/no answer to whether the observable quantity $a_n$ is available as an outcome of measurement of the state $|\psi\rangle$.
Measurements are unitary operations, too, you just don't see it: A measurement is equivalent to some complicated (quantum) operation that acts not just on the system but also on its environment. If one were to model everything as a quantum system (including the environment), one would have unitary operations all the way.
However, usually there is little point in this because we usually don't know the exact action on the environment and typically don't care. If we consider only the system, then the result is the well-known collapse of the wave function, which is indeed a non-unitary operation.
Quantum states can change in two ways: 1. quantumly, 2. classically.
1. All the state changes taking place quantumly, are unitary. All the quantum gates, quantum errors, etc., are quantum changes.
2. There is no obligation on classical changes to be unitary, e.g. measurement is a classical change.
All the more reason, why it is said that the quantum state is 'disturbed' once it's measured.
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$\begingroup$ Why would errors be "quantum"? $\endgroup$ – Norbert Schuch Oct 28 '18 at 22:20
• $\begingroup$ @NorbertSchuch: Some errors could come in the form of the environment "measuring" the state, which could be considered classical in the language of this user, but other errors may come in the form of rotations/transformations in the Bloch sphere which don't make sense classically. Certainly you need to do full quantum dynamics if you want to model decoherence exactly (non-Markovian and non-perturbative ideally, but even Markovian master equations are quantum). $\endgroup$ – user1271772 Oct 29 '18 at 1:05
• $\begingroup$ Surely not all errors are 'quantum', but I meant to say that all 'quantum errors' ($\sigma_x,\sigma_y,\sigma_z$ and their linear combinations) are unitary. Please correct me if I am wrong, thanks. $\endgroup$ – alphaQuant Oct 29 '18 at 5:49
• $\begingroup$ To be more precise, errors which are taken care of by QECCs. $\endgroup$ – alphaQuant Oct 29 '18 at 5:56
• 1
$\begingroup$ I guess I'm not sure what "quantum" and "classical" means. What would a CP map qualify as? $\endgroup$ – Norbert Schuch Oct 29 '18 at 6:45
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16b162052f63d5af | Nat 5 physics equation sheet schrodinger s
Physics schrodinger
Nat 5 physics equation sheet schrodinger s
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bf63b8578a3c9e45 | Quantum aggregates are assemblies of monomers (molecules, atoms, quantum dots...), where the monomers largely keep their individuality. However, interactions between the monomers can lead to collective phenomena, like superradiance or efficient excitation transfer. We study different kinds of aggregates (e.g. light harvesting systems, arrays of Rydberg atoms, self-assembled organic dyes...) Using various methods (ranging from Green-operator methods over stochastic Schroedinger equations to semicalssical surface hopping) we study optical and excitation transport in these systems. Of particular interest is the coupling of the excitation to nuclear degrees of freedom.
The ability of photosynthetic plants, algae and bacteria to efficiently harvest sunlight has attracted researchers for decades and a fairly clear picture of photosynthesis has emerged: Sunlight is absorbed by assemblies of chromophores, e.g. chlorophylls. These assemblies, termed light harvesting complexes, transfer the excitation energy with high efficiency to so-called reaction centers, where the excitation energy is converted into a trans-membrane chemical potential.
Organic Dye Aggregates
Certain molecular aggregates consisting of organic dyes are remarkable in exhibiting an intense and very narrow absorption peak, known as a J-band, which is red-shifted away from the region of monomer absorption. Apart from those dyes showing the J-band on aggregation, there are also dyes where the absorption maximum is shifted to higher energies. The width of the resulting absorption band (called an H-band) is comparable to that of the monomeric dyes and shows a complicated vibrational structure.
Rydberg Aggregates
While in moleculare aggregates there is a strong coupling to vibrational degrees ultracold Rydberg atoms are ideally suited to study the above described phenomena in an clean and controllable environment.
In particular we have found that the coupling of electronic and nuclear degrees of freedoms allows to transport entanglement along a chain of atoms in effecive way. Furthermore we have investigate the effect of conical intersection in ultracold Rydberg gases on the excitation dynamics. We found that it is possible to directly observe the effect of Berry's phase.
Superconducting circuits
Open quantum system approaches are widely used in the description of physical, chemical and biological systems to describe the coupling of electronic degrees of freedom to vibrations. This structured vibrational environment makes simulations on a classical computer very demanding. We propose an analogue quantum simulator of complex open system dynamics which is based on superconducting circuits.
Stochastic Schrödinger equations
We have recently extended the non-Markovian Quantum State Diffusion approach to treat the energy transfer of coupled molecules. The picture shows Transport on a ring, when initially the excitation is localised on monomer 8. (a) No coupling to the environment. (b) coupling to a structured environment. This plot is an average of single trajectories (examples are shown in (c)-(e)).
[1] A short description how to calculate absorption and energy transfer |
75d300be74df9c2d | måndag 29 oktober 2018
Why TGD?
The most frequently asked questions about TGD 2014 . The headlines:
1. Why TGD? (pdf-article)
2. How can TGD help to solve the problems of recent day theoretical physics?
3. What are the basic principles of TGD?
4. What are the basic guidelines in the construction of TGD?
The important is to catch the basic idea: technical details are not important, it is principles and concepts which really matter + Problem solving. TGD itself and almost every new idea in the development of TGD has been inspired by a problem. Standard physics is plagued by problems deeply rooted in basic philosophical - even ideological - assumptions which boil down to -isms like reductionism, materialism, determinism - and locality.
Standard model summarize the recent understanding of physics. The attempts to extrapolate physics beyond standard model are often based on naive length scale reductionism and have produced Grand Unified Theories (GUTs), supersymmetric gauge theories (SUSYs). The attempts to include gravitation under the same theoretical umbrella with electroweak and strong interactions has led to superstring models and M-theory. These programs have not been successful, and the recent dead end culminating in the landscape problem (criticality lost?) of superstring theories and M-theory could have its origins in the basic ontological assumptions about the nature of space-time and quantum.
1. Why TGD?
The question requires an overall view about the recent state of theoretical physics.
• Thermodynamics, special relativity, and general relativity involve also postulates, which can be questioned, as in thermodynamics second law in its recent form, and the assumption about fixed arrow of thermodynamical time, since it is hard to understand biological evolution in this framework.
• In general relativity the symmetries of special relativity are in principle lost and with Noether's theorem this means also the loss of classical conservation laws, even the definitions of energy and momentum are in principle lost.
• In quantum physics the basic problem is that the nondeterminism of quantum measurement theory conflicts with the determinism of Schrödinger equation.
2. How can TGD help? The view about space-time as 4-D surface in fixed 8-D space-time is the starting point.
• motivated by the energy problem of general relativity
• fusion of the basic ideas of special and general relativities (a TOE).
This has led to other ideas:
• dark matter as phases of ordinary matter characterized by non-standard value of Planck constant, this has a strong operative function
• extension of physics in p-adic number fields assumed to describe correlates of cognition and intentionality,
• zero energy ontology (ZEO) - quantum states are identified as counterparts of physical events (two times required).
These new elements generalize considerably the view about space-time and quantum + give possibility to understand living systems and consciousness in physics.
3. TGD as a mathematical theory, basic principles.
A generalization of Einstein's geometrization program from space-time level to the level of "world of classical Worlds" as space of 4-surfaces.
The infinite-dimensional geometry fixes it uniquely.
It is the modes of the classical imbedding space spinor fields - eigenstates of four-momentum and standard model quantum numbers - that define the ground states of the super-conformal representations. It is these modes that correspond to the 4-D spinor modes of QFT limit.
+TGD as a generalized number theory with three separate threads.
• Number theoretical universality from the need to fuse p-adic physics and real physics to a coherent whole.
• Classical number fields (reals, complex numbers, quaternions, and octonions) have dimensions which correspond to those appearing in TGD. Basic laws of both classical and quantum physics could reduce to the requirements of associativity and commutativity.
• The primes (and integer, rational, and algebraic number) can be generalized so infinite primes are possible. Construction of infinite hierarchy of infinite primes using the primes of the previous level as building bricks to new level. This is structurally identical with a repeated second quantization of supersymmetric arithmetic quantum field theory for which elementary bosons and fermions are labelled by primes. Free many-particle states and also the analogs of bound states are obtained. The really hard part of quantum field theories - understanding of bound states - could have number theoretical solution.
• It is not yet clear if both visions (geometrization and number theory) are needed. In any case their combination has provided a lot of insights about quantum TGD.
4. Guidelines in the construction of TGD. The construction of physical theories is nowadays to a high degree guesses about the symmetries and deduction of consequences. The very notion of symmetry has been generalized in this process. Super-conformal symmetries play even more powerful role in TGD than in superstring models and the gigantic symmetries of WCW is a powerful 'proof'.
In TGD context string like objects are not something emerging at Planck length scale but in scales of elementary particle physics. The irony is that although TGD is not string theory, string like objects and genuine string world sheets emerge naturally from TGD in all length scales.
Even TGD view about nuclear physics predicts string like objects.
The most important guidelines:
A kernel of WCW = a conjecture.
M4× CP2 as choice for the imbedding space
Number Theoretical Universality
'Quantum classical correspondence' where classical theory is no approximation but an exact part of quantum theory.
And more technical guidelines:
• Strong form of General Coordinate invariance (GCI) is a very strong assumption, that gives the assumption that Kähler function is Kähler action for a preferred extremal as a counterpart of Bohr orbit. Strict determinism not required. Strong form of GCI requires that the light-like 3-surfaces represent partonic orbits and space-like 3-surfaces at the ends of causal diamonds are physically equivalents as effective 2-D states. The intersections of these two kinds of 3-surfaces and 4-D tangent space data at them code for quantum states.
• Quantum criticality means that Universe is analogous to a critical system with maximal structural richness. Universe is at the boundary line between chaos and order. Quantum criticality fixes the basic coupling constant dictating quantum Dynamics.
• Finite measurement resolution using von Neumann algebras. Usually the measurement problem is a messy, ugly duckling in theoretical physics.
• Strong form of holography from strong form of GCI and TGD reduces to an almost topological QFT= a conjecture. Weak form of electric-magnetic duality is a TGD version of electric-magnetic duality discovered by Olive and Montonen.
• Generalized Feynman diagrams. TGD leads to a realization of counterparts of Feynman diagrams at the level of space-time geometry and topology. The highly non-trivial challenge is to give them precise mathematical content. Twistor revolution has made possible a considerable progress in this respect and led to a vision about twistor Grassmannian description of stringy variants of Feynman diagrams.
• The localization of spinor modes at string world sheets. There are three reasons for the modes of the induced spinor fields to be localized to 2-D string world sheets and partonic 2-surfaces - or to the boundaries of string world sheets at them defining fermionic world lines.
Thanks to holography fermions behave like pointlike particles, which are massless in 8-D sense. General relativity emerges as an approximation due to frustration.
- gives em-charge as eigenstates. Spinor modes can have have well-defined electromagnetic charges - the induced classical W boson fields and perhaps also Z field vanish at string world sheets so that only em field and possibly Z field remain.
- acts as spacetime 'genes'.
- sign problems with partition, not necessary positive always. This is avoided in the 2D World sheet of TGD (many sheeted spacetime).
This is a very compressed text about TGD. Go to the sources for better info.
See the article Why TGD? 2014.
Why TGD and What TGD is? 2018 (with spinor part), 49 pp.
A very short summary, a brief summary about TGD, 2018, 3 pp.
söndag 24 februari 2013
Problems leading to TGD.
Topological Geometro-Dynamics (TGD) is a unified theory of fundamental interactions. Quantum classical correspondence [by dimensional reduction] has been one of the guiding principles.
A different thinking in many questions is very much charachteristic of TGD. Matti himself says that TGD is an oldfashioned quantum hadron model (as instance described by Björken), and the dimensions are emergent from microcosmos and gauge Lorentz invariance with its roots in the vacuum or zero point field. See also The model for hadron masses revisited.
The basic differencies of TGD in relation to other main theories:
• Poincare symmetry, Lorentz invariance, so it is more like a parallell to General Relativity of Einstein, still does not contradict it. TGD is more like a scaled up variant in 8D (GR^2) of GR.
• hadrons, as tripoints, three quarks, or pair of quark-antiquark, instead of strings
• tripoints are 3-surface, non-local points of mostly wave nature as they follow Kähler action
• Planck scale is not the basic scale, and Planck scale can be gigantic
• gravitational Planck constants, gravitational 'waves' (with respect to dark matter)
• there is no cosmological constant
• there is negative energy and magnetism that is vanishing, which make possible the Zero Energy Ontology (ZEO), and because electric currents vanish faster than magnetic dito there are left over magnetic 'bodies' as an effector. This is extremely important in biology
• fields are replaced with effects of spacetime sheets
• actions are made by Noether currents, not Ricci tensors
• time is an active force, creating entanglement and phases, also p-adic time. This is crucial in forming matrices with ZEO.
• the understanding of Feynman diagrams as generalized matrices in 2D (as partonic 2-surfaces) made of 3-surfaces and their matrices/braids. Lightlike 3-surfaces from maxima of Kähler function define the matrices. This can even describe the black hole inside. Partons and partonic 2-surfaces as generalizations too, as are N-atoms and N-particles?
The basic objection against TGD is acc. to Matti, that induced metrics for space-time surfaces in M^4 × CP_2 form an extremely limited set in the space of all space-time metrics appearing in the path integral formulation of General Relativity. Even special metrics like the metric of a rotating black hole fail to be imbeddable as an induced metric. For instance, one can argue that TGD cannot reproduce the post-Newtonian approximation to General Relativity because it involves linear superposition of gravitational fields of massive objects. Holger B. Nielsen made this objection for at least two decades ago. Perhaps the strong objection against TGD is that linear superposition for classical fields is lost.
The linear superposition is however central starting point of field theories. Many-sheeted space-time circumvent this argument. The replacement of linear superposition of fields with the superposition of their effecs meaning that sum is replaced with set theoretic union for space-time sheets. This simple observation has far reaching consequences: it becomes possible to replace the dynamics for a multitude of fields with the dynamics of space-time surfaces with only 4 imbedding space coordinates as primary dynamical variables. See also Standing waves in TGD.
The continuity has also been an obstacle in a world where even the quantum fraction is geometric.
Discrete vs continuous controversy in physics - discrete and continuous features coexist in any natural phenomenon, depending on the scales of observation.
I quote from the TGD Intro (2007) about the main differencies from mainstream:
TGD was originally an attempt to construct a Poincare invariant theory of gravitation. Spacetime, rather than being an abstract manifold endowed with a pseudo-Riemannian structure, is regarded as a 4-surface in the 8-dimensional space.
• H=M^4_+ = the interior of the future light cone of the Minkowski space (to be referred as light cone)
• CP_2= SU(3)/U(2) is the complex projective space of two complex dimensions
The size of CP_2 which is about 10^4 Planck lengths replaces Planck length as a fundamental length scale in TGD Universe.
The identification of the spacetimes as a submanifolds leads to Poincare invariance broken only in cosmological scales and solves the conceptual difficulties related to the definition of the energy-momentum in General Relativity. Even more, sub-manifold geometry, being considerably richer in structure than the abstract manifold geometry behind General Relativity, leads to a geometrization of all basic interactions and elementary particle quantum numbers. In particular, classical electroweak gauge fields are obtained by inducing the spinor curvature of CP_2 to the spacetime surface.
Fig. 1. a) Future light cone of Minkowski space. b) CP_2 is obtained by identifying all points of C^3, space having 3 complex dimensions, which differ by a complex scaling \Lambda: z is identified with \Lambda x z.
This forces a generalization of the conventional spacetime concept to what might be called manysheeted spacetime or 'topological condensate'. The topologically trivial 3-space of General Relativity is replaced with a 'topological condensate' containing matter as particle like 3-surfaces "glued" to the topologically trivial background spacetime sheet by extremely tiny connected sum (wormhole) contacts having CP_2 size connecting the spacetime sheets. End quote.
The criticality.
One big problem in physics is the criticality, how a classic world can come from the quantum uncertainty. This problem does not differ so much from M-theory, but the solution does very much.
TGD can be seen as a model giving rise to GR as a simple 'mirror image', and also there is a double mirror. Time dimension also have this 'mirror image', and magnetism, em-'force' can be vanishing? See How to perform WCW integrations in generalized Feynman diagrams? and "The relationship between TGD and GRT". He writes (from the GRT abstract, and I have filled in other links):
Radically new views about ontology were necessary before it was possible to see what had been there all the time. Zero energy ontology states that all physical states have vanishing net quantum numbers. The hierarchy of dark matter identified as macroscopic quantum phases labeled by arbitrarily large values of Planck constant is second aspect of the new ontology.
1. Equivalence Principle in TGD Universe
2. Zero energy ontology
3. Dark matter hierarchy and hierarchy of Planck constants
4. The problem of cosmological constant
5. The generalized Feynman Diagrams
6. The families and massivation. The symmetries coming out from the microscopic massivation and time distortion or symmetry breaking. This last point I will not take up here.
1. The energy problem, is the equivalence principle holding in TGD?
The source of problems was the attempt to deduce the formulation of Equivalence Principle in the framework provided by General Relativity framework rather than in string model like context. The process shortly summarized:
a) Inertial and gravitational four-momenta are replaced with Super Virasoro generators of two algebras whose differences annihilate physical states = the super-conformal symmetries of quantum TGD.
b) Number theoretical compactification providing a number theoretical interpretation of space spinors, and thus also of standard model quantum numbers.
c) The identification of the preferred extremals of Kähler action and the formulation of quantum TGD in terms of second quantized induced spinors fields. This has turned out to be extremely useful for the development of TGD, made possible the understanding of field equations, and led to a detailed understanding of quantum TGD at the fundamental parton level.
Absolute minimization of so called Kähler action is the fundamental variational principle of TGD and assigns to a given 3-surface X^3 a classical spacetime surface X^4(X^3) which is much like Bohr orbit going through a fixed point in wave mechanics characterized by classical non-determinism caused by enormous vacuum degeneracy and this forces a generalization of the notion of 3-surfaces in order to achieve classical determinism in a more general sense. 3-surfaces are in general unions of disjoint 3-surfaces with timelike separations rather than single time=constant snapshots of the spacetime surface. In particular, spacetime sheets with finite time duration, 'mindlike' spacetime sheets, are possible and are identified as geometric correlates of selves in TGD inspired theory of consciousness
2. Zero energy ontology (S-matrix is replaced with M-matrix definition "square root" of density matrix) allows to avoid the paradox implied in positive energy ontology, by the fact that gravitational energy is not conserved but inertial energy identified as Noether charge is. Energy conservation is always in some length scale in zero energy ontology. This principle is satisfied only by the outcomes of state function reduction.
To sum up, the understanding of Equivalence Principle in TGD context required quite many discoveries of mostly mathematical character: the understanding of the superconformal symmetries of quantum TGD, the discovery of zero energy ontology, the identification of preferred extremals of Kähler action by requiring number theoretical compactification, and the discovery that dimensional reduction allows to formulate quantum in terms of slicing of space-time surface by stringy word sheets. See Tree like structure for the imbedding space
And later...
Gravitational four-momentum can be assigned to the curvature scalar as Noether currents and is thus completely well-defined [but non-conserved] unlike in GRT. Equivalence Principle requires that inertial and gravitational four-momenta are identical. This is satisfied if curvature scalar defines the fundamental action principle crucial for the definition of quantum TGD. Curvature scalar as a fundamental action is however non-physical and had to be replaced with so called Kähler action. The conservation of gravitational four-momentum seems to fail in cosmological scales. Also for vacuum extremals satisfying Einstein's equations gravitational four-momentum fails to be conserved and non-conservation becomes large for small values [lengths] of cosmic time. My basic mistake looks now obvious. I tried to deduce the formulation of Equivalence Principle in the framework provided by General Relativity framework rather than in string model context.
But the conservation laws are questioned by many other too. This frame also gave a new interpretation of time.
The basic prediction of TGD is that the sign of energy depends on the time orientation of the spacetime surface.
Quantum states of 3-D theory in zero energy ontology correspond to generalized S-matrices. M-matrix might be a proper term, and is a "complex square root" of density matrix - matrix valued generalization of Schrodinger amplitude - defining time like entanglement coefficients. Its "phase" is unitary matrix. The counterpart of ordinary S-matrix is between zero energy states. I call it U-matrix. It has nothing to do with particle reactions. It is crucial for understanding consciousness via moment of consciousness as quantum jump identification. See Construction of Quantum Theory: S-matrix.
Wikipedia, Noethers theorem, Constant of motion, conservation law, and conserved current, says;
A conservation law states that some quantity X in the mathematical description of a system's evolution remains constant throughout its motion — it is an invariant. Mathematically, the rate of change of X (its derivative with respect to time) vanishes,
\frac{dX}{dt} = 0 ~.
Such quantities are said to be conserved; they are often called constants of motion (although motion per se need not be involved, just evolution in time). The earliest constants of motion discovered were momentum and energy,
Here are some examples of research about Noether currents by other scientists:
1. Applications of Noether currents. Scale invariance. R. Corrado 1994: To illustrate the use of Noether’s theorem and the currents produced, we examine the case of scale transformations. As we shall see, these are not necessarily invariances of the action and we will have to determine what conditions are necessary for scale transformations to be a symmetry. Only the masses breaks scale invariance. Any operator with a dimensionful coupling constant breaks the scale invariance of the massless theory.
2. Continous symmetries and conserved currents. conservation of the Noether current holds in the quantum theory, with the current inside a correlation function, up to contact terms (that depend on the infinitesimal transformation). Conserved charges associated with this current are generators of the Lorentz group.
3. Symmetries and conservation Laws. Lagrangian density with a symmetry can give 1. time translations, - time translation invariance implies that H is constant. This does not appear to be the case in our Universe, because it is expanding (the cosmological constant). The Hamiltonian generates translations in time. 2. spacetime translations - Noether currents are the components of the stress-energy tensor. The conserved charges (components of the total four-momentum) generate translations. 3. Rotations - specified by a vector ~ pointing in the direction of the axis of rotation. Its magnitude is the angle of rotation. If e is small, under a rotation, the corresponded charges form the angular momentum of the system. The angular momentum generates rotations, rotation 3x3 matrix. 4. Lorentz transformations - In addition to rotations, the group of Lorentz transformations contains boosts (small velocity, corresponding charges are the components of the vector and generates boosts: for infinitesimal velocities and finite vectors where Λ is the corresponding 4×4 Lorentz transformation matrix. show that vectors M and L transform correctly as vectors in R3 under rotations.
4. Noether currents and charges for Maxwell-like Lagrangians, Yakov 2003: Application of the Noether procedure to physical Lagrangians yields, however, meaningful (and measurable) currents. The well-known solution to this 'paradox' is to involve the variation of the metric tensor. The Noether current of the field considered on a variable background coincides with the current treated in a fixed geometry. Consistent description of the canonical energy–momentum current is possible only if the dynamics of the geometry (gravitation) is taken into account.
5. Nonlocal currents as Noether currents, Dolan & Roos 1980: The first two nonlocal currents in the general two-dimensional chiral models are derived as Noether currents. The associated infinitesimal field transformations are shown to obey a group integrability condition. A subset of the structure constants of the symmetry group responsible for these conserved currents is calculated.
6. GAUGE SYMMETRIES AND NOETHER CURRENTS IN OPTIMAL CONTROL. Torres, 2003: extend the second Noether theorem to optimal control problems which are invariant under symmetries depending upon k arbitrary functions of the independent variable and their derivatives up to some order m. As far as we consider a semi-invariance notion, and the transformation group may also depend on the control variables.
3. Dark matter hierarchy.
The dimensional reduction for preferred extremals of Kähler action - if they have the properties required by theoretic compactification - leads to string model with string tension which is however not proportional to the inverse of Newton's constant but to p-adic length scale squared and thus gigantic [and dark], see p-Adic Mass Calculations: New Physics. This allowed to predict the value of Kähler coupling strength by using as input electron mass and p-adic mass calculations. In this framework the role of Planck length as a fundamental length scale is taken by CP2 size so that Planck length scale loses its magic role as a length scale.
The identification of gravitational four-momentum in terms of Einstein tensor makes sense only in long length scales. This resolves the paradoxes associated with objects like cosmic strings.
Dark matter hierarchy corresponds to a hierarchy of conformal symmetries Zn of partonic 2-surfaces and this hierarchy corresponds to an hierarchy of increasingly quantum critical systems in modular degrees of freedom. For a given prime p one has a sub-hierarchy Zp, Zp2=Zp× Zp, etc...
This mapping of integers to quantum critical systems conforms nicely with the general vision that biological evolution corresponds to the increase of quantum criticality as Planck constant increases. The group of conformal symmetries could be also non-commutative discrete group having Zn as a subgroup.
The number theoretically simple ruler-and-compass integers having as factors only first powers of Fermat primes and power of 2 would define a physically preferred sub-hierarchy of quantum criticality for which subsequent levels would correspond to powers of 2: a connection with p-adic length scale hypothesis suggests itself. Updated view here.
Particles of dark matter would reside at the flux tubes but would be delocalized (exist simultaneously at several flux tubes) and belonging to irreducible representations of Ga. What looks weird is that one would have an exact macroscopic or even astroscopic symmetry at the level of generalized imbedding space. Visible matter would reflect this symmetry approximately. This representation would make sense also at the level of biochemistry and predict that magnetic properties of 5- and 6-cycles [pentoses and hexoses] are of special significance for biochemistry. Same should hold true for graphene. Electron pairs are associated with 5- and 6-rings and the hypothesis would be that these pairs are in dark phase with na=5 or 6. Graphene which is a one-atom thick hexagonal lattice could be also an example of (conduction) electronic dark matter with na=6.
The idea about dark matter as a large Planck constant phase, requires na/nb= GMm/v0, v0=2-11 so that the values are gigantic. A possible interpretation is in terms of a dark (gravi)magnetic body assignable to the system playing a key role in TGD inspired quantum biology. See Construction of Elementary Particle Vacuum Functionals.
The fundamental feature of the configuration space is that it has two kinds of degrees of freedom. The degrees of freedom in which metric vanishes correspond to what I call zero modes and are purely TGD based prediction basically due to the non-point like character of particles identified as 3-surfaces. Zero modes are the counterparts of the classical macroscopic variables and in every quantum jump a localization in zero modes occurs; the state function reduction. This also means that the replacement of point like particle with 3-surface means giving up the locality of the physics at spacetime level: physics is however local at the level of configuration space containing 3-surfaces as its points. For instance, classical EPR nonlocality is purely local phenomenon at the level of configuration space. Besides allowing to get rid of the standard infinities of the interacting local field theories, the non-locality explains topologically the generation of structures, in particular biological structures which correspond to spacetime sheets behaving as autonomous units.
4. The cosmological constant.
Astrophysical systems correspond to [relativistic] stationary states analogous to atoms and do not participate [much] to cosmic expansion in a continuous manner but via discrete quantum phase transitions in which gravitational Planck constant increases. This from the dark matter hierarchy.
a) By quantum criticality of these phase transitions critical cosmologies are excellent candidates for the modeling of these transitions. Imbeddable critical (and also over-critical) cosmologies are unique apart from a parameter determining their duration and represent accelerating cosmic expansion so that there is no need to introduce cosmological constant = quantum phase transition increasing the size. See Could the value of fine structure vary in cosmological scales?
b) A possible mechanism driving the strings to the boundaries of large voids could be repulsive interaction, or repulsive gravitational acceleration.
c) Cosmological constant like parameter does not characterize the density of dark energy but that of dark matter identi fiable as quantum phases with large Planck constant.
d) The Lambda problem: large voids arequantum systems which follow the cosmic expansion only during the quantum critical phases.
e) p-Adic fractality predicts that cosmological constant is reduced by a power of 2 in phase transitions occurring at times corresponding to p-adic time scales. These phase transitions would naturally correspond to quantum phase transitions increasing the size of the large voids during which critical cosmology predicting accelerated expansion naturally applies.
f) On the average Lambda (k) behaves as 1/a^2 where a is the light-cone proper time. This predicts correctly the order of magnitude for observed value of Lambda.
g) What empty space is may be a consequence of cosmological constant absence. Such as stochastic quantization and a holography that reduces everything to the level of 3-metrics and more generally, to the level of 3-D eld con figurations. To a given 3-surface the metric of WCW assigns a unique space-time and this space-time serves as the analog of Bohr orbit and allows to realize 4-D general coordinate invariance in the space of 3-surfaces so that classical theory becomes an exact part of quantum theory. Both 4-D path integral and stochastic quantization for gravitation fail in this respect due to the local divergences (insuper-gravity situation might be di fferent). The preferred 3-surfaces circumvent this di fficulty, and give the GR^2. No emergence of space-time, no 'empty' space is there in TGD? In ZEO the S-matrix is replaced with M-matrix de fining a square root of thermodynamics.
Since the space-times allowed by TGD de ne a subset of those allowed by GR one can ask whether the quantization of GRT leads to TGD or at least sub-theory of TGD.
The arguments represented [in the article] however suggest that this is not the case.
A promising signal is that the generalization of Entropic Gravity (Verlinde's) to all interactions in TGD framework leads to a concrete interpretation of gravitational entropy and temperature, to a more precise view about how the arrow of geometric time emerges, to a more concrete realization of the old idea that matter-antimatter asymmetry could relate to di erent arrows of geometric time (not however for matter and antimatter but for space-time sheets mediating attractive and repulsive long range interactions), and to the idea that the small value of cosmological constant could correspond to the small fraction of non-Euclidian regions of space-time with cosmological constant characterized by CP2 size scale.
5. The helicity, vertices or spin.
The basic prediction of TGD is that the sign of energy depends on the time orientation of the spacetime surface ('negative energy' possible as a request or demand?), creating tensions and vortices as an S-matrix.
Generalized Feynman diagrams.
Zero energy ontology (ZEO) has provided profound understanding about how generalizedFeynman diagrams differ from the ordinary ones. The most dramatic prediction is that loop momenta correspond to on mass shell momenta for the two throats of the wormhole contact defining virtual particles: the energies of the energies of on mass shell throats can have both signs in ZEO. This predicts finiteness of Feynman diagrams in the fermionic sector. Even more: the number of Feynman diagrams for a given process is finite if also massless particles receive a small mass by p-adic thermodynamics. See topological torsion and thermodynamic irreversibility, by Kiehn and the TGD version. The mass would be due to IR cutoff provided by the largest CD (causal diamond) involved.
Generalized Feynman Diagrams as Generalized Braids String world sheets and partonic 2-surfaces provide a beatiful visualization of generalized Feynman diagrams as braids and also support for the duality of string world sheets and partonic 2-surfaces as duality of light-like and space-like braids. The dance metaphor.
The TGD inspired proposal (TGD as almost topological QFT) is that generalized Feynman diagrams are in some sense also knot or braid diagrams allowing besides braiding operation also two 3-vertices. The first 3-vertex generalizes the standard stringy 3-vertex but with totally different interpretation having nothing to do with particle decay: rather particle travels along two paths simultaneously after 1→2 decay. Second 3-vertex generalizes the 3-vertex of ordinary Feynman diagram (three 4-D lines of generalized Feynman diagram identified as Euclidian space-time regions meet at this vertex). I have discussed this vision in detail here. The main idea is that in TGD framework knotting and braiding emerges at two levels.
1. At the level of space-time surface string world sheets at which the induced spinor fields (except right-handed neutrino, see this) are localized due to the conservation of electric charge can form 2-knots and can intersect at discrete points in the generic case. The boundaries of strings world sheets at light-like wormhole throat orbits and at space-like 3-surfaces defining the ends of the space-time at light-like boundaries of causal diamonds can form ordinary 1-knots, and get linked and braided. Elementary particles themselves correspond to closed loops at the ends of space-time surface and can also get knotted (for possible effects see this).
2. One can assign to the lines of generalized Feynman diagrams lines in M2 characterizing given causal diamond. Therefore the 2-D representation of Feynman diagrams has concrete physical interpretation in TGD. These lines can intersect and what suggests itself is a description of non-planar diagrams (having this kind of intersections) in terms of an algebraic knot theory. A natural guess is that it is this knot theoretic operation which allows to describe also non-planar diagrams by reducing them to planar ones as one does when one constructs knot invariant by reducing the knot to a trivial one. Scattering amplitudes would be basically knot invariants.
Black holes.
One outcome is a new view about black holes replacing the interior of blackhole with a space-time region of Euclidian signature of induced metric and identifiable as analogs of lines of generalized Feynman diagrams. In fact, black hole interiors are only special cases of Eucdlian regions which can be assigned to any physical system. This means that the description of condensed matter as AdS blackholes is replaced in TGD framework with description using Euclidian regions of space-time.
The effective superposition of the CP2 parts of the induced metrics gives rise to an effective metric which is not in general imbeddable to M4× CP2. Therefore many-sheeted space-time makes possible a rather wide repertoire of 4-metrics realized as effective metrics as one might have expected and the basic objection can be circumvented. In asymptotic regions where one can expect single sheetedness, only a rather narrow repertoire of "archetypal" field patterns of gauge fields and gravitational fields defined by topological field quanta is possible.
The skeptic can argue that this still need not make possible the imbedding of a rotating black hole metric as induced metric in any physically natural manner. This might be the case but need of course not be a catastrophe. We do not really know whether rotating blackhole metric is realized in Nature. I have indeed proposed that TGD predicts new physics new physics in rotating systems. Unfortunately, gravity probe B could not check whether this new physics is there since it was located at equator where the new effects vanish.
Fundamental questions leading to TGD.
Ulla said... Seems this Firewall at the edge of Black holes, by Polchinski, has went through the blogosphere. Here Scott Aaronson.
Lubos saysuncritically promote the views of Joe Polchinski, Leonard Susskind, Raphael Bousso, and a few others. When it comes to the AMPS thought experiment, it just uncritically parrots the wrong statements by Polchinski et al.:
The interior (A) and the near exterior (B) have to be almost maximally entangled for the space near the horizon to feel empty; the near exterior (B) is almost maximally entangled with some qubits inside the Hawking radiation (C) because the Hawking radiation's ability to entangle the infalling and outgoing qubits. Because of the monogamy of the entanglement (at most one maximum entanglement may incorporate (B) at the same time), some assumptions have to be invalid. The unitarity should be preserved which means that the A-B entanglement has to be sacrificed and the space near the horizon isn't empty: it contains a firewall that burns the infalling observer.
That may sound good but, as repeatedly explained on this blog, this argument is wrong for a simple reason. The degrees of freedom in (A) and those in (C) aren't independent and non-overlapping. It is the very point of the black hole complementarity that the degrees of freedom in (A) are a scrambled subset of those in (C). The degrees of freedom in (A) are just another way to pick observable, coarse-grained degrees of freedom and "consistent histories" within the same Hilbert space. So the entanglement of (B) with "both" (A) and (C) isn't contradictory in any sense: it's the entanglement with the same degrees of freedom described twice.
It seems clear to me that this imbalanced perspective was incorporated to the article by the main "informers" among the scientists who communicated with Jennifer. This conclusion of mine partly boils down to the amazing self-glorification of Joe Polchinski in particular. So we're learning that if there's a mistake, the mistake is not obvious, AMPS is a "mighty fine paradox" that is "destined to join the ranks of classic thought experiments in physics" and it's the "most exciting thing that happened since [Bousso] entered physics". Holy cow. The mistake is obvious. AMPS simply assume that complementarity can't hold by insisting on separate parts of the wave function that are responsible for observations inside and outside. That's a wrong assumption, so it's not shocking that various corollaries such as the "firewall" at the horizon are wrong, too. This wrong assumed denial of complementarity is as wrong as the assumption that simultaneity has to be absolute – an assumption made by those who "debunk" Einstein's relativity; the error is in step 1 and means that they just didn't understand the original insights.
Matti Pitkanen said...
Blackholes represent the singularity of general relativity as a theory. What happens for the space-time in the interior of black hole? This should be the difficulty from which to start from. Not the only one.
One could also start from the energy problem of general relativity.
Or from proton instability predicted by GUTs: why quark and lepton numbers seem to be conserved separately?
Or by asking whether it is really true that so many primary fields are needed (superposition of effects of fields replaces superposition of fields in many-sheeted space-time)?
Or what is behind the family replication phenomenon?
Or what is the deeper structure behind standard model symmetries?
I could continue the list: the answer to every question unavoidably leads to TGD.
Superstring theories were claimed to provide quantum theory of gravitation but the outcome was land scape and tinkering with blackholes after it had become clear that superstrings themselves do not tell anything about physics and one must make a lot of ad hoc assumptions to get QFT theory limit. After production of huge amount of literature super stringers are exactly in the same position as before the advent of superstring models.
It would be encouraging if people would gradually realize that we have not made much progress during these four decades. Some really new idea is needed to make genuine progress and we must open our minds for it. Maybe it is here already;-).
Ulla said...
Thanks, this was exactly the kind of list of problems leading to TGD I have asked for. You are welcome to continue on it :)
About Planck units.
Under our current best-guess of a complete theory of physics, the maximum possible temperature is the Planck temperature, or 1.41679 x 10^32 Kelvins. However, it is common knowledge that our current theories of physics are incomplete.
Gustavo Valdiviesso The use of the so called "Planck units" is rather arbitrary, and I will point out why: Every model has its limitations. For instance, Newton's second law breaks down at speeds near c and need to be replaced by a Lorentz invariant version, so that the concept of relativistic energy rises from it. But, you see, the speed of light was known from Maxwell equations well before relativity. It was also known that Maxwell equations and Newton's laws doesn't always get along (there are some situations were the Lorentz force between a point charge and a magnet does not have a action-reaction partner). Also, and more obvious, Maxwell equations are not invariant under Galilean transformations, in which Newton's second law is based. So, we have two models for the same Nature, and they disagree... one of them carries a fundamental constant: the speed of light. Years later, we see that this very same speed is the limit of one of the models: the one that did not care about it.
Now, we have several models (quantum mechanics, general relativity, etc) and we can expect all of them to have a limit, to break down at some value of some physical observable.
So we must have a model based on Lorentz invariance, which is exactly what TGD is.
fredag 7 december 2012
Is it possible to learn TGD?
Saturday, December 01, 2012
This is a blogpost directly from Matti Pitkänens blog. It seems that also Matti thinks that physics is only for physicists, and my efforts to give TGD lessons are in vain. I leave it for the readers to judge.
Matti also complains that the communication does not work, and I thought this would be one way to make things easier, not more difficult? Experts can go to Mattis blog, where they find the math and the physical phrases. I try to avoid them as far as possible here. I use more words, because words are my tool. The reader can determine if they form just a word-salat?
In an earlier blog discussion Hamed asked about some kind of program for learning TGD in roughly the same manner as I did it myself. I decided to write a brief summary about the basic steps leaving aside the worst side tracks since 35 years means too flat learning curve;-).
I wrote a summary about the very first steps, that is the steps made during the four years before my thesis and related to classical dynamics mostly. I could not avoid mentioning and even briefly explaining notions like the "world of classical worlds" (WCW), quantum TGD, Kähler action, modified Dirac equation, zero energy ontology, etc... since I want to relate the problems that I encountered during the first years of TGD to their solutions which came much later, some of them even during this year. I hope that I find time to write similar summaries about later stages in the evolution of TGD and add them to this text.
This summary does not provide any Golden Road to TGD. I do not even know whether it is possible to learn TGD. And certainly it is much more difficult to passively assimilate ideas of others than to actively discover and develop ideas by one self. The authority of the original discoverer - such as that of Witten's - can help enormously but I do not possess this kind of authority so that I must trust only on the power of the ideas themselves.
Since the text consists of five pages it is more practical to give only a link to the pdf file containing it.
Ulla said... I have stranded on the spacetime itself. I cannot decide which is the most easy way into TGD, and I think today it is wrong to start with introducing the classic concept of spacetime. There is clearly an interest in this, because my small 3 texts has got quite many visits. I have 30 texts written, but because of the uncertainty I have not published them yet.
Maybe the intro of some important problems would be a more logical way? As the three body problem?
Zero Energy Ontology etc. To link these to mainstream physics is also difficult for me.
Matti Pitkanen said...
The fact is that understanding of TGD requires understanding of basics of physics and mathematics. As a referee I have read so many unified theories by people who have read a couple of popular science books and got the impression that building a theory of everything requires just "creativity" that unified theories cannot be built in garage.
Physics and theoretical physics are disciplines, whose development has taken a about 500 hundred years, a life work of totally devoted brilliant people. Nowadays these 500 years can be compressed to 3-4 years in university classes. This is wonderful but during this time one of course gets only some important impressions, not much more. There is no hope of compressing 500 years to a couple of web articles. This would be however needed as a background to develop introduction to TGD for dummies;-).
Anyone can learn macro-economics but theoretical physics is a REALLY difficult discipline. Think only that the best mathematical brains have worked for 28 years in vain with superstring models. They did not get anywhere. We still have Einstein's theory as THE theory of gravitation.
The following old saying still applies. "God give me the wisdom to see what I cannot do and give strength to do what I can;-)". In my case this means that I cannot write a five page essay leading the reader to enlightment but I can improve endlessly the articulations of TGD so that who have the needed background can easily understand it when the Zeitgeist allows them to read an article about TGD in presence of colleagues.
Ulla said...
I did not talk of a 5 page essay. I talked about the basic questions leading to TGD. It is so enormously complex by itself. I have met so many questions now that need explanation in terms of mainstream thinking. The implicit part of TGD is one big obstacle. Also the different parts are so intwined in each other that it is almost impossible to start somewhere simple. This made me stop for a while, and now I have so little time for this. My aim is to continue, but I need advice what the best path would be. To simply repeat what you have said is nonsense. I need to UNDERSTAND it. As Hamed said, the most interesting part is the biology, but to reach there I need the physics first. Without university physics and most of the math :) I have only the words. And it is only an intro.
One thing that maybe went wrong is the Kaluza-Klein thinking about the cosmological constant? It led to string theory, but are there other ways out? Or is the try to mathematize the unknown wrong in itself?
I know you have the hierarchy.
Matti Pitkanen said...
Unfortunately words are not enough when one is trying to talk about quantum physics or mathematics. In mathematics words are only a shorthand - program calls initiating processes in the brain of mathematician but not in brain of a layman. This mathematics is sometimes very simple but difficult to grasp without background. There is no concept so boringly simple as finite-D Hilbert space, but when you try to understand quantum theory without it you encounter mission impossible. Quantum superposition, quantum entanglement, and quantum jump: here are three notions whose understanding without Hilbert space is exceedingly difficult.
Complexity is very relative notion. Basic principles are simple but once you start to really develop and apply the theory, things become complex. So it is also with TGD: TGD is a TOE covering everything form CP_2 length scale to cosmology and one cannot expect simplicity at the level of implications.
The theory of von Neumann algebras is excellent example of this: the axioms look trivial but the mathematics generated by them looks formidable and fascinating at the same time.
If you look about text book about QED, something relatively simple by recent standards, you get absolutely scared by the complexity of the formulas. And they are only for electron-photon system!
I am sorry, but this is the situation. It is very very lonely here and also the air is very cold and thin;-). And it took 35 years to climb here;-). Maybe I should have thought twice.
Santeri Satama said...
Matti and Hamed, a learning strategy suggestion: to my knowledge there is no better way to learn a thing than to internalize it by trying to teach it to somebody else, "learn one thing, teach one other" as the saying goes; so if Hamed finds suitable "victim" at some stage of this process he could start trying to teach TGD - while simultaneously learning it with the help that Matti can can provide.
Good to see thins happening. The real test of TGD is can it be communicated to other theoreticians and even laymen, or will it remain the "Lonely God". ;)
PS: it's becoming almost impossible to post on this blog by passing the "not a robot" test. Which is both sad and funny.
Matti Pitkanen said... To Santeri:
You are right about learning. Unfortunately too often only the teacher learns;-).
11 said...
Dear Matti,
Did you think about future experiments to support or falsify your TGD?
Matti Pitkanen said...
To 11:
Experimental tests are very important. TGD makes a lot of predictions. Many of them are acutally successfully tested already. Mention only p-adic mass calculations which are based on extremely general assumptions.
a) No standard SUSY is one prediction but no one takes this as interesting because standard SUSY is already excluded in practice.
b) My hope was that the absence of Higgs and identification of Higgs like state as pion of M_89 physics could provide a killer test. It however turned out that TGD is consistent with Higgslike state allowing at QFT limit effective description of particle masses: this follows more or less from the existence of QFT limit. Experimentally the situation is still unsettled. Decays in two-gamma channel and to fermion pairs are both decisive.
c) An excellent candidate for a breakthrough prediction is M_89 hadron physics. The prediction of entire new hadron physics is sensational. The recent observations from LHC (made for the first time already two yeas ago and commented also here) have simplest interpretation as decay of string like magnetic flux tubes to partons.
This kind of objects should not appear at ultra high energies since they relate to low energy hadron physics. The only possibility is that hadrons of new hadron physics with large mass scale are in question. M_89 hadron physics is of course the natural candidate for this hadron physics.
Already RHIC observed these events and QCD definitely does not predict them. Therefore the notion of color glass condensate was introduced to save the situation but it is not QCD prediction if we are honest. Quark gluon plasma is the prediction of QCD.
More generally fractal copies of hadron physics and also leptohadron physics predicting pion like states consisting of color octet charged leptons are predicted. These states have been observed for all lepton families but since they do not fit with standard model the observations have been put under the rug.
Also "infrared" Regge trajectories for ordinary hadrons are possible and there is recent evidence in case of ordinary hadrons for them: the scale of mass splittins is about 20-40 MeV.
d) TGD explains family replication in terms of the topology of partonic 2-surface and this also means predictions of new physics. Do gauge bosons have analog of family replication meaning an exotic octet of mesons besides singlet for dynamical symmetry SU(3) assigned to the 3 families? And how massive are the fermions corresponding to higher genus: here there is a good argument supporting the guess that they are massive.
These are just few predictions and related to particle physics. There are myriads of predictions in cosmology and astrophysics and also in biology. This because TGD Universe is fractal. Basic quantitative tool is p-adic length scale hypothesis predicting a hierarchy of length scales coming as powers of sqrt(2).
The problem is the communication barrier due to the extremely arrogant attitudes of the academic researcher. For this I cannot do much. It would be a job of psychiatrist.
Fractality said...
Matti: Noble attempt at helping us laymen out in cultivating understanding of TGD. It is much appreciated. Do you think you could do something similar for TGD theory of consciousness?
ThePeSla said... Matti,
This post is an excellent attempt at trying to communicate these frontier ideas. I downloaded it in hard copy of 8 pages and read it carefully. If you are interested in my impressions where we may share some evolution in our approaches I have stated it there.
Professors, like Hoyle when I had tea with him- well he said students are always coming to him to comment on their theories-of course we talked about other things, down to earth and I mentioned supporting him briefly if I did not have other directions but certainly not the new Big Bang cosmology.
Still feel free to comment on my system if I have in discussing yours made it a little easier- then you can get down to maybe something more useful from my own.
I mentioned the scientific american article ulla had posted in facebook- there was a time when I did break from just those two ways to apply and see dimensions and it was an awakening moment.
I think your more careful formal approach is much more difficult than freely allowing the intuition and poetic magic to flow.
But where you say you cannot understand some things even in the asking clearly that is an achievement and perhaps some things in the context were not an error (holographic stuff and surfaces for example) but the context is such an error.
As to why the advanced culture of Finnish science does not support your project that is like some form of economics it would take another Gauss to begin to phantom although I made meager suggestions.
All in all a great posting, thank you. I do wish I had the training in the exponential type notations but perhaps they slow us down.
Matti Pitkanen said...
To Fractality:
I hope I can find time and energy to continue the summaries, also about TGD inspired theory of consciousness and quantum biology.
Thanks for Pesla for encouraging comments.
Fractality said...
Matti: Excellent. You are inducing quantum jump to the order of Einstein ;) Have you ever theorized as to the roles of biologically active compounds like serotonin and certain tryptamine alkaloids?
Matti Pitkanen said...
Not seriously. Just some general ideas about what makes information molecule information molecule.
Santeri Satama said... Not sure how relevant this is to above questions, but watching this vid about psychological time and internal "clock" so far unexplained (http://www.youtube.com/watch?feature=player_embedded&v=DKPhPz5Hqqc)
gave the idea that internal clock or time-sense could be based on Shnoll effect. Through "information molecules"?.
Would that be the expectation of your general view about the relation of geometric times and psychological times?
Matti Pitkanen said...
Thank you for asking. I cannot answer without bringing in magnetic bodies.
One of their basic tasks is go generate EEG rhythms which define the ticks of clocks with different basic units of time. It has been observer that the period of EEG decomposes to two parts such that during first half there is coherence and second half non-coherence.
Maybe this means that during the first half period "alpha" subself with say average lifetime/wakeup time of .05 seconds wakes up and dies in the beginning of second half period (roughly). We ourselves would do the same in time scale of 12+12 hours. Wake-sleep cycle would define a universal clock.
In standard picture one tries to understand basic EEG rhythms in terms of various brain circuits. I see this as hopeless project as trying to find standard SUSY at LHC;-) but modern Big Science is full of this kind of desperate projects.
I have talked in http://matpitka.blogspot.fi/2012/11/quantum-dynamics-for-moduli-associated.html about the recent views concerning the generation of the arrow of time. I do not bother to type the recent view about time.
Perhaps it is enought to say that in zero energy ontology the analog of clock pendulum emerges at very abstract level. The state function reductions at upper and lower boundary of causal diamond take place alternatively. This has as an analog the motion of clock pendulum: the highest position at right- that at left- that at right-..... This would correspond at the level of self sleep-wake-up cycles which would be universal aspect of consciousness.
As always I think that the understanding is rather satisfactory now;-) . I must of course confess that the understanding of the arrow of time has been an Odysseia similar to the understanding of the nature of possibly existing Higgs in TGD Universe;-).
Ulla said...
Matti, fortunately I have absolutely no intention to make things more complex than they need to be. I only try to make some introduction, nothing else.
I have seen enough of nonsense to realize this is difficult, and that's why I usually ask. I WANT to do this because TGD is really a good way to understand things. But the details necessary are troublesome, and they force me to read lots and lots of articles. I don't write for experts, but for common people with common knowledge. For them words like math phrases are just garbage and say absolutely nothing. I have to 'translate' TGD.
Cold and thin air is good for the brain. Why on Earth did you have to come in my way? You should not have had, if things worked well, as T said. Dammit.
I have no problems with the test, which maybe says something about me?
Ulla said...
Can I publish this discussion on my TGD Lessons?
Hamed said...
Dear Matti, and the readers,
When I see the posting I encounter with many comments. I know more of these persons are very interested to understand TGD, but there is a big problem for them that are a lot of physics and mathematics for those that don’t know about basic physics and mathematics.
I have some suggestions for the readers and I request from Matti if anything between them is wrong, he correct and complete it.
Generally learning science is a process that needs patient but it is very enjoyable ;-). About TGD this seems hard, because TGD is not only a theory but a program, a lot of mathematical and physical ideas and principles had evolved more than 30 years. Although learning TGD is hard but it is possible if one has enough patient.
I encourage everyone even laymen that are working in other fields of science to learn TGD. Because the worldview of TGD makes deep influence on their thinking and this leads to progress and evolution in other fields of science too.
How the laymen can learn TGD? I try to answer it.
in learning consciousness and biological parts of TGD, although at first it seems that these parts doesn’t need mathematics or physics but as I tried it, when one goes further and asks some whys in the bases of them, at end it will be Revealed that it needs understanding Quantum TGD. Similarly understanding the bases of Quantum TGD without classical TGD is impossible.
But for a beginner it is useful at first to understanding the definitions of concepts of TGD from classical to Quantum and after it to biology and consciousness at the introductory level without going further. For this the overview articles of TGD is very good.
After this, I encourage them to learn basic concepts of physics and mathematics. For laymen I think it is possible to learn them without calculations because it is enough for their purposes. Unfortunately it is not enough for me ;-). The book “The road to reality” of Penrose is very good for this purpose.(for example it gives some good mathematical intuitions about Hilbert space for learning Quantum too) The prerequisite for learning the book is physics and mathematics of high school. The physical intuitions in this book are very useful even for teachers in physics.
Santeri Satama suggested to me to teach for better understanding. Yes, It will be useful for my learning and I will try as I can. I remember the quoting of Einstein “You do not really understand something unless you can explain it to your grandmother.” ;-). I am certain that I can’t explain anything of physics to my grandmother ;-) but I think I can do it at least for the readers of the blog and request from Matti to correct it.
Matti Pitkanen said... To Ulla:
Nothing against your proposal.
To Hamed:
Thank you for a perspective of a person who is really working hard to understand the ideas of TGD. It would be nice to have the "Road to Reality" in bookshelf. This kind of books are God's gifts to human kind. Maybe someone writes someday this kind of book explaining hyperfinite factors, Kac Moody algebras and all that stuff which makes me feel unpleasant;-).
I feel that technical side is not terribly important but maybe this is illusion: to learn the conceptual thinking one must perhaps learn first the basic techniques such as the mathematics learned in theoretical physical classes during first few years.
Maybe this relates to basic fact about language: words as such have no meaning, they only induce self-organization patterns giving rise to the experience of meaning. The meaning of the word is quite different for a person with and for a person without the adequate background.
Ulla said...
Thanks Matti,
Hamed is right in the fact that consciousness needs the physical background to be right understood. Also every biological event need a physical explanation, and that is alone a huge task. We have as instance with Matti discussed the meaning of endorphins and serotonins, but without the physical background his explanations seems meaningless, even nonsensical. This is exactly what ordinary physicists encounter too, and this is why they say TGD is rubbish. They simply have not the patience to learn it from basics. Many times I feel I know more than them when I have tried to discuss things, but then again I have too many empty boxes of knowledge. The mainstream physicist have maybe what seems a coherent knowledge, but when I ask deep enough it turns out they too know very little. This is why I asked for a list of problems that show TGD as a possible solution.
Hamed is really very good for TGD. I hope time is ripe for it now, and he is not marginalized for it.
Matti Pitkanen said...
To Ulla;
Every generation of scientists plays again the evergreen "Emperor's new clothes" by H. C. Andersen.
tisdag 24 januari 2012
Geometrodynamics from wikipedia (short variant here) generally denotes a program of reformulation and unification which was enthusiastically promoted by John Archibald Wheeler in the 1960s and is today rather loosely used as a synonym for GR, and some authors use the phrase Einstein's geometrodynamics to denote the initial value formulation. Spacetimes are sliced up into spatial hyperslices, and the vacuum Einstein field equation is reformulated as an evolution equation describing how, given the geometry of an initial hyperslice (the "initial value"), the geometry evolves over "time". This is also what distinguishes TGD and GR, in the big structure, says Matti.
As described by Wheeler in the early 1960s, geometrodynamics attempts to realize three catchy slogans
• mass without mass,
• charge without charge,
• field without field.
Just LOOK. This is a pointing to thinking today?
"The vision of Clifford and Einstein can be summarized in a single phrase, 'a geometrodynamical universe': a world whose properties are described by geometry, and a geometry whose curvature changes with time – a dynamical geometry." The geometry of the Reissner-Nordström electrovacuum solution suggests that the symmetry between electric (which "end" in charges) and magnetic field lines (which never end) could be restored if the electric field lines do not actually end but only go through a wormhole to some distant location. He searched the momentum constraint in geometry and wanted to show that GR is emergent, like a logical necessity; he talked of spacetime foam; requres the Einstein-Yang-Mills-Dirac System."
Scattering and virtual particles are similar modern notions? A dynamic metric.
"Geometrodynamics also attracted attention from philosophers intrigued by the suggestion that geometrodynamics might eventually realize mathematically some of the ideas of Descartes and Spinoza concerning the nature of space."
Is this a bad way to say that Matti Pitkänens work is too spiritual? Not even the name mentioned. Still he is 'famous'. See Topological Geometrodynamics: What Might Be the Basic Principles.
Modern geometrodynamics.
Christopher Isham, (he got the Dirac medal 2011), Jeremy Butterfield, (his homepage here) + students have continued to develop quantum geometrodynamics.
Addendum: About the Dirac medal, se all Dirac Medal winners here, note the many famous names:
Professor Christopher Isham
Imperial College London
For his major contributions to the search for a consistent quantum theory of gravity and to the foundations of quantum mechanics.
Chris Isham is a worldwide authority in the fields of quantum gravity and the foundations of quantum theory. Few corners of these subjects have escaped his penetrating mathematical investigations and few workers in these areas have escaped the influence of his fundamental contributions. Isham was one of the first to put quantum field theory on a curved background into a proper mathematical form and his work on anti-de Sitter space is now part of the subject’s standard toolkit.
His early work on conformal anomalies has similarly gone from “breakthrough to calibration”, as all good physics does. He invented the concept of twisted fields which encode topological aspects of the spacetime into quantum theory, and which have found wide application. He did pioneering work on global aspects of quantum theory, developing a group-theoretic approach to quantization, now widely regarded as the “gold standard” of sophisticated quantization techniques. This work laid some of the foundations for the subsequent development of loop-space quantum gravity of Ashtekar and collaborators (the only well-developed possible alternative to string theory). He has also made significant contributions to quantum cosmology and especially the notoriously conceptually difficult “problem of time”.
On the foundations of quantum theory, Isham has made many contributions to the decoherent histories approach to quantum theory (of Gell-Mann and Hartle, Griffiths, Omnes and others), a natural extension of Copenhagen quantum mechanics which lessens dependence on notions of classicality and measurement in the quantum formalism. In particular, using a novel temporal form of quantum logic, he established the axiomatic underpinnings of the decoherent histories approach, crucial to its generalization and application to the quantization of gravity and cosmology.
His recent work has been concerned with the very innovative application of topos theory, a generalization of set theory, into theoretical physics. He showed how it could be used to give a new logical interpretation of standard quantum theory, and also to extend the notion of quantization, giving a firm footing to ideas such as “quantum topology” or “quantum causal sets”. Isham’s contributions to all of these areas, and in particular his continual striving to expose the underlying mathematical and conceptual structures, form an essential part of almost all approaches to quantum gravity.
From Wikipedia cont. "Topological ideas in the realm of gravity date back to Riemann, Clifford and Weyl and found a more concrete realization in the wormholes of Wheeler characterized by the Euler-Poincare invariant. They result from attaching handles to black holes.
Observationally, Einstein's general relativity (GR) is rather well established for the solar system and double pulsars. However, in GR the metric plays a double role: Measuring distances in spacetime and serving as a gravitational potential for the Christoffel connection. This dichotomy seems to be one of the main obstacles for quantizing gravity. Eddington suggested already 1924 in his book `The Mathematical Theory of Relativity' (2nd Edition) to regard the connection as the basic field and the metric merely as a derived concept.
Consequently, the primordial action in four dimensions should be constructed from a metric-free topological action such as the Pontrjagin invariant of the corresponding gauge connection. Similarly as in the Yang-Mills theory, a quantization can be achieved by amending the definition of curvature and the Bianchi identities via topological ghosts. In such a graded Cartan formalism, the nilpotency of the ghost operators is on par with the Poincare lemma for the exterior derivative. Using a BRST antifield formalism with a duality gauge fixing, a consistent quantization in spaces of double dual curvature is obtained. The constraint imposes instanton type solutions on the curvature-squared `Yang-Mielke theory' of gravity, proposed in its affine form already by Weyl 1919 and by Yang in 1974. However, these exact solutions exhibit a `vacuum degeneracy'. One needs to modify the double duality of the curvature via scale breaking terms, in order to retain Einstein's equations with an induced cosmological constant of partially topological origin as the unique macroscopic `background'.
Such scale breaking terms arise more naturally in a constraint formalism, the so-called BF scheme, in which the gauge curvature is denoted by F. In the case of gravity, it departs from the meta-linear group SL(5,R) in four dimensions, thus generalizing (Anti-)de Sitter gauge theories of gravity. After applying spontaneous symmetry breaking to the corresponding topological BF theory, again Einstein spaces emerge with a tiny cosmological constant related to the scale of symmetry breaking. Here the `background' metric is induced via a Higgs-like mechanism. The finiteness of such a deformed topological scheme may convert into asymptotic safeness after quantization of the spontaneously broken model."
LinkAddendum, Wheeler in wikipedia:
During the 1950s, Wheeler formulated geometrodynamics, a program of physical and ontological reduction of every physical phenomenon, such as gravitation and electromagnetism, to the geometrical properties of a curved space-time. Aiming at a systematical identification of matter with space, geometrodynamics was often characterized as a continuation of the philosophy of nature as conceived by Descartes and Spinoza. Wheeler's geometrodynamics, however, failed to explain some important physical phenomena, such as the existence of fermions (electrons, muons, etc.) or that of gravitational singularities. Wheeler therefore abandoned his theory as somewhat fruitless during the early 1970s.
Maybe he used the wrong concept for the unification? Why are forces making qubits?
Addendum: Wikipedia, the talk page for discussing improvements to the John Archibald Wheeler article.
information regarding geometrodynamics is not accurate
This is a good article on J.A. Wheeler. However, the information regarding geometrodynamics is not accurate, especially the following statement: "Wheeler abandoned it as fruitless in the 1970s".As a matter of fact, Wheeler kept using the term "geometrodynamics" to describe Einstein's theory of general relativity till his last days. For example, in Gravitation and Inertia, a book written with the Italian physicist I.Ciufolini in 1995(and which was missing from the bibliography), the authors keep referring to "Einstein Geometrodynamics"(the title of Chapter 2) throughout the the book: Chapter 3 is entitled " Tests of Einstein Geometrodynamics", Chapter 5 is "The Initial-Value Problem in Einstein Geometrodynamics" and Chapter 7:"Some Highlights of the past and a Summary of Geometrodynamics and Inertia".This proves that Wheeler did not abandon the concept at all in the 1970s!
Addendum about quantum geometrodynamics, hard linked to time and quantum gravity: Claus Kiefer, 2008. Quantum geometrodynamics: whence, whither? Total search here. Abstract:
Quantum geometrodynamics is canonical quantum gravity with the three-metric as the configuration variable. Its central equation is the Wheeler--DeWitt equation. Here I give an overview of the status of this approach. The issues discussed include the problem of time, the relation to the covariant theory, the semiclassical approximation as well as applications to black holes and cosmology. I conclude that quantum geometrodynamics is still a viable approach and provides insights into both the conceptual and technical aspects of quantum gravity.
And this is actually published; Gen.Rel.Grav.41:877-901, 2009 DOI:10.1007/s10714-008-0750-1
See also: Interpretation of the triad orientations in loop quantum cosmology
Scalar perturbations in cosmological models with dark energy - dark matter interaction
Look: Does time exist in quantum gravity?
Comments: 10 pages, second prize of the FQXi "The Nature of Time" essay contest
Cosmological constant as result of decoherence. This means non-commutative geometry?
An earlier article (Adrian P. Gentle, Nathan D. George, Arkady Kheyfets, Warner A. Millerfrom 2004; Constraints in quantum geometrodynamics, http://arxiv.org/abs/gr-qc/0302044
And about time and geometrodynamics, by the same authors
A geometric construction of the Riemann scalar curvature in Regge calculus. Jonathan R. McDonald, Warner A. Miller http://arxiv.org/abs/0805.2411
A Discrete Representation of Einstein's Geometric Theory of Gravitation: The Fundamental Role of Dual Tessellations in Regge Calculus http://arxiv.org/abs/0804.0279
Quantum Geometrodynamics of the Bianchi IX cosmological model
Arkady Kheyfets, Warner A. Miller, Ruslan Vaulin 2006 http://arxiv.org/abs/gr-qc/0512040
and from 1995,
Quantum Geometrodynamics I: Quantum-Driven Many-Fingered Time
Arkady Kheyfets, Warner A. Miller http://arxiv.org/abs/gr-qc/9406031
All actually published.
• Anderson, E. (2004). "Geometrodynamics: Spacetime or Space?". arXiv:gr-qc/0409123 [gr-qc]. This Ph.D. thesis offers a readable account of the long development of the notion of "geometrodynamics". University of London, Examined in June by Prof Chris Isham and Prof James Vickers. 226 pages including 21 figures. 396 cit.
This thesis concerns the split of Einstein's field equations (EFE's) with respect to nowhere null hypersurfaces. Areas covered include A) the foundations of relativity, deriving geometrodynamics from relational first principles and showing that this form accommodates a sufficient set of fundamental matter fields to be classically realistic, alternative theories of gravity that arise from similar use of conformal mathematics. B) GR Initial value problem (IVP) methods, the badness of timelike splits of the EFE's and studying braneworlds under guidance from GR IVP and Cauchy problem methods.
The work in this thesis concerns the split of Einstein field equations (EFE’s) with respect to nowhere-null hypersurfaces, the GR Cauchy and Initial Value problems (CP and IVP), the Canonical formulation of GR and its interpretation, and the Foundations of Relativity. I address Wheeler’s question about the why of the form of the GR Hamiltonian constraint “from plausible first principles”. I consider Hojman–Kuchar–Teitelboim’s spacetime-based first principles, and especially the new 3-space approach (TSA) first principles studied by Barbour, Foster, ´O Murchadha and myself. The latter are relational, and assume less structure, but from these Dirac’s procedure picks out GR as one of a few consistent possibilities. The alternative possibilities are Strong gravity theories and some new Conformal theories. The latter have privileged slicings similar to the maximal and constant mean curvature slicings of the Conformal IVP method.
The plausibility of the TSA first principles are tested by coupling to fundamental matter. Yang–Mills theory works. I criticize the original form of the TSA since I find that tacit assumptions remain and Dirac fields are not permitted. However, comparison with Kuchaˇr’s hypersurface formalism allows me to argue that all the known fundamental matter fields can be incorporated into the TSA. The spacetime picture appears to possess more kinematics than strictly necessary for building Lagrangians for physically-realized fundamental matter fields. I debate whether space may be regarded as primary rather than spacetime. The emergence (or not) of the Special Relativity Principles and 4-d General Covariance in the various TSA alternatives is investigated, as is the Equivalence Principle, and the Problem of Time in Quantum Gravity.
Further results concern Elimination versus Conformal IVP methods, the badness of the timelike split of the EFE’s, and reinterpreting Embeddings and Braneworlds guided by CP and IVP knowledge.
Mielke, Eckehard W. (2010, July 15). Einsteinian gravity from a topological action. SciTopics. Retrieved January 17, 2012, from http://www.scitopics.com/Einsteinian_gravity_from_a_topological_action.html
Although many details remain to be seen, topological actions are prospective in being renormalizable and, after symmetry breaking, are inducing general relativity as an "emergent phenomenon" for macroscopic spacetime.
Older ref. |
16cbe644528afb89 | Klein-Gordon Equation
Also found in: Wikipedia.
Klein-Gordon equation
[′klīn ′gȯrd·ən i‚kwā·zhən]
(quantum mechanics)
A wave equation describing a spinless particle which is consistent with the special theory of relativity. Also known as Schrödinger-Klein-Gordon equation.
Klein-Gordon Equation
a relativistic (that is satisfying the requirements of the theory of relativity) quantum equation for particles with zero spin.
Historically, the Klein-Gordon equation was the first relativistic equation in quantum mechanics for the wave function ψ of a particle. It was proposed in 1926 by E. Schrödinger as a relativistic generalization of the Schrödinger equation. It was also proposed, independently, by the Swedish physicist O. Klein, the Soviet physicist V. A. Fok, and the German physicist W. Gordon.
For a free particle, the Klein-Gordon equation is written
This equation is associated with the relativistic relationship between the energy £ and the momentum p of a particle £ 2 = p2c2 + m2c4, where m is the mass of the particle and c is the speed of light.
The solution to the equation is the function ψ (x, y, z, t), which is a function only of the coordinates (x, y, z) and the time (t). Consequently, the particles described by this function have no other internal degrees of freedom—that is, they are spinless (the π-meson and the K-meson are of this type). However, analysis of the equation indicated that its solution ψ differed fundamentally in physical meaning from the ordinary wave function, which is considered as the probability amplitude of detecting a particle at a given point in space and a given moment in time: ψ(x, y, z, f) is not determined uniquely by the value of ψ at the initial moment (such an unambiguous relationship is postulated in nonrelativistic quantum mechanics).
Furthermore, the expression for the probability of a given state can take on not only positive values but also negative values, which are devoid of physical meaning. Therefore, the Klein-Gordon equation was at first rejected. However, in 1934, W. Pauli and W. Weisskopf discovered a suitable interpretation for the equation within the scope of quantum field theory; treating it like a field equation analogous to Maxwell’s equations for an electromagnetic field, they quantized it, so that ψ became an operator.
References in periodicals archive ?
1, Klein formulated in his paper (5) (1927) the general method to treat the interaction between charged particles and an electromagnetic field on the basis of the semiclassical treatment; hence he should be called the founder of the semi-classical treatment (the Klein-Gordon equation was also derived here).
The second order time derivatives involved in the linear Klein-Gordon equation were decomposed into the first order derivatives.
Consequently, a free boson [psi](t, x) hold with the Klein-Gordon equation
Masmoudi, Global solutions for a semilinear 2D Klein-Gordon equation with exponential type nonlinearity, Comm.
At this step, the one-dimensional Klein-Gordon Equation (KGE) appears, eventually, which specifies the modal amplitudes.
This article will focus on the application of the Lie group theory to the Klein-Gordon equation
There are three types of models of the nonlinear Klein-Gordon equation, with power law nonlinearity, that are studied in this paper.
1) and its special solutions to solve the generalized Pochhammer-Chree equation and the Klein-Gordon equation.
PURI, A numerical method for computing radially symmetric solutions of a dissipative non-linear modified Klein-Gordon equation.
Numerous examples demonstrate third-order obstacle problems, third-order singularity perturbed problems, Volterra-Fredholm integral equations, an age-structured population model, a damped Klein-Gordon equation, and coefficient inverse problems.
for a free fermion [psi](t, x) and the Klein-Gordon equation
Their topics are Green's functions for ordinary differential equations, the Laplace equation, the static Klein-Gordon equation, higher order equations, multi-point-posed problems, the partial differential equation matrices of Green's type, diffusion equations, and the Black-Scholes equation. |
4050b978b72e7134 | NomadDigest – The Nature of Reality Observations and Their Implications About the Realities of Our Natural World Thu, 14 Dec 2017 13:04:17 +0000 en-US hourly 1 Pilot Wave Theory Mon, 13 Feb 2017 20:40:21 +0000
There’s one interpretation of the meaning of quantum mechanics that somehow manages to skip a lot of the wildly extravagant, or near mystical ideas of the mainstream interpretations: it’s DeBroglie-Bohm Pilot-Wave theory. Despite it’s alluring intuitive nature, for some reason it remains a fringe theory.
Misinterpretation of the ideas of Quantum Mechanics has spawned some of the worst quackery, Pseudo-Science, hoo-ha, and unfounded mystical story telling of any scientific theory. It’s easy to see why, there are far out there explanations for the processes at work behind the incredible successful mathematics of quantum mechanics.
These explanations claim stuff like: things about waves and particles at the same time, the act of observation defines reality, cats supposed to be alive and dead, and even that the universe is constantly splitting into infinite alternate realities. The weird results of quantum experiments seem to demand weird explanations of the nature of reality.
There is one interpretation of quantum mechanics that remains comfortably, almost stodgily, physical, that’s DeBroglie-Bohm Pilot-Wave theory. Pilot-Wave Theory, also known as Bohmian Mechanics stands in striking contrast to the much more main stream ideas, as for example the Copenhagen, and Many-World interpretations.
Copenhagen Interpretation of Quantum Mechanics ↓Many-Worlds Interpretation of Quantum Mechanics ↑
Pilot-Wave Theory is perhaps the most solidly physical, even mundane, of the complete and self-consistent interpretation of quantum mechanics. But at the same time it’s considered one of the least orthodox. Why so? Because orthodoxy equals radicalism plus time. The founding fathers of the Copenhagen interpretation of quantum mechanics, Werner Heisenberg and Niels Bohr were radicals. When quantum theory was coming together in the twenties, the were fervent about the need to reject all classical thinking in interpreting the strange results of early quantum experiments.
(left) – Werner Heisenberg and Niels Bohr – (right)
One aspect of that radical thinking was that the wave function is not a wave in anything physical, but an abstract distribution of probabilities. Bohr and Heisenberg insisted that in the absents of measurement the unobserved universe is only a suite of possibilities of the various states it could take, were a measurement to be made. Then upon measurement, fundamental randomness determines the properties of say, the particle that would emerge from it’s wave function.
This required an almost mystical duality between the wave and the particle like nature of matter. Not everyone was so sure, Einstein famously hated the idea of fundamental randomness, but to counter Bohr and Heisenberg there needed to be a full theory that described how a quantum object could show both wave and particle behavior at the same time without being fundamentally probabilistic.
That theory came from Louis DeBroglie, the guy who originally proposed the idea that matter could be described as waves right at the beginning of the quantum revolution. DeBroglie theory reasons that, there was no need for quantum objects to transition in a mystical way between non-real waves and real particles, why not just have real waves just push around real particles.
This is Pilot-Wave Theory. In it, the wave function describes a real wave of some stuff, this wave guides the motion of a real point-like particle that has a define location at all times. Importantly, the wave function in Pilot-Wave Theory evolves exactly according to the Schrödinger equation. That’s the equation at the heart of all quantum mechanics that tells the wave function how to change across space and time.
Schrödinger Equation: Describes How a Physical System Will Change Over Time
This means that Pilot-Wave Theory makes the same basic predictions as any other breed of quantum mechanics. For example, it’s guiding wave has all the usual wavy stuff, like forming interference patterns when it passes through a pair of slits. Because particles follow the paths etched out by the wave, it will end up landing according to that pattern. The wave defines a set of possible trajectories and the particle takes one of those trajectories. But the choice of paths isn’t random, if you know the exact particle position and velocity at any point you can figure out it’s entire future trajectory.
Apparent randomness arises because we can’t ever have a perfect measurement of initial position, velocity or other properties. This hypothetical predictability means that a Pilot-Wave universe is completely deterministic. When DeBroglie presented his still incomplete theory at the famous Solway conference of 1927 it didn’t go down so well. Technical objections were raised and Niels Bohr doubled down on the probabilistic interpretation. DeBroglie was so convinced, and he dropped Pilot-Waves all together. The idea was forgotten for decades and Copenhagen became the orthodoxy.
Solvay conference (1927): Schrödinger, Bohr, Eisenberg, De Broglie, Dirac, Lorentz, Einstein..
It took until 1952 for another physicist, David Bohm to feel very uncomfortable with some of the wackiness of Copenhagen and to re-discover DeBroglie’s old idea. Bohm took off where DeBroglie left off and completed the theory. The result was Bohmian Mechanics, also known as DeBroglie-Bohm Pilot-Wave theory. These days, more and more serious physicist are favoring Bohm’s ideas. However, it’s far from being broadly accepted. DeBroglie himself remained firmly in the Copenhagen camp even after Bohm’s efforts.
Although Pilot-Wave theory makes all the usual predictions of Quantum Mechanics, it has some really fundamental differences. Those differences are in a sort of “special thinking” you need do, in order to accept Pilot-Waves over other interpretations. In fact most of the arguments fore or against it are about this “special thinking”. Are you more or less comfortable with the oddness of Pilot-Waves versus the oddness say, of Copenhagen, or Many Worlds. So what uncomfortable thinking does Pilot-Wave theory require. For one thong, it needs a teensy bit of extra math that mainstream interpretations don’t. As well as the Schrödinger equation that tells the wave function how to change, it also has a Guiding (Velocity) Equation that tells the particle how to move within that wave function.
Schrödinger Equation and Guiding (Velocity) Equation
That “extra math” is considered un-parsimonious to some, a needles added complexity. However, the Guiding Equation is derived directly from the wave function, so some would argue that it was there all along. A more troubling requirement of Bohmian Mechanics is that it does contain real complexity that’s not uncoded in the wave function. That’s something that Niels Bohr was so fervently against.
Bohmian Mechanics, has so called “hidden variables” details about the state of the particles that are not described by the wave function. According to Pilot-Wave Theory the wave function just describes the possible distribution of those variables given our lack of perfect knowledge. But hidden variables have a bad wrap in quantum mechanics.
Distribution of Variables Inside a Wave Function
Pretty soon after DeBroglie first proposed Pilot-Waves, the revered mathematician John Von Neumann published a proof, showing that hidden variable explanation for the wave function, just couldn’t work. That proclamation contributed to the long shelving of Pilot-Wave Theory. In fact Von Neumann didn’t get the full answer, it turns out that the restriction against hidden variables, only applies to local hidden variables. So, there can’t be extra hidden information about specific region of the wave function that the rest of the wave function doesn’t know.
This was figured out pretty soon after Von Neumann paper, by German mathematician Grete Hermann. Although her repudiation wasn’t noticed until it was re-derived by John Bell in the 1960’s. This helped the resuscitation of Pilot-Wave Theory, because Bohmian Mechanics doesn’t use local hidden variables, it’s hidden variables are global. The entire wave function knows the location, velocity and spin of each particle. This non-locality is another weird thing you have to believe, in order to accept Pilot-Waves.
Grete Hermann John Bell
Not only does the entire wave function knows the properties of the particle, but the entire wave function could be affected instantaneously. So a measurement at one point in the wave function will effect it’s shape elsewhere. This can therefore effect the trajectories and properties of the particles carried by that wave, potentially very far away. Quantum entanglement experiment show this sort of “Spooky Action” at a distance, is a very real phenomena.
Again, I’ve gone into the non-locality of entangle particles in detail before. Also worth a look. It’s a though idea to swallow, but experiments indicate that some type of non locality is real, wether or not we accept Pilot-Waves. It would be remit of me to talk about Pilot-Waves without mentioning the amazing analogy that was discovered in bouncing droplets on a vibration pool of oil.
Quantum Phenomena on the Macroscopic Level
This is pretty amazing, we see many of the familiar quantum phenomena appear in this macroscopic system with suspended oil droplets following it’s own Pilot-Wave. Now we shouldn’t take a macroscopic analogy as proof of microscopic reality. But it does demonstrate that this sort of thing does happen in this universe, at least on some scales. I should also add that DeBroglie-Bohm Pilot-Wave theory is certainly wrong, and I don’t think anyone could deny that, because it doesn’t account for relativity, either special or general.
That means that at best it’s incomplete, while regular mechanics has Quantum Field theory and it’s relativistic version, Pilot-Wave theory hasn’t quite got there yet. Quantum Field theory pretty explicitly requires that all possible particle trajectories be considered equally real. Pilot-Wave theory postulates that the particle really takes a single actual trajectory, the Bohm trajectory. This is not consistent with quantum field theory and so there isn’t a complete relativistic formulation of Bohmian Mechanics, yet. But there is good effort in that direction.
Now let’s not even start talking about gravity, as no version of Quantum Mechanics has that sorted out. Also, we can’t ignore the fact that the initial motivation behind Pilot-Wave theory was to preserve the idea of real particles. We need to be dubious about that sort of classical bias. All that said, Pilot-Wave theory does do something remarkable. It shows as that it’s possible to have a consistent interpretation of Quantum Mechanics, that is both physical and deterministic, no hoo-ha needed. Maybe something like Pilot-Waves really do drive the microscopic mechanics of space-time.
Particle(s) Taking Bohm Trajectory While Surfing Pilot Waves
Man made Climate Change is a Fraud Sat, 04 Feb 2017 02:26:16 +0000
Man made climate change has been disproved.
Instead, what little evidence there is for rising global temperatures points to a well known natural phenomenon within a developing eco-system.
The sea levels are not rising significantly. You can download a related PDF document here (sea level rise is biggest lie ever told)
The polar ice is increasing, not melting away. Polar Bears (polar bears) are increasing in numbers.
Man made climate change is a myth. It has become a political and environment agenda item, but the science is not valid.
The change of climate change within one generation:
Efforts to prove the theory that carbon dioxide is a significant greenhouse gas and pollutant causing significant warming or weather effects have failed. CO2 is not a “well mixed gas”:
One of the least challenged claims of global warming science is that carbon dioxide in the atmosphere is a “well-mixed gas.” A new scientific analysis not only debunks this assertion but also shows that standard climatology calculations, applicable only to temperature changes of the minor gas, carbon dioxide were fraudulently applied to the entire atmosphere to inflate alleged global temperature rises.
Acceptance of the “well-mixed gas” concept is a key requirement for those who choose to believe in the so-called greenhouse gas effect. A rising group of skeptic scientists have put the “well-mixed gas” hypothesis under the microscope and shown it contradicts not only satellite data but also measurements obtained in standard laboratory experiments.
Canadian climate scientist, Dr Tim Ball is a veteran critic of the “junk science” of the International Panel on Climate Change (IPCC) and no stranger to controversy.
Ball is prominent among the group of skeptics and has been forthright in denouncing the IPCC claims; “I think a major false assumption is that CO2 is evenly distributed regardless of its function.“
School Children Prove Carbon Dioxide is Heavier than Air
Dr. Ball and his colleagues appear to be winning converts with their hard-nosed re-examination of the standard myths of climate science and this latest issue is probably one of the easiest for non-scientists to comprehend.
Indeed, even high school children are taught the basic fact that gravity causes objects heavier than air to fall to the ground. And that is precisely what CO2 is – this miniscule trace gas (just a very tiny 0.04% of atmosphere) is heavy and is soon down and out as shown by a simple school lab experiment. Moreover there is also the CO2 fire extinguisher where the heavy gas is used to displace oxygen.
Or, we can look at it another way to make these technical Physics relationships easy. This is because scientists refer to ratios based on common standards. Rather than refer to unit volumes and masses, scientists use the concept of Specific Gravity (SG). Giving standard air a value of 1.0 then the measured SG of CO2 is 1.5 (considerably heavier). [1.]
CO2: The heavy Gas that heats, then cools faster!
The same principle is applied to heat transfer, the Specific Heat (SH) of air is 1.0 and the SH of CO2 is 0.8 (heats and cools faster). Combining these properties allows for thermal mixing. Heavy CO2 warms faster and rises, as in a hot air balloon. It then rapidly cools and falls.
This ‘thermal’ mixing is aided by wind flow patterns, but the ratios of gases in the atmosphere are never static or uniform anywhere on Earth. Without these properties CO2 would fill every low area to dangerously high levels. Not ‘high’ in a toxic sense, only that CO2 would displace enough Oxygen that you could not have proper respiration. Nitrogen is 78% of the atmosphere and totally non-toxic, but if you continue to increase Nitrogen and reduce Oxygen the mixture becomes ‘unbreathable.’
It is only if we buy into the IPCC’s “well mixed gas” fallacy that climate extremists can then proceed to dupe us further with their next claim; that this so-called “well mixed” CO2 then acts as a “blanket” to “trap” the heat our planet receives from the sun.
The cornerstone of the IPCC claims since 1988 is that “trapped” CO2 adds heat because it is a direct consequence of another dubious and unscientific mechanism they call “back radiation.” In no law of science will you have read of the term “back radiation.” It is a speculative and unphysical concept and is the biggest lie woven into the falsity of what is widely known as the greenhouse gas effect.
Professor Nasif Nahle, has proven that application of standard gas equations reveal that, if it were real, any “trapping” effect of the IPCC’s “back radiation” could last not a moment longer than a miniscule five milliseconds [2.]
Climatologist abandons ‘Back Radiation’
Only recently did Professor Claes Johnson persuade long-time greenhouse gas effect believer Dr. Judith Curry to abandon this unscientific term. Curry now admits:
“Back radiation is a phrase, one that I don’t use myself, and it is not a word that is used in technical radiative transfer studies. Lets lose the back radiation terminology, we all agree on that.”
IPCC doomsayers claim it is under this “blanket” of CO2 (and other so-called greenhouse gases) that the energy absorbed by Earth’s surface from incoming sunlight gets trapped.
But one other important fact often glossed over is that CO2 comprises a tiny 0.4% of all the gases above our heads. Nasif Nahle reminds us that this is a crucial point when considering the claims of the “grandfather” of the greenhouse gas hypothesis (GHE), Svente Arrhenius.
Change in CO2 Temperature is NOT a Change in Atmospheric Temp
When applying the GHE formula devised by Arrhenius, IPCC scientists appear to have forgotten that we must consider the proportion of carbon dioxide in the atmosphere, not the proportion of the whole mixture of gases.
Even if Arrhenius was right about the GHE any change of temperature obtained from his formula is exclusively a change of temperature of the mass of carbon dioxide, not of the atmosphere.
The trick of climate doomsayers is that they draw their conclusions obtained from the Arrhenius formula for CO2 (only 0.04% of atmosphere), then apply that change of temperature to the WHOLE Earth; this is bad science, or possibly fraud.
Nahle poses this question for GHE believers:
“Is the atmosphere composed only of carbon dioxide? Why calculate the change of temperature of a mass of carbon dioxide and then after say it is the change of temperature of this trace gas that now becomes the temperature of the whole Earth?”
Astrophysicist and climate researcher, Joe Postma similarly comments:
“No one seems to have realized that any purported increase in temperature of CO2 due to CO2 absorption is APPLIED TO CO2, not the whole atmosphere! Again, just a slight tweak in comprehending the reality makes a whole paradigm of difference.”
NASA Data confirm CO2 not a Well Mixed Gas
Professor Nahle and his colleagues insist that in addition to the above facts the proven varying density of atmospheric CO2 also needs to be taken into account to show how IPCC scientists are guilty of the greatest scientific swindle ever perpetrated.
From the NASA graph below (verify with link here) we can discern distinct and measurable regional variations in CO2 ppmv. So even NASA data itself further puts paid to the bizarre notion that this benign trace gas is “well-mixed” around the globe.
NASA’s diagram thus not only proves CO2 isn’t a well mixed gas but also demonstrates that there is no link between regions of highest CO2 concentration and areas of highest human industrial emissions.
Groundbreaking Science Trumps IPCC Junk Claims
Both Postma and Nahle have recently published groundbreaking papers discrediting the GHE. Professor Nahle analyzed the thermal properties of carbon dioxide, exclusively, and found that 0.3 °C would be the change of temperature of CO2, also exclusively, not of the whole atmosphere. Nasif pointedly observes:
“Such change of temperature would not affect in absolute the whole mixture of gas because of the thermal diffusivity of carbon dioxide.”
Additionally, Nahle and his team demonstrate that carbon dioxide loses the energy it absorbs almost instantaneously, so there is no place for any kind of storage of thermal energy by carbon dioxide. To the more technically minded what Nahle and his colleagues say is that the release of a quantum/wave, at a different wavelength and frequency, lasts the time an excited electron takes to get back to its base state.
Thus the IPCC’s CO2 “sky blanket” is shot full of holes as rational folk are increasingly abandoning the unphysical nonsense that carbon dioxide “traps” heat and raises global temperatures. Policymakers may be the last to wise up but they, too, must nonetheless consign the man-made global warming sham to the trash can marked “junk science.”
[1.] In our “current environment,” atmospheric nitrogen and oxygen vastly outweigh CO2. Nitrogen: 3,888,899 Gigatons; Oxygen: 1,191,608 Gigatons; Carbon Dioxide: 3,051 Gigatons. On a weight basis the specific heat of nitrogen and oxygen together is approximately 1 per kilogram, whereas CO2’s is about 0.844. Thus it’s clear that everyday air has a better ability to hold onto heat.
[2.] Professor Nahle, N., ‘Determination of Mean Free Path of Quantum/Waves and Total Emissivity of the Carbon Dioxide Considering the Molecular Cross Section’ (2011), Biology Cabinet, (Peer Reviewed by the Faculty of Physics of the University of Nuevo Leon, Mexico).
There has been no warming over 18 years. You can download the Complete PDF document here (man made climate change is fraud)
NASA Exposed: A Nother Fraud Sat, 28 Jan 2017 05:09:50 +0000
NASA is a global warming data factory used by the U S as their credibal source of proof of climate change. Now, faced with too much contradictory information from published research, has actually admitted that there may be a link between the solar climate and the earth climate. Note: “there may be a link“. World wide research, specially in the last ten years, have pointed to the strong possibility that the sun plays a much larger role in global warming then we, here on earth do. After all, the sun is the main source of heat for our planet,” Nasa confirmed. Despite the constant stories of how recent years have been the hottest, historically, NASA now conceeds that four of the 10 hottest years in the U.S. were actually during the 1930s, with 1934 the hottest of all. This was the Dust Bowl; the combination of vast dust storms created by drought and hot weather.
NASA, over the last two decades, using clever statistical skewing, managed to lowered temperature data of the past century significantly. By using different statistical methods for this century, then by comparison, temperature readings for this century turns out much hotter then any time in the past. Cleverly deceitful as in the past.
The branch of research looking at the ice core samples to document climate for thousands of years has established the major solar cycle of about 300 years. Carbon Dioxide Information Analysis Center (CDIAC), which has the ice core data back 800,000 years, is being shut down as of September 2017 (800,000-year Ice-Core Records of Atmospheric Carbon Dioxide (CO2)).
The data clearly establishes that there has always been a cycle to CO2 long before man’s industrial age. This is data government wants to hide. As long as they can pretend CO2 has never risen in the past before 1950, then they can tax the air and pretend it’s to prevent climate change. Moreover, while we can clean the air with regulation as we have done, under global warming, they allow “credits” to pollute as long as you pay the government. It is the ultimate scam where they get to tax pollution and people cheer rather than clean up anything.
NASA has reported: “Indeed, the sun could be on the threshold of a mini-Maunder event right now. Ongoing Solar Cycle 24 is the weakest in more than 50 years. Moreover, there is (controversial) evidence of a long-term weakening trend in the magnetic field strength of sunspots. Matt Penn and William Livingston of the National Solar Observatory predict that by the time Solar Cycle 25 arrives, magnetic fields on the sun will be so weak that few if any sunspots will be formed. Independent lines of research involving helioseismology and surface polar fields tend to support their conclusion. (Note: Penn and Livingston were not participants at the NRC workshop.)
If the sun really is entering an unfamiliar phase of the solar cycle, then we must redouble our efforts to understand the sun-climate link…”
Additionally, the pretense of global warming prevents us from preparing for a sharp decline in cold weather that will be dangerous to society to say the least. This winter is colder than last year and last year was colder than the previous. It has snowed in Japan down to Athens. Even Corsica where winter highs are typically in the mid-50s, lows around 40, saw snow. We need to pay attention to Climate Change but stop blaming man. Something far more significant is developing and handing academics $100 billion to prove global warming is an absolute joke.
The Scale of the Universe Sat, 28 Jan 2017 05:08:16 +0000 Mass, Length and Time. Then ask what kind of mass, length and time can we define?]]>
Start with no tools and just using our senses look at the most basic concept of Physics: Mass, Length and Time. Then ask what kind of mass, length and time can we define? No math, no instruments, and no analysis of any kind, only our bare senses. Using standard international units of M for mass (in kilograms), L for length (in meters), and T for time (in seconds), what are the smallest and largest units we can perceive?
What is the smallest unit of mass perceivable: certainly we can tell the difference between a gram and a kilogram. So a safe bet would be a fraction of a gram. What is the largest unit of mass we can perceive? We can definitely tell the difference between 10 and 100 kilogram, maybe for weight lifters the upper limit is probably 1000 kilogram.
In scientific terms our perception range for mass lies between {{10}^{-4}} <> {{10}^{3}} kilogram.
What is the smallest unit of length perceivable: a millimeter is a safe bet. maybe a fraction of a millimeter. What is the largest unit of length we can perceive? We can definitely tell the difference between 1 and 10 kilometer, maybe the upper limit is probably 100 kilometer.
In scientific terms our perception range for length lies between {{10}^{-4}} <> {{10}^{4}} meters.
What is the smallest unit of time perceivable: a blink of an eyelid, a fraction of a second. What is the largest unit of time we can perceive? In a sensory deprivation environment you loose time perception beyond 100 days.
In scientific terms our perception range for time lies between {{10}^{-1}} <> {{10}^{7}} seconds.
This is the world of intuition. This is the scale at which we are hard wired to operate. This is the universe for people operating on their intuition. Call this the world of middle dimension.
But now ask, at what scale does the Universe itself operates at? Are you ready for a surprised? Using our intelligence coupled with instruments, here is what we know.
What is the smallest unit of mass measurable, the electron. The electron mass is measured at {{10}^{-35}} kilograms.
What is the largest mass we can think of, The mass of the known universe. Take an average star like our sun and multiply it by the estimate of the number of stars in our galaxy and multiply that by the estimated number of galaxies.
Our sun is {{10}^{30}} kilograms There are {{10}^{11}} stars in an average galaxy and there are {{10}^{11}} galaxies. The estimate for mass of the universe then is at least {{10}^{52}} kilogram. Nature’s mass scale is between {{10}^{-35}} <> {{10}^{52}} kilogram, over 80 orders of magnitude.
What is the smallest length know, Plank’s length. This is constructed from the three fundamental constants of nature. Plank’s constant h, the speed of light in vacuum c and Newton’s gravitational constant g. These are natural constants, measured, not something we created, and coincidently their combinations have units of mass length and time.
What is the largest length we can think of, The radius of the known universe.
This implies that nature operates on length scale between {{10}^{-35}} <> {{10}^{26}} meters. That’s almost 60 orders of magnitude.
What about time, again Plank’s time is the smallest unit we can construct. What is the longest time, the age of the universe 13.7 billion years.
Again, nature operates between {{10}^{-42}} <> {{10}^{17}} seconds, that’s also almost 60 orders of magnitude.
Now remember these are orders of magnitudes, the range of middle dimensions is about 7 to 8 orders of magnitude. On the other hand nature operates on a scale of 60 to 80 orders of magnitude, maybe more. What does all this mean?
Take a sheet of paper and draw a pencil line on it. That 1mm wide line illustrates the size our middle dimension, the extent of our experience, a width of a pencil line. How big would a sheet of paper need be to encompass the scale that nature operates on? That sheet of paper would be bigger than the size of our galaxy. In fact it would need to be two to three times that size.
Our experience walks on a pencil thin line on a sheet of paper twice the size of our galaxy. Our experience see’s so little of nature that for practical purposes it’s zero. Yet everyone perceives an understanding of nature to some extent. This is based on the false conjecture that the working of a narrow 1 millimeter middle dimensional experience can be extrapolated to twice the size of our galaxy.
It’s important to understand that you can’t feel, touch, or sense, the scales that the universe operates on. You have to think and use mathematics. To think mathematically, you must abandon subjective notions and opinions and the reliance on the hard wired genetically developed instincts for perception. The use of intellect and the knowledge of mathematics is a way to gain this understanding.
Classical Mechanics Sat, 28 Jan 2017 05:05:59 +0000
All of our physical theories are based on fundamental laws formulated in a mathematical framework and on rules mapping the elements of the mathematical theory to physical objects. Millennia of observations were fully understood only when the seminal ideas of Galileo, Kepler, and Newton were impacted by mathematics. The theoretical elaborations of the laws of motion by Lagrange, Hamilton, and others are at the basis of all mechanical devices that affect modern life in essentially all its practical aspects.
World View of CMClassical mechanics stands as an example of the scientific method organizing a “complex” collection of information into theoretically rigorous, unifying principles; in this sense, mechanics represents one of the highest forms of mathematical modeling. The elegance and depth of the theoretical thinking coupled with its ubiquitous applications make it comparable, in applied sciences, to Euclid’s geometry. There are three formulation of of classical mechanics — Newtonian, Lagrangian, Hamiltonian.
Image Source: Jonathan J. Dickau, Going beyond
Newtonian mechanics The laws of physics that describe the way things move and interact in accordance with Newton’s laws of motion. it deals with continuously changing variables (such as momentum or energy) It works very nicely in Cartesian coordinates, but it’s difficult to switch to a different coordinate system. Newtonian mechanics develops equations of motion based on forces and acceleration as described by Newton’s three laws. In general, the Newtonian formulation is the easiest to understand. You start with a force function, typically expressed as a function of position. The equations of motion come from using Newton’s second law, Force is equal to mass times acceleration (F=ma).
Newtonian mechanics operates in terms of forces in contrast Lagrangian and Hamiltonian mechanics operate in terms of energies. They are obviously related in that they all describe nature. You can think of the relationship as being that forces indicate how the energies are flowing in the system. Newtonian mechanics has more general practical application insofar as you can include friction in it quite easily.
Lagrangian mechanics develops equations of motion by minimizing the integral of a function of system’s coordinates and the time derivatives of those coordinates. For this, one requirement is a basic understanding of how velocity and acceleration are derived from position.
For Lagrangian mechanics you need a Lagrangian function which is typically a function of position and velocity. The Lagrangian is equal to the kinetic energy minus the potential energy of the object at any point along the path. If you have a force as a function of position then it would characterizes the path (trajectory) of an object. The moving object always moves in such a way that the sum of the values of the Lagrangian along the entire path (the integral of the Lagrangian) is a minimum.
Lagrangian formulation generalizes very nicely to handle situations which are outside the realm of basic Newtonian mechanics, including electromagnetism and relativity. But primarily used when one is concerned with the coordinate-independent feature of a solution.
The Lagrangian formulation assumes that in a system, the forces of constraints don’t do any work, they only reduce the number degrees of freedom of the system. So one need not know the form of force the constraint forces have unlike Newtonian mechanics!!!
At the end the equations of motion should be the same for Newtonian and Lagrangian mechanics, although Lagrangian mechanics allows constraints to be expressed more naturally. An accurate description of nature is all about a Frame of reference. In Newtonian physics you stand at a point and watch something move in relation to the stationary observation point based on forces applied. In Lagrangian physics you frame of reference is the object that’s moving and experiences the forces.
Hamiltonian mechanics is a refinement of Lagrangian mechanics in which the time derivatives of a system’s coordinates are replaced by “generalized momenta” (time derivatives of the system’s Lagrangian).
A function (sometimes called the Hamiltonian function) which expresses the energy of a system in terms of momentum and position. In the simplest situations, the total energy is just the sum of the kinetic and potential energies. For more complicated systems, ‘the Hamiltonian’ becomes a set of differential equations describing the system that is being investigated. Mechanics using Hamiltonian equations, representing interactions in terms of changes in momentum, instead of in terms of forces (the way things are described in Newtonian mechanics), and solving those differential equations as necessary.
This approach is conceptually similar to using the equations of motion based on Newton’s laws to describe the way in which the position of a ball flying through the air changes from one instant to the next as it moves along its trajectory. The three formulations are 100% equivalent, but one or the other may be more convenient to use, depending on the system you wish to describe. Generally, a set of 1-dimensional springs & masses is best approached with Newtonian mechanics. Anything else, best with one of the other formulations.
To go beyond Newtonian mechanics you’ll need a foundation in systems of ordinary differential equations in order to understand any of the formulations. It is best to have a good grounding in Newtonian mechanics before approaching Lagrangian or Hamiltonian mechanics; Lagrangian and Hamiltonian mechanics require a foundation in systems of ordinary differential, knowledge of partial differential equations and variational calculus.
Is Quantum Physics Rocket Science? Sat, 28 Jan 2017 05:04:09 +0000 Rocket Science has become a byword in recent times for something really difficult. Rocket scientists require a detailed knowledge of the properties of the materials used in the construction of spacecraft;]]>
Rocket Science has become a byword in recent times for something really difficult. Rocket scientists require a detailed knowledge of the properties of the materials used in the construction of spacecraft; they have to understand the potential and danger of the fuels used to power the rockets and they need a detailed understanding of how planets and satellites move under the influence of gravity. Quantum physics has a similar reputation for difficulty, and a detailed understanding of the behavior of many quantum phenomena certainly presents a considerable challenge – even to many highly trained physicists.
The greatest minds in the physics community are probably those working on the unresolved problem of how quantum physics can be applied to the extremely powerful forces of gravity that are believed to exist inside black holes, and which played a vital part in the early evolution of our universe.
However, the fundamental ideas of quantum physics are really not rocket science: their challenge is more to do with their unfamiliarity than their intrinsic difficulty. We have to abandon some of the ideas of how the world works that we have all acquired from our observation and experience, but once we have done so, replacing them with the new concepts required to understand quantum physics is more an exercise for the imagination than the intellect.
I just finish reading the book In Search of the Ultimate Building Blocks by Gerard Hooft. The book is a first hand account of the most creative and exciting period in the history of Physics. Covering the period from 1965 to 1990 where physicist came to understand the structure of matter and the development of what is now referred to as the Standard Model.
The most interesting aspects of the book is the historic account of all the competing theories of the period and how rules were developed that all these theories had to comply with to survive and not be in conflict with experimental observations. Reading the book, it is quite possible to understand how the principles of quantum mechanics underlie many everyday phenomena, without using the complex mathematical analysis needed for a full professional treatment.
Are Physics & Math Tough Subjects? Sat, 28 Jan 2017 04:54:56 +0000
Why do people feel that Physics and Math are hard to understand? Could the answer be simple? What if people are not blank slates and come equipped with all sorts of naive intuitions about the world, many of which are untrue and hard to change. If that’s true, it means understanding these disciplines are not simply a matter of learning new ideas. Rather, it also requires unlearning instincts and shedding false beliefs. Evolution seems sadly ironic: While the Universe evolves our views about our own development don’t seem to change much.
What about formal education, how does it impact this process? Step back and examine why anyone really goes to school to get an education. First it may be the belief in a good story, which is, if you work hard and do well you will come away with a college degree and end up with a job. The second may just be cultural, to develop a sense of cultural identity to pass on. In effect the understanding of Physics and Math would be highly depended on the approach of the education system in question.
Does anyone realize that the current system of education was designed, conceived, and structured during the Enlightenment for the economic circumstances of the Industrial Revolution. You would think education has evolved since then, in a way you would be right. Education today is a perfect reflection of our modern lives. It’s predicated on convenience and optimized for entertainment, but remains immersed in the Enlightenment view of intelligence.
This intelligence we’ve come to think of as and associate with academic ability. It’s easy to see that this is spliced into the gene pool of formal education. As a result, there are really two types of people, academic and non academic; smart people and non smart people.
Real intelligence, on the other hand, consists of a capacity for a certain type of deductive reasoning. The consequence of that is that most people think they are not bright or smart enough to understand the concepts of Physics and Math, because they’ve been judged against this particular view of the mind.
I can illustrate this using a few current scientific laws that have universal acceptance, and applicability, they are the laws of thermodynamics whose development dates back to the 19th century.
The laws in simple terms state the following:
“You Can’t Win”
“You Can’t Break Even”
“You Can’t Get Out of the Game”
Are these concepts hard to understand? Not really, but how many people believes these statements or accept them psychologically? So, if Physics and Math concepts are as simple as the above ideas and don’t require “supper intelligence”, what then is require for their understanding? Maybe just the development of a new ability, the capacity to unlearn.
Spiritual Healing: Todays Snake Oil Fri, 27 Jan 2017 15:17:10 +0000
We’ve all seen the old Western movies where the charlatan rolls into town the day of a hanging (or Presidential inauguration), puts a sign on the side of his horse drawn wagon and starts hawking bottles of his cure-all Snake Oil to the gullible crowd of onlookers. But that’s just in the movies, right?
Hardly!! Charlatans walk the earth in ever growing numbers, but they no longer sport black hats or greasy handlebar mustaches. Modern day villains attach their nameplates to doors on Wall Street, Sedona, Munich, Toronto, Buenos Aires, and Rio or right down the street in the local alternative medicine clinic. Here’s how to spot them:
Targeted Marketing: Make no mistake; these folks are in business to make money. Who has money? Not the poor and downtrodden. You won’t find these guys in the ghetto. Instead, the pseudo health care worker offers his or her medical marvels in upscale venues. You will find modern day Snake Oil vendors in affluent neighborhoods, resorts, and areas famous for “natural healing.”
It Ain’t Cheep: Snake Oil – no matter what it’s called these days, costs of LOT of money. These “healers” charge a fortune for their products and services and most insist that a series of treatments (cash only please) is required for a full recovery.
Ancient Secrets: Be on the look out for cures that link their efficacy to medical practices from ancient China, India, Atlantis, Aztecs, Mayans and Native Americans. The new Snake Oil vendor presents himself as an explorer who has discovered lost secrets that only he/she can use to cure what ails you.
Medical/Scientific Terminology: Doctors and nurses occasionally use unfamiliar multi-syllable vocabulary words to describe bodily functions, tests or pharmaceuticals. Pseudo science health “professionals” always do. Keep your ears open for words like Homunculus, high-energy chelation balance, chi and chakra interventions, quantum photonic therapy and targeted DNA manipulation.
Highfalutin Names: Another dead give away for a Snake Oil con is a business name that promises otherworldly success like the Dalai Lama Healing Center, New Science Alternative Medicine Center, or Holistic Energy Treatment Guru. If it sounds to good to be true – it is.
Spinning Statistics: If you want proof of the efficacy of $nake oil, you have only to ask your new health care provider – and you will be inundated with testimonials from cured patients, “scientific” study results, and expert opinions to back up the miracle effects of their products or services. Please note: There will be NO peer reviews, no articles from any journal with a reputation for truth, and no newspaper articles about the health benefits of the treatments. Also the self-published statistics will never include charts for a “control group.”
Conspiracy: But you will hear all about the government, establishment, medical community, and drug company attempts to discredit Snake Oil and keep it’s benefits from getting to those who need it. Listen for this conspiracy argument. It’s another clue in the deception plan.
Not all medical doctors are scientists with sincere interest in your health and well-being. Not all alternative medicine practitioners are liars and thieves. But the charlatans do adhere to the above. Be warned! And enjoy your Snake Oil.
The Changing Landscape of Reality Fri, 27 Jan 2017 15:12:50 +0000
Ontology is the study of being, and the central topic of the field is couched, variously, in terms of being, existence, and reality.
When philosophers study reality is’s called metaphysics. Although they have in the past discussed metaphysical questions from a purely conceptual standpoint, advances in science have forced a reevaluation of some traditional metaphysical views.
The discovery of harmful bacteria causing disease, forced us to reassess how the human body works and how it relates to the surrounding world. This will highlight the connection between scientific theories and our picture of reality as found in the work of historian and philosopher Thomas Kuhn, whose concept of a paradigm illustrates the idea that every scientific theory contains a worldview within it.
Metaphysics: The Study of Reality
René Descartes, known as the father of modern philosophy, began his intellectual odyssey with this question: How do we know that there is a reality outside our own minds? We each know that we have experiences, and we can be sure of these experiences; therefore, each of us can be sure that we exist. But how do we know that the internal experiences we have correspond to objects outside our minds?
You can see, smell, touch, and taste a loaf of bread, but those experiences are in your mind, not out in the world. How, then, do you know that there even is a world out there, and if there is, how do you know that it resembles the world of your internal experience? If all of your experiences are in your mind, how do you know that the thing giving you the bread experiences is, in fact, bread?
Perhaps, Descartes considers, you are merely dreaming or there is an evil damon artificially feeding experiences into your mind, creating a false universe that you wrongly believe is real. Descartes ultimately rejects this hypothesis, in part because there are surprising regularities in our experiences that are beyond our ability to control or create. When we keep careful track of our observations, intricate patterns emerge that can be generalized to systems we had never previously known or imagined.
The study of these patterns of observations is science. We look at patterns and create theories to explain their appearance. These theories, in turn, posit mechanisms that are supposed to be in the world and are responsible for creating the patterns. We can use those theories not only to explain what we have already seen but to predict new observations we have yet to make. If those predictions come true, we take it as evidence that the mechanisms in the theory are likely an actual part of the real world.
In this way, we use our best scientific theories to define reality. When we have new theories that replace our old ones, we not only gain new understandings about how our observations relate to each other, but we conceive of the world itself in new and strange ways. This is where science and philosophy meet. Scientists give us new accounts of how the universe works, and philosophers unpack those theories to see what they tell us about what is real.
The Germ Theory of Illness
According to Descartes, we are made up of two parts, a body and a mind. The body is mechanical and runs according to the laws of physics. The mind (for Descartes, the soul) is non-material and is where the will resides. For centuries, medical science was based entirely on this picture of the human body as a machine.
In the 1840s, Ignaz Semmelweis was an Austrian doctor working in the First Maternity Ward at Vienna General Hospital.
Semmelweis noticed that the incidence of childbed or puerperal fever was quite high in his ward and, schooled in Descartes’s machine view of the body, considered a number of causes for the illness that accorded with that picture.
But when another doctor died from childbed fever after being accidentally cut with a scalpel during an autopsy, Semmelweis considered the possibility that the doctors themselves were a vehicle for the ailment.
He then demanded that people working on his ward clean their smocks and wash their hands with a chlorine solution before assisting with a birth. Childbed fever cases fell dramatically. Semmelweis had identified the cause of the illness; a type of bacteria present in both dead and infected tissue.
This idea, however, was widely rejected. The human was thought to be a machine, and all problems with the machine could only be caused by parts of the machine malfunctioning. The idea that there were tiny animals inside us making us sick seemed silly.
After Frenchman Louis Pasteur’s work, the germ theory of illness became accepted, and we had to change our picture of reality. We were no longer “ghosts in a machine” but castles surrounded by hostile one-celled barbarians. Blood was no longer thought of as oil or hydraulic fluid flowing through our veins; white blood cells were now thought of as armed guards doing battle with tiny invaders. This was a completely different foundation from which to understand human physiology and health.
In the century that followed Semmelweis, this standpoint led to vaccines that eradicated diseases from smallpox to polio and to new ways of life. Cleanliness was not just next to godliness but the source of continued life itself.
Not long after the work of Semmelweis and Pasteur, we find the writings of Jules Verne and H. G. Wells, who posited new worlds at the bottom of the ocean, on the Moon, and back in time. It was a period of great discoveries, and the uncertainty associated with seeing that the world contains things we had never seen before was jarring. The entire genre of science fiction grew out of – and was informed by – the need to redefine reality with our scientific advances.
Kuhn’s View of Scientific Theories
In his 1962 book The Structure of Scientific Revolutions, Thomas Kuhn notes that when we think of the archetype of the scientist, the people who come to mind are Newton, Darwin, and Einstein. But science as it is done on a daily basis by working scientists is not at all like what those towering figures did. Newton, Darwin, and Einstein are revolutionary scientists whose work is different in kind from what Kuhn calls “normal science.” Normal science is quite the opposite of revolutionary.
Normal scientists work within a paradigm that tells them what counts as a legitimate scientific question, what tools they can use to answer such questions, and what counts as a legitimate answer. In other words, normal science occurs when someone poses a question deemed meaningful by the paradigm and uses the tools prescribed by the paradigm to find an answer that is acceptable within the paradigm.
Scientists do not question the paradigm. They teach their students how to act according to it, and challenging it is considered a challenge to rationality itself. Rationality, Kuhn argues, exists only within the paradigm. Because the paradigm tells us what is real and how it works, to question it is to question the structure of the world itself; according to those within the paradigm, that leads to nonsense.
Occasionally, however, anomalies pop up. There are questions the paradigm accepts as legitimate, but when the proper tools are applied, the answer fails to be one the paradigm recognizes. The first reaction is to assume that the normal scientist made an error, but sometimes, even after checking the calculations, the anomaly remains unanswered.
The unanswered anomalies, according to Kuhn, are ignored for as long as possible, until there are enough of them or they are so significant that they can no longer be discounted. At this point, the scientific community is thrust into a crises and is forced to reevaluate the paradigm. If the crisis is severe enough, some will consider the unthinkable: using a different paradigm, a new set of basic concepts, a new structure to reality itself.
These people are seen as nonscientific by those in the community because scientific thought is defined by the paradigm. But if the normal science within the new paradigm starts to look good, some will leave the old way for the new. If a critical mass adopts the new paradigm, the result is a scientific revolution. In the same way that a political revolution completely changes the system of government—that is, one legislative reality is replaced with another – so, too, does a scientific revolution replace one reality with another. Scientific revolutions, according to Kuhn, force us to redefine reality.
The revolution that Semmelweis began seems to have triumphed. Bacteria are outside invaders whose penetration into the interlocked systems of the body must be stopped. But there is also a new paradigm emerging, one in which our bodies are no longer seen as metaphorical castles with bacteria as the bad guys.
As scientists have started looking at the interaction between medications and the digestive system, they have discovered about 100 trillion bacteria that naturally live in the gut and play crucial roles in the body’s ability to function properly. These bacteria break down certain chemicals in our food to new forms that the body can use and create an environment that allows for the immune system to work properly. This ecosystem inside of our bodies, called the micro-biome, is essential to good health. Bacteria are not evil invaders; some of them are our partners.
Some bacteria are harmful, and they need to be stopped to cure some ailments, but our weapons, antibiotics, have also been eliminating the bacteria populating the micro-biome. We have been harming ourselves by not realizing that we are not just ourselves. We are not individuals but walking communities.
Soon, we realize that we cannot understand reality entirely by looking at the pieces; rather, we need to see the pieces in relation to one another. Thus, we begin to look at a more complicated reality in which there is interaction between elements.
Eventually, we find that what we have is not a set of individual atomic entities but a complex interrelated system, a web of interdependence.
Carter, Childbed Fever.
Descartes, Discourse on Methods
Kuhn, The Structure of Scientific Revolutions
Correlations Fri, 27 Jan 2017 15:07:10 +0000
Roughly forty years ago, space biologists made a rather important discovery, which in our today’s world seems to be widely forgotten: When these scientists grew plants in atmospheres of artificial, “non-earthly” composition, their charges didn’t at all thrive best in common air, we all breathe on earth, but in an experimentally generated gas mixture. Tomatoes, flowers and other ordinary plants spread richest when the oxygen supply was reduced to slightly less than half and when simultaneously the CO2 portion – normally only 0,025% – was strongly increased.
To begin with, this result appears in so far remarkable that it unmasks a common view, often considered self-evident, as a prejudice, namely the opinion, the conditions on earth would be most favorable for all existing forms of life on our planet.
But the significance of the biologists’ findings exceeds this recognition by far: When viewed more closely, their experiment shows itself as an example for the fact, not yet seen by many contemporaries, that mankind only begins to understand earth when this might already be too late, or in other words, only the occupation with extra-terrestrial topics gives us the possibility to better understand our own environment.
During the process of photosynthesis plants release oxygen. Without this planet’s flora the earth’s atmospheric oxygen supply would be consumed within more or less three centuries and after this time, earth would be uninhabitable for man and beast. However, these space biologists’ experiments now remind us, that the contrary is also true.
Before plants appeared on the surface of earth, the atmosphere was practically free of oxygen, and when the plants started to produce it, nobody yet existed for whom it could have been useful. It was just waste, nothing else, but it increased in the atmosphere continuously up to a concentration which provoked the risk for the plants to suffocate in the oxygen created by themselves. The a.m. experiment shows impressively how close this danger’s threshold had already advanced.
In exactly this critical situation nature got ready for an immense effort: It let emerge a kind of life- form, whose metabolism was exactly right for the consumption of oxygen.
While we are used to regard plants quite one-sidedly as the suppliers of oxygen for man and animals, space research has provided us with a perspective showing the usual picture from a completely different point of view:
On our part we are at the service of plant life, which in a short time would be extinct, if we and all animals wouldn’t attend to the removal of the oxygen waste which photosynthesis creates.
Once having become aware of this aspect, one believes to discover another strange correlation:
The stability of the mutual partnership between plant and animal life certainly is not as great as the fact suggests that it has apparently been existing for at least a billion of years. There are many factors which indeed endanger it:
One of them is the circumstance, that a substantial amount of carbon – as necessary for the cycle as oxygen – has been lost from the beginning, because huge quantities of plant substance have not been eaten by animals but have been deposited in the earth’s crust under and within sediments. This part has continuously been withdrawn from the cycle, to be more precise, as one should think, for ever and irretrievably, and the end seemed to be only another question of time.
But once more something very remarkable is happening:
Exactly at the moment – in proportion with geological timing, of course – when the system’s inherent defect begins to take effect, another new form of life emerged and unfolded an activity, whose effects straighten things out again: Homo faber appeared and drilled shafts deep into the earth’s crust in order to bring back carbon to the surface and to recycle it once more by combustion. Sometimes you would really like to know, who is actually programming the whole.
Or, why it is so difficult for many to understand these comparatively simple correlations. |
1048c7aefed32e76 | KIT | KIT-Bibliothek | Impressum | Datenschutz
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DOI: 10.5445/IR/1000086905
Veröffentlicht am 24.10.2018
Stochastic Galerkin-collocation splitting for PDEs with random parameters
Jahnke, Tobias; Stein, Benny
We propose a numerical method for time-dependent, semilinear partial differential equations (PDEs) with random parameters and random initial data. The method is based on an operator splitting approach. The linear part of the right-hand side is discretized by a stochastic Galerkin method in the stochastic variables and a pseudospectral method in the physical space, whereas the nonlinear part is approximated by a stochastic collocation method in the stochastic variables. In this setting both parts of the random PDE can be propagated very efficiently. The Galerkin method and the collocation method are combined with sparse grids in order to reduce the computational costs. This approach is discussed in detail for the Lugiato-Lefever equation, which serves as a motivating example throughout, but also applies to a much larger class of random PDEs. For such problems our method is computationally much cheaper than standard stochastic Galerkin methods, and numerical tests show that it outperforms standard stochastic collocation methods, too.
Sonderforschungsbereich 1173 (SFB 1173)
Publikationstyp Forschungsbericht
Jahr 2018
Sprache Englisch
Identifikator ISSN: 2365-662X
URN: urn:nbn:de:swb:90-869057
KITopen-ID: 1000086905
Verlag KIT, Karlsruhe
Umfang 32 S.
Serie CRC 1173 ; 2018/28
Schlagworte Uncertainty quantification, splitting methods, spectral methods, (generalized) polynomial chaos, Lugiato-Lefever equation, nonlinear Schrödinger equation, sparse grids, stochastic Galerkin method, stochastic collocation
KIT – Die Forschungsuniversität in der Helmholtz-Gemeinschaft KITopen Landing Page |
fe74d39de5893bca | The God Within
2012, Conspiracy - 314 Comments
Ratings: 5.82/10 from 287 users.
Mike Adams, the author of this documentary, says he always admired physicists. He says physicists seek answers by asking questions of nature and when they follow with rigorous scientific approach to the quest for knowledge, they refuse to be sidelined by dogma, personal belief or trickery. Science, in its most pure form, is about the search for truth.
Mike is not referring to the bastardization of science by modern corporations, which use the language of science to push a kind of intellectual tyranny involving for profit GMO's, vaccines and pharmaceuticals. He's talking about pure non-corporate driven science and the quest for human understanding.
This search for human understanding has lead him through a number of fascinating areas of study, but he found the most fertile ground for exploration in the fields of quantum physics, the many-worlds interpretation and the study of consciousness. Along that path, he decided to read a book by famed physicist Stephen Hawking and co-author Leonard Mlodinow.
As a fan of Hawking's work over the years, Mike relished the idea of reading his explanations of the theory of everything, The Grand Design, the invisible hand behind it all. What he found in his book, however, rather surprised him. On the very first page of the book, Mike found himself quite disappointed in the apparent lack of understanding of the universe from someone as intellectually capable as Hawking.
His words reflect what can only be called "the great failing" of modern day physics, to address the meaning behind the math. Far too many mainstream physicists seem stuck in what can only be called the Newtonian era of consciousness, that is, they don't yet grasp the idea that consciousness exists at all.
Hawking's book, The Grand Design, did serve another useful purpose in Mike's search for understanding. It nicely summarized the outmoded view of conventional physics. This mainstream view of physics is to reality what conventional medicine is to healing. In other words, it has all the technical jargon but none of the soul, and so it misses the whole point.
According to Mike, conventional physics is the clever conglomeration of high level mathematics desperately seeking to avoid any discussion of what it all means. You're not allowed to talk about consciousness or free will or the spooky connectedness that has been experimentally demonstrated to exist between all things in the universe, because that brings up too many questions that make conventional physicists uncomfortable... questions about God or the intersection of intention with the physical universe or free will.
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314 Comments / User Reviews
Leave a Reply to Luyang Han Cancel reply
1. A doc with a low rating sure brings out a rather long list of comments?.......really!? I'd call that very misrated. I loved it. I've thought along these same lines decades before ever hearing these things discussed by others. Just makes sense. The same way new scientific theories are discovered. They are first perceived.
2. This contains wisdom. Minus the idea that there is an intelligent, interested creator of our universe. Why should we not be simply a minuscule part of one of that creators cells that he/she/ (more likely IT) pays no attention to? The point is, science will not cease to be naive until it realizes it will ALWAYS be a child (& of course there is no literal father or mother) and must always continue to grow, much like our selves.
Perhaps there is a race of super intelligent, massive beings, and our universe is simply a cell in something around them, like a blade of grass or a mushroom.
3. This is awful. Don't waste your time watching or trying to make sense of this nonsense
4. JNANI is what this man is trying to explain.
5. I think it is interesting how emotional the comments are. Most of them are critical of Mr. Adams and try to discredit him by using "rational" logic, but it is evident that their reasoning is cramped within a larger emotional package of contempt and anger. I am not vested in any one side of the argument so I am NOT discounting any one's reasoning, but simply pointing out how emotional the responses have been. Why is that?
6. This guy is flat out can he say Stephen Hawking determenistic whose work is intricately connected to quantum physics and thermodynamics. Tweaking facts here and there, he is suggesting us to toss out everything what we have known so in Sundays and we will understand the fact that is how we have come so far as humans
7. Mr. Adams could have saved us all some time had he defined and proven the existence of all those concepts such as soul, divine, God or such. Gee, maybe real scientists don't 'believe' in those ideas because there is no evidence to prove they exist, could that be it? For example, what is the Being of God? Our being is atoms, molecules and such which we can measure and describe using the scientific method, but what is God's being, what is God made of? What is a soul made of? Does it take up space? Where is any evidence for anything except what we call natural?
Look, let's face it, everyone is guessing. No one knows all the answers, obviously. Mike Adams is guessing and making it sound as if he knows something, then putting down the men who are doing science, all to support his religious suppositions, his guesses. Scientists work with what we call reality, not unreality. They are the cutting edge of modern science. But Mike Adams, from his laboratory at Divinity Now, what ideas does he have, or is he just BSing everyone. It's that edge of arrogance in his voice that has got me going........
I would suggest interested thinkers read Marvin Minsky's wonderful book "The Society of Mind" in which he describes his ideas of how mind works. For example, see if you can follow this......
"Everything that happens in our universe is either completely determined by what has already happened in the past, or else depends in part on random chance. Everything, including that which happens in our brains, depends on these and only on these:
There is no room on either side for any third alternative. Whatever actions we may 'choose', it cannot make the slightest change in what might other wise have been -- because those rigid, natural laws already caused the states of mind that caused us to decide that way. And if that choice was in part made by chance -- it still leaves nothing for us to decide."
When I was young they talked about a 'steady state' universe, that is until larger telescopes were invented and the concept of 'Big Bang' came along. So what does anyone know about a Big Bang? Will it still be talked about in 50 or 100 years? Isn't it more logical to think that the 'universe', or 'multiverse', wasn't created but has existed forever? And that it goes on forever?
8. Fell asleep after 12 minutes of the most arrogant monologue I've ever heard.
9. Now that's how you shave off the edges of a square peg to make it fit through a round hole....or at least attempt to do so.
10. Hilariously wrong. If you have no desire to actually learn anything about quantum physics this is the documentary for you. Narrator claims that if God isn't real then genocide is ok.
(Exact words: "The belief that human beings lack souls or conciousness is dangerous for a far more serious reason. It can provide a scientific basis for heinous crimes against humanity including genocide. From Hawkings' point of view of soul-less determinism there is no reason why, the United Nations for example, can't reduce region overpopulation by simply committing genocide against human beings through population reduction programs. Because humans aren't 'real people with souls and conciousness' the poisoning of them does not violate any real ethical boundaries - according to that line of thinking.")
Uhm... Sorry, Mike Adams, but that logic doesn't make one lick of sense to the rational human beings of the world who aren't Christian and don't believe in souls. Maybe you personally would suddenly start thinking genocide is ok, but that says something about you personally, it says nothing about humanity as a whole.
He then goes on to say that the government is already euthenizing millions of animals every year (he doesn't say why - but it's to control animal populations from getting out of control. When animal populations get out of control 90% of their population will starve to death in a single year when food becomes scarce - If he doesn't understand that I have no idea where he gets off talking about quantum physics) but more importantly, Adams states that the government justifies these actions by claiming that animals don't have conciousness. (He wanted to say souls but he knew everyone would call him on his shit if he did so) Nope. Sorry Adams, that's just wrong and I feel bad for you if you actually believe your own words.
Adams seems to think his incredibly simple philosophical statements put holes through Steven Hawkings' interpretations of conciousness, which is actually kind of sad to listen to.
Honestly, this doc should be called "The Pseudo-Intellectual Layman's Guide to Misinterpreting Everything"
1. Adams actually makes a valid point. He states that modern physicists have essentially defined human beings as biological robots, i.e. our behaviour, thoughts and feelings are ultimately the result of biochemistry, that we are complex machines. Hawking has actually suggested that the human mind may one day be uploaded into a computer!
What Adams is suggesting, is if our thoughts and behaviour is simply the product of a biochemical reaction, then human beings are basically machines with no soul or free will. Surely that divests us of any particular rights? The only difference between a human and a rodent is that the former has a more complex brain, but both are essentially 'machines'.
If you created a robot with artificial intelligence, and an IQ of 130, how would that robot be any different to a human being? Its power source may be different - electricity instead of food - but if it has genuine intelligence and is sentient, then would it be any more immoral to terminate it, than it would to kill a human being? If so, why?
11. "Oh boy", another (born again) trained circus monkey taught to do cute tricks. What whacko group of religious lunies, and delusional fanatics funded this silly tripe? You have grossly misinterpreted the intent, and meaning of Physics professionals, the intent, and meaning of science itself, and the meaning of determinism. The only "dangerous" person I see here is you, disingenuous religious clods, and nincompoops who use cheep circus monkey tricks to wow the uneducated with your silly assertions, and gratuitous misuse of scientific jargon. It's just preconceived conclusions of amateurs, and con artists.
Any damned fool can criticize the top physicists when they aren't there to correct you on your misinterpretations of their scientific, systems of observation, experimentation, and hypothesis. You've done a great job distorting the true conclusions of theoretical physics which by no means imply a state of total awareness, and total knowledge, about the universe or it's mechanisms. I actually heard you talk about the term (spirituality). "Really", GMAB. That term may suit your fearful clueless abdication of adult professional responsibility, and your desire to justify your obvious motivations to push the deity, or baby Jesus theory, but it has no place in the world of science ,which seeks neither to prove, or disprove the existence of a deity. Science seeks only to explain phenomena by testing, experimentation, and repeated retesting, and repeatability of those experiments, and their conclusions until they collapse in falsehood, or stand like a wall of undeniable scientific theory, and law.
Spirituality, is merely an expression of "mystery" aka "ignorance", concerning phenomena that is not understood by the individual. It is a clear sign that you are willing to abdicate your responsibility as a true scientist, and student of theoretical physics for the world of conjecture, mysticism, and primitive psychotic wanderings in the world of "currently" unprovable, untestable fantasy. There is no doubt many scientists have made the mistake of (false omniscience) whether by innocent false conclusion, or by wanton arrogance, only to be made fools of later on. That doesn't imply however that science, or serious dedicated scientists, of integrity, and honesty believe they have reached the zenith, of all knowledge. That is BS. The thing you repeatedly hear from every Physics professional, every physics professor, is a full open ,and unequivocal admission that the universe is still full of unknowns, and contradictions that we are still in pursuit of: Dark energy, dark matter, multiverses, pre expansion realities, even the existence of the big bang, or the singularity. These things are openly described by the community as the logical conclusion of our (current mathematical models), and not in any way to be taken as an end of research, experimentation, and peer review, an endless ongoing process.
You imply that the quantum realities of the universe are somehow denied by some scientists, when in reality you and 99.999% of the worlds people wouldn't have any idea of quantum theory unless those scientists you criticize, discovered the properties, and realities of the quantum world for you to explore, and discuss in the first place. One of the top theoretical physicists Leonard Susskind who countered critical aspects of Hawking's assertions, states that in the gravitational nether world of black holes (Information, like energy, and momentum is conserved)
Your assessment of (biological determinism) is grossly unrealistic, and totally wrong headed. The idea that somehow we have slipped into your approaching Armageddon world of social collapse is pure childish misinformed, untrained absurdity. That's why people go to law school ,to learn about silly theories like yours, that completely distort the world of jurisprudence, human psychology, and the realities of life in the real world. The idea of democracy, and a jury of your peers implies that human organisms, have the power of observation, and reason, based on the needs of the organism, and the greater society as an extension of that organism for a species that is a social being. Laws, as any criminal, or good lawyer will tell you, were made to be broken, and interpreted. It is the purpose of a jury, and a judge to determine the meaning, and application of law, based on the needs of the biological beings on the jury, and the society as a whole. The person committing a heinous crime in one time, and place may hang, yet in another venue, may walk away as an innocent person by virtue of (perceived insanity), or be freed by virtue of the whims, and local beliefs of the jury. The condition implies that the biological needs of the wider society, and the individuals on the jury make all the difference in who is guilty, and who is innocent, a matter of local culture, and a matter of individual understanding, and perception. That is the nature of justice in a democracy, for good, or ill, and will always be so. There is no cause to determine the collapse of reason based on a deterministic approach to justice. The old expression goes, "beauty is in the eye of the beholder", or put another way, my lawyer, can beat up your lawyer.
Your entire effort here, it seems to me. is mere discount store tripe. I think, not really worth the effort of serious students to consider. Those of us who know the ancient history of mystical predictions, and conjecture about the mysteries of the universe, as spirits wandering in the ether, are rightfully wary, suspicious, and disdainful of silly conjectures and adherence to mystical non reason. We remember the inquisitions, and debauchery of religious sooth sayers, clergymen, papal despots, religious fundamentalist defilers of truth, and accusatory self serving lunacy that put Galileo Galilei in the hands of the ecclesiastical court. We despise the judgement of innocent victims of religious zealots, and torturers from the Catholic inquisitions, to the lunacy of ISIS fundamentalist psychopaths.
I do not, conclude, or deny, that some form of deity, or cognizant being may be found at the heart of all existence, and reality someday, in some way, because for now it is beyond our dimensional understanding. Like millions of people I would love to believe that there is a benevolent, loving deity out there, to welcome me home when I die. Unfortunately, there is absolutely no evidence for that happening, and no reason for me to believe it will ever be so. I will keep an open mind, given the fact that I do not posses all knowledge. Until some tangible, vision, appears in my dimension of reality, and perception, I shall remain skeptical of any such claim, not out of spite, or disrespect, but by adherence to the realities given to me by the gifts of human perception, sight, sound, touch taste,and smell, interpreted, accepted, or rejected by a rational biological brain, in the dimension I currently inhabit. I suggest we all do the same, and not be conned by the self serving delusions, of charlatans, and priests who seek to control , deceive, and yes profit from the fears, and frightened dreams of the innocents.
12. First a confession: Try as I might, I only made it to 31 mins.
Modern computer processors and operating systems are modeled on the brain. While there is no 'consciousness' present, someone stepping out of 1970 and into 2014, upon encountering a modern computer, would be at the least astounded. Some might even call them, to borrow a term from Mr. Adams, "spooky". In fact, given the level of intuition and interactivity available from current operating systems, the more superstitious time traveler might even perceive a level of consciousness.
They would be wrong, but they wouldn't think so.
I wonder what their narrative would be.
13. wow this guys agrees with almost all of what i believe, *sigh* too bad he isnt of my religion
14. kudos.. Sooner or later.. We,.. We'll figure it out and then realize!
15. Mr. Adams
You have repeatedly said that humans are 'made in God's image' but you don't explain in what ways we are made in 'God's' image. The National Catholic Almanac says that God is "infinite, immortal, holy, eternal, immutable, omnipotent and omniscient, perfect, supreme", and more, but none of these apply to humans. What is 'God's' Being? Our Being is flesh and blood and cells, but this is not God's Being. What could a God be made of, a God that shows up 2000 years ago with his 'son', who say they are coming back, but don't…….
I suggest you do this: Since the film has shown us what Mr. Adams has imagined to be true, but which he has not proven in any way, perhaps he should do another video, such as "Where did God come from"? Science is an effort, an ongoing attempt to describe what we see around us, and to verify by experimentation and confirmation of others. Religion and God belief are guesses, myths and stories from the past with no possible verification, only individual reports that change as one goes from Christianity to Judaism to Hinduism, etc.
The other side of the story is presented by George H. Smith, in his book "Atheism - The Case Against God", 1989, which should be read by all, as it explains why the 'God' concept is impossible.
Another part of my difficulty with religion is the existence of evil in the world. Does God not have responsibility for the Tsunami's that inflict pain and suffering upon children who have not yet begun to live? Or the famines where humans are reduced to eating their children, such as in 2 Kings 6:29? Or the two headed children born into a world of pain?
Your statement that 'You're not allowed to talk about consciousness or free will…' is absurd, consciousness and free will are being studied intensely by neuroscientists are they not, but grounded in reality, not in imagination. Scientists study what they can agree upon is the procedure I believe, called peer review. Is that not sufficient?
16. This is nothing more than a Steven Hawkings bashing party. I do not believe in any supernatural beings. Any more than i believe there is a real daffy duck Nice try. !!!Peace!!
17. Hawking's book, The Grand Design, did serve another useful purpose in Mike's search for understanding."??? don't make me laugh. You can't even call it science, it's sciencefiction, where he makes us believe a nebulous set of theories and no observable proofs are possible. a theory with no experimental support whatsoever, hence not a theory of physics at all.
18. To truly understand science one must accept that the answers to how we came to be are irrelevent .Accept this reality and work towards making life more tollerable for the current group of living beings around you . Self absorbed in trying to explain existance does little to accomplish the task at hand . Dark matter , string theory , multiple dimensions , all fantasy --- weave a rug for "Gods sake" .
To create a being as the explanation for that which is observable and assume it cares about or even knows our level of awareness , is as rediculous as the self absorbed scientists wasting time trying to messure a particle which may or may not be in a particuler place at any given moment.
Learn to be happy in the moment and plan for the future today to ensure survival of our species. Plant a crop , build a shelter, connect communities , and make love as often and with as many as life permits . Your actions today determine the future shared by those in it including yourself if you survive to be in it .
A sexually repressive fundamentalist view espousing monogomus relationships is insane .
Diversity is key to spicies survival .
19. Animals are people too.. So say the bumper stickers.. I think it should be.. People are animals too..!and when the religions that say we are above other creatures, created by God in His image, should look back three and a half billion years to when we all looked like a single cell bacteria.. And not because of book written 2000 years ago says different.. But because a 3 billion years Plus of rock strata has left evidence showing a timeline from where we came from..and where we are now.I can deal with the concept of God. I just can't deal with the Self righteous dogma many spew as a feeble attempt to challenge valid scientific findings.if God is responsible for all things...? He lit a match 14.7 billion years ago.. Turned.. And has not look back on "us" since.
20. i find it ridiculous just how closed minded most pro science advocates have become. seriously what matters more ? the universal pursuit of truth or the thunderous applause of your intellectual peers. many seem more concerned with appearing to have all the answers rather than what answers they actually have.
1. I think what's frighteing is how few answers (precisely ZERO) religion has. It's just pure dogma and ignorance that, thankfully, people are starting to see for what it is - i.e. a fraud
2. i'm not anti science but i'm also not prepared to blindly choose to believe 100 % of everything from a bunch of people who think their own sparkling logical intellect is the mightiest thing in the known universe, especially when the journey of science has always been a process of unfolding truth and revision, hence incomplete. religion is flawed but the intention is not ( sure, some set out with intent to rob a naive congregation ). it is foolish to think one has to be either in one camp or the other 100% when neither has all the answers. as far as ZERO answers is concerned, well the fact these institutions and beliefs have endured for thousands of years, even into the era of modern logic does prove that our connection with such beliefs on some level may be more fundamentally linked to our being,( hence it exist for a reason we don't yet understand) than your probably willing to will never rid itself of it's unwanted guest that is spiritual ideas. to believe that after death there is nothing is quiet unscientific as the is no realm of nothing as everything is something, what that after death something might well be.. i don't know ... but that is more scientific than saying " i know ! there is nothing !" when nobody has returned to confirm such.
21. Blind faith in scientism has led many, including myself, to a delusional state that can only be held up by keeping oneself ignorant of any dissenting viewpoints. If anyone questions scientism, they are marginalized as being creationist or conspiracy theorists or r*tarded. Ad hominem attacks come about because those who defend things like the current model of our universe cannot defend something they don't even understand themselves.
1. You may understand scientism but you don't understand science. Science must be corroborated, and it must be capable of being repeated by others. Scientists aren't ignorant of other possibilities, far from it, but they must assess the evidence that is presented for opposing theories, such as Creationism. When you introduce theories of 'gods' and 'goddesses' and 'devils' that have no referents in reality, yes there is opposition, and rightly so. Religion looks backwards in time, science looks forward, with intention. And beliefs are the social 'glue' that identifies and holds people together in a tight group, they are a socializing tool, and dissenters are punished or ostracized.
Science has learned that the 'being' of humans is cellular, flesh and blood', but what is the 'Being' of God? We can describe ourselves in scientific language, but to describe us as made in 'God's image' requires not science, but simply belief.
The purpose of science is to carry us forward, to understand reality and the future, while the purpose of religion is to promote group cohesion by inculcating a belief system that may or may not have any similarity to the world around us.
2. I understand your perspective but it is a bit naive. Cosmology, for one, is an area that relies on theoretical manifestations with no repeatable scientific experiments to verify (for the most part, that is). For example, black holes are created out of fallacious maths that no one in the field seems to mind. Any beginning math student knows you cannot divide by zero and yet theoretical mathemagicians feel that is it OK to do if it suits their ideas about black holes. It's utter lunacy, and yet most believe in black holes. They are everywhere and in all different sizes, or so we are told. Yet, there is no proof they exist, none. There are other areas of science that do not rely on repeatable experimentation to get the results too.
The problem is that we believe in the basic objectivity of science and we project this idealistic notion on the whole of science. It's as though it is seen as the holiest of holys. This is where science becomes "scientism" -when authority replaces rational thought and dogma covers up gaping holes in theory.
3. Yes, naive perhaps, but what is the alternative? One or a hundred of the various gods and goddesses proposed by this or that group throughout history and prehistory? If you can believe in gods and devils without a bit of evidence, why should you not do the same with science. Surely you see that religion looks backwards, to a 'god' or a goddess which was invented by the tribe. Cosmology looks forward too, perhaps getting beyond the scope of our intelligence, but science begins with some evidence to suggest a direction for study. Gravity is both theory and fact, just as is evolution, are they not? So if you cannot divide by zero you can't just throw up your hands and say that's it, I think I'll believe in a god that has zero evidence of Being, except hearsay and tenth hand written accounts.
Science is the future, why drag religion along? I personally wonder if there never was a 'creation', or if matter and energy have always existed. Why should 'nothing' be the ground of existence, where would 'nothing' have originated, and how much nothing would there have to be to contain 'everything'?
22. This doc is not worth watching unless you are a christian.
23. applying Occum's Razor I'd say this reasoning just doesn't "make the cut."
1. Then try again without applying Occum's Razor. Just because you saw the movie Contact does not make you an authority on life and the goings on in the Cosmos. Try a little intuition.
2. I didn't see "Contact" and I don't get my analysis tools from the movies. I don't pretend to be an authority on life or the cosmos, which incidentally you implicitly do.
You can do a little research on the quirks of the human brain that give rise to beliefs in magic of all sorts (religion being one), but I doubt that anything based on fact will move you beyond your "dogmatic slumber" . It appears to me that you have decided on an epistemology that is not subject to factual verification.
There was a time when that way of thinking dominated Western Civilization. That time is known as The Dark Ages.
You will forgive me for not wishing to join you there,
3. I would not be so quick to judge. Scientism is being projected by headlines as truth when some of the core beliefs of established science are nothing more than many layers of assumptions. The Big Bang, dark matter, dark energy, black holes and neutron stars are some examples of the many assumptions in cosmology with no empirical evidence to support them. Instead, they are inventions created as a result of faulty models running into road blocks. Rather than scrap theories that empirical evidence contradicts, ad hoc remedies are interjected using untested theoretical mathematics as pseudo-evidence. Cosmology is but one area of science tainted by delusional people who call themselves scientists. Tha being said, I do not throw the baby out with the bath water. Science in its ideal form is a wonderful thing. In practice, however, this can be a much less benevolent force.
4. A great response to Michael Jay Burns truthseekah. I am quite amazed at how science has become the 'new religion' for some people, where they don't dare question the validity of what they are being told. Like yourself, I believe science has much to offer in terms of directive thinking, but follow Ernst Mach's advice to Einstein regarding 'intellectual skepticism'. Keep up the good work, and best wishes!
5. I will die.. that's it..if it gives comfort to some thinking they'll live in eternity they are the " lucky" ones. I can't rationalize what they are selling..if if its in your mind it is" truth":)
6. You cannot rationalize open mindedness, if you cannot accept the logic that we will never know everything. (not even with science)
You can't fill a cup that is already full!
The 'truth' is there is no difference between a faith denying atheist, and a science denying theist. Asking for 'proof' regarding that which is unknowable is to imply disparity where none exists.
P.S: To dismiss the 'eternal' 1st law of thermodynamics (energy cannot be created, or destroyed, only converted) is contradictory/irrational for someone who can only rationalize science. (just saying)
7. Hmmmm......... 'delusional people who call themselves scientists'? What's the beef? You would prefer science to exist only in its' 'ideal form'? How curious...... Do you feel the same about religion, that it should only exist in it's 'ideal form'?
Out of curiosity, do you call yourself a 'scientist'?
24. i think he misunderstands much
1. I think what he (understands) is that you don't have to be afraid of consciousness or God Tom. It won't take your dominion of the universe away from you. It will allow you to share and be one with it.
2. you don't have to be afraid of something that doesn't exist...
3. I'm glad you cleared that all up for us. We can all stop thinking now, and let you take care of the big stuff.
25. so boring and preachy in delivery regardless of its veracity
26. This documentary went a little over board on the hawking bashing. It presented really great/interesting points, but lost focus a few times, could have been presented a little more appropriately
1. Far be it from me to judge Hawking's motive and or analysis of consciousness, but it's possible he has a bit of a chip on his shoulder toward a god or an external consciousness because of his physical condition. We all might have misgivings if we were in his position. Don't you think.
27. We have not sent probes to even 1% of our known universe and we are writing a book called "Theory of Everything"? We are like a baby, having explored the 4 corners of his cradle and then deciding that this is what the entire world must look like. Hundreds of years from now, if we ever survive that long, we would look back at this book and smile at how ignorant, arrogant, and narrow-minded we were.
We would realize that many of the constants that we know of in physics today are not as constant as we think they are throughout the universe. And there are still gapping holes in our sciences and in our understanding of our universe, for example, in the study of emergence from seemingly chaotic systems which would explain some of the questions that we have always been wondering about ourselves and our universe.
1. Everything you just said made basically no sense, "seemingly chaotic systems" is not the same at "chaotic systems" and you're purposely being vague because you don't want to held accountable for your claims.
2. Seems to me..(being vague ) is all we can be at this point, unless you know something the rest of us don't.
28. Biased with an agenda, muddled and weak arguments, dishonest and at times contemptuous of matters that I suspect he may not fully understand. It's that type of documentary where the arguments presented reveal how the information was never learned with any openness first day, that the presenters mind was already looking for some way of abusing the facts from the moment he encountered them.
I actually got to the stage where I was expecting him to start telling me how bananas were shaped to fit human hands. I had to stop it around the 25 minute mark as it made me want to pierce my eye with a spoon.
1. Couldn't agree more. Horrifying.
29. Oooh, “Come let us reason together” before my head explodes (see my AVATAR please) how can I understand all this? Cosmologists (and their other academic peers/buddies) say that the universe is made up of 70% Dark Energy and 26% Dark Matter (or there a bouts) and that the remaining 4% (which is called Baryonic matter) is
everything that reflects light (interacts with photon energy) to reveal our world (empirical reality). Then chemists and physicists come along say that all elements (that’s the Baryonic stuff, again) are (at the atomic level) 99.9999% empty space (leaving 0.0001% ‘real’ stuff/things). So with ‘that’ (science folks)…are going to be able to tell us that they know what reality is?! The majority of it you/we cannot even see and that which we can see is mostly empty of stuff? Then they pop to the FACT that all periodic table elements (such as: iron, carbon, oxygen, nitrogen especially the major eight that make up us and all life) are cooked from hydrogen by fusion (gravity and heat) then are exploded out into the vastness of space (which by-the-way is expanding…but it-space-is not expanding into anything). How can anyone grab such ‘knowledge’ and get it to stick inside their noodle? How about ya say: “We really don’t have much of a clue what’s going on (empirically)…but whatever reality is, it (and life) does not revolve around ‘stuff’?
I’m also not overly happy with Darwin: If we are just evolved ‘bugs’ and fit somewhere on a branch of the evolutionary tree then why
do we have the size brains we have? Evolution should not have wasted its ‘natural selection’ energy (our brains are too big for our environment)…and as poorly as we use it…it’s a wonder it hasn’t shrunk). There is no evolutionary survival value associated with us trying to learn particle physics, or build LHC, or to leave footprints and a couple dune buggies on the moon (etc etc). Frankly, I don’t even know how Natural Selection is able to allow us to have this discussion/talk/thought? For none of it, has the slightest evolutionary (survival) value! Natural selection (Darwinism) should not have cared (to allow us such imagination and abilities), for all we need to ‘do’ to fit in the ‘Tree’ is just survive, and that means: eat while not being eaten, adapt to the environment, and produce offspring. Oddly, I think we are more alien to this orb than all the other creatures here
on it…we are as if “foreigners in a strange land, on a temporary journey in time, and all the while trying to get home”. And what is it with morals; they do not dice well with Darwinism? Where does curiosity come from…and how do we know how to use conscience to guide our actions…even to the point of seeking an apology should we wrong someone? Seagulls don’t apologize for stealing food
from their peers, why then do we understand stealing as wrong (after all isn’t it biology/science that makes us animals surviving in Darwin’s Tree)??! We are also concerned for the other creatures (all life) on this planet, as if we were put here to care for them and it (the planet) as if we were/are to “steward a garden”. Heck, scientifically we KNOW that if we do not care for this place that it will no longer support us…so why don’t we DO IT?! ANS: Perhaps, we are
‘poisoned’ “we know what we ought to do, but just always do the opposite and then regret it; yet do it wrong again and again”. Why then is the following statement (morally) wrong: Why do we Darwinian types have concern for endangered species or (in fact) any species—survival is about YOU/ME (aaah that’s ME first then
you) not them!?? Why should we care?
We value abstract qualities like fairness and perfection and trust and we hate the insidious itch of time (can’t scratch it away from bothering us)… yet we have this sick hope for the future…even to the point of terraforming Mars and colonizing some far-off exo-planet (that’s laughable – what are we going to do if that place is inhabited – save the poor retched heathens like the Spanish did to the Aztecs
or we did to the American Indians!!?). We can’t even take care of our own planet never mind get along with each other. We are a mess ("we fall short") and this exo-planet (Earth) would fare better if we went extinct or went back to where (ever) we came from. We are probably the colonizing force that left some other world in search of a home (exo-planet) and found Earth and now we have settled in but
ODDLY we seem to have forgotten where we came from (and none of that works). If we did figure it out, it would be/seem so weird that we would think them gods—it’s like we have been on an extended six thousand-year camping trip and long for home where all the abstract desires which our inner spirit longs for work correctly, such as: fairness, perfection, love, honesty, and there is no time and no stuff: there just is…eternity). CS Lewis put it this way: ‘If we find within
I’m convinced that the only way we can ever understand the physical world is by the non-physical. Frankly, I’m concerned that the word ‘thing’ is totally incorrect—there is no such thing as a thing. Nothing IS nothing. We are as if in a giant hologram and all is energy to include us (E=MC^2). Matter is energy gone berserk; And life is matter infused with intelligent code seeking Home. One thing seems certain. If one finds information (and we never make it—we discover
it) it never points to confusion or chaos or chance or mistake. Information always point to some kind of intelligence which is responsible for its cause. And we are VERY good (almost too good at finding information)--at pattern recognition, empirical abilities,
seeing symmetry, and doing math and science yet we are also conscious observers and moral agents…and we do not like being insulted, mislead, or abused. We do not come here with a blank slate rather we come into ‘life’ as if hard-wired with some basic relational skills! Could life be all about relationship? Isn’t all: science, philosophy and even religion basically about relationship (the inter-actions at the micro and macro levels the plank limit that ultimately reveal to us what we call reality)?
Perhaps we are intelligent energy (spirit), patterned after some grander Intelligence (sort of like “in its image”) yet we do not operate quite as we should (we’re a bit broken, we function poorly—we relate incorrectly). We reside within these electro- chemical bio-mechanical Earthsuits (called bodies) in order to care for this ‘Spaceship Earth’ yet we are “foreigners here”. Perhaps we graduate life through death and that is how this intelligent energy gets to return home (for energy is never destroyed): the elements of our body go back into the Earth and the essence that is the self, “the spirit returns to whence it came“?
Life begets life. It comes with a code (DNA/RNA), it has a purpose (least you be just stuff – a meat robot). Life comes from intelligence because life has within it information and nowhere does information come from chaos. Life does not ‘like’ death (ya think?) yet life is caught in time and “in the bondage of decay” or entropy (the second Law of thermodynamics). Death is not normal to energy (that almost seems a contradiction)…rather energy just changes—it moves on. And wouldn’t it be true that is energy moved at the speed of light that all distance would shrink to zero and all time would stop or be non-existent…or be eternal?!
Our existence and all life smacks of conspiracy and not coincidence!!
When I consider the small span of my life absorbed in the eternity of all time or the small span of space which I can touch or see engulfed by the infinite immensity of spaces that I know not and that know me not, I am frightened and astonished to see myself here instead of there, now instead of then, me instead of you, even why instead of just because. By whose command and act were this time and
place allotted to me!? (Blaise Pascal)
BTW all the full quotes are snippets from the Bible! OUCH!!
1. In light of the ignorance of chemistry, physics ("Matter is energy gone berserk.") and biology (especially evolution "[Evolution should not have wasted its ‘natural selection’ energy (our brains are too big for our environment"]) expressed in your rambling and at times incoherent post, what is your scientific background?
2. lol - wow - you could possibly have begun the writing of 'god within 2', you seem to have a lot in common.
"Our existence and all life smacks of conspiracy and not coincidence!" you say - ah well, perhaps after all, this place is hell - (many writers would agree)
Good quote from Pascal, and a great philosophic question. In my opinion, asking the questions is honest and honourable, but an open mind and courage are required if one would accept the answers - peace
3. Hey Jo,
If you are speaking of the Blaise Pascal the scientist, you may like to know he was a Godly man converted into Christom.
4. good words are good words, i sincerely have no problem with 'whom' is the motivator or intended. " When I consider the small span of my life absorbed in the eternity of all time or the small span of space which I can touch or see engulfed by the infinite immensity of spaces that I know not and that know me not, I am frightened and astonished to see myself here instead of there, now instead of then, me instead of you, even why instead of just because"...I would have stopped there, but, it's still good, honest, human - I like that.
5. And you might note that Isaac Newton believed that the basis of Christianity, the concept of the Trinity, was blasphemy and a violation of the :"one god" rule.
6. Sorry, that's because Newton did not understand the trinity is three persons, but one essence., Not 3 separate gods. This does not make sence from a human perspective, but is perfectly logical from God's extradimensional existence.
7. If it doesn't make sense from a human perspective, how can a human articulate it? Or are you divine? What exists on the spiritual plane, if existent beyond the unseen microcosmic or uninterpreted macro-cosmic, is simply beyond us. And don't give me that God can will a human to understand or one with faith can understand. Because that experience still can't be verified and only happening in one human's head as no two have the same experience. The Scientific method is likely the best interface God would utilize to introduce the divine to man. As the 'method' is without politic.
8. We are all thinking too hard....
I am that I am.
Heaven is not somewhere you go too, if you are good when you die. "That's a lie!"
Heaven is here in the now. It has always been. We never left the "Garden of Eden" only tricked to believe we have!
Our creator is talking to us all the time; most of us just are not listening, . . Meditating is the door.
If your are seeking the answers through science, religion, or any other way, you will never find
You will always be seeking!
Be still do not seek! ... . Listen"
Creation, consciousness is speaking. right now! this very moment. Are you hearing me?
become one with the "I am" in the "now"
That is where you will find me. . . . . Artywayne
9. G; your words have filled me with so much optimism and love.. Inspired would be an understatement, so rare do I feel proud to be human as I do now! time and time again when I feel exhausted by the lack of positive curiosity and insight, which at times overcomes me, I will read them and be ready to continue my journey... Thankyou for your courage to express... and thanks be to god for sharing your beautiful mind/words with us all! :) p.s any recommendations of favourite books would be greatly appreciated if you have the time... kind regards, romy rose
Well, some noodles are stickier than others.
With my medium-sticky noodle I take the position that the things there are to understand will always be greater than the human capacity to grasp them, but it is our nature to keep working at it anyway.
That's one of the things that I like about us.
11. you know that "dark matter" idea always struck me as being cut from the same cloth as the "cosmological constant" i.e. something that you just make up so that you don't have to discard your favorite theory. Accountants call that "plugging" most of us just call it "cheating"
30. I don't like this Doc.
1. In order to give your potentially profound statement meaning, what Docs do you like?
2. one's not full of sh1t
31. I have found a philosophical reflection from Richard Dawkins, to his ten year old daughter, that reveals how feelings can be implemented legitimately by scientists. It comes from brainpickings org
"Inside feelings are valuable in science too, but only for giving you ideas that you later test by looking for evidence. A scientist can have a ‘hunch’ about an idea that just ‘feels’ right. In itself, this is not a good reason for believing something. But it can be a good reason for spending some time doing a particular experiment, or looking in a particular way for evidence. Scientists use inside feelings all the time to get ideas. But they are not worth anything until they are supported by evidence." - an excerpt (edit: not paraphrased) from Richard Dawkins. (copy paste in google for the whole article).
So there we have it...feelings are important on our path to understanding the truth, from the great man himself. Who'd have guessed? Inspiration, free will, abstraction, intuition and deduction... the biological robot hypothesis seems more and more ill conceived. Instead, ALL things considered, it seems to me that there is a hidden motive of provoking certain counter beliefs in claiming such a thing. Which is fine...but let's "call it by it's right name" - Chris McCandless (aka Alexander Supertramp).
1. "Scientists use inside feelings all the time to get ideas. BUT THEY ARE NOT WORTH ANYTHING UNTIL THEY ARE SUPPORTED BY EVIDENCE." (emphasis added)
In short, the beginning of your second paragraph is simply a paraphrase out of context and a distorted one at that. Second, there is nothing about philosophy in the entire quote. Just another display of your inherent dishonesty.
2. I'm afraid, but not sorry, you are incorrect on all counts Mr Allen. And it's very essence is touchingly philosophical - he is sharing his thoughts and ideas with his daughter on how best to think critically about the world around her.
3. Has nothing to do with philosophy, but rather with hunches based upon accumulated knowledge. Once again, another display of your dishonesty.
4. As I have shown you, with evidence of you being dishonest on a few occasions, I would ask you to point out where I have been dishonest, or cease calling the kettle black. So far your claims of dishonesty are unsubstantiated (obviously).
My point has been made so...
*** End of My Discussion With You **
5. Once again, Dr. Dawkins' statement has nothing to do with philosophy, but rather with scientific hunches and trying to pinhole it into philosophy exhibits a desperation on your part amounting to dishonesty.
6. Yeah... good luck with that.
7. So every time a scientist gets a hunch, he becomes a philosopher or he practices philosophy. Won't wash.
8. Digi just thank "God" that you don't manifest his mindset. I can't imagine living with myself like that. I know your not a "dishonest" person and he does too. He just wants to hurt you.
I apologize for him. Lets wish him the best and move on.
9. I don't accuse you of dishonesty, but I agree with robert's view that Dawkins is referring to the subconscious promptings based on accumulated knowledge, which, after all, is what imagination of any sort is based on, no matter how farfetched the result of the creative process.
Dawkins has made it very clear that he does not believe in the supernatural, even though he's willing to change his mind should some convincing proof offer itself.
10. Dawkins career would go down the drain if he was to say that awareness "could be" the mother of matter.
In fact Dawkins's career is fueled on the base that matter is all there is, ANY ONE who says it is not so, is a fool to him.
Nothing could convince him to probe deeper into awareness. He would vanish if he did, in other words he would become a joke...something he is not about to allow under any circumstance of his own doing.
32. This guy is not very smart. His arguments don't stack up. Just to pick 1 thing: he seems completely oblivious to the effect of environment:
he says if you can know the molecular state of a brain you can predict its future actions. WRONG! You would have to know the molecular state of the entire environment as well (physical world and interactions with other beings), environment plays a massive role in the actions of living beings.
It seems to me that genetics and environment are at least 99% responsible for our actions.
What we like or don't like for example, these are like commandments given unto us from the mix of genes and environment. Nobody DECIDES freely what they like and don't like.
Are serial killers like you and me? Do they suddenly of their own free will decide to commit gross acts of murder? Or are they unable to resist the murderous urges that 'normal' people simply do not have?
If there is such a thing as free will, then it is a very, very small thing indeed.
33. Even though Adams is not a credible source of valid information, I think his main point is valid: scientists are over-reaching and corrupting science into scientism.
1. Corrupting? Over-reaching? How about providing a few examples.
34. If you assume science is the only game in town, you are locked into a false assumption. that's like Michael Jordan telling Tom Brady football is a fraud because it is not based in reality.
1. Science isn't the only game in town. It is, however, the only honest game.
2. Says who? A bunch of scientists?
3. Can you name anything better than science for determining scientific matters?
4. no, but scientists insist on pontificating on matters beyond their scope. When they confront a mystery they insist on battering it with scientific nonsense.
5. Such as?
6. Will said jerrymack. As a scientists i agree with you. By the way there are a few "honest" scientists out there
7. When it comes to honesty, it certainly has religion beat.
8. Can you make any examples??
9. What is more honest about it then religion?
10. When it comes to something scientific, can you name any other games?
35. I endured the first 5 minutes with an open mind hoping for a constructive premise besides disparaging remarks about physics and Hawkins from Adams. The proceeding 5 minutes only served to convince me how little Adams knows or how closed minded he is about science, physics, and consciousness. Ironically, I find Adams fitting the profile of the corporate protagonists he describes at the beginning of the video i.e. hiding behind the veil of science to promote an agenda. I apologize if he has something constructive beyond 10mins but that is where I stopped watching because it did not seem to be leading anywhere besides physics and Hawkins bashing. Adams deserves credit for effort albeit seemingly in search of content to confirm own set bias.
36. Physics does not address "consciousness". Neuropsychology addresses that. In turn, neuropsychology does not address quantum relativity.
And you don't usually go to a plumber to get your piano tuned.
Adams seems to be one of those who argue: I don't understand it; they don't understand it; therefore: God did it!
He doesn't understand quantum mechanics* and he doesn't understand consciousness, so they must be related.
Physics has no soul? Neither has it leprechauns, elves, or angels. Science can study primitive mythology. It has done so, and dismissed it. Get over it!
*"If anyone claims to understand quantum mechanics, they don't understand quantum mechanics." -- Richard Feynman
37. I agree with Adams. Hawkings is a dangerous man because his ideas are not only wrong, they are taken seriously by many people. Determinism is a product of reductionism, which states that man is nothing but a sophisticated bag of biochemicals with no apparent purpose or meaning. How can you believe this without embracing despair? Without wallowing in cynicism? And when you consider the arrogance of Hawkings' statements, it's hard not to feel outrage..
1. On whose behalf do you feel outraged?
2. the human race.
3. aam641 On Your Behalf And Yours Alone.
4. I have been following Hawking for years and have read some of his books and observed some of his public statements. I really feel sad for that man for making so many illogical statements. By the way, Hawking may be a popular scientist. But did not even make the top 10 list for best physicists in the 20th century.
38. I am not a scientist, but if what Mike Adam's says is true, then we should all fear the future if it to to remain in the hands Frankensteins. These people have no idea what it means to be human.
1. Please educate us. What does it mean to be human??
2. Philosophy is split, arguably, in to 4 main areas...your question takes up one entire area on its own.
...could be a long response...
3. Who cares how a bunch of philosophers define what it means to be human? What do they know?
4. what the hell does anybody know? We are like dogs trying to grasp calculus.
5. We send men to the moon, construct atomic colliders and increase life expectancy. We must know something.
39. this guy is a joke , apparently he has a gay school girl crush on hawkings . i mean seriously the guy seems so mad at him , he has no actual arguments besides making lame jokes in an eloquent manner at hawkings and other physicists works . which is quite sad .
1. So called science is actually often times a so called "joke" by being stuck in a one sided indoctrinated materialistic world conception.
2. This approach produces actual results. Praying, groveling, and sacrificing to a magic sky-daddy does not.
3. Thanks samir, for your eloquence.
40. Maybe it's just my perspective, but this seems like a documentary of thinly veiled religious apologeticts.
A virtual fallacy feeding frenzy to nourish the ignorant.
41. Is Stephen Hawking real or is he just a science robot? Would certainly be easy to make him represent and follow an agenda.
I know, how dare I say?
1. Yep...better not say. Stephen has been wrong and noted for his stubbornness of assertions even when he was. Being wrong in his field can be a bit like a pop star having a flop - you're only as good as your last contribution. So there is a clear incentive to be stubborn.
...hang on a sec...STUBBORN? What place does such an emotional characteristic response have in the upper echelons of science? That's blatant hypocrisy in employing the scientific method! Or is it just that the 'alpha' scientist rules with the greatest influence?
Incidentally...Stephen Hawkings' PhD was on the assertion/discovery that the Big Bang originated in a singularity (such as that found at the centre of a black hole). But...a singularity is simply a point of infinite, zero, -1 or undefined depending how you look at it. "The point where all calculations, physics and maths can go no further based on our current understandings" would be more accurate. So the massive Stephen Hawkings proved in his PhD that the Big Bang came out of...something we can not understand or work with in our current science....
...And the crowds went mad, applauding with emotional reverence, respect and utter wonder... :-D
Present day science says that those areas of singularity merely point to where we need new ideas and input, because they are the points where our tools no longer work.
...And the crowd...sat down again quietly. :-/
Biological robots? From a man without a full working set of tools? Sounds to me like we need a trip to a hardware store!
2. And your point is? And by the way, what are your scientific credentials? Do they in any way match Hawkings'?
3. He has said: "Brain could exist outside the body".
His might already be the case.
His brain expresses words using the computer-generated voice he controls with a facial muscle and a blink from one eye. Let's see science do it that well (in such complexity) with someone who does not have his condition.
Makes me wonder how much of his talks are filled in by the controllers of that computer. It's not like he could argue the result like all scientists would if anything was said or written that they didn't agree with fully.
I know, how dare I say?
4. I'll be happy to compare his scientific qualifications with yours any day.
5. Woo-hoo *** POST 1000 **** ...made it!
He said that?! Well I guess that fits really - in that he is saying the brain (and thus mind, consciousness and whatever else is up there) is nothing special.
...In a way... I'm not surprised really - it's a case of if he didn't say that, someone would say that for him about his philosophy through a derogatory it helps, perhaps, to claim such a thing for himself first, before it is later used against him in some ridicule argument. Just a thought.
6. It's after 1000 that you really start noticing your opponents . LOL
7. Oh Gawd*!
*A way of writing "Oh my God." while denoting that you're rolling your eyes at the same time.
- urban dictionary com.
8. Like Oh shoot! A way of writing.... denoting that you feel you may be in deep trouble.
9. This is going in to the realms of A.I. and quantum computing, which I love. There was a fantastic book in 2005 from Raymond Kurzweil called "The Singularity is near". In it he optimistically states (overly so most agree) we will equal human brain computation and complexity by around 2035. The following year or will be twice as complex (Moore's Law).
But... to my mind, as exciting as that may be to see ...the essence of mind has yet to be bottled.
10. The essence of mind would have to be bottled by the essence of a mind.
If the complexity of a mind aka consciousness was to be fully understood (I imagine it would be an instant realization, a conscious big bang sort of thing), would that mind find a way to share such finding and have it accepted by others. Perhaps each mind has to find it's own path, if the "terminal" is "existable" (sorry just invented this word).
11. Yes. There is some anecdotal evidence that suggests - for the mind to be fully aware of itself, would require, at the very least, a slightly bigger mind. To understand this, think about a computer program being aware of its programming. That program would need some extra lines of code to understand it's code. But what would understand that extra code? Some more code of course.
...And so on. There is little logical reason to suggest the same would not be the case for any form of full awareness. This is, of course, philosophical though. ;-)
12. Perhaps not a bigger brain in size but a mind, free as in unobstructed.
How does one live unobstructed in our world, perhaps by chance or by choice.
13. That's true oQ, a bigger mind to observe that mind. But since the mind is internal to the brain...that would mean a bigger brain... unless you mean Stephen's external box of tricks?
All things being connected with eternal influences through the sharing of quantum information, infers that you could not 'get away' from the obstructions. Unless you could somehow step outside the universe(s). However does this apply to consciousness? Consciousness is affected by experience, environment and in the same way, unless you could step outside your mind through some sort of barbaric electro therapy lobotomy (not recommended), the same is probably true.
So what's the moral of the story? Matter, energy, mind...if you can't beat em join em.
14. And your point is?
15. Why are you interested, bored?
16. No, I hate phony intellectualism.
17. We know. Take a break from us and participate on some doc in a positive way, it will make you feel better.
18. Do you not get it ?
19. Thanks for all your supportive comments, much appreciated :-)
20. I find it strange that your file's activity is private but on the other hand your home address is in plain sight. I could google earth right above your house or in front of the blue garage door???
21. Go for your life, I could probably do the same for you, but the thought would not ever occur to me, but then I don't hide behind a shield of obscurity like most bloggers etc. If you want any further details just ask, i'm not afraid of justifiable, or constructive criticism. Unlike some people who seem to want to control these comments as if they own them. It is only opinions based on whatever facts we feel we know of. It would seem that some want to infer that their facts are correct, & the rest can change or be damned. Sorry discussion is just that, not lecturing !
22. 1i think you misunderstood me. When I click on your photograph (or name) it opens your file on TDF which shows your full address and says that your activity is private although I see that you have made 14 comments. I have never seen someone include their address information here. I have been on TDF for many years and I am often curious about new comers, that's what drove me to look up your file.
You are free to portray yourself anyway you want, no lecturing.
23. I have no respect for gibberish (read pseudointellectualism).
24. May be you don't need to step outside of the universe, you may only need the universe to step outside of your brain.
Deep Deep and deeper meditation perhaps.
25. And just how does one step outside the universe, as if you know?
26. "you may only need the universe to step outside of your brain".
not the other way around.
27. And just does the universe step outside of the brain? Complete gibberish. You're not fooling anyone.
28. I'm not even fooling myself...that's why I use may be.
29. Ok, let's consider what you're saying...
1) Energy and matter is 'entangled' (this is what I meant by "All things being connected with eternal influences through the sharing of quantum information"
2) Consciousness may be entangled with it. The double slit experiment of 'observing' affects the quantum state. (Even if the observer is an indirect camera).
Because of entanglement, it may not be possible to 'step outside' at all, since all will remain entangled. We could however, in theory, observe our minds fully aware since a universal mind would be bigger than our own, potentially.
In order to keep this on topic, that is where we can use philosophy and where science begins to fall by the way side. Science gives us a map, but the map is not the reality. Reality is looking out the window. This is where the line from the matrix has its roots: "Welcome to the desert of the real." - it means we have a map so real and precise that we forget to look out the window and the map becomes reality. How's that for DEEP!? lol.
30. Do you have any hard evidence to back up this drivel? If not, then you can't claim that science which demands hard evidence begins to fall by the wayside (note spelling).
31. Your entanglement post is as clear as mud, don't know what you are talking about. Will enter a "Quantum Entanglement" post without the math.
32. That's a very, very good link Achems, that explains entanglement and much more besides very clearly.
In a BBC Doc, I don't have the source, with Prof. Cox, he explains that the 'instantaneous communication' is felt across the entire universe. He finds it fascinating too that further, any influence on one single electron has an instantaneous affect on every other electron in the entire universe...through entanglement of quantum information (and energy conservation) as explained in your link. Thanks for the info.
33. 1i think it's a cool link.
34. You are deep in to Philosophical discussion now !
35. The mind may not be internal to the brain.
36. Of course!! Oops. Thanks for the nudge ;-)
37. And just what is that supposed to mean?
38. Unobstructed by what? Sheer gibberish.
39. Ooh, we are getting in DEEP here !
40. When I first heard about this, I couldn't 'get it', but the 'program code' example formed a clear easy picture in my mind as a good example - since I'm a programmer. Hope it helps.
41. I think the mind is well aware of itself but the brain is not aware of the mind's potentiality. I would tend to think that mind/consciousness is size less.
It just is.
The brain that perceives consciousness is sizeable.
42. Yep. In Buddhism it is the Universal Mind. A power house. Our brains are 'used' like light bulbs drawing a trickle from a large power station, but it does not mean we can't tap in to more power if we somehow choose to. That's the analogy anyway. This is also where the power of positive thinking draws its ideology, by tapping in to the Universal Mind. However... I've tried it. I have a friend who's been on Ted Talks who teaches it - privately we both agree it's probably b0llocks - having a positive outlook if you are negative will improve your well being, but beyond that? Getting rich and finding you soul mate, etc. ...well maybe, but not for any magical reasons. If you simply persevere with an idea, relentlessly, you will likely succeed at some point to some degree (imho).
43. Sheer gibberish.
44. Just how do you know this or is it simply more gibberish?
45. Just what are you talking about?
46. What if, being made from universe and being bound to become the dust of it again some day, we are not ever to be capable of understanding our own minds? What if each of us is not a complete mind, just one of seven billion human shaped cells thrown into the mix with a trillion others with differing roles to play? As a species we are endlessly inquisitive, we explore, study, imagine, create, dismantle, even put ourselves in danger, all so that we can learn more. Might be that we are the eye of a navel gazing universe! ;)
47. One does not have to be fully aware of all of the processes of the mind to be aware, any more than one has to consciously beat one's own heart, or breathe.
This would be apparent to you if you were reasoning rather than rationalizing.
48. Sheer gibberish.
49. I am sure to you it is pure gibberish, nice to see that you read it anyway.
50. It is pure gibberish in that it says absolutely nothing.
51. Still you had to read it to come to your conclusion....or perhaps you didn't read it, the avatar is enough for you to conclude.
52. Which humans brain though?
53. Ahhh....your lightest touch :-D
Indeed, indeed!
54. May be yours. Have you noticed that you can never foresee what your mind will think until it does.
Perhaps no one can plan the realization of consciousness...only try and try...which may be in the way.
55. How do you know this? More gibberish.
56. A gibberish you seem to enjoy or are you just lynching your mind with my words?
Notice... may be and perhaps and then may be again.
57. And just how does one lynch ones mind with someone's words? More gibberish!
58. I'm playing with you like a cat except the mouse is not trying to get away, it keeps coming back.
59. Ah, I see the lovebirds are at it again, I hear there are some nice mansions up in the hill in Los angeles where you can share your lives in bliss?
60. It would be nice if you deleted the love birds quarrel and left the conversation between Digi and is on topic and would like to see if anyone wants to add to it.
thanks gave me a good laugh.
61. It would be even nicer if you learned to use the objective after a preposition, just as in French.
62. I'm particular about the people with whom I associate.
63. A computer built around my mind? That would be pen and paper :) If we knew what we were about to think, all the world's thinking would already have been done and our thoughts would be mundane. Rather like being able to read but being allowed only one book for the whole of your life. Knowing how to read would become pointless, as would the book - nothing new would ever come from it. I'd much prefer to be given an empty page, just keep scribbling away 'til something beautiful happens ;)
64. I guess what I was trying to say is that it is perplexing how thoughts come from nowhere, the good ones and the bad ones. We can't control the qualities of thoughts that enter our mind although we can instantly decide which ones to put aside. The thought of thoughts coming from knowhere is equally perplexing.
65. Ohhh, I see! I misunderstood :) I love that about thoughts, some come marauding and unruly, only to be tempered by those that lurk in the back of your head. I guess quality control has its use, Imagine how noisy it could get between your ears if all thoughts had equal standing. Maybe we are just random thought generators, monkeys with typewriters. I think my daft thoughts get recycled as dreams, had some wild ones lately - catpeoplebabies! ;)
66. Random thought generators? could be right! Who would have thought the pinnacle of intelligent inspiration would be the mastery of becoming a good filing clerk!?
That said, however, there is a fascinating (and indeed insightful) BBC Horizon on YouTube called "The Creative Brain – How Insight Works" (2013) that has a permanent place on my shelf.
67. If by filing you mean stuffing everything in a drawer 'til the bottom drops out, I am already a master! Will check that doc, have a feeling I might have seen it... and then filed it with the important stuff ;)
68. You've hit an important point about what life is 'for'
Otherwise, if we all simultaneously understood that it all equates to an answer of 42, we might as well just pack up and go home ...where? lol.
69. Most people, most of the time, make their decisions in the amygdala, and only then rationalize them in the anterior cingulate cortex. Actual reasoning takes care, self-discipline and training that most people cannot or will not exercise. Hence, we have religion and patriotism and other destructive delusions. Science works by subjecting conclusions to actual observation and reasoning.
70. Interesting, where can I read more on this process?
71. you remind me of someone a lot of us miss around here ;)
72. Morning edge, I'll be sure to let her know ;))
73. Hi girl, I missed you too.
74. Thank you Blue :)
75. I read your opinions with interest. So help me out. Can you give me an accurate definition of "infinite"?,that is, something that is NOT finite?
76. Four (4) is finite. Four divided by zero (4/0) is infinite/indeterminate. (But the arctangent of infinity is Pi/2!)
42. Okay yea Stephen Hawking is a little extreme but hes stuck with nothing to do but think so i think you would be a little crazy in the same situation.
Free will has its limits no one can consciously stop breathing that's why people hang themselves, take pills or slash their wrists.
Most physicists say they don't know what came before the the big bang not that it spontaneously erupted, oh and concerning the big bang it is a theory not fact the most likely theory but just a theory all the same.
Its not our consciousness per say that affects the quantum experiments that the guy vaguely refers to its direct observation you can still indirectly observe the experiments (through cameras) even though you consciously watching still the experiment is no longer affected.
dark energy and dark matter mean by definition i dont know what the hell it is but something has to be there otherwise the observed universe wouldn't act the way it does.
Besides that i agree that alot of science these days is theoretical they do the math and when all the math is solid they come up with a theory that best fits, the mistake is that people hear scientific theory and it somehow means scientific fact even scientists do this is very annoying.
overall good documentary gets people thinking.
1. You obviously have no idea what a scientific theory is and I suggest that you find out before causing yourself further embarrassment.
2. please explain scientific theory to me then, please dont say im wrong without correcting me i like to know when im wrong. ive seen alot of your comments and they tend to be very demeaning and then you offer no advice i dont know why you do this but hey to each thier own.
3. (A scientific theory is a well-substantiated explanation of some aspect of the natural world, based on knowledge that has been repeatedly confirmed through observation and experimentation.) I conclude that the big bang is a hypothesis because you cant test or observe this hypothesis thats all it will ever be.
43. Religions and theological concepts are man made.However, there is nothing wrong with contemplating the awesomeness of nature and our perceived reality by diving into the pool that certainly exists outside of our very left brained square.xx
1. Just what is this pool "that certainly exists outside of our very left brained square" and how do you know that it exists?
2. Well said norlavine! The pool, Robert, is ...the frontier of the unknown ...the expanse of questions that remain unanswered, and those which have not yet been asked... and all of which you know little to nothing about because you care not to enquire...
You would deny philosophy, just as you would deny religion (without seemingly understanding the difference - you just bundle them and others up in a bucket of trash). But sadly, you go further, you condemn those who practice philosophical contemplation, as estoterical nonsense, including every other thing that has no perceived practical use or evidence from your perspective.
You have demonstrated over and over, a philosophy of existentialism (or possibly even nihilism) should seriously read the wiki entries on them.
[From Wiki: "Existentialism is different from Nihilism, but there is a similarity. Nihilists believe that human life does not have a meaning (or a purpose) at all; existentialism says that people must choose their own purpose."
After you have looked at your own philosophical reflections, ask me again what answers have philosophy provided. If you are not willing to look, don't ask me again.
Philosophy has much to teach, and contrary to your assertions, many answers to provide. Especially for a person such as yourself. Trouble is, you may not be able or willing to even ask the questions.
The sad part (another demonstrated part) is your condemnation of those who are willing to investigate (and learn something deep and meaningful...i.e. wisdom). Is there resentment on your part towards philosophy, because of what it says? And what is says about you?
You should not resent the love of wisdom - "love of wisdom" is what philosophy originally translates from. Indeed, how can you be against a love of wisdom? Rhetorical question Robert...that means no need to answer, just think about the question....philosophically.
3. So philosophy has much to teach and many answers to provide. Like what? And etymology proves nothing.
4. Philosophy teaches us, for example, that existentialists often have a sad, lonely outlook as it sees humans, with will and consciousness, as being in a world of objects which do not have those qualities.
By identifying with a philosophical outlook, we can better see why we think the way we do, and perhaps learn to see things in a more positive, compassionate way through changing our philosophy to something more 'preferable'.
Questions... answers... suggestions... meaning... wisdom... understanding. It's all there for the learning, through Philosophy.
5. "Philosophy teaches us, for example, that existentialists often have a sad, lonely outlook as it sees humans, with will and consciousness, as being in a world of objects which do not have those qualities." Source? Even if true, is this philosophy or psychology?
So all we have to do is choose one "philosophical outlook" from the many and we stand a wonderful chance of seeing things in a more positive compassionate way. Tell that to Hitler, Pol Pot, Stalin, Idi Amin and Attila the Hun.
6. lol...I would tell them, but they're all dead.
Yes it's philosophy. Psychology being a scientific branch of it.
The source? You didn't look up existentialism on wiki after all did you... oh well, I gave you some philosophical answers any way. Even though you didn't do what you were supposed to... couldn't be free will could it?
7. "Yes it's philosophy. Psychology being a scientific branch of it." Source?
You're the claimant; you're the one charged with providing a source which you failed to do. Contending that somehow I should have figured out that you were quoting or drawing from Wikipedia is patently dishonest.
8. Ok my last post on this thread...
"After you have looked at your own philosophical reflections [on the wiki entries suggested], ask me again what answers have philosophy provided. If you are not willing to look, don't ask me again." - I said.
to which you replied...
"So philosophy has much to teach and many answers to provide. Like what?"
From this, I cordially assumed you did look it up, as requested, and so replied with an answer. You had not. Patently dishonest? Me? You are not fooling anyone.
9. You "cordially" assumed (whatever cordially is supposed to mean in this context). That was your downfall. You're the loser.
10. You sound like one of my 'teachers' at the WSP !
44. Mike Adams misrepresents what doesn't fit into his believer's mold. He states that those who don't believe in a 'free will' or a 'soul' are mindless and without conscience or morals, etc. Nonsense. The prisons are filled with believers, not atheists.
And he's given no evidence to support any belief in a 'God', which I notice he's called 'him' at the end of his presentation. Now where would, or could, a 'God' have come from? His Father, another god? Seriously, if you were a god, an omniscient, omnipotent god, would you create a world such as this, where everything eats everything else, usually alive? Did God sit idly by, as Christopher Hitchens loved to say, for a hundred thousand years as evolving humans suffered and died from childbirth and tooth infections, before finally revealing Himself to an illiterate group of goat herders in an obscure corner of the middle east?
You might try reading Marvin Minskey's book "The Society of Mind" to get a feel for the complexity of mind. And from our earliest days we experience bodily homeostasis which may in fact be a major determinant of our paths.
Does consciousness control, or does it simply observe and record?
1. For this reason, long time ago, I stopped calling this world civilized. On a higher spiritual level the very idea of a necessity of nourishing oneself with some others living beings (plants included) seems to me a violence.
2. Indeed! I see our world of today as being just one step ahead of our thousands of years of 'primitive' past. And the curious phenomenon of every living creature requiring sustenance from outside itself ....... well, what can be said of that? Down to the molecular level even, the oxygen in water or the specific wave lengths of light utilized for the continuance of another species. I mean no offense to any religious belief, but what does it mean to say that 'humans are made in the image of God?' What could that possibly mean? In so many attributes humans are the exact opposite of the image of God, you can go on and on with them, such as not having everlasting life, need other creatures flesh and blood to sustain us, have limited knowledge and wisdom, are made of flesh, bone and blood, and on and on. And the National Catholic Almanac states specifically that God is "almighty, eternal, holy, immortal, invisible, omniscient, ..... perfect, provident, supreme....", none of which could be applied to a human. Tell me, please, anyone, in what way humans are made in the image of a god?
3. You're right..... And what can one say regarding a 'perfect God' who creates a world where every living creature must eat some other living creature, usually alive, in order to survive.
45. It's very easy to comprehend a computer program as having no free will. It is very easy to comprehend that there is a difference between our brain function and a computer program. That difference is free will (imo).
As a programmer, I know that at a crossroads (of equal consideration) I have to invoke a random number to make a decision or choice. But existence of a random number, inside a computer, does not exist: only a psuedo-random number based on a math algorithm 'seed'. It is, perhaps, possible to create a 'true' random number in machine code though. One such method uses a 'seed' taken from atmospheric noise. Truly random though? Or just an extreme degree of complexity? Think about it.
Which brings me to how science employs randomness. Think about genetic mutations of DNA. A 'mistake' in the copying process which has led to all the diversity of life we see. Science STATES it is random. Philosophy can ASK what if it is not? What if those random mistakes were not truly random? What if they were subtly infuenced by complexity. A mistake would become an influence. Just because science declares a process as random, through possibly not understanding the process, does not make it so. Yet the consequences on philosophical interpretation can be enormous. I'm not saying DNA mutations are influenced, I'm saying consider the consequences of such a small nuance and how Science and Philosophy would each interpret such a discovery.
1. The difference is that when a scientist asks the question, he will research the hard evidence, whereas the philosopher merely asks the question and then contemplates. This clearly puts science ahead of philosophy.
2. Why do we have a theory called the big bang, instead of a simple "we don't know"? Why do we even discuss such a thing? Why does science allow for consensus of ideas and beliefs, when all the facts are not in? Why is conjecturing permitted in science? The reason, which has deluded you thus far, is that scientists like to ask philosophical questions too. Any question that leads to a subjective answer is philosophical by nature.
For example:
"Do milkmaids get smallpox less often than others?" was philosophical, but ceased to be on discovering a definitive answer. "How much is 2 + 2?" is not, though even it may have been?
Do you understand the difference between objective and subjective Science? Even scientific consensus is not objective, but poses philosophical questions. Evidence without proof is conjecture and theory. Only objective science is not philosophical. Scientists, therefore, practice philosophy all the time in their research, as a tool to discover understanding. So, to make this clear...
You wish to separate scientist from philosopher in to two people. When really it's the fields that are separate. Indeed, you should be aware that every major science, including physics, biology, and chemistry are all disciplines that originally were considered philosophy. This literally puts philosophy "ahead" of science. ;-)
It is the analysis and speculation of philosophical thought that eventually develops the branches of science.
3. So what if several hundred years ago physics, biology and chemistry were all considered philosophy? We've learned a lot since then.
"The reason, which has deluded [SIC] you thus far, is that scientists like to ask philosophical questions too." Well, so do milkmen and garbage collectors, so what?
"You wish to separate scientist from philosopher in to two people. When really it's the fields that are separate." So every scientist is a philosopher? No more than every composer, painter, architect, etc. is one.
"It is the analysis and speculation of philosophical thought that eventually develops the branches of science." Really? What about the field of genetics which was developed solely through hard hands-on endeavor as was immunology, microbiology, quantum physics, not do-nothing philosophy.
"Scientists, therefore, practice philosophy all the time in their research, as a tool to discover understanding." So those researching cures for certain forms of cancer are actually practicing philosophy? So Werner von Braun, Neils Bohr and Jonas Salk actually practiced philosophy and not science. What abysmal crap!!!
Now how about providing some examples of subjective science and while you're at it explain how string theory which is in its conjectural stage is really philosophy.
4. try some metgodology of science. phylosophy of science. the oppinions you say are valid but only partially.
5. i like your views man but i just want to inform you the reason the big bang theory is the most probable explanation is because the universe is expanding so at one point its likely it started in one place. As for saying why not say we just dont know? if we always said we just dont know then we probably wouldnt learn alot that way. not trying to put you down i just thought that was important.
6. Wrong. Stating that you don't know is not only honest, but a fine prelude to scientific inquiry.
7. Now this is starting ti make some sense !
46. Some cool comments here, some outright angry ones, and a few mundane dismissals. ...but addressing the importance of Philosophy in our understanding? Missing :-/
Philosophy and Science are the yin and yang of our search for truth and understanding. They are often incompatible, yet one without the other can only hinder that search. Science STATES: 'This is the evidence'. Philosophy ASKS, 'What does that mean?'
Because scientists are not responsible for the consequences of their discoveries, there is rightly no place in Science for philosophical reflection (concerning ethics and morality).
But for Science to dismiss Philosophy would be akin to saying there are no consequences, ethics and morality worth considering (a philosophical perspective in itself). Midway through he states '...the philosophy that we have no free will'. Indeed, believing we have no free will is also a philosophy in itself, as there is no proof for denying free will (and arguably evidence for both persuasions).
Philosophy only questions Science when it attempts to be philosophical. Science therefore has no place being philosophical about its discoveries.
One, it seems, can not escape the other, and for good reason - they are necessarily entwined. Yet only one attacks the other with the intention to destroy it: If science wants to kill off philosophy, then it must first show it is responsible for its actions, measure and weigh up consequences and consider ethics. This documentary shows how ill equipped Science would be in doing so, and Science has openly declared itself unwilling to do so.
Science is the one on the attack. Science is the one that has to be kept in check, ethically. Philosophy provides the means to do so, but only if we don't allow Science to convince us it is dead.
Ok, hands up, I asked for this doc to be posted up. I found it interesting and sarcastically humorous, with some great moments of truth. I hope at least some of you can take something positive away from its message. P.S. I had no idea who Mike Adams was (certainly no foul intended).
1. If philosophy is so important to science, how many philosophers are employed by CERN, NASA, Merck and other mainstream, hands-on, scientific organizations? How many philosophy courses not required of the general student population must science majors take?
And just how does philosophy manage to keep science in check ethically, assuming that this is needed? By some armchair philosopher warning of the dangers of nuclear weapons and chemical warfare?
Philosophy might ask what something means which any child of two can do, but it has yet to provide anything approaching answers and thus is useless.
2. As soon as a philosophical question is answered, it becomes science and ceases to be philosophy. I could say to you science has yet to ask anything of interest, and it would be just as inaccurate as your misleading claim.
3. Once again, if philosophy is so important to science, how many philosophers are employed by CERN, NASA, Merck and other mainstream, hands-on, scientific organizations? How many philosophy courses not required of the general student population must science majors take? The proof is in the pudding.
How about a few examples of some answered philosophical questions (not scientific ones relying on hard evidence).
4. May be they(ScO's, should employ & consult Philos, so then they have another perspective of the reasoning & moralistic queries philosophers pose .
5. A philosopher is not heeded to help build a Hadron Collider or to discover cures for various types of cancer or for that matter anything scientific.
Philosophy accomplishes absolutely nothing.
6. I sent your oft-repeated question to NASA. If or when they answer, I will post it. Either way, the answer will be interesting!
7. Would you mind posting your question?
8. No problem. First I quoted you, and then made sure the question was qualified as you did in another post - a philosopher doing philosophy, not science. See below.
A fellow on a documentary site that allows comments is of the same ilk as Richard Feynman, et al, in that he totally dismisses philosophy as useless. I know that Professor Feynman was on the team that investigated one of the shuttle accidents, but that is not on topic, just an aside.
This fellow says this, in fact copies it over and over - "If philosophy is so important to science, how many philosophers are employed by CERN, NASA, Merck and other mainstream, hands-on, scientific organizations? How many philosophy courses not required of the general student population must science majors take?"
Do you have an answer to that for NASA or any of the other agencies that do hard science? Can you give a number of philosophers that do philosophy and are employed by NASA for that purpose? It would be interesting either way you answer. I will look elsewhere for the info, but I think I need an insider to answer that question.
9. Fair enough. I'd be interested in the answer. Did you write to NASA in general or to some specific department such as personnel?
10. No one specific. I went to the Contact Us link and it went to Public-Inquiries. At hq dot Nasa dot gov.
Could be a long wait.
11. Again, fair enough, but perhaps this should be taken further and CERN, Merck, the Venter Institute and other hard scientific entities should be contacted--and perhaps I should assist because it is not fair for you to do all the work.
12. Okay, I'll take CERN and Fermilab for good measure. My employer's USA home office isn't too far from there in Illinois and I'm told it is a fascinating place to visit.
I thought NASA had the best chance of having a philosopher on staff. I suppose I was thinking of things like sending people to Mars, and such. Perhaps a Psychologist or Psychiatrist would be more appropriate as there are lots of people issues to solve along with the issues of getting there and doing stuff.
13. Psychologists or psychiatrists I can see as far as NASA, but these people are not philosophers.
14. No, I didn't mean to imply that at all. I should have written that so it was clear.
However, I don't know if I would call psychiatry or psychology a hard science. Not yet, anyway.
15. And neither do I. Every psychiatrist I have spoken to has described his profession as an art form. However, as I stated, I can envision NASA hiring a few--but this is far different from philosophy.
16. They do use psychologists to vet their ideas already.
17. Source? One way or the other, a psychologist/psychiatrist is not a philosopher.
18. I received a reply from Fermilab today.
Hello Paul,
No, we don't employ philosophers here at Fermilab. Our budget covers scientific research only. Thanks for the question.
Andre Salles
Fermilab Office of Communication
19. Hey decaf, not quite the same thing but CERN has Artists in Residence :)
20. Yes, I spent quite a bit of time on their site and noticed that. Looks like a fascinating place to punch in every morning. That is, unless one has chosen a career in philosophy.
21. Slap on a wide-eyed and toothy grin, mumble something about aesthetics and skip away, they'll put it down to artistic temperament, all good! ;)
22. As you've pointed out, artists in residence are not philosophers. At least artists in residence produce something.
23. True, but if they hope to bring understanding to us with wandering pencils and abstract photography, why not also through philosophy. Some of it is beautiful enough to be poetry. They have given Art a place in their world, all or none, surely? Art is subjective, until the beauty of a piece becomes a consensus, then it's beauty is objective - but that's just philosophy of art. As The Mighty Achem would say, 'bakes no bread' ;)
24. Art is simply the appreciation of something for its own sake. So if you want to consider philosophy an art form, go right ahead, but it's no more than that. However, philosophy itself has no place in any art form. It can't tell you how to paint a picture, compose a symphony or write a novel and when it's starts dictating on these which all too often it does, it falls flat on its face.
25. From Wiki Answers: Why philosophy is a science and art?
"Philosophy is a science because it systematically develops a hypothesis on a premise with analytical tools to resolve the problem through logical reasoning (induced or deduced). It is always open for debates as a human endeavour to seek the truth through learnt knowledge. [edit: exactly what I have been saying all along about scientists employing philosophy as a tool for understanding]
Philosophy is an art because you require inherent skills & natural ability to apply the philosophical principles."
From the Guardian: "Studying philosophy will [teach] you to think logically and critically about issues, to analyse and construct arguments and to be open to new ways of thinking."
Once again, you're dismissing what you do not understand, which unfortunately reveals you to be unscientific in your approach.
26. First of all, you need to comprehend what you read. I wrote, "So if you want to consider philosophy an art form, go right ahead, but it's no more than that."
Once again, philosophy does not help to compose a symphony, write a novel,chisel a sculpture or direct a movie. I hope by now you've read the response which Pwndecaf received from Fermilabs--i.e., no philosophers on staff. So much for philosophy's "importance" to science, so much for philosophy's "importance" to art and so much for you.
27. Fair enough, philosophy can't tell art what it is. It can play with the idea of what makes art, art though :)
28. And play is all it does. Knowing "what makes art" (assuming that this is possible) is quite different from actually doing art which makes philosophy valueless.
29. Philosophy is the art of thinking around an idea.
30. "Around" is the operative word.
31. 1i thinks, without it's surrounding, a thought would remain asleep.
32. "Around" is still the operative word.
33. ..and from Wikipedia itself:
"Philosophy is the study of humans and the world by thinking and asking questions. It is a science and an art. Philosophy tries to answer important questions by coming up with answers about real things and asking why?"
34. Science tries to answer important questions by coming up with answers about real things (and nothing but) and asking why; only science often succeeds whereas philosophy generally fails.
35. Thanks for responding. "Our budget covers scientific research only" says a lot.
36. I finally got a response from Cern. They are succinct!
Sorry , we don’t employ philosophers.
Best regards,
Recruitment Service
HR Department, CERN
CH-1211 Geneva 23
TAKE PART ! My opportunities at CERN
47. Another layman who cannot solve Schrödinger equation is interpreting quantum mechanics. Seriously, be an expert first and then start to discuss such issues. Any yet, if the discussed issue cannot be falsified, it does not matter anyway.
48. To say free will is an illusion is a bit contradictory, isn't it? Whether it is or not, do we not require consciousness to interpret that? What is it that is able to recognize an illusion? What faculty decides the truth or untruth of anything? If not consciousness, what?
1. How about hard evidence and hard results?
2. ALL OF US know there is something beyond the love, you have dreams, visions, a bunch of intangibles. Only sociopaths and bafoons who are too cowardly to look at reality say stuff like this. Go ahead, live in your no-consciousness, no intangible world. Don't love anyong, don't dream, don't think or do intangible stuff cuz you NEED HARD EVIDENCE before you believe any of that exists. My sister with down sydrome has more common sense than you and some of the "intellectuals" on here who intellectualize their way into denial of reality.
3. This is one of the greatest tradegies of this species. A cold, calculative, indifferent mind. Converting everything into numbers, formulas, theories, hard evidence or lack of thereof. And then they wonder why everything around is so dull and grey and frozen.
4. ..there's plenty of evidence for a God-designed world, just like there's plenty of evidence for the invisible force of gravity (which is only theory) cuz you, me and others cannot explain the origin or essence of it, we can only describe the RESULTS or repercussions of it. This is why bafoons like you are such i*iots. You pretend that everything in the world is math and formulas and that you DON'T believe in anything invisible, like love, dreams or even gravity, the magnetic field or dark matter. then you mock folks who believe in something that's invisible because they are only going off of evidence for it and you sheeple demand hard core empirical evidence when you don't even do that for the aforementioned invisible things. What a pathetic hypocrite.
5. Since you are making a claim of "there's plenty of evidence for a God-designed world" please show us your evidence, without resorting to any circular logic etc: thank you.
49. What does it all mean? It doesn't matter. It makes no difference. If you really need there to be a "purpose," then do like Mike did: make it up.
50. What is this physical brain that so many scientists give all power to? What is it made of? What is its foundation?
Science itself seems to tell us that physical-matter is not made of physical-matter;
but rather indefinable vibrations of “pure energy”.
What is this thing we call energy?
No one knows.
Until we know, what do we really know?....if anything?
I only know that I don't know.
1. How about the capacity of a system to perform work for starters?
Among other things, we certainly know enough to send men to the moon and build the Hadron Collider?
2. See my last post/reply to Bob Trees.
51. lol. haha hahaha. heh heh heh... I'm gonna watch it again.
1. Thank goodness! Someone at least found this funny. Good on you Jo. I thought the humour was fantastic.
2. it was fun. I did almost feel sorry for this guy 'Mike'; he 'almost' gets some things right, and then he goes so far wrong the incredulity becomes satirical. I was thinking, someone (maybe the BBC) could have a bit of fun creating a satire - hell - based on this spot of fluff, the script is practically written, HA...peace
3. :-( oh dear, seems I missed your point and you mine. Did you see on RT today frozen light? Photons frozen in time for up to 60 seconds...apparently. I think 'Mike' might be on to something.
52. With all the quality Docs on this site that discuss the topic of consciousness how on earth did this one wiggle in? There may be some interesting points that are fertile ground for comment but anyone familiar with this Mike Adams knows not to bother.
53. Ahem. Someone sends me Mike Adams's Natural News newsletter several times a month. Everything he writes supports his bizarre worldviews and hucksters for his "snake oil" product that "cures everything." There's a "conspiracy" to prevent the world from knowing about all these cures his snake oil performs.
54. I liked this doc! I wished I was paying more attention to it while doing other stuff, but focused more about half way thorough. I liked the different ideas presented on the big bang theory but there was one that I kinda lean more to. I'm thinking that the big bang is somewhat of a pulse or cycle. The universe is expanding but at some point it contracts to the point of becoming a big bang again. I've got absolutely nothing to point that this may be the scenario.
Which brings me to another point on this. Why can't others come to the relization that we don't know somethings? Why can't "we don't know" be an acceptable answer for some things? If physicists offer that as an answer, it seems like they are deemed to be less "knowledgeable".
1. When I was spending time at the The School of Philosophy, in Wellington NZ, (NOT Victoria Uni., But a private institution) we were learning what is called 'Practical Philosophy', ( to some people that will be an anachronism), About 2 years in we were discussing Knowledge, where does it come from ? One of the end products of this discussion was the thought that "You Don't Know, What You Don't Know !". And this I now use to discuss these very subjects & their like !
55. Mike Adams is the owner of Natural News, a website dedicated to alternative medicine and various conspiracy theories, such as chemtrails, the alleged dangers of fluoride in drinking water and health problems caused by "toxic" ingredients in vaccines, including the now-discredited link to autism. In addition, Adams is an AIDS denialist, an promoter of conspiracy theories surrounding the Sandy Hook Elementary School shooting and has endorsed Burzynski: Cancer Is Serious Business. I believe that says everything.
1. Wow. "Him smart".
2. damn...just needed Anti-Semite as well and that would have been Bingo!
56. I love when people are all lie "The best part of this film was the part about the universe being a simulation" Lol if your going to refute all of the other BS why not refute the while matrix idea as well.
57. I had to stop when i realized he does not understand what the "theory of everything" is meant do do or explain. The author is just seeing blindly what HE wants to think of it. not worth watching.
1. What hypocrite....You are complaining that the author is seeing "what HE thinks of it".
THAT'S what folks do, you id**t. A person analyzes info and then draws his/her conclusions after doing so. And here you are drawing your conclusions, being the typical arrogant hypocrite who wants to shut out other opinions.
58. Here's an idea, we need to extract an exotic particle, so we take a cylinder and fill it with plasma, we then compress it so it starts to form into particles, it creates about a hundred particles with a number of isotopes that are particles changing from one thing to another, we then insert an organism to extract the most stable of those particles. standard process for filtering what we want.
The universe is not big it is just you are very small, welcome to the cylinder, now do as your suppose to do and extract the most stable particle in this soup, gold. We are farmed so therefor created to complete a task, when done we will be flushed like all other processes when done.
59. We have a money system that creates money and then charges interest that is not created , so the only way to pay that interest is to create things of value, the only thing the bankers will take is gold, so therefore the money system was created to force us to dig holes in the ground to collect gold, anybody in the last 5000 years that attempted to change the system was done away with, so it is clear why man is here, don't see lions, tigers or sparrows digging holes to collect gold.
The concept of god along with physics were created so to give the illusion of some power over the environment, the fact that there is a harmonic relationship between things shows it is one organism.
We all have a liver, to the liver we are god but can any of us really understand a liver, can we mod it, do we really know how it works, but if it is our liver then we must have knowledge of it to make it, but we don't, what does that tell you.
In just a few years man gained virtual worlds and used them to escape, and within those looks for other social structures to escape, so you could say it is within the nature of life to escape.
The best explanation I ever saw for god and physics came from a very old text, it said "god was lonely so fragmented himself to have some fun", seems being fragmented makes you forget what you knew when whole.
What ever organism we are part of seems to need gold, as the only intelligent life in existence that we know of, digs holes in the ground and converts rock into gold bars, what does that tell you.
In fact if you read the oldest text they tell you that and why, but why let an old story get in the way.
Gold is of no use to us, so why do we dig it up.
60. free will exists and awareness exists and science cannot fully understand that it is okay science can eventually
61. I noticed the analysis of physics. And the role they play in determining our different life sciences.
This doc is great affirmation of general evidences we hold, in understanding our existence.
The God Within is a title that evokes religious implication. It is not; more so a "transcendental abstract" term.
62. how could the universe not be a simulation? that is what else could it be? what are the other options?
1. Brings up the age old 'who created the creator' argument. If our universe is a simulation, its being simulated on a computer in a universe built by someone or something. Not impossible, but highly unlikely. There still needs to be a universe.
2. We don't have a clue what anything could be. Time may only be a factor in the manifest world? In our world there is a start and an ending. Why is this? This must be an interlude a stop off on the way somewhere else? Why even bother with one time around? does it stand to reason there would be an abstract life without reason to be. There are many beings on Earth we discount them as being sole less? No creator just are? I think we are from another "planet" and came here. We are so different then all the rest of the life forms on this Earth? And in other ways almost the same, like procreation? The basic things caring for our young and so on. The rest of life doesn't have a Bible or Koran etc.? Very complex?
3. Are humans more like a snake than a bat is like a snake?
4. That's a question for an evolutionary biologist.
5. "We [I assume you mean homo sapiens] are so different then [SIC] all the rest of the life forms on this Earth?[SIC]." Just how?
The remainder of your post is sheer gibberish ("We don't have a clue what anything could be. Time may only be a factor in the manifest world?[SIC] There must be an interlude a stop off on the way somewhere else?[SIC] . . . does it stand to reason there would be an abstract life without reason to be.]
Try again when you have something to say and express it in clear, conventional English.
63. You'll be a fool if you take any of this man's definitions and assertions seriously. He has an agenda, and the thing is therefore fallacious from start to finish. Having said that much, it is nevertheless interesting to listen to in parts, in my opinion, especially that regarding whether or not the universe is a simulation.
1. i agree
2. So were all robots? LOL. This a perfectly coherent alternative viewpoint to modern physics..
3. Biological Robots!
4. Please explain to me the "agenda" this man has?
5. He demonstrates an agenda within the first two minutes of the doc. He is disappointed with Hawking's first page of the book "The Theory of Everything". Apparently, Hawking did not validate this guy's beliefs, so he is summarily dismissed. He concedes Hawking's brilliance, but then says that Hawking lacks the insight that he, Mike Adams, has. I'm paraphrasing, of course but he is a self-proclaimed genius. "Them scientists, they smart, but not so smart like me."
I wonder if the simulator of the simulation, has his own creator. Maybe this is the nature of the universe. One simulation built onto another one. We are the leading edge, preparing to create the next simulation. Will this be a human collective effort or maybe millions of single creators, writing the code for their own private simulations...billions of universes, created just so that new simulators can create the next generation of simulations? This could lead to the implication that God has his own God, which has his own God, into infinity. This philosophy thing can be so much fun. Put out an idea, without proof, and then bask in my own brilliance. I love it.
6. You certainly have his number. Just a suggestion though, in the last line, change "my" to "one's."
7. Good suggestion but upon reflection, I think I'll let it stand. The use of "my" demonstrates how self indulgent this brand of philosophy can be. After all, it is me speaking and it is "my" beliefs (truth?). It is the branch of "science" (don't laugh) where any speculation can be the truth because its very vagueness is its defence.
8. O.K., but do change the two "it's" in the last line of your previous post.
9. The 'Matryoshka Doll' theory of the universe. These types of bullsh-t-session speculations can be fun, but I don't take them too seriously, at this point. It's just entertainment, like smoking dope with Donald Sutherland's character in 'Animal House'. But it's conceivable that there could be something to them, I suppose. As a matter of fact, physicists are set to run an experiment shortly to see whether there is any validity to the thought-experiment in the real world (I loved saying that). Here's a section from an article on Huffington Post:
wondered, too.
Professor Martin Savage at the University of Washington says while
an atom's nucleus, there are already "signatures of resource
constraints" which could tell us if larger models are possible.
This is where it gets complex.
Essentially, Savage said that computers used to build simulations
perform "lattice quantum chromodynamics calculations" - dividing space
force which binds subatomic particles together into neutrons and
including the development of complex physical "signatures", that
researchers don't program directly into the computer. In looking for
they hope to find similarities within our own universe.
Whoa. Whoa, dude! (lol)
My 10 year old son checked out a youtube video about some of Nick Bostrom's ideas with me the night before last, and dismissed any idea of a simulated universe with the heady words, "Daaad, I eat! So I'm not a simulation."
Sounds just a little bit like Descartes, doesn't it?
10. I suggest Thomas Campbell's trilogy "My Big TOE"
64. What is this... an argument against the 'no free will' hypothesis ? a pseudo scientific argument for an external self or duality ? an attempt to lever the words of a respected man of science like Hawking to support religion ?
Consciousness and free will is not external, Have you ever taken a hard look at artificial intelligence and neural networks ? Its pretty spooky stuff when you discover that some simple math compounded a billion times over can create the infinite possibilities we call choice and the ability to learn.
Most of the things those physicists have 'calculated' have been observed. The 'God particle' aka the higgs boson was predicted mathematically, we just didn't have the technology to observe it at the time. Our mathematical predictions tell us the universe had a point where it began, we can only theorize at this point how it began. We have no solid evidence of what the 'nothing' origin of the universe looked like or what anything outside what we perceive as an infinite universe could really be.... yet. If physicists say there's dark matter and dark energy out there, believe it. There's something they don't understand out there, it accounts for gaps in the calculations and one day they'll come up with a way to explore the questions it raises and answer just what it is.
Science is admirable in the fact that it seeks its answers without being sidetracked by dogma and bias. Theres gaps in our knowledge for sure, we find more at every step, but lets not fill them with god. Take a second look at your work author !
1. In what way is the "God Particle" "God"?
2. Name only. It was thought it would usher in the unified theory with a testable and observable answer to all the questions about the formation of the universe. We will wait and see if it delivers.
3. It got the name because it's supposed to be what imparts mass to certain particles, without which, of course, there would be no matter, and hence no universe as we know it.
4. That's funny its a "particle", itself, the "god particle". So its a particle bring mass to other particles, right?
5. Actually, it's the Higgs field that the particles move through that enable certain of them to acquire mass. The Higgs boson, or God particle, is what confirms the existence of the field.
6. Where did the "Higg's field" emanate from? What does that "field" consist of? Where does that consistence derive from and so on and so forth? etc.........
7. Higgs mechanism...
8. Sidetracked by dogma and bias? Dogma is bias...
65. What a pile of c***. Mike Adams is a fraud and an i****. |
61f8ac1d26c54277 | Solution of a Nonlinear Schrödinger Equation
Requires a Wolfram Notebook System
Requires a Wolfram Notebook System
The nonlinear Schrödinger equation can be applied to describe nonlinear systems such as fiber optics, water waves, quantum condensates, nonlinear acoustics, and many others. This Demonstration solves the specific case of a soliton profile perturbed by a periodic potential.
Contributed by: Enrique Zeleny (May 2012)
Based on a program by: Stephen Wolfram and Rob Knapp
Open content licensed under CC BY-NC-SA
The nonlinear Schrödinger equation has the general form
In this Demonstration we consider an inhomogeneous modified form
where is the solution range.
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8690e0ef4da4d67e | Topics in Quantum Dynamics
A. Jadczyk
Published in: INFINITE DIMENSIONAL GEOMETRY, NONCOMMUTATIVE GEOMETRY, OPERATOR ALGEBRAS AND FUNDAMENTAL INTERACTIONS Proceedings of the First Caribbean Spring School of Mathematics and Theoretical Physics Saint-François-Guadeloupe 30 May - 13 June 1993 edited by R Coquereaux (CNRS, CPT-Marseille), M Dubois-Violette & P Flad (CNRS, Univ. Paris XI)
The Two Kinds of Evolution
In these lectures I will discuss the two kinds of evolution of quantum systems: The first type concerns evolution of closed, isolated1 quantum systems that evolve under the action of prescribed external forces, but are not disturbed by observations, and are not coupled thermodynamically or in some other irreversible way to the environment. This evolution is governed by the Schrödinger equation and is also known as a unitary or, more generally, as an automorphic evolution. In contrast to this idealized case (only approximately valid, when irreversible effects can be neglected), quantum theory is also concerned with a different kind of change of state. It was first formulated by J. von Neumann (cf. [30, Ch. V. 1] and is known as von Neumann - Lüders projection postulate. It tells us roughly this: if some quantum mechanical observable is being measured, then - as a consequence of this measurement - the actual state of the quantum system jumps into one of the eigenstates of the measured observable. 2 This jump was thought to be abrupt and take no time at all , it is also known as reduction of the wave packet. Some physicists feel quite uneasy about this von Neumann's postulate, to the extent that they reject it either as too primitive (not described by dynamical equations) or as unnecessary. We will come to this point later, in Chapter 3, when we will discuss piecewise deterministic stochastic processes that unite both kinds of evolution.
The Schrödinger Equation
It is rather easy to explain to the mathematician the Dirac equation - it became already a part of the mathematical folklore. But Dirac's equation belongs to field theory rather than to quantum theory. 3 Physicists are being taught in the course of their education rather early that every attempt at a sharp localization of a relativistic particle results in creation and annihilation processes. Sometimes it is phrased as: "there is no relativistic quantum mechanics - one needs to go directly to Relativistic Quantum Field Theory". Unfortunately we know of none non-trivial, finite, relativistic quantum field theory in four dimensional space-time continuum. Thus we are left with the mathematics of perturbation theory. Some physicists believe that the physical ideas of relativistic quantum field theory are sound, that it is the best theory we ever had, that it is "the exact description of nature", that the difficulties we have with it are only temporary, and that they will be overcomed one day - the day when bright mathematicians will provide us with new, better, more powerful tools. Some other say: perturbation theory is more than sufficient for all practical purposes, no new tools are needed, that is how physics is - so mathematicians better accept it, digest it, and help the physicists to make it more rigorous and to understand what it is really about. Still some other4, a minority, also believe that it is only a temporary situation, which one day will be resolved. But the resolution will come owing to essentially new physical ideas, and it will result in a new quantum paradigm, more appealing than the present one. It should perhaps be not a surprise if, in an appropriate sense, all these points of view will turn out to be right. In these lectures we will be concerned with the well established Schrödinger equation, which is at the very basis of the current quantum scheme, and with its dissipative generalization - the Liouville equation. In these equations we assume that we know what the time is. Such a knowledge is negated in special relativity, 5 and this results in turn in all kinds of troubles that we are facing since the birth of Einstein's relativity till this day. 6
The Schrödinger equation is more difficult than the Dirac one, and this for two reasons: first, it lives on the background of Galilean relativity - which have to deal with much more intricate geometric structures than Einstein relativity. 7 Second, Schrödinger's equation is about Quantum Mechanics and we have to take care about probabilistic interpretation, observables, states etc. - which is possible for Schrödinger equation but faces problems in the first-quantized Dirac theory.
Let me first make a general comment about Quantum Theory. There are physicists who would say: quantum theory is about calculating of Green's Functions - all numbers of interest can be obtained from these functions and all the other mathematical constructs usually connected with quantum theory are superfluous and unnecessary! It is not my intention to depreciate the achievements of Green's Function Calculators. But for me quantum theory - as any other physical theory - should be about explaining things that happen outside of us - as long as such explanations are possible. The situation in Quantum Theory today, more than 60 years after its birth, is that Quantum Theory explains much less than we would like to have been explained. To reduce quantum theory to Green's function calculations is to reduce its explanatory power almost to zero. It may of course be that in the coming Twenty First Century humanity will unanimously recognize the fact that `understanding'was a luxury, a luxury of the "primitive age" that is gone for ever. But I think it is worth while to take a chance and to try to understand as much as can be understood in a given situation. Today we are trying to understand Nature in terms of geometrical pictures and random processes8 More specifically, in quantum theory, we are trying to understand in terms of observables, states, complex Feynman amplitudes etc. In the next chapter, we will show the way that leads to the Schrödinger Equation using geometrical language as much as possible. However, we will not use the machinery of geometrical quantization because it treats time simply as a parameter, and space as absolute and given once for all. 9On the other hand, geometrical quantization introduces many advanced tools that are unnecessary for our purposes, while at the same time it lacks the concepts which are important and necessary.
Before entering the subject let me tell you the distinguishing feature of the approach that I am advocating, and that will be sketched below in Ch. 2: one obtains a fibration of Hilbert spaces ${\cal H}=\bigcup {\cal H}
_t$ over time. There is a distinguished family of local trivializations, a family parameterized by
For each $t$, the Hilbert space ${\cal H}_t$ is a Hilbert space of sections of a complex line bundle over $E_t$. A space-time observer (that is, a reference frame) allows us to identify the spaces $E_t$ for different $t$-s, while a $U (1)$ gauge allows us to identify the fibers. Schrödinger's dynamics of a particle in external gravitational and electromagnetic fields is given by a Hermitian connection in ${\cal H}. $ Solutions of Schrödinger equation are parallel sections of ${\cal H}. $ Thus Schrödinger equation can be written as
\nabla \Psi = 0 \end{displaymath} (1)
or, in a local trivialization, as
{\frac{{\partial \Psi (t)}}{{\partial t}}} +
{i\over\hbar}H (t)\Psi (t) =
0, \end{displaymath} (2)
where $H (t)$ will be a self-adjoint operator in ${\cal H}_t$. 10Gravitational and electromagnetic forces are coded into this Schrödinger's connection. Let us discuss the resulting structure. First of all there is not a single Hilbert space but a family of Hilbert spaces. These Hilbert spaces can be identified either using an observer and gauge or,better, by using a background dynamical connection. It is only after so doing that one arrives at single-Hilbert-space picture of the textbook Quantum Mechanics - a picture that is the constant source of lot of confusion. 11
In Quantum Mechanics we have a dual scheme - we use the concepts of observables and states. We often use the word measurement in a mutilated sense of simply pairing an observable $a$ with a state $\phi $ to get the expected result - a number $
<\phi , a>. $
One comment is in place here: to compare the results of actual measurements with predictions of the theory - one needs only real numbers. However experience proved that quantum theory with only-real-numbers is inadequate. So, even if the fundamental role of $\sqrt{-1}$ in Quantum Theory is far from being fully understood - we use in Quantum Theory only complex Hilbert spaces, complex algebras etc.12However, usually, only real numbers are at the end interpreted.
Now, it is not always clear what is understood by states and observables. There are several possibilities:
Figure: There are several possibilities of understanding of state and observables. They can be instant , and thus time-dependent, or they can be time-sections - thus time- independent
\begin{picture}(100, 120)
%\{ special\{psfile=a...
As it was already said, it is usual in standard presentations of the quantum theory to identify the Hilbert spaces ${\cal H}_t$. There are several options there. Either we identify them according to an observer (+ gauge) or according to the dynamics. If we identify according to the actual dynamics, then states do not change in time - it is always the same state-vector, but observables (like position coordinates) do change in time - we have what is called the Heisenberg picture. If we identify them according to some background "free" dynamics - we have so called interaction picture. Or, we can identify Hilbert spaces according to an observer - then observables do not change in time, but state vector is changing - we get the Schrödinger picture.
...uml {o}dinger picture}
\end{array} \right.
However, there is no reason at all to identify the ${\cal H}_t$-s. Then dynamics is given by parallel transport operators:
U_{t, s}: {\sl H}_s\rightarrow {\sl H}_t, \hspace{1cm}s\leq t
U_{t, s}U_{s, r}=U_{t, r}
U_{t, t}=id_{{\cal H}_t}.
Dissipative Dynamics
The Schrödinger equation describes time evolution of pure states of a quantum system, for instance evolution of pure states of a quantum particle, or of a many body system. Even if these states contain only statistical information about most of the physical quantities, the Schrödinger evolution of pure states is continuous and deterministic . Under this evolution Hilbert space vector representing the actual state of the system13 changes continuously with time, and with it there is a continuous evolution of probabilities or potentialities, but nothing happens - the formalism leaves no place for events. Schrödinger equation helps us very little, or nothing at all, to understand how potential becomes real. So, if we aim at understanding of this process of becoming, if we want to describe it by mathematical equations and to simulate it with computers - we must go beyond Schrödinger's dynamics. As it happens, we do not have to go very far - it is sufficient to relax only one (but important) property of Schrödinger's dynamics and to admit that pure states can evolve into mixtures. Instead of Schrödinger equation we have then a so called Liouville equation that describes time evolution of mixed states. It contains Schrödinger equation as a special case.14It was shown in [11] that using the Liouville type of dynamics it is possible to describe coupling between quantum systems and classical degrees of freedom of measurement devices. One can derive also a piecewise deterministic random process that takes place on the manifold of pure states. In this way one obtains a minimal description of "quantum jumps" (or "reduction of wave packets") and accompanying, directly observable jumps of the coupled classical devices. In Ch. 3 simple models of such couplings will be discussed. The interested reader will find more examples in Refs. [8]-[11].15In particular in [11] the most advanced model of this kind, the SQUID-tank model is discussed in details.
Geometry of Schrödinger's Equation
Galilean General Relativity is a theory of space-time structure, gravitation and electromagnetism based on the assumption of existence of an absolute time function. Many of the cosmological models based on Einstein`s relativity admit also a distinguished time function. Therefore Galilean physics is not evidently wrong. Its predictions must be tested by experiments. Galilean relativity is not that elegant as the one of Einstein. This can be already seen from the group structures: the homogeneous Lorentz group is simple, while the homogeneous Galilei group is a semidirect product of the rotation group and of three commuting boosts. Einstein`s theory of gravitation is based on one metric tensor, while Galilean gravity needs both: space metric and space-time connection. Similarly for quantum mechanics: it is rather straightforward to construct generally covariant wave equations for Einstein`s relativity, while general covariance and geometrical meaning of the Schrödinger equation was causing problems, and it was not discussed in textbooks. In the following sections we will present a brief overview of some of these problems.
Galilean General Relativity
Let us discuss briefly geometrical data that are needed for building up generally covariant Schrödinger's equation. More details can be found in Ref. [21].16
Our space-time will be a refined version of that of Galilei and of Newton, i.e. space-time with absolute simultaneouity. Four dimensional space-time $E$ is fibrated over one-dimensional time $B . $ The fibers $E_t$ of $E$ are three-dimensional Riemannian manifolds, while the basis $B$ is an affine space over ${\bf R} . $ By a coordinate system on $E$ we will always mean a coordinate system $x^\mu= (x^0, x^i) , $ $i=1, 2, 3\, , $ adapted to the fibration. That means: any two events with the same coordinate $x^0$ are simultaneous, i.e. in the same fibre of $E . $
Coordinate transformations between any two adapted coordinate systems are of the form:
x^{0'}=x^0+const ,
x^{i'}=x^{i'}\left (x^0, x^{i}\right) .
We will denote by $\beta$ the time form $dx^0 . $ Thus in adapted coordinates $\beta_0=1, \beta_i=0$.
$E$ is equipped with a contravariant degenerate metric tensor which, in adapted coordinates, takes the form
\begin{displaymath}\pmatrix{0 &\vrule height 15pt depth 5pt& \phantom{0 }& 0 &\p...
& \vrule height 15pt depth 5pt&& } \ . \end{displaymath}
where $g^{ij}$ is of signature $ (+++) . $ We denote by $g_{ij}$ the inverse $3\times 3$ matrix. It defines Riemannian metric in the fibers of $E . $
We assume a torsion-free connection in $E$ that preserves the two geometrical objects $g^{\mu\nu}$ and $\beta$. 17The condition $\nabla\beta =0$ is equivalent to the conditions $\Gamma^0_{\mu\nu}=0$ on the connection coefficients. Let us introduce the notation $\Gamma_{\mu\nu, i}=g_{ij}\Gamma^j_{\mu\nu}. $ Then $\nabla g^{\mu\nu}=0$ is equivalent to the equations:
\partial_\mu g_{ij}=\Gamma_{\mu i, j}+\Gamma_{\mu j, i} .
\end{displaymath} (3)
Then, because of the assumed zero torsion, the space part of the connection can be expressed in terms of the space metric in the Levi-Civita form:
\Gamma_{ij, k}={1\over2}\left ( \partial_i g_{jk}+
\partial_j g_{ik}-
\partial_k g_{ij}\right) .
\end{displaymath} (4)
>From the remaining equations:
\partial_0 g_{ij}= \Gamma_{0i, j} + \Gamma_{0j, i}
\end{displaymath} (5)
we find that the $(i,j)$-symmetric part of $\Gamma_{0i, j}$ is equal to ${1\over2}\partial_0 g_{ij} , $ otherwise the connection is undetermined. We can write it, introducing a new geometrical object $\Phi$, as
\Gamma_{i0, j}={1\over2}\left (\partial_0 g_{ij}+\Phi_{ij}\right) ,
\end{displaymath} (6)
\Gamma_{00, j}=\Phi_{0j} ,
\end{displaymath} (7)
where $\Phi_{\mu\nu}=-\Phi_{\mu\nu}$ is antisymmetric. Notice that $\Phi$ is not a tensor, except for pure space transformations or time translations.
The Bundle of Galilei Frames
A basis $e_{\mu}$ in $TE$ is called a Galilei frame if $e_0=\partial_0$, and if $e_i$ are unit space-like vectors. If $e_\mu$ and $\tilde{e}_\mu$ are two Galilei frames at the same space-time point, then they are related by a transformation of the homogeneous Galilei group $G$:
\tilde{e}_0=e_0+{\bf e\cdot v} ,
\end{displaymath} (8)
{\bf\tilde{e}}={\bf e}{\bf\Lambda} ,
\end{displaymath} (9)
where ${\bf v}\in {\bf R}^3$ and ${\bf\Lambda}$ is an orthogonal $3\times 3$ matrix. The bundle of Galilei frames is a principal $G$ bundle.
The Bundle of Velocities
The homogeneous Galilei group $G$ acts on ${\bf R}^3$ in two natural ways: by linear and by affine transformations. The first action is not effective one - it involves only the rotations:
({\bf\Lambda}, {\bf v}): {\bf x}\mapsto {\bf\Lambda x}.
\end{displaymath} (10)
The bundle associated to this action can be identified with the vertical subbundle of $TE$ - i.e. with the bundle $VE$ of vectors tangent to the fibers of $E\rightarrow B$.
$G$ acts also on ${\bf R}^3$ by affine isometries :
({\bf\Lambda}, {\bf v}): {\bf y}\mapsto {\bf\Lambda y}+{\bf v} .
\end{displaymath} (11)
To this action there corresponds an associated bundle, which is an affine bundle over the vector bundle $VE$. It can be identified with the subbundle of $TE$ consisting of vectors $\xi$ tangent to $E$, and such that $\beta (\xi )=1$ or, equivalently, as the bundle of first jets of sections of $E\rightarrow B$. We will call it $J_1E$.
We will denote by $ (x^0, x^{i}, y_0^{i})$ the coordinates in $J_1E$ corresponding to coordinates $x^{\mu}$ of $E$.
The Presymplectic Form
The connection $\Gamma$ can be also considered as a principal connection in the bundle of Galilei frames. It induces an affine connection in the affine bundle $J_1E\stackrel{\pi}{\longrightarrow}E$. As a result, it defines a natural $VE$-valued one-form $\nu_\Gamma$ on $J_1E$. It can be described as follows: given a vector $\xi$ tangent to $J_1E$ at $ (x, y_0)$ it projects onto $d\pi (\xi )$. Then $\nu_\Gamma (\xi) $ is defined as the difference of $\xi$ and the horizontal lift of $d\pi (\xi )$. It is a vertical tangent vector to $J_1E$, and can be identified with an element of $VE$. In coordinates:
\nu_\Gamma^{i}=dy_0^i +\left (\Gamma_{\mu j}^iy_0^j+\Gamma_{\mu
0}^i \right)dx^\mu .
\end{displaymath} (12)
There is another $VE$-valued one-form on $J_1E$, namely the canonical form $\theta$. Given $\xi$ at $ (x, y_0)$, we can decompose $\xi$ into space- and time-component along $y_0$. Then $\theta
(\xi)$ is defined as its space component. In coordinates:
\theta^i=dx^i -y_0^i \, dx^0 .
\end{displaymath} (13)
Then, because the fibers of $VE$ are endowed with metric $g^{mn}$, we can build out the following important two-form $\Omega$ on $J_1E$:
\Omega=g_{mn}\nu_\Gamma^m\wedge\theta^n . \end{displaymath} (14)
\Omega = g_{lm} \left[ dy^{l}_{0} \wedge
\end{array}\end{displaymath} (15)
The following theorem, proven in [21], gives a necessary and sufficient condition for $\Omega$ to be closed.
Theorem 1 The following conditions (i-iii) are equivalent:
$d\Omega=0 , $
$R^{\mu \phantom{\nu}
\sigma}_{\phantom{\mu} \nu \phantom{\sigma} \rho} =
R^{\sigma\phantom{\rho} \mu}_{ \phantom{\sigma} \rho\phantom{\mu} \nu}$ where $R_{\mu\nu \phantom{\sigma} \rho}^{\phantom{\mu \nu} \sigma}$ is the curvature tensor of $\Gamma$ and $R^{\mu\ \phantom{\nu}\sigma}_{\phantom{\mu}
\nu \phantom{\sigma} \rho}
R_{\lambda \nu \phantom{\sigma} \rho}^{\phantom{\mu \nu}\sigma} $
$\partial_{[ \mu} \Phi_{\nu\sigma]}=0 $. 18
Quantum Bundle and Quantum Connection
Let $Q$ be a principal $U (1)$ bundle over $E$ and let $Q^{\uparrow}$ be its pullback to $J_{1}E$. We denote by $P$ and $P^{\uparrow}$ the associated Hermitian line bundles corresponding to the natural action of $U (1)$ on ${\kern .1em {\raise .47ex \hbox
\kern -.4em {\rm C}}$. There is a special class of principal connections on $Q^{\uparrow}$, namely those whose connection forms vanish on vectors tangent to the fibers of $Q^{\uparrow} \to
Q$. As has been discussed in [29] specifying such a connection on $Q^{\uparrow}$ is equivalent to specifying a system of connections on $Q$ parameterized by the points in the fibers of $Q^{\uparrow} \to
Q$. Following the terminology of [29] we call such a connection universal.19The fundamental assumption that leads to the Schrödinger equations reads as follows:
Quantization Assumption: There exists a universal connection $\omega$ $Q^{\uparrow}$ whose curvature is $i\Omega$.20
We call such an $\omega$ a quantum connection. >From the explicit form of $\Omega$ one can easily deduce that $\omega$ is necessarily of the form
\omega = i\left ( d \phi + a_{\mu} dx^{\mu}\right)
where $0\leq \phi \leq 2\pi$ parameterizes the fibres of $Q$,
a_{0} = - {1\over 2} \ y^{2}_{0} + \alpha_{0} , \cr\cr
a_{i} = g_{ij} y^{j}_{0} + \alpha_{i} ,
and $\alpha^{\nu} = \left (\alpha^{0}, \alpha^{i}\right)$ is a local potential for $\Phi$.
Schrödinger's Equation and Schrödinger's Bundle
As it is shown in [21], there exists a natural $U (1)$-invariant metric on $P$ of signature $ (++++-)$. Explicitly
& =\ d\...
...^{i}\otimes dx^{0} + dx^{0}\otimes dx^{i} \right) .
Using this metric we can build out a natural Lagrangian for equivariant functions $\psi : P \rightarrow {\kern .1em {\raise .47ex \hbox
\kern -.4em {\rm C}}$ or, equivalently, for sections of the line bundle $Q$. The Euler-Cartan equation for this Lagrangian will prove to be nothing but the Schrödinger equation. Notice that the action of $U (1)$ group on $P$ defines an Killing vector field for $\mathop{g}\limits^{5}$ which is isotropic. Therefore the above construction can explain why the approach of [19] works.
More precisely, the construction leading to the generally covariant Schrödinger-Pauli equation for a charged spin $1/2$ particle in external gravitational and electromagnetic field can be described as follows.
The contravariant metric $\mathop{g}\limits^{5}{}^{-1}
=\left ( g^{\alpha \beta}\right)$, $\alpha, \beta = 0, 1, 2, 3, 5$,
= \pmatrix{ 0 & 0 & 1 \cr\c...
... -g^{ij}a_{j} \cr\cr
1 & -g^{ij}a_{j} & \b{a}^{2}-2a_{0} }
\ .
\end{displaymath} (16)
can be obtained from the following Clifford algebra of $4\times 4$ complex matrices:
\gamma^{0} = \pmatrix{0 & 0 \cr
1 & 0}
... -2a_{0} & \underline{\sigma} \cdot \underline{a} }
\end{array}\end{displaymath} (17)
One takes then 5-dimensional charged Dirac operator $\gamma^{\alpha}
\nabla_{\alpha}$ and considers spinors that are equivariant with respect to the fifth coordinate $x^{5} = \phi: $
{\partial \psi \over \partial \phi} = - i \psi \ .
\end{displaymath} (18)
This first-order, four-component spinor (called Lévy-Leblond equation in Ref. [24]) equation reduces then easily to the second-order, two-component Schrödinger Pauli equation with the correct Landé factor.21
We finish this section with pointing to the Ref. [21], where Schrödinger's quantization is discussed in details and where a probabilistic interpretation of generally covariant Schrödinger equation is given using the bundle $L^{2} (E_{t})$ of Hilbert spaces. The parallel transport induced by the quantum connection is shown in [21] to be directly related to Feynman amplitudes.
Coupled Quantum and Classical Systems
Replacing Schrödinger's evolution, which governs the dynamics of pure states, by an equation of the Liouville type, that describes time evolution of mixed states, is a necessary step - but it does not suffice for modeling of real world events. One must take, to this end, two further steps. First of all we should admit that in our reasoning, our communication, our description of facts - we are using classical logic. Thus somewhere in the final step of transmission of information from quantum systems to macroscopic recording devices and further, to our senses and minds, a translation between quantum and classical should take place. That such a translation is necessary is evident also when we consider the opposite direction: to test a physical theory we perform controlled experiments. But some of the controls are always of classical nature - they are external parameters with concrete numerical values. So, we need to consider systems with both quantum and classical degrees of freedom, and we need evolution equations that enable communication in both directions, i.e. :
Completely Positive Maps
We begin with a brief recall of relevant mathematical concepts. Let ${\cal A}$ be a $C^{\star}$ - algebra. We shall always assume that ${\cal A}$ has unit $I$. An element $A\in {\cal A}$ is positive, $A\geq 0$, iff it is of the form $B^\star B$ for some $B\in {\cal A}$. Every element of a $C^{\star}$-algebra is a linear combination of positive elements. A linear functional $\phi : {\cal A} \rightarrow {\kern .1em {\raise .47ex \hbox
\kern -.4em {\rm C}}$ is positive iff $A\geq 0$ implies $\phi (A) \geq 0$. Every positive functional on a $C^{\star}$ -algebra is continuous and $\Vert \phi\Vert = \phi (I). $ Positive functionals of norm one are called states. The space of states is a convex set. Its extremal points are called pure states. The canonical GNS construction allows one to associate with each state $\omega$ a representation $
\pi_\omega $ of ${\cal A}$ on a Hilbert space ${\cal H}_\omega$, and a cyclic vector $\Psi_\omega\in{\cal H}_\omega$ such that $ (\Psi_\omega
, \pi_\omega (A)\Psi_\omega ) = \omega (A) , \, A\in {\cal A} . $ Irreducibility of $
\pi_\omega $ is then equivalent to purity of $\omega . $
Quantum theory gives us a powerful formal language and statistical algorithms for describing general physical systems. Physical quantities are coded there by Hermitian elements of a $C^{\star}$-algebra ${\cal A}$ of observables, while information about their values (quantum algorithms deal, in general, only with statistical information) is coded in states of ${\cal A}$ . Pure states correspond to a maximal possible information. For each state $\omega , $ and for each $A=A^{\star }\in {\cal A}$ the (real) number $\omega
(A)$ is interpreted as expectation value of observable $A$ in state $\omega , $ while
\delta _\omega ^2 (A)\doteq \omega ( (A-\omega (A))^2)=\omega
(A^2)- (\omega (A))^2
is the quadratic dispersion of $A$ in the state $\omega . $ It is assumed that repeated measurements of $A$ 22made on systems prepared in a state $\omega$ will give a sequence of values $a_1, \ldots , a_n$ so that approximately ${\frac
1N}\sum_{i=1}^Na_i\approx \omega (A), $ and ${\frac 1N}\sum (a_i)^2- ({\frac 1N
}\sum a_i)^2\approx \delta _\omega ^2. $ If ${\cal A}$ is Abelian, then it is isomorphic to an algebra of functions ${\cal A}\approx C (X). $ Then pure states of ${\cal A}$ are dispersion free - they are parameterized by points $
x\in X$ and we have $\omega _x (A)=A (x). $ This corresponds to a classical theory: all observables mutually commute and maximal possible information is without any statistical dispersion. In the extreme opposition to that is pure quantum theory - here defined as that one when ${\cal A}$ is a factor, that is has a trivial centre. The centre ${\cal Z (A)}$ of a $C^\star$-algebra ${\cal A}$ is defined as ${\cal Z (A)}=\{C\in {\cal A}
: AC=CA\, , \, A\in {\cal A}\}. $ In general ${\kern .1em {\raise .47ex \hbox
\kern -.4em {\rm C}}\cdot I\subset {\cal Z (A)}
{\cal A}. $ If Z (A) = A - we have pure classical theory. If ${\cal
Z (A)}={\kern .1em {\raise .47ex \hbox
\kern -.4em {\rm C}}\cdot I$ - we have pure quantum theory. In between we have a theory with superselection rules. Many physicists believe that the ''good theory'' should be a ''pure quantum'' theory. But I know of no one good reason why this should be the case. In fact, we will see that cases with a nontrivial ${\cal Z (A)}$ are interesting ones. Of course, one can always argue that whenever we have an algebra with a nontrivial centre - it is a subalgebra of an algebra with a trivial one, for instance of $B ({\cal H})$ - the algebra of all bounded operators on some Hilbert space. This is, however, not a good argument - one could argue as well that we do not need to consider different groups as most of them are subgroups of ${\cal U (H)}$ - the unitary group of an infinite dimensional Hilbert space - so why to bother with others?
Let ${\cal A, B}$ be $C^{\star}$-algebras. A linear map $\phi : {\cal A}
\rightarrow {\cal B}$ is Hermitian if $\phi (A^{\star}) = \phi
(A)^{\star }. $ It is positive iff $A\geq 0 , $ $A\in {\cal A}$ implies $
\phi (A)\geq 0 . $ Because Hermitian elements of a $C^{\star}$-algebra are differences of two positive ones - each positive map is automatically Hermitian. Let ${\cal M}_n$ denote the $n$ by $n$ matrix algebra, and let $
{\cal M}_n ({\cal A}) = {\cal M}_n \otimes {\cal A}$ be the algebra of $n\times n$ matrices with entries from ${\cal A}. $ Then $
{\cal M}_n ({\cal A})$ carries a natural structure of a $C^{\star}$-algebra. With respect to this structure a matrix ${\bf A}= (A_{ij})$ from $
{\cal M}_n ({\cal A})$ is positive iff it is a sum of matrices of the form $ (A_{ij}) = (A_i^{\star } A_j ), \, $ $A_i\in {\cal A} . $ If ${\cal A}$ is an algebra of operators on a Hilbert space ${\cal H}$, then $
{\cal M}_n ({\cal A})$ can be considered as acting on ${\cal H}^n \doteq {\cal H}
\otimes C^n = \oplus_{i=1}^n {\cal H} . $ Positivity of ${\bf A}= (A_{ij})$ is then equivalent to $ ({\bf\Psi}, {\bf A} {\bf\Psi} )\geq 0\, , \, {\bf\Psi}
\in {\cal H}^n , $ or equivalently, to $\sum_{i, j} (\Psi_i , A_{ij} \Psi_j
) \geq 0 $ for all $\Psi_1, \ldots , \Psi_n \in {\cal H} . $
A positive map $\phi $ is said to be completely positive or, briefly, CP iff $\phi \otimes id_n
: {\cal A }\otimes {\cal M}_n \rightarrow {\cal B}\otimes {\cal M}_n $ defined by $ (\phi\otimes id_n) (A\otimes M ) = \phi (A)\otimes M , \, M\in {\cal M}_n
$, is positive for all $n=2, 3, \ldots . $ When written explicitly, complete positivity is equivalent to
\begin{displaymath}\sum_{i, j=1}^n B_i^{\star}\phi
(A_i^{\star}A_j)B_j \geq 0 \end{displaymath} (19)
for every $A_1, \ldots , A_n \in {\cal A}$ and $B_1, \ldots , B_n \in {\cal B} . $ In particular every homomorphism of $C^{\star}$ algebras is completely positive. One can also show that if either ${\cal A}$ or ${\cal B}$ is Abelian, then positivity implies complete positivity. Another important example: if ${\cal A}$ is a $C^{\star}$ algebra of operators on a Hilbert space ${\cal H}$, and if $V\in {\cal B} ({\cal H}) , $ then $\phi (A) = VAV^{\star}$ is a CP map $\phi : {\cal A}
\rightarrow \phi ({\cal A}) . $ The celebrated Stinespring theorem gives us a general form of a CP map. Stinespring's construction can be described as follows. Let $\phi : {\cal A}
\rightarrow {\cal B}$ be a CP map. Let us restrict to a case when $\phi (I)=I . $ Let ${\cal B}$ be realized as a norm closed algebra of bounded operators on a Hilbert space ${\cal H}$. One takes then the algebraic tensor product ${\cal A}\otimes
{\cal H}$ and defines on this space a sesquilinear form $ <\, , \, >$ by
\begin{displaymath}<A\otimes \Psi , A^{\prime}\otimes
\Psi^{\prime}>= (\Psi , \phi (A^{\star}A^{\prime} ) \Psi^{\prime}) . \end{displaymath} (20)
This scalar product is then positive semi-definite because of complete positivity of $\phi . $ Indeed, we have
\begin{displaymath}<\sum_i A_i\otimes\Psi_i , \sum_j
A_j\otimes\Psi_j > = \sum_{i, j} (\Psi_i, \phi (A_i^{\star } A_j ) \Psi_j )
\geq 0 . \end{displaymath} (21)
Let ${\cal N}$ denote the kernel of $ <\, , \, >$ . Then ${\cal A}\otimes {\cal H} / {\cal N} $ is a pre-Hilbert space. One defines a representation $\pi$ of ${\cal A}$ on ${\cal A}\otimes
{\cal H}$ by $\pi (A) : \, A^{\prime}\otimes\Psi\longmapsto
AA^{\prime}\otimes\Psi . $ One shows then that ${\cal N}$ is invariant under $
\pi ({\cal A})$, so that $\pi$ goes to the quotient space. Similarly, the map ${\cal H}\ni\Psi \mapsto I\otimes \Psi \in {\cal A}\otimes{\cal H}$ defines an isometry $V : {\cal H}
\rightarrow {\cal A}\otimes {\cal H} / {\cal N} . $ We get then $
\phi (A) = V^{\star}\pi (A) V $ on the completion ${\cal H}_{\phi } $ of $
Theorem 2 (Stinespring's Theorem) Let ${\cal A}$ be a $C^{\star}$-algebra with unit and let $\phi : {\cal A}\rightarrow
{\cal B} ({\cal H})$ be a CP map. Then there exists a Hilbert space ${\cal H}_\phi , $ a representation $\pi_\phi$ of ${\cal A}$ on ${\cal H}_\phi , $ and a bounded linear map $V : {\cal
H}\rightarrow {\cal H}_\phi $ such that
\begin{displaymath}\phi (A) = V^{\star}\pi_\phi
(A) V .
\end{displaymath} (22)
$V$ is an isometry iff $\phi $ is unital i.e. iff $\phi $ maps the unit of ${\cal A}$ into the identity operator of ${\cal H}. $ If ${\cal A}$ and ${\cal H}$ are separable, then ${\cal H}_\phi $ can be taken separable.
The space of CP maps from ${\cal A}$ to ${\cal B} ({\cal H})$ is a convex set. Arveson [2] proved that $\phi $ is an extremal element of this set iff the representation $\pi_\phi$ above is irreducible.
Dynamical Semigroups
A dynamical semigroup on a $C^{\star}$-algebra of operators ${\cal A}$ is a strongly continuous semigroup of CP maps of A into itself. A semigroup $\alpha _t , $ is norm continuous iff its infinitesimal generator $L$ is bounded as a linear map $L: {\cal A}\rightarrow {\cal A} . $ We then have
\alpha _t=\exp (t L) \thinspace , \thinspace t\geq 0 . \end{displaymath} (23)
The right hand side is, in this case, a norm convergent series for all real values of $t, $ however for $t$ negative the maps $\exp (tL): {\cal A}
\rightarrow {\cal A}$, although Hermitian, need not be positive.
Evolution of observables gives rise, by duality, to evolution of positive functionals. One defines $
\alpha ^t (\phi ) (A)=\phi (\alpha _t (A)). $ Then $\alpha _t$ preserves the unit of ${\cal A}$ iff $\alpha ^t$ preserves normalization of states. A general form of a generator of a dynamical semigroup in finite dimensional Hilbert space has been derived by Gorini, Kossakowski and Sudarshan [22], and Lindblad [27] gave a general form of a bounded generator of a dynamical semigroup acting on the algebra of all bounded operators ${\sl B}
({\cal H}). $ It is worthwhile to cite, after Lindblad, his original motivation:
"The dynamics of a finite closed quantum system is conventionally represented by a one-parameter group of unitary transformations in Hilbert space. This formalism makes it difficult to describe irreversible processes like the decay of unstable particles, approach to thermodynamic equilibrium and measurement processes [$\ldots $]. It seems that the only possibility of introducing an irreversible behaviour in a finite system is to avoid the unitary time development altogether by considering non-Hamiltonian systems. "
In a recent series of papers [7,8,9,10] Ph. Blanchard and the present author were forced to introduce dynamical semigroups because of another difficulty, namely because of impossibility of obtaining a nontrivial Hamiltonian coupling of classical and quantum degrees of freedom in a system described by an algebra with a non-trivial centre. We felt that lack of a dynamical understanding of quantum mechanical probabilistic postulates is more than annoying. We also believed that the word ''measurement'' instead of being banned, as suggested by J. Bell [5,6], can be perhaps given a precise and acceptable meaning. We suggested that a measurement process is a coupling of a quantum and of a classical system, where information about quantum state is transmitted to the classical recording device by a dynamical semigroup of the total system . It is instructive to see that such a transfer of information can not indeed be accomplished by a Hamiltonian or, more generally, by any automorphic evolution23. To this end consider an algebra ${\cal A}$ with centre ${\cal Z}. $ Then ${\cal Z}$ describes classical degrees freedom. Let $\omega$ be a state of ${\cal A}, $ then ${\omega}\vert _Z$ denotes its restriction to ${\cal Z}. $ Let $\alpha _t$ be an automorphic evolution of ${\cal A}, $ and denote $\omega _t=\alpha ^t (\omega ). $ Each $\alpha _t$ is an automorphism of the algebra ${\cal A}, $ and so it leaves its centre invariant: $\alpha _t: {\cal Z}\rightarrow {\cal Z}. $ The crucial observation is that, because of that fact, the restriction ${\omega _t}\vert _{{\cal Z}}$ depends only on ${
\omega _0}\vert _{{\cal Z}}, $ as the evolution of states of ${\cal Z}$ is dual to the evolution of the observables in ${\cal Z}. $ This shows that information transfer from the total algebra ${\cal A}$ to its centre ${\cal Z}$ is impossible - unless we use more general, non-automorphic evolutions.
>From the above reasoning it may be seen that the Schrödinger picture, when time evolution is applied to states, is better adapted to a discussion of information transfer between different systems. The main properties that a dynamical semigroup $\alpha ^t$ describing time evolution of states should have are: $\alpha ^t$ should preserve convex combinations, positivity and normalization. One can demand even more - it is reasonable to demand a special kind of stability: that it should be always possible to extend the system and its evolution in a trivial way, by adding extra degrees of freedom that do not couple to our system.24That is exactly what is assured by complete positivity of the maps $\alpha_t. $ One could also think that we should require even more, namely that $\alpha ^t$ transforms pure states into pure states. But to assume that would be already too much, as one can prove that then $\alpha ^t$ must be dual to an automorphic evolution. It appears that information gain in one respect (i.e. learning about the actual state of the quantum system) must be accompanied by information loss in another one - as going from pure states to mixtures implies entropy growth.
We will apply the theory of dynamical semigroup to algebras with a non-trivial centre. In all our examples we will deal with tensor products of ${\cal B} ({\cal H})$ and an Abelian algebra of functions. The following theorem by Christensen and Evans [14] generalizes the results of Gorini, Kossakowski and Sudarshan and of Lindblad to the case of arbitrary $C^{\star}$-algebra.
Theorem 3 (Christensen - Evans) Let $\alpha_t = \exp (L t)$ be a norm-continuous semigroup of CP maps of a $C^{\star}$- algebra of operators ${\cal A}\subset {\cal B} ({\cal
H}) . $ Then there exists a CP map $\phi $ of ${\cal A}$ into the ultraweak closure ${\bar {\cal A}}$ and an operator $K\in {\bar {\cal A}}$ such that the generator $L$ is of the form:
L (A) = \phi (A) + K^{\star }A + AK \, .
\end{displaymath} (24)
We will apply this theorem to the cases of ${\cal A}$ being a von Neumann algebra, and the maps $\alpha _t$ being normal. Then $\phi $ can be also taken normal. We also have ${\bar {{\cal A}}} = {\cal A} , $ so that $K\in {\cal A} . $ We will always assume that $\alpha_t (I) = I $ or, equivalently, that $L (I)=0 . $ Moreover, it is convenient to introduce $
H=i (K-K^{\star })/2 \in {\cal A}, $ then from $L (I)=0$ we get $K+K^{\star
}=-\phi (I) , $ and so $K=-iH-\phi (1)/2 . $ Therefore we have
L (A) = i\left[H, A\right]+\phi (A) -\{ \phi (1) , A\}/2 , \end{displaymath} (25)
where $\{\, , \, \}$ denotes anticommutator. Of particular interest to us will be generators $L$ for which $\phi $ is extremal 25. By the already mentioned result of Arveson [2] this is the case when $\phi $ is of the form
\phi (A)=V^{\star }\pi (A) V \, , \end{displaymath} (26)
where $\pi$ is an irreducible representation of ${\cal A}$ on a Hilbert space ${\cal K} , $ and $V: {\cal H}\rightarrow {\cal K}$ is a bounded operator (it must be, however, such that $V^{\star}{\cal A} V \subset
{\cal A
Coupling of Classical and Quantum Systems
We consider a model describing a coupling between a quantum and a classical system. To concentrate on main ideas rather than on technical details let us assume that the quantum system is described in an $n$-dimensional Hilbert space ${\cal H}_q , $ and that it has as its algebra of observables ${\cal B} ({\cal H}_q) \approx {\cal M}_n . $ Similarly, let us assume that the classical system has only a finite number of pure states ${\cal S}=\{ s_1, \ldots , s_m\} . $ Its algebra of observables ${\cal A}_{cl}$ is then isomorphic to ${\kern .1em {\raise .47ex \hbox
\kern -.4em {\rm C}}^m . $ For the algebra of the total system we take ${\cal A}_{tot}={\cal A}_q \otimes {\cal A}_{cl}$ which is isomorphic to the diagonal subalgebra of ${\cal M}_m ({\cal A}_q) . $ Observables of the total system are block diagonal matrices:
\begin{displaymath}{\bf A}=diag (A_{\alpha}) = \pmatrix{A_1&0&0&\ldots&0\cr
0&0&0&\ldots&A_m\cr}, \end{displaymath}
where $A_{\alpha}, (\alpha=1, \ldots , m)$ are operators in ${\cal H}_q
. $ 26Both ${\cal A}_q$ and ${\cal A}_{cl}$ can be considered as subalgebras of ${\cal A}_{tot}$ consisting respectively of matrices of the form $diag (A, \ldots, A)\, , A\in {\cal A}_q$ and $diag (\lambda_1
I_n, \ldots, \lambda_m I_n)\, , \lambda_{\alpha}\in{\kern .1em {\raise .47ex \hbox
\kern -.4em {\rm C}}. $ States of the quantum system are represented by positive, trace one, operators on ${\cal B} ({\cal H}_q) . $ States of the classical system are $m$-tuples of non-negative numbers $p_1, \ldots, p_m , $ with $\sum_{\alpha} p_{\alpha}=1. $ States of the total system are represented by block diagonal matrices $ \rho =diag (\rho_1, \ldots, \rho_m) , $ with ${\cal B} ({\cal
H}_q)\ni \rho_{\alpha}\geq 0$ and $\sum_{\alpha} Tr (\rho_{\alpha})=1
. $ For the expectation value we have $\rho ({\bf A}) =\sum_{\alpha}
Tr (\rho_\alpha A_\alpha ) . $ Given a state $\rho$ of the total system, we can trace over the quantum system to get an effective state of the classical system $p_\alpha =Tr (\rho_\alpha ) , $ or we can trace over the classical system to get the effective state of the quantum system ${\hat\rho}=\sum_\alpha\rho_\alpha . $ 27
Let us consider dynamics. Since the classical system has a discrete set of pure states, there is no non-trivial and continuous time evolution for the classical system that would map pure states into pure states. As for the quantum system, we can have a Hamiltonian dynamics, with the Hamiltonian possibly dependent on time and on the state of the classical system $H (t) = diag (H (\alpha, t)) . $ As we already know a non-trivial coupling between both systems is impossible without a dissipative term, and the simplest dissipative coupling is of the form $L ({\bf A})=V\pi ({\bf A})V^\star , $ where $\pi$ is an irreducible representation of the algebra ${\cal A}_{tot}$ in a Hilbert space ${\cal H}_\pi , $ and $V: {\cal H}_q\rightarrow{\cal H}_\pi$ is a linear map. It is easy to see that such an $L (A)$ is necessarily of the form:
\begin{displaymath}L ({\bf A})={\bf V}^\star {\bf A} {\bf V} , \end{displaymath}
where ${\bf V}$ is an $m\times m$ block matrix with only one non-zero entry. A more general CP map of ${\cal A}_{tot}$ is of the same form, but with ${\bf V}$ having at most one non-zero element in each of its rows.
Let us now discuss desired couplings in somewhat vague, but more intuitive, physical terms. We would like to write down a coupling that enables transfer of information from quantum to classical system. There may be many ways of achieving this aim - the subject is new and there is no ready theory that fits. We will see however that a naive description of a coupling works quite well in many cases. The idea is that the simplest coupling associates to a property of the quantum system a transformation of the actual state of the classical system.
Properties are, in quantum theory, represented by projection operators. Sometimes one considers also more general, unsharp or fuzzy properties. They are represented by positive elements of the algebra which are bounded by the unit. A measurement should discriminate between mutually exclusive and exhaustive properties. Thus one usually considers a family of mutually orthogonal projections $e_i$ of sum one. With an unsharp measurement one associates a family of positive elements $a_i$ of sum one.
As there is no yet a complete, general, theory of dissipative couplings of classical and quantum systems, the best we can do is to show some characteristic examples. It will be done in the following section. For every example a piecewise deterministic random process will be described that takes place on the space of pure states of the total system 28 and which reproduces the Liouville evolution of the total system by averaging over the process. A theory of piecewise deterministic (PD) processes is described in a recent book by M. H. Davis [15]. Processes of that type, but without a non-trivial evolution of the classical system, were discussed also in physical literature - cf. Refs [13,17,18,20]. 29We will consider Liouville equations of the form
\begin{displaymath}{\dot \rho} (t) = -i[H , \rho (t)] + {\sum}_{i} \left ( V_i
...\star} - {1 \over 2}
\{V_i^{\star} V_i , \rho (t)\}
\right ) ,
\end{displaymath} (27)
where in general $H$ and the $V_i$ can explicitly depend on time. The $V_i$ will be chosen as tensor products $V_i=\sqrt{\kappa} e_i\otimes\phi_i$, where $\phi_i$ act as transformations 30 on classical (pure) states.
Examples of Classical-Quantum Couplings
The Simplest Coupling
First, we consider only one orthogonal projector $e$ on the two-dimensional Hilbert space ${\cal H}_q={\kern .1em {\raise .47ex \hbox
\kern -.4em {\rm C}}^2. $ To define the dynamics we choose the coupling operator $V$ in the following way:
\begin{displaymath}V = \sqrt{\kappa} \pmatrix{0, & e \cr
e, & 0 } . \end{displaymath} (28)
The Liouville equation ([*] ) for the density matrix $\rho = diag (\rho_1, \rho_2)$ of the total system reads now
\displaystyle {{\dot \rho}_1 = }\! &\displ...
...+\kappa (e\rho_1e-{1\over
2}\{e, \rho_2\}). }\cr\cr
\end{array}\end{displaymath} (29)
For this particularly simple coupling the effective quantum state $\hat{\rho}={\pi}_q (\rho )=\rho_1+\rho_2$ evolves independently of the state of the classical system. One can say that here we have only transport of information from the quantum system to the classical one. We have:
\begin{displaymath}{\dot {\hat\rho }} = -i[H, {\hat\rho}]+\kappa
(e{\hat\rho}e-{1\over 2}\{e, {\hat\rho}\}).
\end{displaymath} (30)
The Liouville equation ([*] ) describes time evolution of statistical states of the total system.
Let us describe now a the PD process associated to this equation. Let $T_t$ be a one-parameter semigroup of (non-linear) transformations of rays in ${\kern .1em {\raise .47ex \hbox
\kern -.4em {\rm C}}^2$ given by
\begin{displaymath}T (t)\phi ={\phi (t) \over \Vert \phi (t) \Vert},
\end{displaymath} (31)
\phi (t)=\exp\left ({-iHt-{\kappa\over 2}et}\right)
\phi .
\end{displaymath} (32)
Suppose we start with the quantum system in a pure state $\phi_0$, and the classical system in a state $s_1$ (resp. $s_2$). Then $\phi_0$ starts to evolve according to the deterministic (but non-linear Schrödinger) evolution $T (t)\phi_0 $ until a jump occurs at time $t_1$. The time $t_1$ of the jump is governed by an inhomogeneous Poisson process with the rate function $\lambda (t) = \kappa \Vert e T (t)\phi_0 \Vert^2$. Classical system switches from $s_1$ to $s_2$ (resp. from $s_2$ to $s_1$), while $T (t_1)\phi_0$ jumps to $\phi_1=eT (t_1)\phi_0/\Vert eT (t_1)\phi_0 \Vert$, and the process starts again. With the initial state being an eigenstate of $e$, $e\phi_0=\phi_0$, the rate function $\lambda$ is approximately constant and equal to $\kappa$. Thus ${1/\kappa}$ can be interpreted as the expected time interval between the successive jumps.
More details about this model illustrating the quantum Zeno effect can be found in Ref. [8].
Simultaneous "Measurement" of Several Noncommuting Observables
Using somewhat pictorial language we can say that in the previous example each actualization of the property $e$ was causing a flip in the classical system. In the present example, which is a non-commutative and fuzzy generalization of the model discussed in [7], we consider $n , $ in general fuzzy, properties $a_i=a_i^\star , \, i=1, \ldots , n. $ The Hilbert space ${\cal H}_q$ can be completely arbitrary, for instance $2$-dimensional. We will denote $a_0^2\doteq\sum_{i=1}^n a_i^2 . $ The $a_i$-s need not be projections, and the different $a_i$-s need not to commute. The classical system is assumed to have $n+1$ states $s_0, s_1, \ldots , s_n$ with $s_0$ thought of as an initial, neutral state. To each actualization of the property $a_i$ there will be associated a flip between $s_0$ and $s_i . $ Otherwise the state of the classical system will be unchanged. To this end we take
\begin{displaymath}V_{1} ={\sqrt\kappa}
\pmatrix{0, & a_{1} , &0 , &\ldots , &0...
...&\ldots &\ldots&\ldots\cr
0 , & 0 , &0 , &\ldots , &0 , &0} , \end{displaymath}
\begin{displaymath}V_{2} = {\sqrt\kappa}\pmatrix{0, & 0 , & a_{2} , &\ldots , &0...
\begin{displaymath}V_{n} = {\sqrt\kappa}\pmatrix{0, & 0 , & 0 , &\ldots , &0 ,
... &\ldots , &0 , &0\cr
a_{n} , & 0 , &0 , &\ldots , &0 , &0} . \end{displaymath}
The Liouville equation takes now the following form:
\begin{displaymath}{\dot \rho_0}= -i[H, \rho_0]+
\kappa \sum_{i=1}^n a_i\rho_i a_i-{\kappa\over 2} \{ a_0^2, \rho_0\} ,
\end{displaymath} (33)
\begin{displaymath}{\dot \rho_i}=-i[H, \rho_i]+
\kappa a_i\rho_0 a_i -{\kappa\over 2}\{ a_i^2, \rho_i\} . \end{displaymath} (34)
We will derive the PD process for this example in some more details, so that a general method can be seen. First of all we transpose the Liouville equation so as to get time evolution of observables; we use the formula
\sum_\alpha Tr ({\dot A_\alpha}\rho_\alpha ) = \sum_\alpha
Tr (A_\alpha{\dot \rho_\alpha}) .
\end{displaymath} (35)
In the particular case at hand the evolution equation for observables looks almost exactly the same as that for states:
\begin{displaymath}{\dot A_0}= i[H, A_0]+
\kappa \sum_{i=1}^n a_iA_i a_i-{\kappa\over 2} \{ a_0^2, A_0\} ,
\end{displaymath} (36)
\begin{displaymath}{\dot A_i}=i[H, A_i]+
\kappa\, a_iA_0 a_i -{\kappa\over 2}\{ a_i^2, A_i\} . \end{displaymath} (37)
Each observable ${\bf A}$ of the total system defines now a function $f_{\bf A} (\psi, \alpha )$ on the space of pure states of the total system
f_{\bf A} (\psi, \alpha ) = (\psi , A_\alpha \psi ) .
\end{displaymath} (38)
We have to rewrite the evolution equation for observables in terms of the functions $f_{\bf A} . $ To this end we compute the expressions $ (\psi , {\dot A}_\alpha \psi) . $ Let us first introduce the Hamiltonian vector field $X_H$ on the manifold of pure states of the total system:
(X_H f) (\psi, \alpha)={d\over dt}f (e^{-iHt}\psi )\vert_{t=0}.
\end{displaymath} (39)
Then the terms $ (\psi , i[H, A_\alpha ] \psi )$ can be written as $ (X_H f_{\bf A} ) (\psi, \alpha) . $ We also introduce vector field $X_D$ corresponding to non-linear evolution:
(X_D f) (\psi, \alpha ) = {d\over dt}f
\left ( {
\exp ({-\k...
...exp ({-\kappa t a_\alpha^2}/2)\psi\Vert}}\right) \vert_{t=0} . \end{displaymath} (40)
Then evolution equation for observables can be written in a Davis form:
{d\over dt}f_{\bf A} (\psi ...
(\phi, \beta)-f_{\bf A} (\psi, \alpha)\right) ,
\end{array}\end{displaymath} (41)
where $Q$ is a matrix of measures, whose non-zero entries are:
Q (\psi, 0 ;d\phi, i) = {\Vert a_i\psi\Vert^2\over \Vert a_0...
...eft (
\phi - {a_i\psi \over \Vert a_i\psi\Vert }\right)d\phi ,
\end{displaymath} (42)
Q (\psi, i ;d\phi, 0) = \delta \left (
\end{displaymath} (43)
\lambda (\psi , \alpha )=\kappa \Vert a_\alpha\psi\Vert^2 .
\end{displaymath} (44)
The symbol $\delta \left (\phi - \psi\right)d\phi$ denotes here the Dirac measure concentrated at $\psi . $
We describe now PD process associated to the above semigroup. There are $n$ one-parameter (non-linear) semigroups $T_\alpha (s)$ acting on the space of pure states of the quantum system via
\begin{displaymath}\psi \mapsto T_\alpha (t) \psi ={ W_\alpha (t) \psi \over
\Vert W_\alpha (t)\psi \Vert}, \end{displaymath}
\begin{displaymath}W_\alpha (t)=\exp[-iHt-{\kappa\over 2} a_\alpha^2 t]. \end{displaymath}
If initially the classical system is in a pure state $\alpha$, and quantum system in a pure state $\psi$, then quantum system evolves deterministically according to the semigroup $T_\alpha$: $\psi (t)\mapsto T_\alpha (t)\psi$. The classical system then jumps at the time instant $t_1$, determined by the inhomogeneous Poisson process with rate function $\lambda_\alpha = \lambda(\psi,\alpha)$. If the classical system was in one of the states $j=1, 2, \ldots , n$, then it jumps to $0$ with probability one, the quantum state jumps at the same time to the state $a_j\psi (t_1) / \Vert a_j\psi (t_1)\Vert$. If, on the other hand, it was in the state $0$, then it jumps to one of the states $j$ with probability $\Vert a_j\psi (t_1)\Vert^2/\Vert a_0 \psi (t_1)\Vert^2$. The quantum state jumps at the same time to $a_j\psi (t_1)/\Vert a_j\psi (t_1)\Vert. $ Let
\begin{displaymath}F_\alpha (t)=\exp[-\int_0^t \lambda_\alpha (T_\alpha (s)\psi)ds]. \end{displaymath}
Then $F_\alpha$ is the distribution of $t_1$ - the first jump time. More precisely, $F_\alpha (t)$ is the survival function for the state $\alpha$:
\begin{displaymath}F_\alpha (t) = P[t_1> t]. \end{displaymath}
Thus the probability distribution of the jump is $p (t)=-dF_\alpha (t)/dt , $ and the expected jump time is $\int_0^{+\infty } t\,
p (t) dt. $ The probability that the jump will occur between $t$ and $t+dt$, provided it did not occur yet/, is equal to $1-\exp\left(\int_t^{t+dt}\lambda_{\alpha}(s)ds\right)\approx
\lambda_{\alpha}(t)dt$. Notice that this depends on the actual state $(\psi,\alpha)$. However, as numerical computation show, the dependence is negligible and approximately jumps occur always after time $t_1=1/\kappa$. 31
Coupling to All One-Dimensional Projections
In the previous example the coupling between classical and quantum systems involved a finite set of non-commuting observables. In the present one we will go to the extreme - we will use all one-dimensional projections in the coupling. One can naturally discover such a model when looking for a precise answer to the question:
how to determine state of an individual quantum system ?
For some time I was sharing the predominant opinion that a positive answer to this question can not be given, as there is no observable to be measured that answers the question: what state our system is in ?. Recently Aharonov and Vaidman [1] discussed this problem in some details. 32The difficulty here is in the fact that we have to discriminate between non-orthogonal projections (because different states are not necessarily orthogonal), and this implies necessity of simultaneous measuring of non-commuting observables. There have been many papers discussing such measurements, different authors taking often different positions. However they all seem to agree on the fact that predictions from such measurements are necessarily fuzzy. This fuzziness being directly related to the Heisenberg uncertainty relation for non-commuting observables. Using methods and ideas presented in the previous sections of this chapter it is possible to build models corresponding to the intuitive idea of a simultaneous measurement of several non-commuting observables, like, for instance, different spin components, positions and momenta etc. A simple example of such a model was given in the previous section. After playing for a while with similar models it is natural to think of a coupling between a quantum system and a classical device that will result in a determination of the quantum state by the classical device. Ideally, after the interaction, the classical "pointer" should point at some vector in a model Hilbert space. This vector should represent (perhaps, with some uncertainty) the actual state of the quantum system. The model that came out of this simple idea, and which we will now discuss, does not achieve this goal. But it is instructive, as it shows that models of this kind are possible. I believe that one day somebody will invent a better model, a model that can be proven to be optimal, giving the best determination with the least disturbance. Then we will learn something important about the nature of quantum states.
Our model will be formulated for a $2$-state quantum system. It is rather straightforward to rewrite it for an arbitrary $n$-state system, but for $n=2$ we can be helped by our visual imagination. Thus we take ${\cal H}_q={\kern .1em {\raise .47ex \hbox
\kern -.4em {\rm C}}^2$ for the Hilbert space of our quantum system. We can think of it as pure spin $1/2$. Pure states of the system form up the manifold ${\cal S}_q\equiv{\kern .1em {\raise .47ex \hbox
\kern -.4em {\rm C}}P^2$ which is isomorphic to the $2$-sphere $S^2=\{ {\bf n}\in {\bf R}^3: {\bf n}^2=1\}$. Let ${\bf\sigma}=\{\sigma_i\}$, $i=1, 2, 3$ denote the Pauli $\sigma$-matrices. Then for each ${\bf n}\in S^2$ the operator $\sigma ({\bf n})={\bf\sigma\cdot n}$ has eigenvalues $\{+1, -1\} $. We denote by $e ({\bf n})= (I+\sigma ({\bf n}))/2$ the projection onto the $+1$-eigenspace.
For the space ${\cal S}_{cl}$ of pure states of the classical system we take also $S^2$ - a copy of ${\cal S}_q$. Notice that $S^2$ is a homogeneous space for $U (2)$. Let $\mu$ be the $U (2)$ invariant measure on $S^2$ normalized to $\mu (S^2)=1$. In spherical coordinates we have $d\mu=sin (\theta)d\phi d\theta /4\pi$. We denote ${\cal H}_{tot}=L^2 ({\cal S}_{cl}, {\cal H}_q, d\mu )$ the Hilbert space of the total system, and by ${\cal A}_{tot}=L^{\infty} ({\cal S}_{cl},
{\cal L} ({\cal H}_q), d\mu )$ its von Neumann algebra of observables. Normal states of ${\cal A}_{tot}$ are of the form
\begin{displaymath}\rho: {\bf A}\mapsto\int Tr (A ({\bf n})\rho ({\bf n}))d\mu ({\bf n}), \end{displaymath}
where $\rho\in L^{\infty} ({\cal S}_{cl}, {\cal L} ({\cal H}_q), d\mu )$ satisfies
\begin{displaymath}\rho ({\bf n})\geq 0 , \, {\bf n}\in {\cal S}_{cl} , \end{displaymath}
\begin{displaymath}\int Tr \left (\rho ({\bf n})\right) d\mu ({\bf n}) =1 . \end{displaymath}
We proceed now to define the coupling of the two systems. There will be two constants: The idea is that if the quantum system is at some pure state ${\bf n}_q$, and if the classical system is in some pure states ${\bf n}_{cl}$, then ${\bf n}_{cl}$ will cause the Hamiltonian rotation of ${\bf n}_{q}$ around ${\bf n}_{cl}$ with frequency $\omega$, while ${\bf n}_q$ will cause, after a random waiting time $t_1$ proportional to $1/\kappa$, a jump, along geodesics, to the "other side" of ${\bf n}_q$. The classical transformation involved is nothing but a geodesic symmetry on the symmetric space ${\kern .1em {\raise .47ex \hbox
\kern -.4em {\rm C}}P^2=U (2)/ (U (1)\times U (1))$. It has the advantage that it is a measure preserving transformation. It has a disadvantage because ${\bf n}_{cl}$ overjumps ${\bf n}_q$.
We will use the notation ${\bf n} ({\bf n}')$ to denote the $\pi$ rotation of ${\bf n}'$ around ${\bf n}$. Explicitly:
\begin{displaymath}{\bf n} ({\bf n}')=2 ({\bf n\cdot n}'){\bf n}-{\bf n}' . \end{displaymath}
For each ${\bf n}$ we define $V_{\bf n}\in {\cal L} ({\cal H}_{tot})$ by
\left ( V_{\bf n}\Psi\right)
({\bf n}')=\sqrt{\kappa} e ({\bf n})\Psi ({\bf n} (
{\bf n}')).
\end{displaymath} (45)
Using $V_{\bf n}$-s we can define Lindblad-type coupling between the quantum system and the classical one. To give our model more flavor, we will introduce also a quantum Hamiltonian that depends on the actual state of the classical system; thus we define
\left ( H\Psi\right) ({\bf n})=
H ({\bf n})\Psi ({\bf n}) = {\omega\over2}\sigma ({\bf n})\Psi ({\bf n}).
\end{displaymath} (46)
Our coupling is now given by
{\cal L}_{cq} \rho = -i[H, \rho ] +\int_{{\cal S}_{cl}} \lef...
...\bf n}}^{\star} V_{\bf n}, \rho\right\}\right) d\mu ({\bf n}).
\end{displaymath} (47)
Notice that ${V_{\bf n}}^{\star}=
V_{\bf n}$ and $V_{\bf n}^2=\kappa\, e ({\bf n})$. Now, $\int e ({\bf n}) d\mu ({\bf n})$ being $U (2)$-invariant, it must be proportional to the identity. Taking its trace we find that
\begin{displaymath}\int e ({\bf n}) d\mu ({\bf n}) = I/2 , \end{displaymath}
and therefore
{\cal L}_{cq}\rho=-i[H, \rho]+\int V_{\bf n}\rho V_{\bf n}\, d\mu ({\bf n}) -{\kappa\over 2} \rho .
\end{displaymath} (48)
Explicitly, using the definition of $V_{\bf n}$, we have
\left ({\cal L}_{cq}\rho\right) ({\bf n}) =
...)e ({\bf n'}) d\mu ({\bf n'}) -{\kappa\over 2} \rho ({\bf n}).
\end{displaymath} (49)
Notice that for each operator $a\in{\cal L} ({\cal H}_q)$ we have the following formula: 33
\int e ({\bf n})\, a\, e ({\bf n}) d\mu ({\bf n})={1\over 6} (a+Tr (a)I).
\end{displaymath} (50)
If $\omega=0$, that is if we neglect the Hamiltonian part, then using this formula we can integrate over $ {\bf n'}$ to get the effective Liouville operator for the quantum state ${\hat\rho}=\int \rho ({\bf n}) d\mu ({\bf n})$:
{\cal L}_{cq}{\hat\rho}={\kappa\over6}\left (I-2{\hat\rho}\right),
\end{displaymath} (51)
with the solution
{\hat\rho} (t)=\exp\left ({-\kappa t}/3\right)\rho (0)+{1-\exp\left ({-\kappa
t}/3\right)\over 2}I.
\end{displaymath} (52)
It follows that, as the result of the coupling, the effective quantum state undergoes a rather uninteresting time-evolution: it dissipates exponentially towards the totally mixed state ${I\over2}$, and this does not depend on the initial state of the classical system.
Returning back to the case of non-zero $\omega$ we discuss now the piecewise deterministic random process of the two pure states ${\bf n}_q$ and ${\bf n}_{cl}$. To compute it we proceed as in the previous example, with the only change that now pure states of the quantum and of the classical system are parameterized by the same set - $S^2$ in our case. To keep track of the origin of each parameter we will use subscripts as in ${\bf n}_{cl}$ and ${\bf n}_{q}$. As in the previous example each observable ${\bf A}$ of the total system determines a function $f_{\bf A}: S^2\times S^2\rightarrow{\kern .1em {\raise .47ex \hbox
\kern -.4em {\rm C}}$ by
\begin{displaymath}f_{\bf A}\left ({\bf n}_q, {\bf n}_{cl}\right)=Tr\left (e\left ({\bf n}_q\right)A\left ({\bf n}_{cl}\right)\right). \end{displaymath}
The Liouville operator ${\cal L}_{cq}$, acting on observables, can be then rewritten it terms of the functions $f_{\bf A}$:
\left ({\cal L}_{cq}f_{\bf A}\right)\left...
... 2} f_{\bf A}\left ({\bf n}_q, {\bf n}_{cl}\right),
\end{array}\end{displaymath} (53)
where $X_H$ is the Hamiltonian vector field
(X_H f) ({\bf n}_q, {\bf n}_{cl})={d\over dt}f\left (e^{-iH ({\bf n}_{cl})t}
\cdot {\bf n}_{q}
\right)\bigg\vert_{t=0} \ ,
\end{displaymath} (54)
p ({\bf n}, {\bf n'}) = Tr\left (e\left ({\bf n}\right)e\left ({\bf n'}\right)\right)= (1+{\bf n\cdot n'})/2
\end{displaymath} (55)
is known as the transition probability between the two quantum states.
The PD process on ${\cal S}_q\times {\cal S}_{cl}$ can now be described as follows. Let ${\bf n}_q (0)$ and ${\bf n}_{cl} (0)$ be the initial states of the quantum and of the classical system. Then the quantum system evolves unitarily according to the quantum Hamiltonian $H ({\bf n}_{cl})$ until at a time instant $t_1$ a jump occurs. The time rate of jumps is governed by the homogeneous Poisson process with rate $\kappa/2$. The quantum state ${\bf n}_q (t_1)$ jumps to a new state ${{\bf n}'}_q$ with probability distribution $p ({\bf n}_q (t_1), {{\bf n}'}_q)$ while ${\bf n}_{cl}$ jumps to ${{\bf n}'}_q ({\bf n}_q (t_1))$ and the process starts again (see Fig. 2).
Figure: The quantum state ${\bf n}_{q} (t_{1})$ jumps to a new state ${\bf n}^{\prime}_{q}$ with probability distribution $p ({\bf n}_{q} (t_{1}), {\bf n}^{\prime}_{q}$ while ${\bf n}_{cl}$ jumps to ${\bf n}^{\prime} ({\bf n}_{q} (t_{1}))$.
%%%%%%%%%%%%%%%%%% put (80, 190)\{\}
...25, -50)
%%% put (0, 40)
Acknowledgements: This paper is partially based on a series of publications that were done in a collaboration with Ph. Blanchard and M. Modugno. Thanks are due to the Humboldt Foundation and Italian CNR that made this collaboration possible. The financial support for the present work was provided by the Polish KBN grant No PB 1236 on one hand, and European Community program "PECO" handled by the Centre de Physique Theorique (Marseille Luminy) on the other. Parts of this work have been written while I was visiting the CPT CNRS Marseille. I would like to thank Pierre Chiapetta and all the members of the Lab for their kind hospitality. Thanks are due to the Organizers of the School for invitation and for providing travel support. I would like to express my especially warm thanks to Robert Coquereaux for his constant interest, criticism and many fruitful discussions, and for critical reading of much of this text. Thanks are also due to Philippe Blanchard for reading the paper, many discussions and for continuous, vital and encouraging interaction.
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About this document ...
Topics in Quantum Dynamics
The command line arguments were:
latex2html -split 0 -local_icons 9406204.tex
... isolated1
Emphasized style will be used in these notes for concepts that are important, but will not be explained. Sometimes explanation would need too much space, but sometimes because these are either primitive or meta-language notions.
... observable.2
Lüders [28] noticed that this formulation is ambiguous in case of degenerate eigenvalues, and generalized it to cover also this situation.
... theory.3
In these lectures, "quantum theory" usually means "quantum mechanics", although much of the concepts that we discuss are applicable also to systems with infinite number of degrees of freedom and in particular to quantum field theory.
... other4
Including the author of these notes.
... relativity,5
There exists however so called "relativistic Fock-Schwinger proper time formalism" [23, Ch. 2-5-4] where one writes Schrödinger equation with Hamiltonian replaced by "super-Hamiltonian, and time replaced by "proper time"
... day.6
One could try to "explain" time by saying that there is a preferred time direction selected by the choice of a thermal state of the universe. But that is of no help at all, until we are told how it happens that a particular thermal state is being achieved.
... relativity.7
In group theoretical terms: the proper Lorentz group is simple, while the proper Galilei group is not.
... processes8
The paradigm may however change in no so distant future - we may soon try to understand the Universe as a computing machine, with geometry replaced by geometry of connections, and randomness replaced by a variant of algorithmic complexity.
... all.9
Cf. e. g. Ref. [33, Ch. 9].
.... 10
A similar idea was mentioned in [3]. For a detailed description of all the constructions - see the forthcoming book [21]
... confusion.11
Notable exceptions can be found in publications from the Genevé school of Jauch and Piron.
... etc.12
Quaternionic structures, on the other hand, can be always understood as complex ones with an extra structure - they are unnecessary.
... system13
Some physicists deny "objectivity" of quantum states - they would say that Hilbert space vectors describe not states of the system, but states of knowledge or information about the system. In a recent series of papers (see [1] and references therein) Aharonov and Vaidman [1] attempt to justify objectivity of quantum states. Unfortunately their arguments contain a loophole.
... case.14
It should be noted, however, that Schrödinger equation describes evolution of state vectors. and thus contains direct information about phases. This information is absent in the Liouville equation, and its restoration (e. g. as it is with the Berry phase) may sometimes create a non-trivial task.
Cf. also the recent (June 1994) paper "Particle Tracks, Events and Quantum Theory", by the author.
The reader may also consult [19], where a different approach, using dimensional reduction along a null Killing vector, is discussed.
.... 17
Some of these assumptions are superfluous as they would follow anyhow from the assumption $d\Omega=0$ in the next paragraph.
.... 18
Notice that because $\Phi$ is not a tensor, the last condition need not be, a priori, generally covariant.
... universal.19
i.e. universal for the system of connections.
... $i\Omega$.20
We choose the physical units in such a way that the Planck constant $\hbar$ and mass of the quantum particle $m$ are equal to 1.
... factor.21
For an alternative detailed derivation see [12]
... 22
Our point is that "measurement" is an undefined concept in standard quantum theory, and that the probabilistic interpretation must be, because of that, brought from outside. What we propose is to define measurement as a CP semigroup coupling between a classical and a quantum system and to derive the probabilistic interpretation of the quantum theory from that of the classical one.
... evolution23
For a discussion of this fact in a broader context of algebraic theory of superselection sectors - cf. Landsman [25, Sec. 4. 4]. Cf. also the no-go result by Ozawa [31]
... system.24
That requirement is also necessary to guarantee physical consistency of the whole framework, as we always neglect some degrees of freedom as either irrelevant or yet unknown to us.
... extremal25
It should be noticed, however, that splitting of $L$ into $\phi $ and $K$ is, in general, not unique - cf. e. g. Refs [15] and [32, Ch. III. 29-30].
... 26
It is useful to have the algebra ${\cal A}_{tot}$ represented in such a form, as it enables us to apply the theorem of Christensen-Evans.
... 27
One can easily imagine a more general situation when tracing over the classical system will not be meaningful. This can happen if we deal with several phases of the quantum system, parameterized by the classical parameter $\alpha$. It may then happen that the total algebra is not the tensor product algebra. For instance, instead of one Hilbert space ${\cal H}_q , $ we may have, for each value of $\alpha , $ a Hilbert space ${\cal H}_{q, \alpha}$ of dimension $n_\alpha$ .
... system28
One may wonder what does that mean mathematically, as the space of pure states of a $C^{\star}$ algebra is, from measure-theoretical point of view, a rather unpleasant object. The answer is that the only measures on the space of pure states of the quantum algebra will be the Dirac measures.
...car,dal,dum,gar. 29
Thanks are due to N. Gisin for pointing out these references.
... transformations30
Or, more precisely, as Frobenius-Perron operators. Cf. Ref.[26] for definition and examples of Frobenius-Perron and dual to them Koopman operators.
.... 31
This sequence of transformation on the space of pure states of the quantum system can be thought of as a nonlinear version of Barnsley's Iterated Function System (cf. e. g. [4]
... details.32
I do not think that they found the answer, as their arguments are circular, and they seem to be well aware of this circularity.
... formula:33
The formula is easily established for $a$ of the form $e ({\bf n'})$, and then extended to arbitrary operators by linearity. |
0a041aa2ed32d84c | My problem involves the solution of a second-order ODE with a fixed-step (input and output). Specifically, this ODE is the radial part of Dirac and Schrödinger equation for a spherical symmetric potential.
This is for example the ODE of the radial part Schrödinger equation:
$$\left(\frac{d^2}{dr^2} +2r\frac{d}{dr} -\frac{l(l+1)}{r^2} \right) R(r)+V(r)P(r)=E\,R(r)$$
or by substituing R(r) = P(r)*r:
$$\left(\frac{d^2}{dr^2}-\frac{l(l+1)}{r^2} \right) P(r)+V(r)P(r)=E\,P(r)$$
$V(r)$ is defined on a fixed grid, that why I need a fixed-step solver. $E$, $l$ are numerical predefined values. $r$ can have values in the interval from $[0,\infty]$, but usually it is only interesting up to an certain $r_{max}$. The initial condition are $P(0) = 0$ and $P'(0) = 0$.
Commonly used solvers are multistep methods like Adams–Bashfort or also extrapolation methods like the GBS method. I already have some implementation exactly using these aforementioned methods, which were specifically created to deal with this certain problem.
For comparison, I am searching for a software package or library, which has some evolved fixed-step solver implemented.
Are there libraries that contain general-purpose fixed-step ODE solvers?
• $\begingroup$ Could you specify the range of $r$ ? Any boundary/initial conditions ? Even if $V(r)$ is known only at some points, you can construct an interpolant of this function and evaluate it anywhere you want. You do not have to use the same step size as the data of $V(r)$. $\endgroup$ – cfdlab Dec 5 '19 at 16:19
• $\begingroup$ I tried this approach too. Interpolating the potential piecewise and then solve it. Actually one of my solvers is doing the same thing too. But the question arises, if this makes it actually better. $\endgroup$ – Franco M. Dec 5 '19 at 16:41
I think DifferentialEquations.jl in Julia has a very comprehensive suite of ODE solvers, including the ones you mentioned (Adams-Bashfort and GBS) and many others. This Julia library is becoming more and more popular nowadays, is well documented, and has quite the coverage here.
Note: Chris Rackauckas is a core contributor to this project and is pretty active on Computational Science SE, and he should be able to answer really in-depth questions if those arise.
• 1
$\begingroup$ Yes, if you do solve(prob,alg,adaptive=false,dt=x) it'll turn any native Julia method into one that's doing fixed time stepping. So all of the methods mentioned here are all available with fixed time stepping via the OrdinaryDiffEq.jl module of DifferentialEquations.jl. Whether it's a good idea is a different story, but you can do it! I'd like to see if in your specific case if it does better than interpolating. $\endgroup$ – Chris Rackauckas Dec 5 '19 at 16:40
• $\begingroup$ Interpolation can be quite tricky. I have only tried cubic splines so far. But this didn't give the best results. $\endgroup$ – Franco M. Dec 5 '19 at 16:50
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0b095f33a0a5cf57 | When developing algorithms in quantum computing, I've noticed that there are two primary models in which this is done. Some algorithms - such as for the Hamiltonian NAND tree problem (Farhi, Goldstone, Guttman) - work by designing a Hamiltonian and some initial state, and then letting the system evolve according to the Schrödinger equation for some time $t$ before performing a measurement.
Other algorithms - such as Shor's Algorithm for factoring - work by designing a sequence of Unitary transformations (analogous to gates) and applying these transformations one at a time to some initial state before performing a measurement.
My question is, as a novice in quantum computing, what is the relationship between the Hamiltonian model and the Unitary transformation model? Some algorithms, like for the NAND tree problem, have since been adapted to work with a sequence of Unitary transformations (Childs, Cleve, Jordan, Yonge-Mallo). Can every algorithm in one model be transformed into a corresponding algorithm in the other? For example, given a sequence of Unitary transformations to solve a particular problem, is it possible to design a Hamiltonian and solve the problem in that model instead? What about the other direction? If so, what is the relationship between the time in which the system must evolve and the number of unitary transformations (gates) required to solve the problem?
I have found several other problems for which this seems to be the case, but no clear cut argument or proof that would indicate that this is always possible or even true. Perhaps it's because I don't know what this problem is called, so I am unsure what to search for.
• 3
$\begingroup$ Every polynomial-time algorithm in one corresponds to a polynomial-time algorithm in the other, but it's not clear the degree of the polynomial will be the same. Hopefully somebody will come up with references. These results were proved in the early days of quantum computation, and there should be better proofs of these theorems now. $\endgroup$ – Peter Shor Jul 6 '14 at 1:25
• $\begingroup$ does this relate to what is known as the Heisenberg vs Schroedinger picture of QM which relates to how the operators are defined? also if it isnt covered in Nielsen & Chuang then that would seem to be a major oversight! the NAND tree paper uses "hamiltonian oracles" which seem to be introduced by Farhi/Gutmann 1998. here is a nice survey article on Hamiltonian oracles by Mochon 2007 $\endgroup$ – vzn Jul 6 '14 at 15:58
• $\begingroup$ The book link you provided is actually the textbook we used in my undergraduate course in Quantum Information Processing. The book is really geared towards the Unitary approach (within the context of oracles as well), but not so much in the context of Hamiltonians. My undergrad course was focused from a cs perspective and not a physics perspective, which is why I am most familiar with the Unitary model. $\endgroup$ – user340082710 Jul 8 '14 at 15:47
• $\begingroup$ The paper you provided as well is a good reference in general, but I don't believe it addresses my question either. Lastly, I've taken a look at the Heisenberg vs Schroedinger picture of QM, and it does look related, but I believe my question is different (though I could be wrong - It was a hard to follow the Wikipedia entries). $\endgroup$ – user340082710 Jul 8 '14 at 15:49
• $\begingroup$ I think there are different ways to interpret your question and instead of answering all interpretations, I'd like to ask you the following: Could you be more precise about the version of the Hamiltonian model you have in mind? What is the measure of complexity in this model? (i.e., what is it that counts how difficult it is to solve a problem in the Hamiltonian model?) How is the input to the problem given? Is it given explicitly or do you have to query the input via an oracle? $\endgroup$ – Robin Kothari Jul 10 '14 at 0:27
To show that Hamiltonian evolution can simulate the circuit model, one can use the paper Universal computation by multi-particle quantum walk, which shows that a very specific kind of Hamiltonian evolution (multi-particle quantum walks) is BQP complete, and thus can simulate the circuit model.
Here is a survey paper on simulating quantum evolution on a quantum computer. One can use the techniques in this paper to simulate the Hamiltonian evolution model of quantum computers. To do this, one needs to use "Trotterization", which substantially decreases the efficiency of the simulation (although it only introduces a polynomial blowup in computation time).
• $\begingroup$ Thanks! These references look quite good and should be able to give me an idea of how this is done. $\endgroup$ – user340082710 Jul 10 '14 at 20:10
Your Answer
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bbac5fc78e9775be | Schrödinger Equation for a Dirac Bubble Potential
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The Schrödinger equation has been solved in closed form for about 20 quantum-mechanical problems. This Demonstration describes one such example published some time ago. A particle moves in a potential that is zero everywhere except on a spherical bubble of radius , drawn as a red circle in the contour plots. This result has been applied to model the buckminsterfullerene molecule and also to approximate the interatomic potential in the helium van der Waals dimer .
The relevant Schrödinger equation is given by , in units with , and in bohrs, and in hartrees. For , the equation has separable continuum solutions , where the are spherical harmonics. The radial function has the form for and for . Here and are spherical Bessel functions and the are phase shifts. For each value of , a single bound state will exist, provided that . The bound-state radial function is , where and are the greater and lesser of and , and is a Hankel function. The energy is given by , with determined by the transcendental equation . Both the bound and continuum wavefunctions are continuous at but have discontinuous first derivatives. The produces a deltafunction in the second derivative.
This Demonstration shows plots of the radial functions and a cross section of the density plots of for . The wavefunction is positive in the blue regions and negative in the white regions. Be cautioned that the density plots might take some time to complete.
Contributed by: S. M. Blinder (March 2011)
Open content licensed under CC BY-NC-SA
Snapshot 1: contour plot of a continuum state
Snapshot 2: radial function for a bound state
Snapshot 3: contour plot of a bound state
Reference: S. M. Blinder, "Schrödinger Equation for a Dirac Bubble Potential," Chemical Physics Letters, 64(3), 1979 pp. 485–486.
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8e97ea13e3d50bbd | Bose–Einstein condensate
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A Bose–Einstein condensate (BEC) is a state of matter of bosons confined in an external potential and cooled to temperatures very near to absolute zero (0 K or −273.15 °C). Under such supercooled conditions, a large fraction of the atoms collapse into the lowest quantum state of the external potential, at which point quantum effects become apparent on a macroscopic scale.
This state of matter was first predicted by Satyendra Nath Bose in 1925. Bose submitted a paper to the Zeitschrift für Physik but was turned down by the peer review. Bose then took his work to Einstein who recognized its merit and had it published under the names Bose and Einstein, hence the acronymn.
Seventy years later, the first gaseous condensate was produced by Eric Cornell and Carl Wieman in 1995 at the University of Colorado at Boulder NIST-JILA lab, using a gas of rubidium atoms cooled to 170 nanokelvin (nK)[1] (1.7×10−7 K or −273.14999983 °C). Eric Cornell, Carl Wieman and Wolfgang Ketterle at MIT were awarded the 2001 Nobel Prize in Physics in Stockholm, Sweden[2].
"Condensates" are extremely low-temperature fluids which contain properties and exhibit behaviors that are currently not completely understood, such as spontaneously flowing out of their containers. The effect is the consequence of quantum mechanics, which states that systems can only acquire energy in discrete steps. If a system is at such a low temperature that it is in the lowest energy state, it is no longer possible for it to reduce its energy, not even by friction. Without friction, the fluid will easily overcome gravity because of adhesion between the fluid and the container wall, and it will take up the most favorable position, all around the container.[citation needed]
Bose-Einstein condensation is an exotic quantum phenomenon that was observed in dilute atomic gases for the first time in 1995, and is now the subject of intense theoretical and experimental study.
The slowing of atoms by use of cooling apparatuses produces a singular quantum state known as a Bose condensate or Bose–Einstein condensate. This phenomenon was predicted in 1925 by generalizing Satyendra Nath Bose's work on the statistical mechanics of (massless) photons to (massive) atoms. (The Einstein manuscript, believed to be lost, was found in a library at Leiden University in 2005.[3]) The result of the efforts of Bose and Einstein is the concept of a Bose gas, governed by the Bose–Einstein statistics, which describes the statistical distribution of identical particles with integer spin, now known as bosons. Bosonic particles, which include the photon as well as atoms such as helium-4, are allowed to share quantum states with each other. Einstein demonstrated that cooling bosonic atoms to a very low temperature would cause them to fall (or "condense") into the lowest accessible quantum state, resulting in a new form of matter.
is the critical temperature,
is the particle density,
is the mass per boson,
is Planck's constant,
is the Boltzmann constant, and
is the Riemann zeta function; Template:OEIS
Einstein's Argument
Consider a collection of N noninteracting particles which can each be in one of two quantum states, |0> and |1>. If the two states are equal in energy, each different configuration is equally likely.
If we can tell which particle is which, there are different configurations, since each particle can be in |0> or |1> independently. In almost all the configurations, about half the particles are in |0> and the other half in |1>. The balance is a statistical effect--- the number of configurations is largest when the particles are divided equally.
If the particles are indistinguishable, however, there are only N+1 different configurations. If there are K particles in state |0>, there are N-K particles in state |1>. Whether any particular particle is in state |0> or in state |1> can't be determined, so each value of K determines a unique quantum state for the whole system. If all these states are equally likely, there is no statistical spreading out--- it is just as likely for all the particles to sit in |0> as for the particles to be split half and half.
Supposing now that the energy of state |1> is slightly greater than the energy of state |0> by an amount E. At temperature T, a particle will have a lesser probability to be in state |1> by exp(-E/T). In the distinguishable case, the particle distribution will be biased slightly towards state |0> and the distribution will be slightly different from half and half. But in the indistinguishable case, since there is no statistical pressure toward equal numbers, the most likely outcome is that all the particles will collapse into state |0>.
In the distinguishable case, for large N, the fraction in state |0> can be computed. It is the same as coin flipping with a coin which has probability p=exp(-E/T) to land tails. The fraction of heads is 1/(1+p), which is a smooth function of p, of the energy.
For large N, the normalization constant C is (1-p). The expected total number of particles which are not in the lowest energy state, in the limit that , is equal to . It doesn't grow when N is large, it just approaches a constant. This will be a negligible fraction of the total number of particles. So a collection of enough bose particles in thermal equilibrium will mostly be in the ground state, with only a few in any excited state, no matter how small the energy difference.
Consider now a gas of particles, which can be in different momentum states labelled |k>. If the number of particles is less than the number of thermally accessible states, for high temperatures and low densities, the particles will all be in different states. In this limit the gas is classical. As the density increases or the temperature decreases, the number of accessible states per particle becomes smaller, and at some point more particles will be forced into a single state than the maximum allowed for that state by statistical weighting. From this point on, any extra particle added will go into the ground state.
To calculate the transition temperature at any density, integrate over all momentum states the expression for maximum number of excited particles p/1-p:
The integral, when evaluated, with the factors of kB and restored by dimensional analysis, gives the critical temperature formula of the preceding section. It can be seen that this integral defines the critical temperature and particle number corresponding to the conditions of zero chemical potential (μ=0 in the Bose–Einstein statistics distribution).
The Gross-Pitaevskii equation
The state of the BEC can be described by the wavefunction of the condensate . For a system of this nature, is interpreted as the particle density, so the total number of atoms is
Provided essentially all atoms are in the condensate (that is, have condensed to the ground state), and treating the bosons using Mean field theory, the energy (E) associated with the state is:
Minimising this energy with respect to infinitesimal variations in , and holding the number of atoms constant, yields the Gross-Pitaevski equation (GPE) (also a non-linear Schrödinger equation):
is the mass of the bosons,
is the external potential,
is representative of the
inter-particle interactions.
The GPE provides a good description the behavior of the BEC's and is the approach often applied to their theoretical analysis.
Velocity-distribution data graph
Velocity-distribution data of a gas of rubidium atoms, confirming the discovery of a new phase of matter, the Bose–Einstein condensate. Left: just before the appearance of the Bose–Einstein condensate. Center: just after the appearance of the condensate. Right: after further evaporation, leaving a sample of nearly pure condensate.
In the image accompanying this article, the velocity-distribution data confirms the discovery of the Bose–Einstein condensate out of a gas of rubidium atoms. The false colors indicate the number of atoms at each velocity, with red being the fewest and white being the most. The areas appearing white and light blue are at the lowest velocities. The peak is not infinitely narrow because of the Heisenberg uncertainty principle: since the atoms are trapped in a particular region of space, their velocity distribution necessarily possesses a certain minimum width. This width is given by the curvature of the magnetic trapping potential in the given direction. More tightly confined directions have bigger widths in the ballistic velocity distribution. This anisotropy of the peak on the right is a purely quantum-mechanical effect and does not exist in the thermal distribution on the left. This famous graph served as the cover-design for 1999 textbook Thermal Physics by Ralph Baierlein[4].
As in many other systems, vortices can exist in BECs. These can be created, for example, by 'stirring' the condensate with lasers, or rotating the confining trap. The vortex created will be a quantum vortex. These phenomena are allowed for by the non-linear term in the GPE (the term, that is). As the vortices must have quantised angular momentum, the wavefunction will be of the form where and are as in the cylindrical coordinate system, and is the angular number. To determine , the energy of must be minimised, according to the constraint . This is usually done computationally, however in a uniform medium the analytic form
is density far from the vortex,
is healing length of the condensate.
demonstrates the correct behavior, and is a good approximation.
A singly-charged vortex () is in the ground state, with its energy given by
is the farthest distance from the vortex considered.
(to obtain an energy which is well defined it is necessary to include this boundary b)
For multiply-charged vortices () the energy is approximated by
which is greater than that of singly-charged vortices, indicating that these multiply-charged vortices are unstable to decay. Research has, however, indicated they are metastable states, so may have relatively long lifetimes.
Unusual characteristics
File:Normal bose-einstein.PNG
On the left is what atoms normally look like in a solid. On the right is the atoms in a Bose-Einstein condensate. Note: These illustrations are representations/models, not the real thing.
Further experimentation by the JILA team in 2000 uncovered a hitherto unknown property of Bose–Einstein condensates. Cornell, Wieman, and their coworkers originally used rubidium-87, an isotope whose atoms naturally repel each other, making a more stable condensate. The JILA team instrumentation now had better control over the condensate so experimentation was made on naturally attracting atoms of another rubidium isotope, rubidium-85 (having negative atom-atom scattering length). Through a process called Feshbach resonance involving a sweep of the magnetic field causing spin flip collisions, the JILA researchers lowered the characteristic, discrete energies at which the rubidium atoms bond into molecules making their Rb-85 atoms repulsive and creating a stable condensate. The reversible flip from attraction to repulsion stems from quantum interference among condensate atoms which behave as waves.
Because supernova explosions are implosions, the explosion of a collapsing Bose–Einstein condensate was named "bosenova", a pun on the musical style bossa nova.
Current research
Compared to more commonly-encountered states of matter, Bose–Einstein condensates are extremely fragile. The slightest interaction with the outside world can be enough to warm them past the condensation threshold, forming a normal gas and losing their interesting properties. It is likely to be some time before any practical applications are developed.
Nevertheless, they have proved to be useful in exploring a wide range of questions in fundamental physics, and the years since the initial discoveries by the JILA and MIT groups have seen an explosion in experimental and theoretical activity. Examples include experiments that have demonstrated interference between condensates due to wave-particle duality,[6] the study of superfluidity and quantized vortices,[7] and the slowing of light pulses to very low speeds using electromagnetically induced transparency.[8] Vortices in Bose-Einstein condensates are also currently the subject of analogue-gravity research, studying the possibility of modeling black holes and their related phenomena in such environments in the lab. Experimentalists have also realized "optical lattices", where the interference pattern from overlapping lasers provides a periodic potential for the condensate. These have been used to explore the transition between a superfluid and a Mott insulator,[9] and may be useful in studying Bose–Einstein condensation in fewer than three dimensions, for example the Tonks-Girardeau gas.
Bose–Einstein condensates composed of a wide range of isotopes have been produced.[10]
In 1999, Danish physicist Lene Vestergaard Hau led a team from Harvard University who succeeded in slowing a beam of light to about 17 metres per second and, in 2001, was able to momentarily stop a beam. She was able to achieve this by using a superfluid. Hau and her associates at Harvard University have since successfully transformed light into matter and back into light using Bose-Einstein condensates.[citation needed] Details of the experiment are discussed in an article in the journal Nature, 8 February 2007. [12]
Use in popular science
A prominent example of the use of Bose-Einstein condensation in popular science is at the Physics 2000 web site developed at the University of Colorado at Boulder. In the context of popularizations, atomic BEC is sometimes called a Super Atom.[13]
In popular culture
The game Mass Effect which is developed by BioWare has a weapon upgrade called Cryo Rounds. The description states that "Cooling lasers collapse ammunition into small Bose-Einstein condensate - a mass of super-cooled subatomic particles - capable of snap-freezing impacted objects."[14]
See also
• S. N. Bose, Z. Phys. 26, 178 (1924)
• A. Einstein, Sitz. Ber. Preuss. Akad. Wiss. (Berlin) 1, 3 (1925)
• L.D. Landau, J. Phys. USSR 5, 71 (1941)
• L. Landau (1941). "Theory of the Superfluidity of Helium II". Physical Review. 60: 356–358.
• M.H. Anderson, J.R. Ensher, M.R. Matthews, C.E. Wieman, and E.A. Cornell (1995). "Observation of Bose–Einstein Condensation in a Dilute Atomic Vapor". Science. 269: 198–201.
• C. Barcelo, S. Liberati and M. Visser (2001). "Analogue gravity from Bose-Einstein condensates". Classical and Quantum Gravity. 18: 1137–1156.
• P.G. Kevrekidis, R. Carretero-Gonzlaez, D.J. Frantzeskakis and I.G. Kevrekidis (2006). "Vortices in Bose-Einstein Condensates: Some Recent Developments". Modern Physics Letters B. 5 (33).
• K.B. Davis, M.-O. Mewes, M.R. Andrews, N.J. van Druten, D.S. Durfee, D.M. Kurn, and W. Ketterle (1995). "Bose–Einstein condensation in a gas of sodium atoms". Physical Review Letters. 75: 3969–3973..
• D. S. Jin, J. R. Ensher, M. R. Matthews, C. E. Wieman, and E. A. Cornell (1996). "Collective Excitations of a Bose–Einstein Condensate in a Dilute Gas". Physical Review Letters. 77: 420–423.
• M. R. Andrews, C. G. Townsend, H.-J. Miesner, D. S. Durfee, D. M. Kurn, and W. Ketterle (1997). "Observation of interference between two Bose condensates". Science. 275: 637–641. doi:10.1126/science.275.5300.637..
• M. R. Matthews, B. P. Anderson, P. C. Haljan, D. S. Hall, C. E. Wieman, and E. A. Cornell (1999). "Vortices in a Bose–Einstein Condensate". Physical Review Letters. 83: 2498–2501.
• E.A. Donley, N.R. Claussen, S.L. Cornish, J.L. Roberts, E.A. Cornell, and C.E. Wieman (2001). "Dynamics of collapsing and exploding Bose–Einstein condensates". Nature. 412: 295–299.
• M. Greiner, O. Mandel, T. Esslinger, T. W. Hänsch, I. Bloch (2002). "Quantum phase transition from a superfluid to a Mott insulator in a gas of ultracold atoms". Nature. 415: 39–44. doi:10.1038/415039a..
• S. Jochim, M. Bartenstein, A. Altmeyer, G. Hendl, S. Riedl, C. Chin, J. Hecker Denschlag, and R. Grimm (2003). "Bose–Einstein Condensation of Molecules". Science. 302: 2101–2103. doi:10.1126/science.1093280.
• Markus Greiner, Cindy A. Regal and Deborah S. Jin (2003). "Emergence of a molecular Bose−Einstein condensate from a Fermi gas". Nature. 426: 537–540.
• M. W. Zwierlein, C. A. Stan, C. H. Schunck, S. M. F. Raupach, S. Gupta, Z. Hadzibabic, and W. Ketterle (2003). "Observation of Bose–Einstein Condensation of Molecules". Physical Review Letters. 91: 250401. doi:10.1126/science.1093280.
• C. A. Regal, M. Greiner, and D. S. Jin (2004). "Observation of Resonance Condensation of Fermionic Atom Pairs". Physical Review Letters. 92: 040403.
• C. J. Pethick and H. Smith, Bose–Einstein Condensation in Dilute Gases, Cambridge University Press, Cambridge, 2001.
• Lev P. Pitaevskii and S. Stringari, Bose–Einstein Condensation, Clarendon Press, Oxford, 2003.
• Amandine Aftalion, Vortices in Bose–Einstein Condensates, PNLDE Vol.67, Birkhauser, 2006.
• Mackie M, Suominen KA, Javanainen J., "Mean-field theory of Feshbach-resonant interactions in 85Rb condensates." Phys Rev Lett. 2002 Oct 28;89(18):180403.
External links
Template:State of matter
bg:Бозе-Айнщайнова кондензация ca:Condensat de Bose-Einstein cs:Bose-Einsteinův kondenzát da:Bose-Einstein-kondensat de:Bose-Einstein-Kondensat el:Συμπύκνωμα Bose-Einstein fa:چگالش بوز-اینشتین ko:보즈-아인슈타인 응축 id:Kondensat Bose-Einstein it:Condensato di Bose - Einstein he:עיבוי בוז-איינשטיין nl:Bose-Einsteincondensaat no:Bose-Einstein-kondensasjon sk:Boseho-Einsteinov kondenzát sl:Bose-Einsteinov kondenzat fi:Bosen–Einsteinin kondensaatti sv:Bose–Einstein-kondensat uk:Конденсат Бозе-Ейнштейна |
1c4eb53cd057eec4 | Cutting-Edge Computational Chemistry Enabled by Deep Learning
Machine learning is becoming a bigger part of chemistry as of the last two or three years. Industries need to have people trained in both fields, and it’s taken time for them to make their way into this sector. Olexandr Isayev is at the forefront of that wave, and he talks to us about what he’s done while melding deep learning and chemistry together and his vision of where he sees this field going with this new tech.
Olexandr Isayev: Historically, chemistry was empirical science. It’s been driven by experiment. So, you find the observation, you formulate a hypothesis, you make a prediction, and do a test, so it’s the standard scientific method. Now, those new machine learning methods allow us to do a data-driven discovery.
Ginette: I’m Ginette.
Curtis: And I’m Curtis.
Ginette: And you are listening to Data Crunch.
Curtis: A podcast about how applied data science, machine learning, and artificial intelligence are changing the world.
Ginette: A Vault Analytics production.
Ginette: Tableau is the leading software in data analysis, preparation, and interactive analytics, and we’re huge fans of it because we’ve seen how it facilitates quickly finding business value from data—helping you do this faster than anything else can. If you and your team have recently purchased Tableau licenses, you’re off to a great start, but in our experience working with companies over the years, many deployments of Tableau fail to realize their potential value because of a lack of training and understanding of how to use it well—which is a shame because it can truly transform your business when used well.
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We train on Tableau at basic, intermediate, and advanced levels. We’d love to hear from you and help you transform your business with Tableau with an onsite training—send us an email at [email protected] or visit our site at, and we’ll be in touch!
Curtis: Chemistry—while it can conjure up images of begoggled scientists in white coats donning blue rubber gloves in sterile laboratories—it touches more of our lives than we probably give it credit for. Think of lithium batteries that power our electronics, plastics that exist almost ubiquitously around us, dyes that color much of your world, jet engines, among so many other things. And as a personal note, I happen to find the science fascinating. I majored in chemistry in college.
Ginette: Today we speak with a man who is at the forefront of connecting AI with chemical sciences. His work has been published in some well known journals, and he’s headed up some impressive projects.
Olexandr Isayev: My name is Olexandr Isayev, or Olis for short. I grew up in Ukraine then I move to US. Did my PhD in computational chemistry, and I also have a minor in computer science. So I’ve worked in a lot of different topics in chemical sciences, and in particular, we use a computer simulation, like high performance computing simulation, and how we can address the challenges of chemical sciences, and recently, I started the faculty position at UNC, so I’m an assistant professor at the University of North Carolina in Chapel Hill, and basically where current focus of our work is connecting AI and machine learning with chemical and biological sciences and how does technology, data-driven technologies, could help us solve some of the fundamental problems in chemistry and biology.
So chemistry was kind of lagging behind some other field, right, because deep learning was the revolution in vision, in speech, in text, right? So in life sciences, it’s kind of on the back, so I think neural network come—the modern neural networks come—two or three years ago to chemical sciences. Interesting, so there was the old types of neural networks in 90s, so you can still find those papers where people use the neural networks in the 80s and 90s. Then you know, then there was a period of winter and no one used them, but now we see this new way for past 34 years, probably, and so those are very emergent applications, and not that many people work there, but new students come in and been trained in both computer science and chemistry, so the trend is accelerating, so we see this wave of applications last year and this year.
Ginette: He’s been at the forefront of the chemistry–artificial intelligence wave, helping augment traditional chemistry methods with machine learning to bring about what he thinks might be a leap-frog moment for the discipline.
Olexandr: We can make a map of chemical or material space, so this is not a physical space like, you know, like on the globe, but it’s imaginative, you know, high dimensional space where materials or molecules are and then we can use some data-driven methods to actually navigate us as a chemist or material scientist and look for a specific properties or functions of materials, and those types of maps help us design better high performance material, better drugs and stuff like that.
All our chemistry happens in the computer, so we’re computational people, but eventually you know it’s all ultimately in the hands of experimental—so people who go to the lab and synthesize and make actual material and test it, and we’ve worked with several different experimental groups, and we help them to design new material, for example for solar cell applications. We also worked with a lot of organic chemists, medicinal chemists, who do drug discovery, so design new, better drugs to treat disease like cancer or Alzheimer’s, and so this is a work in progress now.
Curtis: Olexandr’s computer simulations that use machine learning to predict chemical reactions saves the experimental chemists lots of time because he can tell them, based on his neural net’s computations, which combination of chemicals are worth testing. And this isn’t trivial. This can save chemists hundreds of hours of lab time because there’s so many possibilities of what you could test.
Olexandr: Some of those experimental methods are very expensive because of nature of the process, and they’re slow. It’s it’s really laborious work in the lab, but now we can do a simulation on the computer, and we can use is a physics-based simulation or data driven, like a machine-learning methods, to guide a chemical experiment, and we can predict and navigate them and say okay, so if you have the options to test thousand different things and instead of running a thousand different experiments, we help them prioritize and say, “oh, don’t do this 990, but this ten can be precious.”
Ginette: So, in what other ways is machine learning fundamentally changing chemistry?
Olexandr: Historically, chemistry was empirical science. It’s been driven by experiment. You find the observation, you formulate a hypothesis and you make a prediction and do a test, so it’s the standard scientific method from the Newton times, but now, you know, those new machine learning methods allow us to do a data-driven discovery. Given your historical data, you can train a machine-learning model and predict a sort of properties or a character or a feature of interest for particular molecule and then you can drive an experiment from that based on your machine learning models.
So there are a lot of physics-based methods, which mean you rely on some kind of fundamental principle, for example quantum mechanics, and you can solve the quantum mechanical problem for a particular molecule, and you can understand the properties of it, but the problem is this is a very computer-intensive process, and typically you have to run like a super computer, and it takes for a lot of time, so once machine learning kicks in, it allows us to do faster and accurate approximation of this problem, and this is one of the project in my lab, so we use neural networks, you know, deep learning, to approximate solution of Schrödinger equation, and this gives us a speedup of up to 6 order of magnitudes and so the supercomputer, you can run essentially on a laptop. So this give you a tremendous speed up.
Instead of doing expensive simulation on a supercomputer, neural net approximate this, and you can use a standard linux box and a GPU and get a answer much faster. Again there are certain approximation because it’s not, you know, exact solution, you know, a space of standard inorganic molecules, drug-like molecules, basically we have a very, very nice accuracy, and the solution is almost exact.
Ginette: Olexandr points out that his team can essentially do the work of a supercomputer with some neural networks, a linux box, and a GPU. By learning or refining your skills with neural networks, there’s a world of possibilities in every field.
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Curtis: So not only can machine learning help target the right experiments to solve a problem, it can also help solve equations that use huge computational resources faster than traditional methods by several orders of magnitude. That’s like riding on a jet instead of on the back of a giant snail. You can go a lot more places on the jet. The data Olexandr uses with his models include properties and structures of molecules already know from scientific literature, as well as solutions to the computationally intensive equations we mentioned earlier.
Olexandr: Either work is experimental data, so either go to a history of literature, some databases, so we collect experimental measurements or properties of molecules, and those properties can be anything, you know, efficiency of your lithium battery or efficiency of your solar cell that can be binding to a specific protein. It can be any kind of useful property, and then we connect this property with a structure of the molecules and materials by using some kind of features and apply machine learning methods.
The second approach is when we approximate the physics-based simulation instead of having this experimental results, we use the solution of this very expensive calculations as the ultimate target, and then neural net would approximate the solution of this equation. That can be energy of the molecule or it can be a computed property of the molecule, that’s the ban gaffe for example, or some other useful properties.
Ginette: So what else has Olexandr and his team been working on that’s caught the attention of so many people?
Olexandr: So we have a code on Github, and a couple of publication, so any curious reader could go to a technical detail sort of play by him- or herself. So basically what we did, we invested a lot of computational resources to solve the Shonier equation or organic molecules, and we generated a gigantic database for pairs of molecule and energy and some other properties, for example, and then we train a deep neural network that would predict that, so now when we did this hard work and very least a trained neural network so everyone can go, and then instead of, you know, plugging their own molecule and get the solution and get the energy and the structure of the molecule and use for their own projects, and now we collaborate with a lot of experiment lab and people who do drug discovery because, you know, those methods are used for many different applications. They are probably would revolutionize the field of computational chemistry soon, at least we hope so.
Curtis: So with this code, Olexandr and his team help many different experimental labs speed up their processes of finding what works well for their particular experiments. But this isn’t the only project he’s been working on.
Olexandr: Probably most of your readers know the Alpha Go, these reinforcement learning that beat the best players of the game Go. And the game Go is super complex. And actually what we built in the same analogy so the game Go has two pieces: one that play there, you know, make a decision, movement of the of the checkers on the board, and the other one to score, and basically they work together, so what we did, we essentially designed an Alpha Go, that you know . . . a machine suggest a molecule for a particular biological application, and then the place is itself and learns chemistry. And then it can suggest to us a molecule with specific desired function, so for example, what we show, we pick a particular protein called on Janus kinase, check 2, and it’s an important protein implicated in implicated in cancer and some other diseases, and what we show is that a machine can design an inhibitor for this protein, and therefore we can we can we can envision that’s fully a machine driven design of new drugs.
So you teach a machine to generate molecules and we use reinforcement learning to reward to make only useful molecule, you know, it’s like a carrot and stick, essentially. Our score system part is a different neural network. They take a structure of the molecule and predict binding to this specific enzyme. And basically it gives you an approximation to an experiment. And basically when you train them in the loop so the scoring part teach the generative part to generate only molecules and then we can maximize, you know, we can maximize binding and we can minimize binding and we can do a combination of different things. So eventually I think this would be a new way to how drugs are discovered instead of a chemist, you know serendipitously goes one by one to a molecule, here a machine could get a pool of useful molecule for you.
Ginette: In addition to helping experimental chemists limit what experiments they need to conduct in the lab, he’s suggesting that machine learning methods can actually help design new drugs by recommending creation of specific molecules. This would speed up getting medicine to the market for various illnesses.
Olexandr: It’s very interesting, so you see this wave of creativity. People use GANS use different types of neural networks, you know, game theory. So essentially what you see, those new interesting ideas and algorithms start coming to chemical sciences and biological sciences, so I’m really happy. I’m really excited about what’s come out of this.
I’m optimistic, but also I’m a little bit worried about, you know, the hype, right? So if you overhype, people get, you know, disappointed so people may have yet another winter, but I am optimistic that you know those those methods would significantly transform chemical sciences, so you’ll see faster drugs in the market, so you can treat more, you know, disease like cancer faster. You know, hopefully we will see personalized medicine when for example your own genome would be sequenced, and then we can design a specific treatment for your particular condition, and that would be possible as a combination of cutting-edge science and data-driven methods, and also design of a new materials, like steels, alloys would be accelerated as well, so I’m very optimistic. I’m very happy we live in this age. It’s very interesting to see this transformation.
Ginette: A huge thank you to Olexandr for speaking with us, and as mentioned in our podcast, if you want some better insights into your business data by training your team in Tableau, go to or email us at [email protected]. We’ll teach you how to find insights and share them effectively, creating improvements for your company and greater success for you.
And as always, for the transcript and links for this podcast, you can go to, and you’ll find the links at the bottom of the show transcript. If you like what you’re learning here with us, please share our podcast with your coworkers and friends and go to iTunes or your favorite podcast playing platform and leave us a review.
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77e40f56bd3c287c | If an EM wave only gives us a probability of where a photon may be at a given moment, and the HUP tells us that we can't know the exact location of the photon. Then would it be correct to say that a photon does not travel in a straight line?
If this is true, wouldn't the photon's crooked path mean that it must travel faster than $c$ for us to measure its straight line speed at $c$?
Here's a comparison to explain my question further: If two sprinters run a 100 yard dash in 10 seconds, but one sprinter is required to run in a zigzag manner wouldn't that runner need to run faster than the straight line sprinter for both runners to run the race in 10 seconds. Does the crooked traveling photon need to exceed $c$ for us to measure its straight line speed at $c$?
• $\begingroup$ the HUP tells us that we can't know the exact location of the photon. No, that is not true. Also, the photon does have not any definite path, crooked or straight. $\endgroup$ – user140606 Feb 8 '17 at 16:59
• $\begingroup$ When we measure c don't we assume it is traveling in a straight line? $\endgroup$ – Lambda Feb 8 '17 at 17:04
• $\begingroup$ We assume it, in order for it to correspond with what we expect classically, but in Q.M, the laws are different. The second answer on this question says it better than I can: physics.stackexchange.com/q/186170 $\endgroup$ – user140606 Feb 8 '17 at 17:11
• $\begingroup$ Great photo of the interferogram by the way. Lovely fringe visibility: I could look at interferograms all day, especially if they are taken with my favorite colors. $\endgroup$ – WetSavannaAnimal Feb 9 '17 at 0:01
• $\begingroup$ WetSavannaAnimal aka Rod Vance: Thank you. I have some shots where I put a paper cone under the image. It produces a cool 3d look. The interferometer is mounted onto steel tubing and is rock solid. I can carry it around and the fringes for the most part stay stationary. $\endgroup$ – Lambda Feb 9 '17 at 1:15
First of all, as Countto10 pointed out, in this other question you can find a nice discussion on the meaning of trajectories in quantum mechanics.
Heisenberg's uncertainty principle tells you that when you measure the photon's position or momentum (or any other pair of complementary variables), you will find the results to always affect each other. In other words, a photon cannot have a definite value of position and momentum at the same time.
This does not really come into play if you are trying to measure the speed of light using single photons. Let us imagine for the purpose an experiment in which a single photon is emitted at point $A$ at a known time $t$, and travels towards the point $B$. After the photon is emitted, its wavefunction will expand and evolve according to a variety of factors. For example, the smaller the error in the transverse position when it is emitted, the faster its wavefunction will disperse and so the harder it will be to detect it at $B$. This kind of things can however be taken care of relatively easily, so to make the probability of detecting the photon in $B$ high enough. When this detection event happens, we can measure the time it took and derive the speed of light accordingly.
The photon did not follow a zig zag path going from $A$ to $B$. It didn't simply because that is not how a photon usually travels, and nothing in Heisenberg's uncertainty principle says it should. Its wavefunction did evolve in the process, but it is wrong to think of this in terms of a point particle jiggling around. It means instead that the probabilities of detecting the photon in the various transverse points vary with the longitudinal position.
As a final note, it is worth noting that there should be a lot of buts and ifs in the above argument. For example, I assumed that the emitted photon can be more or less be thought of as a particle-like thing, in the sense of it being relatively well localized. A photon can however also be highly delocalized, or have a complex inner structure. This kind of things can matter: an interesting example is a recent experiment by Bareza and Hermosa in which it was shown that photons carrying an orbital angular momentum can have (in the vacuum) a group velocity smaller than $c$. This is an interesting reminder that the speed of light being $c$ only strictly holds for plane waves in the vacuum, not really for light with finite extent and complex inner structure.
Your crooked path argument holds for a classical electromagnetic wave going through a medium. That is why the speed of light in a medium can be less than c, the speed of light in vacuum. At the photon level it is that photons effectively go a larger distance at velocity c, as they interact individually with the lattice of the material, and a lower group velocity is measured.
In vacuum there are no interactions for the photon, which is a point particle, and its velocity is c and the same is true for the light beam. The Heisenberg uncertainty does not enter.
The validation of this is the validation of the special theory of relativity which has been tested by the myriads of experiments in physics.
From the point of view of worry that the HUP might contradict the existence of the speed limit $c$, one can assuage your worries by pointing out that:
1. $c$ does not primarily mean the speed of light, rather it is a geometric constant that defines our spacetime and experimentally light is observed to move at that speed. See more about this in my answer here and
2. The ban on speeds amounts to a limit to the speed at which information, or a cause-effect relationship can propagate. So an inferred speed exceeding $c$ is not needfully a problem: it only violates known physics if the speed can be interpreted as a propagation speed for a cause-effect relationship. Thus, for example, phase speeds of waves can happily exceed $c$, e.g. at anomalous dispersion wavelengths for light propagation in dielectric materials. Moreover, gainsaying many texts, even group velocities can exceed $c$ in some wavelength bands and for slightly weird materials: that's not even a problem because, although the group velocity approximates the signal speed propagation, it's not exact and the step response of the medium still propagates at less than $c$.
Any speed inferred from the kind of reasoning you are making falls into the category in my point (2) above. There are two reasons for this:
1. Lone photons without quantum measurement cannot be used to transmit information. In particle physics, the propagation of the electromagnetic field is calculated from a superposition of paths of photons - and so called "off-shell" photons (those with speeds other than $c$ - above or below)- enter this calculation. This is not a problem, because information-bearing aspects of the field are still calculated to propagate at $c$;
2. I don't think you have a sound conception of a photon. Please don't take that as an attack as there are a great many education resources out there that give people these kinds of ideas (I was 35 before I shifted on from these ideas). The modern idea is subtler, but much simpler and much sounder logically than the 1920s Bohrian ideas that are still taught. I recommend you begin with Daniel Sank's glorious answer here. Inspired by his answer, I followed up with this one (which may also be useful to you). Indeed, for massless particles like the photon, one cannot even define a sound notion of "position" as one can for nonrelativistic electrons following the nonrelativistic Schrödinger equation.
Light does travel in a straight line, or else there wouldn't be shades!! lol
The EM Wave
The EM wave is used to describe the possibilities of finding a particle at a certain location. But we can't say that the particle is at some particular point in the wave because then we lose the wave analogy. We can only say that the particle exists somewhere in the wave, and by that, we mean the particle is simultaneously everywhere on the wave.
So if the wave continues throughout the universe, that means at the next point in time, the particle could appear from anywhere throughout the universe.
So does that mean an electron near you could end up being at the edge of the universe at the very next moment? YES!!
But basically, the consequences of particles ending up at a very far distance is extremely low that it's considered never to happen.
The Action
In the Uncertainty Principle, there is this thing called action.
The action calculates the crest and troughs in a wave using the mass, time, and distance between the starting and final position of a particle or an object.
If say the final position of a particle is 10 units away(large distance for the particle say), the action of this would correspond to the crest of the wave(skipping the math); simultaneously, the same particle would also appear somewhere on that original wave, say, its 10.1 units away from the final position, the action would correspond to the trough of the wave. Lastly, we only have to add these waves together in order to get the final wave function for the final position of the wave.
The crest and the troughs cancel each other out, and so we get a final wave function of nearly zero possibility of finding the particle at such a large distance.
On the other hand, the possibilities of a particle ending up at a close location are huge. Because instead of canceling each other out, the waves stacks themselves up into a larger wave with much more amplitude, which means a higher chance of finding the particle at that location.
That's why the particles can't get far from their original position, and therefore you shouldn't be worried about its speed exceeding the speed of light.
protected by Qmechanic Feb 8 '17 at 23:59
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792bf91cb845a469 | Atomic Theory and Structure
Atomic Theory III: Wave-Particle Duality and the Electron
by Adrian Dingle, B.Sc., Anthony Carpi, Ph.D.
As discussed in our Atomic Theory II module, at the end of 1913 Niels Bohr facilitated the leap to a new paradigm of atomic theoryquantum mechanics. Bohr’s new idea that electrons could only be found in specified, quantized orbits was revolutionary (Bohr, 1913). As is consistent with all new scientific discoveries, a fresh way of thinking about the universe at the atomic level would only lead to more questions, the need for additional experimentation and collection of evidence, and the development of expanded theories. As such, at the beginning of the second decade of the 20th century, another rich vein of scientific work was about to be mined.
Periodic trends lead to the distribution of electrons
In the late 19th century, the father of the periodic table, Russian chemist Dmitri Mendeleev, had already determined that the elements could be grouped together in a manner that showed gradual changes in their observed properties. (This is discussed in more detail in our module The Periodic Table of Elements.) By the early 1920s, other periodic trends, such as atomic volume and ionization energy, were also well established.
The Periodic Table of Elements
The Periodic Table of Elements
The German physicist Wolfgang Pauli made a quantum leap by realizing that in order for there to be differences in ionization energies and atomic volumes among atoms with many electrons, there had to be a way that the electrons were not all placed in the lowest energy levels. If multi-electron atoms did have all of their electrons placed in the lowest energy levels, then very different periodic patterns would have resulted from what was actually observed. However, before we reach Pauli and his work, we need to establish a number of more fundamental ideas.
Wave-particle duality
The development of early quantum theory leaned heavily on the concept of wave-particle duality. This simultaneously simple and complex idea is that light (as well as other particles) has properties that are consistent with both waves and particles. The idea had been first seriously hinted at in relation to light in the late 17th century. Two camps formed over the nature of light: one in favor of light as a particle and one in favor of light as a wave. (See our Light I: Particle or Wave? module for more details.) Although both groups presented effective arguments supported by data, it wasn’t until some two hundred years later that the debate was settled.
At the end of the 19th century the wave-particle debate continued. James Clerk Maxwell, a Scottish physicist, developed a series of equations that accurately described the behavior of light as an electromagnetic wave, seemingly tipping the debate in favor of waves. However, at the beginning of the 20th century, both Max Planck and Albert Einstein conceived of experiments which demonstrated that light exhibited behavior that was consistent with it being a particle. In fact, they developed theories that suggested that light was a wave-particle – a hybrid of the two properties. By the time of Bohr’s watershed papers, the time was right for the expansion of this new idea of wave–particle duality in the context of quantum theory, and in stepped French physicist Louis de Broglie.
de Broglie says electrons can act like waves
In 1924, de Broglie published his PhD thesis (de Broglie, 1924). He proposed the extension of the wave-particle duality of light to all matter, but in particular to electrons. The starting point for de Broglie was Einstein’s equation that described the dual nature of photons, and he used an analogy, backed up by mathematics, to derive an equation that came to be known as the “de Broglie wavelength” (see Figure 1 for a visual representation of the wavelength).
The de Broglie wavelength equation is, in the grand scheme of things, a profoundly simple one that relates two variables and a constant: momentum, wavelength, and Planck's constant. There was support for de Broglie’s idea since it made theoretical sense, but the very nature of science demands that good ideas be tested and ultimately demonstrated by experiment. Unfortunately, de Broglie did not have any experimental data, so his idea remained unconfirmed for a number of years.
De Broglie Wavelength
Figure 1: Two representations of a de Broglie wavelength (the blue line) using a hydrogen atom: a radial view (A) and a 3D view (B).
It wasn’t until 1927 that de Broglie’s hypothesis was demonstrated via the Davisson-Germer experiment (Davisson, 1928). In their experiment, Clinton Davisson and Lester Germer fired electrons at a piece of nickel metal and collected data on the diffraction patterns observed (Figure 2). The diffraction pattern of the electrons was entirely consistent with the pattern already measured for X-rays and, since X-rays were known to be electromagnetic radiation (i.e., waves), the experiment confirmed that electrons had a wave component. This confirmation meant that de Broglie’s hypothesis was correct.
Davisson-Germer Experiment
Figure 2: A drawing of the experiment conducted by Davisson and Germer where they fired electrons at a piece of nickel metal and observed the diffraction patterns. image © Roshan220195
Interestingly, it was the (experimental) efforts of others (Davisson and Germer), that led to de Broglie winning the Nobel Prize in Physics in 1929 for his theoretical discovery of the wave-nature of electrons. Without the proof that the Davisson-Germer experiment provided, de Broglie’s 1924 hypothesis would have remained just that – a hypothesis. This sequence of events is a quintessential example of a theory being corroborated by experimental data.
Comprehension Checkpoint
Theories must be backed up by
Schrödinger does the math
In 1926, Erwin Schrödinger derived his now famous equation (Schrödinger, 1926). For approximately 200 years prior to Schrödinger’s work, the infinitely simpler F = ma (Newton’s second law) had been used to describe the motion of particles in classical mechanics. With the advent of quantum mechanics, a completely new equation was required to describe the properties of subatomic particles. Since these particles were no longer thought of as classical particles but as particle-waves, Schrödinger’s partial differential equation was the answer. In the simplest terms, just as Newton’s second law describes how the motion of physical objects changes with changing conditions, the Schrödinger equation describes how the wave function (Ψ) of a quantum system changes over time (Equation 1). The Schrödinger equation was found to be consistent with the description of the electron as a wave, and to correctly predict the parameters of the energy levels of the hydrogen atom that Bohr had proposed.
Schrodinger equation
Equation 1: The Schrödinger equation.
Schrödinger’s equation is perhaps most commonly used to define a three-dimensional area of space where a given electron is most likely to be found. Each area of space is known as an atomic orbital and is characterized by a set of three quantum numbers. These numbers represent values that describe the coordinates of the atomic orbital: including its size (n, the principal quantum number), shape (l, the angular or azimuthal quantum number), and orientation in space (m, the magnetic quantum number). There is also a fourth quantum number that is exclusive to a particular electron rather than a particular orbital (s, the spin quantum number; see below for more information).
Schrödinger’s equation allows the calculation of each of these three quantum numbers. This equation was a critical piece in the quantum mechanics puzzle, since it brought quantum theory into sharp focus via what amounted to a mathematical demonstration of Bohr’s fundamental quantum idea. The Schrödinger wave equation is important since it bridges the gap between classical Newtonian physics (which breaks down at the atomic level) and quantum mechanics.
The Schrödinger equation is rightfully considered to be a monumental contribution to the advancement and understanding of quantum theory, but there are three additional considerations, detailed below, that must also be understood. Without these, we would have an incomplete picture of our non-relativistic understanding of electrons in atoms.
Comprehension Checkpoint
Max Born further interprets the Schrödinger equation
German mathematician and physicist Max Born made a very specific and crucially important contribution to quantum mechanics relating to the Schrödinger equation. Born took the wave functions that Schrödinger produced, and said that the solutions to the equation could be interpreted as three-dimensional probability “maps” of where an electron may most likely be found around an atom (Born, 1926). These maps have come to be known as the s, p, d, and f orbitals (Figure 3).
Atomic orbitals
Figure 3: Based on Born's theories, these are representations of the three-dimensional probabilities of an electron's location around an atom. The four orbitals, in increasing complexity, are: s, p, d, and f. Additional information is given about the orbital's magnetic quantum number (m). image © UC Davis/ChemWiki
Comprehension Checkpoint
Werner Heisenberg’s Uncertainty Principle
In the year following the publication of Schrödinger’s work, the German physicist Werner Heisenberg published a paper that outlined his uncertainty principle (Heisenberg, 1927). He realized that there were limitations on the extent to which the momentum of an electron and its position could be described. The Heisenberg Uncertainty Principle places a limit on the accuracy of simultaneously knowing the position and momentum of a particle: As the certainty of one increases, then the uncertainty of other also increases.
The crucial thing about the uncertainty principle is that it fits with the quantum mechanical model in which electrons are not found in very specific, planetary-like orbits – the original Bohr model – and it also dovetails with Born’s probability maps. The two contributions (Born and Heisenberg’s) taken together with the solution to the Schrödinger equation, reveal that the position of the electron in an atom can only be accurately predicted in a statistical way. That is to say, we know where the electron is most likely to be found in the atom, but we can never be absolutely sure of its exact position.
Comprehension Checkpoint
The Heisenberg uncertainty principle concerning the position and momentum of a particle states that as the certainty of one increases, the _____ of the other increases.
Angular momentum, or "Spin"
In 1922 German physicists Otto Stern, an assistant of Born’s, and Walther Gerlach conducted an experiment in which they passed silver atoms through a magnetic field and observed the deflection pattern. In simple terms, the results yielded two distinct possibilities related to the single, 5s valence electron in each atom. This was an unexpected observation, and implied that a single electron could take on two, very distinct states. At the time, nobody could explain the phenomena that the experiment had demonstrated, and it took a number of scientists, working both independently and in unison with earlier experimental observations, to work it out over a period of several years.
In the early 1920s, Bohr’s quantum model and various spectra that had been produced could be adequately described by the use of only three quantum numbers. However, there were experimental observations that could not be explained via only three mathematical parameters. In particular, as far back as 1896, the Dutch physicist Pieter Zeeman noted that the single valence electron present in the sodium atom could yield two different spectral lines in the presence of a magnetic field. This same phenomenon was observed with other atoms with odd numbers of valence electrons. These observations were problematic since they failed to fit the working model.
In 1925, Dutch physicist George Uhlenbeck and his graduate student Samuel Goudsmit proposed that these odd observations could be explained if electrons possessed angular momentum, a concept that Wolfgang Pauli later called “spin.” As a result, the existence of a fourth quantum number was revealed, one that was independent of the orbital in which the electron resides, but unique to an individual electron.
By considering spin, the observations by Stern and Gerlach made sense. If an electron could be thought of as a rotating, electrically-charged body, it would create its own magnetic moment. If the electron had two different orientations (one right-handed and one left-handed), it would produce two different ‘spins,’ and these two different states would explain the anomalous behavior noted by Zeeman. This observation meant that there was a need for a fourth quantum number, ultimately known as the “spin quantum number,” to fully describe electrons. Later it was determined that the spin number was indeed needed, but for a different reason – either way, a fourth quantum number was required.
Comprehension Checkpoint
Some experimental observations could not be explained mathematically using three parameters because
Spin and the Pauli exclusion principle
In 1922, Niels Bohr visited his colleague Wolfgang Pauli at Göttingen where he was working. At the time, Bohr was still wrestling with the idea that there was something important about the number of electrons that were found in ‘closed shells’ (shells that had been filled).
In his own later account (1946), Pauli describes how building upon Bohr’s ideas and drawing inspiration from others’ work, he proposed the idea that only two electrons (with opposite spins) should be allowed in any one quantum state. He called this ‘two-valuedness’ – a somewhat inelegant translation of the German zweideutigkeit (Pauli, 1925). The consequence was that once a pair of electrons occupies a low energy quantum state (orbitals), any subsequent electrons would have to enter higher energy quantum states, also restricted to pairs at each level.
Using this idea, Bohr and Pauli were able to construct models of all of the electronic structures of the atoms from hydrogen to uranium, and they found that their predicted electronic structures matched the periodic trends that were known to exist from the periodic table – theory met experimental evidence once again.
Pauli ultimately formed what came to be known as the exclusion principle (1925), which used a fourth quantum number (introduced by others) to distinguish between the two electrons that make up the maximum number of electrons that could be in any given quantum level. In its simplest form, the Pauli exclusion principle states that no two electrons in an atom can have the same set of four quantum numbers. The first three quantum numbers for any two electrons can be the same (which places them in the same orbital), but the fourth number must be either +½ or -½, i.e., they must have different ‘spins’ (Figure 4). This is what Uhlenbeck and Goudsmit’s research suggested, following Pauli’s original publication of his theories.
Spin angular momentum
Figure 4: A model of the fourth quantum number, spin (s). Shown here are models for particles with spin (s) of ½, or half angular momentum.
The period described here was rich in the development of the quantum theory of atomic structure. Literally dozens of individuals, some mentioned throughout this module and others not, contributed to this process by providing theoretical insights or experimental results that helped shape our understanding of the atom. Many of the individuals worked in the same laboratories, collaborated together, or communicated with one another during the period, allowing the rapid transfer of ideas and refinements that would shape modern physics. All these contributions can certainly been seen as an incremental building process, where one idea leads to the next, each adding to the refinement of thinking and understanding, and advancing the science of the field.
The 20th century was a period rich in advancing our knowledge of quantum mechanics, shaping modern physics. Tracing developments during this time, this module covers ideas and refinements that built on Bohr’s groundbreaking work in quantum theory. Contributions by many scientists highlight how theoretical insights and experimental results revolutionized our understanding of the atom. Concepts include the Schrödinger equation, Born’s three-dimensional probability maps, the Heisenberg uncertainty principle, and electron spin.
Key Concepts
• Electrons, like light, have been shown to be wave-particles, exhibiting the behavior of both waves and particles.
• The Schrödinger equation describes how the wave function of a wave-particle changes with time in a similar fashion to the way Newton’s second law describes the motion of a classic particle. Using quantum numbers, one can write the wave function, and find a solution to the equation that helps to define the most likely position of an electron within an atom.
• Max Born’s interpretation of the Schrödinger equation allows for the construction of three-dimensional probability maps of where electrons may be found around an atom. These ‘maps’ have come to be known as the s, p, d, and f orbitals.
• The Heisenberg Uncertainty Principle establishes that an electron’s position and momentum cannot be precisely known together, instead we can only calculate statistical likelihood of an electron’s location.
• NGSS
• References
• Bohr, N. (1913). On the constitution of atoms and molecules. Philosophical Magazine (London), Series 6, 26, 1–25.
• Born, M. (1926). Zur Quantenmechanik der Stoßvorgänge. Zeitschrift für Physik, 37(12), 863–867.
• Davisson, C. J. (1928). Are electrons waves? Franklin Institute Journal, 205(5), 597-623.
• de Broglie, L. (1924). Recherches sur la théorie des quanta. Annales de Physique, 10(3), 22-128.
• Heisenberg, W. (1927). Über den anschaulichen Inhalt der quantentheoretischen Kinematik und Mechanik. Zeitschrift für Physik, 43(3-4), 172-198.
• Pauli, W. (1925). Ueber den Einfluss der Geschwindigkeitsabhaengigkeit der Elektronenmasse auf den Zeeman-Effekt. Zeitschrift für Physik, 31(1), 373-385.
• Pauli, W. (1946). Remarks on the history of the exclusion principle. Science, New Series, 103(2669), 213-215.
• Schrödinger, E. (1926). Quantisierung als Eigenwertproblem. Annalen der Physik, 384(4), 273–376.
• Stoner, E. C. (1924). The distribution of electrons among atomic energy levels. The London, Edinburgh and Dublin Philosophical Magazine (6th series), 48(286), 719-736
Adrian Dingle, B.Sc., Anthony Carpi, Ph.D. “Atomic Theory III” Visionlearning Vol. CHE-3 (6), 2015. |
1f8c98765f5cd924 | modern physics 342 n.
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1. Modern Physics 342 References : • Modern Physics by Kenneth S. Krane. , 2nd Ed. John Wiley & Sons, Inc. • Concepts of Modern Physics by A. Beiser, 6th Ed. (2002), McGraw Hill Com. • Modern Physics for Scientists and Engineers by J. Taylor, C. Zafiratos and M. Dubson, 2nd Ed, 2003. Chapters: 5 (Revision), 6 (6.4), 7, 8, 10, 11 and 12
2. Ch. 5 (Revision) Schrödinger Equation
3. Schrödinger Equation Requirements • Conservation of energy is necessary: Kinetic energy Total energy Potential energy
4. The kinetic energy K is conveniently given by Where P is the momentum = m v
5. Schrödinger Equation Requirements (continued) • Consistency with de Broglie hypothesis Where, λ andkare, respectively, the wavelength and the wave number.
6. Schrödinger Equation Requirements (continued) • Validity of the equation The solution of this equation must be valid everywhere, single valued and linear. By linear we meant that the equation must allow de Broglie waves to superimpose properly. The following is a mathematical form of the wave associating the particle.
7. To make sure that the solution is continuous, its derivative must have a value everywhere.
8. Time-Independent Schrödinger Equation
9. Probability, Normalization and Average • The probability density is given by Which is the probability of finding a particle in the space dx. The probability of finding this particle in a region between x1 and x2 is
10. By normalization we mean the total probability allover the space is 1, so that,
11. The mean value ( the expectation value) of x, If the wave function is normalized, therefore,
12. Applications • The Free Particle ( a particle is moving with no forces acting on it) • U(x)= constant anywhere=0 (arbitrarily) The solution of such deferential equation is given in the form Ψ(x)=A sin(kx)+B cos(kx) (8)
13. Particle in a one dimensional box Finding A Ψ(0)=A sin(kx) + B cos(kx)=0 and for this, B=0 Therefore, ψ(x)=A sin(kx) (9)
14. Ψ(L)=0, therefore A sin(kL)=0 since A≠0, sin(kL)=0 kL=p, 2p, 3p, .. .., np (10) Normalization condition
15. The wave function is now given by, The energy is given by using (8) and (10) together, (11) (12)
16. The ground state (lowest ) energy EO is given by We used n=1 The allowed energies for this particle are
17. The wave function is shown here for n=1 and n=2 Ψ(x) n=1 x n=2
18. Example 5.2 P149An electron is trapped in a one-dimensional region of length 1X10-10 m. How much energy must be supplied to excite the electron from the ground state to the first excited state? In the ground state, what is the probability of finding the electron in the region from 0.09 X 10-10 m to 0.11 X 10-10 m? In the first excited state, what is the probability of finding the electron between x=0 and x=0.25 X 10-10 m?
19. Example 5.3 P151Show that the average value of x is L/2, for a particle in a box of length L, independent of the quantum state (not quantized). Since the wave function is And the average value is defined by
20. A particle in a two dimensional box The Schrödinger equation in two dimensions is U(x,y)=0 inside the box (0≤x≤L) & (0≤y≤L) U(x,y)=∞ outside the box
21. The wave function ψ(x,y) is written as a product of two functions in x and y, Since ψ(x,y) must be zero at the boundaries, ψ(0,y) =0 ψ(L,y) =0 ψ(x,0) =0 ψ(x,L) =0
22. Therefore, A sin kx (0)+ B coskx (0)=0 which requires B=0 In the same way For x=L, f(x)=0 and y=L, g(y)=0 This requires kx L=nxpwith n=1,2,3 and
23. To find the constant A’, the wave function should be normalized This integration gives
24. The energy states of a particle in a two dimensional box Substituting about the wave function ψ(x,y) in Schrödinger equation, we find Which after simplification becomes
25. Chapter 7The Hydrogen Atom Wave Functions • The Schrödinger Equation in Spherical Coordinates The Schrödinger equation in three dimensions is The potential energy for the force between the nucleus and the electron is This form does not allow to separate wave function Ψ into functions in terms of x, y and z, so we have to express the whole equation of Schrödinger in terms of spherical coordinates, r, θ, and φ.
26. Cartesian and spherical coordinates z electron r cosθ θ r r sin θ sin φ y φ r sin θcosφ r sin θ x
27. x= r sin θcosφ y= r sin θ sin φ z= r cosθ And Schrödinger equation becomes This wave function can be written in terms of 3 functions in their corresponding variables, r, θ and φ
28. Hydrogen wave functions in spherical coordinates R(r) is called radial function Θ(θ ) is called polar function and Φ(φ) is called azimuthal function when solving the three differential equations in R(r), Θ(θ ) and Φ(φ), l and ml quantum numbers were obtained in addition to the previous principal quantum number n obtained before.
29. n the principal quantum number 1, 2, 3, … l angular momentum quantum number 0, 1, 2, ……±(n-1) ml magnetic quantum number 0, ±1, ±2, ……± l n 1 2 2 2
30. The energy levels of the hydrogen atom The allowed values of the radius r around the nucleus are given by Bohr radius (r at n=1) is denoted by aO and is given by
31. The Radial Probability Density P(r) The radial probability density of finding the electron at a given location is determined by The total probability of finding the electron anywhere around the nucleus is The limits of the integration depend on the conditions of the problem
32. Example 7.1 Prove that the most likely distance from the origin of an electron in the n=2, l=1 state is 4aO . At n=2 and l =1, R2,1 (r) is given by The most likely distance means the most probable position. The maximum value of the probability is obtained if r=4aO . To prove that, the first derivative of P(r) with respect to r is zero at this value.
33. Simplifying this result we get
34. Example 7.2 An electron in the n=1, l=0 state. What is the probability of finding the electron closer to the nucleus than the Bohr radius aO ? The probability is given by 32.3 % of the time the electron is closer than 1 Bohr radius to the nucleus.
35. Angular Momentum We discussed the radial part R(r) of Schrodinger equation. In this section we will discuss the angular parts of the Schrodinger equation. The classical angular momentum vector is given by During the variables separation of wave functions in Schrodinger equation, angular momentum quantum number l was produced. The length of the angular momentum vector L is given by The z-components of L are given by where ml is the magnetic quantum number 0, ±l
36. The angular momentum vector components For l=2, ml =0, ±1, ±2 The angle q is given by
37. Intrinsic Spin Angular momentum vector Magnetic moment due to electric current i Using q=-e the charge of the electron, and rp= L , we get The negative singe indicates that µL and L work in opposite directions.
38. When the angular momentum vector L is inclined to the direction of the z-axis, the magnetic moment µL has a z-component given by Remember, ml=0, ±l
39. An electric dipole in a uniform and non-uniform electric fieldA magnetic dipole in a non-uniform magnetic field The electric dipole has its moment p rotates to align with the direction of the electric field
40. Two opposite dipoles in the same non-uniform electric field are affected by opposite net forces that lead to displacing each dipole up and down according to their respective alignments.
41. Similarly, the magnetic dipoles are affected in the same way. When an electron with an angular momentum inclined to the magnetic filed, it may move up or down according to the direction of rotation around the nucleus.
42. Stern-Gerlach Experiment A beam of hydrogen atoms is in the n=2, l= 1 state. The beam contains equal numbers of atoms in the ml= -1, 0, and +1 states. When the beam passes a region of non-uniform magnetic field, the atoms with ml=+1 experience a net upward force and are deflected upward, the atoms with ml=-1 are deflected downward, while the atoms with ml=0 are undeflected.
43. After passing through the field, the beam strikes a screen where it makes a visible image. When the filed is off, we expect to see one image of the slit in the center of the screen When the field is on, three images of the slit on the screen were expected – one in the center, one above the center (ml=+1 ) and one below (ml=-1). The number of images is the number of mlvalues = 2l+1= 3 in our example. • In the Stern - Gerlach experiment, a beam, of silver atoms is used instead of hydrogen. • While the field is off, and instead of observing a single image of the slit, they observed two separate images.
44. The experiment The magnitude of S, the spin angular momentum vector is given by Where s is the spin quantum number = ±½
45. Example 7.6 In a Stern – Gerlach type of experiment, the magnetic field varies with distance in the z direction according to The silver atoms travel a distance x=3.5 cm through the magnet. The most probable speed of the atoms emerging from the oven is v=750 m/s. Find the separation of the two beams as they leave the magnet. The mass of a silver atom is 1.8 X 10-25 kg, and its magnetic moment is about 1 Bohr magneton.
46. 3.5 cm vO =750 m/s. The force applied to the beam must be obtained
47. The force is the change of potential energy U with distance z. The potential energy U is given by
48. Problem 22 p 233A hydrogen atom is in an excited 5g state, from which it makes a series of transitions, ending in the 1s state. Show on an energy levels diagram the sequence of transitions that can occur. Repeat the last steps if the atom begins in the 5d state.
49. Problem 23 p 233Consider the normal Zeeman effect applied to 3d to 2p transition. (a) sketch an energy-level diagram that shows the splitting of the 3d and 2p levels in an external magnetic field. Indicate all possible transitions from each ml state of the 3d level to each ml state of the 2p level. (b) which transitions satisfy the Dml =±1 or 0 selection rule? (c) that there are only three different transitions energies emitted.
50. For ℓ=2, mℓ=+2,+1,0,-1,-2 |
eb53f6395974df79 | Signatures of the Helical Phase in the Critical Fields at Twin Boundaries of Non-Centrosymmetric Superconductors
Kazushi Aoyama The Hakubi Center for Advanced Research, Kyoto University, Kyoto 606-8501, Japan Department of Physics, Kyoto University, Kyoto 606-8502, Japan Lucile Savary Department of Physics, University of California, Santa Barbara, California 93106-9530, USA Manfred Sigrist Institute for Theoretical Physics, ETH Zurich, Zurich 8093, Switzerland
Domains in non-centrosymmetric materials represent regions of different crystal structure and spin-orbit coupling. Twin boundaries separating such domains display unusual properties in non-centrosymmetric superconductors (NCS), where magneto-electric effects influence the local lower and upper critical magnetic fields. As a model system, we investigate NCS with tetragonal crystal structure and Rashba spin-orbit coupling (RSOC), and with twin boundaries parallel to their basal planes. There, we report that there are two types of such twin boundaries which separate domains of opposite RSOC. In a magnetic field parallel to the basal plane, magneto-electric coupling between the spin polarization and supercurrents induces an effective magnetic field at these twin boundaries. We show this leads to unusual effects in such superconductors, and in particular to the modification of the upper and lower critical fields, in ways that depend on the type of twin boundary, as analyzed in detail, both analytically and numerically. Experimental implications of these effects are discussed.
I introduction
Spin-orbit coupling is the cause of many extraordinary properties of materials, such as the anomalous and the spin Hall effects, topological insulators and superconductors AHE ; SHE ; TPI ; TPS . In the past decade, triggered by the discovery of the heavy Fermion superconductor CePtSi which lacks inversion symmetry CePt3Si , studies of spin-orbit coupling effects on superconductivity have attracted much attention Springer . Moreover, in the context of topological phases local properties of these non-centrosymmetric superconductors (NCS), like the subgap states appearing at sample edges TPS ; Vorontsov ; Iniotakis_PRB and domain boundaries Iniotakis ; Arahata , have been discussed. In our study, we address special properties of NCS with Rashba spin-orbit coupling (RSOC), which possess twin domains of opposite RSOC. In particular, we show that certain twin boundaries separating such domains can influence the superconducting (SC) properties of type-II superconductors in magnetic fields.
The Rashba-type spin-orbit interaction Rashba is inherent to systems lacking certain mirror symmetries. If is not a crystal symmetry then RSOC takes the basic form , with momentum , spin and coupling constant . The NCS CePtSi CePt3Si , and - and -electron NCS with the BaNiSn-type crystal structure such as CeTSi (T=Rh, Ir) CeRhSi3 ; CeIrSi3 , BaPtSi BaPtSi3 , and CaMSi (M=Pt, Ir) CaIrSi3 ; CaPtSi3 belong to this class of Rashba-type superconductors. One intriguing feature of Rashba-type NCS is the magneto-electric effect, which couples the spin polarization to supercurrents through spin-orbit coupling normal_jM ; SC_jM ; SC_Bj ; Dimitrova ; Samokhin ; Kaur ; Fujimoto . A Zeeman field polarising electron spins thereby results in a spatial dependence of the phase of the SC order parameter following . In this sense, this phase-modulated SC state is similar to a Fulde-Ferrell-Larkin-Ovchinnikov state FF ; LO and is known as helical SC phase Kaur . The corresponding wave vector is oriented perpendicularly to both the magnetic field and the direction of the mirror symmetry-breaking (here the -axis) if the electronic structure is nearly isotropic in --direction. Despite the non-vanishing phase gradient there are no supercurrents flowing in the bulk of the system due to gauge invariance Kaur ; Springer . Therefore, the helical phase is generally difficult to detect. It has been proposed, however, that for inhomogeneous systems the helical phase could give rise to observable features. In two-dimensional NCS, such as the LaAlO-SrTiO SC interface interface-SC ; tunable-RSO , where, for inplane fields, orbital depairing is suppressed, inhomogeneities can host magnetic flux patterns pointing perpendicular to the SC film and the applied field in the helical phase AS . Also, in three-dimensional bulk materials, inhomogeneities can generate an unusual flux response to an external field via the helical phase, although in the latter case, vortices and orbital depairing effects could disturb the observation Kaur ; Ikeda .
In our study, we address superconducting properties which are typical for certain twin boundaries in Rashba-type NCS with tetragonal crystal symmetry lacking the mirror symmetry, like in CePtSi and the CeTSi family. Twin domains in such materials have RSOC of opposite signs (in a sense we specify below). We consider here the case of domains which are stacked along the -axis, separated by twin boundaries parallel to the basal plane of the crystal, as shown in Fig. 1(a). For magnetic fields in the basal plane, the wave vector of the helical phase has opposite signs in the two twin domains, following the change of signs of the RSOC. The mismatch of the helical structures at the twin boundaries leads locally to supercurrents which cannot be screened completely, unlike in the bulk of the domains, as mentioned above. The resulting effective field influences the behavior of type-II superconductors in the mixed phase, i.e. between the lower and upper critical magnetic fields, and , respectively. In particular, this magneto-electric effect actually affects the lower and upper critical fields, a phenomenon we address here. It is important to notice that, for domains stacked along the -axis, there are two types of twin boundaries (see Fig.1), which behave differently in a magnetic field. As we will find below the critical fields are shifted in opposite way at these two types of twin boundaries, in one case, being higher, and in the other, lower than the bulk value (see Fig.1(b)).
The remainder of this paper is organized as follows. We first define the minimal model appropriate to eventually describe the novel features we report, and relevant to the bulk of a non-centrosymmetric superconductor with tetragonal symmetry. We then describe the different types of twin boundaries and the modifications we use to implement the existence of each type of twin-boundary. In the following section we thoroughly investigate the upper critical field . There, we show that the effect of twin boundaries can be quite striking, and exhibit the different consequences of “opposite” types of twin boundaries. We then turn to the case of the lower critical field, and argue that the twin boundaries may act as pinning planes for vortices. In both cases, namely and , we show both an analytical and a numerical analysis. Finally, we conclude and discuss experimental consequences.
Ii Model
(a) Crystal twin domains (white and gray regions) inside a single-crystal sample of a non-centrosymmetric superconductor, where the triangles denote the orientations of the axis of RSOC. The out (resp. in)-type twin boundary (parallel to the basal plane (
Figure 1: (a) Crystal twin domains (white and gray regions) inside a single-crystal sample of a non-centrosymmetric superconductor, where the triangles denote the orientations of the axis of RSOC. The out (resp. in)-type twin boundary (parallel to the basal plane (-)) is described as a boundary with a positive (resp. negative) value of (or equivalently ) in Eq. (7). An external magnetic field applied parallel to the twin boundaries yields local internal fields due to a mismatch of magneto-electric currents (blue arrows). (b) Schematic phase diagram of this system. Both upper and lower critical fields are shifted at the twin boundaries from their bulk values, suggesting that physical and curves (solid curves) are determined at the out- and in-type boundaries, respectively.
Superconductivity in twinned materials has drawn much interest for a long time in part because the SC transition temperature can be enhanced at twin boundaries due to soft phonons along the boundary plane or distinct two-dimensional electronic states Buzdin_rev . With such a enhancement, the upper and lower critical fields at twin boundaries should also locally be higher than the corresponding bulk values. In our study, we ignore the possibility of an enhanced SC critical temperature at the twin boundary, and assume a spatially uniform . We focus, rather, on the influence of magneto-electric effects in NCS in a magnetic field. The only feature of sample twinning which we take into account is the sign change of the coupling constant at the twin boundary. Moreover, we restrict the discussion to the case of a dominant -wave SC channel and, in particular, for simplicity, we ignore odd-parity components which, on symmetry grounds, could be admixed Springer .
The relevant Ginzburg-Landau (GL) theory can be derived from the BCS Hamiltonian including RSOC Kaur ; Samokhin ; Ikeda ; AS . The corresponding functional is obtained as usual as an expansion in the -wave order parameter ,
where the covariant gradient is defined as , where is the vector potential satisfying with the internal magnetic field, and where
with the bulk critical temperature, the Bohr magneton, the gyromagnetic ratio, where denote the densities of states of the two bands split by the RSOC (see Appendix), and with , , , and given in the Appendix which also explains details of our notations. The second term in (see Eq. (2)) includes the paramagnetic pair-breaking effect through the Zeeman field , and the last gradient term in Eq. (1) involving introduces the magneto-electric effect which couples the spin polarization to the supercurrent. This term changes signs under the mirror inversion . Thus, we emphasise, it is only allowed in systems where is not a symmetry operation, and is therefore quite specific to NCS. Its coefficient, , is connected to the RSOC and can be expressed as
where is the in-plane Fermi velocity. Note that the sign of is directly connected to the sign of the RSOC.
For the following discussion, we introduce three characteristic length scales: the SC coherence length , the magnetic length , and the London penetration depth , defined as
where , the uniform zero-field order parameter from the GL equations. For the in-plane field configuration, the bulk orbital-limiting and the paramagnetic-limiting (Pauli-limiting) fields at are given by
respectively, where is Euler’s constant, is the magnetic flux quantum, is the in-plane SC coherence length at , and parametrizes the anisotropy of the Fermi surface. The strength of the Pauli-paramagnetic effect is quantified by the Maki parameter
In the following, for concreteness, we apply the magnetic field along the -axis and assume no spatial dependence along this direction.
We turn now to a system with twin domains of ’up’ ( or ) and ’down’ ( or ) characters separated by twin boundaries with a geometry as shown in Fig. 1. The twin boundaries we consider are parallel to the --plane. As mentioned in the introduction, we distinguish two types of twin boundaries, the ‘top-up bottom-down’ (out-type) and ‘top-down bottom-up’ (in-type) twin boundaries. It will become clear below that the two behave differently in a magnetic field parallel to the twin boundary plane. Within our GL model, only the sign of distinguishes the twin domains, as is reflected by (see Eq. (II)). In practice, we implement the existence of twin boundaries by a sharp sign change of a space-dependent coefficient :
Because the change in the RSOC coefficient at the twin boundary happens on atomic length scales, the spatial variation of occurs on a much shorter length scale than the coherence length of the superconductor, so that the infinitely-abrupt change in implemented in Eq. (7) should therefore be qualitatively valid. Moreover, the existence of a sign change in in Eq. (7) can be understood from the viewpoint of symmetry. If we take the twin boundary plane as a mirror reflection plane, the twin domain system is invariant under the corresponding mirror operation. Correspondingly, the magneto-electric term involving with the space dependent described by Eq. (7) does not change signs under this symmetry operation, leaving the free energy Eq. (1) invariant.
Throughout this paper, positive and negative values of will be assigned to crystal domains of ‘up’ and ‘down’ characters, respectively. Therefore, the out (resp. in)-type twin boundary in Fig. 1(a) is described by positive (resp. negative) values of in Eq. (7).
Iii The upper critical field
First we address the nucleation of superconductivity in high magnetic fields, in the presence of twin boundaries parallel to the basal plane. This can be discussed using the linearized GL equations with an unscreened external field parallel to the twin boundary: the derivation of the instability condition of the normal state, which yields the upper critical field necessitates no more. Therefore we need only consider the terms quadratic in in Eq. (1). This quadratic form will be denoted in what follows.
We choose the gauge such that the vector potential is for a field along the -direction, and we impose periodic boundary conditions along the -direction. This allows us to represent the order parameter as
with the linear extension of the system in the -direction. First we tackle the problem variationally to obtain insight into the role of the twin boundary on . The validity of the variational approach will be confirmed later by the comparison to a numerical solution of the linearized GL equation.
iii.1 Variational approximation
The standard way to determine the upper critical field is equivalent to finding and solving the ground state of the Schrödinger equation for a one-dimensional harmonic oscillator introduced by the vector potential . For our gauge choice, this harmonic potential confines the order parameter along the -axis with its center at the twin boundary. However, here, the potential is modified through the additional term in which effectively introduces a small shift of the center in opposite directions on either side of the twin boundary. Still, at large distances away from the twin boundary the potential looks essentially harmonic and the following variational ansatz for the order parameter is therefore justified,
where the length scales are variational parameters which will be determined so as to minimize the free energy, . Inserting Eq. (9) into , we obtain
In the absence of the twin boundary, is just a constant . Then, as we will see in Eq. (III.2), the last term in Eq. (III.1) only yields an overall shift of the center of the harmonic potential and therefore has no effect on the orbital depairing field. (We will also find –see the right-hand side of Eq. (III.2)– that the paramagnetic depairing is suppressed by Kaur .) With the twin boundary, however, we encounter a real deformation of the potential. We can evaluate the integral Eq. (III.1) analytically,
The different Fourier components remain decoupled and we see immediately that only minimizes the variational free energy, resulting in
where . For fixed values of the field , we minimize with respect to , and then, the SC transition point (the highest transition temperature) is determined by the condition , i.e.,
where is the minimum value of the function . The contribution of the magneto-electric effect is incorporated in
(color online) The SC instability in the bulk (green dotted curves) and at the twin boundaries with
Figure 2: (color online) The SC instability in the bulk (green dotted curves) and at the twin boundaries with (red solid lines) and (blue dashed lines) for Maki parameter . (a) Temperature dependence of the upper critical field and (b) the corresponding behavior of the effective magnetic length , which, as depicted in (c), measures the extent of the SC pairing function along the axis centered at the twin boundary. This length scale is normalized by its bulk value.
Now we address the two types of twin boundaries, distinguished here by the sign of , corresponding to the out ()- or in-type () twin boundary as shown in Fig.1. Figure 2 (a) displays curves for the nucleation of the superconducting order parameter at the twin boundary with a moderate paramagnetic effect. For positive values of (out-type), the upper critical field at the twin boundary is enhanced compared to the bulk value, while for negative values of (in-type), it is lower than the bulk . In the latter case, superconductivity would surely appear first in the bulk and would be rather suppressed at the twin boundary. To understand why is enhanced or suppressed at the twin boundaries, we examine the effective magnetic length .
Fig. 2 (b) shows the temperature dependence of (for which the free energy is minimized), which measures the extent of the order parameter along the -axis. For positive , the effective magnetic length is larger than the corresponding bulk value, so that is more extended. This can be interpreted in terms of an effective magnetic field at the twin boundary, lower than the applied field: . In contrast, for negative , the effective field is enhanced at the twin boundary, suppressing the nucleation of SC there. This is consistent with the picture that the mismatch of the helical modulations in the two adjacent domains is compensated by an internal field which is added to or subtracted from the external field. Note that this magneto-electric effect depends on the Zeeman coupling and the stronger the paramagnetic limiting effect, the more pronounced it is. In Fig. 3 we show curves for a stronger paramagnetic effect, i.e. with a larger Maki parameter . There, besides the relative enhancement of the shift of the local , we also observe that the temperature dependence is different from the basically linear increase below in Fig.2. The rather strongly bent curve of seen here originates from the dominant paramagnetic-limiting compared to the orbital-limiting regime Adachi ; Mineev_para ; CeCoIn5_kappa .
Temperature dependence of the upper critical field
Figure 3: Temperature dependence of the upper critical field and the effective magnetic length (inset) for large Maki parameter , with the same notations as in Fig.2.
iii.2 Numerical solution of the GL equation
Now we turn to the numerical evaluation of the linearized GL equations, which allows us to assess the validity of our variational approach. We determine from the differential equation obtained by variationally differentiating with respect to the order parameter,
with a dimensionless coordinate. Because the solution of interest is symmetric under , we choose . This eigenvalue equation is most-efficiently solved by expanding in the basis of wave functions of the harmonic oscillator
where are the Hermite polynomials. Since satisfies the eigenvalue equation
the GL equation can be rewritten as,
where the relation has been used. Note that is symmetric, . The problem is reduced to finding the eigenvalues of the matrix . The superconducting instability follows from the equation , such that
where is the minimal eigenvalue of . At this point, we notice that in Eq. (20) corresponds to in Eq. (14), so that the validity of the variational approach can be checked by comparing and . As one can see in Fig.4, the two values and coincide well at all temperatures, suggesting that our variational approach is a good approximation and also validating the interpretation.
Comparison between the result obtained by the variational method
Figure 4: Comparison between the result obtained by the variational method (dashed curves) and the corresponding numerical result (circles) for (a) and (b). The same Maki parameter as Fig. 3, , is used.
Iv The lower critical field
In this section we address the effect of twin boundaries on the lower critical field. For this purpose we investigate the line energy of a single vortex on the twin boundary. Contrary to the previous section, we consider first the numerical solution, and then turn to a variational discussion in the London limit to give some insight into the mechanism. In order to simplify the discussion, and because we expect the results to not be qualitatively affected by this restriction, we assume an isotropic situation by setting . This allows us to formulate the problem simply in cylindrical coordinates with the magnetic field pointing, again, along the -axis.
iv.1 Magnetic flux distribution and
For the following discussion it will be convenient to express the order parameter and the vector potential in their Fourier expansion with respect to ,
Here, is related to the vector potential in the Cartesian coordinate system through the equation , and both and are assumed to take real values only. By substituting these expressions into , Eq. (1), and carrying out the integral with respect to , we obtain the GL free energy density per unit length in the direction, defined through,
Here, and
where the upper (resp. lower) case is for a vortex far from (resp. right on) the twin boundary. The magnetic field is given by
and holds. Note that, therefore, in the bulk without twin boundaries, the magneto-electric term proportional to vanishes and does not affect the line energy of the vortex.
Now, since a single vortex with its singularity at contains the total flux , we have the limiting conditions, for one vortex centered at ,
for and as well as,
Note that, because the magnetic field vanishes far from the vortex core ( for ), the magneto-electric term proportional to is not active at large distances from the vortex center, and thus, there, the condition for a usual single vortex , Eq. (27), can be used even in the case with the twin boundary. Now, the above constraints lead to the boundary conditions on and ,
The single vortex energy per unit length along the vortex axis is given by
and leads to the lower critical field,
Here, the Zeeman term in has been dropped because it is negligibly small at low fields, near , for any reasonable value of the Maki parameter.
Radial dependences of
Figure 5: Radial dependences of (a) and (b) for the twin boundaries with (red solid curves) and (blue dashed ones) at , where the parameters and are used. Without twin boundaries, only and are nonvanishing with almost the same spatial dependences as displayed here. All the components except and are multiplied by 30.
By numerically solving the GL equations and under the constraints of Eq. (IV.1), we investigate the spatial structure of and . As a typical example, in Fig.5 we plot spatial profiles of and for the large Maki parameter . One can see that, in contrast to the bulk case, where only and are nonvanishing, additional components and appear near the vortex center induced by the twin boundary. Since involves the phase factor , finite values of these components suggest the occurrence of a deformation of the magnetic flux distribution on the twin boundary. Also note that the sign of depends on the sign of .
Figure 6 (a) shows the curves at the two twin boundaries and in the bulk. The effect of the twin boundary on the temperature dependence of is qualitatively the same as that for : the lower critical field is enhanced (suppressed) for positive (negative) values of . The -dependent behavior of is natural because, as we have discussed in the previous section, positive yields a counter vortex field, while negative effectively strengthens the magnetic field stabilizing the vortex. This effect of the twin boundary can be also seen in the magnetic flux distribution. We introduce two length scales measuring the extension of the flux distribution in the - and -directions, and , which are defined by
with and .
The single-vortex instability in the bulk (green dotted curves) and at the twin boundaries with
Figure 6: The single-vortex instability in the bulk (green dotted curves) and at the twin boundaries with (red solid lines) and (blue dashed lines) for . (a) Temperature dependence of the lower critical field and (b) the corresponding behavior of the spatial extent of the magnetic flux along the -axis (upper panel) and along the -axis (lower panel). The magnetic flux distribution of a single-vortex with its core located at the twin boundary is sketched in (c). The inset of (a) shows the ratio of at the twin boundary to its bulk value.
Figure 6 (b) shows the temperature dependence of and normalized by the bulk value . In the bulk, is satisfied because we assumed isotropy. For positive , the magnetic flux is extended in the -direction and squeezed in the -direction, leaving the total flux to be . This anisotropy is caused by the magnetic field induced through the magneto-electric coupling. For positive the effective field on the twin boundary is smaller than the bare field of the vortex, so that the stability of superconductivity against the bare field is higher on the twin boundary than away from it. Thus, the magnetic flux extends along the twin boundary (-direction) to lower the energy. Conversely, for negative the induced field is opposite, leading to a flux distribution compressed along the -direction.
iv.2 Extended London model
We will now focus on the line energy of a vortex on a twin boundary using an extended London theory incorporating the magneto-electric coupling. For this purpose we fix the shape of the vortex in the London limit as with the radius and the step function taking care of the fact that the vortex core extends over a coherence length , and a smooth real function of space coordinates. In this limit, the magnetic field , the SC current , and the vortex-line energy for an ordinary -wave superconductor are given by
where is a modified Bessel function Tinkham . Using the expression of Eq. (IV.2), we evaluate variationally the change of the vortex-line energy due to the magneto-electric coupling by simply adding the integral of in Eq. (1), which leads to
The total vortex energy in the presence of the twin boundaries, , is then
where . Equation (36) shows good agreement with the numerical result shown in the inset of Fig.6 (a), with the dependence, as well as with the rather small difference . The shift of due to the twin boundaries increases with increasing RSOC, i.e. with increasing , and with increasing Pauli-paramagnetic effect quantified by the Maki parameter , but is diminishes with increasing GL parameter .
We may also view as the potential energy of a vortex, which is zero in the bulk, but varies smoothly as the twin boundary is approached. This potential is repulsive for positive and attractive for negative . In the latter case vortices can more easily penetrate the sample along the twin boundary than into the bulk. Thus, vortices should line up on this type of twin boundary. Conversely, when is positive, vortices avoid twin boundaries, which are then (weak) barriers for the crossing of vortices. Quantitatively, however, this local shift of the lower critical field is much weaker than that of the upper critical field and is most likely not of experimental relevance.
V Conclusion
We have examined the influence of magneto-electric effects on the upper and lower critical fields in a non-centrosymmetric superconductor with twin boundaries. Considering the case of tetragonal crystal symmetry with Rashba spin-orbit coupling, appropriate for example for twin boundaries in CePtSi, we found that two types of twin boundaries parallel to the basal plane exist, which separate domains of opposite RSOC. Magneto-electric effects which are irrelevant for the behavior in the bulk, enhance or reduce the upper and lower critical fields at the twin boundaries depending on the type of the latter. Although our analysis is based on a Ginzburg-Landau formulation for an -wave order parameter and ignores the admixture of an odd-parity pairing component, the results obtained should be qualitatively valid beyond the temperature range where the GL theory presented here is valid.
We found that the effect on the lower critical field is most likely too small to be observed, but the fact that for one type of twin boundary the upper critical field is enhanced could indeed be of experimental relevance. Since the volume fraction of the crystal that is actually influenced by the twin boundaries is generally small, experimental probes quite sensitive to superconductivity such as magnetic torque and AC susceptibility should provide the best tools to detect the enhanced at the twin boundary. As we have seen, some twin boundaries suppress and and, thus, may act as pinning planes for vortices in the mixed phase. Any non-centrosymmetric material, as discussed in our model, should display alternating in- and out-type twin boundaries, such that both kinds of observable features, i.e., the enhanced (out-type), as well as the vortex pinning (in-type) due to a reduced , could be potentially seen in such a single sample. Together with the observation of these features, detecting crystal domains directly with a real space imaging method would also provide important information to investigate further novel effects due to twin boundaries, as addressed here.
Finally, we would like to note that one may create a twin-boundary-like structure by contacting two crystals of opposite RSOC to one another along the -axis, forming a planar Josephson junction. In that case also, two types of Josephson junctions exist, and, in particular, the Josephson vortices do display distinct features Savary .
We are grateful to G. Eguchi, S. Yonezawa and Y. Maeno for motivating discussions. This work is supported by a Grant-in-Aid for Scientific Research (Grant No. 25800194) and a grant by the Swiss Nationalfonds. Moreover, K.A. thanks to the Pauli Centre for Theoretical Studies of ETH Zurich for hospitality during his stay.
Vi Appendix
vi.1 GL coefficients in Eq. (1)
The GL coefficients in Eq. (1) have been derived elsewhere Kaur ; Samokhin ; Ikeda ; AS and are given by
where denote the density of states of the two bands split by the RSOC, denotes the Fermi velocity in the direction, represents the angle average of on the Fermi surface, and . In deriving Eq. (1), we restrict ourselves to and .
vi.2 GL equations for Eq. (22)
The saddle point equations with respect to and , and , yield the GL equations |
c51755511d586999 | Project B1 • Klein-Gordon-Zakharov systems in high-frequency regimes (7/2015 - 6/2019)
Principal investigators
Prof. Dr. Guido Schneider (7/2015 - 6/2019)
JProf. Dr. Katharina Schratz (7/2015 - 6/2019)
Project summary
The aim of this project is to analyze the Klein-Gordon-Zakharov system in high plasma frequency and (simultaneous) subsonic regimes. Due to the highly oscillatory nature of the problem classical numerical schemes break down, in particular in certain singular limits. Therefore, we need to develop new analytic techniques and efficient numerical schemes to overcome severe step size restrictions and huge computational costs.
Numerical challenges of highly oscillatory problems
If a solution of the underlying equation is highly oscillatory, it becomes very challenging for numerical methods to resolve the oscillations and to yield a good numerical approximation. In order to compute a numerical approximation of a highly oscillatory function (see Figure 1 below) we have to discretize the interval with a finite number of points. At these grid points we compute approximations of the function values.
Figure 1 Plot of a highly oscillatory function in blue. Grid points of the discretization in black. Approximation in red.
Afterwards we construct our numerical solution by interpolating between the approximated values. The figure shows an example where the approximation fails completely even though we evaluate the function values exactly at the discrete (finite) points. We can improve the approximation by introducing a finer grid, which, however, requires much more computational and memory costs and therefore will not yield an efficient numerical scheme. This makes the efficient approximation of highly oscillatory differential equations an ongoing challenge in numerical analysis.
The Klein-Gordon-Zakharov system
The starting point of this project was the Klein Gordon Zakhavov (KGZ) system \[ \begin{equation*} c^{-2}\partial_{tt}^2 z = \Delta z - c^2 z \pm nz, \\ \alpha^{-2}\partial_{tt}^2 n = \Delta n + \Delta \vert z\vert^2\quad\,\,\, \end{equation*} \] in certain high frequency regimes, i.e., for large \(c\) and/or large \(\alpha\), where the solution becomes highly oscillatory in time (see the Figure 2 below).
Plot of the solution z of the Klein-Gordon-Zakharov system for different values of c at a fixed spatial point.
Figure 2 Plot of the solution \(z\) of the Klein-Gordon-Zakharov system for different values of \(c\) at a fixed spatial point.
We are interested in a robust analytic and numerical description of the KGZ system for such values of \(c\) and \(\alpha\). Such a description can be obtained via the limit systems for \(c\to\infty\) and/or \(\alpha\to\infty\). This analysis is particularly of great interest from a numerical point of view: Correctly resolving this highly oscillatory behavior in the exact solution in these regimes is numerically very delicate.
Severe time step restrictions need to be imposed and this requires a great deal of computational costs. An asymptotic ansatz allows to construct efficient time integrators as the highly oscillatory parts in the exact solution could be filtered out explicitly. Thus, the numerical task could be reduced to the time integration of the corresponding non-oscillatory limit systems, which can be carried out very efficiently without any additional time step restriction.
The construction of such robust, efficient time integrators involves a careful analysis of the numerical time integrators for the corresponding limit systems and rigorous error estimates between true solutions of the original system and the approximations constructed via the limit system.
The KGZ system turned out to be a well chosen prototype problem since the tools developed for the KGZ system can be used to establish similar results for a wider class of problems. Moreover, extending the original question, uniformly accurate methods for KGZ like systems have been developed, i.e., the numerical costs for such schemes are independent of the \(c\) and \(\alpha\) and not only valid for \(c\) large and/or \(\alpha\) large.
• Analytic results that the various limit systems make correct predictions about the dynamics for \(c\) large and/or \(\alpha\) large can be found in [DSS16] and [BSSZ19].
In the first paper it turned out that limit systems in the limit \(c\) fixed and \(\alpha\to\infty\) depends whether the KGZ system is posed on \(\mathbb{R}^3\) or \(\mathbb{T}^3\). The abstract approximation theorem in the second paper applies to a number of semilinear systems, such as the Dirac-Klein-Gordon system, the Klein-Gordon-Zakharov system, and a mean field polaron model. It extracts the common features of scattered results in the literature, but also gains an approximation result for the Dirac-Klein-Gordon system which has not been documented in the literature before. The abstract approximation theorem is sharp in the sense that there exists a quasilinear system of the same structure, namely the Zakharov system with a 'wrong' sign in the nonlinearity, where the regular limit system, namely the NLS equation, makes wrong predictions.
• We propose asymptotic consistent exponential-type integrators for the Klein-Gordon-Schrödinger system. This novel class of integrators allows us to solve the system from slowly varying relativistic up to challenging highly oscillatory non-relativistic regimes without any step size restriction. In particular, our first- and second-order exponential-type integrators are asymptotically consistent in the sense of asymptotically converging to the corresponding decoupled free Schrödinger limit system. See [BKS18].
• We introduce efficient and robust exponential-type integrators for Klein-Gordon equations which resolve the solution in the relativistic regime as well as in the highly-oscillatory nonrelativistic regime without any step-size restriction under the same regularity assumptions on the initial data required for the integration of the corresponding nonlinear Schrödinger (NLS) limit system. In contrast to previous works we do not employ any asymptotic/multiscale expansion of the solution. This allows us to derive uniform convergent schemes under far weaker regularity assumptions on the exact solution. In addition, the newly derived first- and second-order exponential-type integrators converge to the classical Lie, respectively, Strang splitting in the nonlinear Schrödinger limit. See [BFS18].
• We present a novel class of oscillatory integrators for the KGZ system which are uniformly accurate with respect to the plasma frequency \(c\). Convergence holds from the slowly-varying low-plasma up to the highly-oscillatory high-plasma frequency regimes without any step size restriction and, especially, uniformly in \(c\). The introduced scheme is moreover asymptotic consistent and approximates the solutions of the corresponding Zakharov limit system in the high-plasma frequency limit \(c\to\infty\). We establish rigorous error estimates for the the introduced oscillatory integrator and numerically underline its uniform convergence property. The derivation of uniformly accurate methods for the KGZ system in the subsonic limit regime (\(\alpha\to\infty\)) and also in the simultaneous limit regimes (\((c,\alpha)\to\infty\)) will be the subject of future research. See [BS19].
1. and . Asymptotic preserving trigonometric integrators for the quantum Zakharov system. BIT, 61(1):61–81, March . URL [preprint] [bibtex]
2. , , and . Effective numerical simulation of the Klein–Gordon–Zakharov system in the Zakharov limit. In W. Dörfler, M. Hochbruck, D. Hundertmark, W. Reichel, A. Rieder, R. Schnaubelt, and B. Schörkhuber, editors, Mathematics of Wave Phenomena, Trends in Mathematics, pages 37–48, October . Birkhäuser Basel. [bibtex]
3. , , and . Randomized exponential integrators for modulated nonlinear Schrödinger equations. IMA J. Numer. Anal., 40(4):2143–2162, October . URL [preprint] [bibtex]
4. and . On the comparison of asymptotic expansion techniques for the nonlinear Klein–Gordon equation in the nonrelativistic limit regime. Discrete Contin. Dyn. Syst. Ser. B, 25(8):2841–2865, August . URL [preprint] [bibtex]
5. . The KdV approximation for a system with unstable resonances. Math. Methods Appl. Sci., 43(6):3185–3199, April . URL [preprint] [bibtex]
6. , , and . Splitting methods for nonlinear Dirac equations with Thirring type interaction in the nonrelativistic limit regime. J. Computat. Appl. Math., 112494, September . URL Online first, in press. [preprint] [bibtex]
7. , , , and . Effective slow dynamics models for a class of dispersive systems. J. Dyn. Diff. Equat., 1–33, September . URL Online first. [preprint] [bibtex]
8. , , , , and . Trigonometric integrators for quasilinear wave equations. Math. Comp., 88(316):717–749, March . URL [preprint] [bibtex]
9. and . Uniformly accurate oscillatory integrators for the Klein–Gordon–Zakharov system from low- to high-plasma frequency regimes. SIAM J. Numer. Anal., 57(1):429–457, February . URL [preprint] [bibtex]
10. and . Low regularity exponential-type integrators for semilinear Schrödinger equations. Found. Comput. Math., 18(3):731–755, June . URL [preprint] [bibtex]
11. , , and . Uniformly accurate exponential-type integrators for Klein–Gordon equations with asymptotic convergence to the classical NLS splitting. Math. Comp., 87(311):1227–1254, May . URL [preprint] [bibtex]
12. , , and . Asymptotic consistent exponential-type integrators for Klein–Gordon–Schrödinger systems from relativistic to non-relativistic regimes. Electron. Trans. Numer. Anal., 48:63–80, March . URL [bibtex]
13. and . Trigonometric time integrators for the Zakharov system. IMA J. Numer. Anal., 37(4):2042–2066, October . URL [preprint] [bibtex]
14. and . An exponential-type integrator for the KdV equation. Numer. Math., 136(4):1117–1137, August . URL [preprint] [bibtex]
15. and . Efficient time integration of the Maxwell–Klein–Gordon equation in the non-relativistic limit regime. J. Comput. Appl. Math., 316:247–259, May . URL Selected Papers from NUMDIFF-14. [preprint] [bibtex]
16. , , and . From the Klein–Gordon–Zakharov system to the Klein–Gordon equation. Math. Methods Appl. Sci., 39(18):5371–5380, December . URL [preprint] [bibtex]
1. . Uniformly accurate methods for Klein–Gordon type equations. PhD thesis, Karlsruhe Institute of Technology (KIT), July . [bibtex]
2. . Numerical integrators for Maxwell–Klein–Gordon and Maxwell–Dirac systems in highly to slowly oscillatory regimes. PhD thesis, Karlsruhe Institute of Technology (KIT), August . [bibtex]
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e247886c02cf42a1 | Applied Bohmian Mechanics : From Nanoscale Systems to Cosmology
Regular price $149.95 $149.95 Sale
Most textbooks explain quantum mechanics as a story where each step follows naturally from the one preceding it. However, the development of quantum mechanics was exactly the opposite. It was a zigzag route, full of personal disputes where scientists were forced to abandon well-established classical concepts and to explore new and imaginative pathways. Some of the explored routes were successful in providing new mathematical formalisms capable of predicting experiments at the atomic scale. However, even such successful routes were painful enough, so that relevant scientists like Albert Einstein and Erwin Schrödinger decided not to support them.
In this book, the authors demonstrate the huge practical utility of another of these routes in explaining quantum phenomena in many different research fields. Bohmian mechanics, the formulation of the quantum theory pioneered by Louis de Broglie and David Bohm, offers an alternative mathematical formulation of quantum phenomena in terms of quantum trajectories. Novel computational tools to explore physical scenarios that are currently computationally inaccessible, such as many-particle solutions of the Schrödinger equation, can be developed from it.
EAN: 9781000650105 |
27aa6a48853cf70f | As is well-known, the Maxwell equations can be phrased vectorially as,
\begin{align} \nabla \cdot \mathbf E &= \frac{\rho_f}{\varepsilon}, &\text{Gauss's law,}\\\ \nabla \cdot \mathbf B &= 0, &\text{No-name law (no monopoles),}\\\ \nabla \times \mathbf E &= - \partial_t \mathbf B, &\text{Faraday's law,}\\\ \nabla \times \mathbf B &= \mu \varepsilon\partial_t \mathbf E + \mu \mathbf J_f, &\text{Ampere's law}. \end{align} There are many equivalent formulations, for instance in terms of potentials and Gauges. My question is related to the regularity of the solution pair $(\mathbf E, \mathbf B)$. As the equations are hyperbolic and my knowledge is largely in elliptic equations (which seem to be completely different beasts to handle... I have heard: "Partial differential equations are like a zoo, even if the animals look the same you might have to treat them differently").
Regularity questions:
1. What are the standard references for the regularity (of the solutions) of the Maxwell equations?
2. If we have the equations on domains, what is the dependence of the regularity of the solutions in terms of the regularity of the boundary?
3. Which formulations are most convenient to prove regularity properties for hyperbolic equations? As I have said above, there exist many equivalent ones.
4. Is there any work done, and what work, on the regularity questions for the Maxwell equations in a functional analytic framework? Here I mean phrasing the equations as a ordinary differential equations in a Banach space, just as we would have the analysis of the heat kernel as a convolution-type operator for the heat equation. How about harmonic analysis?
5. Has there been any work done on the Maxwell equations in terms of gradient flows on metric spaces (as in the work of Felix Otto et al., for the Fokker-Planck equation, sorry, the Ornstein-Uhlenbeck process)?
Before the question gets closed before it is "overly broad, rhetoric or whatever", please note that my question is mainly about the regularity for Maxwell equations and if one of the other questions can get answered or get pointed to a reference in the process, that would be nice. My background in PDE is mainly from the elliptic side, I do not have much knowledge about their hyperbolic ones, other than the trivial results.
There are a few, not many, books on hyperbolic equations. You might have a look to that of S. Benzoni-Gavage and myself: Multi-dimensional hyperbolic partial differential equations. First order systems and applications, Oxford Mathematical Monographs, Oxford University Press (2007).
A basic fact of hyperbolic systems of PDEs is that the Cauchy problem is well-posed in both directions of time. Therefore the regularity of the solution cannot be improved as time increase, contrary to the parabolic case. Also, this implies that such systems cannot be recast as gradient flows; instead, some of them can be reformulated as Hamiltonian system (say, if the semi-group is reversible).
That said, there exists nevertheless some regularity properties. On the one hand, the singularities are polarized. This means that the solution is smooth along non-characteristic directions, and most of (but not all) the solution is smooth even in characteristic directions. Let me take the example of the wave equation $$\partial_t^2u=\Delta_x u$$ in ${\mathbb R}^{1+d}$. Then the wave-front set is invariant under the bi-characteristic flow $$\frac{dx}{dt}=p,\qquad\frac{dp}{dt}=-x.$$ A by-product (which can be proved directly by an integral formula of the solution) is that if the initial data $u(t=0,\cdot)$, $\partial_tu(t=0,\cdot)$ is smooth away from $x=0$, then the solution is smooth away from $|x|=|t|$. However the wave-frontset approach tells you much more.
On another hand, the decay of the initial data at infinity implies some space-time integrability of the solution. These properties are not directly related to hyperbolicity. They are consequences of the dispersion. In the case of the wave equation, this is the fact that the characteristic cone $|x|=|t|$ has not flat part. Such integrability statements are know as Strichartz-like inequalities.
Finally, the ODE point of view is adopted by Klainerman, Machedon, Christodoulou and others, mixed with Strichartz inequalities, to prove the well-posedness of the Cauchy problem for semi-linear hyperbolic systems, like Einstein equations of general relativity.
• $\begingroup$ So, the statement you make is that the $C_0$-semigroup would actually be a group? As in the Schrödinger equation, which is a Weyl rotation of the heat equation? I understood the 'cannot be made smoother' in this context as the fact that you can 'go back in time' and would make things less smooth. I guess this is what you are saying but more mathematically. I still have to wrap my head around why a rotation in the complex plane would give such profound consequences in the behavior of the heat equation. Perhaps, that is for another question when I thought more about it. (...) $\endgroup$
– JT_NL
Nov 11 '12 at 17:18
• $\begingroup$ (...) Klainerman is the name I have heard before in this context. I will look them up (I have seen the Strichartz-like estimates before). Thanks for the reference, I will request your book from the library. Nice to have a real expert answering your question online. $\endgroup$
– JT_NL
Nov 11 '12 at 17:19
• $\begingroup$ @Jonas. You interpret correctly. $\endgroup$ Nov 11 '12 at 19:46
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0830f700c39d6d64 | Tag Archives: quantum mechanics
Spacetime Without the Time
anti-de-sitter-spaceSince they were first dreamed up explanations of the very small (quantum mechanics) and the very large (general relativity) have both been highly successful at describing their respective spheres of influence. Yet, these two descriptions of our physical universe are not compatible, particularly when it comes to describing gravity. Indeed, physicists and theorists have struggled for decades to unite these two frameworks. Many agree that we need a new theory (of everything).
One new idea, from theorist Erik Verlinde of the University of Amsterdam, proposes that time is an emergent construct (it’s not a fundamental building block) and that dark matter is an illusion.
From Quanta:
Theoretical physicists striving to unify quantum mechanics and general relativity into an all-encompassing theory of quantum gravity face what’s called the “problem of time.”
In quantum mechanics, time is universal and absolute; its steady ticks dictate the evolving entanglements between particles. But in general relativity (Albert Einstein’s theory of gravity), time is relative and dynamical, a dimension that’s inextricably interwoven with directions x, y and z into a four-dimensional “space-time” fabric. The fabric warps under the weight of matter, causing nearby stuff to fall toward it (this is gravity), and slowing the passage of time relative to clocks far away. Or hop in a rocket and use fuel rather than gravity to accelerate through space, and time dilates; you age less than someone who stayed at home.
Unifying quantum mechanics and general relativity requires reconciling their absolute and relative notions of time. Recently, a promising burst of research on quantum gravity has provided an outline of what the reconciliation might look like — as well as insights on the true nature of time.
As I described in an article this week on a new theoretical attempt to explain away dark matter, many leading physicists now consider space-time and gravity to be “emergent” phenomena: Bendy, curvy space-time and the matter within it are a hologram that arises out of a network of entangled qubits (quantum bits of information), much as the three-dimensional environment of a computer game is encoded in the classical bits on a silicon chip. “I think we now understand that space-time really is just a geometrical representation of the entanglement structure of these underlying quantum systems,” said Mark Van Raamsdonk, a theoretical physicist at the University of British Columbia.
Researchers have worked out the math showing how the hologram arises in toy universes that possess a fisheye space-time geometry known as “anti-de Sitter” (AdS) space. In these warped worlds, spatial increments get shorter and shorter as you move out from the center. Eventually, the spatial dimension extending from the center shrinks to nothing, hitting a boundary. The existence of this boundary — which has one fewer spatial dimension than the interior space-time, or “bulk” — aids calculations by providing a rigid stage on which to model the entangled qubits that project the hologram within. “Inside the bulk, time starts bending and curving with the space in dramatic ways,” said Brian Swingle of Harvard and Brandeis universities. “We have an understanding of how to describe that in terms of the ‘sludge’ on the boundary,” he added, referring to the entangled qubits.
The states of the qubits evolve according to universal time as if executing steps in a computer code, giving rise to warped, relativistic time in the bulk of the AdS space. The only thing is, that’s not quite how it works in our universe.
Here, the space-time fabric has a “de Sitter” geometry, stretching as you look into the distance. The fabric stretches until the universe hits a very different sort of boundary from the one in AdS space: the end of time. At that point, in an event known as “heat death,” space-time will have stretched so much that everything in it will become causally disconnected from everything else, such that no signals can ever again travel between them. The familiar notion of time breaks down. From then on, nothing happens.
On the timeless boundary of our space-time bubble, the entanglements linking together qubits (and encoding the universe’s dynamical interior) would presumably remain intact, since these quantum correlations do not require that signals be sent back and forth. But the state of the qubits must be static and timeless. This line of reasoning suggests that somehow, just as the qubits on the boundary of AdS space give rise to an interior with one extra spatial dimension, qubits on the timeless boundary of de Sitter space must give rise to a universe with time — dynamical time, in particular. Researchers haven’t yet figured out how to do these calculations. “In de Sitter space,” Swingle said, “we don’t have a good idea for how to understand the emergence of time.”
Read the entire article here.
Image: Image of (1 + 1)-dimensional anti-de Sitter space embedded in flat (1 + 2)-dimensional space. The t1- and t2-axes lie in the plane of rotational symmetry, and the x1-axis is normal to that plane. The embedded surface contains closed timelike curves circling the x1 axis, though these can be eliminated by “unrolling” the embedding (more precisely, by taking the universal cover). Courtesy: Krishnavedala. Wikipedia. Creative Commons Attribution-Share Alike 3.0.
The Collapsing Wave Function
Schrodinger-equationOnce in every while I have to delve into the esoteric world of quantum mechanics. So, you will have to forgive me.
Since it was formalized in the mid-1920s QM has been extremely successful at describing the behavior of systems at the atomic scale. Two giants of the field — Niels Bohr and Werner Heisenberg — devised the intricate mathematics behind QM in 1927. Since then it has become known as the Copenhagen Interpretation, and has been widely and accurately used to predict and describe the workings of elementary particles and forces between them.
Yet recent theoretical stirrings in the field threaten to turn this widely held and accepted framework on its head. The Copenhagen Interpretation holds that particles do not have definitive locations until they are observed. Rather, their positions and movements are defined by a wave function that describes a spectrum of probabilities, but no certainties.
Rather understandably, this probabilistic description of our microscopic world tends to unnerve those who seek a more solid view of what we actually observe. Enter Bohmian mechanics, or more correctly, the De BroglieBohm theory of quantum mechanics. An increasing number of present day researchers and theorists are revisiting this theory, which may yet hold some promise.
From Wired:
As with the Copenhagen view, there’s a wave function governed by the Schrödinger equation. In addition, every particle has an actual, definite location, even when it’s not being observed. Changes in the positions of the particles are given by another equation, known as the “pilot wave” equation (or “guiding equation”). The theory is fully deterministic; if you know the initial state of a system, and you’ve got the wave function, you can calculate where each particle will end up.
That may sound like a throwback to classical mechanics, but there’s a crucial difference. Classical mechanics is purely “local”—stuff can affect other stuff only if it is adjacent to it (or via the influence of some kind of field, like an electric field, which can send impulses no faster than the speed of light). Quantum mechanics, in contrast, is inherently nonlocal. The best-known example of a nonlocal effect—one that Einstein himself considered, back in the 1930s—is when a pair of particles are connected in such a way that a measurement of one particle appears to affect the state of another, distant particle. The idea was ridiculed by Einstein as “spooky action at a distance.” But hundreds of experiments, beginning in the 1980s, have confirmed that this spooky action is a very real characteristic of our universe.
Read the entire article here.
Image: Schrödinger’s time-dependent equation. Courtesy: Wikipedia.
Universal Amniotic Fluid
Another day, another physics paper describing the origin of the universe. This is no wonder. Since the development of general relativity and quantum mechanics — two mutually incompatible descriptions of our reality — theoreticians have been scurrying to come up with a grand theory, a rapprochement of sorts. This one describes the universe as a quantum fluid, perhaps made up of hypothesized gravitons.
From Nature Asia:
The prevailing model of cosmology, based on Einstein’s theory of general relativity, puts the universe at around 13.8 billion years old and suggests it originated from a “singularity” – an infinitely small and dense point – at the Big Bang.
To understand what happened inside that tiny singularity, physicists must marry general relativity with quantum mechanics – the laws that govern small objects. Applying both of these disciplines has challenged physicists for decades. “The Big Bang singularity is the most serious problem of general relativity, because the laws of physics appear to break down there,” says Ahmed Farag Ali, a physicist at Zewail City of Science and Technology, Egypt.
In an effort to bring together the laws of quantum mechanics and general relativity, and to solve the singularity puzzle, Ali and Saurya Das, a physicist at the University of Lethbridge in Alberta Canada, employed an equation that predicts the development of singularities in general relativity. That equation had been developed by Das’s former professor, Amal Kumar Raychaudhuri, when Das was an undergraduate student at Presidency University, in Kolkata, India, so Das was particularly familiar and fascinated by it.
When Ali and Das made small quantum corrections to the Raychaudhuri equation, they realised it described a fluid, made up of small particles, that pervades space. Physicists have long believed that a quantum version of gravity would include a hypothetical particle, called the graviton, which generates the force of gravity. In their new model — which will appear in Physics Letters B in February — Ali and Das propose that such gravitons could form this fluid.
To understand the origin of the universe, they used this corrected equation to trace the behaviour of the fluid back through time. Surprisingly, they found that it did not converge into a singularity. Instead, the universe appears to have existed forever. Although it was smaller in the past, it never quite crunched down to nothing, says Das.
“Our theory serves to complement Einstein’s general relativity, which is very successful at describing physics over large distances,” says Ali. “But physicists know that to describe short distances, quantum mechanics must be accommodated, and the quantum Raychaudhui equation is a big step towards that.”
The model could also help solve two other cosmic mysteries. In the late 1990s, astronomers discovered that the expansion of the universe is accelerating due the presence of a mysterious dark energy, the origin of which is not known. The model has the potential to explain it since the fluid creates a minor but constant outward force that expands space. “This is a happy offshoot of our work,” says Das.
Astronomers also now know that most matter in the universe is in an invisible mysterious form called dark matter, only perceptible through its gravitational effect on visible matter such as stars. When Das and a colleague set the mass of the graviton in the model to a small level, they could make the density of their fluid match the universe’s observed density of dark matter, while also providing the right value for dark energy’s push.
Read the entire article here.
The Arrow of Time
Arthur_Stanley_EddingtonEinstein’s “spooky action at a distance” and quantum information theory (QIT) may help explain the so-called arrow of time — specifically, why it seems to flow in only one direction. Astronomer Arthur Eddington first described this asymmetry in 1927, and it has stumped theoreticians ever since.
At a macro-level the classic and simple example is that of an egg breaking when it hits your kitchen floor: repeat this over and over, and it’s likely that the egg will always make for a scrambled mess on your clean tiles, but it will never rise up from the floor and spontaneously re-assemble in your slippery hand. Yet at the micro-level, physicists know their underlying laws apply equally in both directions. Enter two new tenets of the quantum world that may help us better understand this perplexing forward flow of time: entanglement and QIT.
From Wired:
Read the entire article here.
Image: English astrophysicist Sir Arthur Stanley Eddington (1882–1944). Courtesy: George Grantham Bain Collection (Library of Congress).
God Is a Thermodynamicist
Physicists and cosmologists are constantly postulating and testing new ideas to explain the universe and everything within. Over the last hundred years or so, two such ideas have grown to explain much about our cosmos, and do so very successfully — quantum mechanics, which describes the very small, and relativity which describes the very large. However, these two views do no reconcile, leaving theoreticians and researchers looking for a more fundamental theory of everything. One possible idea banishes the notions of time and gravity — treating them both as emergent properties of a deeper reality.
From New Scientist:
In its origins, thermodynamics is a theory about heat: how it flows and what it can be made to do (see diagram). The French engineer Sadi Carnot formulated the second law in 1824 to characterise the mundane fact that the steam engines then powering the industrial revolution could never be perfectly efficient. Some of the heat you pumped into them always flowed into the cooler environment, rather than staying in the engine to do useful work. That is an expression of a more general rule: unless you do something to stop it, heat will naturally flow from hotter places to cooler places to even up any temperature differences it finds. The same principle explains why keeping the refrigerator in your kitchen cold means pumping energy into it; only that will keep warmth from the surroundings at bay.
A few decades after Carnot, the German physicist Rudolph Clausius explained such phenomena in terms of a quantity characterising disorder that he called entropy. In this picture, the universe works on the back of processes that increase entropy – for example dissipating heat from places where it is concentrated, and therefore more ordered, to cooler areas, where it is not.
That predicts a grim fate for the universe itself. Once all heat is maximally dissipated, no useful process can happen in it any more: it dies a “heat death”. A perplexing question is raised at the other end of cosmic history, too. If nature always favours states of high entropy, how and why did the universe start in a state that seems to have been of comparatively low entropy? At present we have no answer, and later I will mention an intriguing alternative view.
Perhaps because of such undesirable consequences, the legitimacy of the second law was for a long time questioned. The charge was formulated with the most striking clarity by the British physicist James Clerk Maxwell in 1867. He was satisfied that inanimate matter presented no difficulty for the second law. In an isolated system, heat always passes from the hotter to the cooler, and a neat clump of dye molecules readily dissolves in water and disperses randomly, never the other way round. Disorder as embodied by entropy does always increase.
Maxwell’s problem was with life. Living things have “intentionality”: they deliberately do things to other things to make life easier for themselves. Conceivably, they might try to reduce the entropy of their surroundings and thereby violate the second law.
Information is power
Such a possibility is highly disturbing to physicists. Either something is a universal law or it is merely a cover for something deeper. Yet it was only in the late 1970s that Maxwell’s entropy-fiddling “demon” was laid to rest. Its slayer was the US physicist Charles Bennett, who built on work by his colleague at IBM, Rolf Landauer, using the theory of information developed a few decades earlier by Claude Shannon. An intelligent being can certainly rearrange things to lower the entropy of its environment. But to do this, it must first fill up its memory, gaining information as to how things are arranged in the first place.
This acquired information must be encoded somewhere, presumably in the demon’s memory. When this memory is finally full, or the being dies or otherwise expires, it must be reset. Dumping all this stored, ordered information back into the environment increases entropy – and this entropy increase, Bennett showed, will ultimately always be at least as large as the entropy reduction the demon originally achieved. Thus the status of the second law was assured, albeit anchored in a mantra of Landauer’s that would have been unintelligible to the 19th-century progenitors of thermodynamics: that “information is physical”.
But how does this explain that thermodynamics survived the quantum revolution? Classical objects behave very differently to quantum ones, so the same is presumably true of classical and quantum information. After all, quantum computers are notoriously more powerful than classical ones (or would be if realised on a large scale).
The reason is subtle, and it lies in a connection between entropy and probability contained in perhaps the most profound and beautiful formula in all of science. Engraved on the tomb of the Austrian physicist Ludwig Boltzmann in Vienna’s central cemetery, it reads simply S = k log W. Here S is entropy – the macroscopic, measurable entropy of a gas, for example – while k is a constant of nature that today bears Boltzmann’s name. Log W is the mathematical logarithm of a microscopic, probabilistic quantity W – in a gas, this would be the number of ways the positions and velocities of its many individual atoms can be arranged.
On a philosophical level, Boltzmann’s formula embodies the spirit of reductionism: the idea that we can, at least in principle, reduce our outward knowledge of a system’s activities to basic, microscopic physical laws. On a practical, physical level, it tells us that all we need to understand disorder and its increase is probabilities. Tot up the number of configurations the atoms of a system can be in and work out their probabilities, and what emerges is nothing other than the entropy that determines its thermodynamical behaviour. The equation asks no further questions about the nature of the underlying laws; we need not care if the dynamical processes that create the probabilities are classical or quantum in origin.
There is an important additional point to be made here. Probabilities are fundamentally different things in classical and quantum physics. In classical physics they are “subjective” quantities that constantly change as our state of knowledge changes. The probability that a coin toss will result in heads or tails, for instance, jumps from ½ to 1 when we observe the outcome. If there were a being who knew all the positions and momenta of all the particles in the universe – known as a “Laplace demon”, after the French mathematician Pierre-Simon Laplace, who first countenanced the possibility – it would be able to determine the course of all subsequent events in a classical universe, and would have no need for probabilities to describe them.
In quantum physics, however, probabilities arise from a genuine uncertainty about how the world works. States of physical systems in quantum theory are represented in what the quantum pioneer Erwin Schrödinger called catalogues of information, but they are catalogues in which adding information on one page blurs or scrubs it out on another. Knowing the position of a particle more precisely means knowing less well how it is moving, for example. Quantum probabilities are “objective”, in the sense that they cannot be entirely removed by gaining more information.
That casts in an intriguing light thermodynamics as originally, classically formulated. There, the second law is little more than impotence written down in the form of an equation. It has no deep physical origin itself, but is an empirical bolt-on to express the otherwise unaccountable fact that we cannot know, predict or bring about everything that might happen, as classical dynamical laws suggest we can. But this changes as soon as you bring quantum physics into the picture, with its attendant notion that uncertainty is seemingly hardwired into the fabric of reality. Rooted in probabilities, entropy and thermodynamics acquire a new, more fundamental physical anchor.
It is worth pointing out, too, that this deep-rooted connection seems to be much more general. Recently, together with my colleagues Markus Müller of the Perimeter Institute for Theoretical Physics in Waterloo, Ontario, Canada, and Oscar Dahlsten at the Centre for Quantum Technologies in Singapore, I have looked at what happens to thermodynamical relations in a generalised class of probabilistic theories that embrace quantum theory and much more besides. There too, the crucial relationship between information and disorder, as quantified by entropy, survives (arxiv.org/1107.6029).
One theory to rule them all
As for gravity – the only one of nature’s four fundamental forces not covered by quantum theory – a more speculative body of research suggests it might be little more than entropy in disguise (see “Falling into disorder”). If so, that would also bring Einstein’s general theory of relativity, with which we currently describe gravity, firmly within the purview of thermodynamics.
Take all this together, and we begin to have a hint of what makes thermodynamics so successful. The principles of thermodynamics are at their roots all to do with information theory. Information theory is simply an embodiment of how we interact with the universe – among other things, to construct theories to further our understanding of it. Thermodynamics is, in Einstein’s term, a “meta-theory”: one constructed from principles over and above the structure of any dynamical laws we devise to describe reality’s workings. In that sense we can argue that it is more fundamental than either quantum physics or general relativity.
If we can accept this and, like Eddington and his ilk, put all our trust in the laws of thermodynamics, I believe it may even afford us a glimpse beyond the current physical order. It seems unlikely that quantum physics and relativity represent the last revolutions in physics. New evidence could at any time foment their overthrow. Thermodynamics might help us discern what any usurping theory would look like.
For example, earlier this year, two of my colleagues in Singapore, Esther Hänggi and Stephanie Wehner, showed that a violation of the quantum uncertainty principle – that idea that you can never fully get rid of probabilities in a quantum context – would imply a violation of the second law of thermodynamics. Beating the uncertainty limit means extracting extra information about the system, which requires the system to do more work than thermodynamics allows it to do in the relevant state of disorder. So if thermodynamics is any guide, whatever any post-quantum world might look like, we are stuck with a degree of uncertainty (arxiv.org/abs/1205.6894).
My colleague at the University of Oxford, the physicist David Deutsch, thinks we should take things much further. Not only should any future physics conform to thermodynamics, but the whole of physics should be constructed in its image. The idea is to generalise the logic of the second law as it was stringently formulated by the mathematician Constantin Carathéodory in 1909: that in the vicinity of any state of a physical system, there are other states that cannot physically be reached if we forbid any exchange of heat with the environment.
James Joule’s 19th century experiments with beer can be used to illustrate this idea. The English brewer, whose name lives on in the standard unit of energy, sealed beer in a thermally isolated tub containing a paddle wheel that was connected to weights falling under gravity outside. The wheel’s rotation warmed the beer, increasing the disorder of its molecules and therefore its entropy. But hard as we might try, we simply cannot use Joule’s set-up to decrease the beer’s temperature, even by a fraction of a millikelvin. Cooler beer is, in this instance, a state regrettably beyond the reach of physics.
God, the thermodynamicist
The question is whether we can express the whole of physics simply by enumerating possible and impossible processes in a given situation. This is very different from how physics is usually phrased, in both the classical and quantum regimes, in terms of states of systems and equations that describe how those states change in time. The blind alleys down which the standard approach can lead are easiest to understand in classical physics, where the dynamical equations we derive allow a whole host of processes that patently do not occur – the ones we have to conjure up the laws of thermodynamics expressly to forbid, such as dye molecules reclumping spontaneously in water.
By reversing the logic, our observations of the natural world can again take the lead in deriving our theories. We observe the prohibitions that nature puts in place, be it on decreasing entropy, getting energy from nothing, travelling faster than light or whatever. The ultimately “correct” theory of physics – the logically tightest – is the one from which the smallest deviation gives us something that breaks those taboos.
There are other advantages in recasting physics in such terms. Time is a perennially problematic concept in physical theories. In quantum theory, for example, it enters as an extraneous parameter of unclear origin that cannot itself be quantised. In thermodynamics, meanwhile, the passage of time is entropy increase by any other name. A process such as dissolved dye molecules forming themselves into a clump offends our sensibilities because it appears to amount to running time backwards as much as anything else, although the real objection is that it decreases entropy.
Apply this logic more generally, and time ceases to exist as an independent, fundamental entity, but one whose flow is determined purely in terms of allowed and disallowed processes. With it go problems such as that I alluded to earlier, of why the universe started in a state of low entropy. If states and their dynamical evolution over time cease to be the question, then anything that does not break any transformational rules becomes a valid answer.
Such an approach would probably please Einstein, who once said: “What really interests me is whether God had any choice in the creation of the world.” A thermodynamically inspired formulation of physics might not answer that question directly, but leaves God with no choice but to be a thermodynamicist. That would be a singular accolade for those 19th-century masters of steam: that they stumbled upon the essence of the universe, entirely by accident. The triumph of thermodynamics would then be a revolution by stealth, 200 years in the making.
Read the entire article here.
Quantum Computation: Spooky Arithmetic
Quantum computation holds the promise of vastly superior performance over traditional digital systems based on bits that are either “on” or “off”. Yet for all the theory, quantum computation still remains very much a research enterprise in its very infancy. And, because of the peculiarities of the quantum world — think Schrödinger’s cat, both dead and alive — it’s even difficult to measure a quantum computer at work.
From Wired:
Most researchers have no access to D-Wave’s proprietary system, so they can’t simply examine its specifications to verify the company’s claims. But even if they could look under its hood, how would they know it’s the real thing?
Verifying the processes of an ordinary computer is easy, in principle: At each step of a computation, you can examine its internal state — some series of 0s and 1s — to make sure it is carrying out the steps it claims.
A quantum computer’s internal state, however, is made of “qubits” — a mixture (or “superposition”) of 0 and 1 at the same time, like Schrödinger’s fabled quantum mechanical cat, which is simultaneously alive and dead. Writing down the internal state of a large quantum computer would require an impossibly large number of parameters. The state of a system containing 1,000 qubits, for example, could need more parameters than the estimated number of particles in the universe.
And there’s an even more fundamental obstacle: Measuring a quantum system “collapses” it into a single classical state instead of a superposition of many states. (When Schrödinger’s cat is measured, it instantly becomes alive or dead.) Likewise, examining the inner workings of a quantum computer would reveal an ordinary collection of classical bits. A quantum system, said Umesh Vazirani of the University of California, Berkeley, is like a person who has an incredibly rich inner life, but who, if you ask him “What’s up?” will just shrug and say, “Nothing much.”
“How do you ever test a quantum system?” Vazirani asked. “Do you have to take it on faith? At first glance, it seems that the obvious answer is yes.”
It turns out, however, that there is a way to probe the rich inner life of a quantum computer using only classical measurements, if the computer has two separate “entangled” components.
In the April 25 issue of the journal Nature, Vazirani, together with Ben Reichardt of the University of Southern California in Los Angeles and Falk Unger of Knight Capital Group Inc. in Santa Clara, showed how to establish the precise inner state of such a computer using a favorite tactic from TV police shows: Interrogate the two components in separate rooms, so to speak, and check whether their stories are consistent. If the two halves of the computer answer a particular series of questions successfully, the interrogator can not only figure out their internal state and the measurements they are doing, but also issue instructions that will force the two halves to jointly carry out any quantum computation she wishes.
“It’s a huge achievement,” said Stefano Pironio, of the Université Libre de Bruxelles in Belgium.
The finding will not shed light on the D-Wave computer, which is constructed along very different principles, and it may be decades before a computer along the lines of the Nature paper — or indeed any fully quantum computer — can be built. But the result is an important proof of principle, said Thomas Vidick, who recently completed his post-doctoral research at the Massachusetts Institute of Technology. “It’s a big conceptual step.”
In the short term, the new interrogation approach offers a potential security boost to quantum cryptography, which has been marketed commercially for more than a decade. In principle, quantum cryptography offers “unconditional” security, guaranteed by the laws of physics. Actual quantum devices, however, are notoriously hard to control, and over the past decade, quantum cryptographic systems have repeatedly been hacked.
The interrogation technique creates a quantum cryptography protocol that, for the first time, would transmit a secret key while simultaneously proving that the quantum devices are preventing any potential information leak. Some version of this protocol could very well be implemented within the next five to 10 years, predicted Vidick and his former adviser at MIT, the theoretical computer scientist Scott Aaronson.
“It’s a new level of security that solves the shortcomings of traditional quantum cryptography,” Pironio said.
Spooky Action
In 1964, the Irish physicist John Stewart Bell came up with a test to try to establish, once and for all, that the bafflingly counterintuitive principles of quantum physics are truly inherent properties of the universe — that the decades-long effort of Albert Einstein and other physicists to develop a more intuitive physics could never bear fruit.
Einstein was deeply disturbed by the randomness at the core of quantum physics — God “is not playing at dice,” he famously wrote to the physicist Max Born in 1926.
But in 1964, Bell realized that the EPR paradox could be used to devise an experiment that determines whether quantum physics or a local hidden-variables theory correctly explains the real world. Adapted five years later into a format called the CHSH game (after the researchers John Clauser, Michael Horne, Abner Shimony and Richard Holt), the test asks a system to prove its quantum nature by performing a feat that is impossible using only classical physics.
The CHSH game is a coordination game, in which two collaborating players — Bonnie and Clyde, say — are questioned in separate interrogation rooms. Their joint goal is to give either identical answers or different answers, depending on what questions the “detective” asks them. Neither player knows what question the detective is asking the other player.
If Bonnie and Clyde can use only classical physics, then no matter how many “hidden variables” they share, it turns out that the best they can do is decide on a story before they get separated and then stick to it, no matter what the detective asks them, a strategy that will win the game 75 percent of the time. But if Bonnie and Clyde share an EPR pair of entangled particles — picked up in a bank heist, perhaps — then they can exploit the spooky action at a distance to better coordinate their answers and win the game about 85.4 percent of the time.
Bell’s test gave experimentalists a specific way to distinguish between quantum physics and any hidden-variables theory. Over the decades that followed, physicists, most notably Alain Aspect, currently at the École Polytechnique in Palaiseau, France, carried out this test repeatedly, in increasingly controlled settings. Almost every time, the outcome has been consistent with the predictions of quantum physics, not with hidden variables.
Aspect’s work “painted hidden variables into a corner,” Aaronson said. The experiments had a huge role, he said, in convincing people that the counterintuitive weirdness of quantum physics is here to stay.
If Einstein had known about the Bell test, Vazirani said, “he wouldn’t have wasted 30 years of his life looking for an alternative to quantum mechanics.” He simply would have convinced someone to do the experiment.
Read the whole article here.
Impossible Chemistry in Space
Combine the vastness of the universe with the probabilistic behavior of quantum mechanics and you get some rather odd chemical results. This includes the spontaneous creation of some complex organic molecules in interstellar space — previously believed to be far too inhospitable for all but the lowliest forms of matter.
From the New Scientist:
Quantum weirdness can generate a molecule in space that shouldn’t exist by the classic rules of chemistry. If interstellar space is really a kind of quantum chemistry lab, that might also account for a host of other organic molecules glimpsed in space.
Interstellar space should be too cold for most chemical reactions to occur, as the low temperature makes it tough for molecules drifting through space to acquire the energy needed to break their bonds. “There is a standard law that says as you lower the temperature, the rates of reactions should slow down,” says Dwayne Heard of the University of Leeds, UK.
Yet we know there are a host of complex organic molecules in space. Some reactions could occur when different molecules stick to the surface of cosmic dust grain. This might give them enough time together to acquire the energy needed to react, which doesn’t happen when molecules drift past each other in space.
Not all reactions can be explained in this way, though. Last year astronomers discovered methoxy molecules – containing carbon, hydrogen and oxygen – in the Perseus molecular cloud, around 600 light years from Earth. But researchers couldn’t produce this molecule in the lab by allowing reactants to condense on dust grains, leaving a puzzle as to how it could have formed.
Molecular hang-out
Another route to methoxy is to combine a hydroxyl radical and methanol gas, both present in space. But this reaction requires hurdling a significant energy barrier – and the energy to do that simply isn’t available in the cold expanse of space.
Heard and his colleagues wondered if the answer lay in quantum mechanics: a process called quantum tunnelling might give the hydroxyl radical a small chance to cheat by digging through the barrier instead of going over it, they reasoned.
So, in another attempt to replicate the production of methoxy in space, the team chilled gaseous hydroxyl and methanol to 63 kelvin – and were able to produce methoxy.
The idea is that at low temperatures, the molecules slow down, increasing the likelihood of tunnelling. “At normal temperatures they just collide off each other, but when you go down in temperature they hang out together long enough,” says Heard.
Impossible chemistry
The team also found that the reaction occurred 50 times faster via quantum tunnelling than if it occurred normally at room temperature by hurdling the energy barrier. Empty space is much colder than 63 kelvin, but dust clouds near stars can reach this temperature, adds Heard.
“We’re showing there is organic chemistry in space of the type of reactions where it was assumed these just wouldn’t happen,” says Heard.
That means the chemistry of space may be richer than we had imagined. “There is maybe a suite of chemical reactions we hadn’t yet considered occurring in interstellar space,” agrees Helen Fraser of the University of Strathclyde, UK, who was not part of the team.
Read the entire article here.
Image: Amino-1-methoxy-4-methylbenzol, featuring methoxy molecular structure, recently found in interstellar space. Courtesy of Wikipedia.
Uncertainty Strikes the Uncertainty Principle
Some recent experiments out of the University of Toronto show for the first time an anomaly in measurements predicted by Werner Heisenberg’s fundamental law of quantum mechanics, the Uncertainty Principle.
[div class=attrib]From io9:[end-div]
Heisenberg’s uncertainty principle is an integral component of quantum physics. At the quantum scale, standard physics starts to fall apart, replaced by a fuzzy, nebulous set of phenomena. Among all the weirdness observed at this microscopic scale, Heisenberg famously observed that the position and momentum of a particle cannot be simultaneously measured, with any meaningful degree of precision. This led him to posit the uncertainty principle, the declaration that there’s only so much we can know about a quantum system, namely a particle’s momentum and position.
Now, by definition, the uncertainty principle describes a two-pronged process. First, there’s the precision of a measurement that needs to be considered, and second, the degree of uncertainty, or disturbance, that it must create. It’s this second aspect that quantum physicists refer to as the “measurement-disturbance relationship,” and it’s an area that scientists have not sufficiently explored or proven.
The researchers, a team led by Lee Rozema and Aephraim Steinberg, experimentally observed a clear-cut violation of Heisenberg’s measurement-disturbance relationship. They did this by applying what they called a “weak measurement” to define a quantum system before and after it interacted with their measurement tools — not enough to disturb it, but enough to get a basic sense of a photon’s orientation.
[div class=attrib]Image: Heisenberg, Werner Karl Prof. 1901-1976; Physicist, Nobel Prize for Physics 1933, Germany. Courtesy of Wikipedia.[end-div]
Something Out of Nothing
The debate on how the universe came to be rages on. Perhaps, however, we are a little closer to understanding why there is “something”, including us, rather than “nothing”.
[div class=attrib]Image: There’s Nothing Out There. Courtesy of Rolfe Kanefsky / Image Entertainment.[end-div]
Spooky Action at a Distance Explained
Chance as a Subjective or Objective Measure
[div class=attrib]From Rationally Speaking:[end-div]
Stop me if you’ve heard this before: suppose I flip a coin, right now. I am not giving you any other information. What odds (or probability, if you prefer) do you assign that it will come up heads?
If you would happily say “Even” or “1 to 1” or “Fifty-fifty” or “probability 50%” — and you’re clear on WHY you would say this — then this post is not aimed at you, although it may pleasantly confirm your preexisting opinions as a Bayesian on probability. Bayesians, broadly, consider probability to be a measure of their state of knowledge about some proposition, so that different people with different knowledge may correctly quote different probabilities for the same proposition.
If you would say something along the lines of “The question is meaningless; probability only has meaning as the many-trials limit of frequency in a random experiment,” or perhaps “50%, but only given that a fair coin and fair flipping procedure is being used,” this post is aimed at you. I intend to try to talk you out of your Frequentist view; the view that probability exists out there and is an objective property of certain physical systems, which we humans, merely fallibly, measure.
My broader aim is therefore to argue that “chance” is always and everywhere subjective — a result of the limitations of minds — rather than objective in the sense of actually existing in the outside world.
[div class=attrib]Much more of this article here.[end-div]
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