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Spherical black holes with regular center: a review of existing models including a recent realization with Gaussian sources

Recently there has been a renewed interest for the search of solutions of Einstein equations, mostly motivated by the study of higher dimensional gravity, for instance in the context of the brane world scenario [ 104 , 105 ] and of string theory [ 110 ] . Some beautiful examples of higher dimensional solutions have appeared (see, e.g., [ 68 , 67 ] ), which feature interesting properties that, in the pure 4 4 4 -dimensional framework, are absent. This renewed interest is also having some influence on a rather more specialized, but very interesting, research area, that of regular black holes. It is of course well-known that, under rather generic conditions on the energy–matter content of spacetime, classical solution of Einstein equations exhibit both, future [ 99 ] and past [ 60 , 61 , 62 , 51 , 63 ] singularities [ 65 , 31 , 14 , 114 , 90 , 129 , 64 ] usually hidden by an event horizon [ 71 ] . This fact, which is clearly exemplified by the first and probably most well-known solution of Einstein equations, i.e. the Schwarzschild solution [ 112 , 111 ] , has been likely fully appreciated only after the study of its global analytic extension [ 44 , 46 , 74 , 125 , 95 , 90 , 64 ] , which gives a consistent picture of its properties, like the existence of the region inside the horizon, in which the radial and time coordinate exchange their character, and the presence of the central singularity. Although the presence of, both, the black hole region and the central singularity has been eventually accepted, i.e. we learned how to classically live with them, especially the presence of a singularity is a recurrent motivation to underline the inadequacy of general relativity as a theory of spacetime below some length scale (apart from the theoretical motivations, it is a fact that, experimentally, gravity can be tested only in a finite range of scales). On one side, maybe the most well known one, this has motivated the search for a more complete theory of gravity including also quantum effects (see, e.g., [ 73 ] for a recent, comprehensive review). On the other side, it has also sustained many efforts to push as much as possible Einstein gravity to its limit, trying to avoid, if not the black hole region, at least the central singularity in a way as consistent as possible with physical requirements 2 2 2 The two points of view just outlined should be seen as complementary and, often, even integrated. . Following some very early ideas that date back to the work of Sakharov [ 109 ] , Gliner [ 53 ] and Bardeen [ 9 ] , solutions having a global structure very similar to the one of black hole spacetimes, but in which the central singularity is absent, have been found (references to them will appear in the rest of the paper). In this contribution we are going to briefly review some of these ideas, but, before concluding this introductory section with the layout of the rest of the paper, we would like to make a couple of remarks. The first of them is that, in contrast to the fact that nowadays the world of theoretical physics witnesses a consistent number of strongly believed (but as yet unproven) conjectures, most of the results about black holes and their properties are, in fact theorems . Theorems usually make some hypotheses, which in this case can be roughly interpreted within a threefold scheme; i) the validity of some geometric properties, usually related to the behavior of geodesics (for instance the existence of trapped surfaces, and so on); ii) the validity of some conditions on the matter fields that are coupled to gravity (energy conditions); iii) the validity of some (more technical) hypotheses about the global/causal structure of spacetime. It is then clear that the possibility of singularity avoidance, within the context defined by general relativity, requires the violation of, at least, one of the above conditions. Since conditions of type iii) are mostly technical, there has been a great effort to make them as general as possible (although sometimes this means less immediate) if not to remove them at all, by generalizing the earliest results [ 65 , 64 , 129 ] . Conditions i) are usually the requirement that some indicator exists, which emphasizes that something a little bit unusual for a “flat” non covariant mind is taking place in spacetime, and are usually related to the existence of horizons, so there is little reason to modify them. It is then natural that, as a possible way to avoid singularities, a relaxation of conditions of type ii) has been advocated. With a strongly conservative attitude, a word of caution should be sounded at this point. It is, in fact, known that matter and energy violating some of the energy conditions, have as yet unobserved properties 3 3 3 We will come back to this point later on, mentioning vacuum and the cosmological constant. A clear discussion of this point can be found in the standard reference [ 64 ] ; see also the early [ 107 ] for a physically oriented discussion of the implications of a violation of the weak energy condition. : this means that we are not yet able to produce them in a laboratory by a well-known, generally reproducible procedure. We have, nevertheless, good candidates to realize these violations when we treat at an effective level the quantum properties of spacetime and matter at some length/energy scales: this is very suggestive, since it directly connects to the, possibly, incomplete character of classical general relativity as a theory of spacetime and with the ongoing, diversified, efforts toward its quantization [ 73 ] . To review in more detail some aspects related to the above reflections, we plan as follows. In section 2 we review various regular models of spacetime, centering our attention, almost exclusively, on regular black holes of a very specific type (specified below). After a review of the earliest ideas (subsection 2.1 ) we analyze their first concrete (and, perhaps, to some extent independent) realization, known as the Bardeen solution: we review also some studies, which appeared much later, discussing its global character (subsection 2.2 ); we then continue with a discussion of black hole interiors (subsection 2.3 ) reporting various early proposals, which adopted spacetime junctions to get rid of singularities; this brings us to the central part of this section (subsection 2.4 ), where some exact solutions are analyzed, together with the possibility of physical realizations for the energy-matter content which should act as their source (subsubsection 2.4.1 ). The solutions that we will have described up to this point are not extemporary realizations, but can be understood in a very interesting, complete and general framework: we thus review the essence of this framework in subsection 2.5 . This section is concluded with a very concise summary of the results that we have reviewed (subsection 2.6 ). Then, in section 3 we use a recently obtained solution, which is another possible realization of the general type of solutions described in subsection 2.5 , to perform a simple exercise, i.e. the study of the violation of one of the energy conditions. For completeness, after introducing the algebraic form of the solution, we quickly construct its global spacetime structure in subsection 3.1 (this result follows immediately from the results reviewed in subsection 2.5 ); we then show which regions of spacetime are filled with matter violating the strong energy condition (subsection 3.2 ). The results of this second part of the paper are summarized in subsection 3.3 . Some general comments and remarks find space in the concise concluding section, i.e. section 4 . We now conclude this introduction by fixing one notation and one naming convention, as below. 1.1 Conventions and notations In what follows we will concentrate on spherically symmetric solutions of Einstein equations and restrict ourself to media which satisfy the condition that the radial pressure equals the opposite of the energy density. We will then use, throughout and unless otherwise stated, the coordinate system ( t , r , ϑ , φ ) 𝑡 𝑟 italic-ϑ 𝜑 (t,r,\vartheta,\varphi) , in which the metric can be written in the static form adapted to the spherical symmetry, i.e. g μ ​ ν = diag ​ ( − f ​ ( r ) , f ​ ( r ) − 1 , r 2 , r 2 ​ sin 2 ⁡ ϑ ) . subscript 𝑔 𝜇 𝜈 diag 𝑓 𝑟 𝑓 superscript 𝑟 1 superscript 𝑟 2 superscript 𝑟 2 superscript 2 italic-ϑ g_{\mu\nu}={\mathrm{diag}}\left(-f(r),f(r)^{-1},r^{2},r^{2}\sin^{2}\vartheta\right). (1) As apparent from the above definition we adopt the signature ( − , + , + , + ) (-,+,+,+) . We occasionally will use the name metric function for the function f ​ ( r ) 𝑓 𝑟 f(r) . We do not spend extra comments about the meaning of the coordinate choice, which is standard and discussed in detail in various textbooks (see for instance [ 90 ] ; any other textbook choice will be equivalent); Thus, without restating every time our coordinate choice, in what follows we will specify various metrics just by specifying the corresponding metric function. In view of the above, when we will have to discuss the maximal extension of solutions that admit an expression of the metric in the form ( 1 ), although we will follow the naming conventions of the standard reference [ 64 ] for boundaries as infinity, only in one point of our discussion we will need a few more global ideas than the one concisely and effectively discussed in [ 130 ] . We will moreover use the standard notation T μ ​ ν subscript 𝑇 𝜇 𝜈 T_{\mu\nu} for the stress-energy tensor which appears on the righthand side of Einstein equations.

We review, in a historical perspective, some results about black hole spacetimes with a regular center. We then see how their properties are realized in a specific solution that recently appeared; in particular we analyze in detail the (necessary) violation of the strong energy condition.

[{"content":"Help me to generate the abstract of a survey paper given the title:*Spherical black holes with regular center: a review of existing models including a recent realization with Gaussian sources* and the introduction:Recently there has been a renewed interest for the search of solutions of Einstein equations, mostly motivated by the study of higher dimensional gravity, for instance in the context of the brane world scenario 104 , 105 and of string theory 110 . Some beautiful examples of higher dimensional solutions have appeared (see, e.g., 68 , 67 ), which feature interesting properties that, in the pure 4 4 4 -dimensional framework, are absent. This renewed interest is also having some influence on a rather more specialized, but very interesting, research area, that of regular black holes. It is of course well-known that, under rather generic conditions on the energy–matter content of spacetime, classical solution of Einstein equations exhibit both, future 99 and past 60 , 61 , 62 , 51 , 63 singularities 65 , 31 , 14 , 114 , 90 , 129 , 64 usually hidden by an event horizon 71 . This fact, which is clearly exemplified by the first and probably most well-known solution of Einstein equations, i.e. the Schwarzschild solution 112 , 111 , has been likely fully appreciated only after the study of its global analytic extension 44 , 46 , 74 , 125 , 95 , 90 , 64 , which gives a consistent picture of its properties, like the existence of the region inside the horizon, in which the radial and time coordinate exchange their character, and the presence of the central singularity. Although the presence of, both, the black hole region and the central singularity has been eventually accepted, i.e. we learned how to classically live with them, especially the presence of a singularity is a recurrent motivation to underline the inadequacy of general relativity as a theory of spacetime below some length scale (apart from the theoretical motivations, it is a fact that, experimentally, gravity can be tested only in a finite range of scales). On one side, maybe the most well known one, this has motivated the search for a more complete theory of gravity including also quantum effects (see, e.g., 73 for a recent, comprehensive review). On the other side, it has also sustained many efforts to push as much as possible Einstein gravity to its limit, trying to avoid, if not the black hole region, at least the central singularity in a way as consistent as possible with physical requirements 2 2 2 The two points of view just outlined should be seen as complementary and, often, even integrated. . Following some very early ideas that date back to the work of Sakharov 109 , Gliner 53 and Bardeen 9 , solutions having a global structure very similar to the one of black hole spacetimes, but in which the central singularity is absent, have been found (references to them will appear in the rest of the paper). In this contribution we are going to briefly review some of these ideas, but, before concluding this introductory section with the layout of the rest of the paper, we would like to make a couple of remarks. The first of them is that, in contrast to the fact that nowadays the world of theoretical physics witnesses a consistent number of strongly believed (but as yet unproven) conjectures, most of the results about black holes and their properties are, in fact theorems . Theorems usually make some hypotheses, which in this case can be roughly interpreted within a threefold scheme; i) the validity of some geometric properties, usually related to the behavior of geodesics (for instance the existence of trapped surfaces, and so on); ii) the validity of some conditions on the matter fields that are coupled to gravity (energy conditions); iii) the validity of some (more technical) hypotheses about the global/causal structure of spacetime. It is then clear that the possibility of singularity avoidance, within the context defined by general relativity, requires the violation of, at least, one of the above conditions. Since conditions of type iii) are mostly technical, there has been a great effort to make them as general as possible (although sometimes this means less immediate) if not to remove them at all, by generalizing the earliest results 65 , 64 , 129 . Conditions i) are usually the requirement that some indicator exists, which emphasizes that something a little bit unusual for a flat non covariant mind is taking place in spacetime, and are usually related to the existence of horizons, so there is little reason to modify them. It is then natural that, as a possible way to avoid singularities, a relaxation of conditions of type ii) has been advocated. With a strongly conservative attitude, a word of caution should be sounded at this point. It is, in fact, known that matter and energy violating some of the energy conditions, have as yet unobserved properties 3 3 3 We will come back to this point later on, mentioning vacuum and the cosmological constant. A clear discussion of this point can be found in the standard reference 64 ; see also the early 107 for a physically oriented discussion of the implications of a violation of the weak energy condition. : this means that we are not yet able to produce them in a laboratory by a well-known, generally reproducible procedure. We have, nevertheless, good candidates to realize these violations when we treat at an effective level the quantum properties of spacetime and matter at some length/energy scales: this is very suggestive, since it directly connects to the, possibly, incomplete character of classical general relativity as a theory of spacetime and with the ongoing, diversified, efforts toward its quantization 73 . To review in more detail some aspects related to the above reflections, we plan as follows. In section 2 we review various regular models of spacetime, centering our attention, almost exclusively, on regular black holes of a very specific type (specified below). After a review of the earliest ideas (subsection 2.1 ) we analyze their first concrete (and, perhaps, to some extent independent) realization, known as the Bardeen solution: we review also some studies, which appeared much later, discussing its global character (subsection 2.2 ); we then continue with a discussion of black hole interiors (subsection 2.3 ) reporting various early proposals, which adopted spacetime junctions to get rid of singularities; this brings us to the central part of this section (subsection 2.4 ), where some exact solutions are analyzed, together with the possibility of physical realizations for the energy-matter content which should act as their source (subsubsection 2.4.1 ). The solutions that we will have described up to this point are not extemporary realizations, but can be understood in a very interesting, complete and general framework: we thus review the essence of this framework in subsection 2.5 . This section is concluded with a very concise summary of the results that we have reviewed (subsection 2.6 ). Then, in section 3 we use a recently obtained solution, which is another possible realization of the general type of solutions described in subsection 2.5 , to perform a simple exercise, i.e. the study of the violation of one of the energy conditions. For completeness, after introducing the algebraic form of the solution, we quickly construct its global spacetime structure in subsection 3.1 (this result follows immediately from the results reviewed in subsection 2.5 ); we then show which regions of spacetime are filled with matter violating the strong energy condition (subsection 3.2 ). The results of this second part of the paper are summarized in subsection 3.3 . Some general comments and remarks find space in the concise concluding section, i.e. section 4 . We now conclude this introduction by fixing one notation and one naming convention, as below. 1.1 Conventions and notations In what follows we will concentrate on spherically symmetric solutions of Einstein equations and restrict ourself to media which satisfy the condition that the radial pressure equals the opposite of the energy density. We will then use, throughout and unless otherwise stated, the coordinate system ( t , r , ϑ , φ ) 𝑡 𝑟 italic-ϑ 𝜑 (t,r,\vartheta,\varphi) , in which the metric can be written in the static form adapted to the spherical symmetry, i.e. g μ ​ ν = diag ​ ( − f ​ ( r ) , f ​ ( r ) − 1 , r 2 , r 2 ​ sin 2 ⁡ ϑ ) . subscript 𝑔 𝜇 𝜈 diag 𝑓 𝑟 𝑓 superscript 𝑟 1 superscript 𝑟 2 superscript 𝑟 2 superscript 2 italic-ϑ g_{\mu\nu}={\mathrm{diag}}\left(-f(r),f(r)^{-1},r^{2},r^{2}\sin^{2}\vartheta\right). (1) As apparent from the above definition we adopt the signature ( − , + , + , + ) (-,+,+,+) . We occasionally will use the name metric function for the function f ​ ( r ) 𝑓 𝑟 f(r) . We do not spend extra comments about the meaning of the coordinate choice, which is standard and discussed in detail in various textbooks (see for instance 90 ; any other textbook choice will be equivalent); Thus, without restating every time our coordinate choice, in what follows we will specify various metrics just by specifying the corresponding metric function. In view of the above, when we will have to discuss the maximal extension of solutions that admit an expression of the metric in the form ( 1 ), although we will follow the naming conventions of the standard reference 64 for boundaries as infinity, only in one point of our discussion we will need a few more global ideas than the one concisely and effectively discussed in 130 . We will moreover use the standard notation T μ ​ ν subscript 𝑇 𝜇 𝜈 T_{\mu\nu} for the stress-energy tensor which appears on the righthand side of Einstein equations.","role":"user"},{"content":"Abstract: We review, in a historical perspective, some results about black hole spacetimes with a regular center. We then see how their properties are realized in a specific solution that recently appeared; in particular we analyze in detail the (necessary) violation of the strong energy condition.","role":"assistant"}]

A Survey of Huebschmann and Stasheff's Paper: Formal Solution of the Master Equation via HPT and Deformation Theory

These notes, based on the paper [ 8 ] by Huebschmann and Stasheff, were prepared for a series of talks at Illinois State University with the intention of applying Homological Perturbation Theory (HPT) to the construction of derived brackets [ 11 , 16 ] , and eventually writing Part II of the paper [ 1 ] . Derived brackets are obtained by deforming the initial bracket via a derivation of the bracket. In [ 3 ] it was demonstrated that such deformations correspond to solutions of the Maurer-Cartan equation, and the role of an “almost contraction” was noted. This technique (see also [ 9 ] ) is very similar to the iterative procedure of [ 8 ] for finding the most general solution of the Maurer-Cartan equation, i.e. the deformation of a given structure in a prescribed direction. The present article, besides providing additional details of the condensed article [ 8 ] , forms a theoretical background for understanding and generalizing the current techniques that give rise to derived brackets. The generalization, which will be the subject matter of [ 2 ] , will be achieved by using Stasheff and Huebschmann’s universal solution. A second application of the universal solution will be in deformation quantization and will help us find the coefficients of star products in a combinatorial manner, rather than as a byproduct of string theory which underlies the original solution given by Kontsevich [ 10 ] . HPT is often used to replace given chain complexes by homotopic, smaller, and more readily computable chain complexes (to explore “small” or “minimal” models). This method may prove to be more efficient than “spectral sequences” in computing (co)homology. One useful tool in HPT is Lemma 1 (Basic Perturbation Lemma (BPL)) . Given a contraction of N 𝑁 N onto M 𝑀 M and a perturbation ∂ \partial of d N subscript 𝑑 𝑁 d_{N} , under suitable conditions there exists a perturbation d ∂ subscript 𝑑 d_{\partial} of d M subscript 𝑑 𝑀 d_{M} such that H ​ ( M , d M + d ∂ ) = H ​ ( N , d N + ∂ ) 𝐻 𝑀 subscript 𝑑 𝑀 subscript 𝑑 𝐻 𝑁 subscript 𝑑 𝑁 H(M,d_{M}+d_{\partial})=H(N,d_{N}+\partial) . The main question is: under what conditions does the BPL allow the preservation of the data structures (DGA’s, DG coalgebras, DGLA’s etc.)? (We will use the self-explanatory abbreviations such as DG for “differential graded”, DGA for “differential graded (not necessarily associative) algebra”, and DGLA for “differential graded Lie algebra”.) Another prominent idea is that of a “(universal) twisting cochain” as a solution of the “master equation”: Proposition 1 . Given a contraction of N 𝑁 N onto M 𝑀 M and a twisting cochain N → A → 𝑁 𝐴 N\rightarrow A ( A 𝐴 A some DGA), there exists a unique twisting cochain M → A → 𝑀 𝐴 M\rightarrow A that factors through the given one and which can be constructed inductively. The explicit formulas are reminiscent of the Kuranishi map [ 13 ] (p.17), and the relationship will be investigated elsewhere. Note: we will assume that the ground ring is a field F 𝐹 F of characteristic zero. We will denote the end of an example with the symbol ◇ ◇ \Diamond and the end of a proof by □ □ \Box .

These notes, based on the paper "Formal Solution of the Master Equation via HPT and Deformation Theory" by Huebschmann and Stasheff, were prepared for a series of talks at Illinois State University with the intention of applying Homological Perturbation Theory to the derived bracket constructions of Kosmann-Schwarzbach and T. Voronov, and eventually writing Part II of the paper "Higher Derived Brackets and Deformation Theory I" by the present authors.

[{"content":"Help me to generate the abstract of a survey paper given the title:*A Survey of Huebschmann and Stasheff's Paper: Formal Solution of the Master Equation via HPT and Deformation Theory* and the introduction:These notes, based on the paper 8 by Huebschmann and Stasheff, were prepared for a series of talks at Illinois State University with the intention of applying Homological Perturbation Theory (HPT) to the construction of derived brackets 11 , 16 , and eventually writing Part II of the paper 1 . Derived brackets are obtained by deforming the initial bracket via a derivation of the bracket. In 3 it was demonstrated that such deformations correspond to solutions of the Maurer-Cartan equation, and the role of an almost contraction was noted. This technique (see also 9 ) is very similar to the iterative procedure of 8 for finding the most general solution of the Maurer-Cartan equation, i.e. the deformation of a given structure in a prescribed direction. The present article, besides providing additional details of the condensed article 8 , forms a theoretical background for understanding and generalizing the current techniques that give rise to derived brackets. The generalization, which will be the subject matter of 2 , will be achieved by using Stasheff and Huebschmanns universal solution. A second application of the universal solution will be in deformation quantization and will help us find the coefficients of star products in a combinatorial manner, rather than as a byproduct of string theory which underlies the original solution given by Kontsevich 10 . HPT is often used to replace given chain complexes by homotopic, smaller, and more readily computable chain complexes (to explore small or minimal models). This method may prove to be more efficient than spectral sequences in computing (co)homology. One useful tool in HPT is Lemma 1 (Basic Perturbation Lemma (BPL)) . Given a contraction of N 𝑁 N onto M 𝑀 M and a perturbation ∂ \partial of d N subscript 𝑑 𝑁 d_{N} , under suitable conditions there exists a perturbation d ∂ subscript 𝑑 d_{\partial} of d M subscript 𝑑 𝑀 d_{M} such that H ​ ( M , d M + d ∂ ) = H ​ ( N , d N + ∂ ) 𝐻 𝑀 subscript 𝑑 𝑀 subscript 𝑑 𝐻 𝑁 subscript 𝑑 𝑁 H(M,d_{M}+d_{\partial})=H(N,d_{N}+\partial) . The main question is: under what conditions does the BPL allow the preservation of the data structures (DGAs, DG coalgebras, DGLAs etc.)? (We will use the self-explanatory abbreviations such as DG for differential graded, DGA for differential graded (not necessarily associative) algebra, and DGLA for differential graded Lie algebra.) Another prominent idea is that of a (universal) twisting cochain as a solution of the master equation: Proposition 1 . Given a contraction of N 𝑁 N onto M 𝑀 M and a twisting cochain N → A → 𝑁 𝐴 N\rightarrow A ( A 𝐴 A some DGA), there exists a unique twisting cochain M → A → 𝑀 𝐴 M\rightarrow A that factors through the given one and which can be constructed inductively. The explicit formulas are reminiscent of the Kuranishi map 13 (p.17), and the relationship will be investigated elsewhere. Note: we will assume that the ground ring is a field F 𝐹 F of characteristic zero. We will denote the end of an example with the symbol ◇ ◇ \Diamond and the end of a proof by □ □ \Box .","role":"user"},{"content":"Abstract: These notes, based on the paper "Formal Solution of the Master Equation via HPT and Deformation Theory" by Huebschmann and Stasheff, were prepared for a series of talks at Illinois State University with the intention of applying Homological Perturbation Theory to the derived bracket constructions of Kosmann-Schwarzbach and T. Voronov, and eventually writing Part II of the paper "Higher Derived Brackets and Deformation Theory I" by the present authors.","role":"assistant"}]

A review of spin determination at the LHC

Most models of TeV scale new physics are motivated by solving the hierarchy problem. Therefore, the most crucial ingredient of all of them is the mechanism of cancelling the quadratically divergent correction to the Higgs mass within the Standard Model. In order to achieve this, a set of new physics particles with the same or similar gauge quantum numbers as the Standard Model particles are introduced, whose couplings to the Higgs are related to those of the Standard Model particles. This “partner”-like structure of new physics is very generic in large classes of new physics scenarios. Well-known examples include the set of superpartners in low energy supersymmetry [ 1 ] (for a recent review see Ref. [ 2 ] ) , KK excitations in extra-dimensional models [ 3 ] , as well as similar states in little Higgs models [ 4 ] (Ref. [ 5 ] provides a brief review). Due to the similarities in gauge quantum numbers, initial LHC signatures of new partners are very similar, as they can decay into the same set of observable final state particles. The mass spectra of different scenarios can be chosen to produce similar dominant kinematical features, such as the p T subscript 𝑝 𝑇 p_{T} distribution of the decay product. For example, a typical gluino decay chain in supersymmetry is g ~ → q ​ q ¯ + N ~ 2 → ~ 𝑔 𝑞 ¯ 𝑞 subscript ~ 𝑁 2 \tilde{g}\rightarrow q\bar{q}+\tilde{N}_{2} followed by N ~ 2 → ℓ ​ ℓ ¯ + N ~ 1 → subscript ~ 𝑁 2 ℓ ¯ ℓ subscript ~ 𝑁 1 \tilde{N}_{2}\rightarrow\ell\bar{\ell}+\tilde{N}_{1} . A similar decay chain in universal extra-dimension models [ 3 ] with KK-gluon ( g ( 1 ) superscript 𝑔 1 g^{(1)} ), KK-W ( W 3 ( 1 ) subscript superscript 𝑊 1 3 W^{(1)}_{3} ) and KK-photon ( γ ( 1 ) superscript 𝛾 1 \gamma^{(1)} ), g ( 1 ) → q ​ q ¯ ​ W 3 ( 1 ) → superscript 𝑔 1 𝑞 ¯ 𝑞 subscript superscript 𝑊 1 3 g^{(1)}\rightarrow q\bar{q}W^{(1)}_{3} followed by W 3 ( 1 ) → ℓ ​ ℓ ¯ ​ γ ( 1 ) → subscript superscript 𝑊 1 3 ℓ ¯ ℓ superscript 𝛾 1 W^{(1)}_{3}\rightarrow\ell\bar{\ell}\gamma^{(1)} , gives identical final states since both N ~ 1 subscript ~ 𝑁 1 \tilde{N}_{1} and γ 1 superscript 𝛾 1 \gamma^{1} are neutral stable particles which escape detection. The mass spectra of both supersymmetry and UED can be adjusted in such a way that the p T subscript 𝑝 𝑇 p_{T} of the jets and leptons are quite similar. Some of the similarities in the LHC signature are actually the result of equivalences in low energy effective theory. For example, it is known that “theory space” motivated little Higgs models are equivalent to extra-dimensional models in which Higgs is a non-local mode in the bulk, via deconstruction [ 6 , 7 , 8 ] . Therefore, they can actually be described by the same set of low energy ( ∼ similar-to \sim TeV) degrees of freedom. An important feature of this class of models is that the partners typically have the same spin as their corresponding Standard Model particles. However, the difference between this set of models and low energy supersymmetry is physical and observable with a sufficiently precise measurement. In particular, the spin of superpartners differ from their Standard Model counter parts by half integers. Therefore, spin measurements are crucial to set these scenarios apart. The conventional way of measuring the spin of a new particle involves reconstruction of its rest frame using its decay products and studying the angular distribution about the polarization axis. For example, in process e + ​ e − → Z → μ + ​ μ − → superscript 𝑒 superscript 𝑒 𝑍 → superscript 𝜇 superscript 𝜇 e^{+}e^{-}\rightarrow Z\rightarrow\mu^{+}\mu^{-} , the 1 + cos 2 ⁡ θ 1 superscript 2 𝜃 1+\cos^{2}\theta distribution of the muon direction in the rest frame of the Z 𝑍 Z reveals its vector nature. However, in most new physics scenarios of interest such a strategy is complicated by the generic existence of undetectable massive particles. Motivated by electroweak precision constraints and the existence of Cold Dark Matter, many such scenarios incorporate some discrete symmetry which guarantees the existence of a lightest stable neutral particle. Well-known examples of such discrete symmetries include R-parity in supersymmetry, KK-parity of universal extra-dimension models (UED) [ 3 ] , or similarly, T-parity in Little Higgs Models [ 9 , 10 , 11 , 12 ] . The existence of such a neutral particle at the end of the decay chain results in large missing energy events in which new physics particles are produced. This fact helps to separate them from the Standard Model background. On the other hand, it also makes the spin measurement more complicated because it is generically impossible to reconstruct the momentum, and therefore the rest frame, of the decaying new physics particles. There are two different approaches to measuring spin. First, given the same gauge quantum numbers, particles with different spin usually have very different production rates, due to the difference between fermionic and bosonic couplings and the number of degrees of freedom. Such an approach could be useful, in particular initially, for colored particles due to their large (hence more measurable) production rates. However, a crucial ingredient in such a strategy is the measurement of the masses of particles produced, as rate can only provide definitive information once the mass is fixed. Such an effort is made more difficult owing to the existence of missing massive particles. There is also some residual model dependence since, for example, a couple of complex scalars can fake a dirac fermion. The second approach, is the direct measurement of spin through its effect on angular correlations in decay products. In the absence of a reconstructed rest frame, one is left to consider Lorentz invariant quantities which encode angular correlations. As we will see later in this review, spin correlations typically only exist in certain type of decays. Furthermore, new physics particles are frequently pair produced with independent decay chains containing similar particles. Therefore, a valid spin correlation measurement requires the ability to identify a relatively pure sample of events where we can isolate certain decay chains and suppress combinatorics effectively. Therefore, except for very special cases, we expect this measurement will require large statistics. At the same time, as will be clear from our discussion, using the appropriate variables and correctly interpreting the measured angular distribution frequently requires at least a partial knowledge of the spectrum and the gauge quantum numbers. Obtaining information about the spectrum and the quantum numbers is likely to require a somewhat lower integrated luminosity than spin measurements do. Therefore, the order with which we uncover the properties of new particles is congruent to the order with which we must proceed in the first place to correctly establish these properties. Thus, we should clearly focus on mass scales, branching ratios and gauge quantum numbers first, once new physics is discovered at the LHC, while keeping an unbiased perspective towards the underlying model. More refined measurements, such as the ones described in this review, will enable us to tell the different models apart thereafter. Such measurements can be useful and even more powerful in a linear collider as was recently proposed in Ref. [ 13 ] . In this review we will concentrate on methods applicable to the LHC. In principle, the production of particles with different spins also leads to distinguishable angular distributions. This was investigated in the context of linear colliders in Ref. [ 14 ] . A similar measurement using the process p ​ p → ℓ ~ ​ ℓ ~ ⋆ → 𝑝 𝑝 ~ ℓ superscript ~ ℓ ⋆ pp\rightarrow\tilde{\ell}\tilde{\ell}^{\star} at the LHC has been studied in Ref. [ 15 ] . An analogues measurement in the production of colored states is more challenging. First, typically several different initial states and partial waves can contribute to the same production process. Therefore, it is difficult to extract spin information from the resulting angular distribution in a model independent way. Second, as commented above it is often difficult to reconstruct the direction of the original particles coming out of the production vertex. As a result, angular correlations are further washed out. In the rest of this review, we will survey both of these approaches with slightly heavier emphasis given to the angular correlation technique. For concreteness, we will compare supersymmetry with another generic scenario in which the partners, such as gluon partner g ′ superscript 𝑔 ′ g^{\prime} , W-partner W ′ superscript 𝑊 ′ W^{\prime} , quark partner q ′ superscript 𝑞 ′ q^{\prime} , etc., have the same spin as their corresponding Standard Model particles. As was pointed out above, this so called same-spin scenario effectively parameterizes almost all non-SUSY models which address the hierarchy problem. Spin measurement at the LHC is still a relatively new field where only first steps towards a comprehensive study have been taken. We will briefly summarize these developments in this review. We will focus here on the theoretical foundations and considerations relevant for the construction of observables. The potential for measuring spin in many new decay channels remains to be studied. Important effects, such as Standard Model background and large combinatorics, deserve careful further consideration. We outline these issues in connection to particular channels below.

We review the prospects of direct spin determination of new particles which may be discovered at the LHC. We discuss the general framework and the different channels which contain spin information. The experimental challenges associated with such measurements are briefly discussed and the most urgent unresolved problems are emphasized.

[{"content":"Help me to generate the abstract of a survey paper given the title:*A review of spin determination at the LHC* and the introduction:Most models of TeV scale new physics are motivated by solving the hierarchy problem. Therefore, the most crucial ingredient of all of them is the mechanism of cancelling the quadratically divergent correction to the Higgs mass within the Standard Model. In order to achieve this, a set of new physics particles with the same or similar gauge quantum numbers as the Standard Model particles are introduced, whose couplings to the Higgs are related to those of the Standard Model particles. This partner-like structure of new physics is very generic in large classes of new physics scenarios. Well-known examples include the set of superpartners in low energy supersymmetry 1 (for a recent review see Ref. 2 ) , KK excitations in extra-dimensional models 3 , as well as similar states in little Higgs models 4 (Ref. 5 provides a brief review). Due to the similarities in gauge quantum numbers, initial LHC signatures of new partners are very similar, as they can decay into the same set of observable final state particles. The mass spectra of different scenarios can be chosen to produce similar dominant kinematical features, such as the p T subscript 𝑝 𝑇 p_{T} distribution of the decay product. For example, a typical gluino decay chain in supersymmetry is g ~ → q ​ q ¯ + N ~ 2 → ~ 𝑔 𝑞 ¯ 𝑞 subscript ~ 𝑁 2 \tilde{g}\rightarrow q\bar{q}+\tilde{N}_{2} followed by N ~ 2 → ℓ ​ ℓ ¯ + N ~ 1 → subscript ~ 𝑁 2 ℓ ¯ ℓ subscript ~ 𝑁 1 \tilde{N}_{2}\rightarrow\ell\bar{\ell}+\tilde{N}_{1} . A similar decay chain in universal extra-dimension models 3 with KK-gluon ( g ( 1 ) superscript 𝑔 1 g^{(1)} ), KK-W ( W 3 ( 1 ) subscript superscript 𝑊 1 3 W^{(1)}_{3} ) and KK-photon ( γ ( 1 ) superscript 𝛾 1 \gamma^{(1)} ), g ( 1 ) → q ​ q ¯ ​ W 3 ( 1 ) → superscript 𝑔 1 𝑞 ¯ 𝑞 subscript superscript 𝑊 1 3 g^{(1)}\rightarrow q\bar{q}W^{(1)}_{3} followed by W 3 ( 1 ) → ℓ ​ ℓ ¯ ​ γ ( 1 ) → subscript superscript 𝑊 1 3 ℓ ¯ ℓ superscript 𝛾 1 W^{(1)}_{3}\rightarrow\ell\bar{\ell}\gamma^{(1)} , gives identical final states since both N ~ 1 subscript ~ 𝑁 1 \tilde{N}_{1} and γ 1 superscript 𝛾 1 \gamma^{1} are neutral stable particles which escape detection. The mass spectra of both supersymmetry and UED can be adjusted in such a way that the p T subscript 𝑝 𝑇 p_{T} of the jets and leptons are quite similar. Some of the similarities in the LHC signature are actually the result of equivalences in low energy effective theory. For example, it is known that theory space motivated little Higgs models are equivalent to extra-dimensional models in which Higgs is a non-local mode in the bulk, via deconstruction 6 , 7 , 8 . Therefore, they can actually be described by the same set of low energy ( ∼ similar-to \sim TeV) degrees of freedom. An important feature of this class of models is that the partners typically have the same spin as their corresponding Standard Model particles. However, the difference between this set of models and low energy supersymmetry is physical and observable with a sufficiently precise measurement. In particular, the spin of superpartners differ from their Standard Model counter parts by half integers. Therefore, spin measurements are crucial to set these scenarios apart. The conventional way of measuring the spin of a new particle involves reconstruction of its rest frame using its decay products and studying the angular distribution about the polarization axis. For example, in process e + ​ e − → Z → μ + ​ μ − → superscript 𝑒 superscript 𝑒 𝑍 → superscript 𝜇 superscript 𝜇 e^{+}e^{-}\rightarrow Z\rightarrow\mu^{+}\mu^{-} , the 1 + cos 2 ⁡ θ 1 superscript 2 𝜃 1+\cos^{2}\theta distribution of the muon direction in the rest frame of the Z 𝑍 Z reveals its vector nature. However, in most new physics scenarios of interest such a strategy is complicated by the generic existence of undetectable massive particles. Motivated by electroweak precision constraints and the existence of Cold Dark Matter, many such scenarios incorporate some discrete symmetry which guarantees the existence of a lightest stable neutral particle. Well-known examples of such discrete symmetries include R-parity in supersymmetry, KK-parity of universal extra-dimension models (UED) 3 , or similarly, T-parity in Little Higgs Models 9 , 10 , 11 , 12 . The existence of such a neutral particle at the end of the decay chain results in large missing energy events in which new physics particles are produced. This fact helps to separate them from the Standard Model background. On the other hand, it also makes the spin measurement more complicated because it is generically impossible to reconstruct the momentum, and therefore the rest frame, of the decaying new physics particles. There are two different approaches to measuring spin. First, given the same gauge quantum numbers, particles with different spin usually have very different production rates, due to the difference between fermionic and bosonic couplings and the number of degrees of freedom. Such an approach could be useful, in particular initially, for colored particles due to their large (hence more measurable) production rates. However, a crucial ingredient in such a strategy is the measurement of the masses of particles produced, as rate can only provide definitive information once the mass is fixed. Such an effort is made more difficult owing to the existence of missing massive particles. There is also some residual model dependence since, for example, a couple of complex scalars can fake a dirac fermion. The second approach, is the direct measurement of spin through its effect on angular correlations in decay products. In the absence of a reconstructed rest frame, one is left to consider Lorentz invariant quantities which encode angular correlations. As we will see later in this review, spin correlations typically only exist in certain type of decays. Furthermore, new physics particles are frequently pair produced with independent decay chains containing similar particles. Therefore, a valid spin correlation measurement requires the ability to identify a relatively pure sample of events where we can isolate certain decay chains and suppress combinatorics effectively. Therefore, except for very special cases, we expect this measurement will require large statistics. At the same time, as will be clear from our discussion, using the appropriate variables and correctly interpreting the measured angular distribution frequently requires at least a partial knowledge of the spectrum and the gauge quantum numbers. Obtaining information about the spectrum and the quantum numbers is likely to require a somewhat lower integrated luminosity than spin measurements do. Therefore, the order with which we uncover the properties of new particles is congruent to the order with which we must proceed in the first place to correctly establish these properties. Thus, we should clearly focus on mass scales, branching ratios and gauge quantum numbers first, once new physics is discovered at the LHC, while keeping an unbiased perspective towards the underlying model. More refined measurements, such as the ones described in this review, will enable us to tell the different models apart thereafter. Such measurements can be useful and even more powerful in a linear collider as was recently proposed in Ref. 13 . In this review we will concentrate on methods applicable to the LHC. In principle, the production of particles with different spins also leads to distinguishable angular distributions. This was investigated in the context of linear colliders in Ref. 14 . A similar measurement using the process p ​ p → ℓ ~ ​ ℓ ~ ⋆ → 𝑝 𝑝 ~ ℓ superscript ~ ℓ ⋆ pp\rightarrow\tilde{\ell}\tilde{\ell}^{\star} at the LHC has been studied in Ref. 15 . An analogues measurement in the production of colored states is more challenging. First, typically several different initial states and partial waves can contribute to the same production process. Therefore, it is difficult to extract spin information from the resulting angular distribution in a model independent way. Second, as commented above it is often difficult to reconstruct the direction of the original particles coming out of the production vertex. As a result, angular correlations are further washed out. In the rest of this review, we will survey both of these approaches with slightly heavier emphasis given to the angular correlation technique. For concreteness, we will compare supersymmetry with another generic scenario in which the partners, such as gluon partner g ′ superscript 𝑔 ′ g^{\prime} , W-partner W ′ superscript 𝑊 ′ W^{\prime} , quark partner q ′ superscript 𝑞 ′ q^{\prime} , etc., have the same spin as their corresponding Standard Model particles. As was pointed out above, this so called same-spin scenario effectively parameterizes almost all non-SUSY models which address the hierarchy problem. Spin measurement at the LHC is still a relatively new field where only first steps towards a comprehensive study have been taken. We will briefly summarize these developments in this review. We will focus here on the theoretical foundations and considerations relevant for the construction of observables. The potential for measuring spin in many new decay channels remains to be studied. Important effects, such as Standard Model background and large combinatorics, deserve careful further consideration. We outline these issues in connection to particular channels below.","role":"user"},{"content":"Abstract: We review the prospects of direct spin determination of new particles which may be discovered at the LHC. We discuss the general framework and the different channels which contain spin information. The experimental challenges associated with such measurements are briefly discussed and the most urgent unresolved problems are emphasized.","role":"assistant"}]

A review of the Statistical Mechanics approach to Random Optimization Problems

I Introduction The connection between the statistical physics of disordered systems and optimization problems in computer science dates back from twenty years at least Mézard et al. ( 1987 ) . In combinatorial optimization one is given a cost function (the length of a tour in the traveling salesman problem (TSP), the number of violated constraints in constraint satisfaction problems, …) over a set of variables and looks for the minimal cost over an allowed range for those variables. Finding the true minimum may be complicated, and requires bigger and bigger computational efforts as the number of variables to be minimized over increases Papadimitriou and Steiglitz ( 1998 ) . Statistical physics is at first sight very different. The scope is to deduce the macroscopic, that is, global properties of a physical system, for instance a gas, a liquid or a solid, from the knowledge of the energetic interactions of its elementary components (molecules, atoms or ions). However, at very low temperature, these elementary components are essentially forced to occupy the spatial conformation minimizing the global energy of the system. Hence low temperature statistical physics can be seen as the search for minimizing a cost function whose expression reflects the laws of Nature or, more humbly, the degree of accuracy retained in its description. This problem is generally not difficult to solve for non disordered systems where the lowest energy conformation are crystals in which components are regularly spaced from each other. Yet the presence of disorder, e.g. impurities, makes the problem very difficult and finding the conformation with minimal energy is a true optimization problem. At the beginning of the eighties, following the works of G. Parisi and others on systems called spin glasses Mézard et al. ( 1987 ) , important progresses were made in the statistical physics of disordered systems. Those progresses made possible the quantitative study of the properties of systems given some distribution of the disorder (for instance the location of impurities) such as the average minimal energy and its fluctuations. The application to optimization problems was natural and led to beautiful studies on (among others) the average properties of the minimal tour length in the TSP, the minimal cost in Bipartite Matching, for some specific instance distributions Mézard et al. ( 1987 ) . Unfortunately statistical physicists and computer scientists did not establish close ties on a large scale at that time. The reason could have been of methodological nature Fu and Anderson ( 1986 ) . While physicists were making statistical statements, true for a given distribution of inputs, computer scientists were rather interested in solving one (or several) particular instances of a problem. The focus was thus on efficient ways to do so, that is, requiring a computational effort growing not too quickly with the number of data defining the instance. Knowing precisely the typical properties for a given, academic distribution of instances did not help much to solve practical cases. At the beginning of the nineties practitionners in artificial intelligence realized that classes of random constraint satisfaction problems used as artificial benchmarks for search algorithms exhibited abrupt changes of behaviour when some control parameter were finely tuned Mitchell et al. ( 1992 ) . The most celebrated example was random k 𝑘 k -Satisfiability, where one looks for a solution to a set of random logical constraints over a set of Boolean variables. It appeared that, for large sets of variables, there was a critical value of the number of constraints per variable below which there almost surely existed solutions, and above which solutions were absent. An important feature was that the performances of known search algorithms drastically worsened in the vicinity of this critical ratio. In addition to its intrinsic mathematical interest the random k 𝑘 k -SAT problem was therefore worth to be studied for ‘practical’ reasons. This critical phenomenon, strongly reminiscent of phase transitions in condensed matter physics, led to a revival of the research at the interface between statistical physics and computer science, which is still very active. The purpose of the present review is to introduce the non physicist reader to some concepts required to understand the literature in the field and to present some major results. We shall in particular discuss the refined picture of the satisfiable phase put forward in statistical mechanics studies and the algorithmic approach (Survey Propagation, an extension of Belief Propagation used in communication theory and statistical inference) this picture suggested. While the presentation will mostly focus on the k 𝑘 k -Satisfiability problem (with random constraints) we will occasionally discuss another computational problem, namely, linear systems of Boolean equations. A good reason to do so is that this problem exhibits some essential features encountered in random k 𝑘 k -Satisfiability, while being technically simpler to study. In addition it is closely related to error-correcting codes in communication theory. The chapter is divided into four main parts. In Section II we present the basic statistical physics concepts necessary to understand the onset of phase transitions, and to characterize the nature of the phases. Those are illustrated on a simple example of decision problem, the so-called perceptron problem. In Section III we review the scenario of the various phase transitions taking place in random k 𝑘 k -SAT. Section IV and V present the techniques used to study various type of algorithms in optimization (local search, backtracking procedures, message passing algorithms). We end up with some conclusive remarks in Sec. VI .

We review the connection between statistical mechanics and the analysis of random optimization problems, with particular emphasis on the random k-SAT problem. We discuss and characterize the different phase transitions that are met in these problems, starting from basic concepts. We also discuss how statistical mechanics methods can be used to investigate the behavior of local search and decimation based algorithms.

[{"content":"Help me to generate the abstract of a survey paper given the title:*A review of the Statistical Mechanics approach to Random Optimization Problems* and the introduction:I Introduction The connection between the statistical physics of disordered systems and optimization problems in computer science dates back from twenty years at least Mézard et al. ( 1987 ) . In combinatorial optimization one is given a cost function (the length of a tour in the traveling salesman problem (TSP), the number of violated constraints in constraint satisfaction problems, …) over a set of variables and looks for the minimal cost over an allowed range for those variables. Finding the true minimum may be complicated, and requires bigger and bigger computational efforts as the number of variables to be minimized over increases Papadimitriou and Steiglitz ( 1998 ) . Statistical physics is at first sight very different. The scope is to deduce the macroscopic, that is, global properties of a physical system, for instance a gas, a liquid or a solid, from the knowledge of the energetic interactions of its elementary components (molecules, atoms or ions). However, at very low temperature, these elementary components are essentially forced to occupy the spatial conformation minimizing the global energy of the system. Hence low temperature statistical physics can be seen as the search for minimizing a cost function whose expression reflects the laws of Nature or, more humbly, the degree of accuracy retained in its description. This problem is generally not difficult to solve for non disordered systems where the lowest energy conformation are crystals in which components are regularly spaced from each other. Yet the presence of disorder, e.g. impurities, makes the problem very difficult and finding the conformation with minimal energy is a true optimization problem. At the beginning of the eighties, following the works of G. Parisi and others on systems called spin glasses Mézard et al. ( 1987 ) , important progresses were made in the statistical physics of disordered systems. Those progresses made possible the quantitative study of the properties of systems given some distribution of the disorder (for instance the location of impurities) such as the average minimal energy and its fluctuations. The application to optimization problems was natural and led to beautiful studies on (among others) the average properties of the minimal tour length in the TSP, the minimal cost in Bipartite Matching, for some specific instance distributions Mézard et al. ( 1987 ) . Unfortunately statistical physicists and computer scientists did not establish close ties on a large scale at that time. The reason could have been of methodological nature Fu and Anderson ( 1986 ) . While physicists were making statistical statements, true for a given distribution of inputs, computer scientists were rather interested in solving one (or several) particular instances of a problem. The focus was thus on efficient ways to do so, that is, requiring a computational effort growing not too quickly with the number of data defining the instance. Knowing precisely the typical properties for a given, academic distribution of instances did not help much to solve practical cases. At the beginning of the nineties practitionners in artificial intelligence realized that classes of random constraint satisfaction problems used as artificial benchmarks for search algorithms exhibited abrupt changes of behaviour when some control parameter were finely tuned Mitchell et al. ( 1992 ) . The most celebrated example was random k 𝑘 k -Satisfiability, where one looks for a solution to a set of random logical constraints over a set of Boolean variables. It appeared that, for large sets of variables, there was a critical value of the number of constraints per variable below which there almost surely existed solutions, and above which solutions were absent. An important feature was that the performances of known search algorithms drastically worsened in the vicinity of this critical ratio. In addition to its intrinsic mathematical interest the random k 𝑘 k -SAT problem was therefore worth to be studied for practical reasons. This critical phenomenon, strongly reminiscent of phase transitions in condensed matter physics, led to a revival of the research at the interface between statistical physics and computer science, which is still very active. The purpose of the present review is to introduce the non physicist reader to some concepts required to understand the literature in the field and to present some major results. We shall in particular discuss the refined picture of the satisfiable phase put forward in statistical mechanics studies and the algorithmic approach (Survey Propagation, an extension of Belief Propagation used in communication theory and statistical inference) this picture suggested. While the presentation will mostly focus on the k 𝑘 k -Satisfiability problem (with random constraints) we will occasionally discuss another computational problem, namely, linear systems of Boolean equations. A good reason to do so is that this problem exhibits some essential features encountered in random k 𝑘 k -Satisfiability, while being technically simpler to study. In addition it is closely related to error-correcting codes in communication theory. The chapter is divided into four main parts. In Section II we present the basic statistical physics concepts necessary to understand the onset of phase transitions, and to characterize the nature of the phases. Those are illustrated on a simple example of decision problem, the so-called perceptron problem. In Section III we review the scenario of the various phase transitions taking place in random k 𝑘 k -SAT. Section IV and V present the techniques used to study various type of algorithms in optimization (local search, backtracking procedures, message passing algorithms). We end up with some conclusive remarks in Sec. VI .","role":"user"},{"content":"Abstract: We review the connection between statistical mechanics and the analysis of random optimization problems, with particular emphasis on the random k-SAT problem. We discuss and characterize the different phase transitions that are met in these problems, starting from basic concepts. We also discuss how statistical mechanics methods can be used to investigate the behavior of local search and decimation based algorithms.","role":"assistant"}]

ZIPF'S LAW AND CITY SIZES: A SHORT TUTORIAL REVIEW ON MULTIPLICATIVE PROCESSES IN URBAN GROWTH

Biological populations –and, among them, human communities– are subject, during their existence, to a multitude of actions of quite disparate origins. Such actions involve a complex interplay between factors endogenous to the population and external effects related to the interaction with the ecosystem and with physical environmental factors. The underlying mechanism governing the growth or decline of the population size (i.e., the number of individuals) is however very simple in essence, since it derives from the elementary events of reproduction: at a given time, the growth rate of the population is proportional to the population itself. This statement must be understood in the sense that two populations formed by the same organisms and under the same ecological conditions, one of them –say– twice as large as the other, will grow by amounts also related by a factor of two. Such proportionality between population and growth rate, which is empirically verified in practically all instances of biological systems, defines a multiplicative process [ 11 ] . Populations whose size is governed by multiplicative processes and which, at the same time, are subject to environmental random-like fluctuations, are known to display universal statistical regularities in the distribution of certain features. Specifically, those traits which are transmitted vertically, from parents to their offspring, exhibit broad, long-tailed distributions with stereotyped shapes –typically, log-normal or power laws. For instance, consider a human society where, except for some unfrequent exceptions, the surname of each individual is inherited from the father. Consider moreover the subpopulations formed by individuals with the same surname. It turns out that the frequency of subpopulations of size n 𝑛 n is approximately proportional to n − 2 superscript 𝑛 2 n^{-2} [ 19 , 4 ] . Or take, from the whole human population, the communities whose individuals speak the same language, which in the vast majority of the cases is learnt from the mother. The sizes of those communities are distributed following a log-normal function [ 12 ] . Such statistical regularities are generally referred to as Zipf’s law [ 18 , 19 ] . The derivation of Zipf’s law from the underlying multiplicative processes was first worked out in detail by the sociologist H. A. Simon, within a set of assumptions which became known as Simon’s model [ 9 ] . A well-documented instance of occurrence of Zipf’s law involves the distribution of city sizes [ 3 , 14 , 1 , 2 ] , where “size” is here identified with the number of inhabitants. In practically any country or region over the globe, the frequency of cities of size n 𝑛 n decays as n − z superscript 𝑛 𝑧 n^{-z} , where the exponent z 𝑧 z is approximately equal to 2 2 2 –as in the case of surnames. The occurrence of Zipf’s law in the distribution of city sizes can be understood in terms of multiplicative processes using Simon’s model. Inspection of current literature on the subject of city size distributions, however, suggests that the potential of Simon’s model as an explanation of Zipf’s law, as well as its limitations, are not well understood. In a recently published handbook on urban economics [ 2 ] , for instance, we read: “Simon’s model encounters some serious problems. In the limit where it can generate Zipf’s law, it … requires that the number of cities grow indefinitely, in fact as fast as the urban population.” It turns out that this assertion is wrong: the truth, in fact, happens to be exactly the opposite! Leaving aside the derivation that may have led to this false conclusion [ 1 ] , we note that such strong statements risk to become dogmatic for the part of the scientific community which does not have the tools for their critical analysis. With this motivation, the present short review will be devoted to give a pedagogical presentation of Simon’s model in the frame of the evolution of city size distributions. The emphasis will be put on a qualitative description of the basic processes involved in the modeling. The explicit statement of the hypotheses that define the model should already expose its limitations but, at the same time, should clarify its flexibility regarding possible generalizations. In the next section, an elementary model for the evolution of a population based on stochastic processes is introduced, and the concurrent role of multiplicative and additive mechanisms in the appearance of power-law distributions is discussed. After an outline of the main features of Zipf’s rank plots in the distribution of city sizes, Simon’s model is presented in its original version, describing its implications as for the population distribution in urban systems. Then, we discuss a few extensions of the model, aimed at capturing some relevant processes not present in its original formulation. Finally, we close with a summary of the main results and some concluding remarks.

We address the role of multiplicative stochastic processes in modeling the occurrence of power-law city size distributions. As an explanation of the result of Zipf’s rank analysis, Simon’s model is presented in a mathematically elementary way, with a thorough discussion of the involved hypotheses. Emphasis is put on the flexibility of the model, as to its possible extensions and the relaxation of some strong assumptions. We point out some open problems regarding the prediction of the detailed shape of Zipf’s rank plots, which may be tackled by means of such extensions.

[{"content":"Help me to generate the abstract of a survey paper given the title:*ZIPF'S LAW AND CITY SIZES: A SHORT TUTORIAL REVIEW ON MULTIPLICATIVE PROCESSES IN URBAN GROWTH* and the introduction:Biological populations –and, among them, human communities– are subject, during their existence, to a multitude of actions of quite disparate origins. Such actions involve a complex interplay between factors endogenous to the population and external effects related to the interaction with the ecosystem and with physical environmental factors. The underlying mechanism governing the growth or decline of the population size (i.e., the number of individuals) is however very simple in essence, since it derives from the elementary events of reproduction: at a given time, the growth rate of the population is proportional to the population itself. This statement must be understood in the sense that two populations formed by the same organisms and under the same ecological conditions, one of them –say– twice as large as the other, will grow by amounts also related by a factor of two. Such proportionality between population and growth rate, which is empirically verified in practically all instances of biological systems, defines a multiplicative process 11 . Populations whose size is governed by multiplicative processes and which, at the same time, are subject to environmental random-like fluctuations, are known to display universal statistical regularities in the distribution of certain features. Specifically, those traits which are transmitted vertically, from parents to their offspring, exhibit broad, long-tailed distributions with stereotyped shapes –typically, log-normal or power laws. For instance, consider a human society where, except for some unfrequent exceptions, the surname of each individual is inherited from the father. Consider moreover the subpopulations formed by individuals with the same surname. It turns out that the frequency of subpopulations of size n 𝑛 n is approximately proportional to n − 2 superscript 𝑛 2 n^{-2} 19 , 4 . Or take, from the whole human population, the communities whose individuals speak the same language, which in the vast majority of the cases is learnt from the mother. The sizes of those communities are distributed following a log-normal function 12 . Such statistical regularities are generally referred to as Zipfs law 18 , 19 . The derivation of Zipfs law from the underlying multiplicative processes was first worked out in detail by the sociologist H. A. Simon, within a set of assumptions which became known as Simons model 9 . A well-documented instance of occurrence of Zipfs law involves the distribution of city sizes 3 , 14 , 1 , 2 , where size is here identified with the number of inhabitants. In practically any country or region over the globe, the frequency of cities of size n 𝑛 n decays as n − z superscript 𝑛 𝑧 n^{-z} , where the exponent z 𝑧 z is approximately equal to 2 2 2 –as in the case of surnames. The occurrence of Zipfs law in the distribution of city sizes can be understood in terms of multiplicative processes using Simons model. Inspection of current literature on the subject of city size distributions, however, suggests that the potential of Simons model as an explanation of Zipfs law, as well as its limitations, are not well understood. In a recently published handbook on urban economics 2 , for instance, we read: Simons model encounters some serious problems. In the limit where it can generate Zipfs law, it … requires that the number of cities grow indefinitely, in fact as fast as the urban population. It turns out that this assertion is wrong: the truth, in fact, happens to be exactly the opposite! Leaving aside the derivation that may have led to this false conclusion 1 , we note that such strong statements risk to become dogmatic for the part of the scientific community which does not have the tools for their critical analysis. With this motivation, the present short review will be devoted to give a pedagogical presentation of Simons model in the frame of the evolution of city size distributions. The emphasis will be put on a qualitative description of the basic processes involved in the modeling. The explicit statement of the hypotheses that define the model should already expose its limitations but, at the same time, should clarify its flexibility regarding possible generalizations. In the next section, an elementary model for the evolution of a population based on stochastic processes is introduced, and the concurrent role of multiplicative and additive mechanisms in the appearance of power-law distributions is discussed. After an outline of the main features of Zipfs rank plots in the distribution of city sizes, Simons model is presented in its original version, describing its implications as for the population distribution in urban systems. Then, we discuss a few extensions of the model, aimed at capturing some relevant processes not present in its original formulation. Finally, we close with a summary of the main results and some concluding remarks.","role":"user"},{"content":"Abstract: We address the role of multiplicative stochastic processes in modeling the occurrence of power-law city size distributions. As an explanation of the result of Zipfs rank analysis, Simons model is presented in a mathematically elementary way, with a thorough discussion of the involved hypotheses. Emphasis is put on the flexibility of the model, as to its possible extensions and the relaxation of some strong assumptions. We point out some open problems regarding the prediction of the detailed shape of Zipfs rank plots, which may be tackled by means of such extensions.","role":"assistant"}]

A mathematical and computational review of Hartree–Fock SCF methods in quantum chemistry

In the hot field of computer simulation of biological macromolecules, available potential energy functions are often not accurate enough to properly describe complex processes such as the folding of proteins [ 1 , 2 , 3 , 4 , 5 , 6 , 7 ] . In order to improve the situation, it is convenient to extract ab initio information from quantum mechanical calculations with the hope of being able to devise less computationally demanding methods that can be used to tackle large systems. In this spirit, the effective potential for the nuclei calculated in the non-relativistic Born-Oppenheimer approximation is typically considered as a good reference to assess the accuracy of cheaper potentials [ 8 , 9 , 10 , 11 , 12 , 13 , 14 ] . The study of molecules at this level of theoretical detail and the design of computationally efficient approximations for solving the demanding equations that appear constitute the major part of the field called quantum chemistry [ 15 , 16 ] . In this work, we voluntarily circumscribe ourselves to the basic formalism needed for the ground-state quantum chemical calculations that are typically performed in this context. For more general expositions, we refer the reader to any of the thorough accounts in refs. [ 17 , 18 , 19 ] . In sec. 2 , we introduce the molecular Hamiltonian and a special set of units (the atomic ones) that are convenient to simplify the equations. In sec. 3 , we present in an axiomatic way the concepts and expressions related to the separation of the electronic and nuclear problems in the Born-Oppenheimer scheme. In sec. 4 , we introduce the variational method that underlies the derivation of the basic equations of the Hartree and Hartree-Fock approximations, discussed in sec. 6 and 7 respectively. The computational implementation of the Hartree-Fock approximation is tackled in sec. 8 , where the celebrated Roothaan-Hall equations are derived. In sec. 9 , the main issues related to the construction and selection of Gaussian basis sets are discussed, and, finally, in sec. 10 , the hottest areas of modern research are briefly reviewed and linked to the issues in the rest of the work, with a special emphasis in the development of linear-scaling methods.

We present a review of the fundamental topics of Hartree–Fock theory in quantum chemistry. From the molecular Hamiltonian, using and discussing the Born–Oppenheimer approximation, we arrive at the Hartree and Hartree–Fock equations for the electronic problem. Special emphasis is placed on the most relevant mathematical aspects of the theoretical derivation of the final equations, and on the results regarding the existence and uniqueness of their solutions. All Hartree–Fock versions with different spin restrictions are systematically extracted from the general case, thus providing a unifying framework. The discretization of the one-electron orbital space is then reviewed and the Roothaan–Hall formalism introduced. This leads to an exposition of the basic underlying concepts related to the construction and selection of Gaussian basis sets, focusing on algorithmic efficiency issues. Finally, we close the review with a section in which the most relevant modern developments (especially those related to the design of linear-scaling methods) are commented on and linked to the issues discussed. The paper is intentionally introductory and rather self-contained, and may be useful for non-experts intending to use quantum chemical methods in interdisciplinary applications. Moreover, much material that can be found scattered in the literature has been put together to facilitate comprehension and to serve as a handy reference.

[{"content":"Help me to generate the abstract of a survey paper given the title:*A mathematical and computational review of Hartree–Fock SCF methods in quantum chemistry* and the introduction:In the hot field of computer simulation of biological macromolecules, available potential energy functions are often not accurate enough to properly describe complex processes such as the folding of proteins 1 , 2 , 3 , 4 , 5 , 6 , 7 . In order to improve the situation, it is convenient to extract ab initio information from quantum mechanical calculations with the hope of being able to devise less computationally demanding methods that can be used to tackle large systems. In this spirit, the effective potential for the nuclei calculated in the non-relativistic Born-Oppenheimer approximation is typically considered as a good reference to assess the accuracy of cheaper potentials 8 , 9 , 10 , 11 , 12 , 13 , 14 . The study of molecules at this level of theoretical detail and the design of computationally efficient approximations for solving the demanding equations that appear constitute the major part of the field called quantum chemistry 15 , 16 . In this work, we voluntarily circumscribe ourselves to the basic formalism needed for the ground-state quantum chemical calculations that are typically performed in this context. For more general expositions, we refer the reader to any of the thorough accounts in refs. 17 , 18 , 19 . In sec. 2 , we introduce the molecular Hamiltonian and a special set of units (the atomic ones) that are convenient to simplify the equations. In sec. 3 , we present in an axiomatic way the concepts and expressions related to the separation of the electronic and nuclear problems in the Born-Oppenheimer scheme. In sec. 4 , we introduce the variational method that underlies the derivation of the basic equations of the Hartree and Hartree-Fock approximations, discussed in sec. 6 and 7 respectively. The computational implementation of the Hartree-Fock approximation is tackled in sec. 8 , where the celebrated Roothaan-Hall equations are derived. In sec. 9 , the main issues related to the construction and selection of Gaussian basis sets are discussed, and, finally, in sec. 10 , the hottest areas of modern research are briefly reviewed and linked to the issues in the rest of the work, with a special emphasis in the development of linear-scaling methods.","role":"user"},{"content":"Abstract: We present a review of the fundamental topics of Hartree–Fock theory in quantum chemistry. From the molecular Hamiltonian, using and discussing the Born–Oppenheimer approximation, we arrive at the Hartree and Hartree–Fock equations for the electronic problem. Special emphasis is placed on the most relevant mathematical aspects of the theoretical derivation of the final equations, and on the results regarding the existence and uniqueness of their solutions. All Hartree–Fock versions with different spin restrictions are systematically extracted from the general case, thus providing a unifying framework. The discretization of the one-electron orbital space is then reviewed and the Roothaan–Hall formalism introduced. This leads to an exposition of the basic underlying concepts related to the construction and selection of Gaussian basis sets, focusing on algorithmic efficiency issues. Finally, we close the review with a section in which the most relevant modern developments (especially those related to the design of linear-scaling methods) are commented on and linked to the issues discussed. The paper is intentionally introductory and rather self-contained, and may be useful for non-experts intending to use quantum chemical methods in interdisciplinary applications. Moreover, much material that can be found scattered in the literature has been put together to facilitate comprehension and to serve as a handy reference.","role":"assistant"}]

SOCIOPHYSICS: A REVIEW OF GALAM MODELS

I Introduction The field of sociophysics has emerged in the 19-seventies with only a very few scarce papers in the hostile environment of the community of physicists. It started to attracted some physicists around the mid nineties and then took over fifteen years to nucleate involving a growing number of physicists. Today it is a recognized field of physics anchored within statistical physics. It is flourishing and expanding with hundreds of papers published in the leading physical journals and quite a few international conferences held each year. The topics covered by sociophysics are becoming numerous and address many different problems including social networks, language evolution, population dynamics, epidemic spreading, terrorism, voting, coalition formation and opinion dynamics. Among these topics the subject of opinion dynamics has become one of the main streams of sociophysics producing a great deal of research papers also in this journal, including this issue. This review does not deal with all of these papers because of the restriction made clear by its title. This does not mean that the other papers are less important or worse than those cited here. But we restrict the presentation to the models introduced by Galam and Galam et al over the last twenty five years, a good part of them being the pioneer works of sociophysics. A Springer book is in preparation on the subject. These models deal with the five subjects of democratic voting in bottom up hierarchical systems, decision making, fragmentation versus coalitions, terrorism and opinion dynamics. The first class of models v1 ; v2 ; v3 ; v4 ; v5 ; v6 ; v7 ; v8 ; v9 ; v10 ; v11 ; v12 ; v13 consider a population, which is a mixture of two species A and B. A bottom up hierarchy is then built from the population using local majority rules with the possibility of some power inertia bias. Tree like networks are thus constructed, which combine a random selection of agents at the bottom from the surrounding population with an associated deterministic outcome at the top. The scheme relates on adapting real space renormalization group technics to build a social and political structure. The second class s1 ; s2 ; s3 ; s4 ; s5 ; s6 ; s7 ; s8 ; s9 ; s10 tackles the problem of decision making in various frames including firms and small committees. It uses ferromagnetic Ising spin Hamiltonians with both external and random quenched fields at both zero and non zero temperatures. The associated phase diagrams are constructed. The effect of reversing an external field on the collective equilibrium state is studied with an emphasis on the existence of nucleation phenomena. Mean field treatment is applied. The third class f1 ; f2 ; f3 ; f4 ; f5 introduces a combination of random bond and random site spins glasses to describe the formation of coalitions as well the dynamics of fragmentation among a group of countrys. External and local fields are also considered together with site dilution effects in mixtures of ferro and anti-ferromagnetic spin Hamiltonians. Ising and Potts variables are used. The fourth class t1 ; t2 ; t3 ; t4 ; t5 ; t6 studies some aspects of terrorism by focusing on the role of passive supporters in the creation of the open social spaces, which are opened to terrorist activities. It relies on the theory of percolation and uses the dependence of the percolation threshold upon the space dimensionality. The fifth class o1 ; o2 ; o3 ; o4 ; o5 ; o6 ; o7 ; o8 ; o9 ; o10 ; o11 ; o12 ; o13 ; o14 ; o15 ; o16 investigates opinion dynamics within reaction-diffusion like models. Two and three states variables are used. Three king of agents are also considered, which are respectively floaters, contrarians and inflexibles. The dynamics operates via local updates and reshuffling. Technics from real space renormalization group approach are used. For each class of models the precise connexion to the original physical model is made. Similarities and differences are outlined emphasizing the eventual novelties with respect to the statistical physics counterparts. The numerous results obtained by each class of models are reviewed enlightening the novel and counterintuitive aspects with respect to the associated social and political framework. In particular several major real political events were successfully predicted using these models. It includes the victory of the French extreme right party in the 2000 first round of French presidential elections vp1 ; vp2 ; vp3 ; vp4 ; vp5 ; vp6 , the voting at fifty - fifty in several democratic countries (Germany, Italy, Mexico) op6 ; op7 ; op8 , and the victory of the no to the 2005 French referendum on the European constitution op9 . To conclude, the perspectives to make sociophysics a predictive solid field of science are discussed, emphasizing both the challenges and the risks.

We review a series of models of sociophysics introduced by Galam and Galam et al. in the last 25 years. The models are divided into five different classes, which deal respectively with democratic voting in bottom-up hierarchical systems, decision making, fragmentation versus coalitions, terrorism and opinion dynamics. For each class the connexion to the original physical model and techniques are outlined underlining both the similarities and the differences. Emphasis is put on the numerous novel and counterintuitive results obtained with respect to the associated social and political framework. Using these models several major real political events were successfully predicted including the victory of the French extreme right party in the 2000 first round of French presidential elections, the voting at fifty–fifty in several democratic countries (Germany, Italy, Mexico), and the victory of the "no" to the 2005 French referendum on the European constitution. The perspectives and the challenges to make sociophysics a predictive solid field of science are discussed.

[{"content":"Help me to generate the abstract of a survey paper given the title:*SOCIOPHYSICS: A REVIEW OF GALAM MODELS* and the introduction:I Introduction The field of sociophysics has emerged in the 19-seventies with only a very few scarce papers in the hostile environment of the community of physicists. It started to attracted some physicists around the mid nineties and then took over fifteen years to nucleate involving a growing number of physicists. Today it is a recognized field of physics anchored within statistical physics. It is flourishing and expanding with hundreds of papers published in the leading physical journals and quite a few international conferences held each year. The topics covered by sociophysics are becoming numerous and address many different problems including social networks, language evolution, population dynamics, epidemic spreading, terrorism, voting, coalition formation and opinion dynamics. Among these topics the subject of opinion dynamics has become one of the main streams of sociophysics producing a great deal of research papers also in this journal, including this issue. This review does not deal with all of these papers because of the restriction made clear by its title. This does not mean that the other papers are less important or worse than those cited here. But we restrict the presentation to the models introduced by Galam and Galam et al over the last twenty five years, a good part of them being the pioneer works of sociophysics. A Springer book is in preparation on the subject. These models deal with the five subjects of democratic voting in bottom up hierarchical systems, decision making, fragmentation versus coalitions, terrorism and opinion dynamics. The first class of models v1 ; v2 ; v3 ; v4 ; v5 ; v6 ; v7 ; v8 ; v9 ; v10 ; v11 ; v12 ; v13 consider a population, which is a mixture of two species A and B. A bottom up hierarchy is then built from the population using local majority rules with the possibility of some power inertia bias. Tree like networks are thus constructed, which combine a random selection of agents at the bottom from the surrounding population with an associated deterministic outcome at the top. The scheme relates on adapting real space renormalization group technics to build a social and political structure. The second class s1 ; s2 ; s3 ; s4 ; s5 ; s6 ; s7 ; s8 ; s9 ; s10 tackles the problem of decision making in various frames including firms and small committees. It uses ferromagnetic Ising spin Hamiltonians with both external and random quenched fields at both zero and non zero temperatures. The associated phase diagrams are constructed. The effect of reversing an external field on the collective equilibrium state is studied with an emphasis on the existence of nucleation phenomena. Mean field treatment is applied. The third class f1 ; f2 ; f3 ; f4 ; f5 introduces a combination of random bond and random site spins glasses to describe the formation of coalitions as well the dynamics of fragmentation among a group of countrys. External and local fields are also considered together with site dilution effects in mixtures of ferro and anti-ferromagnetic spin Hamiltonians. Ising and Potts variables are used. The fourth class t1 ; t2 ; t3 ; t4 ; t5 ; t6 studies some aspects of terrorism by focusing on the role of passive supporters in the creation of the open social spaces, which are opened to terrorist activities. It relies on the theory of percolation and uses the dependence of the percolation threshold upon the space dimensionality. The fifth class o1 ; o2 ; o3 ; o4 ; o5 ; o6 ; o7 ; o8 ; o9 ; o10 ; o11 ; o12 ; o13 ; o14 ; o15 ; o16 investigates opinion dynamics within reaction-diffusion like models. Two and three states variables are used. Three king of agents are also considered, which are respectively floaters, contrarians and inflexibles. The dynamics operates via local updates and reshuffling. Technics from real space renormalization group approach are used. For each class of models the precise connexion to the original physical model is made. Similarities and differences are outlined emphasizing the eventual novelties with respect to the statistical physics counterparts. The numerous results obtained by each class of models are reviewed enlightening the novel and counterintuitive aspects with respect to the associated social and political framework. In particular several major real political events were successfully predicted using these models. It includes the victory of the French extreme right party in the 2000 first round of French presidential elections vp1 ; vp2 ; vp3 ; vp4 ; vp5 ; vp6 , the voting at fifty - fifty in several democratic countries (Germany, Italy, Mexico) op6 ; op7 ; op8 , and the victory of the no to the 2005 French referendum on the European constitution op9 . To conclude, the perspectives to make sociophysics a predictive solid field of science are discussed, emphasizing both the challenges and the risks.","role":"user"},{"content":"Abstract: We review a series of models of sociophysics introduced by Galam and Galam et al. in the last 25 years. The models are divided into five different classes, which deal respectively with democratic voting in bottom-up hierarchical systems, decision making, fragmentation versus coalitions, terrorism and opinion dynamics. For each class the connexion to the original physical model and techniques are outlined underlining both the similarities and the differences. Emphasis is put on the numerous novel and counterintuitive results obtained with respect to the associated social and political framework. Using these models several major real political events were successfully predicted including the victory of the French extreme right party in the 2000 first round of French presidential elections, the voting at fifty–fifty in several democratic countries (Germany, Italy, Mexico), and the victory of the "no" to the 2005 French referendum on the European constitution. The perspectives and the challenges to make sociophysics a predictive solid field of science are discussed.","role":"assistant"}]

Energy-Efficient Resource Allocation in Wireless Networks

I Introduction and Motivation Future wireless networks are expected to support a variety of services with diverse quality-of-service (QoS) requirements. For example, a mixture of delay-sensitive applications (e.g., voice and video teleconferencing) and delay-tolerant ones (e.g., web browsing and file downloading) must be supported. Given that the two principal wireless network resources, i.e., bandwidth and energy , are scarce, the main challenge in designing wireless networks is to use network resources as efficiently as possible while providing the QoS required by the users. Game-theoretic approaches to radio resource allocation have recently attracted much attention and will be the focus of this article. We will show that game theory can be used as a unifying framework to study radio resource management in a variety of wireless networks with different service criteria. Our focus will be on infrastructure networks where users transmit to a common concentration point such as a base station in a cellular network or an access point. Since most of the terminals in a wireless network are battery-powered, energy efficiency is crucial to prolonging the life of the terminals. Also, in most practical scenarios, distributed algorithms are preferred over centralized ones. Centralized algorithms tend to be complex and not easily scalable. Therefore, throughout this article, we focus on distributed algorithms with emphasis on energy efficiency. Using a game-theoretic framework, we demonstrate the impact of advanced signal processing on energy efficiency and network capacity. The tradeoffs among throughput, delay, network capacity and energy efficiency are also discussed. The ideas presented in this paper can also be applied to wireless ad hoc networks, however, the topic is beyond the scope of this article (see [ 1 ] for applications of game theory to ad hoc networks). It should be noted that, recently, tools from optimization theory have also been employed to study resource allocation in wireless networks using the network utility maximization framework proposed in [ 2 ] (see for example [ 3 ] ). While there is considerable overlap between the game-theoretic and optimization-theoretic approaches, game theory tends to focus on the multiuser competitive nature of the problem and on the users’ interaction. The rest of this article is organized as follows. In Section II , we describe how game theory can be used for studying radio resource management in wireless networks. The choice of the utility function is discussed in Section III . In Section IV , we present a family of power control games for energy-efficient resource allocation in wireless CDMA networks. Finally, discussions and conclusions are given in Section V .

This paper shows that game theory can be used as a unifying framework to study radio resource management in a variety of wireless networks. with different service criteria. It focuses on infrastructure networks where users transmit to a common concentration point such as a base station in a cellular network or an access point. Since most of the terminals in a wireless network are battery-powered, energy efficiency is crucial to prolonging the life of the terminals. Also, in most practical scenarios, distributed algorithms are preferred over centralized ones. Throughout this article it focuses on distributed algorithms with emphasis on energy efficiency. A family of power control games is presented for energy-efficient resource allocation in wireless code-division multiple-access (CDMA) networks and give discussions and conclusions

[{"content":"Help me to generate the abstract of a survey paper given the title:*Energy-Efficient Resource Allocation in Wireless Networks* and the introduction:I Introduction and Motivation Future wireless networks are expected to support a variety of services with diverse quality-of-service (QoS) requirements. For example, a mixture of delay-sensitive applications (e.g., voice and video teleconferencing) and delay-tolerant ones (e.g., web browsing and file downloading) must be supported. Given that the two principal wireless network resources, i.e., bandwidth and energy , are scarce, the main challenge in designing wireless networks is to use network resources as efficiently as possible while providing the QoS required by the users. Game-theoretic approaches to radio resource allocation have recently attracted much attention and will be the focus of this article. We will show that game theory can be used as a unifying framework to study radio resource management in a variety of wireless networks with different service criteria. Our focus will be on infrastructure networks where users transmit to a common concentration point such as a base station in a cellular network or an access point. Since most of the terminals in a wireless network are battery-powered, energy efficiency is crucial to prolonging the life of the terminals. Also, in most practical scenarios, distributed algorithms are preferred over centralized ones. Centralized algorithms tend to be complex and not easily scalable. Therefore, throughout this article, we focus on distributed algorithms with emphasis on energy efficiency. Using a game-theoretic framework, we demonstrate the impact of advanced signal processing on energy efficiency and network capacity. The tradeoffs among throughput, delay, network capacity and energy efficiency are also discussed. The ideas presented in this paper can also be applied to wireless ad hoc networks, however, the topic is beyond the scope of this article (see 1 for applications of game theory to ad hoc networks). It should be noted that, recently, tools from optimization theory have also been employed to study resource allocation in wireless networks using the network utility maximization framework proposed in 2 (see for example 3 ). While there is considerable overlap between the game-theoretic and optimization-theoretic approaches, game theory tends to focus on the multiuser competitive nature of the problem and on the users interaction. The rest of this article is organized as follows. In Section II , we describe how game theory can be used for studying radio resource management in wireless networks. The choice of the utility function is discussed in Section III . In Section IV , we present a family of power control games for energy-efficient resource allocation in wireless CDMA networks. Finally, discussions and conclusions are given in Section V .","role":"user"},{"content":"Abstract: This paper shows that game theory can be used as a unifying framework to study radio resource management in a variety of wireless networks. with different service criteria. It focuses on infrastructure networks where users transmit to a common concentration point such as a base station in a cellular network or an access point. Since most of the terminals in a wireless network are battery-powered, energy efficiency is crucial to prolonging the life of the terminals. Also, in most practical scenarios, distributed algorithms are preferred over centralized ones. Throughout this article it focuses on distributed algorithms with emphasis on energy efficiency. A family of power control games is presented for energy-efficient resource allocation in wireless code-division multiple-access (CDMA) networks and give discussions and conclusions","role":"assistant"}]

Prospects for producing ultracold NH 3 molecules by sympathetic cooling: A survey of interaction potentials

I Introduction There is great interest at present in producing samples of cold molecules (below 1 K) and ultracold molecules (below 1 mK). Such molecules have many potential applications. High-precision measurements on ultracold molecules might be used to measure quantities of fundamental physics interest, such as the electric dipole moment of the electron Hudson et al. ( 2002 ) and the time-dependence of fundamental constants such as the electron/proton mass ratio van Veldhoven et al. ( 2004 ) . Ultracold molecules are a stepping stone to ultracold quantum gases Baranov et al. ( 2002 ) and might have applications in quantum information and quantum computing DeMille ( 2002 ) . There are two basic approaches to producing ultracold molecules. In direct methods such as Stark deceleration Bethlem and Meijer ( 2003 ); Bethlem et al. ( 2006 ) and helium buffer-gas cooling Weinstein et al. ( 1998 ) , preexisting molecules are cooled from higher temperatures and trapped in electrostatic or magnetic traps. In indirect methods Hutson and Soldán ( 2006 ) , laser-cooled atoms that are already ultracold are paired up to form molecules by either photoassociation Jones et al. ( 2006 ) or tuning through magnetic Feshbach resonances Köhler et al. ( 2006 ) . Indirect methods have already been used extensively to produce ultracold molecules at temperatures below 1 μ 𝜇 \mu K. However, they are limited to molecules formed from atoms that can themselves be cooled to such temperatures. Direct methods are far more general than indirect methods, and can in principle be applied to a very wide range of molecules. However, at present direct methods are limited to temperatures in the range 10-100 mK, which is outside the ultracold regime. There is much current research directed at finding second-stage cooling methods to bridge the gap and eventually allow directly cooled molecules to reach the region below 1 μ 𝜇 \mu K where quantum gases can form. One of the most promising second-stage cooling methods that has been proposed is sympathetic cooling . The hope is that, if a sample of cold molecules in brought into contact with a gas of ultracold atoms, thermalization will occur and the molecules will be cooled towards the temperature of the atoms. Sympathetic cooling has already been used successfully to cool atomic species such as 6 Li Schreck et al. ( 2001 ) and 41 K Modugno et al. ( 2001 ) but has not yet been applied to neutral molecules. Sympathetic cooling relies on thermalization occurring before molecules are lost from the trap. Thermalization requires elastic collisions between atoms and molecules to redistribute translational energy. However, electrostatic and magnetic traps rely on Stark and Zeeman splittings and trapped atoms and molecules are not usually in their absolute ground state in the applied field. Any inelastic collision that converts internal energy into translational energy is likely to kick both colliding species out of the trap. The ratio of elastic to inelastic cross sections is thus crucial, and a commonly stated rule of thumb is that sympathetic cooling will not work unless elastic cross sections are a factor of 10 to 100 greater than inelastic cross sections for the states concerned. Inelastic cross sections for atom-atom collisions are sometimes strongly suppressed by angular momentum constraints. In particular, for s-wave collisions (end-over-end angular momentum L = 0 𝐿 0 L=0 ), pairs of atoms in spin-stretched states (with the maximum possible values of the total angular momentum F 𝐹 F and its projection | M F | subscript 𝑀 𝐹 |M_{F}| ) can undergo inelastic collisions only by changing L 𝐿 L . Cross sections for such processes are very small because, for atoms in S states, the only interaction that can change L 𝐿 L is the weak dipolar coupling between the electron spins. However, for molecular collisions the situation is different: the anisotropy of the intermolecular potential can change L 𝐿 L , and this is usually much stronger than spin-spin coupling. It is thus crucial to investigate the anisotropy of the interaction potential for systems that are candidates for sympathetic cooling experiments. In experimental terms, the easiest systems to work with are those in which molecules that can be cooled by Stark deceleration (such as NH 3 , OH and NH) interact with atoms that can be laser-cooled (such as alkali-metal and alkaline-earth atoms). There has been extensive work on low-energy collisions of molecules with helium atoms Balakrishnan et al. ( 1997 , 1999 , 2000 ); Bohn ( 2000 ); Balakrishnan et al. ( 2003 ); Krems et al. ( 2003 ); González-Martínez and Hutson ( 2007 ) , but relatively little on collisions with alkali-metal and alkaline-earth atoms. Soldán and Hutson Soldán and Hutson ( 2004 ) investigated the potential energy surfaces for Rb + NH and identified deeply bound ion-pair states as well as weakly bound covalent states. They suggested that the ion-pair states might hinder sympathetic cooling. Lara et al. Lara et al. ( 2006 , 2007 ) subsequently calculated full potential energy surfaces for Rb + OH, for both ion-pair states and covalent states, and used them to investigate low-energy elastic and inelastic cross sections, including spin-orbit coupling and nuclear spin splittings. They found that even for the covalent states the potential energy surfaces had anisotropies of the order of 500 cm -1 and that this was sufficient to make the inelastic cross sections larger than inelastic cross sections at temperatures below 10 mK. Tacconi et al. Tacconi et al. ( 2007 ) have recently carried out analogous calculations on Rb + NH, though without considering nuclear spin. There has also been a considerable amount of work on collisions between alkali metal atoms and the corresponding dimers Soldán et al. ( 2002 ); Quéméner et al. ( 2004 ); Cvitaš et al. ( 2005a , b ); Quéméner et al. ( 2005 ); Hutson and Soldán ( 2007 ) . One way around the problem of inelastic collisions is to work with atoms and molecules that are in their absolute ground state in the trapping field. However, this is quite limiting: only optical dipole traps and alternating current traps van Veldhoven et al. ( 2005 ) can trap such molecules. It is therefore highly desirable to seek systems in which the potential energy surface is only weakly anisotropic. The purpose of the present paper is to survey the possibilities for collision partners to use in sympathetic cooling of NH 3 (or ND 3 ), which is one of the easiest molecules for Stark deceleration. Even if sympathetic cooling proves to be impractical for a particular system, the combination of laser cooling for atoms and Stark deceleration for molecules offers opportunities for studying molecular collisions in a new low-energy regime. For example, experiments are under way at the University of Colorado Lewandowski ( 2008 ) to study collisions between decelerated NH 3 molecules and laser-cooled Rb atoms. There alkali-metal atom + NH 3 systems have not been extensively studied theoretically, though there has been experimental interest in the spectroscopy of Li-NH 3 complex as a prototype metal atom-Lewis base complex Wu et al. ( 2001 ) . Lim et al. Lim et al. ( 2007 ) recently calculated electrical properties and infrared spectra for complexes of NH 3 with alkali-metal atoms from K to Fr and gave the equilibrium structures of their global minima. However, to our knowledge, no complete potential energy surfaces have been published for any of these systems. The alkaline-earth + NH 3 have been studied even less, and except for an early study of the Be-NH 3 system Chalasinski et al. ( 1993 ) there are no previous results available.

We investigate the possibility of producing ultracold NH3 molecules by sympathetic cooling in a bath of ultracold atoms. We consider the interactions of NH3 with alkali-metal and alkaline-earth-metal atoms, and with Xe, using ab initio coupled-cluster calculations. For Rb-NH3 and Xe-NH3 we develop full potential energy surfaces, while for the other systems we characterize the stationary points (global and local minima and saddle points). We also calculate isotropic and anisotropic van der Waals C6 coefficients for all the systems. The potential energy surfaces for interaction of NH3 with alkali-metal and alkaline-earth-metal atoms all show deep potential wells and strong anisotropies. The well depths vary from 887 cm−1 for Mg-NH3 to 5104 cm−1 for Li-NH3. This suggests that all these systems will exhibit strong inelasticity whenever inelastic collisions are energetically allowed and that sympathetic cooling will work only when both the atoms and the molecules are already in their lowest internal states. Xe-NH3 is more weakly bound and less anisotropic.

[{"content":"Help me to generate the abstract of a survey paper given the title:*Prospects for producing ultracold NH 3 molecules by sympathetic cooling: A survey of interaction potentials* and the introduction:I Introduction There is great interest at present in producing samples of cold molecules (below 1 K) and ultracold molecules (below 1 mK). Such molecules have many potential applications. High-precision measurements on ultracold molecules might be used to measure quantities of fundamental physics interest, such as the electric dipole moment of the electron Hudson et al. ( 2002 ) and the time-dependence of fundamental constants such as the electron/proton mass ratio van Veldhoven et al. ( 2004 ) . Ultracold molecules are a stepping stone to ultracold quantum gases Baranov et al. ( 2002 ) and might have applications in quantum information and quantum computing DeMille ( 2002 ) . There are two basic approaches to producing ultracold molecules. In direct methods such as Stark deceleration Bethlem and Meijer ( 2003 ); Bethlem et al. ( 2006 ) and helium buffer-gas cooling Weinstein et al. ( 1998 ) , preexisting molecules are cooled from higher temperatures and trapped in electrostatic or magnetic traps. In indirect methods Hutson and Soldán ( 2006 ) , laser-cooled atoms that are already ultracold are paired up to form molecules by either photoassociation Jones et al. ( 2006 ) or tuning through magnetic Feshbach resonances Köhler et al. ( 2006 ) . Indirect methods have already been used extensively to produce ultracold molecules at temperatures below 1 μ 𝜇 \mu K. However, they are limited to molecules formed from atoms that can themselves be cooled to such temperatures. Direct methods are far more general than indirect methods, and can in principle be applied to a very wide range of molecules. However, at present direct methods are limited to temperatures in the range 10-100 mK, which is outside the ultracold regime. There is much current research directed at finding second-stage cooling methods to bridge the gap and eventually allow directly cooled molecules to reach the region below 1 μ 𝜇 \mu K where quantum gases can form. One of the most promising second-stage cooling methods that has been proposed is sympathetic cooling . The hope is that, if a sample of cold molecules in brought into contact with a gas of ultracold atoms, thermalization will occur and the molecules will be cooled towards the temperature of the atoms. Sympathetic cooling has already been used successfully to cool atomic species such as 6 Li Schreck et al. ( 2001 ) and 41 K Modugno et al. ( 2001 ) but has not yet been applied to neutral molecules. Sympathetic cooling relies on thermalization occurring before molecules are lost from the trap. Thermalization requires elastic collisions between atoms and molecules to redistribute translational energy. However, electrostatic and magnetic traps rely on Stark and Zeeman splittings and trapped atoms and molecules are not usually in their absolute ground state in the applied field. Any inelastic collision that converts internal energy into translational energy is likely to kick both colliding species out of the trap. The ratio of elastic to inelastic cross sections is thus crucial, and a commonly stated rule of thumb is that sympathetic cooling will not work unless elastic cross sections are a factor of 10 to 100 greater than inelastic cross sections for the states concerned. Inelastic cross sections for atom-atom collisions are sometimes strongly suppressed by angular momentum constraints. In particular, for s-wave collisions (end-over-end angular momentum L = 0 𝐿 0 L=0 ), pairs of atoms in spin-stretched states (with the maximum possible values of the total angular momentum F 𝐹 F and its projection | M F | subscript 𝑀 𝐹 |M_{F}| ) can undergo inelastic collisions only by changing L 𝐿 L . Cross sections for such processes are very small because, for atoms in S states, the only interaction that can change L 𝐿 L is the weak dipolar coupling between the electron spins. However, for molecular collisions the situation is different: the anisotropy of the intermolecular potential can change L 𝐿 L , and this is usually much stronger than spin-spin coupling. It is thus crucial to investigate the anisotropy of the interaction potential for systems that are candidates for sympathetic cooling experiments. In experimental terms, the easiest systems to work with are those in which molecules that can be cooled by Stark deceleration (such as NH 3 , OH and NH) interact with atoms that can be laser-cooled (such as alkali-metal and alkaline-earth atoms). There has been extensive work on low-energy collisions of molecules with helium atoms Balakrishnan et al. ( 1997 , 1999 , 2000 ); Bohn ( 2000 ); Balakrishnan et al. ( 2003 ); Krems et al. ( 2003 ); González-Martínez and Hutson ( 2007 ) , but relatively little on collisions with alkali-metal and alkaline-earth atoms. Soldán and Hutson Soldán and Hutson ( 2004 ) investigated the potential energy surfaces for Rb + NH and identified deeply bound ion-pair states as well as weakly bound covalent states. They suggested that the ion-pair states might hinder sympathetic cooling. Lara et al. Lara et al. ( 2006 , 2007 ) subsequently calculated full potential energy surfaces for Rb + OH, for both ion-pair states and covalent states, and used them to investigate low-energy elastic and inelastic cross sections, including spin-orbit coupling and nuclear spin splittings. They found that even for the covalent states the potential energy surfaces had anisotropies of the order of 500 cm -1 and that this was sufficient to make the inelastic cross sections larger than inelastic cross sections at temperatures below 10 mK. Tacconi et al. Tacconi et al. ( 2007 ) have recently carried out analogous calculations on Rb + NH, though without considering nuclear spin. There has also been a considerable amount of work on collisions between alkali metal atoms and the corresponding dimers Soldán et al. ( 2002 ); Quéméner et al. ( 2004 ); Cvitaš et al. ( 2005a , b ); Quéméner et al. ( 2005 ); Hutson and Soldán ( 2007 ) . One way around the problem of inelastic collisions is to work with atoms and molecules that are in their absolute ground state in the trapping field. However, this is quite limiting: only optical dipole traps and alternating current traps van Veldhoven et al. ( 2005 ) can trap such molecules. It is therefore highly desirable to seek systems in which the potential energy surface is only weakly anisotropic. The purpose of the present paper is to survey the possibilities for collision partners to use in sympathetic cooling of NH 3 (or ND 3 ), which is one of the easiest molecules for Stark deceleration. Even if sympathetic cooling proves to be impractical for a particular system, the combination of laser cooling for atoms and Stark deceleration for molecules offers opportunities for studying molecular collisions in a new low-energy regime. For example, experiments are under way at the University of Colorado Lewandowski ( 2008 ) to study collisions between decelerated NH 3 molecules and laser-cooled Rb atoms. There alkali-metal atom + NH 3 systems have not been extensively studied theoretically, though there has been experimental interest in the spectroscopy of Li-NH 3 complex as a prototype metal atom-Lewis base complex Wu et al. ( 2001 ) . Lim et al. Lim et al. ( 2007 ) recently calculated electrical properties and infrared spectra for complexes of NH 3 with alkali-metal atoms from K to Fr and gave the equilibrium structures of their global minima. However, to our knowledge, no complete potential energy surfaces have been published for any of these systems. The alkaline-earth + NH 3 have been studied even less, and except for an early study of the Be-NH 3 system Chalasinski et al. ( 1993 ) there are no previous results available.","role":"user"},{"content":"Abstract: We investigate the possibility of producing ultracold NH3 molecules by sympathetic cooling in a bath of ultracold atoms. We consider the interactions of NH3 with alkali-metal and alkaline-earth-metal atoms, and with Xe, using ab initio coupled-cluster calculations. For Rb-NH3 and Xe-NH3 we develop full potential energy surfaces, while for the other systems we characterize the stationary points (global and local minima and saddle points). We also calculate isotropic and anisotropic van der Waals C6 coefficients for all the systems. The potential energy surfaces for interaction of NH3 with alkali-metal and alkaline-earth-metal atoms all show deep potential wells and strong anisotropies. The well depths vary from 887 cm−1 for Mg-NH3 to 5104 cm−1 for Li-NH3. This suggests that all these systems will exhibit strong inelasticity whenever inelastic collisions are energetically allowed and that sympathetic cooling will work only when both the atoms and the molecules are already in their lowest internal states. Xe-NH3 is more weakly bound and less anisotropic.","role":"assistant"}]

A Review of integrable deformations in AdS/CFT

In recent years a large number of studies have emerged indicating that type IIB string theory on A ​ d ​ S 5 × S 5 𝐴 𝑑 subscript 𝑆 5 superscript 𝑆 5 AdS_{5}\times S^{5} and 𝒩 = 4 𝒩 4 {\cal N}=4 super-Yang-Mills (SYM) theory in four dimensions may be integrable in the planar limit. The technology of integrable systems has therefore become extraordinarily useful in studying the AdS/CFT correspondence in detail. The correspondence equates the spectrum of operator anomalous dimensions of the gauge theory with the energy spectrum of the string theory. The rank of the Yang-Mills gauge group determines the amount of Ramond-Ramond (RR) flux on the S 5 superscript 𝑆 5 S^{5} subspace in the string theory, and in the planar limit this number is scaled to infinity: N c → ∞ → subscript 𝑁 𝑐 N_{c}\to\infty . The string coupling g s subscript 𝑔 𝑠 g_{s} is related to the gauge theory coupling g YM subscript 𝑔 YM g_{\rm YM} via the standard relation, g s = e ϕ 0 = g YM 2 / 4 ​ π subscript 𝑔 𝑠 superscript 𝑒 subscript italic-ϕ 0 subscript superscript 𝑔 2 YM 4 𝜋 g_{s}=e^{\phi_{0}}={g^{2}_{\rm YM}/4\pi} , and the radial scale of both the A ​ d ​ S 5 𝐴 𝑑 subscript 𝑆 5 AdS_{5} and S 5 superscript 𝑆 5 S^{5} spaces is given by R 4 = 4 ​ π ​ g s ​ N c = g YM 2 ​ N c = λ superscript 𝑅 4 4 𝜋 subscript 𝑔 𝑠 subscript 𝑁 𝑐 subscript superscript 𝑔 2 YM subscript 𝑁 𝑐 𝜆 R^{4}=4\pi g_{s}N_{c}=g^{2}_{\rm YM}N_{c}=\lambda (with α ′ = 1 superscript 𝛼 ′ 1 \alpha^{\prime}=1 ). If these theories are indeed integrable, the dynamics should be encoded in a diffractionless scattering matrix S 𝑆 S . On the string side, in the strong-coupling limit ( λ = g YM 2 ​ N c → ∞ ) 𝜆 superscript subscript 𝑔 YM 2 subscript 𝑁 𝑐 → (\lambda=g_{\rm YM}^{2}N_{c}\to\infty) , this S 𝑆 S matrix can be interpreted as describing the two-body scattering of elementary excitations on the worldsheet. As their worldsheet momenta becomes large, these excitations are better described as special types of solitonic solutions, or giant magnons , and the interpolating region is described by the dynamics of the so-called near-flat-space regime. [ 1 , 2 ] On the gauge theory side, the action of the dilatation generator on single-trace operators can be equated with that of a Hamiltonian acting on states of a spin chain. [ 3 ] In this picture, operators in the trace are represented as lattice pseudoparticles that, like their stringy counterparts, experience diffractionless scattering encoded by an S 𝑆 S matrix. Proving that the gauge and string theories are identical in the planar limit therefore amounts to showing that the underlying physics of both theories is governed by the same two-body scattering matrix. In fact, symmetry fixes this S 𝑆 S matrix up to an overall phase σ 𝜎 \sigma , so what remains is to somehow determine σ 𝜎 \sigma from basic principles. [ 4 ] (Unitarity and crossing relations, as they exist in this context, constrain this phase to some extent; see Refs. \refcite Janik:2006dc,Beisert:2006ib,Beisert:2006ez for recent developments.) An impressive amount of evidence exists in favor of the mutual integrability of these two theories. If true, this raises the question of whether these theories can be deformed in a controlled manner while remaining integrable. One class of interesting deformations to consider are the marginal β 𝛽 \beta deformations of 𝒩 = 4 𝒩 4 {\cal N}=4 SYM, also known as Leigh-Strassler deformations. [ 9 ] The resulting theories comprise a one-parameter family of 𝒩 = 1 𝒩 1 {\cal N}=1 conformal gauge theories (in the case of real β 𝛽 \beta deformations). On the gravity side of the correspondence, these correspond to special geometrical deformations of the S 5 superscript 𝑆 5 S^{5} subspace in the string theory background. [ 10 ] In fact, the integrability of the gauge and string theory, to the extent that it is understood in the undeformed cases, seems to persist under these deformations. This problem was studied directly and indirectly, for example, in Refs. \refcite Frolov:2005ty,Plefka:2005bk,Frolov:2005dj,Alday:2005ww,Freyhult:2005ws,Chen:2005sb,Chen:2006bh,Beisert:2005if,Spradlin:2005sv,Bobev:2005cz,Ryang:2005pg (see also references therein). The dynamics of both theories can be captured, at least in certain limits, by twisted Bethe equations. Here we review an analogous class of deformations acting on the A ​ d ​ S 5 𝐴 𝑑 subscript 𝑆 5 AdS_{5} subspace of the string theory background, first studied in Ref. \refcite US. While the corresponding gauge theory is less well understood (it may be a non-commutative or even non-associative theory), the string theory seems to be well defined in the near-pp-wave regime. Furthermore, the string energy spectrum can be computed precisely in this limit from a discrete Bethe ansatz, which lends substantial support to the methodology developed in Refs. \refcite Arutyunov:2004vx,Staudacher:2004tk,Beisert:2005fw. In Section 2 below, TsT deformations of the string background geometry are reviewed in detail. The classical integrability of the string sigma model is discussed in Section 3 . String energy spectra are computed directly from the deformed Green-Schwarz action in the near-pp-wave limit in Section 4 . In Section 5 , the thermodynamic Bethe equations are promoted to discrete Bethe equations that correctly reproduce the deformed energy spectra. A brief discussion and thoughts on further research are given in Section 6 . This letter is a review of a seminar based on Ref. \refcite US given in May, 2006 at the Institute for Advanced Study.

Marginal β deformations of $\mathcal{N}=4$ super-Yang–Mills theory are known to correspond to a certain class of deformations of the S5 background subspace of type IIB string theory in AdS5×S5. An analogous set of deformations of the AdS5 subspace is reviewed here. String energy spectra computed in the near-pp-wave limit of these backgrounds match predictions encoded by discrete, asymptotic Bethe equations, suggesting that the twisted string theory is classically integrable in this regime. These Bethe equations can be derived algorithmically by relying on the existence of Lax representations, and on the Riemann–Hilbert interpretation of the thermodynamic Bethe ansatz. This letter is a review of a seminar given at the Institute for Advanced Study, based on research completed in collaboration with McLoughlin.

[{"content":"Help me to generate the abstract of a survey paper given the title:*A Review of integrable deformations in AdS/CFT* and the introduction:In recent years a large number of studies have emerged indicating that type IIB string theory on A ​ d ​ S 5 × S 5 𝐴 𝑑 subscript 𝑆 5 superscript 𝑆 5 AdS_{5}\times S^{5} and 𝒩 = 4 𝒩 4 {\cal N}=4 super-Yang-Mills (SYM) theory in four dimensions may be integrable in the planar limit. The technology of integrable systems has therefore become extraordinarily useful in studying the AdS/CFT correspondence in detail. The correspondence equates the spectrum of operator anomalous dimensions of the gauge theory with the energy spectrum of the string theory. The rank of the Yang-Mills gauge group determines the amount of Ramond-Ramond (RR) flux on the S 5 superscript 𝑆 5 S^{5} subspace in the string theory, and in the planar limit this number is scaled to infinity: N c → ∞ → subscript 𝑁 𝑐 N_{c}\to\infty . The string coupling g s subscript 𝑔 𝑠 g_{s} is related to the gauge theory coupling g YM subscript 𝑔 YM g_{\rm YM} via the standard relation, g s = e ϕ 0 = g YM 2 / 4 ​ π subscript 𝑔 𝑠 superscript 𝑒 subscript italic-ϕ 0 subscript superscript 𝑔 2 YM 4 𝜋 g_{s}=e^{\phi_{0}}={g^{2}_{\rm YM}/4\pi} , and the radial scale of both the A ​ d ​ S 5 𝐴 𝑑 subscript 𝑆 5 AdS_{5} and S 5 superscript 𝑆 5 S^{5} spaces is given by R 4 = 4 ​ π ​ g s ​ N c = g YM 2 ​ N c = λ superscript 𝑅 4 4 𝜋 subscript 𝑔 𝑠 subscript 𝑁 𝑐 subscript superscript 𝑔 2 YM subscript 𝑁 𝑐 𝜆 R^{4}=4\pi g_{s}N_{c}=g^{2}_{\rm YM}N_{c}=\lambda (with α ′ = 1 superscript 𝛼 ′ 1 \alpha^{\prime}=1 ). If these theories are indeed integrable, the dynamics should be encoded in a diffractionless scattering matrix S 𝑆 S . On the string side, in the strong-coupling limit ( λ = g YM 2 ​ N c → ∞ ) 𝜆 superscript subscript 𝑔 YM 2 subscript 𝑁 𝑐 → (\lambda=g_{\rm YM}^{2}N_{c}\to\infty) , this S 𝑆 S matrix can be interpreted as describing the two-body scattering of elementary excitations on the worldsheet. As their worldsheet momenta becomes large, these excitations are better described as special types of solitonic solutions, or giant magnons , and the interpolating region is described by the dynamics of the so-called near-flat-space regime. 1 , 2 On the gauge theory side, the action of the dilatation generator on single-trace operators can be equated with that of a Hamiltonian acting on states of a spin chain. 3 In this picture, operators in the trace are represented as lattice pseudoparticles that, like their stringy counterparts, experience diffractionless scattering encoded by an S 𝑆 S matrix. Proving that the gauge and string theories are identical in the planar limit therefore amounts to showing that the underlying physics of both theories is governed by the same two-body scattering matrix. In fact, symmetry fixes this S 𝑆 S matrix up to an overall phase σ 𝜎 \sigma , so what remains is to somehow determine σ 𝜎 \sigma from basic principles. 4 (Unitarity and crossing relations, as they exist in this context, constrain this phase to some extent; see Refs. \refcite Janik:2006dc,Beisert:2006ib,Beisert:2006ez for recent developments.) An impressive amount of evidence exists in favor of the mutual integrability of these two theories. If true, this raises the question of whether these theories can be deformed in a controlled manner while remaining integrable. One class of interesting deformations to consider are the marginal β 𝛽 \beta deformations of 𝒩 = 4 𝒩 4 {\cal N}=4 SYM, also known as Leigh-Strassler deformations. 9 The resulting theories comprise a one-parameter family of 𝒩 = 1 𝒩 1 {\cal N}=1 conformal gauge theories (in the case of real β 𝛽 \beta deformations). On the gravity side of the correspondence, these correspond to special geometrical deformations of the S 5 superscript 𝑆 5 S^{5} subspace in the string theory background. 10 In fact, the integrability of the gauge and string theory, to the extent that it is understood in the undeformed cases, seems to persist under these deformations. This problem was studied directly and indirectly, for example, in Refs. \refcite Frolov:2005ty,Plefka:2005bk,Frolov:2005dj,Alday:2005ww,Freyhult:2005ws,Chen:2005sb,Chen:2006bh,Beisert:2005if,Spradlin:2005sv,Bobev:2005cz,Ryang:2005pg (see also references therein). The dynamics of both theories can be captured, at least in certain limits, by twisted Bethe equations. Here we review an analogous class of deformations acting on the A ​ d ​ S 5 𝐴 𝑑 subscript 𝑆 5 AdS_{5} subspace of the string theory background, first studied in Ref. \refcite US. While the corresponding gauge theory is less well understood (it may be a non-commutative or even non-associative theory), the string theory seems to be well defined in the near-pp-wave regime. Furthermore, the string energy spectrum can be computed precisely in this limit from a discrete Bethe ansatz, which lends substantial support to the methodology developed in Refs. \refcite Arutyunov:2004vx,Staudacher:2004tk,Beisert:2005fw. In Section 2 below, TsT deformations of the string background geometry are reviewed in detail. The classical integrability of the string sigma model is discussed in Section 3 . String energy spectra are computed directly from the deformed Green-Schwarz action in the near-pp-wave limit in Section 4 . In Section 5 , the thermodynamic Bethe equations are promoted to discrete Bethe equations that correctly reproduce the deformed energy spectra. A brief discussion and thoughts on further research are given in Section 6 . This letter is a review of a seminar based on Ref. \refcite US given in May, 2006 at the Institute for Advanced Study.","role":"user"},{"content":"Abstract: Marginal β deformations of $\mathcal{N}=4$ super-Yang–Mills theory are known to correspond to a certain class of deformations of the S5 background subspace of type IIB string theory in AdS5×S5. An analogous set of deformations of the AdS5 subspace is reviewed here. String energy spectra computed in the near-pp-wave limit of these backgrounds match predictions encoded by discrete, asymptotic Bethe equations, suggesting that the twisted string theory is classically integrable in this regime. These Bethe equations can be derived algorithmically by relying on the existence of Lax representations, and on the Riemann–Hilbert interpretation of the thermodynamic Bethe ansatz. This letter is a review of a seminar given at the Institute for Advanced Study, based on research completed in collaboration with McLoughlin.","role":"assistant"}]

A brief survey of the renormalizability of four dimensional gravity for generalized Kodama states

: Stability of the pure Kodama state This work continues the line of research from [ 1 ] , [ 2 ] . The main question that would like to formulate is (i) whether or not the pure Kodama state Ψ K ​ o ​ d subscript Ψ 𝐾 𝑜 𝑑 \Psi_{Kod} is a good ground state of general relativity about which quantum fluctuations constitue a renormalizable theory. The second main question we would like to analyse is concerning the relation between the pure and the generalized Kodama states with respect to the vacuum state of quantum general relativity: (ii) Is Ψ G ​ K ​ o ​ d subscript Ψ 𝐺 𝐾 𝑜 𝑑 \Psi_{GKod} in any sense an excited version of Ψ K ​ o ​ d subscript Ψ 𝐾 𝑜 𝑑 \Psi_{Kod} , or is it an independent ground state of the gravity-matter system? As a corollary to (ii), does each model for which a Ψ G ​ K ​ o ​ d subscript Ψ 𝐺 𝐾 𝑜 𝑑 \Psi_{GKod} can be constructed consitute an additional vacuum state of general relativity? In [ 3 ] , Chopin Soo and Lee Smolin present the hypothesis for the treatment of matter fields as a perturbation about DeSitter space satisfying a Schrödinger equation for small fluctuations. In [ 4 ] , Smolin and Friedel expand Ψ K ​ o ​ d subscript Ψ 𝐾 𝑜 𝑑 \Psi_{Kod} in gravitions about an abelian theory. However, since the generalized Kodama states are designed to incorporate the matter effects to all orders and to enforce the proper semiclassical limit, we expect it to be the case that Ψ G ​ K ​ o ​ d subscript Ψ 𝐺 𝐾 𝑜 𝑑 \Psi_{GKod} is to all orders in the expansion nonperturbatively related to Ψ K ​ o ​ d subscript Ψ 𝐾 𝑜 𝑑 \Psi_{Kod} . In this case, one should be able to find a discrete transformation between that maps Ψ K ​ o ​ d subscript Ψ 𝐾 𝑜 𝑑 \Psi_{Kod} into Ψ G ​ K ​ o ​ d subscript Ψ 𝐺 𝐾 𝑜 𝑑 \Psi_{GKod} and more generally, a discrete transformation amongst the generalized Kodama states for different models.The manner in which we address this transformation is to view Ψ K ​ o ​ d subscript Ψ 𝐾 𝑜 𝑑 \Psi_{Kod} as being invariant under a symmetry which is broken due to the presence of matter fields. When one views the effect of the matter fields in terms of backreactions on DeSitter spacetime, then one can see more clearly the link from Ψ G ​ K ​ o ​ d subscript Ψ 𝐺 𝐾 𝑜 𝑑 \Psi_{GKod} to the semiclassical limit below the Planck scale. We provide in this work a brief synopsis of the required transformation in terms of tree networks, and then briefly comment in the discussion on the implications for nonperturbative renormalizability in the Ashtekar variables.The layout of this paper is as follows. In section 2 we review the developments which cast the pure Kodama state into a perspective suitable for posing the question of a stable ground state. In section 3 we discuss in detail the effects and the interpretation of incorporating matter fields, in the good semiclassical limit below the Planck scale, into the fully extrapolated theory of quantized gravity. In section 4 we briefly introduce the quantum theory of fluctuations on DeSitter spacetime and the general structures required. In section 5 we introduce the concept of the tree network, which can be seen as the application of Feynman diagrammatic techniques to the solution of the constraints. We then show how the networks implement the discrete transformation amongst generalized Kodama states. We argue for the interpretation of general relativity as a renormalizable theory due to its tree network structure when expressed in Ashtekar variables.

We continue the line of research from previous works in assessing the suitability of the pure Kodama state both as a ground state for the generalized Kodama states, as well as characteristic of a good semiclassical limit of general relativity. We briefly introduce the quantum theory of fluctuations about DeSitter spacetime, which enables one to examine some perturbative aspects of the state. Additionally, we also motivate the concept of the cubic tree network, which enables one to view the generalized Kodama states in compact form as a nonlinear transformation of the pure Kodama states parametrized by the matter content of the proper classical limit. It is hoped that this work constitutes a first step in addressing the nonperturbative renormalizability of general relativity in Ashtekar variables. Remaining issues to address, including the analysis of specific matter models, include finiteness and normalizability of the generalized Kodama state as well as reality conditions on the Ashtekar variables, which we relegate to separate works.

[{"content":"Help me to generate the abstract of a survey paper given the title:*A brief survey of the renormalizability of four dimensional gravity for generalized Kodama states* and the introduction:: Stability of the pure Kodama state This work continues the line of research from 1 , 2 . The main question that would like to formulate is (i) whether or not the pure Kodama state Ψ K ​ o ​ d subscript Ψ 𝐾 𝑜 𝑑 \Psi_{Kod} is a good ground state of general relativity about which quantum fluctuations constitue a renormalizable theory. The second main question we would like to analyse is concerning the relation between the pure and the generalized Kodama states with respect to the vacuum state of quantum general relativity: (ii) Is Ψ G ​ K ​ o ​ d subscript Ψ 𝐺 𝐾 𝑜 𝑑 \Psi_{GKod} in any sense an excited version of Ψ K ​ o ​ d subscript Ψ 𝐾 𝑜 𝑑 \Psi_{Kod} , or is it an independent ground state of the gravity-matter system? As a corollary to (ii), does each model for which a Ψ G ​ K ​ o ​ d subscript Ψ 𝐺 𝐾 𝑜 𝑑 \Psi_{GKod} can be constructed consitute an additional vacuum state of general relativity? In 3 , Chopin Soo and Lee Smolin present the hypothesis for the treatment of matter fields as a perturbation about DeSitter space satisfying a Schrödinger equation for small fluctuations. In 4 , Smolin and Friedel expand Ψ K ​ o ​ d subscript Ψ 𝐾 𝑜 𝑑 \Psi_{Kod} in gravitions about an abelian theory. However, since the generalized Kodama states are designed to incorporate the matter effects to all orders and to enforce the proper semiclassical limit, we expect it to be the case that Ψ G ​ K ​ o ​ d subscript Ψ 𝐺 𝐾 𝑜 𝑑 \Psi_{GKod} is to all orders in the expansion nonperturbatively related to Ψ K ​ o ​ d subscript Ψ 𝐾 𝑜 𝑑 \Psi_{Kod} . In this case, one should be able to find a discrete transformation between that maps Ψ K ​ o ​ d subscript Ψ 𝐾 𝑜 𝑑 \Psi_{Kod} into Ψ G ​ K ​ o ​ d subscript Ψ 𝐺 𝐾 𝑜 𝑑 \Psi_{GKod} and more generally, a discrete transformation amongst the generalized Kodama states for different models.The manner in which we address this transformation is to view Ψ K ​ o ​ d subscript Ψ 𝐾 𝑜 𝑑 \Psi_{Kod} as being invariant under a symmetry which is broken due to the presence of matter fields. When one views the effect of the matter fields in terms of backreactions on DeSitter spacetime, then one can see more clearly the link from Ψ G ​ K ​ o ​ d subscript Ψ 𝐺 𝐾 𝑜 𝑑 \Psi_{GKod} to the semiclassical limit below the Planck scale. We provide in this work a brief synopsis of the required transformation in terms of tree networks, and then briefly comment in the discussion on the implications for nonperturbative renormalizability in the Ashtekar variables.The layout of this paper is as follows. In section 2 we review the developments which cast the pure Kodama state into a perspective suitable for posing the question of a stable ground state. In section 3 we discuss in detail the effects and the interpretation of incorporating matter fields, in the good semiclassical limit below the Planck scale, into the fully extrapolated theory of quantized gravity. In section 4 we briefly introduce the quantum theory of fluctuations on DeSitter spacetime and the general structures required. In section 5 we introduce the concept of the tree network, which can be seen as the application of Feynman diagrammatic techniques to the solution of the constraints. We then show how the networks implement the discrete transformation amongst generalized Kodama states. We argue for the interpretation of general relativity as a renormalizable theory due to its tree network structure when expressed in Ashtekar variables.","role":"user"},{"content":"Abstract: We continue the line of research from previous works in assessing the suitability of the pure Kodama state both as a ground state for the generalized Kodama states, as well as characteristic of a good semiclassical limit of general relativity. We briefly introduce the quantum theory of fluctuations about DeSitter spacetime, which enables one to examine some perturbative aspects of the state. Additionally, we also motivate the concept of the cubic tree network, which enables one to view the generalized Kodama states in compact form as a nonlinear transformation of the pure Kodama states parametrized by the matter content of the proper classical limit. It is hoped that this work constitutes a first step in addressing the nonperturbative renormalizability of general relativity in Ashtekar variables. Remaining issues to address, including the analysis of specific matter models, include finiteness and normalizability of the generalized Kodama state as well as reality conditions on the Ashtekar variables, which we relegate to separate works.","role":"assistant"}]

Dynamical 3-Space: A Review

We review here some of the new physics emerging from the discovery that there exists a dynamical 3-space. This discovery changes all of physics. While at a deeper level this emerges from the information-theoretic Process Physics [ 2 , 5 , 6 , 7 , 8 , 9 , 10 ] here we focus on the phenomenological description of this 3-space in terms of the velocity field that describes the internal dynamics of this structured 3-space. It is straightforward to construct the minimal dynamics for this 3-space, and it involves two constants: G 𝐺 G - Newton’s gravitational constant, and α 𝛼 \alpha - the fine structure constant. G 𝐺 G quantifies the effect of matter upon the flowing 3-space, while α 𝛼 \alpha describes the self-interaction of the 3-space. Bore hole experiments and black hole astronomical observations give the value of α 𝛼 \alpha as the fine structure constant to within observational errors. A major development is that the Newtonian theory of gravity [ 11 ] is fundamentally flawed - that even in the non-relativistic limit it fails to correctly model numerous gravitational phenomena. So Newton’s theory of gravity is far from being ‘universal’. The Hilbert-Einstein theory of gravity (General Relativity - GR), with gravity being a curved spacetime effect, was based on the assumption that Newtonian gravity was valid in the non-relativistic limit. The ongoing effort to save GR against numerous disagreements with experiment and observation lead to the invention first of ‘dark matter’ and then ‘dark energy’. These effects are no longer required in the new physics. The 3-space velocity field has been directly detected in at least eight experiments including the Michelson-Morley experiment [ 3 ] of 1887, but most impressively by the superb experiment by Miller in 1925/1926 [ 4 ] . The Miller experiment was one of the great physics experiments of the 20th century, but has been totally neglected by mainstream physics. All of these experiments detected the dynamical 3-space by means of the light speed anisotropy - that the speed of light is different in different directions, and the anisotropy is very large, namely some 1 part in a 1000. The existence of this 3-space as a detectable phenomenon implies that a generalisation of all the fundamental theories of physics be carried out. The generalisation of the Maxwell equations leads to a simple explanation for gravitational light bending and lensing effects, the generalisation of the Schrödinger equation leads to the first derivation of gravity - as a refraction effect of the quantum matter waves by the time dependence and inhomogeneities of the 3-space, leading as well to a derivation of the equivalence principle. This generalised Schrödinger equation also explains the Lense-Thirring effect as being caused by vorticity in the flowing 3-space. This effect is being studied by the Gravity Probe B (GP-B) gyroscope precession experiment. The generalisation of the Dirac equation to take account of the interaction of the spinor with the dynamical 3-space results in the derivation of the curved spacetime formalism for the quantum matter geodesics, but without reference to the GR equations for the induced spacetime metric. What emerges from this derivation is that the spacetime is purely a mathematical construct - it has no ontological status. That discovery completely overturns the paradigm of 20th century physics. The dynamical equation for the 3-space has black hole solutions with properties very different from the putative black holes of GR, leading to the verified prediction for the masses of the minimal black holes in spherical star systems. That same dynamics has an expanding 3-space solution - the Hubble effect for the universe. That solution has the expansion mainly determined by space itself. This expansion gives a extremely good account of the supernovae/Gamma-Ray Burst redshift data without the notion of ‘dark energy’ or an accelerating universe. This review focuses on the phenomenological modelling of the 3-space dynamics and its experimental checking. Earlier reviews are available in [ 2 ] (2005) and [ 5 ] (2003). Page limitations mean that some developments have not been discussed herein.

For some 100 years physics has modelled space and time via the spacetime concept, with space being merely an observer dependent perspective effect of that spacetime - space itself had no observer independent existence - it had no ontological status, and it certainly had no dynamical description. In recent years this has all changed. In 2002 it was discovered that a dynamical 3-space had been detected many times, including the Michelson-Morley 1887 light-speed anisotropy experiment. Here we review the dynamics of this 3-space, tracing its evolution from that of an emergent phenomena in the information-theoretic Process Physics to the phenomenological description in terms of a velocity field describing the relative internal motion of the structured 3-space. The new physics of the dynamical 3-space is extensively tested against experimental and astronomical observations, including the necessary generalisation of the Maxwell, Schrodinger and Dirac equations, leading to a derivation and explanation of gravity as a refraction effect of quantum matter waves. The flat and curved spacetime formalisms are derived from the new physics, so explaining their apparent many successes.

[{"content":"Help me to generate the abstract of a survey paper given the title:*Dynamical 3-Space: A Review* and the introduction:We review here some of the new physics emerging from the discovery that there exists a dynamical 3-space. This discovery changes all of physics. While at a deeper level this emerges from the information-theoretic Process Physics 2 , 5 , 6 , 7 , 8 , 9 , 10 here we focus on the phenomenological description of this 3-space in terms of the velocity field that describes the internal dynamics of this structured 3-space. It is straightforward to construct the minimal dynamics for this 3-space, and it involves two constants: G 𝐺 G - Newtons gravitational constant, and α 𝛼 \alpha - the fine structure constant. G 𝐺 G quantifies the effect of matter upon the flowing 3-space, while α 𝛼 \alpha describes the self-interaction of the 3-space. Bore hole experiments and black hole astronomical observations give the value of α 𝛼 \alpha as the fine structure constant to within observational errors. A major development is that the Newtonian theory of gravity 11 is fundamentally flawed - that even in the non-relativistic limit it fails to correctly model numerous gravitational phenomena. So Newtons theory of gravity is far from being universal. The Hilbert-Einstein theory of gravity (General Relativity - GR), with gravity being a curved spacetime effect, was based on the assumption that Newtonian gravity was valid in the non-relativistic limit. The ongoing effort to save GR against numerous disagreements with experiment and observation lead to the invention first of dark matter and then dark energy. These effects are no longer required in the new physics. The 3-space velocity field has been directly detected in at least eight experiments including the Michelson-Morley experiment 3 of 1887, but most impressively by the superb experiment by Miller in 1925/1926 4 . The Miller experiment was one of the great physics experiments of the 20th century, but has been totally neglected by mainstream physics. All of these experiments detected the dynamical 3-space by means of the light speed anisotropy - that the speed of light is different in different directions, and the anisotropy is very large, namely some 1 part in a 1000. The existence of this 3-space as a detectable phenomenon implies that a generalisation of all the fundamental theories of physics be carried out. The generalisation of the Maxwell equations leads to a simple explanation for gravitational light bending and lensing effects, the generalisation of the Schrödinger equation leads to the first derivation of gravity - as a refraction effect of the quantum matter waves by the time dependence and inhomogeneities of the 3-space, leading as well to a derivation of the equivalence principle. This generalised Schrödinger equation also explains the Lense-Thirring effect as being caused by vorticity in the flowing 3-space. This effect is being studied by the Gravity Probe B (GP-B) gyroscope precession experiment. The generalisation of the Dirac equation to take account of the interaction of the spinor with the dynamical 3-space results in the derivation of the curved spacetime formalism for the quantum matter geodesics, but without reference to the GR equations for the induced spacetime metric. What emerges from this derivation is that the spacetime is purely a mathematical construct - it has no ontological status. That discovery completely overturns the paradigm of 20th century physics. The dynamical equation for the 3-space has black hole solutions with properties very different from the putative black holes of GR, leading to the verified prediction for the masses of the minimal black holes in spherical star systems. That same dynamics has an expanding 3-space solution - the Hubble effect for the universe. That solution has the expansion mainly determined by space itself. This expansion gives a extremely good account of the supernovae/Gamma-Ray Burst redshift data without the notion of dark energy or an accelerating universe. This review focuses on the phenomenological modelling of the 3-space dynamics and its experimental checking. Earlier reviews are available in 2 (2005) and 5 (2003). Page limitations mean that some developments have not been discussed herein.","role":"user"},{"content":"Abstract: For some 100 years physics has modelled space and time via the spacetime concept, with space being merely an observer dependent perspective effect of that spacetime - space itself had no observer independent existence - it had no ontological status, and it certainly had no dynamical description. In recent years this has all changed. In 2002 it was discovered that a dynamical 3-space had been detected many times, including the Michelson-Morley 1887 light-speed anisotropy experiment. Here we review the dynamics of this 3-space, tracing its evolution from that of an emergent phenomena in the information-theoretic Process Physics to the phenomenological description in terms of a velocity field describing the relative internal motion of the structured 3-space. The new physics of the dynamical 3-space is extensively tested against experimental and astronomical observations, including the necessary generalisation of the Maxwell, Schrodinger and Dirac equations, leading to a derivation and explanation of gravity as a refraction effect of quantum matter waves. The flat and curved spacetime formalisms are derived from the new physics, so explaining their apparent many successes.","role":"assistant"}]

A survey of hypertoric geometry and topology

1 Definitions and basic properties Hypertoric varieties can be considered either as algebraic varieties or, in the smooth case, as hyperkähler manifolds. In this section we give a constructive definition, with a strong bias toward the algebraic interpretation. Section 1.1 proceeds in greater generality than is necessary for hypertoric varieties so as to unify the theory with that of other algebraic symplectic quotients, most notably Nakajima quiver varieties. 1.1 Algebraic symplectic quotients Let G 𝐺 G be a reductive algebraic group over the complex numbers acting linearly and effectively on a finite-dimensional complex vector space V 𝑉 V . The cotangent bundle T ∗ ​ V ≅ V × V ∗ superscript 𝑇 𝑉 𝑉 superscript 𝑉 T^{*}V\cong V\times V^{*} carries a natural algebraic symplectic form Ω Ω \Omega . The induced action of G 𝐺 G on T ∗ ​ V superscript 𝑇 𝑉 T^{*}V is hamiltonian, with moment map μ : T ∗ ​ V → 𝔤 ∗ : 𝜇 → superscript 𝑇 𝑉 superscript 𝔤 \mu:T^{*}V\to\mathfrak{g}^{*} given by the equation μ ​ ( z , w ) ​ ( x ) = Ω ​ ( x ⋅ z , w ) ​ for all z ∈ V , w ∈ V ∗ , x ∈ 𝔤 . 𝜇 𝑧 𝑤 𝑥 Ω ⋅ 𝑥 𝑧 𝑤 for all z ∈ V , w ∈ V ∗ , x ∈ 𝔤 . \mu(z,w)(x)=\Omega(x\cdot z,w)\,\,\text{ for all $z\in V$, $w\in V^{*}$, $x\in\mathfrak{g}$.} Suppose given an element λ ∈ Z ​ ( 𝔤 ∗ ) 𝜆 𝑍 superscript 𝔤 \lambda\in Z(\mathfrak{g}^{*}) (the part of 𝔤 ∗ superscript 𝔤 \mathfrak{g}^{*} fixed by the coadjoint action of G 𝐺 G ), and a multiplicative character α : G → ℂ × : 𝛼 → 𝐺 superscript ℂ \alpha:G\to{\mathbb{C}}^{\times} , which may be identified with an element of Z ​ ( 𝔤 ℤ ∗ ) 𝑍 subscript superscript 𝔤 ℤ Z(\mathfrak{g}^{*}_{\mathbb{Z}}) by taking its derivative at the identity element of G 𝐺 G . 2 2 2 Strictly speaking, an element of Z ​ ( 𝔤 ℤ ∗ ) 𝑍 subscript superscript 𝔤 ℤ Z(\mathfrak{g}^{*}_{\mathbb{Z}}) only determines a character of the connected component of the identity of G 𝐺 G . It can be checked, however, that the notion of α 𝛼 \alpha -stability defined below depends only on the restriction of α 𝛼 \alpha to the identity component, therefore we will abusively think of α 𝛼 \alpha as sitting inside of Z ​ ( 𝔤 ℤ ∗ ) 𝑍 subscript superscript 𝔤 ℤ Z(\mathfrak{g}^{*}_{\mathbb{Z}}) . The fact that λ 𝜆 \lambda lies in Z ​ ( 𝔤 ∗ ) 𝑍 superscript 𝔤 Z(\mathfrak{g}^{*}) implies that G 𝐺 G acts on μ − 1 ​ ( λ ) superscript 𝜇 1 𝜆 \mu^{-1}(\lambda) . Our main object of study in this survey will be the algebraic symplectic quotient 𝔐 α , λ = T ∗ V / / / / α G = μ − 1 ( λ ) / / α G . \mathfrak{M}_{\alpha,\lambda}=T^{*}V{\!/\!\!/\!\!/\!\!/}_{\!\!\alpha}G=\mu^{-1}(\lambda){/\!\!/}_{\!\!\alpha}G. Here the second quotient is a projective GIT quotient μ − 1 ( λ ) / / α G := Proj ⨁ m = 0 ∞ { f ∈ Fun ( μ − 1 ( λ ) ) | ν ( g ) ∗ f = α ( g ) m f for all g ∈ G } , \mu^{-1}(\lambda){/\!\!/}_{\!\!\alpha}G:=\mathrm{Proj}\,\bigoplus_{m=0}^{\infty}\Big{\{}f\in\operatorname{Fun}\big{(}\mu^{-1}(\lambda)\big{)}\hskip 3.0pt\Big{|}\hskip 3.0pt\nu(g)^{*}f=\alpha(g)^{m}f\hskip 3.0pt\text{ for all }g\in G\hskip 3.0pt\Big{\}}, (1) where ν ​ ( g ) 𝜈 𝑔 \nu(g) is the automorphism of μ − 1 ​ ( λ ) superscript 𝜇 1 𝜆 \mu^{-1}(\lambda) defined by g 𝑔 g . This quotient may be defined in a more geometric way as follows. A point ( z , w ) ∈ μ − 1 ​ ( λ ) 𝑧 𝑤 superscript 𝜇 1 𝜆 (z,w)\in\mu^{-1}(\lambda) is called α 𝛼 \mathbf{\alpha} -semistable if there exists a function f 𝑓 f on μ − 1 ​ ( λ ) superscript 𝜇 1 𝜆 \mu^{-1}(\lambda) and a positive integer m 𝑚 m such that ν ​ ( g ) ∗ ​ f = α ​ ( g ) m ​ f 𝜈 superscript 𝑔 𝑓 𝛼 superscript 𝑔 𝑚 𝑓 \nu(g)^{*}f=\alpha(g)^{m}f for all g ∈ G 𝑔 𝐺 g\in G and f ​ ( z , w ) ≠ 0 𝑓 𝑧 𝑤 0 f(z,w)\neq 0 . It is called α 𝛼 \mathbf{\alpha} -stable if it is α 𝛼 \alpha -semistable and its G 𝐺 G -orbit in the α 𝛼 \alpha -semistable set is closed with finite stabilizers. Then the stable and semistable sets μ − 1 ​ ( λ ) α − s ​ t ⊆ μ − 1 ​ ( λ ) α − s ​ s ⊆ μ − 1 ​ ( λ ) superscript 𝜇 1 superscript 𝜆 𝛼 𝑠 𝑡 superscript 𝜇 1 superscript 𝜆 𝛼 𝑠 𝑠 superscript 𝜇 1 𝜆 \mu^{-1}(\lambda)^{\alpha-st}\subseteq\mu^{-1}(\lambda)^{\alpha-ss}\subseteq\mu^{-1}(\lambda) are nonempty and Zariski open, and there is a surjection μ − 1 ​ ( λ ) α − s ​ s ↠ 𝔐 α , λ ↠ superscript 𝜇 1 superscript 𝜆 𝛼 𝑠 𝑠 subscript 𝔐 𝛼 𝜆 \mu^{-1}(\lambda)^{\alpha-ss}\twoheadrightarrow\mathfrak{M}_{\alpha,\lambda} with ( z , w ) 𝑧 𝑤 (z,w) and ( z ′ , w ′ ) superscript 𝑧 ′ superscript 𝑤 ′ (z^{\prime},w^{\prime}) mapping to the same point if and only if the closures of their G 𝐺 G -orbits intersect in μ − 1 ​ ( λ ) α − s ​ s superscript 𝜇 1 superscript 𝜆 𝛼 𝑠 𝑠 \mu^{-1}(\lambda)^{\alpha-ss} . In particular, the restriction of this map to the stable locus is nothing but the geometric quotient by G 𝐺 G . For an introduction to geometric invariant theory that explains the equivalence of these two perspectives, see [ P2 ] . Remark 1.1.1 The algebraic symplectic quotient defined above may also be interpreted as a hyperkähler quotient. The even dimensional complex vector space T ∗ ​ V superscript 𝑇 𝑉 T^{*}V admits a complete hyperkähler metric, and the action of the maximal compact subgroup G ℝ ⊆ G subscript 𝐺 ℝ 𝐺 G_{\mathbb{R}}\subseteq G is hyperhamiltonian , meaning that it is hamiltonian with respect to all three of the real symplectic forms on T ∗ ​ V superscript 𝑇 𝑉 T^{*}V . Then 𝔐 α , λ subscript 𝔐 𝛼 𝜆 \mathfrak{M}_{\alpha,\lambda} is naturally diffeomorphic to the hyperkähler quotient of T ∗ ​ V superscript 𝑇 𝑉 T^{*}V by G ℝ subscript 𝐺 ℝ G_{\mathbb{R}} , in the sense of [ HKLR ] , at the value ( α , Re ⁡ λ , Im ⁡ λ ) ∈ 𝔤 ℝ ∗ ⊗ ℝ 3 𝛼 Re 𝜆 Im 𝜆 tensor-product subscript superscript 𝔤 ℝ superscript ℝ 3 (\alpha,\operatorname{Re}\lambda,\operatorname{Im}\lambda)\in\mathfrak{g}^{*}_{\mathbb{R}}\otimes{\mathbb{R}}^{3} . This was the original perspective on both hypertoric varieties [ BD ] and Nakajima quiver varieties [ N1 ] . For more on this perspective in the hypertoric case, see Konno’s survey in this volume [ K4 , §3] . We note that if α = 0 𝛼 0 \alpha=0 is the trivial character of G 𝐺 G , then Equation ( 1 ) simplifies to 𝔐 0 , λ = Spec Fun ( μ − 1 ( λ ) ) G . \mathfrak{M}_{0,\lambda}=\operatorname{Spec}\operatorname{Fun}\big{(}\mu^{-1}(\lambda)\big{)}^{G}. Furthermore, since 𝔐 α , λ subscript 𝔐 𝛼 𝜆 \mathfrak{M}_{\alpha,\lambda} is defined as the projective spectrum of a graded ring whose degree zero part is the ring of invariant functions on μ − 1 ​ ( λ ) superscript 𝜇 1 𝜆 \mu^{-1}(\lambda) , we always have a projective morphism 𝔐 α , λ ↠ 𝔐 0 , λ . ↠ subscript 𝔐 𝛼 𝜆 subscript 𝔐 0 𝜆 \mathfrak{M}_{\alpha,\lambda}\twoheadrightarrow\mathfrak{M}_{0,\lambda}. (2) This morphism may also be induced from the inclusion of the inclusion μ − 1 ​ ( λ ) α − s ​ s ⊆ μ − 1 ​ ( λ ) = μ − 1 ​ ( λ ) 0 − s ​ s . superscript 𝜇 1 superscript 𝜆 𝛼 𝑠 𝑠 superscript 𝜇 1 𝜆 superscript 𝜇 1 superscript 𝜆 0 𝑠 𝑠 \mu^{-1}(\lambda)^{\alpha-ss}\subseteq\mu^{-1}(\lambda)=\mu^{-1}(\lambda)^{0-ss}. From this we may conclude that it is generically one-to-one, and therefore a partial resolution. When λ = 0 𝜆 0 \lambda=0 , we have a distinguished point in 𝔐 0 , 0 subscript 𝔐 0 0 \mathfrak{M}_{0,0} , namely the image of 0 ∈ μ − 1 ​ ( 0 ) 0 superscript 𝜇 1 0 0\in\mu^{\!-1}(0) under the map induced by the inclusion of the invariant functions into the coordinate ring of μ − 1 ​ ( 0 ) superscript 𝜇 1 0 \mu^{\!-1}(0) . The preimage of this point under the morphism ( 2 ) is called the core of 𝔐 α , 0 subscript 𝔐 𝛼 0 \mathfrak{M}_{\alpha,0} , and will be further studied (in the case where G 𝐺 G is abelian) in Section 2.1 . On the other extreme, if λ 𝜆 \lambda is a regular value of μ 𝜇 \mu , then G 𝐺 G will act locally freely on μ − 1 ​ ( λ ) superscript 𝜇 1 𝜆 \mu^{-1}(\lambda) . In this case all points will be α 𝛼 \alpha -stable for any choice of α 𝛼 \alpha , and the GIT quotient 𝔐 λ = μ − 1 ( λ ) / / G \mathfrak{M}_{\lambda}=\mu^{-1}(\lambda){/\!\!/}G will simply be a geometric quotient. In particular, the morphism ( 2 ) becomes an isomorphism. Both the case of regular λ 𝜆 \lambda and the case λ = 0 𝜆 0 \lambda=0 will be of interest to us. We call a pair ( α , λ ) 𝛼 𝜆 (\alpha,\lambda) generic if μ − 1 ​ ( λ ) α − s ​ t = μ − 1 ​ ( λ ) α − s ​ s superscript 𝜇 1 superscript 𝜆 𝛼 𝑠 𝑡 superscript 𝜇 1 superscript 𝜆 𝛼 𝑠 𝑠 \mu^{-1}(\lambda)^{\alpha-st}=\mu^{-1}(\lambda)^{\alpha-ss} . In this case the moment map condition tells us that the stable set is smooth, and therefore that the quotient 𝔐 α , λ subscript 𝔐 𝛼 𝜆 \mathfrak{M}_{\alpha,\lambda} by the locally free G 𝐺 G -action has at worst orbifold singularities. Using the hyperkähler quotient perspective of Remark 1.1.1 , one can prove the following Proposition. (See [ K3 , 2.6] or [ HP1 , 2.1] in the hypertoric case, and [ N1 , 4.2] in the case of quiver varieties; the general case is no harder than these.) Proposition 1.1.2 If ( α , λ ) 𝛼 𝜆 (\alpha,\lambda) and ( α ′ , λ ′ ) superscript 𝛼 ′ superscript 𝜆 ′ (\alpha^{\prime},\lambda^{\prime}) are both generic, then the two symplectic quotients 𝔐 α , λ subscript 𝔐 𝛼 𝜆 \mathfrak{M}_{\alpha,\lambda} and 𝔐 α ′ , λ ′ subscript 𝔐 superscript 𝛼 ′ superscript 𝜆 ′ \mathfrak{M}_{\alpha^{\prime},\lambda^{\prime}} are diffeomorphic. Remark 1.1.3 If G 𝐺 G is semisimple, then Z ​ ( 𝔤 ∗ ) = { 0 } 𝑍 superscript 𝔤 0 Z(\mathfrak{g}^{*})=\{0\} , and (unless G 𝐺 G is finite) it will not be possible to choose a regular value λ ∈ Z ​ ( 𝔤 ∗ ) 𝜆 𝑍 superscript 𝔤 \lambda\in Z(\mathfrak{g}^{*}) , nor a nontrivial character α 𝛼 \alpha . We will very soon specialize, however, to the case where G 𝐺 G is abelian. In this case Z ​ ( 𝔤 ∗ ) = 𝔤 ∗ 𝑍 superscript 𝔤 superscript 𝔤 Z(\mathfrak{g}^{*})=\mathfrak{g}^{*} , the regular values form a dense open set, and the characters of G 𝐺 G form a full integral lattice 𝔤 ℤ ∗ ⊆ 𝔤 ∗ subscript superscript 𝔤 ℤ superscript 𝔤 \mathfrak{g}^{*}_{\mathbb{Z}}\subseteq\mathfrak{g}^{*} . 1.2 Hypertoric varieties defined Let 𝔱 n superscript 𝔱 𝑛 \mathfrak{t}^{n} be the coordinate complex vector space of dimension n 𝑛 n with basis { ε 1 , … , ε n } subscript 𝜀 1 … subscript 𝜀 𝑛 \{\varepsilon_{1},\ldots,\varepsilon_{n}\} , and let 𝔱 d superscript 𝔱 𝑑 \mathfrak{t}^{d} be a complex vector space of dimension d 𝑑 d with a full lattice 𝔱 ℤ d subscript superscript 𝔱 𝑑 ℤ \mathfrak{t}^{d}_{{\mathbb{Z}}} . Though 𝔱 ℤ d subscript superscript 𝔱 𝑑 ℤ \mathfrak{t}^{d}_{{\mathbb{Z}}} is isomorphic to the standard integer lattice ℤ d superscript ℤ 𝑑 {\mathbb{Z}}^{d} , we will not choose such an isomorphism. Let { a 1 , … , a n } ⊂ 𝔱 ℤ d subscript 𝑎 1 … subscript 𝑎 𝑛 subscript superscript 𝔱 𝑑 ℤ \{a_{1},\ldots,a_{n}\}\subset\mathfrak{t}^{d}_{{\mathbb{Z}}} be a collection of nonzero vectors such that the map 𝔱 n → 𝔱 d → superscript 𝔱 𝑛 superscript 𝔱 𝑑 \mathfrak{t}^{n}\to\mathfrak{t}^{d} taking ε i subscript 𝜀 𝑖 \varepsilon_{i} to a i subscript 𝑎 𝑖 a_{i} is surjective. Let k = n − d 𝑘 𝑛 𝑑 k=n-d , and let 𝔱 k superscript 𝔱 𝑘 \mathfrak{t}^{k} be the kernel of this map. Then we have an exact sequence 0 ⟶ 𝔱 k ⟶ ι 𝔱 n ⟶ 𝔱 d ⟶ 0 , ⟶ 0 superscript 𝔱 𝑘 superscript ⟶ 𝜄 superscript 𝔱 𝑛 ⟶ superscript 𝔱 𝑑 ⟶ 0 \displaystyle 0\longrightarrow\mathfrak{t}^{k}\stackrel{{\scriptstyle\iota}}{{\longrightarrow}}\mathfrak{t}^{n}\longrightarrow\mathfrak{t}^{d}\longrightarrow 0, (3) which exponentiates to an exact sequence of tori 0 ⟶ T k ⟶ T n ⟶ T d ⟶ 0 . ⟶ 0 superscript 𝑇 𝑘 ⟶ superscript 𝑇 𝑛 ⟶ superscript 𝑇 𝑑 ⟶ 0 \displaystyle 0\longrightarrow T^{k}\longrightarrow T^{n}\longrightarrow T^{d}\longrightarrow 0. (4) Here T n = ( ℂ × ) n superscript 𝑇 𝑛 superscript superscript ℂ 𝑛 T^{n}=({\mathbb{C}}^{\times})^{n} , T d superscript 𝑇 𝑑 T^{d} is a quotient of T n superscript 𝑇 𝑛 T^{n} , and T k = ker ⁡ ( T n → T d ) superscript 𝑇 𝑘 kernel → superscript 𝑇 𝑛 superscript 𝑇 𝑑 T^{k}=\ker\!\big{(}T^{n}\to T^{d}\big{)} is a subgroup with Lie algebra 𝔱 k superscript 𝔱 𝑘 \mathfrak{t}^{k} , which is connected if and only if the vectors { a i } subscript 𝑎 𝑖 \{a_{i}\} span the lattice 𝔱 ℤ d subscript superscript 𝔱 𝑑 ℤ \mathfrak{t}^{d}_{{\mathbb{Z}}} over the integers. Note that every algebraic subgroup of T n superscript 𝑇 𝑛 T^{n} arises in this way. The torus T n superscript 𝑇 𝑛 T^{n} acts naturally via coordinatewise multiplication on the vector space ℂ n superscript ℂ 𝑛 {\mathbb{C}}^{n} , thus so does the subtorus T k superscript 𝑇 𝑘 T^{k} . For α ∈ ( 𝔱 k ) ℤ ∗ 𝛼 subscript superscript superscript 𝔱 𝑘 ℤ \alpha\in(\mathfrak{t}^{k})^{*}_{\mathbb{Z}} a multiplicative character of T k superscript 𝑇 𝑘 T^{k} and λ ∈ ( 𝔱 k ) ∗ 𝜆 superscript superscript 𝔱 𝑘 \lambda\in(\mathfrak{t}^{k})^{*} arbitrary, the algebraic symplectic quotient 𝔐 α , λ = T ∗ ℂ n / / / / α T k \mathfrak{M}_{\alpha,\lambda}=T^{*}{\mathbb{C}}^{n}{\!/\!\!/\!\!/\!\!/}_{\!\!\alpha}T^{k} is called a hypertoric variety . The hypertoric variety 𝔐 α , λ subscript 𝔐 𝛼 𝜆 \mathfrak{M}_{\alpha,\lambda} is a symplectic variety of dimension 2 ​ d 2 𝑑 2d which admits a complete hyperkähler metric. The action of the quotient torus T d = T n / T k superscript 𝑇 𝑑 superscript 𝑇 𝑛 superscript 𝑇 𝑘 T^{d}=T^{n}/T^{k} on 𝔐 α , λ subscript 𝔐 𝛼 𝜆 \mathfrak{M}_{\alpha,\lambda} is hamiltonian with respect to the algebraic symplectic form, and the action of the maximal compact subtorus T ℝ d subscript superscript 𝑇 𝑑 ℝ T^{d}_{\mathbb{R}} is hyperhamiltonian. In the original paper of Bielawski and Dancer [ BD ] the hyperkähler perspective was stressed, and the spaces were referred to as “toric hyperkähler manifolds”. However, since we have worked frequently with singular reductions as well as with fields of definition other than the complex numbers (see for example [ HP1 , P3 , PW ] ), we prefer the term hypertoric varieties. Remark 1.2.1 In the hypertoric case, the diffeomorphism of Proposition 1.1.2 can be made T ℝ d subscript superscript 𝑇 𝑑 ℝ T^{d}_{\mathbb{R}} -equivariant [ HP1 , 2.1] . 1.3 Hyperplane arrangements The case in which λ = 0 𝜆 0 \lambda=0 will be of particular importance, and it is convenient to encode the data that were used to construct the hypertoric variety 𝔐 α , 0 subscript 𝔐 𝛼 0 \mathfrak{M}_{\alpha,0} in terms of an arrangement of affine hyperplanes with some additional structure in the real vector space ( 𝔱 d ) ℝ ∗ = ( 𝔱 d ) ℤ ∗ ⊗ ℤ ℝ superscript subscript superscript 𝔱 𝑑 ℝ subscript tensor-product ℤ superscript subscript superscript 𝔱 𝑑 ℤ ℝ (\mathfrak{t}^{d})_{{\mathbb{R}}}^{*}=(\mathfrak{t}^{d})_{{\mathbb{Z}}}^{*}\otimes_{\mathbb{Z}}{\mathbb{R}} . A weighted, cooriented, affine hyperplane H ⊆ ( 𝔱 d ) ℝ ∗ 𝐻 superscript subscript superscript 𝔱 𝑑 ℝ H\subseteq(\mathfrak{t}^{d})_{{\mathbb{R}}}^{*} is an affine hyperplane along with a choice of nonzero integer normal vector a ∈ 𝔱 ℤ d 𝑎 subscript superscript 𝔱 𝑑 ℤ a\in\mathfrak{t}^{d}_{{\mathbb{Z}}} . Here “affine” means that H 𝐻 H need not pass through the origin, and “weighted” means that a 𝑎 a is not required to be primitive. Let r = ( r 1 , … , r n ) ∈ ( 𝔱 n ) ∗ 𝑟 subscript 𝑟 1 … subscript 𝑟 𝑛 superscript superscript 𝔱 𝑛 r=(r_{1},\ldots,r_{n})\in(\mathfrak{t}^{n})^{*} be a lift of α 𝛼 \alpha along ι ∗ superscript 𝜄 \iota^{*} , and let H i = { x ∈ ( 𝔱 d ) ℝ ∗ ∣ x ⋅ a i + r i = 0 } subscript 𝐻 𝑖 conditional-set 𝑥 superscript subscript superscript 𝔱 𝑑 ℝ ⋅ 𝑥 subscript 𝑎 𝑖 subscript 𝑟 𝑖 0 H_{i}=\{x\in(\mathfrak{t}^{d})_{{\mathbb{R}}}^{*}\mid x\cdot a_{i}+r_{i}=0\} be the weighted, cooriented, affine hyperplane with normal vector a i ∈ ( 𝔱 d ) ℤ ∗ subscript 𝑎 𝑖 superscript subscript superscript 𝔱 𝑑 ℤ a_{i}\in(\mathfrak{t}^{d})_{{\mathbb{Z}}}^{*} . (Choosing a different r 𝑟 r corresponds to simultaneously translating all of the hyperplanes by a vector in ( 𝔱 d ) ℤ ∗ superscript subscript superscript 𝔱 𝑑 ℤ (\mathfrak{t}^{d})_{{\mathbb{Z}}}^{*} .) We will denote the collection { H 1 , … , H n } subscript 𝐻 1 … subscript 𝐻 𝑛 \{H_{1},\ldots,H_{n}\} by 𝒜 𝒜 \mathcal{A} , and write 𝔐 ​ ( 𝒜 ) = 𝔐 α , 0 𝔐 𝒜 subscript 𝔐 𝛼 0 \mathfrak{M}(\mathcal{A})=\mathfrak{M}_{\alpha,0} for the corresponding hypertoric variety. We will refer to 𝒜 𝒜 \mathcal{A} simply as an arrangement , always assuming that the weighted coorientations are part of the data. Remark 1.3.1 We note that we allow repetitions of hyperplanes in our arrangement ( 𝒜 𝒜 \mathcal{A} may be a multi-set), and that a repeated occurrence of a particular hyperplane is not the same as a single occurrence of that hyperplane with weight 2. On the other hand, little is lost by restricting one’s attention to arrangements of distinct hyperplanes of weight one. Since each hyperplane H i subscript 𝐻 𝑖 H_{i} comes with a normal vector, it seems at first that it would make the most sense to talk about an arrangement of half-spaces, where the i th superscript 𝑖 th i^{\text{th}} half-space consists of the set of points that lie on the positive side of H i subscript 𝐻 𝑖 H_{i} with respect to a i subscript 𝑎 𝑖 a_{i} . The reason that we talk about hyperplanes rather than half-spaces is the following proposition, proven in [ HP1 , 2.2] . Proposition 1.3.2 The T d superscript 𝑇 𝑑 T^{d} -variety 𝔐 ​ ( 𝒜 ) 𝔐 𝒜 \mathfrak{M}(\mathcal{A}) does not depend on the signs of the vectors a i subscript 𝑎 𝑖 a_{i} . In other words, if we make a new hypertoric variety with the same arrangement of weighted hyperplanes but with some of the coorientations flipped, it will be T d superscript 𝑇 𝑑 T^{d} -equivariantly isomorphic to the hypertoric variety with which we started. 3 3 3 In [ HP1 ] we consider an extra ℂ × superscript ℂ {\mathbb{C}}^{\times} action on 𝔐 ​ ( 𝒜 ) 𝔐 𝒜 \mathfrak{M}(\mathcal{A}) that does depend on the coorientations. We call the arrangement 𝒜 𝒜 \mathcal{A} simple if every subset of m 𝑚 m hyperplanes with nonempty intersection intersects in codimension m 𝑚 m . We call 𝒜 𝒜 \mathcal{A} unimodular if every collection of d 𝑑 d linearly independent vectors { a i 1 , … , a i d } subscript 𝑎 subscript 𝑖 1 … subscript 𝑎 subscript 𝑖 𝑑 \{a_{i_{1}},\ldots,a_{i_{d}}\} spans 𝔱 d superscript 𝔱 𝑑 \mathfrak{t}^{d} over the integers. An arrangement which is both simple and unimodular is called smooth . The following proposition is proven in [ BD , 3.2 & \& 3.3] . Proposition 1.3.3 The hypertoric variety 𝔐 ​ ( 𝒜 ) 𝔐 𝒜 \mathfrak{M}(\mathcal{A}) has at worst orbifold (finite quotient) singularities if and only if 𝒜 𝒜 \mathcal{A} is simple, and is smooth if and only if 𝒜 𝒜 \mathcal{A} is smooth. For the remainder of the paper, Let 𝒜 = { H 1 , … , H n } 𝒜 subscript 𝐻 1 … subscript 𝐻 𝑛 \mathcal{A}=\{H_{1},\ldots,H_{n}\} be a central arrangement, meaning that r i = 0 subscript 𝑟 𝑖 0 r_{i}=0 for all i 𝑖 i , so that all of the hyperplanes pass through the origin. Then 𝔐 ​ ( 𝒜 ) 𝔐 𝒜 \mathfrak{M}(\mathcal{A}) is the singular affine variety 𝔐 0 , 0 subscript 𝔐 0 0 \mathfrak{M}_{0,0} . Let 𝒜 ~ = { H ~ 1 , … , H ~ n } ~ 𝒜 subscript ~ 𝐻 1 … subscript ~ 𝐻 𝑛 \tilde{\mathcal{A}}=\{\tilde{H}_{1},\ldots,\tilde{H}_{n}\} be a simplification of 𝒜 𝒜 \mathcal{A} , by which we mean an arrangement defined by the same vectors { a i } ⊂ 𝔱 d subscript 𝑎 𝑖 superscript 𝔱 𝑑 \{a_{i}\}\subset\mathfrak{t}^{d} , but with a different choice of r ∈ ( 𝔱 n ) ∗ 𝑟 superscript superscript 𝔱 𝑛 r\in(\mathfrak{t}^{n})^{*} , such that 𝒜 ~ ~ 𝒜 \tilde{\mathcal{A}} is simple. This corresponds to translating each of the hyperplanes in 𝒜 𝒜 \mathcal{A} away from the origin by some generic amount. Then 𝔐 ​ ( 𝒜 ~ ) 𝔐 ~ 𝒜 \mathfrak{M}(\tilde{\mathcal{A}}) maps T 𝑇 T -equivariantly to 𝔐 ​ ( 𝒜 ) 𝔐 𝒜 \mathfrak{M}(\mathcal{A}) by Equation ( 2 ), and Proposition 1.3.3 tell us that it is in fact an “orbifold resolution”, meaning a projective morphism, generically one-to-one, in which the source has at worst orbifold singularities. The structure of this map is studied extensively in [ PW ] . 1.4 Toward an abstract definition The definition of a hypertoric variety in Section 1.2 is constructive, modeled on the definition of toric varieties as GIT quotients of the form ℂ n / / α T k {\mathbb{C}}^{n}{/\!\!/}_{\!\!\alpha}T^{k} , or equivalently as symplectic quotients by compact tori. In the case of toric varieties, there are also abstract definitions. In the symplectic world, one defines a toric orbifold to be a symplectic orbifold of dimension 2 ​ d 2 𝑑 2d along with an effective Hamiltonian action of a compact d 𝑑 d -torus, and proves that any connected, compact toric orbifold arises from the symplectic quotient construction [ De , LT ] . In the algebraic world, one defines a toric variety to be a normal variety admitting a torus action with a dense orbit, and then proves that any semiprojective 4 4 4 Hausel and Sturmfels call a toric variety semiprojective if it is projective over its affinization and has at least one torus fixed point. toric variety with at worst orbifold singularities arises from the GIT construction. This idea goes back to [ Co ] , and can be found in this language in [ HSt , 2.6] . It is natural to ask for such an abstract definition and classification theorem for hypertoric varieties, either from the standpoint of symplectic algebraic geometry or that of hyperkähler geometry. In the hyperkähler setting, such a theorem was proven in [ Bi , 3,4] . Theorem 1.4.1 Any complete, connected, hyperkähler manifold of real dimension 4 ​ d 4 𝑑 4d which admits an effective, hyperhamiltonian action of the compact torus T ℝ d subscript superscript 𝑇 𝑑 ℝ T^{d}_{\mathbb{R}} is T ℝ d subscript superscript 𝑇 𝑑 ℝ T^{d}_{\mathbb{R}} -equivariantly diffeomorphic, and Taub-NUT deformation equivalent, to a hypertoric variety. Any such manifold with Euclidean volume growth is T ℝ d subscript superscript 𝑇 𝑑 ℝ T^{d}_{\mathbb{R}} -equivariantly isometric to a hypertoric variety. An analogous algebraic theorem has not been proven, but it should look something like the following. Conjecture 1.4.2 Any connected, symplectic, algebraic variety which is projective over its affinization and admits an effective, hamiltonian action of the algebraic torus T d superscript 𝑇 𝑑 T^{d} is equivariantly isomorphic to a Zariski open subset of a hypertoric variety.

Hypertoric varieties are quaternionic analogues of toric varieties, important for their interaction with the combinatorics of matroids as well as for their prominent place in the rapidly expanding field of algebraic symplectic and hyperkahler geometry. The aim of this survey is to give clear definitions and statements of known results, serving both as a reference and as a point of entry to this beautiful subject.

[{"content":"Help me to generate the abstract of a survey paper given the title:*A survey of hypertoric geometry and topology* and the introduction:1 Definitions and basic properties Hypertoric varieties can be considered either as algebraic varieties or, in the smooth case, as hyperkähler manifolds. In this section we give a constructive definition, with a strong bias toward the algebraic interpretation. Section 1.1 proceeds in greater generality than is necessary for hypertoric varieties so as to unify the theory with that of other algebraic symplectic quotients, most notably Nakajima quiver varieties. 1.1 Algebraic symplectic quotients Let G 𝐺 G be a reductive algebraic group over the complex numbers acting linearly and effectively on a finite-dimensional complex vector space V 𝑉 V . The cotangent bundle T ∗ ​ V ≅ V × V ∗ superscript 𝑇 𝑉 𝑉 superscript 𝑉 T^{*}V\cong V\times V^{*} carries a natural algebraic symplectic form Ω Ω \Omega . The induced action of G 𝐺 G on T ∗ ​ V superscript 𝑇 𝑉 T^{*}V is hamiltonian, with moment map μ : T ∗ ​ V → 𝔤 ∗ : 𝜇 → superscript 𝑇 𝑉 superscript 𝔤 \mu:T^{*}V\to\mathfrak{g}^{*} given by the equation μ ​ ( z , w ) ​ ( x ) = Ω ​ ( x ⋅ z , w ) ​ for all z ∈ V , w ∈ V ∗ , x ∈ 𝔤 . 𝜇 𝑧 𝑤 𝑥 Ω ⋅ 𝑥 𝑧 𝑤 for all z ∈ V , w ∈ V ∗ , x ∈ 𝔤 . \mu(z,w)(x)=\Omega(x\cdot z,w)\,\,\text{ for all $z\in V$, $w\in V^{*}$, $x\in\mathfrak{g}$.} Suppose given an element λ ∈ Z ​ ( 𝔤 ∗ ) 𝜆 𝑍 superscript 𝔤 \lambda\in Z(\mathfrak{g}^{*}) (the part of 𝔤 ∗ superscript 𝔤 \mathfrak{g}^{*} fixed by the coadjoint action of G 𝐺 G ), and a multiplicative character α : G → ℂ × : 𝛼 → 𝐺 superscript ℂ \alpha:G\to{\mathbb{C}}^{\times} , which may be identified with an element of Z ​ ( 𝔤 ℤ ∗ ) 𝑍 subscript superscript 𝔤 ℤ Z(\mathfrak{g}^{*}_{\mathbb{Z}}) by taking its derivative at the identity element of G 𝐺 G . 2 2 2 Strictly speaking, an element of Z ​ ( 𝔤 ℤ ∗ ) 𝑍 subscript superscript 𝔤 ℤ Z(\mathfrak{g}^{*}_{\mathbb{Z}}) only determines a character of the connected component of the identity of G 𝐺 G . It can be checked, however, that the notion of α 𝛼 \alpha -stability defined below depends only on the restriction of α 𝛼 \alpha to the identity component, therefore we will abusively think of α 𝛼 \alpha as sitting inside of Z ​ ( 𝔤 ℤ ∗ ) 𝑍 subscript superscript 𝔤 ℤ Z(\mathfrak{g}^{*}_{\mathbb{Z}}) . The fact that λ 𝜆 \lambda lies in Z ​ ( 𝔤 ∗ ) 𝑍 superscript 𝔤 Z(\mathfrak{g}^{*}) implies that G 𝐺 G acts on μ − 1 ​ ( λ ) superscript 𝜇 1 𝜆 \mu^{-1}(\lambda) . Our main object of study in this survey will be the algebraic symplectic quotient 𝔐 α , λ = T ∗ V / / / / α G = μ − 1 ( λ ) / / α G . \mathfrak{M}_{\alpha,\lambda}=T^{*}V{\!/\!\!/\!\!/\!\!/}_{\!\!\alpha}G=\mu^{-1}(\lambda){/\!\!/}_{\!\!\alpha}G. Here the second quotient is a projective GIT quotient μ − 1 ( λ ) / / α G := Proj ⨁ m = 0 ∞ { f ∈ Fun ( μ − 1 ( λ ) ) | ν ( g ) ∗ f = α ( g ) m f for all g ∈ G } , \mu^{-1}(\lambda){/\!\!/}_{\!\!\alpha}G:=\mathrm{Proj}\,\bigoplus_{m=0}^{\infty}\Big{\{}f\in\operatorname{Fun}\big{(}\mu^{-1}(\lambda)\big{)}\hskip 3.0pt\Big{|}\hskip 3.0pt\nu(g)^{*}f=\alpha(g)^{m}f\hskip 3.0pt\text{ for all }g\in G\hskip 3.0pt\Big{\}}, (1) where ν ​ ( g ) 𝜈 𝑔 \nu(g) is the automorphism of μ − 1 ​ ( λ ) superscript 𝜇 1 𝜆 \mu^{-1}(\lambda) defined by g 𝑔 g . This quotient may be defined in a more geometric way as follows. A point ( z , w ) ∈ μ − 1 ​ ( λ ) 𝑧 𝑤 superscript 𝜇 1 𝜆 (z,w)\in\mu^{-1}(\lambda) is called α 𝛼 \mathbf{\alpha} -semistable if there exists a function f 𝑓 f on μ − 1 ​ ( λ ) superscript 𝜇 1 𝜆 \mu^{-1}(\lambda) and a positive integer m 𝑚 m such that ν ​ ( g ) ∗ ​ f = α ​ ( g ) m ​ f 𝜈 superscript 𝑔 𝑓 𝛼 superscript 𝑔 𝑚 𝑓 \nu(g)^{*}f=\alpha(g)^{m}f for all g ∈ G 𝑔 𝐺 g\in G and f ​ ( z , w ) ≠ 0 𝑓 𝑧 𝑤 0 f(z,w)\neq 0 . It is called α 𝛼 \mathbf{\alpha} -stable if it is α 𝛼 \alpha -semistable and its G 𝐺 G -orbit in the α 𝛼 \alpha -semistable set is closed with finite stabilizers. Then the stable and semistable sets μ − 1 ​ ( λ ) α − s ​ t ⊆ μ − 1 ​ ( λ ) α − s ​ s ⊆ μ − 1 ​ ( λ ) superscript 𝜇 1 superscript 𝜆 𝛼 𝑠 𝑡 superscript 𝜇 1 superscript 𝜆 𝛼 𝑠 𝑠 superscript 𝜇 1 𝜆 \mu^{-1}(\lambda)^{\alpha-st}\subseteq\mu^{-1}(\lambda)^{\alpha-ss}\subseteq\mu^{-1}(\lambda) are nonempty and Zariski open, and there is a surjection μ − 1 ​ ( λ ) α − s ​ s ↠ 𝔐 α , λ ↠ superscript 𝜇 1 superscript 𝜆 𝛼 𝑠 𝑠 subscript 𝔐 𝛼 𝜆 \mu^{-1}(\lambda)^{\alpha-ss}\twoheadrightarrow\mathfrak{M}_{\alpha,\lambda} with ( z , w ) 𝑧 𝑤 (z,w) and ( z ′ , w ′ ) superscript 𝑧 ′ superscript 𝑤 ′ (z^{\prime},w^{\prime}) mapping to the same point if and only if the closures of their G 𝐺 G -orbits intersect in μ − 1 ​ ( λ ) α − s ​ s superscript 𝜇 1 superscript 𝜆 𝛼 𝑠 𝑠 \mu^{-1}(\lambda)^{\alpha-ss} . In particular, the restriction of this map to the stable locus is nothing but the geometric quotient by G 𝐺 G . For an introduction to geometric invariant theory that explains the equivalence of these two perspectives, see P2 . Remark 1.1.1 The algebraic symplectic quotient defined above may also be interpreted as a hyperkähler quotient. The even dimensional complex vector space T ∗ ​ V superscript 𝑇 𝑉 T^{*}V admits a complete hyperkähler metric, and the action of the maximal compact subgroup G ℝ ⊆ G subscript 𝐺 ℝ 𝐺 G_{\mathbb{R}}\subseteq G is hyperhamiltonian , meaning that it is hamiltonian with respect to all three of the real symplectic forms on T ∗ ​ V superscript 𝑇 𝑉 T^{*}V . Then 𝔐 α , λ subscript 𝔐 𝛼 𝜆 \mathfrak{M}_{\alpha,\lambda} is naturally diffeomorphic to the hyperkähler quotient of T ∗ ​ V superscript 𝑇 𝑉 T^{*}V by G ℝ subscript 𝐺 ℝ G_{\mathbb{R}} , in the sense of HKLR , at the value ( α , Re ⁡ λ , Im ⁡ λ ) ∈ 𝔤 ℝ ∗ ⊗ ℝ 3 𝛼 Re 𝜆 Im 𝜆 tensor-product subscript superscript 𝔤 ℝ superscript ℝ 3 (\alpha,\operatorname{Re}\lambda,\operatorname{Im}\lambda)\in\mathfrak{g}^{*}_{\mathbb{R}}\otimes{\mathbb{R}}^{3} . This was the original perspective on both hypertoric varieties BD and Nakajima quiver varieties N1 . For more on this perspective in the hypertoric case, see Konnos survey in this volume K4 , §3 . We note that if α = 0 𝛼 0 \alpha=0 is the trivial character of G 𝐺 G , then Equation ( 1 ) simplifies to 𝔐 0 , λ = Spec Fun ( μ − 1 ( λ ) ) G . \mathfrak{M}_{0,\lambda}=\operatorname{Spec}\operatorname{Fun}\big{(}\mu^{-1}(\lambda)\big{)}^{G}. Furthermore, since 𝔐 α , λ subscript 𝔐 𝛼 𝜆 \mathfrak{M}_{\alpha,\lambda} is defined as the projective spectrum of a graded ring whose degree zero part is the ring of invariant functions on μ − 1 ​ ( λ ) superscript 𝜇 1 𝜆 \mu^{-1}(\lambda) , we always have a projective morphism 𝔐 α , λ ↠ 𝔐 0 , λ . ↠ subscript 𝔐 𝛼 𝜆 subscript 𝔐 0 𝜆 \mathfrak{M}_{\alpha,\lambda}\twoheadrightarrow\mathfrak{M}_{0,\lambda}. (2) This morphism may also be induced from the inclusion of the inclusion μ − 1 ​ ( λ ) α − s ​ s ⊆ μ − 1 ​ ( λ ) = μ − 1 ​ ( λ ) 0 − s ​ s . superscript 𝜇 1 superscript 𝜆 𝛼 𝑠 𝑠 superscript 𝜇 1 𝜆 superscript 𝜇 1 superscript 𝜆 0 𝑠 𝑠 \mu^{-1}(\lambda)^{\alpha-ss}\subseteq\mu^{-1}(\lambda)=\mu^{-1}(\lambda)^{0-ss}. From this we may conclude that it is generically one-to-one, and therefore a partial resolution. When λ = 0 𝜆 0 \lambda=0 , we have a distinguished point in 𝔐 0 , 0 subscript 𝔐 0 0 \mathfrak{M}_{0,0} , namely the image of 0 ∈ μ − 1 ​ ( 0 ) 0 superscript 𝜇 1 0 0\in\mu^{\!-1}(0) under the map induced by the inclusion of the invariant functions into the coordinate ring of μ − 1 ​ ( 0 ) superscript 𝜇 1 0 \mu^{\!-1}(0) . The preimage of this point under the morphism ( 2 ) is called the core of 𝔐 α , 0 subscript 𝔐 𝛼 0 \mathfrak{M}_{\alpha,0} , and will be further studied (in the case where G 𝐺 G is abelian) in Section 2.1 . On the other extreme, if λ 𝜆 \lambda is a regular value of μ 𝜇 \mu , then G 𝐺 G will act locally freely on μ − 1 ​ ( λ ) superscript 𝜇 1 𝜆 \mu^{-1}(\lambda) . In this case all points will be α 𝛼 \alpha -stable for any choice of α 𝛼 \alpha , and the GIT quotient 𝔐 λ = μ − 1 ( λ ) / / G \mathfrak{M}_{\lambda}=\mu^{-1}(\lambda){/\!\!/}G will simply be a geometric quotient. In particular, the morphism ( 2 ) becomes an isomorphism. Both the case of regular λ 𝜆 \lambda and the case λ = 0 𝜆 0 \lambda=0 will be of interest to us. We call a pair ( α , λ ) 𝛼 𝜆 (\alpha,\lambda) generic if μ − 1 ​ ( λ ) α − s ​ t = μ − 1 ​ ( λ ) α − s ​ s superscript 𝜇 1 superscript 𝜆 𝛼 𝑠 𝑡 superscript 𝜇 1 superscript 𝜆 𝛼 𝑠 𝑠 \mu^{-1}(\lambda)^{\alpha-st}=\mu^{-1}(\lambda)^{\alpha-ss} . In this case the moment map condition tells us that the stable set is smooth, and therefore that the quotient 𝔐 α , λ subscript 𝔐 𝛼 𝜆 \mathfrak{M}_{\alpha,\lambda} by the locally free G 𝐺 G -action has at worst orbifold singularities. Using the hyperkähler quotient perspective of Remark 1.1.1 , one can prove the following Proposition. (See K3 , 2.6 or HP1 , 2.1 in the hypertoric case, and N1 , 4.2 in the case of quiver varieties; the general case is no harder than these.) Proposition 1.1.2 If ( α , λ ) 𝛼 𝜆 (\alpha,\lambda) and ( α ′ , λ ′ ) superscript 𝛼 ′ superscript 𝜆 ′ (\alpha^{\prime},\lambda^{\prime}) are both generic, then the two symplectic quotients 𝔐 α , λ subscript 𝔐 𝛼 𝜆 \mathfrak{M}_{\alpha,\lambda} and 𝔐 α ′ , λ ′ subscript 𝔐 superscript 𝛼 ′ superscript 𝜆 ′ \mathfrak{M}_{\alpha^{\prime},\lambda^{\prime}} are diffeomorphic. Remark 1.1.3 If G 𝐺 G is semisimple, then Z ​ ( 𝔤 ∗ ) = { 0 } 𝑍 superscript 𝔤 0 Z(\mathfrak{g}^{*})=\{0\} , and (unless G 𝐺 G is finite) it will not be possible to choose a regular value λ ∈ Z ​ ( 𝔤 ∗ ) 𝜆 𝑍 superscript 𝔤 \lambda\in Z(\mathfrak{g}^{*}) , nor a nontrivial character α 𝛼 \alpha . We will very soon specialize, however, to the case where G 𝐺 G is abelian. In this case Z ​ ( 𝔤 ∗ ) = 𝔤 ∗ 𝑍 superscript 𝔤 superscript 𝔤 Z(\mathfrak{g}^{*})=\mathfrak{g}^{*} , the regular values form a dense open set, and the characters of G 𝐺 G form a full integral lattice 𝔤 ℤ ∗ ⊆ 𝔤 ∗ subscript superscript 𝔤 ℤ superscript 𝔤 \mathfrak{g}^{*}_{\mathbb{Z}}\subseteq\mathfrak{g}^{*} . 1.2 Hypertoric varieties defined Let 𝔱 n superscript 𝔱 𝑛 \mathfrak{t}^{n} be the coordinate complex vector space of dimension n 𝑛 n with basis { ε 1 , … , ε n } subscript 𝜀 1 … subscript 𝜀 𝑛 \{\varepsilon_{1},\ldots,\varepsilon_{n}\} , and let 𝔱 d superscript 𝔱 𝑑 \mathfrak{t}^{d} be a complex vector space of dimension d 𝑑 d with a full lattice 𝔱 ℤ d subscript superscript 𝔱 𝑑 ℤ \mathfrak{t}^{d}_{{\mathbb{Z}}} . Though 𝔱 ℤ d subscript superscript 𝔱 𝑑 ℤ \mathfrak{t}^{d}_{{\mathbb{Z}}} is isomorphic to the standard integer lattice ℤ d superscript ℤ 𝑑 {\mathbb{Z}}^{d} , we will not choose such an isomorphism. Let { a 1 , … , a n } ⊂ 𝔱 ℤ d subscript 𝑎 1 … subscript 𝑎 𝑛 subscript superscript 𝔱 𝑑 ℤ \{a_{1},\ldots,a_{n}\}\subset\mathfrak{t}^{d}_{{\mathbb{Z}}} be a collection of nonzero vectors such that the map 𝔱 n → 𝔱 d → superscript 𝔱 𝑛 superscript 𝔱 𝑑 \mathfrak{t}^{n}\to\mathfrak{t}^{d} taking ε i subscript 𝜀 𝑖 \varepsilon_{i} to a i subscript 𝑎 𝑖 a_{i} is surjective. Let k = n − d 𝑘 𝑛 𝑑 k=n-d , and let 𝔱 k superscript 𝔱 𝑘 \mathfrak{t}^{k} be the kernel of this map. Then we have an exact sequence 0 ⟶ 𝔱 k ⟶ ι 𝔱 n ⟶ 𝔱 d ⟶ 0 , ⟶ 0 superscript 𝔱 𝑘 superscript ⟶ 𝜄 superscript 𝔱 𝑛 ⟶ superscript 𝔱 𝑑 ⟶ 0 \displaystyle 0\longrightarrow\mathfrak{t}^{k}\stackrel{{\scriptstyle\iota}}{{\longrightarrow}}\mathfrak{t}^{n}\longrightarrow\mathfrak{t}^{d}\longrightarrow 0, (3) which exponentiates to an exact sequence of tori 0 ⟶ T k ⟶ T n ⟶ T d ⟶ 0 . ⟶ 0 superscript 𝑇 𝑘 ⟶ superscript 𝑇 𝑛 ⟶ superscript 𝑇 𝑑 ⟶ 0 \displaystyle 0\longrightarrow T^{k}\longrightarrow T^{n}\longrightarrow T^{d}\longrightarrow 0. (4) Here T n = ( ℂ × ) n superscript 𝑇 𝑛 superscript superscript ℂ 𝑛 T^{n}=({\mathbb{C}}^{\times})^{n} , T d superscript 𝑇 𝑑 T^{d} is a quotient of T n superscript 𝑇 𝑛 T^{n} , and T k = ker ⁡ ( T n → T d ) superscript 𝑇 𝑘 kernel → superscript 𝑇 𝑛 superscript 𝑇 𝑑 T^{k}=\ker\!\big{(}T^{n}\to T^{d}\big{)} is a subgroup with Lie algebra 𝔱 k superscript 𝔱 𝑘 \mathfrak{t}^{k} , which is connected if and only if the vectors { a i } subscript 𝑎 𝑖 \{a_{i}\} span the lattice 𝔱 ℤ d subscript superscript 𝔱 𝑑 ℤ \mathfrak{t}^{d}_{{\mathbb{Z}}} over the integers. Note that every algebraic subgroup of T n superscript 𝑇 𝑛 T^{n} arises in this way. The torus T n superscript 𝑇 𝑛 T^{n} acts naturally via coordinatewise multiplication on the vector space ℂ n superscript ℂ 𝑛 {\mathbb{C}}^{n} , thus so does the subtorus T k superscript 𝑇 𝑘 T^{k} . For α ∈ ( 𝔱 k ) ℤ ∗ 𝛼 subscript superscript superscript 𝔱 𝑘 ℤ \alpha\in(\mathfrak{t}^{k})^{*}_{\mathbb{Z}} a multiplicative character of T k superscript 𝑇 𝑘 T^{k} and λ ∈ ( 𝔱 k ) ∗ 𝜆 superscript superscript 𝔱 𝑘 \lambda\in(\mathfrak{t}^{k})^{*} arbitrary, the algebraic symplectic quotient 𝔐 α , λ = T ∗ ℂ n / / / / α T k \mathfrak{M}_{\alpha,\lambda}=T^{*}{\mathbb{C}}^{n}{\!/\!\!/\!\!/\!\!/}_{\!\!\alpha}T^{k} is called a hypertoric variety . The hypertoric variety 𝔐 α , λ subscript 𝔐 𝛼 𝜆 \mathfrak{M}_{\alpha,\lambda} is a symplectic variety of dimension 2 ​ d 2 𝑑 2d which admits a complete hyperkähler metric. The action of the quotient torus T d = T n / T k superscript 𝑇 𝑑 superscript 𝑇 𝑛 superscript 𝑇 𝑘 T^{d}=T^{n}/T^{k} on 𝔐 α , λ subscript 𝔐 𝛼 𝜆 \mathfrak{M}_{\alpha,\lambda} is hamiltonian with respect to the algebraic symplectic form, and the action of the maximal compact subtorus T ℝ d subscript superscript 𝑇 𝑑 ℝ T^{d}_{\mathbb{R}} is hyperhamiltonian. In the original paper of Bielawski and Dancer BD the hyperkähler perspective was stressed, and the spaces were referred to as toric hyperkähler manifolds. However, since we have worked frequently with singular reductions as well as with fields of definition other than the complex numbers (see for example HP1 , P3 , PW ), we prefer the term hypertoric varieties. Remark 1.2.1 In the hypertoric case, the diffeomorphism of Proposition 1.1.2 can be made T ℝ d subscript superscript 𝑇 𝑑 ℝ T^{d}_{\mathbb{R}} -equivariant HP1 , 2.1 . 1.3 Hyperplane arrangements The case in which λ = 0 𝜆 0 \lambda=0 will be of particular importance, and it is convenient to encode the data that were used to construct the hypertoric variety 𝔐 α , 0 subscript 𝔐 𝛼 0 \mathfrak{M}_{\alpha,0} in terms of an arrangement of affine hyperplanes with some additional structure in the real vector space ( 𝔱 d ) ℝ ∗ = ( 𝔱 d ) ℤ ∗ ⊗ ℤ ℝ superscript subscript superscript 𝔱 𝑑 ℝ subscript tensor-product ℤ superscript subscript superscript 𝔱 𝑑 ℤ ℝ (\mathfrak{t}^{d})_{{\mathbb{R}}}^{*}=(\mathfrak{t}^{d})_{{\mathbb{Z}}}^{*}\otimes_{\mathbb{Z}}{\mathbb{R}} . A weighted, cooriented, affine hyperplane H ⊆ ( 𝔱 d ) ℝ ∗ 𝐻 superscript subscript superscript 𝔱 𝑑 ℝ H\subseteq(\mathfrak{t}^{d})_{{\mathbb{R}}}^{*} is an affine hyperplane along with a choice of nonzero integer normal vector a ∈ 𝔱 ℤ d 𝑎 subscript superscript 𝔱 𝑑 ℤ a\in\mathfrak{t}^{d}_{{\mathbb{Z}}} . Here affine means that H 𝐻 H need not pass through the origin, and weighted means that a 𝑎 a is not required to be primitive. Let r = ( r 1 , … , r n ) ∈ ( 𝔱 n ) ∗ 𝑟 subscript 𝑟 1 … subscript 𝑟 𝑛 superscript superscript 𝔱 𝑛 r=(r_{1},\ldots,r_{n})\in(\mathfrak{t}^{n})^{*} be a lift of α 𝛼 \alpha along ι ∗ superscript 𝜄 \iota^{*} , and let H i = { x ∈ ( 𝔱 d ) ℝ ∗ ∣ x ⋅ a i + r i = 0 } subscript 𝐻 𝑖 conditional-set 𝑥 superscript subscript superscript 𝔱 𝑑 ℝ ⋅ 𝑥 subscript 𝑎 𝑖 subscript 𝑟 𝑖 0 H_{i}=\{x\in(\mathfrak{t}^{d})_{{\mathbb{R}}}^{*}\mid x\cdot a_{i}+r_{i}=0\} be the weighted, cooriented, affine hyperplane with normal vector a i ∈ ( 𝔱 d ) ℤ ∗ subscript 𝑎 𝑖 superscript subscript superscript 𝔱 𝑑 ℤ a_{i}\in(\mathfrak{t}^{d})_{{\mathbb{Z}}}^{*} . (Choosing a different r 𝑟 r corresponds to simultaneously translating all of the hyperplanes by a vector in ( 𝔱 d ) ℤ ∗ superscript subscript superscript 𝔱 𝑑 ℤ (\mathfrak{t}^{d})_{{\mathbb{Z}}}^{*} .) We will denote the collection { H 1 , … , H n } subscript 𝐻 1 … subscript 𝐻 𝑛 \{H_{1},\ldots,H_{n}\} by 𝒜 𝒜 \mathcal{A} , and write 𝔐 ​ ( 𝒜 ) = 𝔐 α , 0 𝔐 𝒜 subscript 𝔐 𝛼 0 \mathfrak{M}(\mathcal{A})=\mathfrak{M}_{\alpha,0} for the corresponding hypertoric variety. We will refer to 𝒜 𝒜 \mathcal{A} simply as an arrangement , always assuming that the weighted coorientations are part of the data. Remark 1.3.1 We note that we allow repetitions of hyperplanes in our arrangement ( 𝒜 𝒜 \mathcal{A} may be a multi-set), and that a repeated occurrence of a particular hyperplane is not the same as a single occurrence of that hyperplane with weight 2. On the other hand, little is lost by restricting ones attention to arrangements of distinct hyperplanes of weight one. Since each hyperplane H i subscript 𝐻 𝑖 H_{i} comes with a normal vector, it seems at first that it would make the most sense to talk about an arrangement of half-spaces, where the i th superscript 𝑖 th i^{\text{th}} half-space consists of the set of points that lie on the positive side of H i subscript 𝐻 𝑖 H_{i} with respect to a i subscript 𝑎 𝑖 a_{i} . The reason that we talk about hyperplanes rather than half-spaces is the following proposition, proven in HP1 , 2.2 . Proposition 1.3.2 The T d superscript 𝑇 𝑑 T^{d} -variety 𝔐 ​ ( 𝒜 ) 𝔐 𝒜 \mathfrak{M}(\mathcal{A}) does not depend on the signs of the vectors a i subscript 𝑎 𝑖 a_{i} . In other words, if we make a new hypertoric variety with the same arrangement of weighted hyperplanes but with some of the coorientations flipped, it will be T d superscript 𝑇 𝑑 T^{d} -equivariantly isomorphic to the hypertoric variety with which we started. 3 3 3 In HP1 we consider an extra ℂ × superscript ℂ {\mathbb{C}}^{\times} action on 𝔐 ​ ( 𝒜 ) 𝔐 𝒜 \mathfrak{M}(\mathcal{A}) that does depend on the coorientations. We call the arrangement 𝒜 𝒜 \mathcal{A} simple if every subset of m 𝑚 m hyperplanes with nonempty intersection intersects in codimension m 𝑚 m . We call 𝒜 𝒜 \mathcal{A} unimodular if every collection of d 𝑑 d linearly independent vectors { a i 1 , … , a i d } subscript 𝑎 subscript 𝑖 1 … subscript 𝑎 subscript 𝑖 𝑑 \{a_{i_{1}},\ldots,a_{i_{d}}\} spans 𝔱 d superscript 𝔱 𝑑 \mathfrak{t}^{d} over the integers. An arrangement which is both simple and unimodular is called smooth . The following proposition is proven in BD , 3.2 & \& 3.3 . Proposition 1.3.3 The hypertoric variety 𝔐 ​ ( 𝒜 ) 𝔐 𝒜 \mathfrak{M}(\mathcal{A}) has at worst orbifold (finite quotient) singularities if and only if 𝒜 𝒜 \mathcal{A} is simple, and is smooth if and only if 𝒜 𝒜 \mathcal{A} is smooth. For the remainder of the paper, Let 𝒜 = { H 1 , … , H n } 𝒜 subscript 𝐻 1 … subscript 𝐻 𝑛 \mathcal{A}=\{H_{1},\ldots,H_{n}\} be a central arrangement, meaning that r i = 0 subscript 𝑟 𝑖 0 r_{i}=0 for all i 𝑖 i , so that all of the hyperplanes pass through the origin. Then 𝔐 ​ ( 𝒜 ) 𝔐 𝒜 \mathfrak{M}(\mathcal{A}) is the singular affine variety 𝔐 0 , 0 subscript 𝔐 0 0 \mathfrak{M}_{0,0} . Let 𝒜 ~ = { H ~ 1 , … , H ~ n } ~ 𝒜 subscript ~ 𝐻 1 … subscript ~ 𝐻 𝑛 \tilde{\mathcal{A}}=\{\tilde{H}_{1},\ldots,\tilde{H}_{n}\} be a simplification of 𝒜 𝒜 \mathcal{A} , by which we mean an arrangement defined by the same vectors { a i } ⊂ 𝔱 d subscript 𝑎 𝑖 superscript 𝔱 𝑑 \{a_{i}\}\subset\mathfrak{t}^{d} , but with a different choice of r ∈ ( 𝔱 n ) ∗ 𝑟 superscript superscript 𝔱 𝑛 r\in(\mathfrak{t}^{n})^{*} , such that 𝒜 ~ ~ 𝒜 \tilde{\mathcal{A}} is simple. This corresponds to translating each of the hyperplanes in 𝒜 𝒜 \mathcal{A} away from the origin by some generic amount. Then 𝔐 ​ ( 𝒜 ~ ) 𝔐 ~ 𝒜 \mathfrak{M}(\tilde{\mathcal{A}}) maps T 𝑇 T -equivariantly to 𝔐 ​ ( 𝒜 ) 𝔐 𝒜 \mathfrak{M}(\mathcal{A}) by Equation ( 2 ), and Proposition 1.3.3 tell us that it is in fact an orbifold resolution, meaning a projective morphism, generically one-to-one, in which the source has at worst orbifold singularities. The structure of this map is studied extensively in PW . 1.4 Toward an abstract definition The definition of a hypertoric variety in Section 1.2 is constructive, modeled on the definition of toric varieties as GIT quotients of the form ℂ n / / α T k {\mathbb{C}}^{n}{/\!\!/}_{\!\!\alpha}T^{k} , or equivalently as symplectic quotients by compact tori. In the case of toric varieties, there are also abstract definitions. In the symplectic world, one defines a toric orbifold to be a symplectic orbifold of dimension 2 ​ d 2 𝑑 2d along with an effective Hamiltonian action of a compact d 𝑑 d -torus, and proves that any connected, compact toric orbifold arises from the symplectic quotient construction De , LT . In the algebraic world, one defines a toric variety to be a normal variety admitting a torus action with a dense orbit, and then proves that any semiprojective 4 4 4 Hausel and Sturmfels call a toric variety semiprojective if it is projective over its affinization and has at least one torus fixed point. toric variety with at worst orbifold singularities arises from the GIT construction. This idea goes back to Co , and can be found in this language in HSt , 2.6 . It is natural to ask for such an abstract definition and classification theorem for hypertoric varieties, either from the standpoint of symplectic algebraic geometry or that of hyperkähler geometry. In the hyperkähler setting, such a theorem was proven in Bi , 3,4 . Theorem 1.4.1 Any complete, connected, hyperkähler manifold of real dimension 4 ​ d 4 𝑑 4d which admits an effective, hyperhamiltonian action of the compact torus T ℝ d subscript superscript 𝑇 𝑑 ℝ T^{d}_{\mathbb{R}} is T ℝ d subscript superscript 𝑇 𝑑 ℝ T^{d}_{\mathbb{R}} -equivariantly diffeomorphic, and Taub-NUT deformation equivalent, to a hypertoric variety. Any such manifold with Euclidean volume growth is T ℝ d subscript superscript 𝑇 𝑑 ℝ T^{d}_{\mathbb{R}} -equivariantly isometric to a hypertoric variety. An analogous algebraic theorem has not been proven, but it should look something like the following. Conjecture 1.4.2 Any connected, symplectic, algebraic variety which is projective over its affinization and admits an effective, hamiltonian action of the algebraic torus T d superscript 𝑇 𝑑 T^{d} is equivariantly isomorphic to a Zariski open subset of a hypertoric variety.","role":"user"},{"content":"Abstract: Hypertoric varieties are quaternionic analogues of toric varieties, important for their interaction with the combinatorics of matroids as well as for their prominent place in the rapidly expanding field of algebraic symplectic and hyperkahler geometry. The aim of this survey is to give clear definitions and statements of known results, serving both as a reference and as a point of entry to this beautiful subject.","role":"assistant"}]

Black hole entropy from entanglement: a review

One of the most remarkable features of black hole physics is the realization that black holes behave as thermodynamic systems and possess entropy and temperature. The pioneering works in the field of black hole thermodynamics started with Bekenstein [ 1 ] , who argued that the universal applicability of the second law of thermodynamics rests on the fact that a black hole must possess an entropy ( S BH subscript 𝑆 BH S_{{}_{\rm BH}} ) proportional to the area ( 𝒜 H subscript 𝒜 H \mathcal{A}_{\rm H} ) of its horizon. The macroscopic properties of black holes were subsequently formalized by Bardeen, Carter and Hawking [ 2 ] as the four laws of black hole mechanics, in analogy with ordinary thermodynamics. They showed that (i) the surface gravity κ 𝜅 \kappa , which is the force applied by an observer at spatial infinity to hold a particle of unit mass in place at the location of the horizon, is same everywhere on the horizon for a stationary black hole — the statement of the zeroth law of black hole physics. (ii) The surface gravity κ 𝜅 \kappa of the black hole analogically resembles the temperature ( T H subscript 𝑇 H T_{\rm H} ) of the hole, in accordance with the interpretation of the horizon area as the black hole entropy. This may be conceived from the first law, which states that the change in mass (energy) of the black hole is proportional to the surface gravity times the change in horizon area. Hawking’s demonstration of black hole thermal radiation [ 3 ] paved the way to understand the physical significance of the temperature T H subscript 𝑇 H T_{\rm H} (and hence the entropy-area proportionality). Hawking showed that quantum effects in the background of a body collapsing to a Schwarzschild black hole leads to the emission of a thermal radiation at a characteristic temperature: T H = ( ℏ ​ c k B ) ​ κ 2 ​ π = ( ℏ ​ c 3 G ​ k B ) ​ 1 8 ​ π ​ M , subscript 𝑇 H Planck-constant-over-2-pi 𝑐 subscript 𝑘 𝐵 𝜅 2 𝜋 Planck-constant-over-2-pi superscript 𝑐 3 𝐺 subscript 𝑘 𝐵 1 8 𝜋 𝑀 T_{\rm H}=\left(\frac{\hbar c}{k_{{}_{B}}}\right)\frac{\kappa}{2\pi}=\left(\frac{\hbar c^{3}}{Gk_{{}_{B}}}\right)\frac{1}{8\pi M}\,, (1) where G 𝐺 G is the Newton’s constant in four dimensions, k B subscript 𝑘 𝐵 k_{{}_{B}} is the Boltzmann constant, and M 𝑀 M is the mass of the black hole. The factor of proportionality between temperature and surface gravity (and as such between entropy and area) gets fixed in Hawking’s derivation [ 3 ] , thus leading to the Bekenstein-Hawking area law: S BH = ( k B 4 ) ​ 𝒜 H ℓ Pl 2 , subscript 𝑆 BH subscript 𝑘 𝐵 4 subscript 𝒜 H superscript subscript ℓ Pl 2 S_{{}_{\rm BH}}=\left(\frac{k_{{}_{B}}}{4}\right)\frac{\mathcal{A}_{\rm H}}{\ell_{{}_{\rm Pl}}^{2}}\quad\,, (2) where ℓ Pl = G ​ ℏ / c 3 subscript ℓ Pl 𝐺 Planck-constant-over-2-pi superscript 𝑐 3 {\ell_{{}_{\rm Pl}}}=\sqrt{G\hbar/c^{3}} is the four dimensional Planck length. Black-hole thermodynamics and, in particular, black-hole entropy raises several important questions which can be broadly classified into two categories: • Gravitational collapse leading to black-hole formation (i) What is the dynamical mechanism that makes S BH subscript 𝑆 BH S_{{}_{\rm BH}} a universal function, independent of the black-hole’s past history and detailed internal condition? (ii) How does a pure state evolve into a mixed (thermal) state? Is there a information loss due to the formation of black-hole and Hawking process? Does the usual quantum mechanics need to be modified in the context of black-holes? (iii) Can quantum theory of gravity remove the formation of space-time singularity due to the gravitational collapse? • Near thermodynamical equilibrium (iv) Unlike other thermodynamical systems, why is black-hole entropy non-extensive? i. e. why S BH subscript 𝑆 BH S_{{}_{\rm BH}} is proportional to area and not volume? (v) Why is the black-hole entropy large? 4 4 4 In order to see that, let us compare S BH subscript 𝑆 BH S_{{}_{\rm BH}} with the entropy of the current universe. The entropy of the universe within our horizon today [ 4 ] is S Univ ∼ 10 87 ​ ( T ​ h − 1 2.75 ​ K ) similar-to subscript 𝑆 Univ superscript 10 87 𝑇 superscript ℎ 1 2.75 𝐾 S_{{}_{\rm Univ}}\sim 10^{87}\left(\frac{Th^{-1}}{2.75K}\right) (3) where T 𝑇 T is the temperature of the universe, h ℎ h is of the order unity. On the other hand the entropy of a Schwarzschild black-hole is S BH ∼ 10 77 ​ ( M M ⊙ ) 2 similar-to subscript 𝑆 BH superscript 10 77 superscript 𝑀 subscript 𝑀 direct-product 2 S_{{}_{\rm BH}}\sim 10^{77}\left(\frac{M}{M_{\odot}}\right)^{2} (4) where M ⊙ subscript 𝑀 direct-product M_{\odot} is the solar mass. Hence, a couple of hundred thousand solar mass black holes can contain as much entropy as is free in the entire universe. There is increasing evidence that super-massive black holes exist at the center of many galaxies. By now we know that a large fraction of galaxies — of the total (of order) 10 11 superscript 10 11 10^{11} — contain such super-massive black holes with mass range 10 6 < M BH / M ⊙ < 10 10 superscript 10 6 subscript 𝑀 BH subscript 𝑀 direct-product superscript 10 10 10^{6}<M_{{}_{\rm BH}}/M_{\odot}<10^{10} . This implies that the entropy of the black holes dominates all other sources of entropy. Hence, understanding the origins of black-hole entropy may help us explain the entropy budget of the universe. (vi) How S BH subscript 𝑆 BH S_{{}_{\rm BH}} concords with the standard view of the statistical origin? What are the black-hole microstates? S = ? k B ​ ln ⁡ ( # of microstates ) superscript ? 𝑆 subscript 𝑘 𝐵 # of microstates S\stackrel{{\scriptstyle?}}{{=}}k_{{}_{B}}\ln\left({\mbox{\# of microstates}}\right) (vii) Are there corrections to S BH subscript 𝑆 BH S_{{}_{\rm BH}} ? If there are, how generic are they? (viii) Where are the microscopic degrees of freedom responsible for black-hole entropy located? These questions often seem related, which a correct theory of quantum gravity is expected to address. In the absence of a workable theory of quantum gravity, there have been several approaches which address one or several of the above questions. Most of the effort in the literature, as in this review, has been to understand the microscopic statistical mechanical origin of S BH subscript 𝑆 BH S_{{}_{\rm BH}} assuming that the black-hole is in a (near) thermal equilibrium or not interacting with surroundings. The various approaches may broadly be classified into two categories [ 5 ] : (a) the ones that associate S BH subscript 𝑆 BH S_{{}_{\rm BH}} with fundamental states such as strings, D 𝐷 D -Branes, spin-networks, etc. [ 6 , 7 ] , and (b) the other that associate S BH subscript 𝑆 BH S_{{}_{\rm BH}} with quantum fields in a fixed BH background, like the brick-wall model [ 8 ] , the quantum entanglement of modes inside and outside of the horizon [ 9 , 10 , 11 , 12 ] and the Noether charge [ 13 ] . As mentioned above, although, none of these approaches can be considered to be complete; all of them — within their domains of applicability — by counting certain microscopic states yield ( 2 ). This is in complete contrast to other physical systems, such as ideal gas, where quantum degrees of freedom (DOF) are uniquely identified and lead to the classical thermodynamic entropy. The above discussion raises three important questions which we try to address in this review: 1. Is it sufficient for an approach to reproduce ( 2 ) or need to go beyond S BH subscript 𝑆 BH S_{{}_{\rm BH}} ? As we know, S BH subscript 𝑆 BH S_{{}_{\rm BH}} is a semi-classical result and there are strong indications that Eq. ( 2 ) is valid for large black holes [i.e. 𝒜 H ≫ ℓ Pl much-greater-than subscript 𝒜 H subscript ℓ Pl \mathcal{A}_{\rm H}\gg\ell_{{}_{\rm Pl}} ]. However, it is not clear, whether this relation will continue to hold for the Planck-size black-holes. Besides, there is no reason to expect that S BH subscript 𝑆 BH S_{{}_{\rm BH}} to be the whole answer for a correct theory of quantum gravity. In order to have a better understanding of black-hole entropy, it is imperative for any approach to go beyond S BH subscript 𝑆 BH S_{{}_{\rm BH}} and identify the subleading corrections. 2. Are the quantum DOF that contribute to S BH subscript 𝑆 BH S_{{}_{\rm BH}} and its subleading corrections, identical or different? In general, the quantum DOF can be different. However, several approaches in the literature [ 14 ] that do lead to subleading corrections either assume that the quantum DOF are identical or do not disentangle DOF contribution to S BH subscript 𝑆 BH S_{{}_{\rm BH}} and the subleading corrections. 3. Can one locate the quantum DOF that give rise to S BH subscript 𝑆 BH S_{{}_{\rm BH}} and its subleading corrections? More specifically, can we determine to what extend do the quantum DOF close to the horizon or far from the horizon contribute to the S BH subscript 𝑆 BH S_{{}_{\rm BH}} and its corrections. Depending on the approach, one either counts certain DOF on the horizon, or abstract DOF related to the black hole, and there does not appear to be a consensus about which DOF are relevant or about their precise location [ 5 ] . In this review, using the approach of entanglement of modes across the black-hole horizon, we address the above three issues and show that: 1. the entanglement leads to generic power-law corrections to the Bekenstein-Hawking entropy, of the form [ 15 ] : S = σ 0 ​ ( 𝒜 a 2 ) + σ 1 ​ ( 𝒜 a 2 ) − ν ; ( 0 < ν < 1 ) , 𝑆 subscript 𝜎 0 𝒜 superscript 𝑎 2 subscript 𝜎 1 superscript 𝒜 superscript 𝑎 2 𝜈 0 𝜈 1 S=\sigma_{0}\left(\frac{\mathcal{A}}{a^{2}}\right)+\sigma_{1}\left(\frac{\mathcal{A}}{a^{2}}\right)^{-\nu}~{};~{}~{}~{}~{}(0<\nu<1)\,, (5) where s 0 , σ 1 subscript 𝑠 0 subscript 𝜎 1 s_{0},\sigma_{1} are constants and a 𝑎 a is the lattice spacing, 2. the quantum degrees of freedom that lead to S BH subscript 𝑆 BH S_{{}_{\rm BH}} and subleading corrections are different, and 3. the contribution to S BH subscript 𝑆 BH S_{{}_{\rm BH}} comes from the region close to the horizon while the subleading corrections have a larger contribution from the region far from the horizon [ 16 ] . This review is organized as follows: in the next section we briefly review the basic features of quantum entanglement and the concept of entanglement entropy. In sec. ( 3 ), we provide a heuristic picture of the link between the entanglement entropy and black-hole entropy. In sec. 4 we discuss the procedure and assumptions to compute the entanglement entropy of a scalar field in black-hole space-times. In sec. 5 we review the cases where the scalar field is either in its ground state, or in a generalized coherent state, or in a class of squeezed states — for all these cases the area law is found to hold. In sec. 6 , we show that the for the superposition of ground and 1-particle state, the entanglement entropy has a subleading power-law corrections to the AL. In sec. 7 we study the locations scalar field DOF that are responsible for the entanglement entropy. In sec. 8 we examine the entanglement entropy due to a massive scalar field and compare it with that obtained for a massless scalar field. We conclude with a summary and open questions in sec. 9 . In Appendix A we discuss the relevance for considering massless or massive scalar field for computing the entanglement entropy of black holes, from the perspective of gravitational perturbations in static black hole space-times. In Appendix B we discuss the steps to obtain the Hamiltonian of a scalar field in the static black-hole background, which in Lemaître coordinates, and at a fixed Lemaître time, reduces to the scalar field Hamiltonian in flat space-time. Before we proceed, we outline our conventions and notations: We work in four-dimensions and our signature for the metric is ( − , + , + , + ) (-,+,+,+) . Hereafter, we use units with k B = c = ℏ = 1 subscript 𝑘 𝐵 𝑐 Planck-constant-over-2-pi 1 k_{{}_{B}}=c=\hbar=1 and set the Planck mass M Pl 2 = 1 / ( 16 ​ π ​ G ) superscript subscript 𝑀 Pl 2 1 16 𝜋 𝐺 M_{{}_{\rm Pl}}^{2}=1/(16\pi G) . The quantum field φ 𝜑 \varphi is a minimally coupled scalar field.

We review aspects of the thermodynamics of black holes and in particular take into account the fact that the quantum entanglement between the degrees of freedom of a scalar field, traced inside the event horizon, can be the origin of black hole entropy. The main reason behind such a plausibility is that the well-known Bekenstein-Hawking entropy-area proportionality -- the so-called `area law' of black hole physics -- holds for entanglement entropy as well, provided the scalar field is in its ground state, or in other minimum uncertainty states, such as a generic coherent state or squeezed state. However, when the field is either in an excited state or in a state which is a superposition of ground and excited states, a power-law correction to the area law is shown to exist. Such a correction term falls off with increasing area, so that eventually the area law is recovered for large enough horizon area. On ascertaining the location of the microscopic degrees of freedom that lead to the entanglement entropy of black holes, it is found that although the degrees of freedom close to the horizon contribute most to the total entropy, the contributions from those that are far from the horizon are more significant for excited/superposed states than for the ground state. Thus, the deviations from the area law for excited/superposed states may, in a way, be attributed to the far-away degrees of freedom. Finally, taking the scalar field (which is traced over) to be massive, we explore the changes on the area law due to the mass. Although most of our computations are done in flat space-time with a hypothetical spherical region, considered to be the analogue of the horizon, we show that our results hold as well in curved space-times representing static asymptotically flat spherical black holes with single horizon.

[{"content":"Help me to generate the abstract of a survey paper given the title:*Black hole entropy from entanglement: a review* and the introduction:One of the most remarkable features of black hole physics is the realization that black holes behave as thermodynamic systems and possess entropy and temperature. The pioneering works in the field of black hole thermodynamics started with Bekenstein 1 , who argued that the universal applicability of the second law of thermodynamics rests on the fact that a black hole must possess an entropy ( S BH subscript 𝑆 BH S_{{}_{\rm BH}} ) proportional to the area ( 𝒜 H subscript 𝒜 H \mathcal{A}_{\rm H} ) of its horizon. The macroscopic properties of black holes were subsequently formalized by Bardeen, Carter and Hawking 2 as the four laws of black hole mechanics, in analogy with ordinary thermodynamics. They showed that (i) the surface gravity κ 𝜅 \kappa , which is the force applied by an observer at spatial infinity to hold a particle of unit mass in place at the location of the horizon, is same everywhere on the horizon for a stationary black hole — the statement of the zeroth law of black hole physics. (ii) The surface gravity κ 𝜅 \kappa of the black hole analogically resembles the temperature ( T H subscript 𝑇 H T_{\rm H} ) of the hole, in accordance with the interpretation of the horizon area as the black hole entropy. This may be conceived from the first law, which states that the change in mass (energy) of the black hole is proportional to the surface gravity times the change in horizon area. Hawkings demonstration of black hole thermal radiation 3 paved the way to understand the physical significance of the temperature T H subscript 𝑇 H T_{\rm H} (and hence the entropy-area proportionality). Hawking showed that quantum effects in the background of a body collapsing to a Schwarzschild black hole leads to the emission of a thermal radiation at a characteristic temperature: T H = ( ℏ ​ c k B ) ​ κ 2 ​ π = ( ℏ ​ c 3 G ​ k B ) ​ 1 8 ​ π ​ M , subscript 𝑇 H Planck-constant-over-2-pi 𝑐 subscript 𝑘 𝐵 𝜅 2 𝜋 Planck-constant-over-2-pi superscript 𝑐 3 𝐺 subscript 𝑘 𝐵 1 8 𝜋 𝑀 T_{\rm H}=\left(\frac{\hbar c}{k_{{}_{B}}}\right)\frac{\kappa}{2\pi}=\left(\frac{\hbar c^{3}}{Gk_{{}_{B}}}\right)\frac{1}{8\pi M}\,, (1) where G 𝐺 G is the Newtons constant in four dimensions, k B subscript 𝑘 𝐵 k_{{}_{B}} is the Boltzmann constant, and M 𝑀 M is the mass of the black hole. The factor of proportionality between temperature and surface gravity (and as such between entropy and area) gets fixed in Hawkings derivation 3 , thus leading to the Bekenstein-Hawking area law: S BH = ( k B 4 ) ​ 𝒜 H ℓ Pl 2 , subscript 𝑆 BH subscript 𝑘 𝐵 4 subscript 𝒜 H superscript subscript ℓ Pl 2 S_{{}_{\rm BH}}=\left(\frac{k_{{}_{B}}}{4}\right)\frac{\mathcal{A}_{\rm H}}{\ell_{{}_{\rm Pl}}^{2}}\quad\,, (2) where ℓ Pl = G ​ ℏ / c 3 subscript ℓ Pl 𝐺 Planck-constant-over-2-pi superscript 𝑐 3 {\ell_{{}_{\rm Pl}}}=\sqrt{G\hbar/c^{3}} is the four dimensional Planck length. Black-hole thermodynamics and, in particular, black-hole entropy raises several important questions which can be broadly classified into two categories: • Gravitational collapse leading to black-hole formation (i) What is the dynamical mechanism that makes S BH subscript 𝑆 BH S_{{}_{\rm BH}} a universal function, independent of the black-holes past history and detailed internal condition? (ii) How does a pure state evolve into a mixed (thermal) state? Is there a information loss due to the formation of black-hole and Hawking process? Does the usual quantum mechanics need to be modified in the context of black-holes? (iii) Can quantum theory of gravity remove the formation of space-time singularity due to the gravitational collapse? • Near thermodynamical equilibrium (iv) Unlike other thermodynamical systems, why is black-hole entropy non-extensive? i. e. why S BH subscript 𝑆 BH S_{{}_{\rm BH}} is proportional to area and not volume? (v) Why is the black-hole entropy large? 4 4 4 In order to see that, let us compare S BH subscript 𝑆 BH S_{{}_{\rm BH}} with the entropy of the current universe. The entropy of the universe within our horizon today 4 is S Univ ∼ 10 87 ​ ( T ​ h − 1 2.75 ​ K ) similar-to subscript 𝑆 Univ superscript 10 87 𝑇 superscript ℎ 1 2.75 𝐾 S_{{}_{\rm Univ}}\sim 10^{87}\left(\frac{Th^{-1}}{2.75K}\right) (3) where T 𝑇 T is the temperature of the universe, h ℎ h is of the order unity. On the other hand the entropy of a Schwarzschild black-hole is S BH ∼ 10 77 ​ ( M M ⊙ ) 2 similar-to subscript 𝑆 BH superscript 10 77 superscript 𝑀 subscript 𝑀 direct-product 2 S_{{}_{\rm BH}}\sim 10^{77}\left(\frac{M}{M_{\odot}}\right)^{2} (4) where M ⊙ subscript 𝑀 direct-product M_{\odot} is the solar mass. Hence, a couple of hundred thousand solar mass black holes can contain as much entropy as is free in the entire universe. There is increasing evidence that super-massive black holes exist at the center of many galaxies. By now we know that a large fraction of galaxies — of the total (of order) 10 11 superscript 10 11 10^{11} — contain such super-massive black holes with mass range 10 6 < M BH / M ⊙ < 10 10 superscript 10 6 subscript 𝑀 BH subscript 𝑀 direct-product superscript 10 10 10^{6}<M_{{}_{\rm BH}}/M_{\odot}<10^{10} . This implies that the entropy of the black holes dominates all other sources of entropy. Hence, understanding the origins of black-hole entropy may help us explain the entropy budget of the universe. (vi) How S BH subscript 𝑆 BH S_{{}_{\rm BH}} concords with the standard view of the statistical origin? What are the black-hole microstates? S = ? k B ​ ln ⁡ ( # of microstates ) superscript ? 𝑆 subscript 𝑘 𝐵 # of microstates S\stackrel{{\scriptstyle?}}{{=}}k_{{}_{B}}\ln\left({\mbox{\# of microstates}}\right) (vii) Are there corrections to S BH subscript 𝑆 BH S_{{}_{\rm BH}} ? If there are, how generic are they? (viii) Where are the microscopic degrees of freedom responsible for black-hole entropy located? These questions often seem related, which a correct theory of quantum gravity is expected to address. In the absence of a workable theory of quantum gravity, there have been several approaches which address one or several of the above questions. Most of the effort in the literature, as in this review, has been to understand the microscopic statistical mechanical origin of S BH subscript 𝑆 BH S_{{}_{\rm BH}} assuming that the black-hole is in a (near) thermal equilibrium or not interacting with surroundings. The various approaches may broadly be classified into two categories 5 : (a) the ones that associate S BH subscript 𝑆 BH S_{{}_{\rm BH}} with fundamental states such as strings, D 𝐷 D -Branes, spin-networks, etc. 6 , 7 , and (b) the other that associate S BH subscript 𝑆 BH S_{{}_{\rm BH}} with quantum fields in a fixed BH background, like the brick-wall model 8 , the quantum entanglement of modes inside and outside of the horizon 9 , 10 , 11 , 12 and the Noether charge 13 . As mentioned above, although, none of these approaches can be considered to be complete; all of them — within their domains of applicability — by counting certain microscopic states yield ( 2 ). This is in complete contrast to other physical systems, such as ideal gas, where quantum degrees of freedom (DOF) are uniquely identified and lead to the classical thermodynamic entropy. The above discussion raises three important questions which we try to address in this review: 1. Is it sufficient for an approach to reproduce ( 2 ) or need to go beyond S BH subscript 𝑆 BH S_{{}_{\rm BH}} ? As we know, S BH subscript 𝑆 BH S_{{}_{\rm BH}} is a semi-classical result and there are strong indications that Eq. ( 2 ) is valid for large black holes i.e. 𝒜 H ≫ ℓ Pl much-greater-than subscript 𝒜 H subscript ℓ Pl \mathcal{A}_{\rm H}\gg\ell_{{}_{\rm Pl}} . However, it is not clear, whether this relation will continue to hold for the Planck-size black-holes. Besides, there is no reason to expect that S BH subscript 𝑆 BH S_{{}_{\rm BH}} to be the whole answer for a correct theory of quantum gravity. In order to have a better understanding of black-hole entropy, it is imperative for any approach to go beyond S BH subscript 𝑆 BH S_{{}_{\rm BH}} and identify the subleading corrections. 2. Are the quantum DOF that contribute to S BH subscript 𝑆 BH S_{{}_{\rm BH}} and its subleading corrections, identical or different? In general, the quantum DOF can be different. However, several approaches in the literature 14 that do lead to subleading corrections either assume that the quantum DOF are identical or do not disentangle DOF contribution to S BH subscript 𝑆 BH S_{{}_{\rm BH}} and the subleading corrections. 3. Can one locate the quantum DOF that give rise to S BH subscript 𝑆 BH S_{{}_{\rm BH}} and its subleading corrections? More specifically, can we determine to what extend do the quantum DOF close to the horizon or far from the horizon contribute to the S BH subscript 𝑆 BH S_{{}_{\rm BH}} and its corrections. Depending on the approach, one either counts certain DOF on the horizon, or abstract DOF related to the black hole, and there does not appear to be a consensus about which DOF are relevant or about their precise location 5 . In this review, using the approach of entanglement of modes across the black-hole horizon, we address the above three issues and show that: 1. the entanglement leads to generic power-law corrections to the Bekenstein-Hawking entropy, of the form 15 : S = σ 0 ​ ( 𝒜 a 2 ) + σ 1 ​ ( 𝒜 a 2 ) − ν ; ( 0 < ν < 1 ) , 𝑆 subscript 𝜎 0 𝒜 superscript 𝑎 2 subscript 𝜎 1 superscript 𝒜 superscript 𝑎 2 𝜈 0 𝜈 1 S=\sigma_{0}\left(\frac{\mathcal{A}}{a^{2}}\right)+\sigma_{1}\left(\frac{\mathcal{A}}{a^{2}}\right)^{-\nu}~{};~{}~{}~{}~{}(0<\nu<1)\,, (5) where s 0 , σ 1 subscript 𝑠 0 subscript 𝜎 1 s_{0},\sigma_{1} are constants and a 𝑎 a is the lattice spacing, 2. the quantum degrees of freedom that lead to S BH subscript 𝑆 BH S_{{}_{\rm BH}} and subleading corrections are different, and 3. the contribution to S BH subscript 𝑆 BH S_{{}_{\rm BH}} comes from the region close to the horizon while the subleading corrections have a larger contribution from the region far from the horizon 16 . This review is organized as follows: in the next section we briefly review the basic features of quantum entanglement and the concept of entanglement entropy. In sec. ( 3 ), we provide a heuristic picture of the link between the entanglement entropy and black-hole entropy. In sec. 4 we discuss the procedure and assumptions to compute the entanglement entropy of a scalar field in black-hole space-times. In sec. 5 we review the cases where the scalar field is either in its ground state, or in a generalized coherent state, or in a class of squeezed states — for all these cases the area law is found to hold. In sec. 6 , we show that the for the superposition of ground and 1-particle state, the entanglement entropy has a subleading power-law corrections to the AL. In sec. 7 we study the locations scalar field DOF that are responsible for the entanglement entropy. In sec. 8 we examine the entanglement entropy due to a massive scalar field and compare it with that obtained for a massless scalar field. We conclude with a summary and open questions in sec. 9 . In Appendix A we discuss the relevance for considering massless or massive scalar field for computing the entanglement entropy of black holes, from the perspective of gravitational perturbations in static black hole space-times. In Appendix B we discuss the steps to obtain the Hamiltonian of a scalar field in the static black-hole background, which in Lemaître coordinates, and at a fixed Lemaître time, reduces to the scalar field Hamiltonian in flat space-time. Before we proceed, we outline our conventions and notations: We work in four-dimensions and our signature for the metric is ( − , + , + , + ) (-,+,+,+) . Hereafter, we use units with k B = c = ℏ = 1 subscript 𝑘 𝐵 𝑐 Planck-constant-over-2-pi 1 k_{{}_{B}}=c=\hbar=1 and set the Planck mass M Pl 2 = 1 / ( 16 ​ π ​ G ) superscript subscript 𝑀 Pl 2 1 16 𝜋 𝐺 M_{{}_{\rm Pl}}^{2}=1/(16\pi G) . The quantum field φ 𝜑 \varphi is a minimally coupled scalar field.","role":"user"},{"content":"Abstract: We review aspects of the thermodynamics of black holes and in particular take into account the fact that the quantum entanglement between the degrees of freedom of a scalar field, traced inside the event horizon, can be the origin of black hole entropy. The main reason behind such a plausibility is that the well-known Bekenstein-Hawking entropy-area proportionality -- the so-called `area law' of black hole physics -- holds for entanglement entropy as well, provided the scalar field is in its ground state, or in other minimum uncertainty states, such as a generic coherent state or squeezed state. However, when the field is either in an excited state or in a state which is a superposition of ground and excited states, a power-law correction to the area law is shown to exist. Such a correction term falls off with increasing area, so that eventually the area law is recovered for large enough horizon area. On ascertaining the location of the microscopic degrees of freedom that lead to the entanglement entropy of black holes, it is found that although the degrees of freedom close to the horizon contribute most to the total entropy, the contributions from those that are far from the horizon are more significant for excited/superposed states than for the ground state. Thus, the deviations from the area law for excited/superposed states may, in a way, be attributed to the far-away degrees of freedom. Finally, taking the scalar field (which is traced over) to be massive, we explore the changes on the area law due to the mass. Although most of our computations are done in flat space-time with a hypothetical spherical region, considered to be the analogue of the horizon, we show that our results hold as well in curved space-times representing static asymptotically flat spherical black holes with single horizon.","role":"assistant"}]

A review of wildland fire spread modelling, 1990-present 3: Mathematical analogues and simulation models

1.1 History The ultimate aim of any prediction system is to enable an end user to carry out useful predictions. A useful prediction is one that helps the user achieve a particular aim. In the field of wildland fire behaviour, that aim is primarily to stop the spread of the fire or to at least reduce its impact on life and property. The earliest efforts at wildland fire behaviour prediction concentrated on predicting the likely danger posed by a particular fire or set of conditions prior to the outbreak of a fire. These fire danger systems were used to set the level of preparedness of suppression resources or to aid in the identification of the onset of bad fire weather for the purpose of calling total bans on intentionally lit fires. In addition to a subjective index of fire danger, many of early fire danger systems also provided a prediction of the likely spread of a fire, as a prediction of the rate of forward spread of the fire, the rate of perimeter increase or rate of area increase. In many cases, these predictions were used by users to plot the likely spread of the fire on a map, thereby putting the prediction in context with geographic features or resource locations, and constituted the first form of fire spread simulation. Because much of the development of the early wildland fire behaviour models was carried out by those organisations intended to use the systems, the level of sophistication of the model tended to match the level of sophistication of the technology used to implement it. Thus, the early fire spread models provided only a single dimension prediction (generally the forward rate of spread of the headfire) which could be easily plotted on a map and extrapolated over time. While modern wildland fire spread modelling has expanded to include physical approaches ( Sullivan, 2007a, ) , all modern operational fire spread models have continued this empirical approach in the form of one-dimensional spread models ( Sullivan, 2007b, ) . Much of the development of technology for implementing the models in a simulation environment has concentrated on methods for converting the one-dimensional linear model of fire spread to that of two-dimensional planar models of fire spread. In parallel with approaches to implement existing empirical models of fire spread have been efforts to approach the simulation of fire spread across the landscape from a holistic perspective. This has resulted in the use of methods other than those directly related to the observation, measurement and modelling of fire behaviour. These methods are mathematical in nature and provide an analogue of fire behaviour. Many of these approaches have also paralleled the development of the computer as a computational device to undertake the calculations required to implement the mathematical concepts. An increase in the capabilities of remote sensing, geographical information systems and computing power during the 1990s resulted in a revival in the interest of fire behaviour modelling as applied to the prediction of spread across the landscape. 1.2 Background This series of review papers endeavours to comprehensively and critically review the extensive range of modelling work that has been conducted in recent years. The range of methods that have been undertaken over the years represents a continuous spectrum of possible modelling (Karplus,, 1977 ) , ranging from the purely physical (those that are based on fundamental understanding of the physics and chemistry involved in the behaviour of a wildland fire) through to the purely empirical (those that have been based on phenomenological description or statistical regression of fire behaviour). In between is a continuous meld of approaches from one end of the spectrum or the other. Weber, ( 1991 ) in his comprehensive review of physical wildland fire modelling proposed a system by which models were described as physical, empirical or statistical, depending on whether they account for different modes of heat transfer, make no distinction between different heat transfer modes, or involve no physics at all. Pastor et al., ( 2003 ) proposed descriptions of theoretical, empirical and semi-empirical, again depending on whether the model was based on purely physical understanding, of a statistical nature with no physical understanding, or a combination of both. Grishin, ( 1997 ) divided models into two classes, deterministic or stochastic-statistical. However, these schemes are rather limited given the combination of possible approaches and, given that describing a model as semi-empirical or semi-physical is a ‘glass half-full or half-empty’ subjective issue, a more comprehensive and complete convection was required. Thus, this review series is divided into three broad categories: Physical and quasi-physical models; Empirical and quasi-empirical models; and Simulation and Mathematical analogous models. In this context, a physical model is one that attempts to represent both the physics and chemistry of fire spread; a quasi-physical model attempts to represent only the physics. An empirical model is one that contains no physical basis at all (generally only statistical in nature), a quasi-empirical model is one that uses some form of physical framework upon which to base the statistical modelling chosen. Empirical models are further subdivided into field-based and laboratory-based. Simulation models are those that implement the preceding types of models in a simulation rather than modelling context. Mathematical analogous models are those that utilise a mathematical precept rather than a physical one for the modelling of the spread of wildland fire. Since 1990, there has been rapid development in the field of spatial data analysis, e.g. geographic information systems and remote sensing. As a result, I have limited this review to works published since 1990. However, as much of the work that will be discussed derives or continues from work carried out prior to 1990, such work will be included much less comprehensively in order to provide context. 1.3 Previous reviews Many of the reviews that have been published in recent years have been for audiences other than wildland fire researchers and conducted by people without an established background in the field. Indeed, many of the reviews read like purchase notes by people shopping around for the best fire spread model to implement in their part of the world for their particular purpose. Recent reviews (e.g. Perry, ( 1998 ); Pastor et al., ( 2003 ) ; etc), while endeavouring to be comprehensive, have offered only superficial and cursory inspections of the models presented. Morvan et al., ( 2004 ) takes a different line by analysing a much broader spectrum of models in some detail and concludes that no single approach is going to be suitable for all uses. While the recent reviews provide an overview of the models and approaches that have been undertaken around the world, mention must be made of significant reviews published much earlier that discussed the processes in wildland fire propagation themselves. Foremost is the work of Williams, ( 1982 ) , which comprehensively covers the phenomenology of both wildland and urban fire, the physics and chemistry of combustion, and is recommended reading for the beginner. The earlier work of Emmons, ( 1963 , 1966 ) and Lee, ( 1972 ) provides a sound background on the advances made during the post-war boom era. Grishin, ( 1997 ) provides an extensive review of the work conducted in Russia in the 1970s, 80s and 90s. The first paper in this series discussed those models based upon the fundamental principles of the physics and chemistry of wildland fire behaviour. The second paper in the series discussed those models based directly upon only statistical analysis of fire behaviour observations or models that utilise some form of physical framework upon which the statistical analysis of observations have been based. Particular distinction was made between observations of the behaviour of fires in the strictly controlled and artificial conditions of the laboratory and those observed in the field under more naturally occurring conditions. This paper, the final in the series, focuses upon models concerned only with the simulation of fire spread over the landscape and models that utilise mathematical conceits analogous to fire spread but which have no real-world connection to fire. The former generally utilise a pre-existing fire spread model (which can be physical, quasi-physical, quasi-empirical or empirical) and implements it in such a way as to simulate the spread of fire across a landscape. As such, it is generally based upon a geographic information system (GIS) of some description to represent the landscape and uses a propagation algorithm to spread the fire perimeter across it. The latter models are for the most part based upon accepted mathematical functions or concepts that have been applied to wildland fire spread but are not derived from any understanding of wildland fire behaviour. Rather, these models utilise apparent similarities between wildland fire behaviour and the behaviour of these concepts within certain limited contexts. Because of this, these mathematical concepts could equally be applied to other fields of endeavour and, for the most part have been, to greater or lesser success. Unlike the preceding entries in this series, this paper is segmented by the approaches taken by the various authors, not by the authors or their organisations, given the broad range of authors that in some instances have taken similar approaches.

In recent years, advances in computational power and spatial data analysis (GIS, remote sensing, etc) have led to an increase in attempts to model the spread and behvaiour of wildland fires across the landscape. This series of review papers endeavours to critically and comprehensively review all types of surface fire spread models developed since 1990. This paper reviews models of a simulation or mathematical analogue nature. Most simulation models are implementations of existing empirical or quasi-empirical models and their primary function is to convert these generally one dimensional models to two dimensions and then propagate a fire perimeter across a modelled landscape. Mathematical analogue models are those that are based on some mathematical conceit (rather than a physical representation of fire spread) that coincidentally simulates the spread of fire. Other papers in the series review models of an physical or quasi-physical nature and empirical or quasi-empirical nature. Many models are extensions or refinements of models developed before 1990. Where this is the case, these models are also discussed but much less comprehensively.

[{"content":"Help me to generate the abstract of a survey paper given the title:*A review of wildland fire spread modelling, 1990-present 3: Mathematical analogues and simulation models* and the introduction:1.1 History The ultimate aim of any prediction system is to enable an end user to carry out useful predictions. A useful prediction is one that helps the user achieve a particular aim. In the field of wildland fire behaviour, that aim is primarily to stop the spread of the fire or to at least reduce its impact on life and property. The earliest efforts at wildland fire behaviour prediction concentrated on predicting the likely danger posed by a particular fire or set of conditions prior to the outbreak of a fire. These fire danger systems were used to set the level of preparedness of suppression resources or to aid in the identification of the onset of bad fire weather for the purpose of calling total bans on intentionally lit fires. In addition to a subjective index of fire danger, many of early fire danger systems also provided a prediction of the likely spread of a fire, as a prediction of the rate of forward spread of the fire, the rate of perimeter increase or rate of area increase. In many cases, these predictions were used by users to plot the likely spread of the fire on a map, thereby putting the prediction in context with geographic features or resource locations, and constituted the first form of fire spread simulation. Because much of the development of the early wildland fire behaviour models was carried out by those organisations intended to use the systems, the level of sophistication of the model tended to match the level of sophistication of the technology used to implement it. Thus, the early fire spread models provided only a single dimension prediction (generally the forward rate of spread of the headfire) which could be easily plotted on a map and extrapolated over time. While modern wildland fire spread modelling has expanded to include physical approaches ( Sullivan, 2007a, ) , all modern operational fire spread models have continued this empirical approach in the form of one-dimensional spread models ( Sullivan, 2007b, ) . Much of the development of technology for implementing the models in a simulation environment has concentrated on methods for converting the one-dimensional linear model of fire spread to that of two-dimensional planar models of fire spread. In parallel with approaches to implement existing empirical models of fire spread have been efforts to approach the simulation of fire spread across the landscape from a holistic perspective. This has resulted in the use of methods other than those directly related to the observation, measurement and modelling of fire behaviour. These methods are mathematical in nature and provide an analogue of fire behaviour. Many of these approaches have also paralleled the development of the computer as a computational device to undertake the calculations required to implement the mathematical concepts. An increase in the capabilities of remote sensing, geographical information systems and computing power during the 1990s resulted in a revival in the interest of fire behaviour modelling as applied to the prediction of spread across the landscape. 1.2 Background This series of review papers endeavours to comprehensively and critically review the extensive range of modelling work that has been conducted in recent years. The range of methods that have been undertaken over the years represents a continuous spectrum of possible modelling (Karplus,, 1977 ) , ranging from the purely physical (those that are based on fundamental understanding of the physics and chemistry involved in the behaviour of a wildland fire) through to the purely empirical (those that have been based on phenomenological description or statistical regression of fire behaviour). In between is a continuous meld of approaches from one end of the spectrum or the other. Weber, ( 1991 ) in his comprehensive review of physical wildland fire modelling proposed a system by which models were described as physical, empirical or statistical, depending on whether they account for different modes of heat transfer, make no distinction between different heat transfer modes, or involve no physics at all. Pastor et al., ( 2003 ) proposed descriptions of theoretical, empirical and semi-empirical, again depending on whether the model was based on purely physical understanding, of a statistical nature with no physical understanding, or a combination of both. Grishin, ( 1997 ) divided models into two classes, deterministic or stochastic-statistical. However, these schemes are rather limited given the combination of possible approaches and, given that describing a model as semi-empirical or semi-physical is a glass half-full or half-empty subjective issue, a more comprehensive and complete convection was required. Thus, this review series is divided into three broad categories: Physical and quasi-physical models; Empirical and quasi-empirical models; and Simulation and Mathematical analogous models. In this context, a physical model is one that attempts to represent both the physics and chemistry of fire spread; a quasi-physical model attempts to represent only the physics. An empirical model is one that contains no physical basis at all (generally only statistical in nature), a quasi-empirical model is one that uses some form of physical framework upon which to base the statistical modelling chosen. Empirical models are further subdivided into field-based and laboratory-based. Simulation models are those that implement the preceding types of models in a simulation rather than modelling context. Mathematical analogous models are those that utilise a mathematical precept rather than a physical one for the modelling of the spread of wildland fire. Since 1990, there has been rapid development in the field of spatial data analysis, e.g. geographic information systems and remote sensing. As a result, I have limited this review to works published since 1990. However, as much of the work that will be discussed derives or continues from work carried out prior to 1990, such work will be included much less comprehensively in order to provide context. 1.3 Previous reviews Many of the reviews that have been published in recent years have been for audiences other than wildland fire researchers and conducted by people without an established background in the field. Indeed, many of the reviews read like purchase notes by people shopping around for the best fire spread model to implement in their part of the world for their particular purpose. Recent reviews (e.g. Perry, ( 1998 ); Pastor et al., ( 2003 ) ; etc), while endeavouring to be comprehensive, have offered only superficial and cursory inspections of the models presented. Morvan et al., ( 2004 ) takes a different line by analysing a much broader spectrum of models in some detail and concludes that no single approach is going to be suitable for all uses. While the recent reviews provide an overview of the models and approaches that have been undertaken around the world, mention must be made of significant reviews published much earlier that discussed the processes in wildland fire propagation themselves. Foremost is the work of Williams, ( 1982 ) , which comprehensively covers the phenomenology of both wildland and urban fire, the physics and chemistry of combustion, and is recommended reading for the beginner. The earlier work of Emmons, ( 1963 , 1966 ) and Lee, ( 1972 ) provides a sound background on the advances made during the post-war boom era. Grishin, ( 1997 ) provides an extensive review of the work conducted in Russia in the 1970s, 80s and 90s. The first paper in this series discussed those models based upon the fundamental principles of the physics and chemistry of wildland fire behaviour. The second paper in the series discussed those models based directly upon only statistical analysis of fire behaviour observations or models that utilise some form of physical framework upon which the statistical analysis of observations have been based. Particular distinction was made between observations of the behaviour of fires in the strictly controlled and artificial conditions of the laboratory and those observed in the field under more naturally occurring conditions. This paper, the final in the series, focuses upon models concerned only with the simulation of fire spread over the landscape and models that utilise mathematical conceits analogous to fire spread but which have no real-world connection to fire. The former generally utilise a pre-existing fire spread model (which can be physical, quasi-physical, quasi-empirical or empirical) and implements it in such a way as to simulate the spread of fire across a landscape. As such, it is generally based upon a geographic information system (GIS) of some description to represent the landscape and uses a propagation algorithm to spread the fire perimeter across it. The latter models are for the most part based upon accepted mathematical functions or concepts that have been applied to wildland fire spread but are not derived from any understanding of wildland fire behaviour. Rather, these models utilise apparent similarities between wildland fire behaviour and the behaviour of these concepts within certain limited contexts. Because of this, these mathematical concepts could equally be applied to other fields of endeavour and, for the most part have been, to greater or lesser success. Unlike the preceding entries in this series, this paper is segmented by the approaches taken by the various authors, not by the authors or their organisations, given the broad range of authors that in some instances have taken similar approaches.","role":"user"},{"content":"Abstract: In recent years, advances in computational power and spatial data analysis (GIS, remote sensing, etc) have led to an increase in attempts to model the spread and behvaiour of wildland fires across the landscape. This series of review papers endeavours to critically and comprehensively review all types of surface fire spread models developed since 1990. This paper reviews models of a simulation or mathematical analogue nature. Most simulation models are implementations of existing empirical or quasi-empirical models and their primary function is to convert these generally one dimensional models to two dimensions and then propagate a fire perimeter across a modelled landscape. Mathematical analogue models are those that are based on some mathematical conceit (rather than a physical representation of fire spread) that coincidentally simulates the spread of fire. Other papers in the series review models of an physical or quasi-physical nature and empirical or quasi-empirical nature. Many models are extensions or refinements of models developed before 1990. Where this is the case, these models are also discussed but much less comprehensively.","role":"assistant"}]

FPCP Theory Overview

I Introduction and UT Theory In this first section of this talk I will skim over the determination of sides and angles of the unitarity triangle (UT). I do not pretend to make a complete review or even an overview. I picked topics on the basis of where I thought we should be weary overly cautions of theorists “predictions.” In subsequent sections I attempt to get some perspective on the field, and will ask and try to answer the questions of what we have learned in FP and CP physics and where should we go from here. I.1 | V t ​ d / V t ​ s | subscript 𝑉 𝑡 𝑑 subscript 𝑉 𝑡 𝑠 |V_{td}/V_{ts}| The magnitudes of V t ​ d subscript 𝑉 𝑡 𝑑 V_{td} and V t ​ s subscript 𝑉 𝑡 𝑠 V_{ts} are determined from measurements of neutral B d subscript 𝐵 𝑑 B_{d} and B s subscript 𝐵 𝑠 B_{s} oscillations, respectively. The big news last year was the precise measurement of the B s subscript 𝐵 𝑠 B_{s} mixing rate at Tevatron experiments Abulencia:2006mq ; Abulencia:2006ze . While | V t ​ s | subscript 𝑉 𝑡 𝑠 |V_{ts}| does not provide direct information on the apex of the unitarity triangle, the ratio | V t ​ d / V t ​ s | subscript 𝑉 𝑡 𝑑 subscript 𝑉 𝑡 𝑠 |V_{td}/V_{ts}| does. The interest in the ratio stems from the cancellation of hadronic uncertainties: | V t ​ d | | V t ​ s | = ξ ​ Δ ​ m s ​ m B s Δ ​ m d ​ m B d , where ξ 2 ≡ B B s ​ f B s 2 B B d ​ f B d 2 . formulae-sequence subscript 𝑉 𝑡 𝑑 subscript 𝑉 𝑡 𝑠 𝜉 Δ subscript 𝑚 𝑠 subscript 𝑚 subscript 𝐵 𝑠 Δ subscript 𝑚 𝑑 subscript 𝑚 subscript 𝐵 𝑑 where superscript 𝜉 2 subscript 𝐵 subscript 𝐵 𝑠 superscript subscript 𝑓 subscript 𝐵 𝑠 2 subscript 𝐵 subscript 𝐵 𝑑 superscript subscript 𝑓 subscript 𝐵 𝑑 2 \frac{|V_{td}|}{|V_{ts}|}=\xi\sqrt{\frac{\Delta m_{s}\,m_{B_{s}}}{\Delta m_{d}\,m_{B_{d}}}},\quad\text{where}\quad\xi^{2}\equiv\frac{B_{B_{s}}f_{B_{s}}^{2}}{B_{B_{d}}f_{B_{d}}^{2}}. (1) The hadronic parameter ξ 𝜉 \xi would be unity in the flavor- S ​ U ​ ( 3 ) 𝑆 𝑈 3 SU(3) symmetry limit. Lattice QCD gives Aoki:2003xb ξ = 1.21 ​ + 0.047 − 0.035 𝜉 1.21 FRACOP 0.047 0.035 \xi=1.21{\textstyle{+0.047\atop-0.035}} , and combining with the experimental result | V t ​ d | | V t ​ s | = 0.2060 ± 0.0007 ​ (exp) ​ + 0.0081 − 0.0060 ​ (theory) subscript 𝑉 𝑡 𝑑 subscript 𝑉 𝑡 𝑠 plus-or-minus 0.2060 0.0007 (exp) FRACOP 0.0081 0.0060 (theory) \frac{|V_{td}|}{|V_{ts}|}=0.2060\pm 0.0007\text{(exp)}{\textstyle{+0.0081\atop-0.0060}}\text{(theory)} The error, approximately 3%, is dominated by theory, which comes solely from the error in ξ 𝜉 \xi . There aren’t many examples of quantities that the lattice has post-dicted (let alone predicted) with this sort of accuracy. So can the rest of us, non-latticists, trust it? On the one hand, because this result is protected by symmetry the required precision is not really 3%. The quantity one must measure is the deviation from the symmetry limit, ξ 2 − 1 superscript 𝜉 2 1 \xi^{2}-1 , for which the error is about 25% and perhaps we should be confident that the lattice result is correct at this level. On the other hand, this also tells us that other methods can be competitive at this level. The leading chiral log calculation Grinstein:1992qt gives ξ ≈ 1.15 𝜉 1.15 \xi\approx 1.15 , and the error in ξ 2 − 1 superscript 𝜉 2 1 \xi^{2}-1 is estimated from naive dimensional analysis as m K 2 / Λ χ 2 ∼ 24 % similar-to superscript subscript 𝑚 𝐾 2 superscript subscript Λ 𝜒 2 percent 24 m_{K}^{2}/\Lambda_{\chi}^{2}\sim 24\% , comparable to the lattice result. Moreover, the lattice determination has been made with only one method (staggered fermions) and it would be reassuring to see the same result from other methods. For the lattice to achieve the 0.35% accuracy in ξ 𝜉 \xi needed to match the experimental error in | V t ​ d / V t ​ s | subscript 𝑉 𝑡 𝑑 subscript 𝑉 𝑡 𝑠 |V_{td}/V_{ts}| a precision of 2% in the determination of ξ 2 − 1 superscript 𝜉 2 1 \xi^{2}-1 is required. Before we, skeptics, trust any significant improvement in this determination, other independent lattice QCD post-dictions of similar accuracy are necessary. I.2 | V c ​ b | subscript 𝑉 𝑐 𝑏 |V_{cb}| Inclusive The method of moments gives a very accurate determination of | V c ​ b | subscript 𝑉 𝑐 𝑏 |V_{cb}| from inclusive semileptonic B 𝐵 B decays. In QCD, the rate d ​ Γ ​ ( B → X c ​ ℓ ​ ν ) / d ​ x ​ d ​ y = | V c ​ b | 2 ​ f ​ ( x , y ) d Γ → 𝐵 subscript 𝑋 𝑐 ℓ 𝜈 d 𝑥 d 𝑦 superscript subscript 𝑉 𝑐 𝑏 2 𝑓 𝑥 𝑦 {\rm d}\Gamma(B\to X_{c}\ell\nu)/{\rm d}x\,{\rm d}y=|V_{cb}|^{2}f(x,y) , where x 𝑥 x and y 𝑦 y are the invariant lepton pair mass and energy in units of m B subscript 𝑚 𝐵 m_{B} , is given in terms of four parameters: | V c ​ b | subscript 𝑉 𝑐 𝑏 |V_{cb}| , α s subscript 𝛼 𝑠 \alpha_{s} , m c subscript 𝑚 𝑐 m_{c} and m b subscript 𝑚 𝑏 m_{b} . | V c ​ b | subscript 𝑉 𝑐 𝑏 |V_{cb}| , which is what we are after, drops out of normalized moments. Since α s subscript 𝛼 𝑠 \alpha_{s} is well known, the idea is to fix m c subscript 𝑚 𝑐 m_{c} and m b subscript 𝑚 𝑏 m_{b} from normalized moments and then use them to compute the normalization, hence determining | V c ​ b | subscript 𝑉 𝑐 𝑏 |V_{cb}| . In reality we cannot solve QCD to give the moments in terms of m c subscript 𝑚 𝑐 m_{c} and m b subscript 𝑚 𝑏 m_{b} , but we can use a 1 / m Q 1 subscript 𝑚 𝑄 1/m_{Q} expansion to write the moments in terms of m c subscript 𝑚 𝑐 m_{c} , m b subscript 𝑚 𝑏 m_{b} and a few constants that parametrize our ignorance Falk:1995kn . These constants are in fact matrix elements of operators in the 1 / m Q 1 subscript 𝑚 𝑄 1/m_{Q} expansion. If terms of order 1 / m Q 3 1 superscript subscript 𝑚 𝑄 3 1/m_{Q}^{3} are retained in the expansion one needs to introduce five such constants; and an additional two are determined by meson masses. All five constants and two quark masses can be over-determined from a few normalized moments that are functions of E cut subscript 𝐸 cut E_{\rm cut} , the lowest limit of the lepton energy integration. The error in the determination of | V c ​ b | subscript 𝑉 𝑐 𝑏 |V_{cb}| is a remarkably small 2% Bauer:2004ve . But even most remarkable is that this estimate for the error is truly believable. It is obtained by assigning the last term retained in the expansion to the error, as opposed to the less conservative guessing of the next order not kept in the expansion. Since there is also a perturbative expansion, the assigned error is the combination of the last term kept in all expansions, of order β 0 ​ α s 2 subscript 𝛽 0 superscript subscript 𝛼 𝑠 2 \beta_{0}\alpha_{s}^{2} , α s ​ Λ QCD / m b subscript 𝛼 𝑠 subscript Λ QCD subscript 𝑚 𝑏 \alpha_{s}\Lambda_{\rm QCD}/m_{b} and ( Λ QCD / m b ) 3 superscript subscript Λ QCD subscript 𝑚 𝑏 3 (\Lambda_{\rm QCD}/m_{b})^{3} . There is only one assumption in the calculation that is not fully justified from first principles. The moment integrals can be computed perturbatively (in the 1 / m Q 1 subscript 𝑚 𝑄 1/m_{Q} expansion) only because the integral can be turned into a contour over a complex E 𝐸 E away from the physical region Chay:1990da . However, the contour is pinned at the minimal energy, E cut subscript 𝐸 cut E_{\rm cut} , on the real axis, right on the physical cut. So there is a small region of integration where quark-hadron duality cannot be justified and has to be invoked. Parametrically this region of integration is small, a fraction of order Λ / m Q Λ subscript 𝑚 𝑄 \Lambda/m_{Q} of the total. But this is a disaster because this is parametrically much larger than the claimed error of order ( Λ / m Q ) 3 superscript Λ subscript 𝑚 𝑄 3 (\Lambda/m_{Q})^{3} . However, this is believed not to be a problem. For one thing, the fits to moments as functions of E cut subscript 𝐸 cut E_{\rm cut} are extremely good: the system is over-constrained and these internal checks work. And for another, it has been shown Boyd:1995ht that duality works exactly in the Shifman-Voloshin (small velocity) limit, to order 1 / m Q 2 1 superscript subscript 𝑚 𝑄 2 1/m_{Q}^{2} . It seems unlikely that the violation to local quark-hadron duality mainly changes the normalization and has mild dependence on E cut subscript 𝐸 cut E_{\rm cut} , and that this effect only shows up away from the SV limit. Exclusive The exclusive determination of | V c ​ b | subscript 𝑉 𝑐 𝑏 |V_{cb}| is in pretty good shape theoretically, but is not competitive with the inclusive one. So it provides a sanity check, but not an improvement. The semileptonic rates into either D 𝐷 D or D ∗ superscript 𝐷 D^{*} are parametrized by functions ℱ ℱ {\cal F} , ℱ ∗ subscript ℱ {\cal F}_{*} , of the rapidity of the charmed meson in the B 𝐵 B rest-frame, w 𝑤 w . Luke’s theorem Luke:1990eg states ℱ = ℱ ∗ = 1 + 𝒪 ​ ( Λ QCD / m c ) 2 ℱ subscript ℱ 1 𝒪 superscript subscript Λ QCD subscript 𝑚 𝑐 2 {\cal F}={\cal F}_{*}=1+{\cal O}(\Lambda_{\rm QCD}/m_{c})^{2} at w = 1 𝑤 1 w=1 . The rate is measured at w > 1 𝑤 1 w>1 and extrapolated to w = 1 𝑤 1 w=1 . The extrapolation is made with a first principles calculation to avoid introducing extraneous errors Boyd:1997kz . The result has a 4% error dominated by the uncertainty in the determination of ℱ ℱ {\cal F} , ℱ ∗ subscript ℱ {\cal F}_{*} at w = 1 𝑤 1 w=1 . There is some tension between theory and experiment in these exclusive decays that needs attention. The ratios of form factors R 1 , 2 subscript 𝑅 1 2 R_{1,2} are at variance from theory by three and two sigma respectively Aubert:2006cx . Also, in the heavy quark limit the slopes ρ 2 superscript 𝜌 2 \rho^{2} of ℱ ℱ {\cal F} and ℱ ∗ subscript ℱ {\cal F}_{*} should be equal. One can estimate symmetry violations and obtains Grinstein:2001yg ρ ℱ 2 − ρ ℱ ∗ 2 ≃ 0.19 similar-to-or-equals subscript superscript 𝜌 2 ℱ subscript superscript 𝜌 2 subscript ℱ 0.19 \rho^{2}_{{\cal F}}-\rho^{2}_{{\cal F}_{*}}\simeq 0.19 , while experimentally this is − 0.22 ± 0.20 plus-or-minus 0.22 0.20 -0.22\pm 0.20 , a deviation in the opposite direction. This is a good place for the lattice to make post-dictions at the few percent error level that may lend it some credibility in other areas where it is needed to determine a fundamental parameter. I.3 | V u ​ b | subscript 𝑉 𝑢 𝑏 |V_{ub}| The magnitude | V u ​ b | subscript 𝑉 𝑢 𝑏 |V_{ub}| determines the rate for B → X u ​ ℓ ​ ν → 𝐵 subscript 𝑋 𝑢 ℓ 𝜈 B\to X_{u}\ell\nu . The well known experimental difficulty is that since | V u ​ b | ≪ | V c ​ b | much-less-than subscript 𝑉 𝑢 𝑏 subscript 𝑉 𝑐 𝑏 |V_{ub}|\ll|V_{cb}| the semileptonic decay rate is dominated by charmed final states. To measure a signal it is necessary to either look at exclusive final states or suppress charm kinematically. The interpretation of the measurement requires, in the exclusive case, knowledge of hadronic matrix elements parametrized in terms of form-factors, and for inclusive decays, understanding of the effect of the kinematic cuts on the the perturbative expansion and quark-hadron duality. Inclusive This has been the method of choice until recently, since it was thought that the perturbative calculation was reliable and systematic and hence could be made sufficiently accurate. However it has become increasingly clear of late that the calculation cannot be made arbitrarily precise. The method uses effective field theories to expand the amplitude systematically in inverse powers of a large energy, either the heavy mass or the energy of the up-quark (or equivalently, of the hadronic final state). One shows that in the restricted kinematic region needed for experiment (to enhance the up-signal to charm-background) the inclusive amplitude is governed by a non-perturbative “shape function,” which is, however, universal: it also determines other processes, like the radiative B → X s ​ γ → 𝐵 subscript 𝑋 𝑠 𝛾 B\to X_{s}\gamma . So the strategy has been to eliminate this unknown, non-perturbative function from the rates for semileptonic and radiative decays. Surprisingly, most analysis do not eliminate the shape function dependence between the two processes. Instead, practitioners commonly use parametrized fits that unavoidably introduce uncontrolled errors. It is not surprising that errors quoted in the determination of | V u ​ b | subscript 𝑉 𝑢 𝑏 |V_{ub}| are smaller if by a parametrized fit than by the elimination method of Leibovich:1999xf . The problem is that parameterized fits introduce systematic errors that are unaccounted for. Parametrized fits aside, there is an intrinsic problem with the method. Universality is violated by sub-leading terms brickwall in the large energy expansion (“sub-leading shape functions”). One can estimate this uncontrolled correction to be of order α s ​ Λ / m b subscript 𝛼 𝑠 Λ subscript 𝑚 𝑏 \alpha_{s}\Lambda/m_{b} , where Λ Λ \Lambda is hadronic scale that characterizes the sub-leading effects (in the effective theory language: matrix elements of higher dimension operators). We can try to estimate these effects using models of sub-leading shape functions but then one introduces uncontrolled errors into the determination. At best one should use models to estimate the errors. I think it is fair, albeit unpopular, to say that this method is limited to a precision of about 15%: since there are about 10 sub-leading shape functions, I estimate the precision as 10 ​ α s ​ Λ / m b 10 subscript 𝛼 𝑠 Λ subscript 𝑚 𝑏 \sqrt{10}\,\alpha_{s}\Lambda/m_{b} . This is much larger than the error commonly quoted in the determination of | V u ​ b | subscript 𝑉 𝑢 𝑏 |V_{ub}| . This is just as well, since the value of | V u ​ b | subscript 𝑉 𝑢 𝑏 |V_{ub}| from inclusives is in disagreement not only with the value from exclusives but also with the global unitarity triangle fit. You can quantify this if you like, but it is graphically obvious when you see plots of the fit in the ρ 𝜌 \rho - η 𝜂 \eta plane that use only some inputs inputs and contrast those with the remaining inputs of the global fit. At this conference last year, Jerome Charles presented Charles:2006yw three pairs of fits contrasting measurements: tree vs. loop, CP violating vs. CV conserving, and theory free vs. QCD based (see also slide 25 of Heiko Lacker, this conference). In all these it is evident to the naked eye that | V u ​ b | subscript 𝑉 𝑢 𝑏 |V_{ub}| (the dark green circle’s radius) is too large; the input used is dominated by inclusives. Exclusives The branching fraction ℬ ​ ( B → π ​ ℓ ​ ν ) ℬ → 𝐵 𝜋 ℓ 𝜈 {\cal B}(B\to\pi\ell\nu) is known Abe:2004zm to 8%. A comparable determination of | V u ​ b | subscript 𝑉 𝑢 𝑏 |V_{ub}| requires knowledge of the B → π → 𝐵 𝜋 B\to\pi form factor f + ​ ( q 2 ) subscript 𝑓 superscript 𝑞 2 f_{+}(q^{2}) to 4%. There are some things we do know about f + subscript 𝑓 f_{+} : (i)The shape is constrained by dispersion relations Boyd:1994tt . This means that if we know f + subscript 𝑓 f_{+} at a few well spaced points we can pretty much determine the whole function f + subscript 𝑓 f_{+} . (ii)We can get a rough measurement of the form factor at q 2 = m π 2 superscript 𝑞 2 superscript subscript 𝑚 𝜋 2 q^{2}=m_{\pi}^{2} from the rate for B → π ​ π → 𝐵 𝜋 𝜋 B\to\pi\pi Bauer:2004tj . This requires a sophisticated effective theory (SCET) analysis which both shows that the leading order contains a term with f + ​ ( m π 2 ) subscript 𝑓 superscript subscript 𝑚 𝜋 2 f_{+}(m_{\pi}^{2}) and systematically characterizes the corrections to the lowest order SCET. It is safe to assume that this determination of f + ​ ( m π 2 ) subscript 𝑓 superscript subscript 𝑚 𝜋 2 f_{+}(m_{\pi}^{2}) will not improve beyond the 10% mark. Lattice QCD can determine the form factor, at least over a limited region of large q 2 superscript 𝑞 2 q^{2} . At the moment there is some disagreement between the best two lattice calculations, which however use the same method Shigemitsu:2004ft . A skeptic would require not only agreement between the two existing calculations but also with other methods, not to mention a set of additional independent successful post-dictions, before the result can be trusted for a precision determination of | V u ​ b | subscript 𝑉 𝑢 𝑏 |V_{ub}| . The experimental and lattice measurements can be combined using constraints from dispersion relations and unitarity Arnesen:2005ez . Because these constraints follow from fundamentals, they do not introduce additional uncertainties. They improve the determination of | V u ​ b | subscript 𝑉 𝑢 𝑏 |V_{ub}| significantly. The lattice determination is for the q 2 superscript 𝑞 2 q^{2} -region where the rate is smallest. This is true even if the form factor is largest there, because in that region the rate is phase space suppressed. But a rough shape of the spectrum is experimentally observed, through a binned measurement Abe:2004zm , and the dispersion relation constraints allows one to combine the full experimental spectrum with the restricted- q 2 superscript 𝑞 2 q^{2} lattice measurement. The result of this analysis gives a 13% error in | V u ​ b | subscript 𝑉 𝑢 𝑏 |V_{ub}| , completely dominated by the lattice errors. Alternatives Exclusive and inclusive determinations of | V u ​ b | subscript 𝑉 𝑢 𝑏 |V_{ub}| have comparable precisions. Neither is very good and the prospect for significant improvement is limited. Other methods need be explored, if not to improve on existing | V u ​ b | subscript 𝑉 𝑢 𝑏 |V_{ub}| to lend confidence to the result. A lattice-free method would be preferable. A third method, proposed a while ago Ligeti:1995yz , uses the idea of double ratios Grinstein:1993ys to reduce hadronic uncertainties. Two independent approximate symmetries protect double ratios from deviations from unity, which are therefore of the order of the product of two small symmetry breaking parameters. For example, the double ratio ( f B s / f B d ) / ( f D s / f D d ) = ( f B s / f D s ) / ( f B d / f D d ) = 1 + 𝒪 ​ ( m s / m c ) subscript 𝑓 subscript 𝐵 𝑠 subscript 𝑓 subscript 𝐵 𝑑 subscript 𝑓 subscript 𝐷 𝑠 subscript 𝑓 subscript 𝐷 𝑑 subscript 𝑓 subscript 𝐵 𝑠 subscript 𝑓 subscript 𝐷 𝑠 subscript 𝑓 subscript 𝐵 𝑑 subscript 𝑓 subscript 𝐷 𝑑 1 𝒪 subscript 𝑚 𝑠 subscript 𝑚 𝑐 (f_{B_{s}}/f_{B_{d}})/(f_{D_{s}}/f_{D_{d}})=(f_{B_{s}}/f_{D_{s}})/(f_{B_{d}}/f_{D_{d}})=1+{\cal O}(m_{s}/m_{c}) because f B s / f B d = f D s / f D d = 1 subscript 𝑓 subscript 𝐵 𝑠 subscript 𝑓 subscript 𝐵 𝑑 subscript 𝑓 subscript 𝐷 𝑠 subscript 𝑓 subscript 𝐷 𝑑 1 f_{B_{s}}/f_{B_{d}}=f_{D_{s}}/f_{D_{d}}=1 by S ​ U ​ ( 3 ) 𝑆 𝑈 3 SU(3) flavor, while f B s / f D s = f B d / f D d = m c / m b subscript 𝑓 subscript 𝐵 𝑠 subscript 𝑓 subscript 𝐷 𝑠 subscript 𝑓 subscript 𝐵 𝑑 subscript 𝑓 subscript 𝐷 𝑑 subscript 𝑚 𝑐 subscript 𝑚 𝑏 f_{B_{s}}/f_{D_{s}}=f_{B_{d}}/f_{D_{d}}=\sqrt{m_{c}/m_{b}} by heavy flavor symmetry. One can extract | V u ​ b / V t ​ s ​ V t ​ b | subscript 𝑉 𝑢 𝑏 subscript 𝑉 𝑡 𝑠 subscript 𝑉 𝑡 𝑏 |V_{ub}/V_{ts}V_{tb}| by measuring the ratio, d ​ Γ ​ ( B ¯ d → ρ ​ ℓ ​ ν ) / d ​ q 2 d ​ Γ ​ ( B ¯ d → K ∗ ​ ℓ + ​ ℓ − ) / d ​ q 2 = | V u ​ b | 2 | V t ​ s ​ V t ​ b | 2 ⋅ 8 ​ π 2 α 2 ⋅ 1 N ​ ( q 2 ) ⋅ R B , d Γ → subscript ¯ 𝐵 𝑑 𝜌 ℓ 𝜈 d superscript 𝑞 2 d Γ → subscript ¯ 𝐵 𝑑 superscript 𝐾 superscript ℓ superscript ℓ d superscript 𝑞 2 ⋅ superscript subscript 𝑉 𝑢 𝑏 2 superscript subscript 𝑉 𝑡 𝑠 subscript 𝑉 𝑡 𝑏 2 8 superscript 𝜋 2 superscript 𝛼 2 1 𝑁 superscript 𝑞 2 subscript 𝑅 𝐵 \frac{{\rm d}\Gamma(\bar{B}_{d}\to\rho\ell\nu)/{\rm d}q^{2}}{{\rm d}\Gamma(\bar{B}_{d}\to K^{*}\ell^{+}\ell^{-})/{\rm d}q^{2}}=\frac{|V_{ub}|^{2}}{|V_{ts}V_{tb}|^{2}}\cdot\frac{8\pi^{2}}{\alpha^{2}}\cdot\frac{1}{N(q^{2})}\cdot R_{B}, (2) where q 2 superscript 𝑞 2 q^{2} is the lepton pair invariant mass, and N ​ ( q 2 ) 𝑁 superscript 𝑞 2 N(q^{2}) is a known function Grinstein:2004vb . When expressed as functions of the rapidity of the vector meson, y = E V / m V 𝑦 subscript 𝐸 𝑉 subscript 𝑚 𝑉 y=E_{V}/m_{V} , the ratios of helicity amplitudes R B = ∑ λ | H λ B → ρ ​ ( y ) | 2 ∑ λ | H λ B → K ∗ ​ ( y ) | 2 , R D = ∑ λ | H λ D → ρ ​ ( y ) | 2 ∑ λ | H λ D → K ∗ ​ ( y ) | 2 , formulae-sequence subscript 𝑅 𝐵 subscript 𝜆 superscript subscript superscript 𝐻 → 𝐵 𝜌 𝜆 𝑦 2 subscript 𝜆 superscript subscript superscript 𝐻 → 𝐵 superscript 𝐾 𝜆 𝑦 2 subscript 𝑅 𝐷 subscript 𝜆 superscript subscript superscript 𝐻 → 𝐷 𝜌 𝜆 𝑦 2 subscript 𝜆 superscript subscript superscript 𝐻 → 𝐷 superscript 𝐾 𝜆 𝑦 2 R_{B}=\frac{\sum_{\lambda}|H^{B\to\rho}_{\lambda}(y)|^{2}}{\sum_{\lambda}|H^{B\to K^{*}}_{\lambda}(y)|^{2}},\quad R_{D}=\frac{\sum_{\lambda}|H^{D\to\rho}_{\lambda}(y)|^{2}}{\sum_{\lambda}|H^{D\to K^{*}}_{\lambda}(y)|^{2}}, are related by a double ratio: R B ​ ( y ) = R D ​ ( y ) ​ ( 1 + 𝒪 ​ ( m s ​ ( m c − 1 − m b − 1 ) ) ) subscript 𝑅 𝐵 𝑦 subscript 𝑅 𝐷 𝑦 1 𝒪 subscript 𝑚 𝑠 superscript subscript 𝑚 𝑐 1 superscript subscript 𝑚 𝑏 1 R_{B}(y)=R_{D}(y)(1+{\cal O}(m_{s}(m_{c}^{-1}-m_{b}^{-1}))) . This measurement could be done today: CLEO has accurately measured the required semileptonic D 𝐷 D decays Adam:2007pv ; Gray:2007pw . A fourth method is available if we are willing to use rarer decays. To extract | V u ​ b | subscript 𝑉 𝑢 𝑏 |V_{ub}| from ℬ ​ ( B + → τ + ​ ν τ ) = ( 0.88 ​ + 0.68 − 0.67 ± 0.11 ) × 10 − 4 ℬ → superscript 𝐵 superscript 𝜏 subscript 𝜈 𝜏 plus-or-minus 0.88 FRACOP 0.68 0.67 0.11 superscript 10 4 {\cal B}(B^{+}\to\tau^{+}\nu_{\tau})=(0.88{\textstyle{+0.68\atop-0.67}}\pm 0.11)\times 10^{-4} Aubert:2004kz one needs a lattice determination of f B subscript 𝑓 𝐵 f_{B} . Since we want to move away from relying on non-perturbative methods (lattice) to extract | V u ​ b | subscript 𝑉 𝑢 𝑏 |V_{ub}| we propose a cleaner but more difficult measurement, the double ratio Γ ​ ( B u → τ ​ ν ) Γ ​ ( B s → ℓ + ​ ℓ − ) Γ ​ ( D d → ℓ ​ ν ) Γ ​ ( D s → ℓ ​ ν ) ∼ | V u ​ b | 2 | V t ​ s ​ V t ​ b | 2 ⋅ π 2 α 2 ⋅ ( f B / f B s f D / f D s ) 2 similar-to Γ → subscript 𝐵 𝑢 𝜏 𝜈 Γ → subscript 𝐵 𝑠 superscript ℓ superscript ℓ Γ → subscript 𝐷 𝑑 ℓ 𝜈 Γ → subscript 𝐷 𝑠 ℓ 𝜈 ⋅ superscript subscript 𝑉 𝑢 𝑏 2 superscript subscript 𝑉 𝑡 𝑠 subscript 𝑉 𝑡 𝑏 2 superscript 𝜋 2 superscript 𝛼 2 superscript subscript 𝑓 𝐵 subscript 𝑓 subscript 𝐵 𝑠 subscript 𝑓 𝐷 subscript 𝑓 subscript 𝐷 𝑠 2 \frac{\frac{\Gamma(B_{u}\to\tau\nu)}{\Gamma(B_{s}\to\ell^{+}\ell^{-})}}{\frac{\Gamma(D_{d}\to\ell\nu)}{\Gamma(D_{s}\to\ell\nu)}}\sim\frac{|V_{ub}|^{2}}{|V_{ts}V_{tb}|^{2}}\cdot\frac{\pi^{2}}{\alpha^{2}}\cdot\left(\frac{f_{B}/f_{B_{s}}}{f_{D}/f_{D_{s}}}\right)^{2} (3) In the SM ℬ ​ ( B s → μ + ​ μ − ) ≈ 3.5 × 10 − 9 ℬ → subscript 𝐵 𝑠 superscript 𝜇 superscript 𝜇 3.5 superscript 10 9 {\cal B}(B_{s}\to\mu^{+}\mu^{-})\approx 3.5\times 10^{-9} × ( f B s / 210 ​ MeV ) 2 ​ ( | V t ​ s | / 0.040 ) 2 absent superscript subscript 𝑓 subscript 𝐵 𝑠 210 MeV 2 superscript subscript 𝑉 𝑡 𝑠 0.040 2 \times(f_{B_{s}}/210\,\text{MeV})^{2}(|V_{ts}|/0.040)^{2} is the only presently unknown quantity in the double ratio and is expected to be well measured at the LHC Schopper:2006he . The ratio Γ ​ ( B + → τ + ​ ν ) / Γ ​ ( B d → μ + ​ μ − ) Γ → superscript 𝐵 superscript 𝜏 𝜈 Γ → subscript 𝐵 𝑑 superscript 𝜇 superscript 𝜇 \Gamma(B^{+}\to\tau^{+}\nu)/\Gamma(B_{d}\to\mu^{+}\mu^{-}) gives us a fifth method. It has basically no hadronic uncertainty, since the hadronic factor f B / f B d = 1 subscript 𝑓 𝐵 subscript 𝑓 subscript 𝐵 𝑑 1 f_{B}/f_{B_{d}}=1 , by isospin. It involves | V u ​ b | 2 / | V t ​ d ​ V t ​ b | 2 superscript subscript 𝑉 𝑢 𝑏 2 superscript subscript 𝑉 𝑡 𝑑 subscript 𝑉 𝑡 𝑏 2 |V_{ub}|^{2}/|V_{td}V_{tb}|^{2} , an unusual combination of CKMs. In the ρ − η 𝜌 𝜂 \rho-\eta plane it forms a circle centered at ∼ ( − 0.2 , 0 ) similar-to absent 0.2 0 \sim(-0.2,0) of radius ∼ 0.5 similar-to absent 0.5 \sim 0.5 . Of course, measuring Γ ​ ( B d → μ + ​ μ − ) Γ → subscript 𝐵 𝑑 superscript 𝜇 superscript 𝜇 \Gamma(B_{d}\to\mu^{+}\mu^{-}) is extremely hard. In a sixth method one studies wrong charm decays B ¯ d , s → D ¯ ​ X → subscript ¯ 𝐵 𝑑 𝑠 ¯ 𝐷 𝑋 \bar{B}_{d,s}\to\bar{D}X (really b ​ q ¯ → u ​ c ¯ → 𝑏 ¯ 𝑞 𝑢 ¯ 𝑐 b\bar{q}\to u\bar{c} ). This can be done both in semi-inclusive decays Falk:1999sa (an experimentally challenging measurement) or in exclusive decays Evans:1999wx (where an interesting connection to B d , s subscript 𝐵 𝑑 𝑠 B_{d,s} mixing matrix elements is involved). I.4 α 𝛼 \alpha from B → π ​ π , π ​ ρ , ρ ​ ρ → 𝐵 𝜋 𝜋 𝜋 𝜌 𝜌 𝜌 B\to\pi\pi,\pi\rho,\rho\rho . In principle the penguin contamination problem Grinstein:1989df requires a full isospin analysis Gronau:1990ka for a theoretically clean determination of the angle α 𝛼 \alpha . The angle determination works slightly better than we had a right to expect a priori. The reason lies in two empirical observation in B → ρ ​ ρ → 𝐵 𝜌 𝜌 B\to\rho\rho . First, the longitudinal polarization dominates, and therefore the final state is to good approximation a CP eigenstate (CP even, in fact). And second, the branching fraction for B → ρ 0 ​ ρ 0 → 𝐵 superscript 𝜌 0 superscript 𝜌 0 B\to\rho^{0}\rho^{0} is small: relative to B → ρ + ​ ρ − → 𝐵 superscript 𝜌 superscript 𝜌 B\to\rho^{+}\rho^{-} it is 6 ± 3 % plus-or-minus 6 percent 3 6\pm 3\% , to be compared with the neutral to charged decay into pions of 23 ± 4 % plus-or-minus 23 percent 4 23\pm 4\% . This means that the contamination from penguin operators is small and one can get a clean measurement of α 𝛼 \alpha . All three decay modes are about equally important in current fits, which give α = 93 ​ + 11 − 9 𝛼 93 FRACOP 11 9 \alpha=93{\textstyle{+11\atop-9}} degrees. I.5 γ 𝛾 \gamma from B ± → D ​ K ± → superscript 𝐵 plus-or-minus 𝐷 superscript 𝐾 plus-or-minus B^{\pm}\to DK^{\pm} . Three different methods are used. They are all based on the interference between Cabibbo-allowed ( e.g., B − → D 0 ​ K − → superscript 𝐵 superscript 𝐷 0 superscript 𝐾 B^{-}\to D^{0}K^{-} ) and suppressed decays ( e.g., B − → D 0 ¯ ​ K − → superscript 𝐵 ¯ superscript 𝐷 0 superscript 𝐾 B^{-}\to\overline{D^{0}}K^{-} ) with D 0 superscript 𝐷 0 D^{0} , D 0 ¯ ¯ superscript 𝐷 0 \overline{D^{0}} decaying to a common state. The GLW Gronau:1990ra method uses decays to a common CP eigenstate. In the ADS method Atwood:1996ci the final state is chosen to be a suppressed D 𝐷 D decay mode if the D 𝐷 D came from an allowed B 𝐵 B decay; for example, the final state in the charm decay can be taken to be K + ​ π − superscript 𝐾 superscript 𝜋 K^{+}\pi^{-} so it is doubly Cabibbo suppressed for a D 0 superscript 𝐷 0 D^{0} decay but allowed for a D 0 ¯ ¯ superscript 𝐷 0 \overline{D^{0}} decay. The efficacy of this method depends sensitively on the ratio of amplitudes, which can be measured separately, r B = | A ​ ( B − → D 0 ¯ ​ K − ) / A ​ ( B − → D 0 ​ K − ) | subscript 𝑟 𝐵 𝐴 → superscript 𝐵 ¯ superscript 𝐷 0 superscript 𝐾 𝐴 → superscript 𝐵 superscript 𝐷 0 superscript 𝐾 r_{B}=|A(B^{-}\to\overline{D^{0}}K^{-})/A(B^{-}\to D^{0}K^{-})| . In the GGSZ method Giri:2003ty the D 0 superscript 𝐷 0 D^{0} and D 0 ¯ ¯ superscript 𝐷 0 \overline{D^{0}} are reconstructed in a common three body final state. The results to date vary depending on which decay mode is actually used, so the determination of γ 𝛾 \gamma from all measurements combined is not very good, γ = 62 ​ + 38 − 24 𝛾 62 FRACOP 38 24 \gamma=62{\textstyle{+38\atop-24}} degrees. More data should improve the determination of γ 𝛾 \gamma . I.6 Are there anomalies? There seem to be as many papers in the literature claiming there is a “ B → K ​ π → 𝐵 𝐾 𝜋 B\to K\pi puzzle” as those that claim it is not a puzzle. It is easy to see why. In order to find a puzzle one must know a priori the hadronic amplitudes. Those who find a puzzle in B → K ​ π → 𝐵 𝐾 𝜋 B\to K\pi make assumptions about hadronic amplitudes that those who find no puzzle think are unwarranted. Moreover, Ref. donoghue showed that soft final state interactions do not disappear in the large m b subscript 𝑚 𝑏 m_{b} limit, and Refs. falkFSI and wolfensteinFSI studied this quantitatively for B → K ​ π → 𝐵 𝐾 𝜋 B\to K\pi and B → π ​ π → 𝐵 𝜋 𝜋 B\to\pi\pi , respectively, and concluded the effects should be expected to be large. For example, the CP asymmetry in B → K ​ π → 𝐵 𝐾 𝜋 B\to K\pi could easily be 20% and the bound sin 2 ⁡ γ ≤ R superscript 2 𝛾 𝑅 \sin^{2}\gamma\leq R , where R = Γ ​ ( B d → π ∓ ​ K ± ) / Γ ​ ( B ± → π ± ​ K ) 𝑅 Γ → subscript 𝐵 𝑑 superscript 𝜋 minus-or-plus superscript 𝐾 plus-or-minus Γ → superscript 𝐵 plus-or-minus superscript 𝜋 plus-or-minus 𝐾 R=\Gamma(B_{d}\to\pi^{\mp}K^{\pm})/\Gamma(B^{\pm}\to\pi^{\pm}K) could easily be violated at the 20% level. The case for new physics in CPV in charmless b → s → 𝑏 𝑠 b\to s decays would seem to be stronger. Regardless of decay mode β e ​ f ​ f subscript 𝛽 𝑒 𝑓 𝑓 \beta_{eff} is predicted by SCET, QCD-factorization and pQCD to deviate from β J / ψ ​ K s subscript 𝛽 𝐽 𝜓 subscript 𝐾 𝑠 \beta_{J/\psi K_{s}} by a small positive amount. Experimentally the deviations vary from mode to mode but are all non-positive and not necessarily small. However, many things have to be checked before one can begin to believe we are seeing new physics here. First, all of the theoretical schemes need to come to terms with the soft final state interactions issue raised in donoghue or show that work is incorrect. Then, also, the fact that all deviations are negative strongly suggests that the measurements have been corrupted by an admixture of the opposite CP final state. In my view there is at present no case for deviations from the standard CKM model of flavor.

We first review some aspects of the determination of the sides and angles of the unitarity triangle. We pay particular attention to theory shortcomings, and present many alternative proposals for the determination of |Vub| (which at present is problematic). We then turn our attention to the more general question: What have we learned so far about flavor physics and where do we go from here? We argue that the aim of Flavor Physics should be to establish or rule out Minimal Flavor Violating interactions up to a scale of 10 TeV.

[{"content":"Help me to generate the abstract of a survey paper given the title:*FPCP Theory Overview* and the introduction:I Introduction and UT Theory In this first section of this talk I will skim over the determination of sides and angles of the unitarity triangle (UT). I do not pretend to make a complete review or even an overview. I picked topics on the basis of where I thought we should be weary overly cautions of theorists predictions. In subsequent sections I attempt to get some perspective on the field, and will ask and try to answer the questions of what we have learned in FP and CP physics and where should we go from here. I.1 | V t ​ d / V t ​ s | subscript 𝑉 𝑡 𝑑 subscript 𝑉 𝑡 𝑠 |V_{td}/V_{ts}| The magnitudes of V t ​ d subscript 𝑉 𝑡 𝑑 V_{td} and V t ​ s subscript 𝑉 𝑡 𝑠 V_{ts} are determined from measurements of neutral B d subscript 𝐵 𝑑 B_{d} and B s subscript 𝐵 𝑠 B_{s} oscillations, respectively. The big news last year was the precise measurement of the B s subscript 𝐵 𝑠 B_{s} mixing rate at Tevatron experiments Abulencia:2006mq ; Abulencia:2006ze . While | V t ​ s | subscript 𝑉 𝑡 𝑠 |V_{ts}| does not provide direct information on the apex of the unitarity triangle, the ratio | V t ​ d / V t ​ s | subscript 𝑉 𝑡 𝑑 subscript 𝑉 𝑡 𝑠 |V_{td}/V_{ts}| does. The interest in the ratio stems from the cancellation of hadronic uncertainties: | V t ​ d | | V t ​ s | = ξ ​ Δ ​ m s ​ m B s Δ ​ m d ​ m B d , where ξ 2 ≡ B B s ​ f B s 2 B B d ​ f B d 2 . formulae-sequence subscript 𝑉 𝑡 𝑑 subscript 𝑉 𝑡 𝑠 𝜉 Δ subscript 𝑚 𝑠 subscript 𝑚 subscript 𝐵 𝑠 Δ subscript 𝑚 𝑑 subscript 𝑚 subscript 𝐵 𝑑 where superscript 𝜉 2 subscript 𝐵 subscript 𝐵 𝑠 superscript subscript 𝑓 subscript 𝐵 𝑠 2 subscript 𝐵 subscript 𝐵 𝑑 superscript subscript 𝑓 subscript 𝐵 𝑑 2 \frac{|V_{td}|}{|V_{ts}|}=\xi\sqrt{\frac{\Delta m_{s}\,m_{B_{s}}}{\Delta m_{d}\,m_{B_{d}}}},\quad\text{where}\quad\xi^{2}\equiv\frac{B_{B_{s}}f_{B_{s}}^{2}}{B_{B_{d}}f_{B_{d}}^{2}}. (1) The hadronic parameter ξ 𝜉 \xi would be unity in the flavor- S ​ U ​ ( 3 ) 𝑆 𝑈 3 SU(3) symmetry limit. Lattice QCD gives Aoki:2003xb ξ = 1.21 ​ + 0.047 − 0.035 𝜉 1.21 FRACOP 0.047 0.035 \xi=1.21{\textstyle{+0.047\atop-0.035}} , and combining with the experimental result | V t ​ d | | V t ​ s | = 0.2060 ± 0.0007 ​ (exp) ​ + 0.0081 − 0.0060 ​ (theory) subscript 𝑉 𝑡 𝑑 subscript 𝑉 𝑡 𝑠 plus-or-minus 0.2060 0.0007 (exp) FRACOP 0.0081 0.0060 (theory) \frac{|V_{td}|}{|V_{ts}|}=0.2060\pm 0.0007\text{(exp)}{\textstyle{+0.0081\atop-0.0060}}\text{(theory)} The error, approximately 3%, is dominated by theory, which comes solely from the error in ξ 𝜉 \xi . There arent many examples of quantities that the lattice has post-dicted (let alone predicted) with this sort of accuracy. So can the rest of us, non-latticists, trust it? On the one hand, because this result is protected by symmetry the required precision is not really 3%. The quantity one must measure is the deviation from the symmetry limit, ξ 2 − 1 superscript 𝜉 2 1 \xi^{2}-1 , for which the error is about 25% and perhaps we should be confident that the lattice result is correct at this level. On the other hand, this also tells us that other methods can be competitive at this level. The leading chiral log calculation Grinstein:1992qt gives ξ ≈ 1.15 𝜉 1.15 \xi\approx 1.15 , and the error in ξ 2 − 1 superscript 𝜉 2 1 \xi^{2}-1 is estimated from naive dimensional analysis as m K 2 / Λ χ 2 ∼ 24 % similar-to superscript subscript 𝑚 𝐾 2 superscript subscript Λ 𝜒 2 percent 24 m_{K}^{2}/\Lambda_{\chi}^{2}\sim 24\% , comparable to the lattice result. Moreover, the lattice determination has been made with only one method (staggered fermions) and it would be reassuring to see the same result from other methods. For the lattice to achieve the 0.35% accuracy in ξ 𝜉 \xi needed to match the experimental error in | V t ​ d / V t ​ s | subscript 𝑉 𝑡 𝑑 subscript 𝑉 𝑡 𝑠 |V_{td}/V_{ts}| a precision of 2% in the determination of ξ 2 − 1 superscript 𝜉 2 1 \xi^{2}-1 is required. Before we, skeptics, trust any significant improvement in this determination, other independent lattice QCD post-dictions of similar accuracy are necessary. I.2 | V c ​ b | subscript 𝑉 𝑐 𝑏 |V_{cb}| Inclusive The method of moments gives a very accurate determination of | V c ​ b | subscript 𝑉 𝑐 𝑏 |V_{cb}| from inclusive semileptonic B 𝐵 B decays. In QCD, the rate d ​ Γ ​ ( B → X c ​ ℓ ​ ν ) / d ​ x ​ d ​ y = | V c ​ b | 2 ​ f ​ ( x , y ) d Γ → 𝐵 subscript 𝑋 𝑐 ℓ 𝜈 d 𝑥 d 𝑦 superscript subscript 𝑉 𝑐 𝑏 2 𝑓 𝑥 𝑦 {\rm d}\Gamma(B\to X_{c}\ell\nu)/{\rm d}x\,{\rm d}y=|V_{cb}|^{2}f(x,y) , where x 𝑥 x and y 𝑦 y are the invariant lepton pair mass and energy in units of m B subscript 𝑚 𝐵 m_{B} , is given in terms of four parameters: | V c ​ b | subscript 𝑉 𝑐 𝑏 |V_{cb}| , α s subscript 𝛼 𝑠 \alpha_{s} , m c subscript 𝑚 𝑐 m_{c} and m b subscript 𝑚 𝑏 m_{b} . | V c ​ b | subscript 𝑉 𝑐 𝑏 |V_{cb}| , which is what we are after, drops out of normalized moments. Since α s subscript 𝛼 𝑠 \alpha_{s} is well known, the idea is to fix m c subscript 𝑚 𝑐 m_{c} and m b subscript 𝑚 𝑏 m_{b} from normalized moments and then use them to compute the normalization, hence determining | V c ​ b | subscript 𝑉 𝑐 𝑏 |V_{cb}| . In reality we cannot solve QCD to give the moments in terms of m c subscript 𝑚 𝑐 m_{c} and m b subscript 𝑚 𝑏 m_{b} , but we can use a 1 / m Q 1 subscript 𝑚 𝑄 1/m_{Q} expansion to write the moments in terms of m c subscript 𝑚 𝑐 m_{c} , m b subscript 𝑚 𝑏 m_{b} and a few constants that parametrize our ignorance Falk:1995kn . These constants are in fact matrix elements of operators in the 1 / m Q 1 subscript 𝑚 𝑄 1/m_{Q} expansion. If terms of order 1 / m Q 3 1 superscript subscript 𝑚 𝑄 3 1/m_{Q}^{3} are retained in the expansion one needs to introduce five such constants; and an additional two are determined by meson masses. All five constants and two quark masses can be over-determined from a few normalized moments that are functions of E cut subscript 𝐸 cut E_{\rm cut} , the lowest limit of the lepton energy integration. The error in the determination of | V c ​ b | subscript 𝑉 𝑐 𝑏 |V_{cb}| is a remarkably small 2% Bauer:2004ve . But even most remarkable is that this estimate for the error is truly believable. It is obtained by assigning the last term retained in the expansion to the error, as opposed to the less conservative guessing of the next order not kept in the expansion. Since there is also a perturbative expansion, the assigned error is the combination of the last term kept in all expansions, of order β 0 ​ α s 2 subscript 𝛽 0 superscript subscript 𝛼 𝑠 2 \beta_{0}\alpha_{s}^{2} , α s ​ Λ QCD / m b subscript 𝛼 𝑠 subscript Λ QCD subscript 𝑚 𝑏 \alpha_{s}\Lambda_{\rm QCD}/m_{b} and ( Λ QCD / m b ) 3 superscript subscript Λ QCD subscript 𝑚 𝑏 3 (\Lambda_{\rm QCD}/m_{b})^{3} . There is only one assumption in the calculation that is not fully justified from first principles. The moment integrals can be computed perturbatively (in the 1 / m Q 1 subscript 𝑚 𝑄 1/m_{Q} expansion) only because the integral can be turned into a contour over a complex E 𝐸 E away from the physical region Chay:1990da . However, the contour is pinned at the minimal energy, E cut subscript 𝐸 cut E_{\rm cut} , on the real axis, right on the physical cut. So there is a small region of integration where quark-hadron duality cannot be justified and has to be invoked. Parametrically this region of integration is small, a fraction of order Λ / m Q Λ subscript 𝑚 𝑄 \Lambda/m_{Q} of the total. But this is a disaster because this is parametrically much larger than the claimed error of order ( Λ / m Q ) 3 superscript Λ subscript 𝑚 𝑄 3 (\Lambda/m_{Q})^{3} . However, this is believed not to be a problem. For one thing, the fits to moments as functions of E cut subscript 𝐸 cut E_{\rm cut} are extremely good: the system is over-constrained and these internal checks work. And for another, it has been shown Boyd:1995ht that duality works exactly in the Shifman-Voloshin (small velocity) limit, to order 1 / m Q 2 1 superscript subscript 𝑚 𝑄 2 1/m_{Q}^{2} . It seems unlikely that the violation to local quark-hadron duality mainly changes the normalization and has mild dependence on E cut subscript 𝐸 cut E_{\rm cut} , and that this effect only shows up away from the SV limit. Exclusive The exclusive determination of | V c ​ b | subscript 𝑉 𝑐 𝑏 |V_{cb}| is in pretty good shape theoretically, but is not competitive with the inclusive one. So it provides a sanity check, but not an improvement. The semileptonic rates into either D 𝐷 D or D ∗ superscript 𝐷 D^{*} are parametrized by functions ℱ ℱ {\cal F} , ℱ ∗ subscript ℱ {\cal F}_{*} , of the rapidity of the charmed meson in the B 𝐵 B rest-frame, w 𝑤 w . Lukes theorem Luke:1990eg states ℱ = ℱ ∗ = 1 + 𝒪 ​ ( Λ QCD / m c ) 2 ℱ subscript ℱ 1 𝒪 superscript subscript Λ QCD subscript 𝑚 𝑐 2 {\cal F}={\cal F}_{*}=1+{\cal O}(\Lambda_{\rm QCD}/m_{c})^{2} at w = 1 𝑤 1 w=1 . The rate is measured at w > 1 𝑤 1 w>1 and extrapolated to w = 1 𝑤 1 w=1 . The extrapolation is made with a first principles calculation to avoid introducing extraneous errors Boyd:1997kz . The result has a 4% error dominated by the uncertainty in the determination of ℱ ℱ {\cal F} , ℱ ∗ subscript ℱ {\cal F}_{*} at w = 1 𝑤 1 w=1 . There is some tension between theory and experiment in these exclusive decays that needs attention. The ratios of form factors R 1 , 2 subscript 𝑅 1 2 R_{1,2} are at variance from theory by three and two sigma respectively Aubert:2006cx . Also, in the heavy quark limit the slopes ρ 2 superscript 𝜌 2 \rho^{2} of ℱ ℱ {\cal F} and ℱ ∗ subscript ℱ {\cal F}_{*} should be equal. One can estimate symmetry violations and obtains Grinstein:2001yg ρ ℱ 2 − ρ ℱ ∗ 2 ≃ 0.19 similar-to-or-equals subscript superscript 𝜌 2 ℱ subscript superscript 𝜌 2 subscript ℱ 0.19 \rho^{2}_{{\cal F}}-\rho^{2}_{{\cal F}_{*}}\simeq 0.19 , while experimentally this is − 0.22 ± 0.20 plus-or-minus 0.22 0.20 -0.22\pm 0.20 , a deviation in the opposite direction. This is a good place for the lattice to make post-dictions at the few percent error level that may lend it some credibility in other areas where it is needed to determine a fundamental parameter. I.3 | V u ​ b | subscript 𝑉 𝑢 𝑏 |V_{ub}| The magnitude | V u ​ b | subscript 𝑉 𝑢 𝑏 |V_{ub}| determines the rate for B → X u ​ ℓ ​ ν → 𝐵 subscript 𝑋 𝑢 ℓ 𝜈 B\to X_{u}\ell\nu . The well known experimental difficulty is that since | V u ​ b | ≪ | V c ​ b | much-less-than subscript 𝑉 𝑢 𝑏 subscript 𝑉 𝑐 𝑏 |V_{ub}|\ll|V_{cb}| the semileptonic decay rate is dominated by charmed final states. To measure a signal it is necessary to either look at exclusive final states or suppress charm kinematically. The interpretation of the measurement requires, in the exclusive case, knowledge of hadronic matrix elements parametrized in terms of form-factors, and for inclusive decays, understanding of the effect of the kinematic cuts on the the perturbative expansion and quark-hadron duality. Inclusive This has been the method of choice until recently, since it was thought that the perturbative calculation was reliable and systematic and hence could be made sufficiently accurate. However it has become increasingly clear of late that the calculation cannot be made arbitrarily precise. The method uses effective field theories to expand the amplitude systematically in inverse powers of a large energy, either the heavy mass or the energy of the up-quark (or equivalently, of the hadronic final state). One shows that in the restricted kinematic region needed for experiment (to enhance the up-signal to charm-background) the inclusive amplitude is governed by a non-perturbative shape function, which is, however, universal: it also determines other processes, like the radiative B → X s ​ γ → 𝐵 subscript 𝑋 𝑠 𝛾 B\to X_{s}\gamma . So the strategy has been to eliminate this unknown, non-perturbative function from the rates for semileptonic and radiative decays. Surprisingly, most analysis do not eliminate the shape function dependence between the two processes. Instead, practitioners commonly use parametrized fits that unavoidably introduce uncontrolled errors. It is not surprising that errors quoted in the determination of | V u ​ b | subscript 𝑉 𝑢 𝑏 |V_{ub}| are smaller if by a parametrized fit than by the elimination method of Leibovich:1999xf . The problem is that parameterized fits introduce systematic errors that are unaccounted for. Parametrized fits aside, there is an intrinsic problem with the method. Universality is violated by sub-leading terms brickwall in the large energy expansion (sub-leading shape functions). One can estimate this uncontrolled correction to be of order α s ​ Λ / m b subscript 𝛼 𝑠 Λ subscript 𝑚 𝑏 \alpha_{s}\Lambda/m_{b} , where Λ Λ \Lambda is hadronic scale that characterizes the sub-leading effects (in the effective theory language: matrix elements of higher dimension operators). We can try to estimate these effects using models of sub-leading shape functions but then one introduces uncontrolled errors into the determination. At best one should use models to estimate the errors. I think it is fair, albeit unpopular, to say that this method is limited to a precision of about 15%: since there are about 10 sub-leading shape functions, I estimate the precision as 10 ​ α s ​ Λ / m b 10 subscript 𝛼 𝑠 Λ subscript 𝑚 𝑏 \sqrt{10}\,\alpha_{s}\Lambda/m_{b} . This is much larger than the error commonly quoted in the determination of | V u ​ b | subscript 𝑉 𝑢 𝑏 |V_{ub}| . This is just as well, since the value of | V u ​ b | subscript 𝑉 𝑢 𝑏 |V_{ub}| from inclusives is in disagreement not only with the value from exclusives but also with the global unitarity triangle fit. You can quantify this if you like, but it is graphically obvious when you see plots of the fit in the ρ 𝜌 \rho - η 𝜂 \eta plane that use only some inputs inputs and contrast those with the remaining inputs of the global fit. At this conference last year, Jerome Charles presented Charles:2006yw three pairs of fits contrasting measurements: tree vs. loop, CP violating vs. CV conserving, and theory free vs. QCD based (see also slide 25 of Heiko Lacker, this conference). In all these it is evident to the naked eye that | V u ​ b | subscript 𝑉 𝑢 𝑏 |V_{ub}| (the dark green circles radius) is too large; the input used is dominated by inclusives. Exclusives The branching fraction ℬ ​ ( B → π ​ ℓ ​ ν ) ℬ → 𝐵 𝜋 ℓ 𝜈 {\cal B}(B\to\pi\ell\nu) is known Abe:2004zm to 8%. A comparable determination of | V u ​ b | subscript 𝑉 𝑢 𝑏 |V_{ub}| requires knowledge of the B → π → 𝐵 𝜋 B\to\pi form factor f + ​ ( q 2 ) subscript 𝑓 superscript 𝑞 2 f_{+}(q^{2}) to 4%. There are some things we do know about f + subscript 𝑓 f_{+} : (i)The shape is constrained by dispersion relations Boyd:1994tt . This means that if we know f + subscript 𝑓 f_{+} at a few well spaced points we can pretty much determine the whole function f + subscript 𝑓 f_{+} . (ii)We can get a rough measurement of the form factor at q 2 = m π 2 superscript 𝑞 2 superscript subscript 𝑚 𝜋 2 q^{2}=m_{\pi}^{2} from the rate for B → π ​ π → 𝐵 𝜋 𝜋 B\to\pi\pi Bauer:2004tj . This requires a sophisticated effective theory (SCET) analysis which both shows that the leading order contains a term with f + ​ ( m π 2 ) subscript 𝑓 superscript subscript 𝑚 𝜋 2 f_{+}(m_{\pi}^{2}) and systematically characterizes the corrections to the lowest order SCET. It is safe to assume that this determination of f + ​ ( m π 2 ) subscript 𝑓 superscript subscript 𝑚 𝜋 2 f_{+}(m_{\pi}^{2}) will not improve beyond the 10% mark. Lattice QCD can determine the form factor, at least over a limited region of large q 2 superscript 𝑞 2 q^{2} . At the moment there is some disagreement between the best two lattice calculations, which however use the same method Shigemitsu:2004ft . A skeptic would require not only agreement between the two existing calculations but also with other methods, not to mention a set of additional independent successful post-dictions, before the result can be trusted for a precision determination of | V u ​ b | subscript 𝑉 𝑢 𝑏 |V_{ub}| . The experimental and lattice measurements can be combined using constraints from dispersion relations and unitarity Arnesen:2005ez . Because these constraints follow from fundamentals, they do not introduce additional uncertainties. They improve the determination of | V u ​ b | subscript 𝑉 𝑢 𝑏 |V_{ub}| significantly. The lattice determination is for the q 2 superscript 𝑞 2 q^{2} -region where the rate is smallest. This is true even if the form factor is largest there, because in that region the rate is phase space suppressed. But a rough shape of the spectrum is experimentally observed, through a binned measurement Abe:2004zm , and the dispersion relation constraints allows one to combine the full experimental spectrum with the restricted- q 2 superscript 𝑞 2 q^{2} lattice measurement. The result of this analysis gives a 13% error in | V u ​ b | subscript 𝑉 𝑢 𝑏 |V_{ub}| , completely dominated by the lattice errors. Alternatives Exclusive and inclusive determinations of | V u ​ b | subscript 𝑉 𝑢 𝑏 |V_{ub}| have comparable precisions. Neither is very good and the prospect for significant improvement is limited. Other methods need be explored, if not to improve on existing | V u ​ b | subscript 𝑉 𝑢 𝑏 |V_{ub}| to lend confidence to the result. A lattice-free method would be preferable. A third method, proposed a while ago Ligeti:1995yz , uses the idea of double ratios Grinstein:1993ys to reduce hadronic uncertainties. Two independent approximate symmetries protect double ratios from deviations from unity, which are therefore of the order of the product of two small symmetry breaking parameters. For example, the double ratio ( f B s / f B d ) / ( f D s / f D d ) = ( f B s / f D s ) / ( f B d / f D d ) = 1 + 𝒪 ​ ( m s / m c ) subscript 𝑓 subscript 𝐵 𝑠 subscript 𝑓 subscript 𝐵 𝑑 subscript 𝑓 subscript 𝐷 𝑠 subscript 𝑓 subscript 𝐷 𝑑 subscript 𝑓 subscript 𝐵 𝑠 subscript 𝑓 subscript 𝐷 𝑠 subscript 𝑓 subscript 𝐵 𝑑 subscript 𝑓 subscript 𝐷 𝑑 1 𝒪 subscript 𝑚 𝑠 subscript 𝑚 𝑐 (f_{B_{s}}/f_{B_{d}})/(f_{D_{s}}/f_{D_{d}})=(f_{B_{s}}/f_{D_{s}})/(f_{B_{d}}/f_{D_{d}})=1+{\cal O}(m_{s}/m_{c}) because f B s / f B d = f D s / f D d = 1 subscript 𝑓 subscript 𝐵 𝑠 subscript 𝑓 subscript 𝐵 𝑑 subscript 𝑓 subscript 𝐷 𝑠 subscript 𝑓 subscript 𝐷 𝑑 1 f_{B_{s}}/f_{B_{d}}=f_{D_{s}}/f_{D_{d}}=1 by S ​ U ​ ( 3 ) 𝑆 𝑈 3 SU(3) flavor, while f B s / f D s = f B d / f D d = m c / m b subscript 𝑓 subscript 𝐵 𝑠 subscript 𝑓 subscript 𝐷 𝑠 subscript 𝑓 subscript 𝐵 𝑑 subscript 𝑓 subscript 𝐷 𝑑 subscript 𝑚 𝑐 subscript 𝑚 𝑏 f_{B_{s}}/f_{D_{s}}=f_{B_{d}}/f_{D_{d}}=\sqrt{m_{c}/m_{b}} by heavy flavor symmetry. One can extract | V u ​ b / V t ​ s ​ V t ​ b | subscript 𝑉 𝑢 𝑏 subscript 𝑉 𝑡 𝑠 subscript 𝑉 𝑡 𝑏 |V_{ub}/V_{ts}V_{tb}| by measuring the ratio, d ​ Γ ​ ( B ¯ d → ρ ​ ℓ ​ ν ) / d ​ q 2 d ​ Γ ​ ( B ¯ d → K ∗ ​ ℓ + ​ ℓ − ) / d ​ q 2 = | V u ​ b | 2 | V t ​ s ​ V t ​ b | 2 ⋅ 8 ​ π 2 α 2 ⋅ 1 N ​ ( q 2 ) ⋅ R B , d Γ → subscript ¯ 𝐵 𝑑 𝜌 ℓ 𝜈 d superscript 𝑞 2 d Γ → subscript ¯ 𝐵 𝑑 superscript 𝐾 superscript ℓ superscript ℓ d superscript 𝑞 2 ⋅ superscript subscript 𝑉 𝑢 𝑏 2 superscript subscript 𝑉 𝑡 𝑠 subscript 𝑉 𝑡 𝑏 2 8 superscript 𝜋 2 superscript 𝛼 2 1 𝑁 superscript 𝑞 2 subscript 𝑅 𝐵 \frac{{\rm d}\Gamma(\bar{B}_{d}\to\rho\ell\nu)/{\rm d}q^{2}}{{\rm d}\Gamma(\bar{B}_{d}\to K^{*}\ell^{+}\ell^{-})/{\rm d}q^{2}}=\frac{|V_{ub}|^{2}}{|V_{ts}V_{tb}|^{2}}\cdot\frac{8\pi^{2}}{\alpha^{2}}\cdot\frac{1}{N(q^{2})}\cdot R_{B}, (2) where q 2 superscript 𝑞 2 q^{2} is the lepton pair invariant mass, and N ​ ( q 2 ) 𝑁 superscript 𝑞 2 N(q^{2}) is a known function Grinstein:2004vb . When expressed as functions of the rapidity of the vector meson, y = E V / m V 𝑦 subscript 𝐸 𝑉 subscript 𝑚 𝑉 y=E_{V}/m_{V} , the ratios of helicity amplitudes R B = ∑ λ | H λ B → ρ ​ ( y ) | 2 ∑ λ | H λ B → K ∗ ​ ( y ) | 2 , R D = ∑ λ | H λ D → ρ ​ ( y ) | 2 ∑ λ | H λ D → K ∗ ​ ( y ) | 2 , formulae-sequence subscript 𝑅 𝐵 subscript 𝜆 superscript subscript superscript 𝐻 → 𝐵 𝜌 𝜆 𝑦 2 subscript 𝜆 superscript subscript superscript 𝐻 → 𝐵 superscript 𝐾 𝜆 𝑦 2 subscript 𝑅 𝐷 subscript 𝜆 superscript subscript superscript 𝐻 → 𝐷 𝜌 𝜆 𝑦 2 subscript 𝜆 superscript subscript superscript 𝐻 → 𝐷 superscript 𝐾 𝜆 𝑦 2 R_{B}=\frac{\sum_{\lambda}|H^{B\to\rho}_{\lambda}(y)|^{2}}{\sum_{\lambda}|H^{B\to K^{*}}_{\lambda}(y)|^{2}},\quad R_{D}=\frac{\sum_{\lambda}|H^{D\to\rho}_{\lambda}(y)|^{2}}{\sum_{\lambda}|H^{D\to K^{*}}_{\lambda}(y)|^{2}}, are related by a double ratio: R B ​ ( y ) = R D ​ ( y ) ​ ( 1 + 𝒪 ​ ( m s ​ ( m c − 1 − m b − 1 ) ) ) subscript 𝑅 𝐵 𝑦 subscript 𝑅 𝐷 𝑦 1 𝒪 subscript 𝑚 𝑠 superscript subscript 𝑚 𝑐 1 superscript subscript 𝑚 𝑏 1 R_{B}(y)=R_{D}(y)(1+{\cal O}(m_{s}(m_{c}^{-1}-m_{b}^{-1}))) . This measurement could be done today: CLEO has accurately measured the required semileptonic D 𝐷 D decays Adam:2007pv ; Gray:2007pw . A fourth method is available if we are willing to use rarer decays. To extract | V u ​ b | subscript 𝑉 𝑢 𝑏 |V_{ub}| from ℬ ​ ( B + → τ + ​ ν τ ) = ( 0.88 ​ + 0.68 − 0.67 ± 0.11 ) × 10 − 4 ℬ → superscript 𝐵 superscript 𝜏 subscript 𝜈 𝜏 plus-or-minus 0.88 FRACOP 0.68 0.67 0.11 superscript 10 4 {\cal B}(B^{+}\to\tau^{+}\nu_{\tau})=(0.88{\textstyle{+0.68\atop-0.67}}\pm 0.11)\times 10^{-4} Aubert:2004kz one needs a lattice determination of f B subscript 𝑓 𝐵 f_{B} . Since we want to move away from relying on non-perturbative methods (lattice) to extract | V u ​ b | subscript 𝑉 𝑢 𝑏 |V_{ub}| we propose a cleaner but more difficult measurement, the double ratio Γ ​ ( B u → τ ​ ν ) Γ ​ ( B s → ℓ + ​ ℓ − ) Γ ​ ( D d → ℓ ​ ν ) Γ ​ ( D s → ℓ ​ ν ) ∼ | V u ​ b | 2 | V t ​ s ​ V t ​ b | 2 ⋅ π 2 α 2 ⋅ ( f B / f B s f D / f D s ) 2 similar-to Γ → subscript 𝐵 𝑢 𝜏 𝜈 Γ → subscript 𝐵 𝑠 superscript ℓ superscript ℓ Γ → subscript 𝐷 𝑑 ℓ 𝜈 Γ → subscript 𝐷 𝑠 ℓ 𝜈 ⋅ superscript subscript 𝑉 𝑢 𝑏 2 superscript subscript 𝑉 𝑡 𝑠 subscript 𝑉 𝑡 𝑏 2 superscript 𝜋 2 superscript 𝛼 2 superscript subscript 𝑓 𝐵 subscript 𝑓 subscript 𝐵 𝑠 subscript 𝑓 𝐷 subscript 𝑓 subscript 𝐷 𝑠 2 \frac{\frac{\Gamma(B_{u}\to\tau\nu)}{\Gamma(B_{s}\to\ell^{+}\ell^{-})}}{\frac{\Gamma(D_{d}\to\ell\nu)}{\Gamma(D_{s}\to\ell\nu)}}\sim\frac{|V_{ub}|^{2}}{|V_{ts}V_{tb}|^{2}}\cdot\frac{\pi^{2}}{\alpha^{2}}\cdot\left(\frac{f_{B}/f_{B_{s}}}{f_{D}/f_{D_{s}}}\right)^{2} (3) In the SM ℬ ​ ( B s → μ + ​ μ − ) ≈ 3.5 × 10 − 9 ℬ → subscript 𝐵 𝑠 superscript 𝜇 superscript 𝜇 3.5 superscript 10 9 {\cal B}(B_{s}\to\mu^{+}\mu^{-})\approx 3.5\times 10^{-9} × ( f B s / 210 ​ MeV ) 2 ​ ( | V t ​ s | / 0.040 ) 2 absent superscript subscript 𝑓 subscript 𝐵 𝑠 210 MeV 2 superscript subscript 𝑉 𝑡 𝑠 0.040 2 \times(f_{B_{s}}/210\,\text{MeV})^{2}(|V_{ts}|/0.040)^{2} is the only presently unknown quantity in the double ratio and is expected to be well measured at the LHC Schopper:2006he . The ratio Γ ​ ( B + → τ + ​ ν ) / Γ ​ ( B d → μ + ​ μ − ) Γ → superscript 𝐵 superscript 𝜏 𝜈 Γ → subscript 𝐵 𝑑 superscript 𝜇 superscript 𝜇 \Gamma(B^{+}\to\tau^{+}\nu)/\Gamma(B_{d}\to\mu^{+}\mu^{-}) gives us a fifth method. It has basically no hadronic uncertainty, since the hadronic factor f B / f B d = 1 subscript 𝑓 𝐵 subscript 𝑓 subscript 𝐵 𝑑 1 f_{B}/f_{B_{d}}=1 , by isospin. It involves | V u ​ b | 2 / | V t ​ d ​ V t ​ b | 2 superscript subscript 𝑉 𝑢 𝑏 2 superscript subscript 𝑉 𝑡 𝑑 subscript 𝑉 𝑡 𝑏 2 |V_{ub}|^{2}/|V_{td}V_{tb}|^{2} , an unusual combination of CKMs. In the ρ − η 𝜌 𝜂 \rho-\eta plane it forms a circle centered at ∼ ( − 0.2 , 0 ) similar-to absent 0.2 0 \sim(-0.2,0) of radius ∼ 0.5 similar-to absent 0.5 \sim 0.5 . Of course, measuring Γ ​ ( B d → μ + ​ μ − ) Γ → subscript 𝐵 𝑑 superscript 𝜇 superscript 𝜇 \Gamma(B_{d}\to\mu^{+}\mu^{-}) is extremely hard. In a sixth method one studies wrong charm decays B ¯ d , s → D ¯ ​ X → subscript ¯ 𝐵 𝑑 𝑠 ¯ 𝐷 𝑋 \bar{B}_{d,s}\to\bar{D}X (really b ​ q ¯ → u ​ c ¯ → 𝑏 ¯ 𝑞 𝑢 ¯ 𝑐 b\bar{q}\to u\bar{c} ). This can be done both in semi-inclusive decays Falk:1999sa (an experimentally challenging measurement) or in exclusive decays Evans:1999wx (where an interesting connection to B d , s subscript 𝐵 𝑑 𝑠 B_{d,s} mixing matrix elements is involved). I.4 α 𝛼 \alpha from B → π ​ π , π ​ ρ , ρ ​ ρ → 𝐵 𝜋 𝜋 𝜋 𝜌 𝜌 𝜌 B\to\pi\pi,\pi\rho,\rho\rho . In principle the penguin contamination problem Grinstein:1989df requires a full isospin analysis Gronau:1990ka for a theoretically clean determination of the angle α 𝛼 \alpha . The angle determination works slightly better than we had a right to expect a priori. The reason lies in two empirical observation in B → ρ ​ ρ → 𝐵 𝜌 𝜌 B\to\rho\rho . First, the longitudinal polarization dominates, and therefore the final state is to good approximation a CP eigenstate (CP even, in fact). And second, the branching fraction for B → ρ 0 ​ ρ 0 → 𝐵 superscript 𝜌 0 superscript 𝜌 0 B\to\rho^{0}\rho^{0} is small: relative to B → ρ + ​ ρ − → 𝐵 superscript 𝜌 superscript 𝜌 B\to\rho^{+}\rho^{-} it is 6 ± 3 % plus-or-minus 6 percent 3 6\pm 3\% , to be compared with the neutral to charged decay into pions of 23 ± 4 % plus-or-minus 23 percent 4 23\pm 4\% . This means that the contamination from penguin operators is small and one can get a clean measurement of α 𝛼 \alpha . All three decay modes are about equally important in current fits, which give α = 93 ​ + 11 − 9 𝛼 93 FRACOP 11 9 \alpha=93{\textstyle{+11\atop-9}} degrees. I.5 γ 𝛾 \gamma from B ± → D ​ K ± → superscript 𝐵 plus-or-minus 𝐷 superscript 𝐾 plus-or-minus B^{\pm}\to DK^{\pm} . Three different methods are used. They are all based on the interference between Cabibbo-allowed ( e.g., B − → D 0 ​ K − → superscript 𝐵 superscript 𝐷 0 superscript 𝐾 B^{-}\to D^{0}K^{-} ) and suppressed decays ( e.g., B − → D 0 ¯ ​ K − → superscript 𝐵 ¯ superscript 𝐷 0 superscript 𝐾 B^{-}\to\overline{D^{0}}K^{-} ) with D 0 superscript 𝐷 0 D^{0} , D 0 ¯ ¯ superscript 𝐷 0 \overline{D^{0}} decaying to a common state. The GLW Gronau:1990ra method uses decays to a common CP eigenstate. In the ADS method Atwood:1996ci the final state is chosen to be a suppressed D 𝐷 D decay mode if the D 𝐷 D came from an allowed B 𝐵 B decay; for example, the final state in the charm decay can be taken to be K + ​ π − superscript 𝐾 superscript 𝜋 K^{+}\pi^{-} so it is doubly Cabibbo suppressed for a D 0 superscript 𝐷 0 D^{0} decay but allowed for a D 0 ¯ ¯ superscript 𝐷 0 \overline{D^{0}} decay. The efficacy of this method depends sensitively on the ratio of amplitudes, which can be measured separately, r B = | A ​ ( B − → D 0 ¯ ​ K − ) / A ​ ( B − → D 0 ​ K − ) | subscript 𝑟 𝐵 𝐴 → superscript 𝐵 ¯ superscript 𝐷 0 superscript 𝐾 𝐴 → superscript 𝐵 superscript 𝐷 0 superscript 𝐾 r_{B}=|A(B^{-}\to\overline{D^{0}}K^{-})/A(B^{-}\to D^{0}K^{-})| . In the GGSZ method Giri:2003ty the D 0 superscript 𝐷 0 D^{0} and D 0 ¯ ¯ superscript 𝐷 0 \overline{D^{0}} are reconstructed in a common three body final state. The results to date vary depending on which decay mode is actually used, so the determination of γ 𝛾 \gamma from all measurements combined is not very good, γ = 62 ​ + 38 − 24 𝛾 62 FRACOP 38 24 \gamma=62{\textstyle{+38\atop-24}} degrees. More data should improve the determination of γ 𝛾 \gamma . I.6 Are there anomalies? There seem to be as many papers in the literature claiming there is a B → K ​ π → 𝐵 𝐾 𝜋 B\to K\pi puzzle as those that claim it is not a puzzle. It is easy to see why. In order to find a puzzle one must know a priori the hadronic amplitudes. Those who find a puzzle in B → K ​ π → 𝐵 𝐾 𝜋 B\to K\pi make assumptions about hadronic amplitudes that those who find no puzzle think are unwarranted. Moreover, Ref. donoghue showed that soft final state interactions do not disappear in the large m b subscript 𝑚 𝑏 m_{b} limit, and Refs. falkFSI and wolfensteinFSI studied this quantitatively for B → K ​ π → 𝐵 𝐾 𝜋 B\to K\pi and B → π ​ π → 𝐵 𝜋 𝜋 B\to\pi\pi , respectively, and concluded the effects should be expected to be large. For example, the CP asymmetry in B → K ​ π → 𝐵 𝐾 𝜋 B\to K\pi could easily be 20% and the bound sin 2 ⁡ γ ≤ R superscript 2 𝛾 𝑅 \sin^{2}\gamma\leq R , where R = Γ ​ ( B d → π ∓ ​ K ± ) / Γ ​ ( B ± → π ± ​ K ) 𝑅 Γ → subscript 𝐵 𝑑 superscript 𝜋 minus-or-plus superscript 𝐾 plus-or-minus Γ → superscript 𝐵 plus-or-minus superscript 𝜋 plus-or-minus 𝐾 R=\Gamma(B_{d}\to\pi^{\mp}K^{\pm})/\Gamma(B^{\pm}\to\pi^{\pm}K) could easily be violated at the 20% level. The case for new physics in CPV in charmless b → s → 𝑏 𝑠 b\to s decays would seem to be stronger. Regardless of decay mode β e ​ f ​ f subscript 𝛽 𝑒 𝑓 𝑓 \beta_{eff} is predicted by SCET, QCD-factorization and pQCD to deviate from β J / ψ ​ K s subscript 𝛽 𝐽 𝜓 subscript 𝐾 𝑠 \beta_{J/\psi K_{s}} by a small positive amount. Experimentally the deviations vary from mode to mode but are all non-positive and not necessarily small. However, many things have to be checked before one can begin to believe we are seeing new physics here. First, all of the theoretical schemes need to come to terms with the soft final state interactions issue raised in donoghue or show that work is incorrect. Then, also, the fact that all deviations are negative strongly suggests that the measurements have been corrupted by an admixture of the opposite CP final state. In my view there is at present no case for deviations from the standard CKM model of flavor.","role":"user"},{"content":"Abstract: We first review some aspects of the determination of the sides and angles of the unitarity triangle. We pay particular attention to theory shortcomings, and present many alternative proposals for the determination of |Vub| (which at present is problematic). We then turn our attention to the more general question: What have we learned so far about flavor physics and where do we go from here? We argue that the aim of Flavor Physics should be to establish or rule out Minimal Flavor Violating interactions up to a scale of 10 TeV.","role":"assistant"}]

Review of factorization breaking in diffractive photoproduction of dijets

It is well known that at high-energy colliders such as the e ​ p 𝑒 𝑝 ep collider HERA at DESY and the p ​ p ¯ 𝑝 ¯ 𝑝 p\bar{p} collider Tevatron at Fermilab, a large fraction of the observed events are diffractive. These events are defined experimentally by the presence of a forward-going hadronic system Y 𝑌 Y with four-momentum p Y subscript 𝑝 𝑌 p_{Y} , low mass M Y subscript 𝑀 𝑌 M_{Y} (typically a proton that remained intact or a proton plus low-lying nucleon resonances), small four-momentum transfer t = ( P − p Y ) 2 𝑡 superscript 𝑃 subscript 𝑝 𝑌 2 t=(P-p_{Y})^{2} , and small longitudinal-momentum transfer x I ​ P = q ​ ( P − p Y ) / ( q ​ P ) subscript 𝑥 𝐼 𝑃 𝑞 𝑃 subscript 𝑝 𝑌 𝑞 𝑃 x_{I\!\!P}=q(P-p_{Y})/(qP) from the incoming proton with four momentum P 𝑃 P to the central hadronic system X 𝑋 X (see Fig. 1 for the case of e ​ p → e ​ X ​ Y → 𝑒 𝑝 𝑒 𝑋 𝑌 ep\rightarrow eXY ). Experimentally, a large Figure 1: Diffractive scattering process e ​ p → e ​ X ​ Y → 𝑒 𝑝 𝑒 𝑋 𝑌 ep\to eXY , where the hadronic systems X 𝑋 X and Y 𝑌 Y are separated by the largest rapidity gap in the final state. rapidity gap separates the hadronic system X 𝑋 X with invariant mass M X subscript 𝑀 𝑋 M_{X} from the final-state system Y 𝑌 Y with invariant mass M Y subscript 𝑀 𝑌 M_{Y} . Theoretically, diffractive interactions are described in the framework of Regge theory [ 1 ] as the exchange of a trajectory with vacuum quantum numbers, the pomeron ( I ​ P 𝐼 𝑃 I\!\!P ) trajectory. Then the object exchanged between the systems X 𝑋 X and Y 𝑌 Y , as indicated in Fig. 1 , is the pomeron (or additional lower-lying Regge poles), and the upper vertex of the process e ​ I ​ P → e ​ X → 𝑒 𝐼 𝑃 𝑒 𝑋 eI\!\!P\rightarrow eX can be interpreted as deep-inelastic scattering (DIS) on the pomeron target for the case that the virtuality of the exchanged photon Q 2 = − q 2 superscript 𝑄 2 superscript 𝑞 2 Q^{2}=-q^{2} is sufficiently large. In analogy to DIS on a proton target, e ​ p → e ​ X → 𝑒 𝑝 𝑒 𝑋 ep\rightarrow eX , the cross section for the process e ​ I ​ P → e ​ X → 𝑒 𝐼 𝑃 𝑒 𝑋 eI\!\!P\rightarrow eX in the DIS region can be expressed as the convolution of partonic cross sections and universal parton distribution functions (PDFs) of the pomeron. The partonic cross sections are the same as for DIS e ​ p 𝑒 𝑝 ep scattering. Usually these pomeron PDFs are multiplied with vertex functions for the lower vertex in Fig. 1 , yielding the diffractive parton distribution functions (DPDFs). The Q 2 superscript 𝑄 2 Q^{2} -evolution of the DPDFs is calculated with the usual DGLAP [ 2 ] evolution equations known from e ​ p → e ​ X → 𝑒 𝑝 𝑒 𝑋 ep\rightarrow eX DIS. Except for their evolution with Q 2 superscript 𝑄 2 Q^{2} , the DPDFs can not be calculated in the framework of perturbative QCD and must be determined from experiment. Such DPDFs [ 3 , 4 , 5 , 6 ] have been obtained from the HERA inclusive measurements of the diffractive structure function F 2 D subscript superscript 𝐹 𝐷 2 F^{D}_{2} [ 3 , 4 ] , defined analogously to the proton structure function F 2 subscript 𝐹 2 F_{2} . Similarly to diffractive DIS, e ​ p → e ​ X ​ Y → 𝑒 𝑝 𝑒 𝑋 𝑌 ep\rightarrow eXY , where the presence of the large scale Q 𝑄 Q allows for the application of perturbative QCD and X 𝑋 X comprises the sum over all possible final states, many other processes with a hard scale provided by specific final states in the central system X 𝑋 X can be predicted using QCD perturbation theory. Such processes, usually called hard diffractive processes, are e.g. dijet production in diffractive photoproduction ( Q 2 ≃ 0 similar-to-or-equals superscript 𝑄 2 0 Q^{2}\simeq 0 ) and DIS ( Q 2 ≠ 0 superscript 𝑄 2 0 Q^{2}\neq 0 ), where the large scale is given by the jet transverse energy E T j ​ e ​ t superscript subscript 𝐸 𝑇 𝑗 𝑒 𝑡 E_{T}^{jet} and possibly Q 𝑄 Q , and diffractive open heavy-flavor production, where the large scale is given by the heavy-flavor mass and possibly E T subscript 𝐸 𝑇 E_{T} and/or Q 𝑄 Q , in photoproduction or DIS and many more diffractive processes induced by p ​ p ¯ 𝑝 ¯ 𝑝 p\bar{p} or p ​ p 𝑝 𝑝 pp collisions. The central problem in hard diffraction is the problem of QCD factorization, i.e. the question whether diffractive cross sections are factorisable into universal DPDFs and partonic cross sections, which are calculable in perturbative QCD. This question is the subject of the current debate in diffractive physics and is of particular interest for the prospects of discovery of new particles such as the Higgs boson in diffractive reactions at the LHC [ 7 , 8 , 9 ] . For the inclusive DIS process, factorization has indeed been proven to hold [ 10 ] , and on this basis DPDFs have been extracted at Q 2 ≠ 0 superscript 𝑄 2 0 Q^{2}\neq 0 [ 3 , 4 , 5 ] from high-precision inclusive measurements of the process e ​ p → e ​ X ​ Y → 𝑒 𝑝 𝑒 𝑋 𝑌 ep\rightarrow eXY using the usual DGLAP evolution equations. The proof of the factorization formula, usually referred to as the validity of QCD factorization in hard diffraction, also appears to be valid for the production of specific final states in DIS, as e.g. the production of jets or heavy-flavor particles, and for the direct part of photoproduction ( Q 2 ≃ 0 similar-to-or-equals superscript 𝑄 2 0 Q^{2}\simeq 0 ) or low- Q 2 superscript 𝑄 2 Q^{2} electroproduction of jets [ 10 ] . However, factorization does not hold for hard processes in diffractive hadron-hadron scattering. The problem is that soft interactions between the ingoing hadrons and/or their remnants occur in both the initial and the final state. This agrees with experimental measurements at the Tevatron [ 11 ] . Predictions of diffractive dijet cross sections for collisions as measured by CDF using DPDFs determined earlier by the H1 collaboration [ 12 ] at HERA overestimate the measured cross section by up to an order of magnitude [ 11 ] . This large suppression of the CDF cross section can be explained by the rescattering of the two incoming hadron beams, which, by creating additional hadrons, destroy the rapidity gap [ 13 ] . Jet production with real photons involves direct interactions of the photon with quarks or gluons originating from the proton or pomeron, respectively, as well as resolved photon contributions leading to parton-parton interactions with an additional remnant jet coming from the photon as reviewed in [ 14 ] (see Fig. 2 ). For the direct interactions, we expect factorization Figure 2: Diffractive production of dijets with invariant mass M 12 subscript 𝑀 12 M_{12} in direct (left) and resolved (right) photon-pomeron collisions, leading to the production of one or two additional remnant jets. to be valid as in the case of inclusive DIS, as already mentioned, whereas we expect it to fail for the resolved process as in hadron-hadron scattering. For this part of photoproduction we would therefore expect a similar suppression factor (sometimes also called rapidity-gap survival probability) due to rescattering effects of the ingoing partons or hadrons. Introducing vector-meson dominance photon fluctuations, such a suppression by about a factor of three was predicted for resolved photoproduction at HERA [ 15 ] . The first measurements of dijet cross sections in diffractive photoproduction have been presented by the H1 collaboration as contributions to two conferences [ 16 ] . The kinematic range for these data were Q 2 < 0.01 superscript 𝑄 2 0.01 Q^{2}<0.01 GeV 2 , x I ​ P < 0.03 subscript 𝑥 𝐼 𝑃 0.03 x_{I\!\!P}<0.03 , E T j ​ e ​ t ​ 1 > 5 superscript subscript 𝐸 𝑇 𝑗 𝑒 𝑡 1 5 E_{T}^{jet1}>5 GeV, E T j ​ e ​ t ​ 2 > 4 superscript subscript 𝐸 𝑇 𝑗 𝑒 𝑡 2 4 E_{T}^{jet2}>4 GeV and 165 < W < 240 165 𝑊 240 165<W<240 GeV, where jets were identified using the inclusive k T subscript 𝑘 𝑇 k_{T} -cluster algorithm (the definitions of these and the following variables will be given in the next section). The measured cross sections as a function of x γ o ​ b ​ s superscript subscript 𝑥 𝛾 𝑜 𝑏 𝑠 x_{\gamma}^{obs} and z I ​ P o ​ b ​ s superscript subscript 𝑧 𝐼 𝑃 𝑜 𝑏 𝑠 z_{I\!\!P}^{obs} were compared to leading-order (LO) QCD predictions, using the RAPGAP Monte Carlo model [ 17 ] . For the DPDFs the LO ‘H1 2002 fit’ was used [ 12 ] . The two cross sections were found to be well described by the predictions in normalization and shape over the whole range of x γ o ​ b ​ s superscript subscript 𝑥 𝛾 𝑜 𝑏 𝑠 x_{\gamma}^{obs} and z I ​ P o ​ b ​ s superscript subscript 𝑧 𝐼 𝑃 𝑜 𝑏 𝑠 z_{I\!\!P}^{obs} , showing no breakdown of factorization neither in resolved nor in direct photoproduction. In addition, normalized cross sections as a function of various other variables were compared to the predictions with the result that all measured distributions were in good agreement. Subsequently we calculated the next-to-leading order (NLO) corrections for the cross section of diffractive dijet production using the same kinematic cuts and with the same DPDFs as in the first H1 analysis [ 16 ] on the basis of our previous work on NLO corrections for inclusive direct [ 18 ] and resolved [ 19 ] dijet photoproduction. While at LO good agreement with the H1 data [ 16 ] was found, consistent with the finding in the H1 analysis [ 16 ] , it was found that the NLO corrections increase the cross section significantly [ 20 , 21 ] and require a suppression factor of the order of R = 0.5 𝑅 0.5 R=0.5 . Since on theoretical grounds only a suppression of the resolved cross section would be acceptable, we demonstrated in [ 20 , 21 ] that by multiplying the resolved cross section with the suppression factor R = 0.34 𝑅 0.34 R=0.34 , reasonably good agreement with the preliminary H1 data [ 16 ] could be achieved. This value for the suppression factor turned out to be in good agreement with the prediction of [ 15 ] . The first experimental data from the ZEUS collaboration were presented at the DIS workshop in 2004 [ 22 ] . The dijet cross sections were obtained in the kinematic range Q 2 < 1 superscript 𝑄 2 1 Q^{2}<1 GeV 2 , x I ​ P < 0.035 subscript 𝑥 𝐼 𝑃 0.035 x_{I\!\!P}<0.035 and E T j ​ e ​ t ​ 1 ​ ( 2 ) > 7.5 superscript subscript 𝐸 𝑇 𝑗 𝑒 𝑡 1 2 7.5 E_{T}^{jet1(2)}>7.5 ( 6.5 ) 6.5 (6.5) GeV. For these kinematic constraints NLO calculations were not available in 2004. So, the measurements were compared to LO calculations, unfortunately with previous H1 DPDFs [ 23 ] with the result, that good agreement in the shape was achieved. But the normalization was off by a factor of 0.6 0.6 0.6 , which was attributed to the older DPDF input [ 24 ] , so that the H1 and ZEUS results were consistent with each other. The situation concerning the agreement of H1 and ZEUS data and the influence of NLO corrections improved already considerably in the fall of 2004. These preliminary data from both HERA collaborations together with comparisons to NLO calculations based on the DPDF fits from [ 12 ] were presented at workshops and conferences in the following years. In 2006 the H1 collaboration published their final DPDF fits from their high-precision measurements using the DGLAP evolution equations [ 4 ] . This analysis was based on the larger data sample of the years 1997-2000 as compared to the earlier preliminary DPDF sets [ 12 ] . In [ 4 ] two DPDF sets, the ’H1 2006 fit A’ and the ’H1 2006 fit B’ were presented, which both give a good description of the inclusive diffractive data. These two sets differ mainly in the gluon density at large fractional parton momenta, which is poorly constrained by the inclusive diffractive scattering data, since there is no direct coupling of the photon to gluons, so that the gluon density is constrained only through the evolution. The gluon density of fit A is peaked at the starting scale at large fractional momenta, whereas fit B is flat in this region. In 2007 the final publications for diffractive dijet production appeared [ 25 ] . The comparison between these experimental results and the NLO theory was based on the new and final DPDFs from H1 [ 4 ] . The differential cross sections as measured by H1 [ 25 ] were compared to NLO predictions obtained with the Frixione program [ 26 ] interfaced to the ‘H1 2006 fit B’ DPDFs. The conclusions deduced earlier from the comparison with the preliminary data and the preliminary ‘H1 2002 fit’ [ 12 ] are fully confirmed in [ 25 ] with the new DPDFs fits [ 4 ] . In particular, a global suppression is obtained, independent of the DPDFs fits used, i.e. fit A or fit B, by considering the ratio of measured dijet cross sections to NLO predictions in photoproduction in relation to the same ratio in DIS. In this comparison the value of the suppression is 0.5 0.5 0.5 . In addition, by using this overall suppression factor, H1 obtained a good description of all the measured distributions in the variables z I ​ P o ​ b ​ s superscript subscript 𝑧 𝐼 𝑃 𝑜 𝑏 𝑠 z_{I\!\!P}^{obs} , x γ o ​ b ​ s superscript subscript 𝑥 𝛾 𝑜 𝑏 𝑠 x_{\gamma}^{obs} , x I ​ P subscript 𝑥 𝐼 𝑃 x_{I\!\!P} , W 𝑊 W , E T j ​ e ​ t ​ 1 superscript subscript 𝐸 𝑇 𝑗 𝑒 𝑡 1 E_{T}^{jet1} , η ¯ j ​ e ​ t ​ s superscript ¯ 𝜂 𝑗 𝑒 𝑡 𝑠 \bar{\eta}^{jets} , | Δ ​ η j ​ e ​ t ​ s | Δ superscript 𝜂 𝑗 𝑒 𝑡 𝑠 |\Delta\eta^{jets}| and M 12 subscript 𝑀 12 M_{12} interfaced with the ‘H1 2006 fit B’ DPDFs and taking into account hadronization corrections [ 25 ] . Finally, the H1 collaboration investigated how well the data are describable under the assumption that in the NLO calculation the cross section for x γ o ​ b ​ s > 0.9 superscript subscript 𝑥 𝛾 𝑜 𝑏 𝑠 0.9 x_{\gamma}^{obs}>0.9 is not suppressed. The best agreement in a fit was obtained for a suppression factor 0.44 for the NLO calculation with x γ o ​ b ​ s < 0.9 superscript subscript 𝑥 𝛾 𝑜 𝑏 𝑠 0.9 x_{\gamma}^{obs}<0.9 , based on fitting the distributions for x γ o ​ b ​ s superscript subscript 𝑥 𝛾 𝑜 𝑏 𝑠 x_{\gamma}^{obs} , W 𝑊 W , η ¯ j ​ e ​ t ​ s superscript ¯ 𝜂 𝑗 𝑒 𝑡 𝑠 \bar{\eta}^{jets} and E T j ​ e ​ t ​ 1 superscript subscript 𝐸 𝑇 𝑗 𝑒 𝑡 1 E_{T}^{jet1} . In this comparison they found disagreement for the largest x γ o ​ b ​ s superscript subscript 𝑥 𝛾 𝑜 𝑏 𝑠 x_{\gamma}^{obs} -bin and the lowest η ¯ j ​ e ​ t ​ s superscript ¯ 𝜂 𝑗 𝑒 𝑡 𝑠 \bar{\eta}^{jets} (which are related), but better agreement in the E T j ​ e ​ t ​ 1 superscript subscript 𝐸 𝑇 𝑗 𝑒 𝑡 1 E_{T}^{jet1} -distribution. In [ 25 ] this leads to the statement, that the assumption that the direct cross section obeys factorization is strongly disfavored by their analysis. In total, it is obvious that in the final H1 analysis [ 25 ] a global suppression in diffractive dijet photoproduction is clearly established and the model with resolved suppression only is not as well supported by the data. Just recently also the ZEUS collaboration presented their final result on diffractive dijet photoproduction [ 27 ] . As in their preliminary analysis, the two jets with the highest transverse energies E T j ​ e ​ t superscript subscript 𝐸 𝑇 𝑗 𝑒 𝑡 E_{T}^{jet} were required to satisfy E T j ​ e ​ t ​ 1 ​ ( 2 ) > 7.5 superscript subscript 𝐸 𝑇 𝑗 𝑒 𝑡 1 2 7.5 E_{T}^{jet1(2)}>7.5 ( 6.5 ) 6.5 (6.5) GeV, which is higher than in the H1 analysis with E T j ​ e ​ t ​ 1 ​ ( 2 ) > 5 superscript subscript 𝐸 𝑇 𝑗 𝑒 𝑡 1 2 5 E_{T}^{jet1(2)}>5 ( 4 ) 4 (4) GeV [ 25 ] . ZEUS compared their measurements with the NLO predictions for diffractive photoproduction of dijets based on our program [ 21 ] . Three sets of DPDFs were used, the ZEUS LPS fit, determined from a NLO analysis of inclusive diffraction and diffractive charm-production data [ 3 ] , and the two H1 fits, H1 2006 fit A and fit B [ 4 ] . The NLO results obtained with the two H1 fits were scaled down by a factor of 0.87 [ 4 ] , since the H1 measurements used to derive the DPDFs include low-mass proton dissociative processes with M Y < 1.6 subscript 𝑀 𝑌 1.6 M_{Y}<1.6 GeV, which increases the photon-diffractive cross section by 1.15 − 0.08 + 0.15 subscript superscript 1.15 0.15 0.08 1.15^{+0.15}_{-0.08} as compared to the pure proton final state as corrected to in the ZEUS analysis. The comparison of the measured cross sections and the theoretical predictions was based on the distributions in the variables y 𝑦 y , M X subscript 𝑀 𝑋 M_{X} , x I ​ P subscript 𝑥 𝐼 𝑃 x_{I\!\!P} , z I ​ P o ​ b ​ s superscript subscript 𝑧 𝐼 𝑃 𝑜 𝑏 𝑠 z_{I\!\!P}^{obs} , E T j ​ e ​ t ​ 1 superscript subscript 𝐸 𝑇 𝑗 𝑒 𝑡 1 E_{T}^{jet1} , η l ​ a ​ b j ​ e ​ t ​ 1 subscript superscript 𝜂 𝑗 𝑒 𝑡 1 𝑙 𝑎 𝑏 \eta^{jet1}_{lab} and x γ o ​ b ​ s superscript subscript 𝑥 𝛾 𝑜 𝑏 𝑠 x_{\gamma}^{obs} . The data were reasonably well described in their shape as a function of these variables and lay systematically below the predictions. The predictions for the three DPDFs differed appreciably. The cross sections for the H1 2006 fit A (fit B) were the highest (lowest) and the one for the ZEUS LPS fit lay between the two, but nearer to the fit A than the fit B predictions. For d ​ σ / d ​ x γ o ​ b ​ s 𝑑 𝜎 𝑑 superscript subscript 𝑥 𝛾 𝑜 𝑏 𝑠 d\sigma/dx_{\gamma}^{obs} ZEUS also showed the ratio of the data and the NLO predictions using the ZEUS LPS fit. It was consistent with a suppression factor of 0.7 independent of x γ o ​ b ​ s superscript subscript 𝑥 𝛾 𝑜 𝑏 𝑠 x_{\gamma}^{obs} . This suppression factor depended on the DPDFs and ranged between 0.6 (H1 2006 fit A) and 0.9 (fit B). Taking into account the scale dependence of the theoretical predictions the ratio was outside the theoretical uncertainty for the ZEUS LPS fit and the H1 2006 fit A, but not for fit B. In their conclusions the authors of the ZEUS analysis [ 27 ] made the statement that the NLO calculations tend to overestimate the measured cross section, which would mean that a suppression is present. Unfortunately, however, they continued, that, within the large uncertainties of the NLO calculations, the data were compatible with the QCD calculations, i.e. with no suppression. Such a statement clearly contradicts the result of the H1 collaboration [ 25 ] and casts doubts on the correctness of the H1 analysis. The authors of [ 27 ] attribute this discrepancy to the fact that the H1 measurements [ 25 ] were carried out in a lower E T j ​ e ​ t superscript subscript 𝐸 𝑇 𝑗 𝑒 𝑡 E_{T}^{jet} and a higher x I ​ P subscript 𝑥 𝐼 𝑃 x_{I\!\!P} range than those in the ZEUS study [ 27 ] . Besides the different E T j ​ e ​ t superscript subscript 𝐸 𝑇 𝑗 𝑒 𝑡 E_{T}^{jet} and x I ​ P subscript 𝑥 𝐼 𝑃 x_{I\!\!P} regions in [ 25 ] and [ 27 ] , the two measurements suffer also from different experimental cuts of some other variables, which makes it difficult to compare the two data sets directly (note also the lower center-of-mass energy for the H1 data). In addition the comparison with NLO theory in [ 25 ] and [ 27 ] was done with two different programs [ 19 ] versus [ 26 ] , which, however agreed quite well with each other [ 25 ] . The rather different conclusions concerning factorization breaking in diffractive dijet photoproduction calls for a new comparative study of the two data sets in [ 25 ] and [ 27 ] . We have therefore performed a new calculation of the NLO cross sections on the basis of our earlier work [ 21 ] with the new H1 2006 DPDFs and revised hadronic corrections as compared to [ 20 ] , in order to see whether we can confirm the very different conclusions achieved in the H1 [ 25 ] and ZEUS [ 27 ] analyses. In the comparison with the new data sets we shall follow more or less the same strategy as in our earlier work [ 20 , 21 ] . We first calculate the unsuppressed NLO cross sections including an error band based on the scale variation and see how much and in which distribution the data points lie inside or outside this error band. Then we determine a global suppression factor by fitting the differential cross section d ​ σ / d ​ E T j ​ e ​ t ​ 1 𝑑 𝜎 𝑑 superscript subscript 𝐸 𝑇 𝑗 𝑒 𝑡 1 d\sigma/dE_{T}^{jet1} at the bin with the lowest E T j ​ e ​ t ​ 1 superscript subscript 𝐸 𝑇 𝑗 𝑒 𝑡 1 E_{T}^{jet1} . With this suppression factor we shall compare to the differential cross sections of all the other measured variables and look for consistency. In this new comparison between the experimental and the theoretical results we shall concentrate on using the H1 2006 fit B [ 4 ] input, since it leads to smaller cross sections than the DPDFs from H1 2006 fit A [ 4 ] or the ZEUS LPS DPDF fit [ 3 ] . Actually the H1 collaboration constructed a third set of DPDFs, which is called the ’H1 2007 fit jets’. This fit is obtained through a simultaneous fit to the diffractive inclusive and DIS dijet cross sections [ 28 ] . It is performed under the assumption that there is no factorization breaking in the diffractive dijet cross sections. Under this assumption, including the diffractive dijet cross sections in the analysis leads to additional constraints, mostly on the diffractive gluon distribution. On average the ’H1 2007 fit jets’ is similar to the ’H1 2006 fit B’ except for the gluon distribution at high momentum fraction and smaller factorization scales. In our analysis we shall disregard this new set of DPDFs, since it would be compatible with the factorization test of the photoproduction data only if we restricted these tests to the case that only the resolved part has this breaking and not the direct part, which has the same structure as the DIS dijet cross section. In Sec. 2 we shall present the complete list of cuts on the experimental variables, give all the input used in the cross section calculations, and present the basic formulæ, from which the dijet cross sections have been calculated. The comparison with the H1 [ 25 ] and the ZEUS [ 27 ] experimental data is presented and discussed in Sec. 3. In this comparison we shall concentrate on the main question, whether there is a suppression in the photoproduction data at all. In addition we shall investigate also whether a reasonable description of the data is possible with suppression of the resolved cross section only, as we studied it already in our previous work in 2004 [ 20 , 21 ] . In Sec. 4 we shall finish with a summary and our conclusions.

After the final analyses of the H1 and ZEUS collaborations for the diffractive photoproduction of dijets have appeared, we have recalculated these cross sections in next-to-leading order (NLO) of perturbative QCD to see whether they can be interpreted consistently. The results of these calculations are compared to the data of both collaborations. We find that at NLO the cross sections disagree with the data, showing that factorization breaking occurs at this order. If direct and resolved contributions are both suppressed by the same amount, the global suppression factor depends on the transverse-energy cut and is 0.42 for the H1 and 0.71 for the ZEUS analysis. However, by suppressing only the resolved contribution by a factor of approximately three, also reasonably good agreement with all the data is found. The size of the factorization breaking effects for resolved photons agrees with absorptive-model predictions.

[{"content":"Help me to generate the abstract of a survey paper given the title:*Review of factorization breaking in diffractive photoproduction of dijets* and the introduction:It is well known that at high-energy colliders such as the e ​ p 𝑒 𝑝 ep collider HERA at DESY and the p ​ p ¯ 𝑝 ¯ 𝑝 p\bar{p} collider Tevatron at Fermilab, a large fraction of the observed events are diffractive. These events are defined experimentally by the presence of a forward-going hadronic system Y 𝑌 Y with four-momentum p Y subscript 𝑝 𝑌 p_{Y} , low mass M Y subscript 𝑀 𝑌 M_{Y} (typically a proton that remained intact or a proton plus low-lying nucleon resonances), small four-momentum transfer t = ( P − p Y ) 2 𝑡 superscript 𝑃 subscript 𝑝 𝑌 2 t=(P-p_{Y})^{2} , and small longitudinal-momentum transfer x I ​ P = q ​ ( P − p Y ) / ( q ​ P ) subscript 𝑥 𝐼 𝑃 𝑞 𝑃 subscript 𝑝 𝑌 𝑞 𝑃 x_{I\!\!P}=q(P-p_{Y})/(qP) from the incoming proton with four momentum P 𝑃 P to the central hadronic system X 𝑋 X (see Fig. 1 for the case of e ​ p → e ​ X ​ Y → 𝑒 𝑝 𝑒 𝑋 𝑌 ep\rightarrow eXY ). Experimentally, a large Figure 1: Diffractive scattering process e ​ p → e ​ X ​ Y → 𝑒 𝑝 𝑒 𝑋 𝑌 ep\to eXY , where the hadronic systems X 𝑋 X and Y 𝑌 Y are separated by the largest rapidity gap in the final state. rapidity gap separates the hadronic system X 𝑋 X with invariant mass M X subscript 𝑀 𝑋 M_{X} from the final-state system Y 𝑌 Y with invariant mass M Y subscript 𝑀 𝑌 M_{Y} . Theoretically, diffractive interactions are described in the framework of Regge theory 1 as the exchange of a trajectory with vacuum quantum numbers, the pomeron ( I ​ P 𝐼 𝑃 I\!\!P ) trajectory. Then the object exchanged between the systems X 𝑋 X and Y 𝑌 Y , as indicated in Fig. 1 , is the pomeron (or additional lower-lying Regge poles), and the upper vertex of the process e ​ I ​ P → e ​ X → 𝑒 𝐼 𝑃 𝑒 𝑋 eI\!\!P\rightarrow eX can be interpreted as deep-inelastic scattering (DIS) on the pomeron target for the case that the virtuality of the exchanged photon Q 2 = − q 2 superscript 𝑄 2 superscript 𝑞 2 Q^{2}=-q^{2} is sufficiently large. In analogy to DIS on a proton target, e ​ p → e ​ X → 𝑒 𝑝 𝑒 𝑋 ep\rightarrow eX , the cross section for the process e ​ I ​ P → e ​ X → 𝑒 𝐼 𝑃 𝑒 𝑋 eI\!\!P\rightarrow eX in the DIS region can be expressed as the convolution of partonic cross sections and universal parton distribution functions (PDFs) of the pomeron. The partonic cross sections are the same as for DIS e ​ p 𝑒 𝑝 ep scattering. Usually these pomeron PDFs are multiplied with vertex functions for the lower vertex in Fig. 1 , yielding the diffractive parton distribution functions (DPDFs). The Q 2 superscript 𝑄 2 Q^{2} -evolution of the DPDFs is calculated with the usual DGLAP 2 evolution equations known from e ​ p → e ​ X → 𝑒 𝑝 𝑒 𝑋 ep\rightarrow eX DIS. Except for their evolution with Q 2 superscript 𝑄 2 Q^{2} , the DPDFs can not be calculated in the framework of perturbative QCD and must be determined from experiment. Such DPDFs 3 , 4 , 5 , 6 have been obtained from the HERA inclusive measurements of the diffractive structure function F 2 D subscript superscript 𝐹 𝐷 2 F^{D}_{2} 3 , 4 , defined analogously to the proton structure function F 2 subscript 𝐹 2 F_{2} . Similarly to diffractive DIS, e ​ p → e ​ X ​ Y → 𝑒 𝑝 𝑒 𝑋 𝑌 ep\rightarrow eXY , where the presence of the large scale Q 𝑄 Q allows for the application of perturbative QCD and X 𝑋 X comprises the sum over all possible final states, many other processes with a hard scale provided by specific final states in the central system X 𝑋 X can be predicted using QCD perturbation theory. Such processes, usually called hard diffractive processes, are e.g. dijet production in diffractive photoproduction ( Q 2 ≃ 0 similar-to-or-equals superscript 𝑄 2 0 Q^{2}\simeq 0 ) and DIS ( Q 2 ≠ 0 superscript 𝑄 2 0 Q^{2}\neq 0 ), where the large scale is given by the jet transverse energy E T j ​ e ​ t superscript subscript 𝐸 𝑇 𝑗 𝑒 𝑡 E_{T}^{jet} and possibly Q 𝑄 Q , and diffractive open heavy-flavor production, where the large scale is given by the heavy-flavor mass and possibly E T subscript 𝐸 𝑇 E_{T} and/or Q 𝑄 Q , in photoproduction or DIS and many more diffractive processes induced by p ​ p ¯ 𝑝 ¯ 𝑝 p\bar{p} or p ​ p 𝑝 𝑝 pp collisions. The central problem in hard diffraction is the problem of QCD factorization, i.e. the question whether diffractive cross sections are factorisable into universal DPDFs and partonic cross sections, which are calculable in perturbative QCD. This question is the subject of the current debate in diffractive physics and is of particular interest for the prospects of discovery of new particles such as the Higgs boson in diffractive reactions at the LHC 7 , 8 , 9 . For the inclusive DIS process, factorization has indeed been proven to hold 10 , and on this basis DPDFs have been extracted at Q 2 ≠ 0 superscript 𝑄 2 0 Q^{2}\neq 0 3 , 4 , 5 from high-precision inclusive measurements of the process e ​ p → e ​ X ​ Y → 𝑒 𝑝 𝑒 𝑋 𝑌 ep\rightarrow eXY using the usual DGLAP evolution equations. The proof of the factorization formula, usually referred to as the validity of QCD factorization in hard diffraction, also appears to be valid for the production of specific final states in DIS, as e.g. the production of jets or heavy-flavor particles, and for the direct part of photoproduction ( Q 2 ≃ 0 similar-to-or-equals superscript 𝑄 2 0 Q^{2}\simeq 0 ) or low- Q 2 superscript 𝑄 2 Q^{2} electroproduction of jets 10 . However, factorization does not hold for hard processes in diffractive hadron-hadron scattering. The problem is that soft interactions between the ingoing hadrons and/or their remnants occur in both the initial and the final state. This agrees with experimental measurements at the Tevatron 11 . Predictions of diffractive dijet cross sections for collisions as measured by CDF using DPDFs determined earlier by the H1 collaboration 12 at HERA overestimate the measured cross section by up to an order of magnitude 11 . This large suppression of the CDF cross section can be explained by the rescattering of the two incoming hadron beams, which, by creating additional hadrons, destroy the rapidity gap 13 . Jet production with real photons involves direct interactions of the photon with quarks or gluons originating from the proton or pomeron, respectively, as well as resolved photon contributions leading to parton-parton interactions with an additional remnant jet coming from the photon as reviewed in 14 (see Fig. 2 ). For the direct interactions, we expect factorization Figure 2: Diffractive production of dijets with invariant mass M 12 subscript 𝑀 12 M_{12} in direct (left) and resolved (right) photon-pomeron collisions, leading to the production of one or two additional remnant jets. to be valid as in the case of inclusive DIS, as already mentioned, whereas we expect it to fail for the resolved process as in hadron-hadron scattering. For this part of photoproduction we would therefore expect a similar suppression factor (sometimes also called rapidity-gap survival probability) due to rescattering effects of the ingoing partons or hadrons. Introducing vector-meson dominance photon fluctuations, such a suppression by about a factor of three was predicted for resolved photoproduction at HERA 15 . The first measurements of dijet cross sections in diffractive photoproduction have been presented by the H1 collaboration as contributions to two conferences 16 . The kinematic range for these data were Q 2 < 0.01 superscript 𝑄 2 0.01 Q^{2}<0.01 GeV 2 , x I ​ P < 0.03 subscript 𝑥 𝐼 𝑃 0.03 x_{I\!\!P}<0.03 , E T j ​ e ​ t ​ 1 > 5 superscript subscript 𝐸 𝑇 𝑗 𝑒 𝑡 1 5 E_{T}^{jet1}>5 GeV, E T j ​ e ​ t ​ 2 > 4 superscript subscript 𝐸 𝑇 𝑗 𝑒 𝑡 2 4 E_{T}^{jet2}>4 GeV and 165 < W < 240 165 𝑊 240 165<W<240 GeV, where jets were identified using the inclusive k T subscript 𝑘 𝑇 k_{T} -cluster algorithm (the definitions of these and the following variables will be given in the next section). The measured cross sections as a function of x γ o ​ b ​ s superscript subscript 𝑥 𝛾 𝑜 𝑏 𝑠 x_{\gamma}^{obs} and z I ​ P o ​ b ​ s superscript subscript 𝑧 𝐼 𝑃 𝑜 𝑏 𝑠 z_{I\!\!P}^{obs} were compared to leading-order (LO) QCD predictions, using the RAPGAP Monte Carlo model 17 . For the DPDFs the LO H1 2002 fit was used 12 . The two cross sections were found to be well described by the predictions in normalization and shape over the whole range of x γ o ​ b ​ s superscript subscript 𝑥 𝛾 𝑜 𝑏 𝑠 x_{\gamma}^{obs} and z I ​ P o ​ b ​ s superscript subscript 𝑧 𝐼 𝑃 𝑜 𝑏 𝑠 z_{I\!\!P}^{obs} , showing no breakdown of factorization neither in resolved nor in direct photoproduction. In addition, normalized cross sections as a function of various other variables were compared to the predictions with the result that all measured distributions were in good agreement. Subsequently we calculated the next-to-leading order (NLO) corrections for the cross section of diffractive dijet production using the same kinematic cuts and with the same DPDFs as in the first H1 analysis 16 on the basis of our previous work on NLO corrections for inclusive direct 18 and resolved 19 dijet photoproduction. While at LO good agreement with the H1 data 16 was found, consistent with the finding in the H1 analysis 16 , it was found that the NLO corrections increase the cross section significantly 20 , 21 and require a suppression factor of the order of R = 0.5 𝑅 0.5 R=0.5 . Since on theoretical grounds only a suppression of the resolved cross section would be acceptable, we demonstrated in 20 , 21 that by multiplying the resolved cross section with the suppression factor R = 0.34 𝑅 0.34 R=0.34 , reasonably good agreement with the preliminary H1 data 16 could be achieved. This value for the suppression factor turned out to be in good agreement with the prediction of 15 . The first experimental data from the ZEUS collaboration were presented at the DIS workshop in 2004 22 . The dijet cross sections were obtained in the kinematic range Q 2 < 1 superscript 𝑄 2 1 Q^{2}<1 GeV 2 , x I ​ P < 0.035 subscript 𝑥 𝐼 𝑃 0.035 x_{I\!\!P}<0.035 and E T j ​ e ​ t ​ 1 ​ ( 2 ) > 7.5 superscript subscript 𝐸 𝑇 𝑗 𝑒 𝑡 1 2 7.5 E_{T}^{jet1(2)}>7.5 ( 6.5 ) 6.5 (6.5) GeV. For these kinematic constraints NLO calculations were not available in 2004. So, the measurements were compared to LO calculations, unfortunately with previous H1 DPDFs 23 with the result, that good agreement in the shape was achieved. But the normalization was off by a factor of 0.6 0.6 0.6 , which was attributed to the older DPDF input 24 , so that the H1 and ZEUS results were consistent with each other. The situation concerning the agreement of H1 and ZEUS data and the influence of NLO corrections improved already considerably in the fall of 2004. These preliminary data from both HERA collaborations together with comparisons to NLO calculations based on the DPDF fits from 12 were presented at workshops and conferences in the following years. In 2006 the H1 collaboration published their final DPDF fits from their high-precision measurements using the DGLAP evolution equations 4 . This analysis was based on the larger data sample of the years 1997-2000 as compared to the earlier preliminary DPDF sets 12 . In 4 two DPDF sets, the H1 2006 fit A and the H1 2006 fit B were presented, which both give a good description of the inclusive diffractive data. These two sets differ mainly in the gluon density at large fractional parton momenta, which is poorly constrained by the inclusive diffractive scattering data, since there is no direct coupling of the photon to gluons, so that the gluon density is constrained only through the evolution. The gluon density of fit A is peaked at the starting scale at large fractional momenta, whereas fit B is flat in this region. In 2007 the final publications for diffractive dijet production appeared 25 . The comparison between these experimental results and the NLO theory was based on the new and final DPDFs from H1 4 . The differential cross sections as measured by H1 25 were compared to NLO predictions obtained with the Frixione program 26 interfaced to the H1 2006 fit B DPDFs. The conclusions deduced earlier from the comparison with the preliminary data and the preliminary H1 2002 fit 12 are fully confirmed in 25 with the new DPDFs fits 4 . In particular, a global suppression is obtained, independent of the DPDFs fits used, i.e. fit A or fit B, by considering the ratio of measured dijet cross sections to NLO predictions in photoproduction in relation to the same ratio in DIS. In this comparison the value of the suppression is 0.5 0.5 0.5 . In addition, by using this overall suppression factor, H1 obtained a good description of all the measured distributions in the variables z I ​ P o ​ b ​ s superscript subscript 𝑧 𝐼 𝑃 𝑜 𝑏 𝑠 z_{I\!\!P}^{obs} , x γ o ​ b ​ s superscript subscript 𝑥 𝛾 𝑜 𝑏 𝑠 x_{\gamma}^{obs} , x I ​ P subscript 𝑥 𝐼 𝑃 x_{I\!\!P} , W 𝑊 W , E T j ​ e ​ t ​ 1 superscript subscript 𝐸 𝑇 𝑗 𝑒 𝑡 1 E_{T}^{jet1} , η ¯ j ​ e ​ t ​ s superscript ¯ 𝜂 𝑗 𝑒 𝑡 𝑠 \bar{\eta}^{jets} , | Δ ​ η j ​ e ​ t ​ s | Δ superscript 𝜂 𝑗 𝑒 𝑡 𝑠 |\Delta\eta^{jets}| and M 12 subscript 𝑀 12 M_{12} interfaced with the H1 2006 fit B DPDFs and taking into account hadronization corrections 25 . Finally, the H1 collaboration investigated how well the data are describable under the assumption that in the NLO calculation the cross section for x γ o ​ b ​ s > 0.9 superscript subscript 𝑥 𝛾 𝑜 𝑏 𝑠 0.9 x_{\gamma}^{obs}>0.9 is not suppressed. The best agreement in a fit was obtained for a suppression factor 0.44 for the NLO calculation with x γ o ​ b ​ s < 0.9 superscript subscript 𝑥 𝛾 𝑜 𝑏 𝑠 0.9 x_{\gamma}^{obs}<0.9 , based on fitting the distributions for x γ o ​ b ​ s superscript subscript 𝑥 𝛾 𝑜 𝑏 𝑠 x_{\gamma}^{obs} , W 𝑊 W , η ¯ j ​ e ​ t ​ s superscript ¯ 𝜂 𝑗 𝑒 𝑡 𝑠 \bar{\eta}^{jets} and E T j ​ e ​ t ​ 1 superscript subscript 𝐸 𝑇 𝑗 𝑒 𝑡 1 E_{T}^{jet1} . In this comparison they found disagreement for the largest x γ o ​ b ​ s superscript subscript 𝑥 𝛾 𝑜 𝑏 𝑠 x_{\gamma}^{obs} -bin and the lowest η ¯ j ​ e ​ t ​ s superscript ¯ 𝜂 𝑗 𝑒 𝑡 𝑠 \bar{\eta}^{jets} (which are related), but better agreement in the E T j ​ e ​ t ​ 1 superscript subscript 𝐸 𝑇 𝑗 𝑒 𝑡 1 E_{T}^{jet1} -distribution. In 25 this leads to the statement, that the assumption that the direct cross section obeys factorization is strongly disfavored by their analysis. In total, it is obvious that in the final H1 analysis 25 a global suppression in diffractive dijet photoproduction is clearly established and the model with resolved suppression only is not as well supported by the data. Just recently also the ZEUS collaboration presented their final result on diffractive dijet photoproduction 27 . As in their preliminary analysis, the two jets with the highest transverse energies E T j ​ e ​ t superscript subscript 𝐸 𝑇 𝑗 𝑒 𝑡 E_{T}^{jet} were required to satisfy E T j ​ e ​ t ​ 1 ​ ( 2 ) > 7.5 superscript subscript 𝐸 𝑇 𝑗 𝑒 𝑡 1 2 7.5 E_{T}^{jet1(2)}>7.5 ( 6.5 ) 6.5 (6.5) GeV, which is higher than in the H1 analysis with E T j ​ e ​ t ​ 1 ​ ( 2 ) > 5 superscript subscript 𝐸 𝑇 𝑗 𝑒 𝑡 1 2 5 E_{T}^{jet1(2)}>5 ( 4 ) 4 (4) GeV 25 . ZEUS compared their measurements with the NLO predictions for diffractive photoproduction of dijets based on our program 21 . Three sets of DPDFs were used, the ZEUS LPS fit, determined from a NLO analysis of inclusive diffraction and diffractive charm-production data 3 , and the two H1 fits, H1 2006 fit A and fit B 4 . The NLO results obtained with the two H1 fits were scaled down by a factor of 0.87 4 , since the H1 measurements used to derive the DPDFs include low-mass proton dissociative processes with M Y < 1.6 subscript 𝑀 𝑌 1.6 M_{Y}<1.6 GeV, which increases the photon-diffractive cross section by 1.15 − 0.08 + 0.15 subscript superscript 1.15 0.15 0.08 1.15^{+0.15}_{-0.08} as compared to the pure proton final state as corrected to in the ZEUS analysis. The comparison of the measured cross sections and the theoretical predictions was based on the distributions in the variables y 𝑦 y , M X subscript 𝑀 𝑋 M_{X} , x I ​ P subscript 𝑥 𝐼 𝑃 x_{I\!\!P} , z I ​ P o ​ b ​ s superscript subscript 𝑧 𝐼 𝑃 𝑜 𝑏 𝑠 z_{I\!\!P}^{obs} , E T j ​ e ​ t ​ 1 superscript subscript 𝐸 𝑇 𝑗 𝑒 𝑡 1 E_{T}^{jet1} , η l ​ a ​ b j ​ e ​ t ​ 1 subscript superscript 𝜂 𝑗 𝑒 𝑡 1 𝑙 𝑎 𝑏 \eta^{jet1}_{lab} and x γ o ​ b ​ s superscript subscript 𝑥 𝛾 𝑜 𝑏 𝑠 x_{\gamma}^{obs} . The data were reasonably well described in their shape as a function of these variables and lay systematically below the predictions. The predictions for the three DPDFs differed appreciably. The cross sections for the H1 2006 fit A (fit B) were the highest (lowest) and the one for the ZEUS LPS fit lay between the two, but nearer to the fit A than the fit B predictions. For d ​ σ / d ​ x γ o ​ b ​ s 𝑑 𝜎 𝑑 superscript subscript 𝑥 𝛾 𝑜 𝑏 𝑠 d\sigma/dx_{\gamma}^{obs} ZEUS also showed the ratio of the data and the NLO predictions using the ZEUS LPS fit. It was consistent with a suppression factor of 0.7 independent of x γ o ​ b ​ s superscript subscript 𝑥 𝛾 𝑜 𝑏 𝑠 x_{\gamma}^{obs} . This suppression factor depended on the DPDFs and ranged between 0.6 (H1 2006 fit A) and 0.9 (fit B). Taking into account the scale dependence of the theoretical predictions the ratio was outside the theoretical uncertainty for the ZEUS LPS fit and the H1 2006 fit A, but not for fit B. In their conclusions the authors of the ZEUS analysis 27 made the statement that the NLO calculations tend to overestimate the measured cross section, which would mean that a suppression is present. Unfortunately, however, they continued, that, within the large uncertainties of the NLO calculations, the data were compatible with the QCD calculations, i.e. with no suppression. Such a statement clearly contradicts the result of the H1 collaboration 25 and casts doubts on the correctness of the H1 analysis. The authors of 27 attribute this discrepancy to the fact that the H1 measurements 25 were carried out in a lower E T j ​ e ​ t superscript subscript 𝐸 𝑇 𝑗 𝑒 𝑡 E_{T}^{jet} and a higher x I ​ P subscript 𝑥 𝐼 𝑃 x_{I\!\!P} range than those in the ZEUS study 27 . Besides the different E T j ​ e ​ t superscript subscript 𝐸 𝑇 𝑗 𝑒 𝑡 E_{T}^{jet} and x I ​ P subscript 𝑥 𝐼 𝑃 x_{I\!\!P} regions in 25 and 27 , the two measurements suffer also from different experimental cuts of some other variables, which makes it difficult to compare the two data sets directly (note also the lower center-of-mass energy for the H1 data). In addition the comparison with NLO theory in 25 and 27 was done with two different programs 19 versus 26 , which, however agreed quite well with each other 25 . The rather different conclusions concerning factorization breaking in diffractive dijet photoproduction calls for a new comparative study of the two data sets in 25 and 27 . We have therefore performed a new calculation of the NLO cross sections on the basis of our earlier work 21 with the new H1 2006 DPDFs and revised hadronic corrections as compared to 20 , in order to see whether we can confirm the very different conclusions achieved in the H1 25 and ZEUS 27 analyses. In the comparison with the new data sets we shall follow more or less the same strategy as in our earlier work 20 , 21 . We first calculate the unsuppressed NLO cross sections including an error band based on the scale variation and see how much and in which distribution the data points lie inside or outside this error band. Then we determine a global suppression factor by fitting the differential cross section d ​ σ / d ​ E T j ​ e ​ t ​ 1 𝑑 𝜎 𝑑 superscript subscript 𝐸 𝑇 𝑗 𝑒 𝑡 1 d\sigma/dE_{T}^{jet1} at the bin with the lowest E T j ​ e ​ t ​ 1 superscript subscript 𝐸 𝑇 𝑗 𝑒 𝑡 1 E_{T}^{jet1} . With this suppression factor we shall compare to the differential cross sections of all the other measured variables and look for consistency. In this new comparison between the experimental and the theoretical results we shall concentrate on using the H1 2006 fit B 4 input, since it leads to smaller cross sections than the DPDFs from H1 2006 fit A 4 or the ZEUS LPS DPDF fit 3 . Actually the H1 collaboration constructed a third set of DPDFs, which is called the H1 2007 fit jets. This fit is obtained through a simultaneous fit to the diffractive inclusive and DIS dijet cross sections 28 . It is performed under the assumption that there is no factorization breaking in the diffractive dijet cross sections. Under this assumption, including the diffractive dijet cross sections in the analysis leads to additional constraints, mostly on the diffractive gluon distribution. On average the H1 2007 fit jets is similar to the H1 2006 fit B except for the gluon distribution at high momentum fraction and smaller factorization scales. In our analysis we shall disregard this new set of DPDFs, since it would be compatible with the factorization test of the photoproduction data only if we restricted these tests to the case that only the resolved part has this breaking and not the direct part, which has the same structure as the DIS dijet cross section. In Sec. 2 we shall present the complete list of cuts on the experimental variables, give all the input used in the cross section calculations, and present the basic formulæ, from which the dijet cross sections have been calculated. The comparison with the H1 25 and the ZEUS 27 experimental data is presented and discussed in Sec. 3. In this comparison we shall concentrate on the main question, whether there is a suppression in the photoproduction data at all. In addition we shall investigate also whether a reasonable description of the data is possible with suppression of the resolved cross section only, as we studied it already in our previous work in 2004 20 , 21 . In Sec. 4 we shall finish with a summary and our conclusions.","role":"user"},{"content":"Abstract: After the final analyses of the H1 and ZEUS collaborations for the diffractive photoproduction of dijets have appeared, we have recalculated these cross sections in next-to-leading order (NLO) of perturbative QCD to see whether they can be interpreted consistently. The results of these calculations are compared to the data of both collaborations. We find that at NLO the cross sections disagree with the data, showing that factorization breaking occurs at this order. If direct and resolved contributions are both suppressed by the same amount, the global suppression factor depends on the transverse-energy cut and is 0.42 for the H1 and 0.71 for the ZEUS analysis. However, by suppressing only the resolved contribution by a factor of approximately three, also reasonably good agreement with all the data is found. The size of the factorization breaking effects for resolved photons agrees with absorptive-model predictions.","role":"assistant"}]

Continuous Opinion Dynamics under Bounded Confidence: A Survey

The term “opinion dynamics” nowadays summarizes a wide class of different models differing in heuristics, formalization as well as in the phenomena of interest. The latter range from emergence of fads, minority opinion spreading, collective decision making, finding and not finding of consensus, emergence of political parties, minority opinion survival, emergence of extremism and so on. This paper deals with these phenomena in models of continuous opinion dynamics under bounded confidence. ‘Continuous’ refers to the opinion issue and not to the time. Thus opinions in continuous opinion dynamics should be expressable in real numbers where compromising in the middle is always possible. Example issues are prices, tax rates or predictions about macroeconomic variables. The political spectrum is also often mapped to a continuum from left to right wing. In opinion dynamics one considers a set of agents where each holds an opinion from a certain opinion space. She may change her opinion when she gets aware of the opinions of others. In the physics literature discrete opinion spaces (classically binary opinions) Galam2002 ; Schweitzer2000 ; Sznajd-Weron2000 have dominated research due to their striking analogy with spin systems. Sometimes they have been extended to more than two spin values, which are ordered and thus get closer to continuous opinion dynamics Stauffer2002a ; Stauffer2002 . In recent years two models of genuinely continuous opinion dynamics under bounded confidence have raised the interest of the sociophysics community: the models of Hegselmann and Krause KrauseEtal1997 ; Krause2000 ; Hegselmann2002 and Deffuant, Weisbuch and others Deffuant2000 ; Weisbuch2002 . Due to physicist’s research some progress in understanding the dynamics of these models has been made, especially by introducing the master equation on the agents density in the opinion space for these type of models. For a recent review in this broader context of opinion dynamics including discrete opinions from a physicist’s perspective see StaufferEtal2006 . Let us consider a population of agents which hold diverse opinions about certain issues expressible in real numbers. Each agent is willing to change her opinion if she hears the opinions of others by adjusting towards those opinions. Every adjustment in terms of averaging is possible due to the continuous nature of the opinions. Further on, consider agents to have bounded confidence. That means an agent is only willing to take those opinions into account, which differ less than a certain bound of confidence ε 𝜀 \varepsilon from her own opinion. The Deffuant-Weisbuch (DW) model and the Hegselmann-Krause (HK) model both rely on the idea of repeated averaging under bounded confidence. They differ in their communication regime. In the DW model agents meet in random pairwise encounters after which they compromise or not. In the HK model, each agent moves to the average opinion of all agents which lie in her area of confidence (including herself). Actually, the DW model contains another parameter which controlls how close an agent moves to the opinion of the other. But it has turned out that this parameter has only an effect on convergence time in the basic model. Therefore we neglect it in the basic analysis and discuss its impact afterwards. The DW model was partly inspired by the famous Axelrod model about the dissemination of culture Axelrod1997 where something similar to the bounded confidence assumption is implemented. It has been developed in a project about improving agri-environmental policies in the European union. The HK model has been presented by Krause KrauseEtal1997 ; Krause2000 in a mathematical context as a nonlinear version of older consensus models DeGroot1974 ; Chatterjee1977 ; Lehrer1981 . It was analyzed through computer simulations by Hegselmann and Krause Hegselmann2002 in the context of social simulation and has gained a lot of attention since. The next section presents the two models in their original agent-based versions and in their density-based formulation inspired by statistical physics. Section 3 shows and explains the bifurcation diagrams of both models which serve as a reference for the review of several extensions in Section 4.

Models of continuous opinion dynamics under bounded confidence have been presented independently by Krause and Hegselmann and by Deffuant et al. in 2000. They have raised a fair amount of attention in the communities of social simulation, sociophysics and complexity science. The researchers working on it come from disciplines such as physics, mathematics, computer science, social psychology and philosophy. In these models agents hold continuous opinions which they can gradually adjust if they hear the opinions of others. The idea of bounded confidence is that agents only interact if they are close in opinion to each other. Usually, the models are analyzed with agent-based simulations in a Monte Carlo style, but they can also be reformulated on the agent's density in the opinion space in a master equation style. The contribution of this survey is fourfold. First, it will present the agent-based and density-based modeling frameworks including the cases of multidimensional opinions and heterogeneous bounds o...

[{"content":"Help me to generate the abstract of a survey paper given the title:*Continuous Opinion Dynamics under Bounded Confidence: A Survey* and the introduction:The term opinion dynamics nowadays summarizes a wide class of different models differing in heuristics, formalization as well as in the phenomena of interest. The latter range from emergence of fads, minority opinion spreading, collective decision making, finding and not finding of consensus, emergence of political parties, minority opinion survival, emergence of extremism and so on. This paper deals with these phenomena in models of continuous opinion dynamics under bounded confidence. Continuous refers to the opinion issue and not to the time. Thus opinions in continuous opinion dynamics should be expressable in real numbers where compromising in the middle is always possible. Example issues are prices, tax rates or predictions about macroeconomic variables. The political spectrum is also often mapped to a continuum from left to right wing. In opinion dynamics one considers a set of agents where each holds an opinion from a certain opinion space. She may change her opinion when she gets aware of the opinions of others. In the physics literature discrete opinion spaces (classically binary opinions) Galam2002 ; Schweitzer2000 ; Sznajd-Weron2000 have dominated research due to their striking analogy with spin systems. Sometimes they have been extended to more than two spin values, which are ordered and thus get closer to continuous opinion dynamics Stauffer2002a ; Stauffer2002 . In recent years two models of genuinely continuous opinion dynamics under bounded confidence have raised the interest of the sociophysics community: the models of Hegselmann and Krause KrauseEtal1997 ; Krause2000 ; Hegselmann2002 and Deffuant, Weisbuch and others Deffuant2000 ; Weisbuch2002 . Due to physicists research some progress in understanding the dynamics of these models has been made, especially by introducing the master equation on the agents density in the opinion space for these type of models. For a recent review in this broader context of opinion dynamics including discrete opinions from a physicists perspective see StaufferEtal2006 . Let us consider a population of agents which hold diverse opinions about certain issues expressible in real numbers. Each agent is willing to change her opinion if she hears the opinions of others by adjusting towards those opinions. Every adjustment in terms of averaging is possible due to the continuous nature of the opinions. Further on, consider agents to have bounded confidence. That means an agent is only willing to take those opinions into account, which differ less than a certain bound of confidence ε 𝜀 \varepsilon from her own opinion. The Deffuant-Weisbuch (DW) model and the Hegselmann-Krause (HK) model both rely on the idea of repeated averaging under bounded confidence. They differ in their communication regime. In the DW model agents meet in random pairwise encounters after which they compromise or not. In the HK model, each agent moves to the average opinion of all agents which lie in her area of confidence (including herself). Actually, the DW model contains another parameter which controlls how close an agent moves to the opinion of the other. But it has turned out that this parameter has only an effect on convergence time in the basic model. Therefore we neglect it in the basic analysis and discuss its impact afterwards. The DW model was partly inspired by the famous Axelrod model about the dissemination of culture Axelrod1997 where something similar to the bounded confidence assumption is implemented. It has been developed in a project about improving agri-environmental policies in the European union. The HK model has been presented by Krause KrauseEtal1997 ; Krause2000 in a mathematical context as a nonlinear version of older consensus models DeGroot1974 ; Chatterjee1977 ; Lehrer1981 . It was analyzed through computer simulations by Hegselmann and Krause Hegselmann2002 in the context of social simulation and has gained a lot of attention since. The next section presents the two models in their original agent-based versions and in their density-based formulation inspired by statistical physics. Section 3 shows and explains the bifurcation diagrams of both models which serve as a reference for the review of several extensions in Section 4.","role":"user"},{"content":"Abstract: Models of continuous opinion dynamics under bounded confidence have been presented independently by Krause and Hegselmann and by Deffuant et al. in 2000. They have raised a fair amount of attention in the communities of social simulation, sociophysics and complexity science. The researchers working on it come from disciplines such as physics, mathematics, computer science, social psychology and philosophy. In these models agents hold continuous opinions which they can gradually adjust if they hear the opinions of others. The idea of bounded confidence is that agents only interact if they are close in opinion to each other. Usually, the models are analyzed with agent-based simulations in a Monte Carlo style, but they can also be reformulated on the agent's density in the opinion space in a master equation style. The contribution of this survey is fourfold. First, it will present the agent-based and density-based modeling frameworks including the cases of multidimensional opinions and heterogeneous bounds o...","role":"assistant"}]

Survey on the Burnside ring of compact Lie groups

1. The Burnside ring of a finite group Let G 𝐺 G be a finite group. The Burnside ring of G 𝐺 G is the Grothendieck group completion of the semiring of isomorphism classes of finite G 𝐺 G -sets. It is denoted by A ​ ( G ) 𝐴 𝐺 A(G) (other notations are B ​ ( G ) 𝐵 𝐺 B(G) and Ω ​ ( G ) Ω 𝐺 \Omega(G) ). Addition is given by disjoint union, and multiplication by Cartesian product. These operations are well defined on isomorphism classed of G 𝐺 G -sets. The Burnside ring of G 𝐺 G is isomorphic, as an abelian group, to the free abelian group with generators the isomorphism classes of transitive G 𝐺 G -sets. Under this identification, the multiplication of the additive generators is given by the double coset formula. The double coset formula says that the G 𝐺 G -set G / H × G / K 𝐺 𝐻 𝐺 𝐾 G/H\times G/K is G 𝐺 G -isomorphic to the disjoint union of the transitive G 𝐺 G -sets G / ( H ∩ g ​ K ​ g − 1 ) 𝐺 𝐻 𝑔 𝐾 superscript 𝑔 1 G/(H\cap gKg^{-1}) where H ​ g ​ K 𝐻 𝑔 𝐾 HgK runs over the double coset H \ G / K \ 𝐻 𝐺 𝐾 H\,\backslash G/K . Let H 𝐻 H be a subgroup of G 𝐺 G and let X 𝑋 X be a G 𝐺 G -set. The H 𝐻 H -fixed point set X H superscript 𝑋 𝐻 X^{H} is the subset { x ∈ X | h ​ x = x , h ∈ H } conditional-set 𝑥 𝑋 formulae-sequence ℎ 𝑥 𝑥 ℎ 𝐻 \{x\in X\,|\,hx=x,h\in H\} of X 𝑋 X . The number of elements in X H superscript 𝑋 𝐻 X^{H} , denoted | X H | superscript 𝑋 𝐻 |X^{H}| , only depends on the G 𝐺 G -isomorphism class of X 𝑋 X and the G 𝐺 G -conjugacy class of H 𝐻 H . For every conjugacy class of a subgroup H 𝐻 H of G 𝐺 G the map X ↦ | X H | maps-to 𝑋 superscript 𝑋 𝐻 X\mapsto|X^{H}| gives a semiring homomorphism from the semiring of isomorphism classes of finite G 𝐺 G -sets to the integers. Hence there is an induced H 𝐻 H -fixed point ring homomorphism ϕ H : A ​ ( G ) → ℤ : subscript italic-ϕ 𝐻 → 𝐴 𝐺 ℤ \phi_{H}\colon A(G)\rightarrow\mathbb{Z} . These H 𝐻 H -fixed point ring homomorphisms ensemble to give a ring homomorphism ϕ : A ​ ( G ) → ∏ ( H ) ℤ , : italic-ϕ → 𝐴 𝐺 subscript product 𝐻 ℤ \phi\colon A(G)\rightarrow\textstyle\prod_{(H)}\mathbb{Z}, where the product is over the G 𝐺 G -conjugacy classes, ( H ) 𝐻 (H) , of subgroups H 𝐻 H of G 𝐺 G . The map ϕ italic-ϕ \phi is sometimes called the mark homomorphism. Choose a linear ordering of the conjugacy classes of subgroups of G 𝐺 G that respects subconjugacy. The matrix with the ( H ) 𝐻 (H) , ( K ) 𝐾 (K) entry given by ϕ K ​ ( [ G / H ] ) subscript italic-ϕ 𝐾 delimited-[] 𝐺 𝐻 \phi_{K}([G/H]) is called the table of marks, or the mark matrix, of G 𝐺 G . The basic properties of the Burnside ring of a finite group are described nicely in the first chapter of [ 10 ] and in [ 16 ] . A recent survey is [ 4 ] .

The definition and basic properties of the Burnside ring of compact Lie groups are presented, with emphasis on the analogy with the construction of the Burnside ring of finite groups.

[{"content":"Help me to generate the abstract of a survey paper given the title:*Survey on the Burnside ring of compact Lie groups* and the introduction:1. The Burnside ring of a finite group Let G 𝐺 G be a finite group. The Burnside ring of G 𝐺 G is the Grothendieck group completion of the semiring of isomorphism classes of finite G 𝐺 G -sets. It is denoted by A ​ ( G ) 𝐴 𝐺 A(G) (other notations are B ​ ( G ) 𝐵 𝐺 B(G) and Ω ​ ( G ) Ω 𝐺 \Omega(G) ). Addition is given by disjoint union, and multiplication by Cartesian product. These operations are well defined on isomorphism classed of G 𝐺 G -sets. The Burnside ring of G 𝐺 G is isomorphic, as an abelian group, to the free abelian group with generators the isomorphism classes of transitive G 𝐺 G -sets. Under this identification, the multiplication of the additive generators is given by the double coset formula. The double coset formula says that the G 𝐺 G -set G / H × G / K 𝐺 𝐻 𝐺 𝐾 G/H\times G/K is G 𝐺 G -isomorphic to the disjoint union of the transitive G 𝐺 G -sets G / ( H ∩ g ​ K ​ g − 1 ) 𝐺 𝐻 𝑔 𝐾 superscript 𝑔 1 G/(H\cap gKg^{-1}) where H ​ g ​ K 𝐻 𝑔 𝐾 HgK runs over the double coset H \ G / K \ 𝐻 𝐺 𝐾 H\,\backslash G/K . Let H 𝐻 H be a subgroup of G 𝐺 G and let X 𝑋 X be a G 𝐺 G -set. The H 𝐻 H -fixed point set X H superscript 𝑋 𝐻 X^{H} is the subset { x ∈ X | h ​ x = x , h ∈ H } conditional-set 𝑥 𝑋 formulae-sequence ℎ 𝑥 𝑥 ℎ 𝐻 \{x\in X\,|\,hx=x,h\in H\} of X 𝑋 X . The number of elements in X H superscript 𝑋 𝐻 X^{H} , denoted | X H | superscript 𝑋 𝐻 |X^{H}| , only depends on the G 𝐺 G -isomorphism class of X 𝑋 X and the G 𝐺 G -conjugacy class of H 𝐻 H . For every conjugacy class of a subgroup H 𝐻 H of G 𝐺 G the map X ↦ | X H | maps-to 𝑋 superscript 𝑋 𝐻 X\mapsto|X^{H}| gives a semiring homomorphism from the semiring of isomorphism classes of finite G 𝐺 G -sets to the integers. Hence there is an induced H 𝐻 H -fixed point ring homomorphism ϕ H : A ​ ( G ) → ℤ : subscript italic-ϕ 𝐻 → 𝐴 𝐺 ℤ \phi_{H}\colon A(G)\rightarrow\mathbb{Z} . These H 𝐻 H -fixed point ring homomorphisms ensemble to give a ring homomorphism ϕ : A ​ ( G ) → ∏ ( H ) ℤ , : italic-ϕ → 𝐴 𝐺 subscript product 𝐻 ℤ \phi\colon A(G)\rightarrow\textstyle\prod_{(H)}\mathbb{Z}, where the product is over the G 𝐺 G -conjugacy classes, ( H ) 𝐻 (H) , of subgroups H 𝐻 H of G 𝐺 G . The map ϕ italic-ϕ \phi is sometimes called the mark homomorphism. Choose a linear ordering of the conjugacy classes of subgroups of G 𝐺 G that respects subconjugacy. The matrix with the ( H ) 𝐻 (H) , ( K ) 𝐾 (K) entry given by ϕ K ​ ( G / H ) subscript italic-ϕ 𝐾 delimited- 𝐺 𝐻 \phi_{K}(G/H) is called the table of marks, or the mark matrix, of G 𝐺 G . The basic properties of the Burnside ring of a finite group are described nicely in the first chapter of 10 and in 16 . A recent survey is 4 .","role":"user"},{"content":"Abstract: The definition and basic properties of the Burnside ring of compact Lie groups are presented, with emphasis on the analogy with the construction of the Burnside ring of finite groups.","role":"assistant"}]

Double beta decay: experiments and theory review

I Introduction The double beta decay is a second order weak transition which can be energetically favored for some even-even nuclei belonging to A 𝐴 A even multiplets. The ( A , Z ) → ( A , Z + 2 ) + 2 ​ e − + 2 ​ ν ¯ e → 𝐴 𝑍 𝐴 𝑍 2 2 superscript 𝑒 2 subscript ¯ 𝜈 𝑒 (A,Z)\rightarrow(A,Z+2)+2e^{-}+2\bar{\nu}_{e} double beta ( β ​ β 𝛽 𝛽 \beta\beta -2 ν 𝜈 \nu ) decay process is allowed by the Standard Model and has been observed for many isotopes with lifetimes longer than 10 19 superscript 10 19 10^{19} y. A more interesting process is the so-called neutrinoless double beta ( β ​ β 𝛽 𝛽 \beta\beta -0 ν 𝜈 \nu ) decay given by ( A , Z ) → ( A , Z + 2 ) + 2 ​ e − → 𝐴 𝑍 𝐴 𝑍 2 2 superscript 𝑒 (A,Z)\rightarrow(A,Z+2)+2e^{-} : this process violates lepton number conservation and is therefore forbidden by the Standard Model. The lifetime for the β ​ β 𝛽 𝛽 \beta\beta -0 ν 𝜈 \nu decay is expected to be longer than 10 25 superscript 10 25 10^{25} y and so far only one evidence has been reported for 76 Ge (see Sec. II.1 ). For recent comprehensive reviews on this topic refer for example to Ref. reviews . I.1 β ​ β 𝛽 𝛽 \beta\beta -0 ν 𝜈 \nu decay and neutrino physics Many mechanisms have been proposed for driving this decay, but the simplest one is the “mass mechanism”, where a light Majorana neutrino is exchanged. Whatever is the mechanism actually causing the β ​ β 𝛽 𝛽 \beta\beta -0 ν 𝜈 \nu decay, its observation would imply that the neutrino is massive and is a Majorana particle (i.e. ν ≡ ν ¯ 𝜈 ¯ 𝜈 \nu\equiv\bar{\nu} ). For a light Majorana neutrino mediate β ​ β 𝛽 𝛽 \beta\beta -0 ν 𝜈 \nu decay, the rate is given by [ τ 1 / 2 0 ​ ν ] − 1 = ⟨ m ν ⟩ 2 ​ F N / m e 2 superscript delimited-[] superscript subscript 𝜏 1 2 0 𝜈 1 superscript delimited-⟨⟩ subscript 𝑚 𝜈 2 subscript 𝐹 𝑁 superscript subscript 𝑚 𝑒 2 [\tau_{1/2}^{0\nu}]^{-1}=\langle m_{\nu}\rangle^{2}F_{N}/m_{e}^{2} , where m e subscript 𝑚 𝑒 m_{e} is electron mass and the nuclear structure factor F N subscript 𝐹 𝑁 F_{N} contains the nuclear matrix element and the phase space. The effective neutrino Majorana mass is given by ⟨ m ν ⟩ = | ∑ k m k ​ η k ​ | U e ​ k | 2 | delimited-⟨⟩ subscript 𝑚 𝜈 subscript 𝑘 subscript 𝑚 𝑘 subscript 𝜂 𝑘 superscript subscript 𝑈 𝑒 𝑘 2 \langle m_{\nu}\rangle=|\sum_{k}m_{k}\eta_{k}|U_{ek}|^{2}| , where m k subscript 𝑚 𝑘 m_{k} are the mass eigenvalues of the three neutrino mass eigenstates | ν k ⟩ ket subscript 𝜈 𝑘 |\nu_{k}\rangle , η k subscript 𝜂 𝑘 \eta_{k} are the CP Majorana phases ( η k = ± 1 subscript 𝜂 𝑘 plus-or-minus 1 \eta_{k}=\pm 1 for CP conservation) and U e ​ k subscript 𝑈 𝑒 𝑘 U_{ek} are the elements of the electron sector of the neutrino mixing matrix. As suggested in mechanism the exact mechanism causing the β ​ β 𝛽 𝛽 \beta\beta -0 ν 𝜈 \nu could be discerned by measuring the decay rates for different isotopes. With the help of the Δ ​ m i ​ k 2 = | m i 2 − m k 2 | Δ superscript subscript 𝑚 𝑖 𝑘 2 superscript subscript 𝑚 𝑖 2 superscript subscript 𝑚 𝑘 2 \Delta m_{ik}^{2}=|m_{i}^{2}-m_{k}^{2}| and sin 2 ⁡ 2 ​ θ i ​ k = f ​ ( | U i ​ k | 2 ) superscript 2 2 subscript 𝜃 𝑖 𝑘 𝑓 superscript subscript 𝑈 𝑖 𝑘 2 \sin^{2}2\theta_{ik}=f(|U_{ik}|^{2}) parameters determined by neutrino flavor oscillation experiments (see Ref. Neutrino2006 for the latest results), it is possible to calculate ⟨ m ν ⟩ delimited-⟨⟩ subscript 𝑚 𝜈 \langle m_{\nu}\rangle as a function of the unknown neutrino absolute mass scale and η k subscript 𝜂 𝑘 \eta_{k} phases osc-bb . From this analysis two possible scenarios can be devised for the upcoming new generation experiments aiming at a 10 meV sensitivity. (1) The β ​ β 𝛽 𝛽 \beta\beta -0 ν 𝜈 \nu decay is discovered with ⟨ m ν ⟩ ≥ 10 delimited-⟨⟩ subscript 𝑚 𝜈 10 \langle m_{\nu}\rangle\geq 10 meV: then the neutrino is a Majorana particle and the masses are either degenerate ( m 1 ≈ m 2 ≈ m 3 subscript 𝑚 1 subscript 𝑚 2 subscript 𝑚 3 m_{1}\approx m_{2}\approx m_{3} ) or follow an inverse hierarchy ( m 3 ≪ m 1 ≈ m 2 much-less-than subscript 𝑚 3 subscript 𝑚 1 subscript 𝑚 2 m_{3}\ll m_{1}\approx m_{2} ). If neutrinos are degenerate (for ⟨ m ν ⟩ ≥ ≈ 0.05 \langle m_{\nu}\rangle\geq\approx 0.05 eV) then the absolute mass scale can be established. (2) The β ​ β 𝛽 𝛽 \beta\beta -0 ν 𝜈 \nu decay is not observed and only an upper limit ⟨ m ν ⟩ ≤ 10 delimited-⟨⟩ subscript 𝑚 𝜈 10 \langle m_{\nu}\rangle\leq 10 meV is set: then, if the neutrino is a Majorana particle, the masses must have a normal hierarchy ( m 1 < m 2 ≪ m 3 subscript 𝑚 1 subscript 𝑚 2 much-less-than subscript 𝑚 3 m_{1}<m_{2}\ll m_{3} ). I.2 β ​ β 𝛽 𝛽 \beta\beta -0 ν 𝜈 \nu decay and nuclear physics To obtain ⟨ m ν ⟩ delimited-⟨⟩ subscript 𝑚 𝜈 \langle m_{\nu}\rangle from the experimental observable τ 1 / 2 0 ​ ν superscript subscript 𝜏 1 2 0 𝜈 \tau_{1/2}^{0\nu} the nuclear structure factor F N ≡ G 0 ​ ν ​ ( Q β ​ β , Z ) ​ | M 0 ​ ν | 2 subscript 𝐹 𝑁 superscript 𝐺 0 𝜈 subscript 𝑄 𝛽 𝛽 𝑍 superscript superscript 𝑀 0 𝜈 2 F_{N}\equiv G^{0\nu}(Q_{\beta\beta},Z)|M^{0\nu}|^{2} must be known. While the phase space G 0 ​ ν ​ ( Q β ​ β , Z ) superscript 𝐺 0 𝜈 subscript 𝑄 𝛽 𝛽 𝑍 G^{0\nu}(Q_{\beta\beta},Z) can be precisely calculated, the nuclear matrix | M 0 ​ ν | superscript 𝑀 0 𝜈 |M^{0\nu}| contains the uncertain details of the nuclear part of the process. In fact there is a large spread in the nuclear matrix elements calculated by different authors with different nuclear models nuclearmatrix (see also references in Ref. reviews ; osc-bb ). Because of this spread also ⟨ m ν ⟩ delimited-⟨⟩ subscript 𝑚 𝜈 \langle m_{\nu}\rangle is affected by large uncertainties (about a factor 3 on the average). Presently these uncertainties are a severe limitation to the potentialities of β ​ β 𝛽 𝛽 \beta\beta -0 ν 𝜈 \nu decay as a tool for neutrino physics: it has been recently suggested rodin ; kortelainen that measured β ​ β 𝛽 𝛽 \beta\beta -2 ν 𝜈 \nu decay lifetimes can be used to reduce the spread in QRPA calculations. Nevertheless, it is important to search for β ​ β 𝛽 𝛽 \beta\beta -0 ν 𝜈 \nu decay of as many as possible candidate isotopes. I.3 β ​ β 𝛽 𝛽 \beta\beta -0 ν 𝜈 \nu decay and CP-violation The observation of β ​ β 𝛽 𝛽 \beta\beta -0 ν 𝜈 \nu could be used to establish CP-violation associated with Majorana neutrinos in the lepton sector due to the η k subscript 𝜂 𝑘 \eta_{k} Majorana phases. This issue has gained interest because it could provide an explanation for the observed baryon asymmetry in the Universe through the leptogenesis theory leptogenesis . The Majorana phases could be constrained by simultaneous precise measurements of ⟨ m ν ⟩ delimited-⟨⟩ subscript 𝑚 𝜈 \langle m_{\nu}\rangle and ∑ i m i subscript 𝑖 subscript 𝑚 𝑖 \sum_{i}m_{i} (from cosmological observation) or m l ​ i ​ g ​ h ​ t ​ e ​ s ​ t subscript 𝑚 𝑙 𝑖 𝑔 ℎ 𝑡 𝑒 𝑠 𝑡 m_{lightest} (from β 𝛽 \beta end-point direct experiments). This possibility has been explored by many authors with opposite conclusions CP-yes ; CP-no . Indeed the task is experimentally very challenging and maybe successful only for some values of the neutrino mixing matrix elements. The possibility of success rely also on a strong reduction of the uncertainties in the mixing matrix elements, in ⟨ m ν ⟩ delimited-⟨⟩ subscript 𝑚 𝜈 \langle m_{\nu}\rangle , in ∑ i m i subscript 𝑖 subscript 𝑚 𝑖 \sum_{i}m_{i} (or m l ​ i ​ g ​ h ​ t ​ e ​ s ​ t subscript 𝑚 𝑙 𝑖 𝑔 ℎ 𝑡 𝑒 𝑠 𝑡 m_{lightest} ), and in the nuclear matrix elements. I.4 Experimental approaches to β ​ β 𝛽 𝛽 \beta\beta -0 ν 𝜈 \nu There are two approaches for direct β ​ β 𝛽 𝛽 \beta\beta -0 ν 𝜈 \nu searches. In the first approach the β ​ β 𝛽 𝛽 \beta\beta -0 ν 𝜈 \nu active source is external to the detector: the experimental configuration usually consists of foil shaped sources with two detectors (e.g. scintillators, TPCs, drift chambers …) analyzing the electrons emerging from the foil. Using tracking detectors a background rejection is possible studying the event topology. The limits of this approach are the energy resolution and the small source mass. In the second approach the source is internal to the detector ( calorimeter ) and only the sum energy of the two electrons is measured. The signature for β ​ β 𝛽 𝛽 \beta\beta -0 ν 𝜈 \nu decay is therefore a peak at the transition energy Q β ​ β subscript 𝑄 𝛽 𝛽 Q_{\beta\beta} . The detector can be a scintillator, a bolometer, a semiconductor diode or a gas chamber. Calorimeters can have large mass and high efficiency. Depending on the technique, high energy resolution and also some tracking are possible. From statistical considerations, the sensitivity Σ ​ ( τ 1 / 2 0 ​ ν ) Σ superscript subscript 𝜏 1 2 0 𝜈 \Sigma(\tau_{1/2}^{0\nu}) of a β ​ β 𝛽 𝛽 \beta\beta -0 ν 𝜈 \nu decay search is given by Σ ​ ( τ 1 / 2 0 ​ ν ) ∝ ϵ ​ i . a . ( M ​ t M / ( Δ ​ E ​ b ​ k ​ g ) ) 1 / 2 formulae-sequence proportional-to Σ superscript subscript 𝜏 1 2 0 𝜈 italic-ϵ 𝑖 𝑎 superscript 𝑀 subscript 𝑡 M Δ 𝐸 𝑏 𝑘 𝑔 1 2 \Sigma(\tau_{1/2}^{0\nu})\propto\epsilon\,i.a.(Mt_{\mathrm{M}}/(\Delta E\,bkg))^{1/2} , where ϵ italic-ϵ \epsilon , i . a . formulae-sequence 𝑖 𝑎 i.a. , M 𝑀 M , t M subscript 𝑡 M t_{\mathrm{M}} , Δ ​ E Δ 𝐸 \Delta E and b ​ k ​ g 𝑏 𝑘 𝑔 bkg are the detector efficiency, the active isotope abundance, the source mass, the measuring time, the energy resolution and specific background at Q β ​ β subscript 𝑄 𝛽 𝛽 Q_{\beta\beta} , respectively. In case no background count is observed in the region of interest, the sensitivity becomes Σ ​ ( τ 1 / 2 0 ​ ν ) ∝ ϵ ​ i . a . M ​ t M formulae-sequence proportional-to Σ superscript subscript 𝜏 1 2 0 𝜈 italic-ϵ 𝑖 𝑎 𝑀 subscript 𝑡 M \Sigma(\tau_{1/2}^{0\nu})\propto\epsilon\,i.a.Mt_{\mathrm{M}} . In any experimental approach the various experimental parameters may be optimized up to some intrinsic technical limit while working on the background level usually offers the best possibility of sensitivity improvement. The background is therefore a fundamental issue in all β ​ β 𝛽 𝛽 \beta\beta -0 ν 𝜈 \nu searches: to reduce it, all passive (e.g. heavy shielding in underground sites, material selection and purification) and active (e.g. Pulse Shape Discrimination, topology analysis through granularity and segmentation) measures must be taken. However the background caused by the high energy tail of the continuous β ​ β 𝛽 𝛽 \beta\beta -2 ν 𝜈 \nu spectrum cannot be avoided and must be minimized by improving the energy resolution.

Neutrinoless double beta decay is one of the most powerful tools to set the neutrino mass absolute scale and establish whether the neutrino is a Majorana particle. After a summary of the neutrinoless double beta decay phenomenology, the present status of the experimental search for this rare decay is reported and the prospects for next generation experiments are reviewed.

[{"content":"Help me to generate the abstract of a survey paper given the title:*Double beta decay: experiments and theory review* and the introduction:I Introduction The double beta decay is a second order weak transition which can be energetically favored for some even-even nuclei belonging to A 𝐴 A even multiplets. The ( A , Z ) → ( A , Z + 2 ) + 2 ​ e − + 2 ​ ν ¯ e → 𝐴 𝑍 𝐴 𝑍 2 2 superscript 𝑒 2 subscript ¯ 𝜈 𝑒 (A,Z)\rightarrow(A,Z+2)+2e^{-}+2\bar{\nu}_{e} double beta ( β ​ β 𝛽 𝛽 \beta\beta -2 ν 𝜈 \nu ) decay process is allowed by the Standard Model and has been observed for many isotopes with lifetimes longer than 10 19 superscript 10 19 10^{19} y. A more interesting process is the so-called neutrinoless double beta ( β ​ β 𝛽 𝛽 \beta\beta -0 ν 𝜈 \nu ) decay given by ( A , Z ) → ( A , Z + 2 ) + 2 ​ e − → 𝐴 𝑍 𝐴 𝑍 2 2 superscript 𝑒 (A,Z)\rightarrow(A,Z+2)+2e^{-} : this process violates lepton number conservation and is therefore forbidden by the Standard Model. The lifetime for the β ​ β 𝛽 𝛽 \beta\beta -0 ν 𝜈 \nu decay is expected to be longer than 10 25 superscript 10 25 10^{25} y and so far only one evidence has been reported for 76 Ge (see Sec. II.1 ). For recent comprehensive reviews on this topic refer for example to Ref. reviews . I.1 β ​ β 𝛽 𝛽 \beta\beta -0 ν 𝜈 \nu decay and neutrino physics Many mechanisms have been proposed for driving this decay, but the simplest one is the mass mechanism, where a light Majorana neutrino is exchanged. Whatever is the mechanism actually causing the β ​ β 𝛽 𝛽 \beta\beta -0 ν 𝜈 \nu decay, its observation would imply that the neutrino is massive and is a Majorana particle (i.e. ν ≡ ν ¯ 𝜈 ¯ 𝜈 \nu\equiv\bar{\nu} ). For a light Majorana neutrino mediate β ​ β 𝛽 𝛽 \beta\beta -0 ν 𝜈 \nu decay, the rate is given by τ 1 / 2 0 ​ ν − 1 = ⟨ m ν ⟩ 2 ​ F N / m e 2 superscript delimited- superscript subscript 𝜏 1 2 0 𝜈 1 superscript delimited-⟨⟩ subscript 𝑚 𝜈 2 subscript 𝐹 𝑁 superscript subscript 𝑚 𝑒 2 \tau_{1/2}^{0\nu}^{-1}=\langle m_{\nu}\rangle^{2}F_{N}/m_{e}^{2} , where m e subscript 𝑚 𝑒 m_{e} is electron mass and the nuclear structure factor F N subscript 𝐹 𝑁 F_{N} contains the nuclear matrix element and the phase space. The effective neutrino Majorana mass is given by ⟨ m ν ⟩ = | ∑ k m k ​ η k ​ | U e ​ k | 2 | delimited-⟨⟩ subscript 𝑚 𝜈 subscript 𝑘 subscript 𝑚 𝑘 subscript 𝜂 𝑘 superscript subscript 𝑈 𝑒 𝑘 2 \langle m_{\nu}\rangle=|\sum_{k}m_{k}\eta_{k}|U_{ek}|^{2}| , where m k subscript 𝑚 𝑘 m_{k} are the mass eigenvalues of the three neutrino mass eigenstates | ν k ⟩ ket subscript 𝜈 𝑘 |\nu_{k}\rangle , η k subscript 𝜂 𝑘 \eta_{k} are the CP Majorana phases ( η k = ± 1 subscript 𝜂 𝑘 plus-or-minus 1 \eta_{k}=\pm 1 for CP conservation) and U e ​ k subscript 𝑈 𝑒 𝑘 U_{ek} are the elements of the electron sector of the neutrino mixing matrix. As suggested in mechanism the exact mechanism causing the β ​ β 𝛽 𝛽 \beta\beta -0 ν 𝜈 \nu could be discerned by measuring the decay rates for different isotopes. With the help of the Δ ​ m i ​ k 2 = | m i 2 − m k 2 | Δ superscript subscript 𝑚 𝑖 𝑘 2 superscript subscript 𝑚 𝑖 2 superscript subscript 𝑚 𝑘 2 \Delta m_{ik}^{2}=|m_{i}^{2}-m_{k}^{2}| and sin 2 ⁡ 2 ​ θ i ​ k = f ​ ( | U i ​ k | 2 ) superscript 2 2 subscript 𝜃 𝑖 𝑘 𝑓 superscript subscript 𝑈 𝑖 𝑘 2 \sin^{2}2\theta_{ik}=f(|U_{ik}|^{2}) parameters determined by neutrino flavor oscillation experiments (see Ref. Neutrino2006 for the latest results), it is possible to calculate ⟨ m ν ⟩ delimited-⟨⟩ subscript 𝑚 𝜈 \langle m_{\nu}\rangle as a function of the unknown neutrino absolute mass scale and η k subscript 𝜂 𝑘 \eta_{k} phases osc-bb . From this analysis two possible scenarios can be devised for the upcoming new generation experiments aiming at a 10 meV sensitivity. (1) The β ​ β 𝛽 𝛽 \beta\beta -0 ν 𝜈 \nu decay is discovered with ⟨ m ν ⟩ ≥ 10 delimited-⟨⟩ subscript 𝑚 𝜈 10 \langle m_{\nu}\rangle\geq 10 meV: then the neutrino is a Majorana particle and the masses are either degenerate ( m 1 ≈ m 2 ≈ m 3 subscript 𝑚 1 subscript 𝑚 2 subscript 𝑚 3 m_{1}\approx m_{2}\approx m_{3} ) or follow an inverse hierarchy ( m 3 ≪ m 1 ≈ m 2 much-less-than subscript 𝑚 3 subscript 𝑚 1 subscript 𝑚 2 m_{3}\ll m_{1}\approx m_{2} ). If neutrinos are degenerate (for ⟨ m ν ⟩ ≥ ≈ 0.05 \langle m_{\nu}\rangle\geq\approx 0.05 eV) then the absolute mass scale can be established. (2) The β ​ β 𝛽 𝛽 \beta\beta -0 ν 𝜈 \nu decay is not observed and only an upper limit ⟨ m ν ⟩ ≤ 10 delimited-⟨⟩ subscript 𝑚 𝜈 10 \langle m_{\nu}\rangle\leq 10 meV is set: then, if the neutrino is a Majorana particle, the masses must have a normal hierarchy ( m 1 < m 2 ≪ m 3 subscript 𝑚 1 subscript 𝑚 2 much-less-than subscript 𝑚 3 m_{1}<m_{2}\ll m_{3} ). I.2 β ​ β 𝛽 𝛽 \beta\beta -0 ν 𝜈 \nu decay and nuclear physics To obtain ⟨ m ν ⟩ delimited-⟨⟩ subscript 𝑚 𝜈 \langle m_{\nu}\rangle from the experimental observable τ 1 / 2 0 ​ ν superscript subscript 𝜏 1 2 0 𝜈 \tau_{1/2}^{0\nu} the nuclear structure factor F N ≡ G 0 ​ ν ​ ( Q β ​ β , Z ) ​ | M 0 ​ ν | 2 subscript 𝐹 𝑁 superscript 𝐺 0 𝜈 subscript 𝑄 𝛽 𝛽 𝑍 superscript superscript 𝑀 0 𝜈 2 F_{N}\equiv G^{0\nu}(Q_{\beta\beta},Z)|M^{0\nu}|^{2} must be known. While the phase space G 0 ​ ν ​ ( Q β ​ β , Z ) superscript 𝐺 0 𝜈 subscript 𝑄 𝛽 𝛽 𝑍 G^{0\nu}(Q_{\beta\beta},Z) can be precisely calculated, the nuclear matrix | M 0 ​ ν | superscript 𝑀 0 𝜈 |M^{0\nu}| contains the uncertain details of the nuclear part of the process. In fact there is a large spread in the nuclear matrix elements calculated by different authors with different nuclear models nuclearmatrix (see also references in Ref. reviews ; osc-bb ). Because of this spread also ⟨ m ν ⟩ delimited-⟨⟩ subscript 𝑚 𝜈 \langle m_{\nu}\rangle is affected by large uncertainties (about a factor 3 on the average). Presently these uncertainties are a severe limitation to the potentialities of β ​ β 𝛽 𝛽 \beta\beta -0 ν 𝜈 \nu decay as a tool for neutrino physics: it has been recently suggested rodin ; kortelainen that measured β ​ β 𝛽 𝛽 \beta\beta -2 ν 𝜈 \nu decay lifetimes can be used to reduce the spread in QRPA calculations. Nevertheless, it is important to search for β ​ β 𝛽 𝛽 \beta\beta -0 ν 𝜈 \nu decay of as many as possible candidate isotopes. I.3 β ​ β 𝛽 𝛽 \beta\beta -0 ν 𝜈 \nu decay and CP-violation The observation of β ​ β 𝛽 𝛽 \beta\beta -0 ν 𝜈 \nu could be used to establish CP-violation associated with Majorana neutrinos in the lepton sector due to the η k subscript 𝜂 𝑘 \eta_{k} Majorana phases. This issue has gained interest because it could provide an explanation for the observed baryon asymmetry in the Universe through the leptogenesis theory leptogenesis . The Majorana phases could be constrained by simultaneous precise measurements of ⟨ m ν ⟩ delimited-⟨⟩ subscript 𝑚 𝜈 \langle m_{\nu}\rangle and ∑ i m i subscript 𝑖 subscript 𝑚 𝑖 \sum_{i}m_{i} (from cosmological observation) or m l ​ i ​ g ​ h ​ t ​ e ​ s ​ t subscript 𝑚 𝑙 𝑖 𝑔 ℎ 𝑡 𝑒 𝑠 𝑡 m_{lightest} (from β 𝛽 \beta end-point direct experiments). This possibility has been explored by many authors with opposite conclusions CP-yes ; CP-no . Indeed the task is experimentally very challenging and maybe successful only for some values of the neutrino mixing matrix elements. The possibility of success rely also on a strong reduction of the uncertainties in the mixing matrix elements, in ⟨ m ν ⟩ delimited-⟨⟩ subscript 𝑚 𝜈 \langle m_{\nu}\rangle , in ∑ i m i subscript 𝑖 subscript 𝑚 𝑖 \sum_{i}m_{i} (or m l ​ i ​ g ​ h ​ t ​ e ​ s ​ t subscript 𝑚 𝑙 𝑖 𝑔 ℎ 𝑡 𝑒 𝑠 𝑡 m_{lightest} ), and in the nuclear matrix elements. I.4 Experimental approaches to β ​ β 𝛽 𝛽 \beta\beta -0 ν 𝜈 \nu There are two approaches for direct β ​ β 𝛽 𝛽 \beta\beta -0 ν 𝜈 \nu searches. In the first approach the β ​ β 𝛽 𝛽 \beta\beta -0 ν 𝜈 \nu active source is external to the detector: the experimental configuration usually consists of foil shaped sources with two detectors (e.g. scintillators, TPCs, drift chambers …) analyzing the electrons emerging from the foil. Using tracking detectors a background rejection is possible studying the event topology. The limits of this approach are the energy resolution and the small source mass. In the second approach the source is internal to the detector ( calorimeter ) and only the sum energy of the two electrons is measured. The signature for β ​ β 𝛽 𝛽 \beta\beta -0 ν 𝜈 \nu decay is therefore a peak at the transition energy Q β ​ β subscript 𝑄 𝛽 𝛽 Q_{\beta\beta} . The detector can be a scintillator, a bolometer, a semiconductor diode or a gas chamber. Calorimeters can have large mass and high efficiency. Depending on the technique, high energy resolution and also some tracking are possible. From statistical considerations, the sensitivity Σ ​ ( τ 1 / 2 0 ​ ν ) Σ superscript subscript 𝜏 1 2 0 𝜈 \Sigma(\tau_{1/2}^{0\nu}) of a β ​ β 𝛽 𝛽 \beta\beta -0 ν 𝜈 \nu decay search is given by Σ ​ ( τ 1 / 2 0 ​ ν ) ∝ ϵ ​ i . a . ( M ​ t M / ( Δ ​ E ​ b ​ k ​ g ) ) 1 / 2 formulae-sequence proportional-to Σ superscript subscript 𝜏 1 2 0 𝜈 italic-ϵ 𝑖 𝑎 superscript 𝑀 subscript 𝑡 M Δ 𝐸 𝑏 𝑘 𝑔 1 2 \Sigma(\tau_{1/2}^{0\nu})\propto\epsilon\,i.a.(Mt_{\mathrm{M}}/(\Delta E\,bkg))^{1/2} , where ϵ italic-ϵ \epsilon , i . a . formulae-sequence 𝑖 𝑎 i.a. , M 𝑀 M , t M subscript 𝑡 M t_{\mathrm{M}} , Δ ​ E Δ 𝐸 \Delta E and b ​ k ​ g 𝑏 𝑘 𝑔 bkg are the detector efficiency, the active isotope abundance, the source mass, the measuring time, the energy resolution and specific background at Q β ​ β subscript 𝑄 𝛽 𝛽 Q_{\beta\beta} , respectively. In case no background count is observed in the region of interest, the sensitivity becomes Σ ​ ( τ 1 / 2 0 ​ ν ) ∝ ϵ ​ i . a . M ​ t M formulae-sequence proportional-to Σ superscript subscript 𝜏 1 2 0 𝜈 italic-ϵ 𝑖 𝑎 𝑀 subscript 𝑡 M \Sigma(\tau_{1/2}^{0\nu})\propto\epsilon\,i.a.Mt_{\mathrm{M}} . In any experimental approach the various experimental parameters may be optimized up to some intrinsic technical limit while working on the background level usually offers the best possibility of sensitivity improvement. The background is therefore a fundamental issue in all β ​ β 𝛽 𝛽 \beta\beta -0 ν 𝜈 \nu searches: to reduce it, all passive (e.g. heavy shielding in underground sites, material selection and purification) and active (e.g. Pulse Shape Discrimination, topology analysis through granularity and segmentation) measures must be taken. However the background caused by the high energy tail of the continuous β ​ β 𝛽 𝛽 \beta\beta -2 ν 𝜈 \nu spectrum cannot be avoided and must be minimized by improving the energy resolution.","role":"user"},{"content":"Abstract: Neutrinoless double beta decay is one of the most powerful tools to set the neutrino mass absolute scale and establish whether the neutrino is a Majorana particle. After a summary of the neutrinoless double beta decay phenomenology, the present status of the experimental search for this rare decay is reported and the prospects for next generation experiments are reviewed.","role":"assistant"}]

Survey on geometric group theory

1. Classical examples A classical example of geometric methods used in group theory is the topological proof of Schreier’s theorem. Theorem 1.1 (Schreier’s Theorem) . Let G 𝐺 G be a free group and H ⊆ G 𝐻 𝐺 H\subseteq G be a subgroup. Then H 𝐻 H is free. If the rank rk ⁡ ( G ) rk 𝐺 \operatorname{rk}(G) and the index [ G : H ] delimited-[] : 𝐺 𝐻 [G:H] are finite, then the rank of H 𝐻 H is finite and satisfies rk ( H ) = [ G : H ] ⋅ ( rk ( G ) − 1 ) + 1 . \operatorname{rk}(H)=[G:H]\cdot\bigl{(}\operatorname{rk}(G)-1\bigr{)}+1. Proof. Let G 𝐺 G be a free group on the set S 𝑆 S . Take the wedge X = ⋁ S S 1 𝑋 subscript 𝑆 superscript 𝑆 1 X=\bigvee_{S}S^{1} of circles, one copy for each element in S 𝑆 S . This is a 1 1 1 -dimensional C ​ W 𝐶 𝑊 CW -complex with π 1 ​ ( X ) ≅ G subscript 𝜋 1 𝑋 𝐺 \pi_{1}(X)\cong G by the Seifert-van Kampen Theorem. Let p : X ¯ → X : 𝑝 → ¯ 𝑋 𝑋 p\colon\overline{X}\to X be the covering associated to H ⊆ G = π 1 ​ ( X ) 𝐻 𝐺 subscript 𝜋 1 𝑋 H\subseteq G=\pi_{1}(X) . We have π 1 ​ ( X ¯ ) ≅ H subscript 𝜋 1 ¯ 𝑋 𝐻 \pi_{1}(\overline{X})\cong H . Since X 𝑋 X is a 1 1 1 -dimensional C ​ W 𝐶 𝑊 CW -complex, X ¯ ¯ 𝑋 \overline{X} is a 1 1 1 -dimensional C ​ W 𝐶 𝑊 CW -complex. If T ⊆ X ¯ 𝑇 ¯ 𝑋 T\subseteq\overline{X} is a maximal tree, then X ¯ ¯ 𝑋 \overline{X} is homotopy equivalent to X ¯ / T = ⋁ S ¯ S 1 ¯ 𝑋 𝑇 subscript ¯ 𝑆 superscript 𝑆 1 \overline{X}/T=\bigvee_{\overline{S}}S^{1} for some set S ¯ ¯ 𝑆 \overline{S} . By the Seifert-van Kampen Theorem H ≅ π 1 ​ ( X ¯ ) 𝐻 subscript 𝜋 1 ¯ 𝑋 H\cong\pi_{1}(\overline{X}) is the free group generated by the set S ¯ ¯ 𝑆 \overline{S} . Suppose that rk ⁡ ( G ) rk 𝐺 \operatorname{rk}(G) and [ G : H ] delimited-[] : 𝐺 𝐻 [G:H] are finite. Since | S | = rk ⁡ ( G ) 𝑆 rk 𝐺 |S|=\operatorname{rk}(G) , the C ​ W 𝐶 𝑊 CW -complex X 𝑋 X is compact. Since [ G : H ] delimited-[] : 𝐺 𝐻 [G:H] is finite, the C ​ W 𝐶 𝑊 CW -complex X ¯ ¯ 𝑋 \overline{X} and hence X ¯ / T ¯ 𝑋 𝑇 \overline{X}/T are compact. Hence rk ⁡ ( H ) = | S ¯ | rk 𝐻 ¯ 𝑆 \operatorname{rk}(H)=|\overline{S}| is finite. We obtain for the Euler characteristics 1 − | S ¯ | = χ ( X ¯ ) = [ G : H ] ⋅ χ ( X ) = [ G : H ] ⋅ ( 1 − | S | ) . 1-|\overline{S}|=\chi(\overline{X})=[G:H]\cdot\chi(X)=[G:H]\cdot\left(1-|S|\right). Since | S | = rk ⁡ ( G ) 𝑆 rk 𝐺 |S|=\operatorname{rk}(G) and | S ¯ | = rk ⁡ ( H ) ¯ 𝑆 rk 𝐻 |\overline{S}|=\operatorname{rk}(H) , the claim follows. ∎ Another example of this type is the topological proof of Kurosh’s Theorem, which can be found for instance in [ 130 , Theorem 14 in I.5 on page 56] . The interpretation of amalgamated products and HNN HNN \operatorname{HNN} -extensions in terms of topological spaces by the Seifert-van Kampen Theorem or actions of groups on trees are in the same spirit (see for instance [ 8 ] , [ 28 ] , [ 35 ] , [ 91 ] , [ 130 ] ).

This article is a survey article on geometric group theory from the point of view of a non-expert who likes geometric group theory and uses it in his own research. The sections are: classical examples, basics about quasiisometry,properties and invariants of groups invariant under quasiisometry, rigidity, hyperbolic spaces and CAT(k)-spaces, the boundary of a hyperbolic space, hyperbolic groups, CAT(0)-groups, classifying spaces for proper actions, measurable group theory, some open problems.

[{"content":"Help me to generate the abstract of a survey paper given the title:*Survey on geometric group theory* and the introduction:1. Classical examples A classical example of geometric methods used in group theory is the topological proof of Schreiers theorem. Theorem 1.1 (Schreiers Theorem) . Let G 𝐺 G be a free group and H ⊆ G 𝐻 𝐺 H\subseteq G be a subgroup. Then H 𝐻 H is free. If the rank rk ⁡ ( G ) rk 𝐺 \operatorname{rk}(G) and the index G : H delimited- : 𝐺 𝐻 G:H are finite, then the rank of H 𝐻 H is finite and satisfies rk ( H ) = G : H ⋅ ( rk ( G ) − 1 ) + 1 . \operatorname{rk}(H)=G:H\cdot\bigl{(}\operatorname{rk}(G)-1\bigr{)}+1. Proof. Let G 𝐺 G be a free group on the set S 𝑆 S . Take the wedge X = ⋁ S S 1 𝑋 subscript 𝑆 superscript 𝑆 1 X=\bigvee_{S}S^{1} of circles, one copy for each element in S 𝑆 S . This is a 1 1 1 -dimensional C ​ W 𝐶 𝑊 CW -complex with π 1 ​ ( X ) ≅ G subscript 𝜋 1 𝑋 𝐺 \pi_{1}(X)\cong G by the Seifert-van Kampen Theorem. Let p : X ¯ → X : 𝑝 → ¯ 𝑋 𝑋 p\colon\overline{X}\to X be the covering associated to H ⊆ G = π 1 ​ ( X ) 𝐻 𝐺 subscript 𝜋 1 𝑋 H\subseteq G=\pi_{1}(X) . We have π 1 ​ ( X ¯ ) ≅ H subscript 𝜋 1 ¯ 𝑋 𝐻 \pi_{1}(\overline{X})\cong H . Since X 𝑋 X is a 1 1 1 -dimensional C ​ W 𝐶 𝑊 CW -complex, X ¯ ¯ 𝑋 \overline{X} is a 1 1 1 -dimensional C ​ W 𝐶 𝑊 CW -complex. If T ⊆ X ¯ 𝑇 ¯ 𝑋 T\subseteq\overline{X} is a maximal tree, then X ¯ ¯ 𝑋 \overline{X} is homotopy equivalent to X ¯ / T = ⋁ S ¯ S 1 ¯ 𝑋 𝑇 subscript ¯ 𝑆 superscript 𝑆 1 \overline{X}/T=\bigvee_{\overline{S}}S^{1} for some set S ¯ ¯ 𝑆 \overline{S} . By the Seifert-van Kampen Theorem H ≅ π 1 ​ ( X ¯ ) 𝐻 subscript 𝜋 1 ¯ 𝑋 H\cong\pi_{1}(\overline{X}) is the free group generated by the set S ¯ ¯ 𝑆 \overline{S} . Suppose that rk ⁡ ( G ) rk 𝐺 \operatorname{rk}(G) and G : H delimited- : 𝐺 𝐻 G:H are finite. Since | S | = rk ⁡ ( G ) 𝑆 rk 𝐺 |S|=\operatorname{rk}(G) , the C ​ W 𝐶 𝑊 CW -complex X 𝑋 X is compact. Since G : H delimited- : 𝐺 𝐻 G:H is finite, the C ​ W 𝐶 𝑊 CW -complex X ¯ ¯ 𝑋 \overline{X} and hence X ¯ / T ¯ 𝑋 𝑇 \overline{X}/T are compact. Hence rk ⁡ ( H ) = | S ¯ | rk 𝐻 ¯ 𝑆 \operatorname{rk}(H)=|\overline{S}| is finite. We obtain for the Euler characteristics 1 − | S ¯ | = χ ( X ¯ ) = G : H ⋅ χ ( X ) = G : H ⋅ ( 1 − | S | ) . 1-|\overline{S}|=\chi(\overline{X})=G:H\cdot\chi(X)=G:H\cdot\left(1-|S|\right). Since | S | = rk ⁡ ( G ) 𝑆 rk 𝐺 |S|=\operatorname{rk}(G) and | S ¯ | = rk ⁡ ( H ) ¯ 𝑆 rk 𝐻 |\overline{S}|=\operatorname{rk}(H) , the claim follows. ∎ Another example of this type is the topological proof of Kuroshs Theorem, which can be found for instance in 130 , Theorem 14 in I.5 on page 56 . The interpretation of amalgamated products and HNN HNN \operatorname{HNN} -extensions in terms of topological spaces by the Seifert-van Kampen Theorem or actions of groups on trees are in the same spirit (see for instance 8 , 28 , 35 , 91 , 130 ).","role":"user"},{"content":"Abstract: This article is a survey article on geometric group theory from the point of view of a non-expert who likes geometric group theory and uses it in his own research. The sections are: classical examples, basics about quasiisometry,properties and invariants of groups invariant under quasiisometry, rigidity, hyperbolic spaces and CAT(k)-spaces, the boundary of a hyperbolic space, hyperbolic groups, CAT(0)-groups, classifying spaces for proper actions, measurable group theory, some open problems.","role":"assistant"}]

A SURVEY ON THE INTERPLAY BETWEEN ARITHMETIC MEAN IDEALS, TRACES, LATTICES OF OPERATOR IDEALS, AND AN INFINITE SCHUR-HORN MAJORIZATION THEOREM

1. Introduction Commutator spaces (also known as commutator ideals) of operator ideals were characterized in [ 24 ] in terms of arithmetic mean operations and this led already in [ 24 ] to the notion of two basic arithmetic mean ideals, the arithmetic mean closure and the arithmetic mean interior of an ideal, and to the notion of arithmetically mean stable ideals - those that coincide with their commutator space and hence support no nonzero trace. Thus arithmetic mean ideals have become an intrinsic part of the theory of operator ideals. To explore their structure and to find how they relate to the existing theory of operator ideals and in particular, how they contribute to it, was the goal of our program. The first results of this work were announced in PNAS-US [ 33 ] and then expanded and developed in a series of papers [ 34 ] - [ 38 ] . Our beginning question, explored in [ 34 ] , was to ask “how many traces can a given ideal support.” To address it, we found that we first needed both to investigate much more systematically the structure of arithmetic mean ideals and to introduce formally the notion of “soft ideals.” From the start we discovered that, to study the two basic arithmetic mean ideals, we had to expand our investigations by adding to the menagerie the smallest am-open ideal containing an ideal and the largest am-closed ideal contained by it. The latter, for instance, provided a useful tool for the study of countably generated am-stable ideals. The arithmetic mean is too crude for distinguishing ideals contained in the trace class ℒ 1 subscript ℒ 1 \mathscr{L}_{1} : for them what is required is the arithmetic mean at infinity. The arithmetic mean at infinity (am- ∞ \infty for short) has substantially different properties from the arithmetic mean, e.g., the am- ∞ \infty of a sequence can fail to satisfy the Δ 1 / 2 subscript Δ 1 2 \Delta_{1/2} -condition whereas an important feature of the arithmetic mean of a sequence is that it always satisfies this condition. The ensuing theory of am- ∞ \infty ideals is considerably more complicated, but in large measure mirrors the arithmetic mean case. This material is presented in Section 2 and was developed in [ 34 ] and [ 35 ] . In Section 3 we present “soft ideals” which are ubiquitous in the literature on operator ideals and which we studied systematically in [ 35 ] . We show that many of the classical ideals are soft and that keeping this in mind can add perspective and simplify the proofs of a number of results in the literature. Our interest in the notion of softness arose from the interplay between softness properties and arithmetic mean operations, which provided the main tool for our work on traces. We explore in this section this interplay and present several open questions. Section 5.2 is devoted to traces and based is on [ 34 ] . The question ”how many traces can a given ideal support” asks about the codimension of the commutator space [ I , B ​ ( H ) ] 𝐼 𝐵 𝐻 [I,B(H)] of an ideal I 𝐼 I . We were led by our work in [ 34 ] to the conjecture that dim ⁡ I [ I , B ​ ( H ) ] ∈ { { 1 , ∞ } when ​ ω ∉ Σ ​ ( I ) { 0 , ∞ } when ​ ω ∈ Σ ​ ( I ) dim 𝐼 𝐼 𝐵 𝐻 cases 1 when 𝜔 Σ 𝐼 0 when 𝜔 Σ 𝐼 \operatorname{dim}\frac{I}{[I,B(H)]}~{}\in\begin{cases}\{1,\infty\}&\text{when}~{}\omega\notin\Sigma(I)\\ \{0,\infty\}&\text{when}~{}\omega\in\Sigma(I)\end{cases} where ω := < 1 , 1 2 , 1 3 , ⋯ > \omega:=\,<1,\frac{1}{2},\frac{1}{3},\dots> is the harmonic sequence and Σ ​ ( I ) Σ 𝐼 \Sigma(I) is the characteristic set of I 𝐼 I . In the case when ω ∉ Σ ​ ( I ) 𝜔 Σ 𝐼 \omega\notin\Sigma(I) , we show that the codimension is 1 1 1 precisely when I 𝐼 I is am- ∞ \infty stable. In the process, we obtain results that can be applied to solve questions on elementary operators. All ideals divide naturally into the classes: • “small ideals,” (those contained in the lower stabilizer at infinity of the principal ideal ( ω ) 𝜔 (\omega) , s ​ t a ∞ ​ ( ω ) 𝑠 subscript 𝑡 subscript 𝑎 𝜔 st_{a_{\infty}}(\omega) , i.e., the ideal with characteristic set { ξ ∈ c o ∗ ∣ ∑ ξ n ​ l ​ o ​ g m ​ n < ∞ } conditional-set 𝜉 superscript subscript c o subscript 𝜉 𝑛 𝑙 𝑜 superscript 𝑔 𝑚 𝑛 \{\xi\in{\text{c}_{\text{o}}^{*}}\mid\sum\xi_{n}log^{m}n<\infty\} ), • “large ideals” (those that contain the upper stabilizer s ​ t a ​ ( ω ) := ⋃ m = 0 ∞ ( ω ​ log m ) assign 𝑠 superscript 𝑡 𝑎 𝜔 subscript superscript 𝑚 0 𝜔 superscript 𝑚 st^{a}(\omega):=\bigcup^{\infty}_{m=0}(\omega\log^{m}) , a directed union of the principal ideals ( ω ​ log m ) 𝜔 superscript 𝑚 (\omega\log^{m}) ) and • “intermediate ideals” (all the remaining ideals) For all intermediate ideals, the codimension of the commutator space is always infinite. Then we show that the above codimension conjecture holds for all soft ideals and present other sufficient conditions for the codimension of the commutator space to be infinite. Section 5 is based on [ 36 ] , a study of the lattice structure of operator ideals focusing in particular on some distinguished sublattices of principal ideals. We investigate density properties, i.e., when between two ideals each in one of two nested lattices lies another from the smaller lattice. These properties are powerful tools for dealing with general ideals. We also study representations of ideals as directed unions of countably generated ideals and of principal ideals. We apply then some of these tools to problems on arithmetic mean “equality cancellations”: when for an ideal I 𝐼 I does J a = I a ⇒ J = I subscript 𝐽 𝑎 subscript 𝐼 𝑎 ⇒ 𝐽 𝐼 J_{a}=I_{a}\Rightarrow J=I ? And “inclusion cancellations”: when for an ideal I 𝐼 I does J a ⊂ I a ⇒ J ⊂ I subscript 𝐽 𝑎 subscript 𝐼 𝑎 ⇒ 𝐽 𝐼 J_{a}\subset I_{a}\Rightarrow J\subset I or J a ⊃ I a ⇒ J ⊃ I superset-of subscript 𝐽 𝑎 subscript 𝐼 𝑎 ⇒ 𝐽 superset-of 𝐼 J_{a}\supset I_{a}\Rightarrow J\supset I ? The most interesting case is the latter and the answer is when I = I ^ 𝐼 ^ 𝐼 I=\widehat{I} . For I 𝐼 I principal and not contained in the trace class we show that I ^ ^ 𝐼 \widehat{I} is principal as well and we prove that for for 0 < p < 1 0 𝑝 1 0<p<1 , ( ω p ) ^ = ( ω p ′ ) ^ superscript 𝜔 𝑝 superscript 𝜔 superscript 𝑝 ′ \widehat{(\omega^{p})}=(\omega^{p^{\prime}}) where 1 p − 1 p ′ = 1 1 𝑝 1 superscript 𝑝 ′ 1 \frac{1}{p}-\frac{1}{p^{\prime}}=1 . We show that the condition ( ξ ) = ( ξ ) ^ 𝜉 ^ 𝜉 (\xi)=\widehat{(\xi)} is strictly stronger than regularity. Second order cancellation properties are considerably different and harder than first order cancellation properties. We found that the cancellation I a 2 = J a 2 ⇒ J a = I a subscript 𝐼 superscript 𝑎 2 subscript 𝐽 superscript 𝑎 2 ⇒ subscript 𝐽 𝑎 subscript 𝐼 𝑎 I_{a^{2}}=J_{a^{2}}\Rightarrow J_{a}=I_{a} does not hold in general even if I 𝐼 I and J 𝐽 J are principal ideals, thus answering a question by M. Wodzicki. Sufficient conditions for this and for the two second order inclusion cancellations to hold are given in Section 6 in terms of the ”ratio of regularity,” r ​ ( ξ a ) := ξ a 2 ξ a assign 𝑟 subscript 𝜉 𝑎 subscript 𝜉 superscript 𝑎 2 subscript 𝜉 𝑎 r(\xi_{a}):=\frac{\xi_{a^{2}}}{\xi_{a}} , where ξ 𝜉 \xi is the generator of I 𝐼 I . This material was developed in [ 37 ] . Finally, Section 7 focuses on majorization theory for infinite sequences. The link with operator ideals is that the definition of am-closure of an ideal can be restated in terms of hereditariness (i.e., solidity) with respect to majorization. Our initial goal was to prove that am-closure for an ideal is equivalent to diagonal invariance, i.e., the property that for any fixed orthonormal basis, if an operator A 𝐴 A belongs to the ideal then its main diagonal, E ​ ( A ) 𝐸 𝐴 E(A) , also belongs to the ideal. To prove this equivalence, we introduced the notions of block majorization and strong majorization of sequences and these, in turn, provided the tools and the stimulus for proving an infinite Schur-Horn majorization theorem (Theorem 7.12 ). The key part of this theorem is that if ξ 𝜉 \xi and η 𝜂 \eta are monotone sequences decreasing to 0 0 and ξ 𝜉 \xi is not summable, then the condition ∑ j = 1 n ξ j ≤ ∑ j = 1 n η j superscript subscript 𝑗 1 𝑛 subscript 𝜉 𝑗 superscript subscript 𝑗 1 𝑛 subscript 𝜂 𝑗 \sum_{j=1}^{n}\xi_{j}\leq\sum_{j=1}^{n}\eta_{j} for every n 𝑛 n is necessary and sufficient for diag ⁡ ξ = E ​ ( U ​ diag ⁡ η ​ U ∗ ) diag 𝜉 𝐸 𝑈 diag 𝜂 superscript 𝑈 \operatorname{diag}\xi=E(U\operatorname{diag}\eta U^{*}) for some unitary operator U 𝑈 U . The material of Section 7 is based on [ 38 ] which is not yet in final form but we expect it to be available by the time this survey is published. No proofs are presented in this survey and many of the basic facts are stated without explicit reference, but both proofs and further background information can be found easily in the respective citations. We include herein ten questions that arose in this project.

The main result in (23) on the structure of commutators showed that arithmetic means play an important role in the study of operator ideals. In this survey we present the notions of arithmetic mean ideals and arithmetic mean at infinity ideals. Then we explore their connections with commutator spaces, traces, elementary operators, lattice and sub- lattice structure of ideals, arithmetic mean ideal cancellation properties of first and second order, and softness properties - a term that we introduced, but a notion that is ubiquitous in operator ideals. Arithmetic mean closure of ideals leads us to investigate majorization for infinite sequences and this in turn leads us to an infinite Schur-Horn majorization theorem that extends recent work by Arveson and Kadison to nonsummable sequences. This survey covers the material announced towards the beginning of the project in PNAS- US (33) and then expanded and developed in a series of papers (34)-(38).

[{"content":"Help me to generate the abstract of a survey paper given the title:*A SURVEY ON THE INTERPLAY BETWEEN ARITHMETIC MEAN IDEALS, TRACES, LATTICES OF OPERATOR IDEALS, AND AN INFINITE SCHUR-HORN MAJORIZATION THEOREM* and the introduction:1. Introduction Commutator spaces (also known as commutator ideals) of operator ideals were characterized in 24 in terms of arithmetic mean operations and this led already in 24 to the notion of two basic arithmetic mean ideals, the arithmetic mean closure and the arithmetic mean interior of an ideal, and to the notion of arithmetically mean stable ideals - those that coincide with their commutator space and hence support no nonzero trace. Thus arithmetic mean ideals have become an intrinsic part of the theory of operator ideals. To explore their structure and to find how they relate to the existing theory of operator ideals and in particular, how they contribute to it, was the goal of our program. The first results of this work were announced in PNAS-US 33 and then expanded and developed in a series of papers 34 - 38 . Our beginning question, explored in 34 , was to ask how many traces can a given ideal support. To address it, we found that we first needed both to investigate much more systematically the structure of arithmetic mean ideals and to introduce formally the notion of soft ideals. From the start we discovered that, to study the two basic arithmetic mean ideals, we had to expand our investigations by adding to the menagerie the smallest am-open ideal containing an ideal and the largest am-closed ideal contained by it. The latter, for instance, provided a useful tool for the study of countably generated am-stable ideals. The arithmetic mean is too crude for distinguishing ideals contained in the trace class ℒ 1 subscript ℒ 1 \mathscr{L}_{1} : for them what is required is the arithmetic mean at infinity. The arithmetic mean at infinity (am- ∞ \infty for short) has substantially different properties from the arithmetic mean, e.g., the am- ∞ \infty of a sequence can fail to satisfy the Δ 1 / 2 subscript Δ 1 2 \Delta_{1/2} -condition whereas an important feature of the arithmetic mean of a sequence is that it always satisfies this condition. The ensuing theory of am- ∞ \infty ideals is considerably more complicated, but in large measure mirrors the arithmetic mean case. This material is presented in Section 2 and was developed in 34 and 35 . In Section 3 we present soft ideals which are ubiquitous in the literature on operator ideals and which we studied systematically in 35 . We show that many of the classical ideals are soft and that keeping this in mind can add perspective and simplify the proofs of a number of results in the literature. Our interest in the notion of softness arose from the interplay between softness properties and arithmetic mean operations, which provided the main tool for our work on traces. We explore in this section this interplay and present several open questions. Section 5.2 is devoted to traces and based is on 34 . The question how many traces can a given ideal support asks about the codimension of the commutator space I , B ​ ( H ) 𝐼 𝐵 𝐻 I,B(H) of an ideal I 𝐼 I . We were led by our work in 34 to the conjecture that dim ⁡ I I , B ​ ( H ) ∈ { { 1 , ∞ } when ​ ω ∉ Σ ​ ( I ) { 0 , ∞ } when ​ ω ∈ Σ ​ ( I ) dim 𝐼 𝐼 𝐵 𝐻 cases 1 when 𝜔 Σ 𝐼 0 when 𝜔 Σ 𝐼 \operatorname{dim}\frac{I}{I,B(H)}~{}\in\begin{cases}\{1,\infty\}&\text{when}~{}\omega\notin\Sigma(I)\\ \{0,\infty\}&\text{when}~{}\omega\in\Sigma(I)\end{cases} where ω := < 1 , 1 2 , 1 3 , ⋯ > \omega:=\,<1,\frac{1}{2},\frac{1}{3},\dots> is the harmonic sequence and Σ ​ ( I ) Σ 𝐼 \Sigma(I) is the characteristic set of I 𝐼 I . In the case when ω ∉ Σ ​ ( I ) 𝜔 Σ 𝐼 \omega\notin\Sigma(I) , we show that the codimension is 1 1 1 precisely when I 𝐼 I is am- ∞ \infty stable. In the process, we obtain results that can be applied to solve questions on elementary operators. All ideals divide naturally into the classes: • small ideals, (those contained in the lower stabilizer at infinity of the principal ideal ( ω ) 𝜔 (\omega) , s ​ t a ∞ ​ ( ω ) 𝑠 subscript 𝑡 subscript 𝑎 𝜔 st_{a_{\infty}}(\omega) , i.e., the ideal with characteristic set { ξ ∈ c o ∗ ∣ ∑ ξ n ​ l ​ o ​ g m ​ n < ∞ } conditional-set 𝜉 superscript subscript c o subscript 𝜉 𝑛 𝑙 𝑜 superscript 𝑔 𝑚 𝑛 \{\xi\in{\text{c}_{\text{o}}^{*}}\mid\sum\xi_{n}log^{m}n<\infty\} ), • large ideals (those that contain the upper stabilizer s ​ t a ​ ( ω ) := ⋃ m = 0 ∞ ( ω ​ log m ) assign 𝑠 superscript 𝑡 𝑎 𝜔 subscript superscript 𝑚 0 𝜔 superscript 𝑚 st^{a}(\omega):=\bigcup^{\infty}_{m=0}(\omega\log^{m}) , a directed union of the principal ideals ( ω ​ log m ) 𝜔 superscript 𝑚 (\omega\log^{m}) ) and • intermediate ideals (all the remaining ideals) For all intermediate ideals, the codimension of the commutator space is always infinite. Then we show that the above codimension conjecture holds for all soft ideals and present other sufficient conditions for the codimension of the commutator space to be infinite. Section 5 is based on 36 , a study of the lattice structure of operator ideals focusing in particular on some distinguished sublattices of principal ideals. We investigate density properties, i.e., when between two ideals each in one of two nested lattices lies another from the smaller lattice. These properties are powerful tools for dealing with general ideals. We also study representations of ideals as directed unions of countably generated ideals and of principal ideals. We apply then some of these tools to problems on arithmetic mean equality cancellations: when for an ideal I 𝐼 I does J a = I a ⇒ J = I subscript 𝐽 𝑎 subscript 𝐼 𝑎 ⇒ 𝐽 𝐼 J_{a}=I_{a}\Rightarrow J=I ? And inclusion cancellations: when for an ideal I 𝐼 I does J a ⊂ I a ⇒ J ⊂ I subscript 𝐽 𝑎 subscript 𝐼 𝑎 ⇒ 𝐽 𝐼 J_{a}\subset I_{a}\Rightarrow J\subset I or J a ⊃ I a ⇒ J ⊃ I superset-of subscript 𝐽 𝑎 subscript 𝐼 𝑎 ⇒ 𝐽 superset-of 𝐼 J_{a}\supset I_{a}\Rightarrow J\supset I ? The most interesting case is the latter and the answer is when I = I ^ 𝐼 ^ 𝐼 I=\widehat{I} . For I 𝐼 I principal and not contained in the trace class we show that I ^ ^ 𝐼 \widehat{I} is principal as well and we prove that for for 0 < p < 1 0 𝑝 1 0<p<1 , ( ω p ) ^ = ( ω p ′ ) ^ superscript 𝜔 𝑝 superscript 𝜔 superscript 𝑝 ′ \widehat{(\omega^{p})}=(\omega^{p^{\prime}}) where 1 p − 1 p ′ = 1 1 𝑝 1 superscript 𝑝 ′ 1 \frac{1}{p}-\frac{1}{p^{\prime}}=1 . We show that the condition ( ξ ) = ( ξ ) ^ 𝜉 ^ 𝜉 (\xi)=\widehat{(\xi)} is strictly stronger than regularity. Second order cancellation properties are considerably different and harder than first order cancellation properties. We found that the cancellation I a 2 = J a 2 ⇒ J a = I a subscript 𝐼 superscript 𝑎 2 subscript 𝐽 superscript 𝑎 2 ⇒ subscript 𝐽 𝑎 subscript 𝐼 𝑎 I_{a^{2}}=J_{a^{2}}\Rightarrow J_{a}=I_{a} does not hold in general even if I 𝐼 I and J 𝐽 J are principal ideals, thus answering a question by M. Wodzicki. Sufficient conditions for this and for the two second order inclusion cancellations to hold are given in Section 6 in terms of the ratio of regularity, r ​ ( ξ a ) := ξ a 2 ξ a assign 𝑟 subscript 𝜉 𝑎 subscript 𝜉 superscript 𝑎 2 subscript 𝜉 𝑎 r(\xi_{a}):=\frac{\xi_{a^{2}}}{\xi_{a}} , where ξ 𝜉 \xi is the generator of I 𝐼 I . This material was developed in 37 . Finally, Section 7 focuses on majorization theory for infinite sequences. The link with operator ideals is that the definition of am-closure of an ideal can be restated in terms of hereditariness (i.e., solidity) with respect to majorization. Our initial goal was to prove that am-closure for an ideal is equivalent to diagonal invariance, i.e., the property that for any fixed orthonormal basis, if an operator A 𝐴 A belongs to the ideal then its main diagonal, E ​ ( A ) 𝐸 𝐴 E(A) , also belongs to the ideal. To prove this equivalence, we introduced the notions of block majorization and strong majorization of sequences and these, in turn, provided the tools and the stimulus for proving an infinite Schur-Horn majorization theorem (Theorem 7.12 ). The key part of this theorem is that if ξ 𝜉 \xi and η 𝜂 \eta are monotone sequences decreasing to 0 0 and ξ 𝜉 \xi is not summable, then the condition ∑ j = 1 n ξ j ≤ ∑ j = 1 n η j superscript subscript 𝑗 1 𝑛 subscript 𝜉 𝑗 superscript subscript 𝑗 1 𝑛 subscript 𝜂 𝑗 \sum_{j=1}^{n}\xi_{j}\leq\sum_{j=1}^{n}\eta_{j} for every n 𝑛 n is necessary and sufficient for diag ⁡ ξ = E ​ ( U ​ diag ⁡ η ​ U ∗ ) diag 𝜉 𝐸 𝑈 diag 𝜂 superscript 𝑈 \operatorname{diag}\xi=E(U\operatorname{diag}\eta U^{*}) for some unitary operator U 𝑈 U . The material of Section 7 is based on 38 which is not yet in final form but we expect it to be available by the time this survey is published. No proofs are presented in this survey and many of the basic facts are stated without explicit reference, but both proofs and further background information can be found easily in the respective citations. We include herein ten questions that arose in this project.","role":"user"},{"content":"Abstract: The main result in (23) on the structure of commutators showed that arithmetic means play an important role in the study of operator ideals. In this survey we present the notions of arithmetic mean ideals and arithmetic mean at infinity ideals. Then we explore their connections with commutator spaces, traces, elementary operators, lattice and sub- lattice structure of ideals, arithmetic mean ideal cancellation properties of first and second order, and softness properties - a term that we introduced, but a notion that is ubiquitous in operator ideals. Arithmetic mean closure of ideals leads us to investigate majorization for infinite sequences and this in turn leads us to an infinite Schur-Horn majorization theorem that extends recent work by Arveson and Kadison to nonsummable sequences. This survey covers the material announced towards the beginning of the project in PNAS- US (33) and then expanded and developed in a series of papers (34)-(38).","role":"assistant"}]

Generalized Kodama partition functions: A preview into normalizability for the generalized Kodama states.

The purpose of this paper is to apply a special technique for computing partition functions to the computation of the norm of the generalized Kodama states of quantum gravity introduced in [ 1 ] . The closest analogy is the partition function for the Chern–Simons functional. There exist in the literature various methods for computing the partition function of the Chern–Simons wavefunction ( [ 2 ] , [ 3 ] , [ 4 ] and refences therein). As a path integral, the nonabelian Chern–Simons theory in three spatial to the present author’s knowledge dimensions produces a renormalizable loop expansion, since any quantum divergences can always be absorbed, as shown by various authors, into a redefinition of tree-level coupling constants. This implies that the classical action and the effective action for the Chern–Simons functional are the same, to within re-definition of these constants.The first attempt to apply the perturbation theory of the Chern–Simons partition function to the fill theory of quantum gravity in the connection representation appears to be [ 5 ] , in which Chopin Soo computes the partition function for the pure Kodama state. We would like to generalize the techniques for computation of the Chern–Simons partition function from the pure Kodama state Ψ K ​ o ​ d subscript Ψ 𝐾 𝑜 𝑑 \Psi_{Kod} to the generalized Kodama states Ψ G ​ K ​ o ​ d subscript Ψ 𝐺 𝐾 𝑜 𝑑 \Psi_{GKod} . In the present work we will apply the background field method for computing the effective action, usually attributed to Wilson and DeWitt, for the task as set out.We would ultimately like to extend the results of this paper to address the normalizability of Ψ G ​ K ​ o ​ d subscript Ψ 𝐺 𝐾 𝑜 𝑑 \Psi_{GKod} . This will depend upon the ability to make sense of the configuration path integral, for the partition function, which will in turn depend upon the finiteness of the generalized Kodama state. We will not focus in this paper on such issues as gauge fixing and the renormalization group theory. The main thrust is to show that the perturbative expansion exists and conforms to the requirements of a perturbatively renormalizable effective action, by direct analogy to the case for Chern–Simons theory which has been thorougly explored by other authors. Our primary contribution is that the principle of the semiclassical-quantum correspondence introduced in [ 1 ] is particularly fortuitous when applied particularly to this special class of quantum states Ψ G ​ K ​ o ​ d subscript Ψ 𝐺 𝐾 𝑜 𝑑 \Psi_{GKod} . We also present a first Ansatz for the norm of Ψ G ​ K ​ o ​ d subscript Ψ 𝐺 𝐾 𝑜 𝑑 \Psi_{GKod} as well as expectation values of operators on the state, not commenting in detail on reality conditions.An important aspect of this paper is that all quantities are computed on the spatial hypersurface Σ T subscript Σ 𝑇 \Sigma_{T} on which which Ψ G ​ K ​ o ​ d subscript Ψ 𝐺 𝐾 𝑜 𝑑 \Psi_{GKod} is evaluated. So while it may appear that the treatment is timeless, we take time to be an implicit label. We will often in this paper not distinguish between spacetime position x = ( 𝒙 , t ) 𝑥 𝒙 𝑡 x=(\boldsymbol{x},t) and spatial position 𝒙 𝒙 \boldsymbol{x} . The partition function can be seen as a unified gravity-matter system with a dimensionless coupling constant. When split into its individual components, the partition function has an interesting geometric interpretation bearing a distinct resemblance to Kaluza–Klein like theories on the infinite dimensional space of fields. We will argue for renormalizability and finiteness of the generalized Kodama partition function in direct analogy to that for the partition function for the pure Kodama state Ψ K ​ o ​ d subscript Ψ 𝐾 𝑜 𝑑 \Psi_{Kod} .The format of this paper is as follows. In section 2 we illustrate in detail the computation of the Chern–Simons partition function via the background field method, to set the stage and to cast the formalism into the language of wavefunctions. In section 3 we motivate the notion of the existence of a wavefunction of the universe whose norm should be computed. In section 4 we outline the computation of some one particle-irreducible (1PI) vertices of Ψ G ​ K ​ o ​ d subscript Ψ 𝐺 𝐾 𝑜 𝑑 \Psi_{GKod} highlighting their geometric interpretation in analogy to Kaluza–Klein theory on infinite dimensional spaces. In section 5 we compute the generalized Kodama partition function, taking into account the Gaussian and all higher order contributions, from these 1PI vertices. In section 6 we write down an expression for the generalized Kodama partition function both perturbatively and as well in terms of an interesting nonperturbative representation 1 1 1 This requires some concepts from quantum field theories on curved spacetimes. obtaining a loop expansion in the dimensionless constant ℏ ​ G ​ Λ ℏ 𝐺 Λ \sqrt{\mathchar 1406\relax{G}\Lambda} . In section 7 we rewrite the quantum Hamiltonian constraint of [ 1 ] in the language of 1PI vertices, which highlights some imprints from the semicalssical limit of gravity below the Planck scale upon the generalized Kodama partition function.

In this paper we outline the computation of the partition function for the generalized Kodama states (GKod) of quantum gravity using the background field method. We show that the coupling constant for GKod is the same dimensionless coupling constant that appears in the partition function of the pure Kodama state (Chern–Simons functional) and argue that the GKod partition function is renormalizable as a loop expansion in direct analogy to Chern–Simons perturbation theory. The GKod partition function contains an infinite set of 1PI vertices uniquely fixed, as a result of the semiclassical-quantum correspondence, by the first-order vertex. This implies the existence of a well-defined effective action for the partition function since the ‘phase’ of the GKod, provided a finite state exists, is equivalent to this effective action. Additionally, the separation of the matter from the gravitational contributions bears a resemblance to the infinite dimensional analogue to Kaluza–Klein theory. Future directions of research include extension of the computations of this paper to the norm of the GKod as well as to examine the analogue of the Chern–Simons Jone’s polynomials and link invariants using the GKod as a measure.

[{"content":"Help me to generate the abstract of a survey paper given the title:*Generalized Kodama partition functions: A preview into normalizability for the generalized Kodama states.* and the introduction:The purpose of this paper is to apply a special technique for computing partition functions to the computation of the norm of the generalized Kodama states of quantum gravity introduced in 1 . The closest analogy is the partition function for the Chern–Simons functional. There exist in the literature various methods for computing the partition function of the Chern–Simons wavefunction ( 2 , 3 , 4 and refences therein). As a path integral, the nonabelian Chern–Simons theory in three spatial to the present authors knowledge dimensions produces a renormalizable loop expansion, since any quantum divergences can always be absorbed, as shown by various authors, into a redefinition of tree-level coupling constants. This implies that the classical action and the effective action for the Chern–Simons functional are the same, to within re-definition of these constants.The first attempt to apply the perturbation theory of the Chern–Simons partition function to the fill theory of quantum gravity in the connection representation appears to be 5 , in which Chopin Soo computes the partition function for the pure Kodama state. We would like to generalize the techniques for computation of the Chern–Simons partition function from the pure Kodama state Ψ K ​ o ​ d subscript Ψ 𝐾 𝑜 𝑑 \Psi_{Kod} to the generalized Kodama states Ψ G ​ K ​ o ​ d subscript Ψ 𝐺 𝐾 𝑜 𝑑 \Psi_{GKod} . In the present work we will apply the background field method for computing the effective action, usually attributed to Wilson and DeWitt, for the task as set out.We would ultimately like to extend the results of this paper to address the normalizability of Ψ G ​ K ​ o ​ d subscript Ψ 𝐺 𝐾 𝑜 𝑑 \Psi_{GKod} . This will depend upon the ability to make sense of the configuration path integral, for the partition function, which will in turn depend upon the finiteness of the generalized Kodama state. We will not focus in this paper on such issues as gauge fixing and the renormalization group theory. The main thrust is to show that the perturbative expansion exists and conforms to the requirements of a perturbatively renormalizable effective action, by direct analogy to the case for Chern–Simons theory which has been thorougly explored by other authors. Our primary contribution is that the principle of the semiclassical-quantum correspondence introduced in 1 is particularly fortuitous when applied particularly to this special class of quantum states Ψ G ​ K ​ o ​ d subscript Ψ 𝐺 𝐾 𝑜 𝑑 \Psi_{GKod} . We also present a first Ansatz for the norm of Ψ G ​ K ​ o ​ d subscript Ψ 𝐺 𝐾 𝑜 𝑑 \Psi_{GKod} as well as expectation values of operators on the state, not commenting in detail on reality conditions.An important aspect of this paper is that all quantities are computed on the spatial hypersurface Σ T subscript Σ 𝑇 \Sigma_{T} on which which Ψ G ​ K ​ o ​ d subscript Ψ 𝐺 𝐾 𝑜 𝑑 \Psi_{GKod} is evaluated. So while it may appear that the treatment is timeless, we take time to be an implicit label. We will often in this paper not distinguish between spacetime position x = ( 𝒙 , t ) 𝑥 𝒙 𝑡 x=(\boldsymbol{x},t) and spatial position 𝒙 𝒙 \boldsymbol{x} . The partition function can be seen as a unified gravity-matter system with a dimensionless coupling constant. When split into its individual components, the partition function has an interesting geometric interpretation bearing a distinct resemblance to Kaluza–Klein like theories on the infinite dimensional space of fields. We will argue for renormalizability and finiteness of the generalized Kodama partition function in direct analogy to that for the partition function for the pure Kodama state Ψ K ​ o ​ d subscript Ψ 𝐾 𝑜 𝑑 \Psi_{Kod} .The format of this paper is as follows. In section 2 we illustrate in detail the computation of the Chern–Simons partition function via the background field method, to set the stage and to cast the formalism into the language of wavefunctions. In section 3 we motivate the notion of the existence of a wavefunction of the universe whose norm should be computed. In section 4 we outline the computation of some one particle-irreducible (1PI) vertices of Ψ G ​ K ​ o ​ d subscript Ψ 𝐺 𝐾 𝑜 𝑑 \Psi_{GKod} highlighting their geometric interpretation in analogy to Kaluza–Klein theory on infinite dimensional spaces. In section 5 we compute the generalized Kodama partition function, taking into account the Gaussian and all higher order contributions, from these 1PI vertices. In section 6 we write down an expression for the generalized Kodama partition function both perturbatively and as well in terms of an interesting nonperturbative representation 1 1 1 This requires some concepts from quantum field theories on curved spacetimes. obtaining a loop expansion in the dimensionless constant ℏ ​ G ​ Λ ℏ 𝐺 Λ \sqrt{\mathchar 1406\relax{G}\Lambda} . In section 7 we rewrite the quantum Hamiltonian constraint of 1 in the language of 1PI vertices, which highlights some imprints from the semicalssical limit of gravity below the Planck scale upon the generalized Kodama partition function.","role":"user"},{"content":"Abstract: In this paper we outline the computation of the partition function for the generalized Kodama states (GKod) of quantum gravity using the background field method. We show that the coupling constant for GKod is the same dimensionless coupling constant that appears in the partition function of the pure Kodama state (Chern–Simons functional) and argue that the GKod partition function is renormalizable as a loop expansion in direct analogy to Chern–Simons perturbation theory. The GKod partition function contains an infinite set of 1PI vertices uniquely fixed, as a result of the semiclassical-quantum correspondence, by the first-order vertex. This implies the existence of a well-defined effective action for the partition function since the phase of the GKod, provided a finite state exists, is equivalent to this effective action. Additionally, the separation of the matter from the gravitational contributions bears a resemblance to the infinite dimensional analogue to Kaluza–Klein theory. Future directions of research include extension of the computations of this paper to the norm of the GKod as well as to examine the analogue of the Chern–Simons Jones polynomials and link invariants using the GKod as a measure.","role":"assistant"}]

A Survey on the Classical Limit of Quantum Dynamical Entropies

The notion of classical chaos is associated with motion on a compact phase–space with high sensitivity to initial conditions: trajectories diverge exponentially fast and nevertheless remain confined to bounded regions [ 1 , 2 , 3 , 4 , 5 , 6 , 7 ] . In an opposite way, quantization on compacts yields discrete energy spectra, which in term entail quasi–periodic time–evolution [ 8 ] . Nevertheless, nature is fundamentally quantal and, according to the correspondence principle, classical behavior must emerge in the limit ℏ → 0 → Planck-constant-over-2-pi 0 \hbar\to 0 . Also, classical and quantum mechanics are expected to overlap over times expected to scale as ℏ − α superscript Planck-constant-over-2-pi 𝛼 \hbar^{-\alpha} for some α > 0 𝛼 0 \alpha>0 [ 7 ] , the so–called semi–classical regime. Actually, it turns out that this is true only for regular classical limits whereas, for chaotic ones, classical and quantum mechanics agree over times which scale as − log ⁡ ℏ Planck-constant-over-2-pi -\log\hbar [ 5 , 6 , 7 ] , and footprints of the exponential separation of classical trajectories are found even on finite dimensional quantization provide that the time does not exceed such a logarithmic upper bound [ 6 , 9 ] . Both time scales diverge when ℏ → 0 → Planck-constant-over-2-pi 0 \hbar\to 0 , but the shortness of the latter means that classical mechanics has to be replaced by quantum mechanics much sooner for quantum systems with chaotic classical behavior. The logarithmic breaking time − log ⁡ ℏ Planck-constant-over-2-pi -\!\log{\hbar} has been considered by some as a violation of the correspondence principle [ 10 , 11 ] and by others, see [ 6 ] and Chirikov in [ 5 ] , as the evidence that time and classical limits do not commute. The analytic studies of logarithmic time scales have been mainly performed by means of semi–classical tools, essentially by focusing, via coherent state techniques, on the phase space localization of specific time evolving quantum observables. In the following, we shall show how they emerge in the context of quantum dynamical entropies. As a particular example, we shall concentrate on finite dimensional quantizations of continuous hyperbolic automorphisms of the 2–torus 𝕋 2 ≔ ℝ 2 / ℤ 2 ≔ superscript 𝕋 2 superscript ℝ 2 superscript ℤ 2 \mathds{T}^{2}\coloneqq\mathds{R}^{2}/\mathds{Z}^{2} (the unit square with opposite sides identified), which are prototypes of chaotic behavior; indeed, their trajectories separate exponentially fast with a Lyapunov exponent log ⁡ λ + > 0 subscript 𝜆 0 \log\lambda_{+}>0 [ 12 , 13 ] . If δ 𝛿 \delta is an initial error along a trajectory, and δ n ≃ δ ​ λ + n similar-to-or-equals subscript 𝛿 𝑛 𝛿 superscript subscript 𝜆 𝑛 \delta_{n}\simeq\delta\lambda_{+}^{n} its classical spreading after n 𝑛 n steps of the (time–stroboscopic) dynamics, then boundness of the motion imposes δ n ⩽ 1 subscript 𝛿 𝑛 1 \delta_{n}\leqslant 1 , where 1 is the diameter of the 2–torus 𝕋 2 superscript 𝕋 2 \mathds{T}^{2} . This explain why the limit δ → 0 → 𝛿 0 \delta\to 0 has necessarily to be performed before the time–limit, and the Lyapunov exponent can be computed as log ⁡ λ + = lim n → ∞ 1 n ​ lim δ → 0 log ⁡ ( δ n δ ) . subscript 𝜆 subscript → 𝑛 1 𝑛 subscript → 𝛿 0 subscript 𝛿 𝑛 𝛿 \log\lambda_{+}=\lim_{n\to\infty}\;\frac{1}{n}\;\lim_{\delta\to 0}\;\log\left({\delta_{n}\over\delta}\right)\quad. (1) Standard quantization, à la Berry, of hyperbolic automorphisms on 𝕋 2 superscript 𝕋 2 \mathds{T}^{2} [ 14 , 15 ] yields Hilbert spaces of a finite dimension N 𝑁 N , this latter variable playing the role of the semi–classical parameter and setting to 1 / N 1 𝑁 1/N the minimal size of the phase–space grain cells. Imposing the latter bound, min ⁡ { δ } ⩾ 1 / N 𝛿 1 𝑁 \min\left\{\delta\right\}\geqslant 1/N , its evident how the conflict between the two limits, emerging once δ n ≃ 1 similar-to-or-equals subscript 𝛿 𝑛 1 \delta_{n}\simeq 1 , can be transferred in the time–step n 𝑛 n as n ≃ log ⁡ N / log ⁡ λ + similar-to-or-equals 𝑛 𝑁 subscript 𝜆 n\simeq\log N/\log\lambda_{+} . In this sense, rather than a violation of the correspondence principle, the logarithmic breaking–time indicates the typical scaling for a joint time–classical limit suited to classically chaotic quantum systems. The K olmogorov– S inai dynamical entropy [ 3 ] ( KS –entropy, for short) is defined by assigning measures to bunches of trajectories and computing the Shannon–entropy per time–step of the ensemble of bunches in the limit of infinitely many time–steps: The more chaotic the time–evolution, the more the possibile bunches and the larger their entropy. The production of different bunches of trajectories issuing from the same bunch is typical of high sensitivity to initial conditions and this is indeed the mechanism at the basis of the theorem of Ruelle and Pesin [ 16 ] , linking KS –entropy of a smooth, classical dynamical systems, to the sum of its positive Lyapunov exponents. In the quantum realm, there are different candidates for non–commutative extensions of the KS –invariant [ 17 , 18 , 19 , 20 , 21 ] : in this paper we shall focus on one of them, called ALF –entropy [ 18 ] , and we shall study its semi–classical limit. The ALF –entropy is based on the algebraic properties of dynamical systems, that is on the fact that they are describable by suitable algebras of observables, their time evolution by linear maps on these algebras, and their states by expectations over them. We show that, while being bounded by log ⁡ N 𝑁 \log N , nevertheless over numbers of time steps 1 ≪ n < log ⁡ N much-less-than 1 𝑛 𝑁 1\ll n<\log N , the entropy content per letter, or entropy production, is log ⁡ λ + subscript 𝜆 \log\lambda_{+} . It thus follows that the joint limit n , N → + ∞ → 𝑛 𝑁 n,N\to+\infty , with n ∝ log ⁡ N proportional-to 𝑛 𝑁 n\propto\log N , yields the K olmogorov– S inai entropy. This confirms the numerical results in [ 22 ] and [ 23 ] , where the dynamical entropy [ 18 ] is applied to the study of the quantum kicked top, respectively to quantum cat maps. In this approach, the presence of logarithmic time scales indicates the typical scaling for a joint time/classical limit suited to preserve positive entropy production in quantized classically chaotic quantum systems. The paper is organized as follows: Section 2 contains a brief review of the algebraic approach to classical and dynamical systems, while Section 3 introduces some basic semi–classical tools. Sections 4 and 5 deal with the quantization of hyperbolic maps on finite dimensional Hilbert spaces and the relation between classical and time limits. Section 6 gives an overview of various models of quantum dynamical entropies present in the literature and particularly focus on the one proposed by Alicki and Fannes [ 24 , 18 ] ( ALF –entropy, where L stands for Lindblad). Finally, in Section 7 , the semi–classical behavior of quantum dynamical entropies is studied and the emergence of a typical logarithmic time scale is showed.

We analyze the behavior of quantum dynamical entropies prod uction from sequences of quantum approximants approaching their (chaoti c) classical limit. The model of the quantized hyperbolic automorphisms of the 2–to rus is examined in detail and a semi–classical analysis is performed on it usin g coherent states, fulfilling an appropriate dynamical localization property. Co rrespondence between quantum dynamical entropies and the Kolmogorov–Sinai inva r ant is found only over time scales that are logarithmic in the quantization pa rameter.

[{"content":"Help me to generate the abstract of a survey paper given the title:*A Survey on the Classical Limit of Quantum Dynamical Entropies* and the introduction:The notion of classical chaos is associated with motion on a compact phase–space with high sensitivity to initial conditions: trajectories diverge exponentially fast and nevertheless remain confined to bounded regions 1 , 2 , 3 , 4 , 5 , 6 , 7 . In an opposite way, quantization on compacts yields discrete energy spectra, which in term entail quasi–periodic time–evolution 8 . Nevertheless, nature is fundamentally quantal and, according to the correspondence principle, classical behavior must emerge in the limit ℏ → 0 → Planck-constant-over-2-pi 0 \hbar\to 0 . Also, classical and quantum mechanics are expected to overlap over times expected to scale as ℏ − α superscript Planck-constant-over-2-pi 𝛼 \hbar^{-\alpha} for some α > 0 𝛼 0 \alpha>0 7 , the so–called semi–classical regime. Actually, it turns out that this is true only for regular classical limits whereas, for chaotic ones, classical and quantum mechanics agree over times which scale as − log ⁡ ℏ Planck-constant-over-2-pi -\log\hbar 5 , 6 , 7 , and footprints of the exponential separation of classical trajectories are found even on finite dimensional quantization provide that the time does not exceed such a logarithmic upper bound 6 , 9 . Both time scales diverge when ℏ → 0 → Planck-constant-over-2-pi 0 \hbar\to 0 , but the shortness of the latter means that classical mechanics has to be replaced by quantum mechanics much sooner for quantum systems with chaotic classical behavior. The logarithmic breaking time − log ⁡ ℏ Planck-constant-over-2-pi -\!\log{\hbar} has been considered by some as a violation of the correspondence principle 10 , 11 and by others, see 6 and Chirikov in 5 , as the evidence that time and classical limits do not commute. The analytic studies of logarithmic time scales have been mainly performed by means of semi–classical tools, essentially by focusing, via coherent state techniques, on the phase space localization of specific time evolving quantum observables. In the following, we shall show how they emerge in the context of quantum dynamical entropies. As a particular example, we shall concentrate on finite dimensional quantizations of continuous hyperbolic automorphisms of the 2–torus 𝕋 2 ≔ ℝ 2 / ℤ 2 ≔ superscript 𝕋 2 superscript ℝ 2 superscript ℤ 2 \mathds{T}^{2}\coloneqq\mathds{R}^{2}/\mathds{Z}^{2} (the unit square with opposite sides identified), which are prototypes of chaotic behavior; indeed, their trajectories separate exponentially fast with a Lyapunov exponent log ⁡ λ + > 0 subscript 𝜆 0 \log\lambda_{+}>0 12 , 13 . If δ 𝛿 \delta is an initial error along a trajectory, and δ n ≃ δ ​ λ + n similar-to-or-equals subscript 𝛿 𝑛 𝛿 superscript subscript 𝜆 𝑛 \delta_{n}\simeq\delta\lambda_{+}^{n} its classical spreading after n 𝑛 n steps of the (time–stroboscopic) dynamics, then boundness of the motion imposes δ n ⩽ 1 subscript 𝛿 𝑛 1 \delta_{n}\leqslant 1 , where 1 is the diameter of the 2–torus 𝕋 2 superscript 𝕋 2 \mathds{T}^{2} . This explain why the limit δ → 0 → 𝛿 0 \delta\to 0 has necessarily to be performed before the time–limit, and the Lyapunov exponent can be computed as log ⁡ λ + = lim n → ∞ 1 n ​ lim δ → 0 log ⁡ ( δ n δ ) . subscript 𝜆 subscript → 𝑛 1 𝑛 subscript → 𝛿 0 subscript 𝛿 𝑛 𝛿 \log\lambda_{+}=\lim_{n\to\infty}\;\frac{1}{n}\;\lim_{\delta\to 0}\;\log\left({\delta_{n}\over\delta}\right)\quad. (1) Standard quantization, à la Berry, of hyperbolic automorphisms on 𝕋 2 superscript 𝕋 2 \mathds{T}^{2} 14 , 15 yields Hilbert spaces of a finite dimension N 𝑁 N , this latter variable playing the role of the semi–classical parameter and setting to 1 / N 1 𝑁 1/N the minimal size of the phase–space grain cells. Imposing the latter bound, min ⁡ { δ } ⩾ 1 / N 𝛿 1 𝑁 \min\left\{\delta\right\}\geqslant 1/N , its evident how the conflict between the two limits, emerging once δ n ≃ 1 similar-to-or-equals subscript 𝛿 𝑛 1 \delta_{n}\simeq 1 , can be transferred in the time–step n 𝑛 n as n ≃ log ⁡ N / log ⁡ λ + similar-to-or-equals 𝑛 𝑁 subscript 𝜆 n\simeq\log N/\log\lambda_{+} . In this sense, rather than a violation of the correspondence principle, the logarithmic breaking–time indicates the typical scaling for a joint time–classical limit suited to classically chaotic quantum systems. The K olmogorov– S inai dynamical entropy 3 ( KS –entropy, for short) is defined by assigning measures to bunches of trajectories and computing the Shannon–entropy per time–step of the ensemble of bunches in the limit of infinitely many time–steps: The more chaotic the time–evolution, the more the possibile bunches and the larger their entropy. The production of different bunches of trajectories issuing from the same bunch is typical of high sensitivity to initial conditions and this is indeed the mechanism at the basis of the theorem of Ruelle and Pesin 16 , linking KS –entropy of a smooth, classical dynamical systems, to the sum of its positive Lyapunov exponents. In the quantum realm, there are different candidates for non–commutative extensions of the KS –invariant 17 , 18 , 19 , 20 , 21 : in this paper we shall focus on one of them, called ALF –entropy 18 , and we shall study its semi–classical limit. The ALF –entropy is based on the algebraic properties of dynamical systems, that is on the fact that they are describable by suitable algebras of observables, their time evolution by linear maps on these algebras, and their states by expectations over them. We show that, while being bounded by log ⁡ N 𝑁 \log N , nevertheless over numbers of time steps 1 ≪ n < log ⁡ N much-less-than 1 𝑛 𝑁 1\ll n<\log N , the entropy content per letter, or entropy production, is log ⁡ λ + subscript 𝜆 \log\lambda_{+} . It thus follows that the joint limit n , N → + ∞ → 𝑛 𝑁 n,N\to+\infty , with n ∝ log ⁡ N proportional-to 𝑛 𝑁 n\propto\log N , yields the K olmogorov– S inai entropy. This confirms the numerical results in 22 and 23 , where the dynamical entropy 18 is applied to the study of the quantum kicked top, respectively to quantum cat maps. In this approach, the presence of logarithmic time scales indicates the typical scaling for a joint time/classical limit suited to preserve positive entropy production in quantized classically chaotic quantum systems. The paper is organized as follows: Section 2 contains a brief review of the algebraic approach to classical and dynamical systems, while Section 3 introduces some basic semi–classical tools. Sections 4 and 5 deal with the quantization of hyperbolic maps on finite dimensional Hilbert spaces and the relation between classical and time limits. Section 6 gives an overview of various models of quantum dynamical entropies present in the literature and particularly focus on the one proposed by Alicki and Fannes 24 , 18 ( ALF –entropy, where L stands for Lindblad). Finally, in Section 7 , the semi–classical behavior of quantum dynamical entropies is studied and the emergence of a typical logarithmic time scale is showed.","role":"user"},{"content":"Abstract: We analyze the behavior of quantum dynamical entropies prod uction from sequences of quantum approximants approaching their (chaoti c) classical limit. The model of the quantized hyperbolic automorphisms of the 2–to rus is examined in detail and a semi–classical analysis is performed on it usin g coherent states, fulfilling an appropriate dynamical localization property. Co rrespondence between quantum dynamical entropies and the Kolmogorov–Sinai inva r ant is found only over time scales that are logarithmic in the quantization pa rameter.","role":"assistant"}]

Decays of B_s Mesons and b Baryons: A Review of Recent First Observations and Branching Fractions

I Introduction Heavier b 𝑏 b -flavoured hadrons represent a fecund source of particle physics. While the rich interplay between electroweak and non-perturbative strong effects typically poses formidable experimental and theoretical challenges, decays of hadrons with masses at the frontiers of Standard Model spectroscopy constitute an exciting proving ground for effective theories, QCD factorization and lattice methods, as well as potential models. Moreover, such heavy hadronic states present opportunities to uncover real or constrain hypothetical new physics lying beyond the Standard Model. The measurement of observables from b 𝑏 b baryons and strange or charmed B 𝐵 B mesons is complementary to the wealth of physics that the BaBar , Belle, and CLEO collaborations have harvested from e + ​ e − superscript 𝑒 superscript 𝑒 e^{+}e^{-} colliders operating at the Υ ​ ( 4 ​ S ) Υ 4 𝑆 \Upsilon(4S) open-beauty threshold. Comparisons of heavy b 𝑏 b -hadron decays to the analogous non-strange B u , d subscript 𝐵 𝑢 𝑑 B_{u,d} ( B + , 0 superscript 𝐵 0 B^{+,0} ) decays can yield advantages that include cancellations of hadronic uncertainties, tests of S ​ U ​ ( 3 ) 𝑆 𝑈 3 SU(3) flavour symmetry, decay-amplitude disentanglement, and improved access to fundamental electroweak parameters of Nature. This paper reviews recent rate measurements of B s 0 superscript subscript 𝐵 𝑠 0 B_{s}^{0} mesons and Λ b 0 superscript subscript Λ 𝑏 0 \Lambda_{b}^{0} baryons produced in s = 1.96 𝑠 1.96 \sqrt{s}=1.96~{} TeV proton-antiproton and Υ ​ ( 5 ​ S ) Υ 5 𝑆 \Upsilon(5S) electron-positron collisions at the Fermilab Tevatron (USA) and KEKB (Japan) accelerator facilities, respectively. Described are first observations of B s 0 → D s + ​ K − → subscript superscript 𝐵 0 𝑠 superscript subscript 𝐷 𝑠 superscript 𝐾 B^{0}_{s}\to D_{s}^{+}\,K^{-} and B s 0 → D s − ​ D s + → superscript subscript 𝐵 𝑠 0 superscript subscript 𝐷 𝑠 superscript subscript 𝐷 𝑠 B_{s}^{0}\to D_{s}^{-}\,D_{s}^{+} decays 1 1 1 Charge conjugate decays are implied throughout. by CDF; recent results on B s 0 → D s ( ∗ ) ​ D s ( ∗ ) → superscript subscript 𝐵 𝑠 0 superscript subscript 𝐷 𝑠 superscript subscript 𝐷 𝑠 B_{s}^{0}\to D_{s}^{(*)}\,D_{s}^{(*)} decays and worldwide status from Belle, CDF, and DZero; the first observation of B s 0 → D s ​ 1 − ​ ( 2536 ) ​ μ + ​ ν μ ​ X → superscript subscript 𝐵 𝑠 0 superscript subscript 𝐷 𝑠 1 2536 superscript 𝜇 subscript 𝜈 𝜇 𝑋 B_{s}^{0}\to D_{s1}^{-}(2536)\,\mu^{+}\,\nu_{\mu}\,X by DZero; the first observation of the B s 0 → ϕ ​ γ → superscript subscript 𝐵 𝑠 0 italic-ϕ 𝛾 B_{s}^{0}\to\phi\,\gamma mode and a search for B s 0 → γ ​ γ → superscript subscript 𝐵 𝑠 0 𝛾 𝛾 B_{s}^{0}\to\gamma\,\gamma decays by Belle; three recently updated measurements of charmless two-body B s 0 superscript subscript 𝐵 𝑠 0 B_{s}^{0} meson decays by CDF; and first observations of Λ b 0 → p ​ K − → superscript subscript Λ 𝑏 0 𝑝 superscript 𝐾 \Lambda_{b}^{0}\to p\,K^{-} and Λ b 0 → p ​ π − → superscript subscript Λ 𝑏 0 𝑝 superscript 𝜋 \Lambda_{b}^{0}\to p\,\pi^{-} decays by CDF.

Recent rate measurements of B{sub s}{sup 0} mesons and {Lambda}{sub b}{sup 0} baryons produced in {radical}s = 1.96 TeV proton-antiproton and {Upsilon}(5S) electron-positron collisions are reviewed, including the first observations of six new decay modes: B{sub s}{sup 0} {yields} D{sub s}{sup +} K{sup -} (CDF), B{sub s}{sup 0} {yields} D{sub s}{sup -} D{sub s}{sup +} (CDF), B{sub s}{sup 0} {yields} D{sub s1}{sup -}(2536){mu}{sup +} {nu}{sub {mu}} X (DZero), B{sub s}{sup 0} {yields} {phi}{gamma} (Belle)< {Lambda}{sub b}{sup 0} {yields} p{pi}{sup -} (CDF), and {Lambda}{sub b}{sup 0} {yields} pK{sup -} (CDF). Also examined are branching-fraction measurements or limits for the B{sub s}{sup 0} {yields} D{sub s}{sup (*)} D{sub s}{sup (*)} modes (Belle, CDF, and DZero), the B{sub s}{sup 0} {yields} {gamma}{gamma} radiative penguin decay (Belle), and three two-body charmless B{sub s}{sup 0} meson decay channels (CDF). Implications for the phenomenology of electroweak and QCD physics, as well as searches for physics beyond the Standard Model, are identified where applicable.

[{"content":"Help me to generate the abstract of a survey paper given the title:*Decays of B_s Mesons and b Baryons: A Review of Recent First Observations and Branching Fractions* and the introduction:I Introduction Heavier b 𝑏 b -flavoured hadrons represent a fecund source of particle physics. While the rich interplay between electroweak and non-perturbative strong effects typically poses formidable experimental and theoretical challenges, decays of hadrons with masses at the frontiers of Standard Model spectroscopy constitute an exciting proving ground for effective theories, QCD factorization and lattice methods, as well as potential models. Moreover, such heavy hadronic states present opportunities to uncover real or constrain hypothetical new physics lying beyond the Standard Model. The measurement of observables from b 𝑏 b baryons and strange or charmed B 𝐵 B mesons is complementary to the wealth of physics that the BaBar , Belle, and CLEO collaborations have harvested from e + ​ e − superscript 𝑒 superscript 𝑒 e^{+}e^{-} colliders operating at the Υ ​ ( 4 ​ S ) Υ 4 𝑆 \Upsilon(4S) open-beauty threshold. Comparisons of heavy b 𝑏 b -hadron decays to the analogous non-strange B u , d subscript 𝐵 𝑢 𝑑 B_{u,d} ( B + , 0 superscript 𝐵 0 B^{+,0} ) decays can yield advantages that include cancellations of hadronic uncertainties, tests of S ​ U ​ ( 3 ) 𝑆 𝑈 3 SU(3) flavour symmetry, decay-amplitude disentanglement, and improved access to fundamental electroweak parameters of Nature. This paper reviews recent rate measurements of B s 0 superscript subscript 𝐵 𝑠 0 B_{s}^{0} mesons and Λ b 0 superscript subscript Λ 𝑏 0 \Lambda_{b}^{0} baryons produced in s = 1.96 𝑠 1.96 \sqrt{s}=1.96~{} TeV proton-antiproton and Υ ​ ( 5 ​ S ) Υ 5 𝑆 \Upsilon(5S) electron-positron collisions at the Fermilab Tevatron (USA) and KEKB (Japan) accelerator facilities, respectively. Described are first observations of B s 0 → D s + ​ K − → subscript superscript 𝐵 0 𝑠 superscript subscript 𝐷 𝑠 superscript 𝐾 B^{0}_{s}\to D_{s}^{+}\,K^{-} and B s 0 → D s − ​ D s + → superscript subscript 𝐵 𝑠 0 superscript subscript 𝐷 𝑠 superscript subscript 𝐷 𝑠 B_{s}^{0}\to D_{s}^{-}\,D_{s}^{+} decays 1 1 1 Charge conjugate decays are implied throughout. by CDF; recent results on B s 0 → D s ( ∗ ) ​ D s ( ∗ ) → superscript subscript 𝐵 𝑠 0 superscript subscript 𝐷 𝑠 superscript subscript 𝐷 𝑠 B_{s}^{0}\to D_{s}^{(*)}\,D_{s}^{(*)} decays and worldwide status from Belle, CDF, and DZero; the first observation of B s 0 → D s ​ 1 − ​ ( 2536 ) ​ μ + ​ ν μ ​ X → superscript subscript 𝐵 𝑠 0 superscript subscript 𝐷 𝑠 1 2536 superscript 𝜇 subscript 𝜈 𝜇 𝑋 B_{s}^{0}\to D_{s1}^{-}(2536)\,\mu^{+}\,\nu_{\mu}\,X by DZero; the first observation of the B s 0 → ϕ ​ γ → superscript subscript 𝐵 𝑠 0 italic-ϕ 𝛾 B_{s}^{0}\to\phi\,\gamma mode and a search for B s 0 → γ ​ γ → superscript subscript 𝐵 𝑠 0 𝛾 𝛾 B_{s}^{0}\to\gamma\,\gamma decays by Belle; three recently updated measurements of charmless two-body B s 0 superscript subscript 𝐵 𝑠 0 B_{s}^{0} meson decays by CDF; and first observations of Λ b 0 → p ​ K − → superscript subscript Λ 𝑏 0 𝑝 superscript 𝐾 \Lambda_{b}^{0}\to p\,K^{-} and Λ b 0 → p ​ π − → superscript subscript Λ 𝑏 0 𝑝 superscript 𝜋 \Lambda_{b}^{0}\to p\,\pi^{-} decays by CDF.","role":"user"},{"content":"Abstract: Recent rate measurements of B{sub s}{sup 0} mesons and {Lambda}{sub b}{sup 0} baryons produced in {radical}s = 1.96 TeV proton-antiproton and {Upsilon}(5S) electron-positron collisions are reviewed, including the first observations of six new decay modes: B{sub s}{sup 0} {yields} D{sub s}{sup +} K{sup -} (CDF), B{sub s}{sup 0} {yields} D{sub s}{sup -} D{sub s}{sup +} (CDF), B{sub s}{sup 0} {yields} D{sub s1}{sup -}(2536){mu}{sup +} {nu}{sub {mu}} X (DZero), B{sub s}{sup 0} {yields} {phi}{gamma} (Belle)< {Lambda}{sub b}{sup 0} {yields} p{pi}{sup -} (CDF), and {Lambda}{sub b}{sup 0} {yields} pK{sup -} (CDF). Also examined are branching-fraction measurements or limits for the B{sub s}{sup 0} {yields} D{sub s}{sup (*)} D{sub s}{sup (*)} modes (Belle, CDF, and DZero), the B{sub s}{sup 0} {yields} {gamma}{gamma} radiative penguin decay (Belle), and three two-body charmless B{sub s}{sup 0} meson decay channels (CDF). Implications for the phenomenology of electroweak and QCD physics, as well as searches for physics beyond the Standard Model, are identified where applicable.","role":"assistant"}]

Theory Review of Exclusive Rare Radiative Decays

I Introduction Experimental measurements of B → V ​ γ → 𝐵 𝑉 𝛾 B\to V\gamma decays, with V 𝑉 V a light vector meson such as K ∗ , ρ , ω , ϕ superscript 𝐾 𝜌 𝜔 italic-ϕ K^{*},\rho,\omega,\phi , have continued to improve and will become more precise at the end of the B 𝐵 B -factories and at LHC-b. Given their rich CKM phenomenology and potential to constrain new physics models, having reliable theory predictions for these decays is increasingly relevant. In this talk I briefly review the status of this area. The starting point is the effective weak Hamiltonian, which for B → V ​ γ → 𝐵 𝑉 𝛾 B\to V\gamma decays is Buchalla:1996vs : ℋ eff = G F 2 ​ ∑ p = u , c λ p ( q ) ​ [ C 1 ​ Q 1 p + C 2 ​ Q 2 p + ∑ i = 3 8 C i ​ Q i ] , subscript ℋ eff subscript 𝐺 𝐹 2 subscript 𝑝 𝑢 𝑐 superscript subscript 𝜆 𝑝 𝑞 delimited-[] subscript 𝐶 1 superscript subscript 𝑄 1 𝑝 subscript 𝐶 2 superscript subscript 𝑄 2 𝑝 superscript subscript 𝑖 3 8 subscript 𝐶 𝑖 subscript 𝑄 𝑖 \mathcal{H}_{\rm eff}=\frac{G_{F}}{\sqrt{2}}\sum_{p=u,c}\lambda_{p}^{(q)}\left[C_{1}Q_{1}^{p}+C_{2}Q_{2}^{p}+\sum_{i=3}^{8}C_{i}Q_{i}\right], (1) where λ p ( q ) = V p ​ q ∗ ​ V p ​ b superscript subscript 𝜆 𝑝 𝑞 subscript superscript 𝑉 𝑝 𝑞 subscript 𝑉 𝑝 𝑏 \lambda_{p}^{(q)}=V^{*}_{pq}V_{pb} . The operators with the largest Wilson coefficients are the four-quark operators Q 1 p superscript subscript 𝑄 1 𝑝 Q_{1}^{p} and Q 2 p superscript subscript 𝑄 2 𝑝 Q_{2}^{p} , which read Q 1 p superscript subscript 𝑄 1 𝑝 \displaystyle Q_{1}^{p} = \displaystyle= ( q ¯ ​ p ) V − A ​ ( p ¯ ​ b ) V − A subscript ¯ 𝑞 𝑝 𝑉 𝐴 subscript ¯ 𝑝 𝑏 𝑉 𝐴 \displaystyle(\bar{q}\,p)_{V-A}\,(\bar{p}\,b)_{V-A} Q 2 p superscript subscript 𝑄 2 𝑝 \displaystyle Q_{2}^{p} = \displaystyle= ( q ¯ i ​ p j ) V − A ​ ( p ¯ j ​ b i ) V − A , subscript subscript ¯ 𝑞 𝑖 subscript 𝑝 𝑗 𝑉 𝐴 subscript subscript ¯ 𝑝 𝑗 subscript 𝑏 𝑖 𝑉 𝐴 \displaystyle(\bar{q}_{i}p_{j})_{V-A}\,(\bar{p}_{j}b_{i})_{V-A}, (2) and the electromagnetic and chromomagnetic penguin operators Q 7 subscript 𝑄 7 Q_{7} and Q 8 subscript 𝑄 8 Q_{8} , which are Q 7 subscript 𝑄 7 \displaystyle Q_{7} = \displaystyle= − e ​ m ¯ b ​ ( μ ) 8 ​ π 2 ​ q ¯ ​ σ μ ​ ν ​ [ 1 + γ 5 ] ​ b ​ F μ ​ ν , 𝑒 subscript ¯ 𝑚 𝑏 𝜇 8 superscript 𝜋 2 ¯ 𝑞 superscript 𝜎 𝜇 𝜈 delimited-[] 1 subscript 𝛾 5 𝑏 subscript 𝐹 𝜇 𝜈 \displaystyle-\frac{e\,\overline{m}_{b}(\mu)}{8\pi^{2}}\,\bar{q}\,\sigma^{\mu\nu}\,[1+\gamma_{5}]\,bF_{\mu\nu}\,, Q 8 subscript 𝑄 8 \displaystyle Q_{8} = \displaystyle= − g ​ m ¯ b ​ ( μ ) 8 ​ π 2 ​ q ¯ ​ σ μ ​ ν ​ [ 1 + γ 5 ] ​ T a ​ b ​ G μ ​ ν a . 𝑔 subscript ¯ 𝑚 𝑏 𝜇 8 superscript 𝜋 2 ¯ 𝑞 superscript 𝜎 𝜇 𝜈 delimited-[] 1 subscript 𝛾 5 superscript 𝑇 𝑎 𝑏 subscript superscript 𝐺 𝑎 𝜇 𝜈 \displaystyle-\frac{g\,\overline{m}_{b}(\mu)}{8\pi^{2}}\,\bar{q}\,\sigma^{\mu\nu}\,[1+\gamma_{5}]\,T^{a}\,bG^{a}_{\mu\nu}. (3) Here q = d 𝑞 𝑑 q=d or s 𝑠 s , and the convention for the sign of the couplings corresponds to the covariant derivative i ​ D μ = i ​ ∂ μ + e ​ Q f ​ A μ + g ​ T a ​ A μ a 𝑖 subscript 𝐷 𝜇 𝑖 subscript 𝜇 𝑒 subscript 𝑄 𝑓 subscript 𝐴 𝜇 𝑔 superscript 𝑇 𝑎 superscript subscript 𝐴 𝜇 𝑎 iD_{\mu}=i\partial_{\mu}+eQ_{f}A_{\mu}+gT^{a}A_{\mu}^{a} , with A μ subscript 𝐴 𝜇 A_{\mu} and A μ a superscript subscript 𝐴 𝜇 𝑎 A_{\mu}^{a} representing the photon and gluon fields respectively, and Q e = − 1 subscript 𝑄 𝑒 1 Q_{e}=-1 etc. The factor m ¯ b ​ ( μ ) subscript ¯ 𝑚 𝑏 𝜇 \overline{m}_{b}(\mu) is the MS ¯ ¯ MS \overline{\rm MS} mass of the b 𝑏 b quark. The main theoretical challenge is to evaluate the hadronic matrix elements of the operators in the effective weak Hamiltonian. Common ways of doing this include the QCD factorization Beneke:2001at ; Bosch:2001gv ; Ali:2001ez and pQCD Keum:2004is ; Lu:2005yz approaches. Both of these rely on the fact that α s ​ ( m b ) subscript 𝛼 𝑠 subscript 𝑚 𝑏 \alpha_{s}(m_{b}) and Λ QCD / m b subscript Λ QCD subscript 𝑚 𝑏 \Lambda_{\rm QCD}/m_{b} can be considered as small expansion parameters. In this talk I will focus exclusively on QCD factorization methods, including in this category also strategies which supply additional information on 1 / m b 1 subscript 𝑚 𝑏 1/m_{b} -suppressed contributions from QCD sum rules Ball:2006nr ; Ball:2006eu , or apply a form of renormalization-group (RG) improved perturbation theory based on soft-collinear effective theory (SCET) Becher:2005fg ; Ali:2007sj . The remainder of the talk is organized as follows. Section II introduces the QCD factorization formalism, Section III gives two sample applications, and Section IV reviews recent results in higher-order perturbative corrections to the hard-scattering kernels. This is followed by a short discussion of 1 / m b 1 subscript 𝑚 𝑏 1/m_{b} power corrections and endpoint divergences in Section V , using isospin violation in B → K ∗ ​ γ → 𝐵 superscript 𝐾 𝛾 B\to K^{*}\gamma decays as an example, and a summary in Section VI .

Experimental measurements of B → V γ decays, with V a light vector meson such as K, ρ, ω, φ, have continued to improve and will become more precise at the end of the B-factories and at LHC-b. Given their rich CKM phenomenology and potential to constrain new physics models, having reliable theory predictions for these decays is increasingly relevant. In this talk I briefly review the status of this area. The starting point is the effective weak Hamiltonian, which for B → V γ decays is [1]:

[{"content":"Help me to generate the abstract of a survey paper given the title:*Theory Review of Exclusive Rare Radiative Decays* and the introduction:I Introduction Experimental measurements of B → V ​ γ → 𝐵 𝑉 𝛾 B\to V\gamma decays, with V 𝑉 V a light vector meson such as K ∗ , ρ , ω , ϕ superscript 𝐾 𝜌 𝜔 italic-ϕ K^{*},\rho,\omega,\phi , have continued to improve and will become more precise at the end of the B 𝐵 B -factories and at LHC-b. Given their rich CKM phenomenology and potential to constrain new physics models, having reliable theory predictions for these decays is increasingly relevant. In this talk I briefly review the status of this area. The starting point is the effective weak Hamiltonian, which for B → V ​ γ → 𝐵 𝑉 𝛾 B\to V\gamma decays is Buchalla:1996vs : ℋ eff = G F 2 ​ ∑ p = u , c λ p ( q ) ​ C 1 ​ Q 1 p + C 2 ​ Q 2 p + ∑ i = 3 8 C i ​ Q i , subscript ℋ eff subscript 𝐺 𝐹 2 subscript 𝑝 𝑢 𝑐 superscript subscript 𝜆 𝑝 𝑞 delimited- subscript 𝐶 1 superscript subscript 𝑄 1 𝑝 subscript 𝐶 2 superscript subscript 𝑄 2 𝑝 superscript subscript 𝑖 3 8 subscript 𝐶 𝑖 subscript 𝑄 𝑖 \mathcal{H}_{\rm eff}=\frac{G_{F}}{\sqrt{2}}\sum_{p=u,c}\lambda_{p}^{(q)}\leftC_{1}Q_{1}^{p}+C_{2}Q_{2}^{p}+\sum_{i=3}^{8}C_{i}Q_{i}\right, (1) where λ p ( q ) = V p ​ q ∗ ​ V p ​ b superscript subscript 𝜆 𝑝 𝑞 subscript superscript 𝑉 𝑝 𝑞 subscript 𝑉 𝑝 𝑏 \lambda_{p}^{(q)}=V^{*}_{pq}V_{pb} . The operators with the largest Wilson coefficients are the four-quark operators Q 1 p superscript subscript 𝑄 1 𝑝 Q_{1}^{p} and Q 2 p superscript subscript 𝑄 2 𝑝 Q_{2}^{p} , which read Q 1 p superscript subscript 𝑄 1 𝑝 \displaystyle Q_{1}^{p} = \displaystyle= ( q ¯ ​ p ) V − A ​ ( p ¯ ​ b ) V − A subscript ¯ 𝑞 𝑝 𝑉 𝐴 subscript ¯ 𝑝 𝑏 𝑉 𝐴 \displaystyle(\bar{q}\,p)_{V-A}\,(\bar{p}\,b)_{V-A} Q 2 p superscript subscript 𝑄 2 𝑝 \displaystyle Q_{2}^{p} = \displaystyle= ( q ¯ i ​ p j ) V − A ​ ( p ¯ j ​ b i ) V − A , subscript subscript ¯ 𝑞 𝑖 subscript 𝑝 𝑗 𝑉 𝐴 subscript subscript ¯ 𝑝 𝑗 subscript 𝑏 𝑖 𝑉 𝐴 \displaystyle(\bar{q}_{i}p_{j})_{V-A}\,(\bar{p}_{j}b_{i})_{V-A}, (2) and the electromagnetic and chromomagnetic penguin operators Q 7 subscript 𝑄 7 Q_{7} and Q 8 subscript 𝑄 8 Q_{8} , which are Q 7 subscript 𝑄 7 \displaystyle Q_{7} = \displaystyle= − e ​ m ¯ b ​ ( μ ) 8 ​ π 2 ​ q ¯ ​ σ μ ​ ν ​ 1 + γ 5 ​ b ​ F μ ​ ν , 𝑒 subscript ¯ 𝑚 𝑏 𝜇 8 superscript 𝜋 2 ¯ 𝑞 superscript 𝜎 𝜇 𝜈 delimited- 1 subscript 𝛾 5 𝑏 subscript 𝐹 𝜇 𝜈 \displaystyle-\frac{e\,\overline{m}_{b}(\mu)}{8\pi^{2}}\,\bar{q}\,\sigma^{\mu\nu}\,1+\gamma_{5}\,bF_{\mu\nu}\,, Q 8 subscript 𝑄 8 \displaystyle Q_{8} = \displaystyle= − g ​ m ¯ b ​ ( μ ) 8 ​ π 2 ​ q ¯ ​ σ μ ​ ν ​ 1 + γ 5 ​ T a ​ b ​ G μ ​ ν a . 𝑔 subscript ¯ 𝑚 𝑏 𝜇 8 superscript 𝜋 2 ¯ 𝑞 superscript 𝜎 𝜇 𝜈 delimited- 1 subscript 𝛾 5 superscript 𝑇 𝑎 𝑏 subscript superscript 𝐺 𝑎 𝜇 𝜈 \displaystyle-\frac{g\,\overline{m}_{b}(\mu)}{8\pi^{2}}\,\bar{q}\,\sigma^{\mu\nu}\,1+\gamma_{5}\,T^{a}\,bG^{a}_{\mu\nu}. (3) Here q = d 𝑞 𝑑 q=d or s 𝑠 s , and the convention for the sign of the couplings corresponds to the covariant derivative i ​ D μ = i ​ ∂ μ + e ​ Q f ​ A μ + g ​ T a ​ A μ a 𝑖 subscript 𝐷 𝜇 𝑖 subscript 𝜇 𝑒 subscript 𝑄 𝑓 subscript 𝐴 𝜇 𝑔 superscript 𝑇 𝑎 superscript subscript 𝐴 𝜇 𝑎 iD_{\mu}=i\partial_{\mu}+eQ_{f}A_{\mu}+gT^{a}A_{\mu}^{a} , with A μ subscript 𝐴 𝜇 A_{\mu} and A μ a superscript subscript 𝐴 𝜇 𝑎 A_{\mu}^{a} representing the photon and gluon fields respectively, and Q e = − 1 subscript 𝑄 𝑒 1 Q_{e}=-1 etc. The factor m ¯ b ​ ( μ ) subscript ¯ 𝑚 𝑏 𝜇 \overline{m}_{b}(\mu) is the MS ¯ ¯ MS \overline{\rm MS} mass of the b 𝑏 b quark. The main theoretical challenge is to evaluate the hadronic matrix elements of the operators in the effective weak Hamiltonian. Common ways of doing this include the QCD factorization Beneke:2001at ; Bosch:2001gv ; Ali:2001ez and pQCD Keum:2004is ; Lu:2005yz approaches. Both of these rely on the fact that α s ​ ( m b ) subscript 𝛼 𝑠 subscript 𝑚 𝑏 \alpha_{s}(m_{b}) and Λ QCD / m b subscript Λ QCD subscript 𝑚 𝑏 \Lambda_{\rm QCD}/m_{b} can be considered as small expansion parameters. In this talk I will focus exclusively on QCD factorization methods, including in this category also strategies which supply additional information on 1 / m b 1 subscript 𝑚 𝑏 1/m_{b} -suppressed contributions from QCD sum rules Ball:2006nr ; Ball:2006eu , or apply a form of renormalization-group (RG) improved perturbation theory based on soft-collinear effective theory (SCET) Becher:2005fg ; Ali:2007sj . The remainder of the talk is organized as follows. Section II introduces the QCD factorization formalism, Section III gives two sample applications, and Section IV reviews recent results in higher-order perturbative corrections to the hard-scattering kernels. This is followed by a short discussion of 1 / m b 1 subscript 𝑚 𝑏 1/m_{b} power corrections and endpoint divergences in Section V , using isospin violation in B → K ∗ ​ γ → 𝐵 superscript 𝐾 𝛾 B\to K^{*}\gamma decays as an example, and a summary in Section VI .","role":"user"},{"content":"Abstract: Experimental measurements of B → V γ decays, with V a light vector meson such as K, ρ, ω, φ, have continued to improve and will become more precise at the end of the B-factories and at LHC-b. Given their rich CKM phenomenology and potential to constrain new physics models, having reliable theory predictions for these decays is increasingly relevant. In this talk I briefly review the status of this area. The starting point is the effective weak Hamiltonian, which for B → V γ decays is [1]:","role":"assistant"}]

From Time-symmetric Microscopic Dynamics to Time-asymmetric Macroscopic Behavior: An Overview

Let me start by stating clearly that I am not going to discuss here—much less claim to resolve—the many complex issues, philosophical and physical, concerning the nature of time, from the way we perceive it to the way it enters into the space-time structure in relativistic theories. I will also not try to philosophize about the “true” nature of probability. My goal here, as in my previous articles [ 1 , 2 ] on this subject, is much more modest. a a a The interested reader may wish to look at the three book reviews of which are contained in [1e], [1f]. These books attempt to deal with some fundamental questions about time. As for the primitive notion of probability I have in mind something like this: the probability that when you next check your mail box you will find a package with a million dollars in it is very small, c.f. section 3. I will take (our everyday notions of) space, time and probability as primitive undefined concepts and try to clarify the many conceptual and mathematical problems encountered in going from a time symmetric Hamiltonian microscopic dynamics to a time asymmetric macroscopic one, as given for example by the diffusion equation. I will also take it for granted that every bit of macroscopic matter is composed of an enormous number of quasi-autonomous units, called atoms (or molecules). The atoms, taken to be the basic entities making up these macroscopic objects, will be simplified to the point of caricature: they will be treated, to quote Feynman [ 3 ] , as “little particles that move around in perpetual motion, attracting each other when they are a little distance apart, but repelling upon being squeezed into one another.” This crude picture of atoms (a refined version of that held by some ancient Greek philosophers) moving according to non-relativistic classical Hamiltonian equations contains the essential qualitative and even quantitative ingredients of macroscopic irreversibility. To accord with our understanding of microscopic reality it must, of course, be modified to take account of quantum mechanics. This raises further issues for the question of irreversibility which will be discussed in section 9. Much of what I have to say is a summary and elaboration of the work done over a century ago, when the problem of reconciling time asymmetric macroscopic behavior with the time symmetric microscopic dynamics became a central issue in physics. To quote from Thomson’s (later Lord Kelvin) beautiful and highly recommended 1874 article [ 4 ] , [ 5 ] “The essence of Joule’s discovery is the subjection of physical [read thermal] phenomena to [microscopic] dynamical law. If, then, the motion of very particle of matter in the universe were precisely reversed at any instant, the course of nature would be simply reversed for ever after. The bursting bubble of foam at the foot of a waterfall would reunite and descend into the water …. Physical processes, on the other hand, are irreversible: for example, the friction of solids, conduction of heat, and diffusion. Nevertheless, the principle of dissipation of energy [irreversible behavior] is compatible with a molecular theory in which each particle is subject to the laws of abstract dynamics.” 1.1 Formulation of Problem Formally the problem considered by Thomson in the context of Newtonian theory, the “theory of everything” at that time, is as follows: The complete microscopic (or micro) state of a classical system of N 𝑁 N particles is represented by a point X 𝑋 X in its phase space Γ Γ \Gamma , X = ( 𝐫 1 , 𝐩 1 , 𝐫 2 , 𝐩 2 , … , 𝐫 N , 𝐩 N ) 𝑋 subscript 𝐫 1 subscript 𝐩 1 subscript 𝐫 2 subscript 𝐩 2 … subscript 𝐫 𝑁 subscript 𝐩 𝑁 X=({\bf r}_{1},{\bf p}_{1},{\bf r}_{2},{\bf p}_{2},...,{\bf r}_{N},{\bf p}_{N}) , 𝐫 i subscript 𝐫 𝑖 {\bf r}_{i} and 𝐩 i subscript 𝐩 𝑖 {\bf p}_{i} being the position and momentum (or velocity) of the i 𝑖 i th particle. When the system is isolated its evolution is governed by Hamiltonian dynamics with some specified Hamiltonian H ​ ( X ) 𝐻 𝑋 H(X) which we will assume for simplicity to be an even function of the momenta. Given H ​ ( X ) 𝐻 𝑋 H(X) , the microstate X ​ ( t 0 ) 𝑋 subscript 𝑡 0 X(t_{0}) , at time t 0 subscript 𝑡 0 t_{0} , determines the microstate X ​ ( t ) 𝑋 𝑡 X(t) at all future and past times t 𝑡 t during which the system will be or was isolated: X ​ ( t ) = T t − t 0 ​ X ​ ( t 0 ) 𝑋 𝑡 subscript 𝑇 𝑡 subscript 𝑡 0 𝑋 subscript 𝑡 0 X(t)=T_{t-t_{0}}X(t_{0}) . Let X ​ ( t 0 ) 𝑋 subscript 𝑡 0 X(t_{0}) and X ​ ( t 0 + τ ) 𝑋 subscript 𝑡 0 𝜏 X(t_{0}+\tau) , with τ 𝜏 \tau positive, be two such microstates. Reversing (physically or mathematically) all velocities at time t 0 + τ subscript 𝑡 0 𝜏 t_{0}+\tau , we obtain a new microstate. If we now follow the evolution for another interval τ 𝜏 \tau we find that the new microstate at time t 0 + 2 ​ τ subscript 𝑡 0 2 𝜏 t_{0}+2\tau is just R ​ X ​ ( t 0 ) 𝑅 𝑋 subscript 𝑡 0 RX(t_{0}) , the microstate X ​ ( t 0 ) 𝑋 subscript 𝑡 0 X(t_{0}) with all velocities reversed: R ​ X = ( 𝐫 1 , − 𝐩 1 , 𝐫 2 , − 𝐩 2 , … , 𝐫 N , − 𝐩 N ) 𝑅 𝑋 subscript 𝐫 1 subscript 𝐩 1 subscript 𝐫 2 subscript 𝐩 2 … subscript 𝐫 𝑁 subscript 𝐩 𝑁 RX=({\bf r}_{1},-{\bf p}_{1},{\bf r}_{2},-{\bf p}_{2},...,{\bf r}_{N},-{\bf p}_{N}) . Hence if there is an evolution, i.e. a trajectory X ​ ( t ) 𝑋 𝑡 X(t) , in which some property of the system, specified by a function f ​ ( X ​ ( t ) ) 𝑓 𝑋 𝑡 f(X(t)) , behaves in a certain way as t 𝑡 t increases, then if f ​ ( X ) = f ​ ( R ​ X ) 𝑓 𝑋 𝑓 𝑅 𝑋 f(X)=f(RX) there is also a trajectory in which the property evolves in the time reversed direction. Thus, for example, if particle densities get more uniform as time increases, in a way described by the diffusion equation, then since the density profile is the same for X 𝑋 X and R ​ X 𝑅 𝑋 RX there is also an evolution in which the density gets more nonuniform. So why is one type of evolution, the one consistent with an entropy increase in accord with the “second law”, common and the other never seen? The difficulty is illustrated by the impossibility of time ordering of the snapshots in Fig. 1 using solely the microscopic dynamical laws: the above time symmetry 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 subscript 𝑡 𝑎 subscript 𝑡 𝑏 subscript 𝑡 𝑐 subscript 𝑡 𝑑 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 solid in which the shading indicates the local temperature. How would you order this sequence in time? 1.2 Resolution of Problem The explanation of this apparent paradox, due to Thomson, Maxwell and Boltzmann, as described in references [ 1 ] – [ 17 ] , which I will summarize in this article, 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 b b b Boltzmann’s early writings on the subject are sometimes unclear, wrong, and even contradictory. His later writings, however, are generally very clear and right on the money (even if a bit verbose for Maxwell’s taste, c.f. [ 8 ] .) The presentation here is not intended to be historical. , there are extremely strong reasons to expect the latter from the former . 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. They are: a) the great disparity between microscopic and macroscopic scales, b) the fact that the events we observe in our world are determined not only by the microscopic dynamics, but also by the initial conditions of our system, which, as we shall see later, in section 6, are very much related to the initial conditions of our universe, and c) the fact that it is not every microscopic state of a macroscopic system that will evolve in accordance with the entropy increase predicted by the second law, but only the “majority” of such states—a majority which however becomes so overwhelming when the number of atoms in the system becomes very large that irreversible behavior becomes effectively a certainty. To make the last statement complete we shall have to specify the assignment of weights, or probabilities, to different microstates consistent with a given macrostate. Note, however, that since we are concerned with events which have overwhelming probability, many different assignments are equivalent and there is no need to worry about them unduly. There is however, as we shall see later, a “natural” choice based on phase space volume (or dimension of Hilbert space in quantum mechanics). These considerations enabled Boltzmann to define the entropy of a macroscopic system in terms of its microstate and to relate its change, as expressed by the second law, to the evolution of the system’s microstate. We detail below how the above explanation works by describing first how to specify the macrostates of a macroscopic system. It is in the time evolution of these macrostates that we observe irreversible behavior [ 1 ] – [ 17 ] . 1.3 Macrostates To describe the macroscopic state of a system of N 𝑁 N atoms in a box V 𝑉 V , say N ≳ 10 20 greater-than-or-equivalent-to 𝑁 superscript 10 20 N\gtrsim 10^{20} , with the volume of V 𝑉 V , denoted by | V | 𝑉 |V| , satisfying | V | ≳ N ​ l 3 greater-than-or-equivalent-to 𝑉 𝑁 superscript 𝑙 3 |V|\gtrsim Nl^{3} , where l 𝑙 l is a typical atomic length scale, we make use of a much cruder description than that provided by the microstate X 𝑋 X , a point in the 6 ​ N 6 𝑁 6N dimensional phase space Γ = V N ⊗ ℝ 3 ​ N Γ tensor-product superscript 𝑉 𝑁 superscript ℝ 3 𝑁 \Gamma=V^{N}\otimes\mathbb{R}^{3N} . We shall denote by M 𝑀 M such a macroscopic description or macrostate. As an example we may take M 𝑀 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 𝑉 V . A more refined macroscopic description would divide V 𝑉 V into K 𝐾 K cells, where K 𝐾 K is large but still K << N much-less-than 𝐾 𝑁 K<<N , and specify the number of particles, the momentum, and the amount of energy in each cell, again with some tolerance. For many purposes it is convenient to consider cells which are small on the macroscopic scale yet contain many atoms. This leads to a description of the macrostate in terms of smooth particle, momentum and energy densities, such as those used in the Navier-Stokes equations [ 18 ] , [ 19 ] . An even more refined description is obtained by considering a smoothed out density f ​ ( 𝐫 , 𝐩 ) 𝑓 𝐫 𝐩 f({\bf r},{\bf p}) in the six-dimensional position and momentum space such as enters the Boltzmann equation for dilute gases [ 17 ] . (For dense systems this needs to be supplemented by the positional potential energy density; see footnote d 𝑑 d and reference [ 2 ] for details.) Clearly M 𝑀 M is determined by X 𝑋 X (we will thus write M ​ ( X ) 𝑀 𝑋 M(X) ) but there are many X 𝑋 X ’s (in fact a continuum) which correspond to the same M 𝑀 M . Let Γ M subscript Γ 𝑀 \Gamma_{M} be the region in Γ Γ \Gamma consisting of all microstates X 𝑋 X corresponding to a given macrostate M 𝑀 M and denote by | Γ M | = ( N ! ​ h 3 ​ N ) − 1 ​ ∫ Γ M Π i = 1 N ​ 𝑑 𝐫 i ​ 𝑑 𝐩 i subscript Γ 𝑀 superscript 𝑁 superscript ℎ 3 𝑁 1 subscript subscript Γ 𝑀 subscript superscript Π 𝑁 𝑖 1 differential-d subscript 𝐫 𝑖 differential-d subscript 𝐩 𝑖 |\Gamma_{M}|=(N!h^{3N})^{-1}\int_{\Gamma_{M}}\Pi^{N}_{i=1}d{\bf r}_{i}d{\bf p}_{i} , its symmetrized 6 ​ N 6 𝑁 6N dimensional Liouville volume (in units of h 3 ​ N superscript ℎ 3 𝑁 h^{3N} ). 1.4 Time Evolution of Macrostates: An Example Consider a situation in which a gas of N 𝑁 N atoms with energy E 𝐸 E (with some tolerance) is initially confined by a partition to the left half of of the box V 𝑉 V , and suppose that this constraint is removed at time t a subscript 𝑡 𝑎 t_{a} , see Fig. 1. The phase space volume available to the system for times t > t a 𝑡 subscript 𝑡 𝑎 t>t_{a} is then fantastically enlarged c c c 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 proton. compared to what it was initially, roughly by a factor of 2 N superscript 2 𝑁 2^{N} . Let us now consider the macrostate of this gas as given by M = ( N L N , E L E ) 𝑀 subscript 𝑁 𝐿 𝑁 subscript 𝐸 𝐿 𝐸 M=\left({N_{L}\over N},{E_{L}\over E}\right) , the fraction of particles and energy in the left half of V 𝑉 V (within some small tolerance). The macrostate at time t a , M = ( 1 , 1 ) subscript 𝑡 𝑎 𝑀 1 1 t_{a},M=(1,1) , will be denoted by M a subscript 𝑀 𝑎 M_{a} . The phase-space region | Γ | = Σ E Γ subscript Σ 𝐸 |\Gamma|=\Sigma_{E} , available to the system for t > t a 𝑡 subscript 𝑡 𝑎 t>t_{a} , i.e., the region in which H ​ ( X ) ∈ ( E , E + δ ​ E ) , δ ​ E << E formulae-sequence 𝐻 𝑋 𝐸 𝐸 𝛿 𝐸 much-less-than 𝛿 𝐸 𝐸 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 𝑋 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 𝑀 M for which | Γ M | subscript Γ 𝑀 |\Gamma_{M}| is increasing. The set of all the phase points X t subscript 𝑋 𝑡 X_{t} , which at time t a subscript 𝑡 𝑎 t_{a} were in Γ M a subscript Γ subscript 𝑀 𝑎 \Gamma_{M_{a}} , forms a region T t ​ Γ M a subscript 𝑇 𝑡 subscript Γ subscript 𝑀 𝑎 T_{t}\Gamma_{M_{a}} whose volume is, by Liouville’s Theorem, equal to | Γ M a | subscript Γ subscript 𝑀 𝑎 |\Gamma_{M_{a}}| . The shape of T t ​ Γ M a subscript 𝑇 𝑡 subscript Γ subscript 𝑀 𝑎 T_{t}\Gamma_{M_{a}} will however change with t 𝑡 t and as t 𝑡 t increases T t ​ Γ M a subscript 𝑇 𝑡 subscript Γ subscript 𝑀 𝑎 T_{t}\Gamma_{M_{a}} will increasingly be contained in regions Γ M subscript Γ 𝑀 \Gamma_{M} corresponding to macrostates with larger and larger phase space volumes | Γ M | subscript Γ 𝑀 |\Gamma_{M}| . This will continue until almost all the phase points initially in Γ M a subscript Γ subscript 𝑀 𝑎 \Gamma_{M_{a}} are contained in Γ M e ​ q subscript Γ subscript 𝑀 𝑒 𝑞 \Gamma_{M_{eq}} , with M e ​ q subscript 𝑀 𝑒 𝑞 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 e ​ q = ( 1 2 , 1 2 ) subscript 𝑀 𝑒 𝑞 1 2 1 2 M_{eq}=({1\over 2},{1\over 2}) i.e. N L / N subscript 𝑁 𝐿 𝑁 N_{L}/N and E L / E subscript 𝐸 𝐿 𝐸 E_{L}/E will each be in an interval ( 1 2 − ϵ , 1 2 + ϵ ) 1 2 italic-ϵ 1 2 italic-ϵ \left({1\over 2}-\epsilon,{1\over 2}+\epsilon\right) , N − 1 / 2 << ϵ << 1 much-less-than superscript 𝑁 1 2 italic-ϵ much-less-than 1 N^{-1/2}<<\epsilon<<1 . M e ​ q subscript 𝑀 𝑒 𝑞 M_{eq} is characterized, in fact defined, by the fact that it is the unique macrostate, among all the M α subscript 𝑀 𝛼 M_{\alpha} , for which | Γ M e ​ q | / | Σ E | ≃ 1 similar-to-or-equals subscript Γ subscript 𝑀 𝑒 𝑞 subscript Σ 𝐸 1 |\Gamma_{M_{eq}}|/|\Sigma_{E}|\simeq 1 , where | Σ E | subscript Σ 𝐸 |\Sigma_{E}| is the total phase space volume available under the energy constraint H ​ ( X ) ∈ ( E , E + δ ​ E ) 𝐻 𝑋 𝐸 𝐸 𝛿 𝐸 H(X)\in(E,E+\delta E) . (Here the symbol ≃ similar-to-or-equals \simeq means equality when N → ∞ → 𝑁 N\to\infty .) That there exists a macrostate containing almost all of the microstates in Σ E subscript Σ 𝐸 \Sigma_{E} is a consequence of the law of large numbers [ 20 ] , [ 18 ] . The fact that N 𝑁 N is enormously large for macroscope systems is absolutely critical for the existence of thermodynamic equilibrium states for any reasonable definition of macrostates, e.g. for any ϵ italic-ϵ \epsilon , in the above example such that N − 1 / 2 << ϵ << 1 much-less-than superscript 𝑁 1 2 italic-ϵ much-less-than 1 N^{-1/2}<<\epsilon<<1 . Indeed thermodynamics does not apply (is even meaningless) for isolated systems containing just a few particles, c.f. Onsager [ 21 ] and Maxwell quote in the next section [ 22 ] . Nanosystems are interesting and important intermediate cases which I shall however not discuss here; see related discussion about computer simulations in footnote e 𝑒 e . After reaching M e ​ q subscript 𝑀 𝑒 𝑞 M_{eq} we will (mostly) see only small fluctuations in N L ​ ( t ) / N subscript 𝑁 𝐿 𝑡 𝑁 N_{L}(t)/N and E L ​ ( t ) / E subscript 𝐸 𝐿 𝑡 𝐸 E_{L}(t)/E , about the value 1 2 1 2 {1\over 2} : typical fluctuations in N L subscript 𝑁 𝐿 N_{L} and E L subscript 𝐸 𝐿 E_{L} being of the order of the square root of the number of particles involved [ 18 ] . (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 [ 6 ] , [ 8 ] .) As already noted earlier the scenario in which | Γ M ​ ( X ​ ( t ) ) | subscript Γ 𝑀 𝑋 𝑡 |\Gamma_{M(X(t))}| increase with time for the M a subscript 𝑀 𝑎 M_{a} shown in Fig.1 cannot be true for all microstates X ⊂ Γ M a 𝑋 subscript Γ subscript 𝑀 𝑎 X\subset\Gamma_{M_{a}} . There will of necessity be X 𝑋 X ’s in Γ M a subscript Γ subscript 𝑀 𝑎 \Gamma_{M_{a}} which will evolve for a certain amount of time into microstates X ​ ( t ) ≡ X t 𝑋 𝑡 subscript 𝑋 𝑡 X(t)\equiv X_{t} such that | Γ M ​ ( X t ) | < | Γ M a | subscript Γ 𝑀 subscript 𝑋 𝑡 subscript Γ subscript 𝑀 𝑎 |\Gamma_{M(X_{t})}|<|\Gamma_{M_{a}}| , e.g. microstates X ∈ Γ M a 𝑋 subscript Γ subscript 𝑀 𝑎 X\in\Gamma_{M_{a}} which have all velocities directed away from the barrier which was lifted at t a subscript 𝑡 𝑎 t_{a} . What is true however is that the subset B 𝐵 B of such “bad” initial states has a phase space volume which is very very small compared to that of Γ M a subscript Γ subscript 𝑀 𝑎 \Gamma_{M_{a}} . This is what I mean when I say that entropy increasing behavior is typical ; a more extensive discussion of typicality is given later.

Time-asymmetric behavior as embodied in the second law of thermodynamics is observed in {\it individual macroscopic} systems. It can be understood as arising naturally from time-symmetric microscopic laws when account is taken of a) the great disparity between microscopic and macroscopic scales, b) a low entropy state of the early universe, and c) the fact that what we observe is the behavior of systems coming from such an initial state--not all possible systems. The explanation of the origin of the second law based on these ingredients goes back to Maxwell, Thomson and particularly Boltzmann. Common alternate explanations, such as those based on the ergodic or mixing properties of probability distributions (ensembles) already present for chaotic dynamical systems having only a few degrees of freedom or on the impossibility of having a truly isolated system, are either unnecessary, misguided or misleading. Specific features of macroscopic evolution, such as the diffusion equation, do however depend on the dynamical instability (deterministic chaos) of trajectories of isolated macroscopic systems. The extensions of these classical notions to the quantum world is in many ways fairly direct. It does however also bring in some new problems. These will be discussed but not resolved.

[{"content":"Help me to generate the abstract of a survey paper given the title:*From Time-symmetric Microscopic Dynamics to Time-asymmetric Macroscopic Behavior: An Overview* and the introduction:Let me start by stating clearly that I am not going to discuss here—much less claim to resolve—the many complex issues, philosophical and physical, concerning the nature of time, from the way we perceive it to the way it enters into the space-time structure in relativistic theories. I will also not try to philosophize about the true nature of probability. My goal here, as in my previous articles 1 , 2 on this subject, is much more modest. a a a The interested reader may wish to look at the three book reviews of which are contained in 1e, 1f. These books attempt to deal with some fundamental questions about time. As for the primitive notion of probability I have in mind something like this: the probability that when you next check your mail box you will find a package with a million dollars in it is very small, c.f. section 3. I will take (our everyday notions of) space, time and probability as primitive undefined concepts and try to clarify the many conceptual and mathematical problems encountered in going from a time symmetric Hamiltonian microscopic dynamics to a time asymmetric macroscopic one, as given for example by the diffusion equation. I will also take it for granted that every bit of macroscopic matter is composed of an enormous number of quasi-autonomous units, called atoms (or molecules). The atoms, taken to be the basic entities making up these macroscopic objects, will be simplified to the point of caricature: they will be treated, to quote Feynman 3 , as little particles that move around in perpetual motion, attracting each other when they are a little distance apart, but repelling upon being squeezed into one another. This crude picture of atoms (a refined version of that held by some ancient Greek philosophers) moving according to non-relativistic classical Hamiltonian equations contains the essential qualitative and even quantitative ingredients of macroscopic irreversibility. To accord with our understanding of microscopic reality it must, of course, be modified to take account of quantum mechanics. This raises further issues for the question of irreversibility which will be discussed in section 9. Much of what I have to say is a summary and elaboration of the work done over a century ago, when the problem of reconciling time asymmetric macroscopic behavior with the time symmetric microscopic dynamics became a central issue in physics. To quote from Thomsons (later Lord Kelvin) beautiful and highly recommended 1874 article 4 , 5 The essence of Joules discovery is the subjection of physical read thermal phenomena to microscopic dynamical law. If, then, the motion of very particle of matter in the universe were precisely reversed at any instant, the course of nature would be simply reversed for ever after. The bursting bubble of foam at the foot of a waterfall would reunite and descend into the water …. Physical processes, on the other hand, are irreversible: for example, the friction of solids, conduction of heat, and diffusion. Nevertheless, the principle of dissipation of energy irreversible behavior is compatible with a molecular theory in which each particle is subject to the laws of abstract dynamics. 1.1 Formulation of Problem Formally the problem considered by Thomson in the context of Newtonian theory, the theory of everything at that time, is as follows: The complete microscopic (or micro) state of a classical system of N 𝑁 N particles is represented by a point X 𝑋 X in its phase space Γ Γ \Gamma , X = ( 𝐫 1 , 𝐩 1 , 𝐫 2 , 𝐩 2 , … , 𝐫 N , 𝐩 N ) 𝑋 subscript 𝐫 1 subscript 𝐩 1 subscript 𝐫 2 subscript 𝐩 2 … subscript 𝐫 𝑁 subscript 𝐩 𝑁 X=({\bf r}_{1},{\bf p}_{1},{\bf r}_{2},{\bf p}_{2},...,{\bf r}_{N},{\bf p}_{N}) , 𝐫 i subscript 𝐫 𝑖 {\bf r}_{i} and 𝐩 i subscript 𝐩 𝑖 {\bf p}_{i} being the position and momentum (or velocity) of the i 𝑖 i th particle. When the system is isolated its evolution is governed by Hamiltonian dynamics with some specified Hamiltonian H ​ ( X ) 𝐻 𝑋 H(X) which we will assume for simplicity to be an even function of the momenta. Given H ​ ( X ) 𝐻 𝑋 H(X) , the microstate X ​ ( t 0 ) 𝑋 subscript 𝑡 0 X(t_{0}) , at time t 0 subscript 𝑡 0 t_{0} , determines the microstate X ​ ( t ) 𝑋 𝑡 X(t) at all future and past times t 𝑡 t during which the system will be or was isolated: X ​ ( t ) = T t − t 0 ​ X ​ ( t 0 ) 𝑋 𝑡 subscript 𝑇 𝑡 subscript 𝑡 0 𝑋 subscript 𝑡 0 X(t)=T_{t-t_{0}}X(t_{0}) . Let X ​ ( t 0 ) 𝑋 subscript 𝑡 0 X(t_{0}) and X ​ ( t 0 + τ ) 𝑋 subscript 𝑡 0 𝜏 X(t_{0}+\tau) , with τ 𝜏 \tau positive, be two such microstates. Reversing (physically or mathematically) all velocities at time t 0 + τ subscript 𝑡 0 𝜏 t_{0}+\tau , we obtain a new microstate. If we now follow the evolution for another interval τ 𝜏 \tau we find that the new microstate at time t 0 + 2 ​ τ subscript 𝑡 0 2 𝜏 t_{0}+2\tau is just R ​ X ​ ( t 0 ) 𝑅 𝑋 subscript 𝑡 0 RX(t_{0}) , the microstate X ​ ( t 0 ) 𝑋 subscript 𝑡 0 X(t_{0}) with all velocities reversed: R ​ X = ( 𝐫 1 , − 𝐩 1 , 𝐫 2 , − 𝐩 2 , … , 𝐫 N , − 𝐩 N ) 𝑅 𝑋 subscript 𝐫 1 subscript 𝐩 1 subscript 𝐫 2 subscript 𝐩 2 … subscript 𝐫 𝑁 subscript 𝐩 𝑁 RX=({\bf r}_{1},-{\bf p}_{1},{\bf r}_{2},-{\bf p}_{2},...,{\bf r}_{N},-{\bf p}_{N}) . Hence if there is an evolution, i.e. a trajectory X ​ ( t ) 𝑋 𝑡 X(t) , in which some property of the system, specified by a function f ​ ( X ​ ( t ) ) 𝑓 𝑋 𝑡 f(X(t)) , behaves in a certain way as t 𝑡 t increases, then if f ​ ( X ) = f ​ ( R ​ X ) 𝑓 𝑋 𝑓 𝑅 𝑋 f(X)=f(RX) there is also a trajectory in which the property evolves in the time reversed direction. Thus, for example, if particle densities get more uniform as time increases, in a way described by the diffusion equation, then since the density profile is the same for X 𝑋 X and R ​ X 𝑅 𝑋 RX there is also an evolution in which the density gets more nonuniform. So why is one type of evolution, the one consistent with an entropy increase in accord with the second law, common and the other never seen? The difficulty is illustrated by the impossibility of time ordering of the snapshots in Fig. 1 using solely the microscopic dynamical laws: the above time symmetry 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 subscript 𝑡 𝑎 subscript 𝑡 𝑏 subscript 𝑡 𝑐 subscript 𝑡 𝑑 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 solid in which the shading indicates the local temperature. How would you order this sequence in time? 1.2 Resolution of Problem The explanation of this apparent paradox, due to Thomson, Maxwell and Boltzmann, as described in references 1 – 17 , which I will summarize in this article, 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 b b b Boltzmanns early writings on the subject are sometimes unclear, wrong, and even contradictory. His later writings, however, are generally very clear and right on the money (even if a bit verbose for Maxwells taste, c.f. 8 .) The presentation here is not intended to be historical. , there are extremely strong reasons to expect the latter from the former . 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. They are: a) the great disparity between microscopic and macroscopic scales, b) the fact that the events we observe in our world are determined not only by the microscopic dynamics, but also by the initial conditions of our system, which, as we shall see later, in section 6, are very much related to the initial conditions of our universe, and c) the fact that it is not every microscopic state of a macroscopic system that will evolve in accordance with the entropy increase predicted by the second law, but only the majority of such states—a majority which however becomes so overwhelming when the number of atoms in the system becomes very large that irreversible behavior becomes effectively a certainty. To make the last statement complete we shall have to specify the assignment of weights, or probabilities, to different microstates consistent with a given macrostate. Note, however, that since we are concerned with events which have overwhelming probability, many different assignments are equivalent and there is no need to worry about them unduly. There is however, as we shall see later, a natural choice based on phase space volume (or dimension of Hilbert space in quantum mechanics). These considerations enabled Boltzmann to define the entropy of a macroscopic system in terms of its microstate and to relate its change, as expressed by the second law, to the evolution of the systems microstate. We detail below how the above explanation works by describing first how to specify the macrostates of a macroscopic system. It is in the time evolution of these macrostates that we observe irreversible behavior 1 – 17 . 1.3 Macrostates To describe the macroscopic state of a system of N 𝑁 N atoms in a box V 𝑉 V , say N ≳ 10 20 greater-than-or-equivalent-to 𝑁 superscript 10 20 N\gtrsim 10^{20} , with the volume of V 𝑉 V , denoted by | V | 𝑉 |V| , satisfying | V | ≳ N ​ l 3 greater-than-or-equivalent-to 𝑉 𝑁 superscript 𝑙 3 |V|\gtrsim Nl^{3} , where l 𝑙 l is a typical atomic length scale, we make use of a much cruder description than that provided by the microstate X 𝑋 X , a point in the 6 ​ N 6 𝑁 6N dimensional phase space Γ = V N ⊗ ℝ 3 ​ N Γ tensor-product superscript 𝑉 𝑁 superscript ℝ 3 𝑁 \Gamma=V^{N}\otimes\mathbb{R}^{3N} . We shall denote by M 𝑀 M such a macroscopic description or macrostate. As an example we may take M 𝑀 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 𝑉 V . A more refined macroscopic description would divide V 𝑉 V into K 𝐾 K cells, where K 𝐾 K is large but still K << N much-less-than 𝐾 𝑁 K<<N , and specify the number of particles, the momentum, and the amount of energy in each cell, again with some tolerance. For many purposes it is convenient to consider cells which are small on the macroscopic scale yet contain many atoms. This leads to a description of the macrostate in terms of smooth particle, momentum and energy densities, such as those used in the Navier-Stokes equations 18 , 19 . An even more refined description is obtained by considering a smoothed out density f ​ ( 𝐫 , 𝐩 ) 𝑓 𝐫 𝐩 f({\bf r},{\bf p}) in the six-dimensional position and momentum space such as enters the Boltzmann equation for dilute gases 17 . (For dense systems this needs to be supplemented by the positional potential energy density; see footnote d 𝑑 d and reference 2 for details.) Clearly M 𝑀 M is determined by X 𝑋 X (we will thus write M ​ ( X ) 𝑀 𝑋 M(X) ) but there are many X 𝑋 X s (in fact a continuum) which correspond to the same M 𝑀 M . Let Γ M subscript Γ 𝑀 \Gamma_{M} be the region in Γ Γ \Gamma consisting of all microstates X 𝑋 X corresponding to a given macrostate M 𝑀 M and denote by | Γ M | = ( N ! ​ h 3 ​ N ) − 1 ​ ∫ Γ M Π i = 1 N ​ 𝑑 𝐫 i ​ 𝑑 𝐩 i subscript Γ 𝑀 superscript 𝑁 superscript ℎ 3 𝑁 1 subscript subscript Γ 𝑀 subscript superscript Π 𝑁 𝑖 1 differential-d subscript 𝐫 𝑖 differential-d subscript 𝐩 𝑖 |\Gamma_{M}|=(N!h^{3N})^{-1}\int_{\Gamma_{M}}\Pi^{N}_{i=1}d{\bf r}_{i}d{\bf p}_{i} , its symmetrized 6 ​ N 6 𝑁 6N dimensional Liouville volume (in units of h 3 ​ N superscript ℎ 3 𝑁 h^{3N} ). 1.4 Time Evolution of Macrostates: An Example Consider a situation in which a gas of N 𝑁 N atoms with energy E 𝐸 E (with some tolerance) is initially confined by a partition to the left half of of the box V 𝑉 V , and suppose that this constraint is removed at time t a subscript 𝑡 𝑎 t_{a} , see Fig. 1. The phase space volume available to the system for times t > t a 𝑡 subscript 𝑡 𝑎 t>t_{a} is then fantastically enlarged c c c 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 proton. compared to what it was initially, roughly by a factor of 2 N superscript 2 𝑁 2^{N} . Let us now consider the macrostate of this gas as given by M = ( N L N , E L E ) 𝑀 subscript 𝑁 𝐿 𝑁 subscript 𝐸 𝐿 𝐸 M=\left({N_{L}\over N},{E_{L}\over E}\right) , the fraction of particles and energy in the left half of V 𝑉 V (within some small tolerance). The macrostate at time t a , M = ( 1 , 1 ) subscript 𝑡 𝑎 𝑀 1 1 t_{a},M=(1,1) , will be denoted by M a subscript 𝑀 𝑎 M_{a} . The phase-space region | Γ | = Σ E Γ subscript Σ 𝐸 |\Gamma|=\Sigma_{E} , available to the system for t > t a 𝑡 subscript 𝑡 𝑎 t>t_{a} , i.e., the region in which H ​ ( X ) ∈ ( E , E + δ ​ E ) , δ ​ E << E formulae-sequence 𝐻 𝑋 𝐸 𝐸 𝛿 𝐸 much-less-than 𝛿 𝐸 𝐸 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 𝑋 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 𝑀 M for which | Γ M | subscript Γ 𝑀 |\Gamma_{M}| is increasing. The set of all the phase points X t subscript 𝑋 𝑡 X_{t} , which at time t a subscript 𝑡 𝑎 t_{a} were in Γ M a subscript Γ subscript 𝑀 𝑎 \Gamma_{M_{a}} , forms a region T t ​ Γ M a subscript 𝑇 𝑡 subscript Γ subscript 𝑀 𝑎 T_{t}\Gamma_{M_{a}} whose volume is, by Liouvilles Theorem, equal to | Γ M a | subscript Γ subscript 𝑀 𝑎 |\Gamma_{M_{a}}| . The shape of T t ​ Γ M a subscript 𝑇 𝑡 subscript Γ subscript 𝑀 𝑎 T_{t}\Gamma_{M_{a}} will however change with t 𝑡 t and as t 𝑡 t increases T t ​ Γ M a subscript 𝑇 𝑡 subscript Γ subscript 𝑀 𝑎 T_{t}\Gamma_{M_{a}} will increasingly be contained in regions Γ M subscript Γ 𝑀 \Gamma_{M} corresponding to macrostates with larger and larger phase space volumes | Γ M | subscript Γ 𝑀 |\Gamma_{M}| . This will continue until almost all the phase points initially in Γ M a subscript Γ subscript 𝑀 𝑎 \Gamma_{M_{a}} are contained in Γ M e ​ q subscript Γ subscript 𝑀 𝑒 𝑞 \Gamma_{M_{eq}} , with M e ​ q subscript 𝑀 𝑒 𝑞 M_{eq} the systems 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 e ​ q = ( 1 2 , 1 2 ) subscript 𝑀 𝑒 𝑞 1 2 1 2 M_{eq}=({1\over 2},{1\over 2}) i.e. N L / N subscript 𝑁 𝐿 𝑁 N_{L}/N and E L / E subscript 𝐸 𝐿 𝐸 E_{L}/E will each be in an interval ( 1 2 − ϵ , 1 2 + ϵ ) 1 2 italic-ϵ 1 2 italic-ϵ \left({1\over 2}-\epsilon,{1\over 2}+\epsilon\right) , N − 1 / 2 << ϵ << 1 much-less-than superscript 𝑁 1 2 italic-ϵ much-less-than 1 N^{-1/2}<<\epsilon<<1 . M e ​ q subscript 𝑀 𝑒 𝑞 M_{eq} is characterized, in fact defined, by the fact that it is the unique macrostate, among all the M α subscript 𝑀 𝛼 M_{\alpha} , for which | Γ M e ​ q | / | Σ E | ≃ 1 similar-to-or-equals subscript Γ subscript 𝑀 𝑒 𝑞 subscript Σ 𝐸 1 |\Gamma_{M_{eq}}|/|\Sigma_{E}|\simeq 1 , where | Σ E | subscript Σ 𝐸 |\Sigma_{E}| is the total phase space volume available under the energy constraint H ​ ( X ) ∈ ( E , E + δ ​ E ) 𝐻 𝑋 𝐸 𝐸 𝛿 𝐸 H(X)\in(E,E+\delta E) . (Here the symbol ≃ similar-to-or-equals \simeq means equality when N → ∞ → 𝑁 N\to\infty .) That there exists a macrostate containing almost all of the microstates in Σ E subscript Σ 𝐸 \Sigma_{E} is a consequence of the law of large numbers 20 , 18 . The fact that N 𝑁 N is enormously large for macroscope systems is absolutely critical for the existence of thermodynamic equilibrium states for any reasonable definition of macrostates, e.g. for any ϵ italic-ϵ \epsilon , in the above example such that N − 1 / 2 << ϵ << 1 much-less-than superscript 𝑁 1 2 italic-ϵ much-less-than 1 N^{-1/2}<<\epsilon<<1 . Indeed thermodynamics does not apply (is even meaningless) for isolated systems containing just a few particles, c.f. Onsager 21 and Maxwell quote in the next section 22 . Nanosystems are interesting and important intermediate cases which I shall however not discuss here; see related discussion about computer simulations in footnote e 𝑒 e . After reaching M e ​ q subscript 𝑀 𝑒 𝑞 M_{eq} we will (mostly) see only small fluctuations in N L ​ ( t ) / N subscript 𝑁 𝐿 𝑡 𝑁 N_{L}(t)/N and E L ​ ( t ) / E subscript 𝐸 𝐿 𝑡 𝐸 E_{L}(t)/E , about the value 1 2 1 2 {1\over 2} : typical fluctuations in N L subscript 𝑁 𝐿 N_{L} and E L subscript 𝐸 𝐿 E_{L} being of the order of the square root of the number of particles involved 18 . (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 6 , 8 .) As already noted earlier the scenario in which | Γ M ​ ( X ​ ( t ) ) | subscript Γ 𝑀 𝑋 𝑡 |\Gamma_{M(X(t))}| increase with time for the M a subscript 𝑀 𝑎 M_{a} shown in Fig.1 cannot be true for all microstates X ⊂ Γ M a 𝑋 subscript Γ subscript 𝑀 𝑎 X\subset\Gamma_{M_{a}} . There will of necessity be X 𝑋 X s in Γ M a subscript Γ subscript 𝑀 𝑎 \Gamma_{M_{a}} which will evolve for a certain amount of time into microstates X ​ ( t ) ≡ X t 𝑋 𝑡 subscript 𝑋 𝑡 X(t)\equiv X_{t} such that | Γ M ​ ( X t ) | < | Γ M a | subscript Γ 𝑀 subscript 𝑋 𝑡 subscript Γ subscript 𝑀 𝑎 |\Gamma_{M(X_{t})}|<|\Gamma_{M_{a}}| , e.g. microstates X ∈ Γ M a 𝑋 subscript Γ subscript 𝑀 𝑎 X\in\Gamma_{M_{a}} which have all velocities directed away from the barrier which was lifted at t a subscript 𝑡 𝑎 t_{a} . What is true however is that the subset B 𝐵 B of such bad initial states has a phase space volume which is very very small compared to that of Γ M a subscript Γ subscript 𝑀 𝑎 \Gamma_{M_{a}} . This is what I mean when I say that entropy increasing behavior is typical ; a more extensive discussion of typicality is given later.","role":"user"},{"content":"Abstract: Time-asymmetric behavior as embodied in the second law of thermodynamics is observed in {\it individual macroscopic} systems. It can be understood as arising naturally from time-symmetric microscopic laws when account is taken of a) the great disparity between microscopic and macroscopic scales, b) a low entropy state of the early universe, and c) the fact that what we observe is the behavior of systems coming from such an initial state--not all possible systems. The explanation of the origin of the second law based on these ingredients goes back to Maxwell, Thomson and particularly Boltzmann. Common alternate explanations, such as those based on the ergodic or mixing properties of probability distributions (ensembles) already present for chaotic dynamical systems having only a few degrees of freedom or on the impossibility of having a truly isolated system, are either unnecessary, misguided or misleading. Specific features of macroscopic evolution, such as the diffusion equation, do however depend on the dynamical instability (deterministic chaos) of trajectories of isolated macroscopic systems. The extensions of these classical notions to the quantum world is in many ways fairly direct. It does however also bring in some new problems. These will be discussed but not resolved.","role":"assistant"}]

Noncommutative generalization of SU(n)-principal fiber bundles: a review

1   Introduction The geometry of fiber bundles is now widely used in the physical literature, mainly through the concept of connections, which are interpreted as gauge fields in particle physics. It is worth to recall why the structure of these gauge theories leads to this mathematical identification. The main points which connect these two concepts are the common expression for gauge transformations and the field strength of the gauge fields recognized as the curvature of the connection. Since the introduction of the Higgs mechanics, some attempts have been made to understand its geometrical origin in a same satisfactory and elegant way as the gauge fields. The reduction of some higher dimensional gauge field theories to some more “conventional” dimensions has been proposed to reproduce the Higgs part of some models. Nevertheless, one of the more convincing constructions from which Higgs fields emerged naturally and without the need to perform some dimensional reduction of some extra ad-hoc afterward arbitrary distortion of the model, was firstly exposed in [ Dubois-Violette et al., 1990a ] , and highly popularized in subsequent work by A. Connes in its noncommutative standard model (see [ Chamseddine and Connes, 2007 ] for a review of the recent developments in this direction). What the pioneer work by Dubois-Violette, Kerner and Madore revealed is that the Higgs fields can be identified with the purely noncommutative part of a noncommutative connection on an noncommutative algebra “containing” an ordinary smooth algebra of functions over a manifold and a purely noncommutative algebra. The algebra used there is the tensor product 𝑪 ∞ ​ ( 𝓜 ) ⊗ 𝑴 𝒏 tensor-product superscript 𝑪 𝓜 subscript 𝑴 𝒏 C^{\infty}({{\mathcal{M}}})\otimes M_{n} of smooth functions on some manifold 𝓜 𝓜 {{\mathcal{M}}} and the matrix algebra of size 𝒏 𝒏 n . This “trivial” product does not reveal the richness of this approach when some more intricate algebra is involved. In this review, we consider the algebra of endomorphisms of a 𝑺 ​ 𝑼 ​ ( 𝒏 ) 𝑺 𝑼 𝒏 SU(n) -vector bundle. This algebra reduces to the previous situation for a trivial vector bundle. This non triviality gives rise to some elegant and powerful constructions we exposed in a series of previous papers, and to some results nowhere published before. The first part deals with some reviews of the ordinary geometry of fiber bundles and connections. We think this is useful to fix notations, but also to highlight what the noncommutative differential geometry defined in the following extends from these constructions. We then define the general settings of our noncommutative geometry, which is based on derivations. The notion of noncommutative connections is exposed, and some important examples are then given to better understand the general situation. The algebra we are interested in is then introduced as the algebra of endomorphisms of a 𝑺 ​ 𝑼 ​ ( 𝒏 ) 𝑺 𝑼 𝒏 SU(n) -vector bundle. We show how it is related to ordinary geometry, and how ordinary connections plays an essential role to study its noncommutative geometry. The noncommutative connections on this algebra are then studied, and here we recall why the purely noncommutative part can be identified with Higgs fields. Then it is shown that this algebra is indeed related, through the algebraic notion of Cartan operations on a bigger algebra, to the geometry of the 𝑺 ​ 𝑼 ​ ( 𝒏 ) 𝑺 𝑼 𝒏 SU(n) -principal fiber bundle underlying the geometry of the 𝑺 ​ 𝑼 ​ ( 𝒏 ) 𝑺 𝑼 𝒏 SU(n) -vector bundle. Some considerations about the cohomology behind the endomorphism algebra are then exposed, in particular a new construction of the Chern classes of the 𝑺 ​ 𝑼 ​ ( 𝒏 ) 𝑺 𝑼 𝒏 SU(n) -vector bundle which are obtained from a short exact sequence of Lie algebras of derivations. The last section is concerned with the symmetric reduction of noncommutative connections, which generalizes a lot of previous works about symmetric reduction of ordinary connections.

This is an extended version of a communication made at the international conference 'Noncommutative Geometry and Physics' held at Orsay in april 2007. In this proceeding, we make a review of some noncommutative constructions connected to the ordinary fiber bundle theory. The noncommutative algebra is the endomorphism algebra of a SU(n)-vector bundle, and its differential calculus is based on its Lie algebra of derivations. It is shown that this noncommutative geometry contains some of the most important constructions introduced and used in the theory of connections on vector bundles, in particular, what is needed to introduce gauge models in physics, and it also contains naturally the essential aspects of the Higgs fields and its associated mechanics of mass generation. It permits one also to extend some previous constructions, as for instance symmetric reduction of (here noncommutative) connections. From a mathematical point of view, these geometrico-algebraic considerations highlight some new point on view, in particular we introduce a new construction of the Chern characteristic classes.

[{"content":"Help me to generate the abstract of a survey paper given the title:*Noncommutative generalization of SU(n)-principal fiber bundles: a review* and the introduction:1   Introduction The geometry of fiber bundles is now widely used in the physical literature, mainly through the concept of connections, which are interpreted as gauge fields in particle physics. It is worth to recall why the structure of these gauge theories leads to this mathematical identification. The main points which connect these two concepts are the common expression for gauge transformations and the field strength of the gauge fields recognized as the curvature of the connection. Since the introduction of the Higgs mechanics, some attempts have been made to understand its geometrical origin in a same satisfactory and elegant way as the gauge fields. The reduction of some higher dimensional gauge field theories to some more conventional dimensions has been proposed to reproduce the Higgs part of some models. Nevertheless, one of the more convincing constructions from which Higgs fields emerged naturally and without the need to perform some dimensional reduction of some extra ad-hoc afterward arbitrary distortion of the model, was firstly exposed in Dubois-Violette et al., 1990a , and highly popularized in subsequent work by A. Connes in its noncommutative standard model (see Chamseddine and Connes, 2007 for a review of the recent developments in this direction). What the pioneer work by Dubois-Violette, Kerner and Madore revealed is that the Higgs fields can be identified with the purely noncommutative part of a noncommutative connection on an noncommutative algebra containing an ordinary smooth algebra of functions over a manifold and a purely noncommutative algebra. The algebra used there is the tensor product 𝑪 ∞ ​ ( 𝓜 ) ⊗ 𝑴 𝒏 tensor-product superscript 𝑪 𝓜 subscript 𝑴 𝒏 C^{\infty}({{\mathcal{M}}})\otimes M_{n} of smooth functions on some manifold 𝓜 𝓜 {{\mathcal{M}}} and the matrix algebra of size 𝒏 𝒏 n . This trivial product does not reveal the richness of this approach when some more intricate algebra is involved. In this review, we consider the algebra of endomorphisms of a 𝑺 ​ 𝑼 ​ ( 𝒏 ) 𝑺 𝑼 𝒏 SU(n) -vector bundle. This algebra reduces to the previous situation for a trivial vector bundle. This non triviality gives rise to some elegant and powerful constructions we exposed in a series of previous papers, and to some results nowhere published before. The first part deals with some reviews of the ordinary geometry of fiber bundles and connections. We think this is useful to fix notations, but also to highlight what the noncommutative differential geometry defined in the following extends from these constructions. We then define the general settings of our noncommutative geometry, which is based on derivations. The notion of noncommutative connections is exposed, and some important examples are then given to better understand the general situation. The algebra we are interested in is then introduced as the algebra of endomorphisms of a 𝑺 ​ 𝑼 ​ ( 𝒏 ) 𝑺 𝑼 𝒏 SU(n) -vector bundle. We show how it is related to ordinary geometry, and how ordinary connections plays an essential role to study its noncommutative geometry. The noncommutative connections on this algebra are then studied, and here we recall why the purely noncommutative part can be identified with Higgs fields. Then it is shown that this algebra is indeed related, through the algebraic notion of Cartan operations on a bigger algebra, to the geometry of the 𝑺 ​ 𝑼 ​ ( 𝒏 ) 𝑺 𝑼 𝒏 SU(n) -principal fiber bundle underlying the geometry of the 𝑺 ​ 𝑼 ​ ( 𝒏 ) 𝑺 𝑼 𝒏 SU(n) -vector bundle. Some considerations about the cohomology behind the endomorphism algebra are then exposed, in particular a new construction of the Chern classes of the 𝑺 ​ 𝑼 ​ ( 𝒏 ) 𝑺 𝑼 𝒏 SU(n) -vector bundle which are obtained from a short exact sequence of Lie algebras of derivations. The last section is concerned with the symmetric reduction of noncommutative connections, which generalizes a lot of previous works about symmetric reduction of ordinary connections.","role":"user"},{"content":"Abstract: This is an extended version of a communication made at the international conference 'Noncommutative Geometry and Physics' held at Orsay in april 2007. In this proceeding, we make a review of some noncommutative constructions connected to the ordinary fiber bundle theory. The noncommutative algebra is the endomorphism algebra of a SU(n)-vector bundle, and its differential calculus is based on its Lie algebra of derivations. It is shown that this noncommutative geometry contains some of the most important constructions introduced and used in the theory of connections on vector bundles, in particular, what is needed to introduce gauge models in physics, and it also contains naturally the essential aspects of the Higgs fields and its associated mechanics of mass generation. It permits one also to extend some previous constructions, as for instance symmetric reduction of (here noncommutative) connections. From a mathematical point of view, these geometrico-algebraic considerations highlight some new point on view, in particular we introduce a new construction of the Chern characteristic classes.","role":"assistant"}]

Noncommutative Black Holes, The Final Appeal To Quantum Gravity: A Review

In 1975 Steven Hawking showed, in a remarkable paper [ 1 ] , that black holes can evaporate, namely that they can emit thermal radiation like a black body. This conjecture opened the window towards the mysteries of Quantum Gravity, since, at that time, the reconciliation between Quantum Mechanics and General Relativity was kept in a lethargic state as a matter of secondary concern. Indeed, theoretical physicists were basically interested in the Standard Model of particle physics and gravity was simply assumed to be too weak with respect to the already unified fundamental interactions. Even strings were employed as a theory of hadrons. Direct quantization of General Relativity into a quantum field theory in which the gravitational force is carried by gravitons led to the unpleasant appearance of infinities. This lack of renormalization was somehow an underestimated problem or at least considered as being too far away, only relevant at a scale 19 orders of magnitude higher than the mainstream high energy physics at the time. Therefore Gravity was kept classical and the major efforts were directed towards the Standard Model of particle physics, calculating Feynman diagrams, cross sections and decay rates. In this spirit, Hawking’s conjecture about black hole evaporation was supported by semiclassical arguments [ 2 ] \cdash [ 4 ] which hold only when the energy of the emitted particle is small compared to the black hole mass and it is possible to neglect the back reaction on the metric during the emission process 1 1 1 The physical picture of Hawking radiation is in terms of particles tunnelling through the horizon. Only recently has a transparent derivation of it been given in terms of a quasi-classical tunnelling calculation (see Refs. \refcite Parikh:1999mf– \refcite Akhmedov:2008ru). . After realizing that black holes can emit photons, gravitons and other elementary particles as products of the decay, Hawking raised a natural and profound question about the final stage of the evaporation and the possible loss of information encoded in quantum states of matter. All of the troubles about the black hole evaporation were basically due to a breakdown of the semiclassical description, which requires that the black hole mass M B ​ H ≫ M P much-greater-than subscript 𝑀 𝐵 𝐻 subscript 𝑀 𝑃 M_{BH}\gg M_{P} , where M P subscript 𝑀 𝑃 M_{P} is Planck mass. As the decay proceeds and the black hole mass decreases, this condition eventually can no longer be met and a quantum theory of gravity must be invoked. Indeed the Planck scale marks the limes between the classical and the quantum behavior of the spacetime. After more than 30 30 30 years of intensive research, the scenario is now drastically different and we have at least two plausible candidate theories of Quantum Gravity: Superstrings 2 2 2 In (Super)String Theory one adopts 1-dimensional quantum objects instead of the more conventional point-like structures [ 23 , 24 ] . This represents the first chapter in a program for studying the physics of extended quantum objects, which also include 2-dimensional branes and 3-dimensional bubbles (for more details see Refs. \refcite Pavsic:2001qh– \refcite Ansoldi:1997hz). and Loop Quantum Gravity [ 25 ] . Both of them have great merits and still-open problems, but at present both suffer from a basic limitation: the absence of any support from experiments. For this reason they cannot be considered as being more than theoretical speculations, even if they are both physically very promising, aesthetically very attractive and mathematically elegant and well defined. The main problem is that we do not yet have any experimental data demonstrating some quantum gravitational effect. From historical, epistemological and philosophical points of view, the long-standing inability to find observable effects of Quantum Gravity is probably the most dramatic crisis in theoretical physics and perhaps in all of science. What makes the situation even more embarrassing is that Quantum Gravity is expected to be the synthesis of two physical theories which are the most successful ever formulated from the point of view of their capacity for reproducing experimental results 3 3 3 The relative error in the General Relativity prediction for the rate of change of the orbital period of the Hulse-Taylor binary pulsar PSR 1913+16 is 10 − 14 superscript 10 14 10^{-14} (see Ref. \refcite Taylor:1989sw). Experimental data and Quantum Field Theory predictions of the value of the anomalous magnetic moment of the electron agree to within a relative error of 10 − 11 superscript 10 11 10^{-11} (see Ref. \refcite Kinoshita_1995). . What is the role of black holes in this context? It is very likely that black holes are fated to provide the final answer about our knowledge of Quantum Gravity and close the logical path which they started in 1975, igniting the initial interest in this area. Indeed, in the next few months 4 4 4 The Large Hadron Collider (LHC) is being built at the CERN lab and is planned to circulate the first beams in August 2008, while the first collisions are expected in October 2008 with the first results arriving soon afterwards [ 37 ] . we could be very close to having revolutionary experimental evidence for both particle phenomenology and gravitational physics: the detection of some signature of the production of mini black holes [ 38 ] \cdash [ 40 ] that are predicted to be produced as a result of high energy hadronic collisions. According to theory, decay of a mini black hole would lead to the sudden emission of a huge quantity of particles independently of their spin, charge, quantum numbers or interaction properties, as long as their rest mass is smaller than the black hole temperature [ 41 ] . Therefore mini black holes may be excellent intermediate states for long-sought but yet undiscovered particles, like the Higgs boson or supersymmetric particles, possibly with cross sections much larger than for the direct processes. For these reasons, the starting-up of the LHC, or of some other future experiments 5 5 5 Formation and detection of mini black holes is expected in both future hadron colliders, such as the Very Large Hadron Collider (VLHC) at Fermilab, and the particle shower of Ultra High Energy Cosmic Rays (UHECR), impacting Earth’s atmosphere [ 42 ] . , will open up very good opportunities for getting information about Quantum Gravity [ 42 ] \cdash [ 52 ] and the very early universe, as well as for solving some basic questions, whose answers are too often taken for granted. First, do (mini) black holes exist? From velocity measurements for the whirlpool of hot gas surrounding it, astronomers have found convincing evidence for the existence of a supermassive black hole in the center of the giant elliptical galaxy M87 M87 \mathrm{M}87 6 6 6 In 1994, Hubble Space Telescope data produced an unprecedented measurement of the mass of an unseen object at the center of M87 M87 \mathrm{M}87 . Based on the kinetic energy of the material whirling about the center, the object is estimated to be about 3 billion times the mass of our Sun and appears to be concentrated within a volume of space smaller than that of our solar system. . Astronomers have also detected radio emission coming from within 30 million kilometers of the dark object SGR A*, thought to be a colossal black hole, that lies at the center of the Milky Way [ 53 ] \cdash [ 56 ] . Previously, X-ray emission from the binary star system Cygnus X-1 convinced many astronomers that it contains a black hole, which is supported by more precise measurements which have recently become available. To this purpose we also have to mention V404 Cygni, one of the most evident cases of stellar black hole: observations at X-ray and optical wavelengths have shown that this is a binary system in which a late yellowish G star or maybe an early orange-red K star revolves, every 6.47 days, around a compact companion with a diameter around 60 − 90 60 90 60-90 km and a probable mass of 8 to 15 solar masses, well above the mass limit at which a collapsed star must become a black hole [ 57 ] \cdash [ 62 ] . In spite of these observations, however, there are still some ranges of mass in which the existence of black holes is unclear, in particular of black holes of less than 3 3 3 solar masses. The relevance of these objects is connected with the possibility of observing the Hawking radiation. Indeed, what we know for sure is that for astrophysical black holes the Hawking radiation is negligible because their temperatures can be at most some tens of nK, far below T CMB ∼ 2.7 similar-to subscript 𝑇 CMB 2.7 T_{\mathrm{CMB}}\sim 2.7 K, the temperature of Cosmic Microwave Background (CMB) radiation. On the other hand tidal effects are significant in the case of mini black holes that could be very hot and very bright if their mass is sufficiently small [ 63 ] . As second point, we might be able to conclude that (mini) black holes really can evaporate. Indeed for the above reasons the detection of these objects is the unique direct way 7 7 7 There exist alternative proposals to observe the Hawking radiation by an “indirect” means in the so-called Analogue Models, namely condensed matter systems which behave, to some extent, like gravitational ones [ 64 ] : in Bose-Einstein condensates the supersonic region is the acoustic analog of a black hole, while the flux of phonons corresponds to the Hawking radiation. The resulting Hawking temperature is T H ∼ 10 similar-to subscript 𝑇 𝐻 10 T_{H}\sim 10 nK as compared with a condensate temperature T c ∼ 100 similar-to subscript 𝑇 𝑐 100 T_{c}\sim\ 100 nK (see Ref. \refcite Balbinot:2006ua for a review and Refs. \refcite Balbinot:2007de– \refcite Carusotto:2008ep for recent results). Another interesting proposal is that concerning the possibility of experimentally detecting the Unruh effect in particle storage rings in terms of the Sokolov–Ternov effect [ 68 , 69 ] . This could be the simplest prototype of vacuum polarization effects in curved space. to have experimental evidence of the Hawking conjecture, one of the most important predictions of Quantum Field Theory in Curved Spacetime and of the associated semiclassical gravity. Thirdly, we could find out what it is the fate of a radiating black hole. If mini black holes can be created in high energy particle collisions, the black holes produced will pass through a number of phases before completing their evaporation. After a loss during the first two phases of their “hair” (i.e. the associated long-range fields) and of their angular momentum (the “balding” and spin-down phases respectively), the picture of the evaporation will be described by the Schwarzschild phase, in which the resulting spherically symmetric black hole loses energy by the emission of Hawking radiation, with a gradual decrease in its mass and an increase in its temperature. Since the Schwarzschild geometry has a curvature singularity at the origin, there would be a divergent behavior of the Hawking temperature if the black hole were to shrink to the origin as a result of losing mass by thermal emission. However, we do not expect that this divergent behavior will, in fact, take place since in the vicinity of the origin, the evaporating black hole will be dramatically disturbed by strong quantum gravitational fluctuations of the spacetime manifold. In other words the black hole will undergo a Plank phase of the evaporation during which a theory of Quantum Gravity must be used. Observations of the final stages of black hole evaporation could provide the profile of the temperature as a function of the mass of the black hole and hence let us pick out, for the first time, the correct quantum gravitational theory. All of this promising and very fascinating program is, on the other hand, subject to a severe and unavoidable constraint: the inferred existence of large spatial extra dimensions. This hypothesis is currently considered to be the unique viable solution to the long-standing hierarchy problem, namely the presence of two fundamental scales in nature, the electroweak scale and the Planck scale, separated by 16 16 16 orders of magnitude [ 70 ] \cdash [ 79 ] . The main point in this potential resolution of the problem is that extra dimensions can be assumed to be as large as around a millimeter, if we suppose that Standard Model fields are constrained in a four dimensional sub-manifold of the higher dimensional spacetime, and that only gravity can probe the additional transverse dimensions. In connection with this, recent experiments involving direct measurement of Newtonian gravity put limits on the size of extra dimensions at length scales below 1 1 1 mm. [ 80 , 81 ] In this type of scenario, we can identify a new scale M ∗ subscript 𝑀 ∗ M_{\ast} , derived from the Planck scale through the following relation M ∗ ( 2 + n ) = M P 2 / R n superscript subscript 𝑀 ∗ 2 𝑛 superscript subscript 𝑀 𝑃 2 superscript 𝑅 𝑛 M_{\ast}^{(2+n)}=M_{P}^{2}/R^{n} (1) where R 𝑅 R is the mean size of each of the n 𝑛 n extra dimensions. If R 𝑅 R is large enough with respect the Planck length ℓ P subscript ℓ 𝑃 \ell_{P} , then M ∗ subscript 𝑀 ∗ M_{\ast} will be much smaller than M P subscript 𝑀 𝑃 M_{P} and we obtain a unique fundamental scale M ∗ ∼ 1 similar-to subscript 𝑀 ∗ 1 M_{\ast}\sim 1 TeV for both the electroweak and gravitational interactions. Conversely, in the absence of any large extra dimensions, Quantum Gravity is lost forever: it is very likely that at the current rate of technological progress, mankind will probably become extinct before any experimental evidence of Quantum Gravity would become accessible. Giving a possible solution of the hierarchy problem is not the only key consequence of the presence of large extra dimensions: there are also further important consequences which we will summarize briefly here. First, any black hole smaller than the size of the extra dimensions can be considered, to a good approximation, as being totally submerged in a 4 + n 4 𝑛 4+n dimensional isotropic spacetime, with one time dimension and 3 + n 3 𝑛 3+n spatial ones. This allows one to use the higher-dimensional Schwarzschild solution to describe at least the neutral non-rotating phase of the black hole’s life. As a result, the cross section for creation of mini black holes due to parton collisions significantly increases in the presence of large extra dimensions. Indeed, the corresponding Schwarzschild radii r H subscript 𝑟 𝐻 r_{H} become of the order of 10 − 4 superscript 10 4 10^{-4} fm, which is large with respect to the parton impact parameter b 𝑏 b . Therefore we can estimate the black hole production cross section by the geometrical approximation σ ∼ π ​ r H 2 ∼ 400 similar-to 𝜎 𝜋 superscript subscript 𝑟 𝐻 2 similar-to 400 \sigma\sim\pi r_{H}^{2}\sim 400 pb and so, at the estimated luminosity for the LHC ( L ∼ 10 33 similar-to 𝐿 superscript 10 33 L\sim 10^{33} cm -2 s -1 ) we find that roughly ten black holes would be created per second [ 82 ] . Another consequence of the introduction of large extra dimensions is that the mini black holes would be colder and thus live longer than their four-dimensional analogs, and so would be more easily detectable once created. Indeed, one typically finds mini black hole temperatures of the order of 100 100 100 GeV and lifetimes of the order of 10 − 26 superscript 10 26 10^{-26} s, that are interpreted as those of resonances. At this point, the game seems to be over: we have candidate theories of Quantum Gravity and forthcoming experiments which are potentially able to check them. We should then try to find some easily testable theoretical predictions and wait for their experimental confirmation. The problem is that the final state of black hole evaporation cannot be efficiently described by means of the aforementioned theories of Quantum Gravity. Indeed (Super)String Theory provides a quantum description of black holes only for a few cases, namely for the extremal (and near-extremal) charged black hole models [ 83 , 84 ] . This is rather unsatisfactory, since the Planck phase, during the terminal stage of the evaporation, occurs in a neutral regime due to the rapid discharge of the black hole in the very initial stages. On the other hand, Loop Quantum Gravity suffers from the absence of a clear semiclassical limit. What is missing is a systematic way of computing scattering amplitudes and cross sections by perturbative techniques, a fact that is the basic limitation for obtaining significant quantitative results [ 25 ] . Therefore, in the absence of a full quantum description of all of the significant black hole evaporation phases, one uses effective theories to describe the quantum gravitational behavior, at least in some regimes. The most common effective tool is Quantum Field Theory in Curved Space, which works efficiently at least until the black hole quantum back reaction destroys the hypothesis of a fixed background spacetime [ 85 ] . Very recently, stimulated by the need for going beyond the rough semiclassical approximation of Quantum Field Theory in Curved Space, significant new approaches have been proposed based on Noncommutative Geometry arguments. Indeed there is a long-held belief that Quantum Gravity should have an uncertainty principle which prevents one from measuring positions to accuracies better than that given by the Planck length: the momentum and energy required to make such a measurement would themselves modify the geometry at these scales [ 87 ] \cdash [ 91 ] . Therefore, one might wish to describe these effects, at least effectively, by a model theory having a new sort of uncertainty principle among the coordinates. In the same way as happens with coordinates and momenta in conventional quantum theory, the uncertainty would come from a noncommutative relation and so one is led to examine the possibility that position measurements might fail to commute, postulating the existence of a noncommutative manifold [ x i , x j ] ≠ 0 . superscript x 𝑖 superscript x 𝑗 0 \left[\textbf{x}^{i},\textbf{x}^{j}\right]\neq 0. (2) A feature of a Noncommutative Geometry would be the presence of quantum fluctuations able to remove the infinities which usually appear and cause the bad short-distance behavior of field theories, including gravity. The purpose of this review is to explore the current status of the physics of Quantum Black Holes from the viewpoint of Noncommutative Geometry, used as an effective tool for modelling the extreme energy quantum gravitational effects of the final phase of the evaporation, which are plagued by singularities at a semiclassical level. In Section 2, we start by reviewing the most popular Noncommutative Geometry models existing in the literature and their role in comparison with General Relativity. In Section 3, we discuss the four dimensional noncommutative Schwarzschild solution, from both geometrical and thermodynamical points of view. In Section 4, we study how noncommutativity affects the Maxwell field and present the four dimensional noncommutative Reissner-Nordström solution, providing a detailed analysis of both the Hawking and Schwinger pair production mechanisms. In Section 5, we consider evaporation in the extra-dimensional scenario, for both neutral and charged solutions, reviewing its phenomenological consequences in view of a possible experimental detection at the LHC. The final Section is devoted to future perspectives.

We present the state of the art regarding the relation between the physics of Quantum Black Holes and Noncommutative Geometry. We start with a review of models proposed in the literature for describing deformations of General Relativity in the presence of noncommutativity, seen as an effective theory of Quantum Gravity. We study the resulting metrics, proposed to replace or at least to improve the conventional black hole solutions of Einstein's equation. In particular, we analyze noncommutative-inspired solutions obtained in terms of quasiclassical noncommutative coordinates: indeed because of their surprising new features, these solutions enable us to circumvent long standing problems with Quantum Field Theory in Curved Space and to cure the singular behavior of gravity at the centers of black holes. As a consequence, for the first time, we get a complete description of what we may call the black hole SCRAM, the shut down of the emission of thermal radiation from the black hole: in place of the conventio...

[{"content":"Help me to generate the abstract of a survey paper given the title:*Noncommutative Black Holes, The Final Appeal To Quantum Gravity: A Review* and the introduction:In 1975 Steven Hawking showed, in a remarkable paper 1 , that black holes can evaporate, namely that they can emit thermal radiation like a black body. This conjecture opened the window towards the mysteries of Quantum Gravity, since, at that time, the reconciliation between Quantum Mechanics and General Relativity was kept in a lethargic state as a matter of secondary concern. Indeed, theoretical physicists were basically interested in the Standard Model of particle physics and gravity was simply assumed to be too weak with respect to the already unified fundamental interactions. Even strings were employed as a theory of hadrons. Direct quantization of General Relativity into a quantum field theory in which the gravitational force is carried by gravitons led to the unpleasant appearance of infinities. This lack of renormalization was somehow an underestimated problem or at least considered as being too far away, only relevant at a scale 19 orders of magnitude higher than the mainstream high energy physics at the time. Therefore Gravity was kept classical and the major efforts were directed towards the Standard Model of particle physics, calculating Feynman diagrams, cross sections and decay rates. In this spirit, Hawkings conjecture about black hole evaporation was supported by semiclassical arguments 2 \cdash 4 which hold only when the energy of the emitted particle is small compared to the black hole mass and it is possible to neglect the back reaction on the metric during the emission process 1 1 1 The physical picture of Hawking radiation is in terms of particles tunnelling through the horizon. Only recently has a transparent derivation of it been given in terms of a quasi-classical tunnelling calculation (see Refs. \refcite Parikh:1999mf– \refcite Akhmedov:2008ru). . After realizing that black holes can emit photons, gravitons and other elementary particles as products of the decay, Hawking raised a natural and profound question about the final stage of the evaporation and the possible loss of information encoded in quantum states of matter. All of the troubles about the black hole evaporation were basically due to a breakdown of the semiclassical description, which requires that the black hole mass M B ​ H ≫ M P much-greater-than subscript 𝑀 𝐵 𝐻 subscript 𝑀 𝑃 M_{BH}\gg M_{P} , where M P subscript 𝑀 𝑃 M_{P} is Planck mass. As the decay proceeds and the black hole mass decreases, this condition eventually can no longer be met and a quantum theory of gravity must be invoked. Indeed the Planck scale marks the limes between the classical and the quantum behavior of the spacetime. After more than 30 30 30 years of intensive research, the scenario is now drastically different and we have at least two plausible candidate theories of Quantum Gravity: Superstrings 2 2 2 In (Super)String Theory one adopts 1-dimensional quantum objects instead of the more conventional point-like structures 23 , 24 . This represents the first chapter in a program for studying the physics of extended quantum objects, which also include 2-dimensional branes and 3-dimensional bubbles (for more details see Refs. \refcite Pavsic:2001qh– \refcite Ansoldi:1997hz). and Loop Quantum Gravity 25 . Both of them have great merits and still-open problems, but at present both suffer from a basic limitation: the absence of any support from experiments. For this reason they cannot be considered as being more than theoretical speculations, even if they are both physically very promising, aesthetically very attractive and mathematically elegant and well defined. The main problem is that we do not yet have any experimental data demonstrating some quantum gravitational effect. From historical, epistemological and philosophical points of view, the long-standing inability to find observable effects of Quantum Gravity is probably the most dramatic crisis in theoretical physics and perhaps in all of science. What makes the situation even more embarrassing is that Quantum Gravity is expected to be the synthesis of two physical theories which are the most successful ever formulated from the point of view of their capacity for reproducing experimental results 3 3 3 The relative error in the General Relativity prediction for the rate of change of the orbital period of the Hulse-Taylor binary pulsar PSR 1913+16 is 10 − 14 superscript 10 14 10^{-14} (see Ref. \refcite Taylor:1989sw). Experimental data and Quantum Field Theory predictions of the value of the anomalous magnetic moment of the electron agree to within a relative error of 10 − 11 superscript 10 11 10^{-11} (see Ref. \refcite Kinoshita_1995). . What is the role of black holes in this context? It is very likely that black holes are fated to provide the final answer about our knowledge of Quantum Gravity and close the logical path which they started in 1975, igniting the initial interest in this area. Indeed, in the next few months 4 4 4 The Large Hadron Collider (LHC) is being built at the CERN lab and is planned to circulate the first beams in August 2008, while the first collisions are expected in October 2008 with the first results arriving soon afterwards 37 . we could be very close to having revolutionary experimental evidence for both particle phenomenology and gravitational physics: the detection of some signature of the production of mini black holes 38 \cdash 40 that are predicted to be produced as a result of high energy hadronic collisions. According to theory, decay of a mini black hole would lead to the sudden emission of a huge quantity of particles independently of their spin, charge, quantum numbers or interaction properties, as long as their rest mass is smaller than the black hole temperature 41 . Therefore mini black holes may be excellent intermediate states for long-sought but yet undiscovered particles, like the Higgs boson or supersymmetric particles, possibly with cross sections much larger than for the direct processes. For these reasons, the starting-up of the LHC, or of some other future experiments 5 5 5 Formation and detection of mini black holes is expected in both future hadron colliders, such as the Very Large Hadron Collider (VLHC) at Fermilab, and the particle shower of Ultra High Energy Cosmic Rays (UHECR), impacting Earths atmosphere 42 . , will open up very good opportunities for getting information about Quantum Gravity 42 \cdash 52 and the very early universe, as well as for solving some basic questions, whose answers are too often taken for granted. First, do (mini) black holes exist? From velocity measurements for the whirlpool of hot gas surrounding it, astronomers have found convincing evidence for the existence of a supermassive black hole in the center of the giant elliptical galaxy M87 M87 \mathrm{M}87 6 6 6 In 1994, Hubble Space Telescope data produced an unprecedented measurement of the mass of an unseen object at the center of M87 M87 \mathrm{M}87 . Based on the kinetic energy of the material whirling about the center, the object is estimated to be about 3 billion times the mass of our Sun and appears to be concentrated within a volume of space smaller than that of our solar system. . Astronomers have also detected radio emission coming from within 30 million kilometers of the dark object SGR A*, thought to be a colossal black hole, that lies at the center of the Milky Way 53 \cdash 56 . Previously, X-ray emission from the binary star system Cygnus X-1 convinced many astronomers that it contains a black hole, which is supported by more precise measurements which have recently become available. To this purpose we also have to mention V404 Cygni, one of the most evident cases of stellar black hole: observations at X-ray and optical wavelengths have shown that this is a binary system in which a late yellowish G star or maybe an early orange-red K star revolves, every 6.47 days, around a compact companion with a diameter around 60 − 90 60 90 60-90 km and a probable mass of 8 to 15 solar masses, well above the mass limit at which a collapsed star must become a black hole 57 \cdash 62 . In spite of these observations, however, there are still some ranges of mass in which the existence of black holes is unclear, in particular of black holes of less than 3 3 3 solar masses. The relevance of these objects is connected with the possibility of observing the Hawking radiation. Indeed, what we know for sure is that for astrophysical black holes the Hawking radiation is negligible because their temperatures can be at most some tens of nK, far below T CMB ∼ 2.7 similar-to subscript 𝑇 CMB 2.7 T_{\mathrm{CMB}}\sim 2.7 K, the temperature of Cosmic Microwave Background (CMB) radiation. On the other hand tidal effects are significant in the case of mini black holes that could be very hot and very bright if their mass is sufficiently small 63 . As second point, we might be able to conclude that (mini) black holes really can evaporate. Indeed for the above reasons the detection of these objects is the unique direct way 7 7 7 There exist alternative proposals to observe the Hawking radiation by an indirect means in the so-called Analogue Models, namely condensed matter systems which behave, to some extent, like gravitational ones 64 : in Bose-Einstein condensates the supersonic region is the acoustic analog of a black hole, while the flux of phonons corresponds to the Hawking radiation. The resulting Hawking temperature is T H ∼ 10 similar-to subscript 𝑇 𝐻 10 T_{H}\sim 10 nK as compared with a condensate temperature T c ∼ 100 similar-to subscript 𝑇 𝑐 100 T_{c}\sim\ 100 nK (see Ref. \refcite Balbinot:2006ua for a review and Refs. \refcite Balbinot:2007de– \refcite Carusotto:2008ep for recent results). Another interesting proposal is that concerning the possibility of experimentally detecting the Unruh effect in particle storage rings in terms of the Sokolov–Ternov effect 68 , 69 . This could be the simplest prototype of vacuum polarization effects in curved space. to have experimental evidence of the Hawking conjecture, one of the most important predictions of Quantum Field Theory in Curved Spacetime and of the associated semiclassical gravity. Thirdly, we could find out what it is the fate of a radiating black hole. If mini black holes can be created in high energy particle collisions, the black holes produced will pass through a number of phases before completing their evaporation. After a loss during the first two phases of their hair (i.e. the associated long-range fields) and of their angular momentum (the balding and spin-down phases respectively), the picture of the evaporation will be described by the Schwarzschild phase, in which the resulting spherically symmetric black hole loses energy by the emission of Hawking radiation, with a gradual decrease in its mass and an increase in its temperature. Since the Schwarzschild geometry has a curvature singularity at the origin, there would be a divergent behavior of the Hawking temperature if the black hole were to shrink to the origin as a result of losing mass by thermal emission. However, we do not expect that this divergent behavior will, in fact, take place since in the vicinity of the origin, the evaporating black hole will be dramatically disturbed by strong quantum gravitational fluctuations of the spacetime manifold. In other words the black hole will undergo a Plank phase of the evaporation during which a theory of Quantum Gravity must be used. Observations of the final stages of black hole evaporation could provide the profile of the temperature as a function of the mass of the black hole and hence let us pick out, for the first time, the correct quantum gravitational theory. All of this promising and very fascinating program is, on the other hand, subject to a severe and unavoidable constraint: the inferred existence of large spatial extra dimensions. This hypothesis is currently considered to be the unique viable solution to the long-standing hierarchy problem, namely the presence of two fundamental scales in nature, the electroweak scale and the Planck scale, separated by 16 16 16 orders of magnitude 70 \cdash 79 . The main point in this potential resolution of the problem is that extra dimensions can be assumed to be as large as around a millimeter, if we suppose that Standard Model fields are constrained in a four dimensional sub-manifold of the higher dimensional spacetime, and that only gravity can probe the additional transverse dimensions. In connection with this, recent experiments involving direct measurement of Newtonian gravity put limits on the size of extra dimensions at length scales below 1 1 1 mm. 80 , 81 In this type of scenario, we can identify a new scale M ∗ subscript 𝑀 ∗ M_{\ast} , derived from the Planck scale through the following relation M ∗ ( 2 + n ) = M P 2 / R n superscript subscript 𝑀 ∗ 2 𝑛 superscript subscript 𝑀 𝑃 2 superscript 𝑅 𝑛 M_{\ast}^{(2+n)}=M_{P}^{2}/R^{n} (1) where R 𝑅 R is the mean size of each of the n 𝑛 n extra dimensions. If R 𝑅 R is large enough with respect the Planck length ℓ P subscript ℓ 𝑃 \ell_{P} , then M ∗ subscript 𝑀 ∗ M_{\ast} will be much smaller than M P subscript 𝑀 𝑃 M_{P} and we obtain a unique fundamental scale M ∗ ∼ 1 similar-to subscript 𝑀 ∗ 1 M_{\ast}\sim 1 TeV for both the electroweak and gravitational interactions. Conversely, in the absence of any large extra dimensions, Quantum Gravity is lost forever: it is very likely that at the current rate of technological progress, mankind will probably become extinct before any experimental evidence of Quantum Gravity would become accessible. Giving a possible solution of the hierarchy problem is not the only key consequence of the presence of large extra dimensions: there are also further important consequences which we will summarize briefly here. First, any black hole smaller than the size of the extra dimensions can be considered, to a good approximation, as being totally submerged in a 4 + n 4 𝑛 4+n dimensional isotropic spacetime, with one time dimension and 3 + n 3 𝑛 3+n spatial ones. This allows one to use the higher-dimensional Schwarzschild solution to describe at least the neutral non-rotating phase of the black holes life. As a result, the cross section for creation of mini black holes due to parton collisions significantly increases in the presence of large extra dimensions. Indeed, the corresponding Schwarzschild radii r H subscript 𝑟 𝐻 r_{H} become of the order of 10 − 4 superscript 10 4 10^{-4} fm, which is large with respect to the parton impact parameter b 𝑏 b . Therefore we can estimate the black hole production cross section by the geometrical approximation σ ∼ π ​ r H 2 ∼ 400 similar-to 𝜎 𝜋 superscript subscript 𝑟 𝐻 2 similar-to 400 \sigma\sim\pi r_{H}^{2}\sim 400 pb and so, at the estimated luminosity for the LHC ( L ∼ 10 33 similar-to 𝐿 superscript 10 33 L\sim 10^{33} cm -2 s -1 ) we find that roughly ten black holes would be created per second 82 . Another consequence of the introduction of large extra dimensions is that the mini black holes would be colder and thus live longer than their four-dimensional analogs, and so would be more easily detectable once created. Indeed, one typically finds mini black hole temperatures of the order of 100 100 100 GeV and lifetimes of the order of 10 − 26 superscript 10 26 10^{-26} s, that are interpreted as those of resonances. At this point, the game seems to be over: we have candidate theories of Quantum Gravity and forthcoming experiments which are potentially able to check them. We should then try to find some easily testable theoretical predictions and wait for their experimental confirmation. The problem is that the final state of black hole evaporation cannot be efficiently described by means of the aforementioned theories of Quantum Gravity. Indeed (Super)String Theory provides a quantum description of black holes only for a few cases, namely for the extremal (and near-extremal) charged black hole models 83 , 84 . This is rather unsatisfactory, since the Planck phase, during the terminal stage of the evaporation, occurs in a neutral regime due to the rapid discharge of the black hole in the very initial stages. On the other hand, Loop Quantum Gravity suffers from the absence of a clear semiclassical limit. What is missing is a systematic way of computing scattering amplitudes and cross sections by perturbative techniques, a fact that is the basic limitation for obtaining significant quantitative results 25 . Therefore, in the absence of a full quantum description of all of the significant black hole evaporation phases, one uses effective theories to describe the quantum gravitational behavior, at least in some regimes. The most common effective tool is Quantum Field Theory in Curved Space, which works efficiently at least until the black hole quantum back reaction destroys the hypothesis of a fixed background spacetime 85 . Very recently, stimulated by the need for going beyond the rough semiclassical approximation of Quantum Field Theory in Curved Space, significant new approaches have been proposed based on Noncommutative Geometry arguments. Indeed there is a long-held belief that Quantum Gravity should have an uncertainty principle which prevents one from measuring positions to accuracies better than that given by the Planck length: the momentum and energy required to make such a measurement would themselves modify the geometry at these scales 87 \cdash 91 . Therefore, one might wish to describe these effects, at least effectively, by a model theory having a new sort of uncertainty principle among the coordinates. In the same way as happens with coordinates and momenta in conventional quantum theory, the uncertainty would come from a noncommutative relation and so one is led to examine the possibility that position measurements might fail to commute, postulating the existence of a noncommutative manifold x i , x j ≠ 0 . superscript x 𝑖 superscript x 𝑗 0 \left\textbf{x}^{i},\textbf{x}^{j}\right\neq 0. (2) A feature of a Noncommutative Geometry would be the presence of quantum fluctuations able to remove the infinities which usually appear and cause the bad short-distance behavior of field theories, including gravity. The purpose of this review is to explore the current status of the physics of Quantum Black Holes from the viewpoint of Noncommutative Geometry, used as an effective tool for modelling the extreme energy quantum gravitational effects of the final phase of the evaporation, which are plagued by singularities at a semiclassical level. In Section 2, we start by reviewing the most popular Noncommutative Geometry models existing in the literature and their role in comparison with General Relativity. In Section 3, we discuss the four dimensional noncommutative Schwarzschild solution, from both geometrical and thermodynamical points of view. In Section 4, we study how noncommutativity affects the Maxwell field and present the four dimensional noncommutative Reissner-Nordström solution, providing a detailed analysis of both the Hawking and Schwinger pair production mechanisms. In Section 5, we consider evaporation in the extra-dimensional scenario, for both neutral and charged solutions, reviewing its phenomenological consequences in view of a possible experimental detection at the LHC. The final Section is devoted to future perspectives.","role":"user"},{"content":"Abstract: We present the state of the art regarding the relation between the physics of Quantum Black Holes and Noncommutative Geometry. We start with a review of models proposed in the literature for describing deformations of General Relativity in the presence of noncommutativity, seen as an effective theory of Quantum Gravity. We study the resulting metrics, proposed to replace or at least to improve the conventional black hole solutions of Einstein's equation. In particular, we analyze noncommutative-inspired solutions obtained in terms of quasiclassical noncommutative coordinates: indeed because of their surprising new features, these solutions enable us to circumvent long standing problems with Quantum Field Theory in Curved Space and to cure the singular behavior of gravity at the centers of black holes. As a consequence, for the first time, we get a complete description of what we may call the black hole SCRAM, the shut down of the emission of thermal radiation from the black hole: in place of the conventio...","role":"assistant"}]

Product Systems:. a Survey with Commutants in View

A product system of Hilbert spaces is a family E ⊗ = ( E t ) t ∈ ℝ + superscript 𝐸 tensor-product subscript subscript 𝐸 𝑡 𝑡 subscript ℝ E^{\otimes}=\bigl{(}E_{t}\bigr{)}_{t\in\mathbb{R}_{+}} of Hilbert spaces that factor as E s + t ≅ E s ⊗ E t subscript 𝐸 𝑠 𝑡 tensor-product subscript 𝐸 𝑠 subscript 𝐸 𝑡 E_{s+t}~{}\cong~{}E_{s}\otimes E_{t} by means of an associative bilinear multiplication E s × E t ∋ ( x s , y t ) ↦ x s ​ y t ∈ E s + t contains subscript 𝐸 𝑠 subscript 𝐸 𝑡 subscript 𝑥 𝑠 subscript 𝑦 𝑡 maps-to subscript 𝑥 𝑠 subscript 𝑦 𝑡 subscript 𝐸 𝑠 𝑡 E_{s}\times E_{t}\ni(x_{s},y_{t})\mapsto x_{s}y_{t}\in E_{s+t} . (Depending on the application, there are also technical conditions about continuity or measurability of sections. We speak about this later on.) The definition of such product systems is due to Arveson [ Arv89a ] . It is motivated by Arveson’s construction that associates with every E 0 subscript 𝐸 0 E_{0} –semigroup (a semigroup of normal unital endomorphisms) of the ℬ ​ ( H ) ℬ 𝐻 \mathscr{B}(H) of all adjointable operators on a Hilbert space H 𝐻 H a product system E A ⊗ superscript superscript 𝐸 𝐴 tensor-product {E^{A}}^{\otimes} . If H 𝐻 H is infinite-dimensional and separable, then the product system determines the E 0 subscript 𝐸 0 E_{0} –semigroup up to cocycle conjugacy . In a series of four articles [ Arv89a , Arv90a , Arv89b , Arv90b ] Arveson showed the fundamental theorem , namely, that every product system of Hilbert spaces is the one associated with a suitable E 0 subscript 𝐸 0 E_{0} –semigroup. Thus, there is a one-to-one correspondence between product systems (up to isomorphism) and E 0 subscript 𝐸 0 E_{0} –semigroups (up to cocycle conjugacy). In the sequel, we will speak about an Arveson system if we intend a product system of Hilbert spaces. In particular, we will speak about the Arveson system associated with an E 0 subscript 𝐸 0 E_{0} –semigroup on ℬ ​ ( H ) ℬ 𝐻 \mathscr{B}(H) . Meanwhile, product systems of Hilbert bimodules or, more fashonably, correspondences made appearance in many contexts. Bhat and Skeide [ BS00 ] constructed a product system of correspondences over a (unital) C ∗ superscript 𝐶 C^{*} –algebra ℬ ℬ \mathcal{B} from a (unital) CP-semigroup on ℬ ℬ \mathcal{B} . (See also the discussion of Muhly and Solel [ MS02 ] in Remark 6.6 .) This construction overcomes constructions by Bhat [ Bha96 ] and by Arveson [ Arv96 ] who construct an Arveson system starting from a CP-semigroup on ℬ ​ ( H ) ℬ 𝐻 \mathscr{B}(H) by, first, dilating in a unique way the CP-semigroup to a minimal E 0 subscript 𝐸 0 E_{0} –semigroup and, then, constructing the Arveson system of that E 0 subscript 𝐸 0 E_{0} –semigroup. The construction of [ BS00 ] , instead, is direct and allows, then, to construct the minimal dilation in a transparent way. Only later, Skeide [ Ske02 , Ske05a , Ske04 ] associated in several ways with an E 0 subscript 𝐸 0 E_{0} –semigroup a product system. Now the E 0 subscript 𝐸 0 E_{0} –semigroup acts on the algebra ℬ a ​ ( E ) superscript ℬ 𝑎 𝐸 \mathscr{B}^{a}(E) of all adjointable maps on a Hilbert ℬ ℬ \mathcal{B} –module E 𝐸 E . Although historically earlier, the approach to product systems from CP-semigroups (that is, irreversible quantum dynamics) has the disadvantage that not all product systems arise in that way. While one of the latest results (Skeide [ Ske07 ] ; still in preparation) asserts that cum grano salis every product system comes from an E 0 subscript 𝐸 0 E_{0} –semigroup (that is, reversible quantum dynamics in a sense we specify later on). So the approach via E 0 subscript 𝐸 0 E_{0} –semigroups allows a more coherent discussion. In this survey we will concentrate on this connection between product systems and E 0 subscript 𝐸 0 E_{0} –semigroups, while we will have no space to discuss also the connections with CP-semigroup and their dilations; see Skeide [ Ske03b ] . Also basic classification of product systems must be sacrificed; see Skeide [ Ske03b , Ske06g ] . The basic factorization property of the symmetric Fock space Γ ​ ( H 1 ⊕ H 2 ) = Γ ​ ( H 1 ) ⊗ Γ ​ ( H 2 ) Γ direct-sum subscript 𝐻 1 subscript 𝐻 2 tensor-product Γ subscript 𝐻 1 Γ subscript 𝐻 2 \Gamma(H_{1}\oplus H_{2})~{}=~{}\Gamma(H_{1})\otimes\Gamma(H_{2}) ( H 1 subscript 𝐻 1 H_{1} and H 2 subscript 𝐻 2 H_{2} some Hilbert spaces) has drawn attention since a long time. In the form Γ ​ ( L 2 ​ ( [ r , t ] , K ) ) = Γ ​ ( L 2 ​ ( [ r , s ] , K ) ) ⊗ Γ ​ ( L 2 ​ ( [ s , t ] , K ) ) , r ≤ s ≤ t formulae-sequence Γ superscript 𝐿 2 𝑟 𝑡 𝐾 tensor-product Γ superscript 𝐿 2 𝑟 𝑠 𝐾 Γ superscript 𝐿 2 𝑠 𝑡 𝐾 𝑟 𝑠 𝑡 \Gamma(L^{2}([r,t],K))~{}=~{}\Gamma(L^{2}([r,s],K))\otimes\Gamma(L^{2}([s,t],K)),~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}r\leq s\leq t ( ∗ * ) ( K 𝐾 K a Hilbert space) it made appearance in the work of Araki [ Ara70 ] and Streater [ Str69 ] on current representations of Lie algebras, in the work of Parthasarathy and Schmidt [ PS72 ] about Lévy processes (culminating in Schürmann’s work [ Sch93 ] on quantum Lévy processes) and in quantum stochastic calculus on the symmetric Fock space initiated by Hudson and Parthasarathy [ HP84 ] . Let us put E t = Γ ​ ( L 2 ​ ( [ 0 , t ] , K ) ) subscript 𝐸 𝑡 Γ superscript 𝐿 2 0 𝑡 𝐾 E_{t}=\Gamma(L^{2}([0,t],K)) . Then, from the beginning, there are two possibilities to use ( ∗ * ‣ 1 ) in order to define an isomorphism E s ⊗ E t ≅ E s + t tensor-product subscript 𝐸 𝑠 subscript 𝐸 𝑡 subscript 𝐸 𝑠 𝑡 E_{s}\otimes E_{t}\cong E_{s+t} , namely, E s ⊗ E t tensor-product subscript 𝐸 𝑠 subscript 𝐸 𝑡 \displaystyle E_{s}\otimes E_{t} ≅ 𝒮 t ​ E s ⊗ E t ≅ E s + t absent tensor-product subscript 𝒮 𝑡 subscript 𝐸 𝑠 subscript 𝐸 𝑡 subscript 𝐸 𝑠 𝑡 \displaystyle~{}\cong~{}\text{\scriptsize$\mathscr{S}$}_{t}E_{s}\otimes E_{t}~{}\cong~{}E_{s+t} and E s ⊗ E t tensor-product subscript 𝐸 𝑠 subscript 𝐸 𝑡 \displaystyle E_{s}\otimes E_{t} ≅ E s ⊗ 𝒮 s ​ E t ≅ E s + t , absent tensor-product subscript 𝐸 𝑠 subscript 𝒮 𝑠 subscript 𝐸 𝑡 subscript 𝐸 𝑠 𝑡 \displaystyle~{}\cong~{}E_{s}\otimes\text{\scriptsize$\mathscr{S}$}_{s}E_{t}~{}\cong~{}E_{s+t}, where 𝒮 t : Γ ​ ( L 2 ​ ( [ 0 , s ] , K ) ) → Γ ​ ( L 2 ​ ( [ t , t + s ] , K ) ) : subscript 𝒮 𝑡 → Γ superscript 𝐿 2 0 𝑠 𝐾 Γ superscript 𝐿 2 𝑡 𝑡 𝑠 𝐾 \text{\scriptsize$\mathscr{S}$}_{t}\colon\Gamma(L^{2}([0,s],K))\rightarrow\Gamma(L^{2}([t,t+s],K)) is the time shift. If we consider the CCR-flow , that is, the E 0 subscript 𝐸 0 E_{0} –semigroup induced on ℬ ​ ( Γ ​ ( L 2 ​ ( ℝ + , K ) ) ) ℬ Γ superscript 𝐿 2 subscript ℝ 𝐾 \mathscr{B}\bigl{(}\Gamma(L^{2}(\mathbb{R}_{+},K))\bigr{)} by the time shift, then the associated Arveson system is E t subscript 𝐸 𝑡 E_{t} with the second choice of an isomorphism, that is, with the time shift acting on the right factor in E s ⊗ E t tensor-product subscript 𝐸 𝑠 subscript 𝐸 𝑡 E_{s}\otimes E_{t} . However, Bhat discovered a second possibility to associate an Arveson with an E 0 subscript 𝐸 0 E_{0} –semigroup. In the case of the CCR-flow one obtains the same Hilbert spaces E t subscript 𝐸 𝑡 E_{t} but with the first choice of an isomorphism, that is, with the time shift acting on the left factor in E s ⊗ E t tensor-product subscript 𝐸 𝑠 subscript 𝐸 𝑡 E_{s}\otimes E_{t} . More generally, the Bhat system associated with any E 0 subscript 𝐸 0 E_{0} –semigroup shows always to be anti-isomorphic to the associated Arveson system. This ambivalence in the tensor product of Hilbert spaces, where we may switch the order of factors without changing (up to canonical isomorphism) the resulting space, is by far less innocent than it appears at the first sight. Nothing like this is true in the module case for the tensor product of correspondences over ℬ ℬ \mathcal{B} . (It is very well possible that in one order their tensor product is { 0 } 0 \{0\} , while in the other order it is not.) In fact, we will see that the construction of a product system of correspondences over ℬ ℬ \mathcal{B} from an E 0 subscript 𝐸 0 E_{0} –semigroup on ℬ a ​ ( E ) superscript ℬ 𝑎 𝐸 \mathscr{B}^{a}(E) for some Hilbert ℬ ℬ \mathcal{B} –module E 𝐸 E corresponds to the construction of the Bhat system of an E 0 subscript 𝐸 0 E_{0} –semigroup on ℬ ​ ( H ) ℬ 𝐻 \mathscr{B}(H) . Also the construction of product system following the ideas of Arveson is still possible. However, it yields a product system of correspondences over the commutant ℬ ′ superscript ℬ ′ \mathcal{B}^{\prime} of ℬ ℬ \mathcal{B} and works nicely only for von Neumann algebras ℬ ℬ \mathcal{B} . The relation between these two product systems is that one is the commutant of the other. The commutant of a correspondence was introduced in Skeide [ Ske03a ] , the conribution to the proceedings of the conference in Mount Holyoke 2002. In the space available we are not able to even scratch the basic classification results for product systems. We refere the reader to the still quite up-to-date survey Skeide [ Ske03b ] in the proceedings of the conference in Burg 2001. The classification is based on spatial product systems and their product in Skeide [ Ske06g ] (preprint 2001). We fix some notations used throughout, and recall very few basics about Hilbert modules in order to make this survey digestable also for nonexperts in Hilbert modules. Let ℬ ℬ \mathcal{B} be a C ∗ superscript 𝐶 C^{*} –algebra. Recall that a pre-Hilbert ℬ ℬ \mathcal{B} –module is a right ℬ ℬ \mathcal{B} –module E 𝐸 E with a sesquilinear inner product ⟨ ∙ , ∙ ⟩ : E × E → ℬ : ∙ ∙ → 𝐸 𝐸 ℬ \langle\bullet,\bullet\rangle\colon E\times E\rightarrow\mathcal{B} satisfying ⟨ x , x ⟩ ≥ 0 𝑥 𝑥 0 \langle x,x\rangle\geq 0 for all x ∈ E 𝑥 𝐸 x\in E (positivity), ⟨ x , y ​ b ⟩ = ⟨ x , y ⟩ ​ b 𝑥 𝑦 𝑏 𝑥 𝑦 𝑏 \langle x,yb\rangle=\langle x,y\rangle b for all x , y ∈ E ; b ∈ ℬ formulae-sequence 𝑥 𝑦 𝐸 𝑏 ℬ x,y\in E;b\in\mathcal{B} (right linearity), and ⟨ x , x ⟩ = 0 ⟹ x = 0 𝑥 𝑥 0 ⟹ 𝑥 0 \langle x,x\rangle=0~{}\Longrightarrow~{}x=0 (definiteness). If definiteness is missing, then E 𝐸 E is a semi-Hilbert ℬ ℬ \mathcal{B} –module . (Properties like ⟨ x , y ⟩ ∗ = ⟨ y , x ⟩ superscript 𝑥 𝑦 𝑦 𝑥 \langle x,y\rangle^{*}=\langle y,x\rangle and ⟨ x ​ b , y ⟩ = b ∗ ​ ⟨ x , y ⟩ 𝑥 𝑏 𝑦 superscript 𝑏 𝑥 𝑦 \langle xb,y\rangle=b^{*}\langle x,y\rangle are automatic.) The most basic property of the inner product in a semi-Hilbert ℬ ℬ \mathcal{B} –module is the following Cauchy-Schwartz inequality ⟨ x , y ⟩ ​ ⟨ y , x ⟩ ≤ ∥ ⟨ y , y ⟩ ∥ ​ ⟨ x , x ⟩ . 𝑥 𝑦 𝑦 𝑥 delimited-∥∥ 𝑦 𝑦 𝑥 𝑥 \langle x,y\rangle\langle y,x\rangle~{}\leq~{}\left\lVert\langle y,y\rangle\right\rVert\langle x,x\rangle. By Cauchy-Schwartz inequality it is possible to quotient out length-zero elements. By Cauchy-Schwartz inequality ∥ x ∥ := ⟨ x , x ⟩ assign delimited-∥∥ 𝑥 𝑥 𝑥 \left\lVert x\right\rVert:=\sqrt{\langle x,x\rangle} defines a norm on the pre-Hilbert module E 𝐸 E . If E 𝐸 E is complete in that norm, then E 𝐸 E is a Hilbert ℬ ℬ \mathcal{B} –module . By Cauchy-Schwartz inequality the operator norm turns the algebra of bounded adjointable operators ℬ a ​ ( E ) superscript ℬ 𝑎 𝐸 \mathscr{B}^{a}(E) on the pre-Hilbert module E 𝐸 E into a pre– C ∗ superscript 𝐶 C^{*} –algebra. Recall that a map a 𝑎 a on E 𝐸 E is adjointable , if it admits an adjoint a ∗ superscript 𝑎 a^{*} such that ⟨ x , a ​ y ⟩ = ⟨ a ∗ ​ x , y ⟩ 𝑥 𝑎 𝑦 superscript 𝑎 𝑥 𝑦 \langle x,ay\rangle=\langle a^{*}x,y\rangle for all x , y ∈ E 𝑥 𝑦 𝐸 x,y\in E . Every adjointable map is closeable. Therefore, by the closed graph theorem , an adjointable map on a Hilbert module is bounded, automatically. In order to speak about product systems we need the (internal) tensor product, and the tensor product is among bimodules or correspondences . If 𝒜 𝒜 \mathcal{A} is another C ∗ superscript 𝐶 C^{*} –algebra, then a correspondence from 𝒜 𝒜 \mathcal{A} to ℬ ℬ \mathcal{B} (or a 𝒜 𝒜 \mathcal{A} – ℬ ℬ \mathcal{B} –correspondence) is a Hilbert ℬ ℬ \mathcal{B} –module with a nondegenerate representation of 𝒜 𝒜 \mathcal{A} by adjointable operators. If 𝒜 = ℬ 𝒜 ℬ \mathcal{A}=\mathcal{B} , then we speak also of a correspondence over ℬ ℬ \mathcal{B} or of a ℬ ℬ \mathcal{B} –correspondence. [1] [1] [1] The nondegeneracy condition is crucial in all what follows. For the right action of ℬ ℬ \mathcal{B} on a Hilbert ℬ ℬ \mathcal{B} –module it is automatic. (Exercise: Why?) But, there are left actions that act degenerately. However, in that case we will never say 𝒜 𝒜 \mathcal{A} – ℬ ℬ \mathcal{B} –module, but rather speak of a (possibly degenerate) representation of 𝒜 𝒜 \mathcal{A} . The ( internal ) tensor product of a correspondence E 𝐸 E from 𝒜 𝒜 \mathcal{A} to ℬ ℬ \mathcal{B} and a correspondence F 𝐹 F from ℬ ℬ \mathcal{B} to 𝒞 𝒞 \mathcal{C} is the unique correspondence E ⊙ F direct-product 𝐸 𝐹 E\odot F from 𝒜 𝒜 \mathcal{A} to 𝒞 𝒞 \mathcal{C} that is generated by elementary tensors x ⊙ y direct-product 𝑥 𝑦 x\odot y with inner product ⟨ x ⊙ y , x ′ ⊙ y ′ ⟩ = ⟨ y ′ , ⟨ x , x ′ ⟩ ​ y ′ ⟩ direct-product 𝑥 𝑦 direct-product superscript 𝑥 ′ superscript 𝑦 ′ superscript 𝑦 ′ 𝑥 superscript 𝑥 ′ superscript 𝑦 ′ \langle x\odot y,x^{\prime}\odot y^{\prime}\rangle~{}=~{}\bigl{\langle}y^{\prime},\langle x,x^{\prime}\rangle y^{\prime}\bigr{\rangle} (1.1) and the obvious bimodule operation. Uniqueness is, in the sense of a universal property, up to canonical isomorphism. (In two realizations, simply identify the elementary tensors. For a construction take the vector space tensor product E ⊗ F tensor-product 𝐸 𝐹 E\otimes F , define a semiinner product by ( 1.1 ) and divide by the length-zero elements.) The tensor product applies also if E 𝐸 E is just a Hilbert ℬ ℬ \mathcal{B} –module, as every Hilbert ℬ ℬ \mathcal{B} –module E 𝐸 E may be viewed as a correspondence from ℬ a ​ ( E ) superscript ℬ 𝑎 𝐸 \mathscr{B}^{a}(E) to ℬ ℬ \mathcal{B} . This also shows that E ⊙ F direct-product 𝐸 𝐹 E\odot F carries a canonical nondegenerate left action of a ∈ ℬ a ​ ( E ) 𝑎 superscript ℬ 𝑎 𝐸 a\in\mathscr{B}^{a}(E) which we denote by a ⊙ id F direct-product 𝑎 subscript id 𝐹 a\odot\operatorname{\text{\small$\textsf{id}$}}_{F} or, sometimes, simply by a 𝑎 a , too. (Attention! The unital embedding ℬ a ​ ( E ) → ℬ a ​ ( E ) ⊙ id F ⊂ ℬ a ​ ( E ⊙ F ) → superscript ℬ 𝑎 𝐸 direct-product superscript ℬ 𝑎 𝐸 subscript id 𝐹 superscript ℬ 𝑎 direct-product 𝐸 𝐹 \mathscr{B}^{a}(E)\rightarrow\mathscr{B}^{a}(E)\odot\operatorname{\text{\small$\textsf{id}$}}_{F}\subset\mathscr{B}^{a}(E\odot F) need not be faithful.) By ℬ a , b ​ i ​ l ​ ( F ) superscript ℬ 𝑎 𝑏 𝑖 𝑙 𝐹 \mathscr{B}^{a,bil}(F) we denote the space of those elements a ∈ ℬ a ​ ( F ) 𝑎 superscript ℬ 𝑎 𝐹 a\in\mathscr{B}^{a}(F) that are bilinear , that is, which fulfill a ​ ( b ​ y ) = b ​ ( a ​ y ) 𝑎 𝑏 𝑦 𝑏 𝑎 𝑦 a(by)=b(ay) for all b ∈ ℬ , y ∈ F formulae-sequence 𝑏 ℬ 𝑦 𝐹 b\in\mathcal{B},y\in F . There is an embedding ℬ a , b ​ i ​ l ​ ( F ) → id E ⊙ ℬ a , b ​ i ​ l ​ ( F ) ⊂ ℬ a ​ ( E ⊙ F ) → superscript ℬ 𝑎 𝑏 𝑖 𝑙 𝐹 direct-product subscript id 𝐸 superscript ℬ 𝑎 𝑏 𝑖 𝑙 𝐹 superscript ℬ 𝑎 direct-product 𝐸 𝐹 \mathscr{B}^{a,bil}(F)\rightarrow\operatorname{\text{\small$\textsf{id}$}}_{E}\odot\mathscr{B}^{a,bil}(F)\subset\mathscr{B}^{a}(E\odot F) . If E 𝐸 E is full , that is, if the range ideal ℬ E := span ¯ ⁡ ⟨ E , E ⟩ assign subscript ℬ 𝐸 ¯ span 𝐸 𝐸 \mathcal{B}_{E}:=\operatorname{\overline{\operatorname{\text{\small$\textsf{span}$}}}}\langle E,E\rangle in ℬ ℬ \mathcal{B} is ℬ ℬ \mathcal{B} , then one may show that this embedding is an isomorphism onto the relative commutant of ℬ a ​ ( E ) ⊙ id F direct-product superscript ℬ 𝑎 𝐸 subscript id 𝐹 \mathscr{B}^{a}(E)\odot\operatorname{\text{\small$\textsf{id}$}}_{F} in ℬ a ​ ( E ⊙ F ) superscript ℬ 𝑎 direct-product 𝐸 𝐹 \mathscr{B}^{a}(E\odot F) . If ( v , w ) ↦ v ⋅ w maps-to 𝑣 𝑤 ⋅ 𝑣 𝑤 (v,w)\mapsto v\cdot w is bilinear or sesquilinear operation, then V ​ W 𝑉 𝑊 VW is the set { v ⋅ w : v ∈ V , w ∈ W } conditional-set ⋅ 𝑣 𝑤 formulae-sequence 𝑣 𝑉 𝑤 𝑊 \{v\cdot w\colon v\in V,w\in W\} . We do not adopt the convention that V ​ W = span ⁡ V ​ W 𝑉 𝑊 span 𝑉 𝑊 VW=\operatorname{\text{\small$\textsf{span}$}}VW or even V ​ W = span ¯ ⁡ V ​ W 𝑉 𝑊 ¯ span 𝑉 𝑊 VW=\operatorname{\overline{\operatorname{\text{\small$\textsf{span}$}}}}VW .

The theory of product systems both of Hilbert spaces (Arveson systems) and product systems of Hilbert modules has reached a status where it seems appropriate to rest a moment and to have a look at what is known so far and what are open problems. However, the attempt to give an approximately complete account in view pages is destined to fail already for Arveson systems since Tsirelson, Powers and Liebscher have discovered their powerful methods to construct large classes of examples. In this survey we concentrate on that part of the theory that works also for Hilbert modules. This does not only help to make a selection among the possible topics, but it also helps to shed some new light on the case of Arveson systems. Often, proofs that work for Hilbert modules also lead to simpler proofs in the case of Hilbert spaces. We put emphasis on those aspects that arise from recent results about commutants of von Neumann correspondences, which, in the case of Hilbert spaces, explain the relation between the Arveson system and the Bhat system associated with an E_0--semigroup on B(H).

[{"content":"Help me to generate the abstract of a survey paper given the title:*Product Systems:. a Survey with Commutants in View* and the introduction:A product system of Hilbert spaces is a family E ⊗ = ( E t ) t ∈ ℝ + superscript 𝐸 tensor-product subscript subscript 𝐸 𝑡 𝑡 subscript ℝ E^{\otimes}=\bigl{(}E_{t}\bigr{)}_{t\in\mathbb{R}_{+}} of Hilbert spaces that factor as E s + t ≅ E s ⊗ E t subscript 𝐸 𝑠 𝑡 tensor-product subscript 𝐸 𝑠 subscript 𝐸 𝑡 E_{s+t}~{}\cong~{}E_{s}\otimes E_{t} by means of an associative bilinear multiplication E s × E t ∋ ( x s , y t ) ↦ x s ​ y t ∈ E s + t contains subscript 𝐸 𝑠 subscript 𝐸 𝑡 subscript 𝑥 𝑠 subscript 𝑦 𝑡 maps-to subscript 𝑥 𝑠 subscript 𝑦 𝑡 subscript 𝐸 𝑠 𝑡 E_{s}\times E_{t}\ni(x_{s},y_{t})\mapsto x_{s}y_{t}\in E_{s+t} . (Depending on the application, there are also technical conditions about continuity or measurability of sections. We speak about this later on.) The definition of such product systems is due to Arveson Arv89a . It is motivated by Arvesons construction that associates with every E 0 subscript 𝐸 0 E_{0} –semigroup (a semigroup of normal unital endomorphisms) of the ℬ ​ ( H ) ℬ 𝐻 \mathscr{B}(H) of all adjointable operators on a Hilbert space H 𝐻 H a product system E A ⊗ superscript superscript 𝐸 𝐴 tensor-product {E^{A}}^{\otimes} . If H 𝐻 H is infinite-dimensional and separable, then the product system determines the E 0 subscript 𝐸 0 E_{0} –semigroup up to cocycle conjugacy . In a series of four articles Arv89a , Arv90a , Arv89b , Arv90b Arveson showed the fundamental theorem , namely, that every product system of Hilbert spaces is the one associated with a suitable E 0 subscript 𝐸 0 E_{0} –semigroup. Thus, there is a one-to-one correspondence between product systems (up to isomorphism) and E 0 subscript 𝐸 0 E_{0} –semigroups (up to cocycle conjugacy). In the sequel, we will speak about an Arveson system if we intend a product system of Hilbert spaces. In particular, we will speak about the Arveson system associated with an E 0 subscript 𝐸 0 E_{0} –semigroup on ℬ ​ ( H ) ℬ 𝐻 \mathscr{B}(H) . Meanwhile, product systems of Hilbert bimodules or, more fashonably, correspondences made appearance in many contexts. Bhat and Skeide BS00 constructed a product system of correspondences over a (unital) C ∗ superscript 𝐶 C^{*} –algebra ℬ ℬ \mathcal{B} from a (unital) CP-semigroup on ℬ ℬ \mathcal{B} . (See also the discussion of Muhly and Solel MS02 in Remark 6.6 .) This construction overcomes constructions by Bhat Bha96 and by Arveson Arv96 who construct an Arveson system starting from a CP-semigroup on ℬ ​ ( H ) ℬ 𝐻 \mathscr{B}(H) by, first, dilating in a unique way the CP-semigroup to a minimal E 0 subscript 𝐸 0 E_{0} –semigroup and, then, constructing the Arveson system of that E 0 subscript 𝐸 0 E_{0} –semigroup. The construction of BS00 , instead, is direct and allows, then, to construct the minimal dilation in a transparent way. Only later, Skeide Ske02 , Ske05a , Ske04 associated in several ways with an E 0 subscript 𝐸 0 E_{0} –semigroup a product system. Now the E 0 subscript 𝐸 0 E_{0} –semigroup acts on the algebra ℬ a ​ ( E ) superscript ℬ 𝑎 𝐸 \mathscr{B}^{a}(E) of all adjointable maps on a Hilbert ℬ ℬ \mathcal{B} –module E 𝐸 E . Although historically earlier, the approach to product systems from CP-semigroups (that is, irreversible quantum dynamics) has the disadvantage that not all product systems arise in that way. While one of the latest results (Skeide Ske07 ; still in preparation) asserts that cum grano salis every product system comes from an E 0 subscript 𝐸 0 E_{0} –semigroup (that is, reversible quantum dynamics in a sense we specify later on). So the approach via E 0 subscript 𝐸 0 E_{0} –semigroups allows a more coherent discussion. In this survey we will concentrate on this connection between product systems and E 0 subscript 𝐸 0 E_{0} –semigroups, while we will have no space to discuss also the connections with CP-semigroup and their dilations; see Skeide Ske03b . Also basic classification of product systems must be sacrificed; see Skeide Ske03b , Ske06g . The basic factorization property of the symmetric Fock space Γ ​ ( H 1 ⊕ H 2 ) = Γ ​ ( H 1 ) ⊗ Γ ​ ( H 2 ) Γ direct-sum subscript 𝐻 1 subscript 𝐻 2 tensor-product Γ subscript 𝐻 1 Γ subscript 𝐻 2 \Gamma(H_{1}\oplus H_{2})~{}=~{}\Gamma(H_{1})\otimes\Gamma(H_{2}) ( H 1 subscript 𝐻 1 H_{1} and H 2 subscript 𝐻 2 H_{2} some Hilbert spaces) has drawn attention since a long time. In the form Γ ​ ( L 2 ​ ( r , t , K ) ) = Γ ​ ( L 2 ​ ( r , s , K ) ) ⊗ Γ ​ ( L 2 ​ ( s , t , K ) ) , r ≤ s ≤ t formulae-sequence Γ superscript 𝐿 2 𝑟 𝑡 𝐾 tensor-product Γ superscript 𝐿 2 𝑟 𝑠 𝐾 Γ superscript 𝐿 2 𝑠 𝑡 𝐾 𝑟 𝑠 𝑡 \Gamma(L^{2}(r,t,K))~{}=~{}\Gamma(L^{2}(r,s,K))\otimes\Gamma(L^{2}(s,t,K)),~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}r\leq s\leq t ( ∗ * ) ( K 𝐾 K a Hilbert space) it made appearance in the work of Araki Ara70 and Streater Str69 on current representations of Lie algebras, in the work of Parthasarathy and Schmidt PS72 about Lévy processes (culminating in Schürmanns work Sch93 on quantum Lévy processes) and in quantum stochastic calculus on the symmetric Fock space initiated by Hudson and Parthasarathy HP84 . Let us put E t = Γ ​ ( L 2 ​ ( 0 , t , K ) ) subscript 𝐸 𝑡 Γ superscript 𝐿 2 0 𝑡 𝐾 E_{t}=\Gamma(L^{2}(0,t,K)) . Then, from the beginning, there are two possibilities to use ( ∗ * ‣ 1 ) in order to define an isomorphism E s ⊗ E t ≅ E s + t tensor-product subscript 𝐸 𝑠 subscript 𝐸 𝑡 subscript 𝐸 𝑠 𝑡 E_{s}\otimes E_{t}\cong E_{s+t} , namely, E s ⊗ E t tensor-product subscript 𝐸 𝑠 subscript 𝐸 𝑡 \displaystyle E_{s}\otimes E_{t} ≅ 𝒮 t ​ E s ⊗ E t ≅ E s + t absent tensor-product subscript 𝒮 𝑡 subscript 𝐸 𝑠 subscript 𝐸 𝑡 subscript 𝐸 𝑠 𝑡 \displaystyle~{}\cong~{}\text{\scriptsize$\mathscr{S}$}_{t}E_{s}\otimes E_{t}~{}\cong~{}E_{s+t} and E s ⊗ E t tensor-product subscript 𝐸 𝑠 subscript 𝐸 𝑡 \displaystyle E_{s}\otimes E_{t} ≅ E s ⊗ 𝒮 s ​ E t ≅ E s + t , absent tensor-product subscript 𝐸 𝑠 subscript 𝒮 𝑠 subscript 𝐸 𝑡 subscript 𝐸 𝑠 𝑡 \displaystyle~{}\cong~{}E_{s}\otimes\text{\scriptsize$\mathscr{S}$}_{s}E_{t}~{}\cong~{}E_{s+t}, where 𝒮 t : Γ ​ ( L 2 ​ ( 0 , s , K ) ) → Γ ​ ( L 2 ​ ( t , t + s , K ) ) : subscript 𝒮 𝑡 → Γ superscript 𝐿 2 0 𝑠 𝐾 Γ superscript 𝐿 2 𝑡 𝑡 𝑠 𝐾 \text{\scriptsize$\mathscr{S}$}_{t}\colon\Gamma(L^{2}(0,s,K))\rightarrow\Gamma(L^{2}(t,t+s,K)) is the time shift. If we consider the CCR-flow , that is, the E 0 subscript 𝐸 0 E_{0} –semigroup induced on ℬ ​ ( Γ ​ ( L 2 ​ ( ℝ + , K ) ) ) ℬ Γ superscript 𝐿 2 subscript ℝ 𝐾 \mathscr{B}\bigl{(}\Gamma(L^{2}(\mathbb{R}_{+},K))\bigr{)} by the time shift, then the associated Arveson system is E t subscript 𝐸 𝑡 E_{t} with the second choice of an isomorphism, that is, with the time shift acting on the right factor in E s ⊗ E t tensor-product subscript 𝐸 𝑠 subscript 𝐸 𝑡 E_{s}\otimes E_{t} . However, Bhat discovered a second possibility to associate an Arveson with an E 0 subscript 𝐸 0 E_{0} –semigroup. In the case of the CCR-flow one obtains the same Hilbert spaces E t subscript 𝐸 𝑡 E_{t} but with the first choice of an isomorphism, that is, with the time shift acting on the left factor in E s ⊗ E t tensor-product subscript 𝐸 𝑠 subscript 𝐸 𝑡 E_{s}\otimes E_{t} . More generally, the Bhat system associated with any E 0 subscript 𝐸 0 E_{0} –semigroup shows always to be anti-isomorphic to the associated Arveson system. This ambivalence in the tensor product of Hilbert spaces, where we may switch the order of factors without changing (up to canonical isomorphism) the resulting space, is by far less innocent than it appears at the first sight. Nothing like this is true in the module case for the tensor product of correspondences over ℬ ℬ \mathcal{B} . (It is very well possible that in one order their tensor product is { 0 } 0 \{0\} , while in the other order it is not.) In fact, we will see that the construction of a product system of correspondences over ℬ ℬ \mathcal{B} from an E 0 subscript 𝐸 0 E_{0} –semigroup on ℬ a ​ ( E ) superscript ℬ 𝑎 𝐸 \mathscr{B}^{a}(E) for some Hilbert ℬ ℬ \mathcal{B} –module E 𝐸 E corresponds to the construction of the Bhat system of an E 0 subscript 𝐸 0 E_{0} –semigroup on ℬ ​ ( H ) ℬ 𝐻 \mathscr{B}(H) . Also the construction of product system following the ideas of Arveson is still possible. However, it yields a product system of correspondences over the commutant ℬ ′ superscript ℬ ′ \mathcal{B}^{\prime} of ℬ ℬ \mathcal{B} and works nicely only for von Neumann algebras ℬ ℬ \mathcal{B} . The relation between these two product systems is that one is the commutant of the other. The commutant of a correspondence was introduced in Skeide Ske03a , the conribution to the proceedings of the conference in Mount Holyoke 2002. In the space available we are not able to even scratch the basic classification results for product systems. We refere the reader to the still quite up-to-date survey Skeide Ske03b in the proceedings of the conference in Burg 2001. The classification is based on spatial product systems and their product in Skeide Ske06g (preprint 2001). We fix some notations used throughout, and recall very few basics about Hilbert modules in order to make this survey digestable also for nonexperts in Hilbert modules. Let ℬ ℬ \mathcal{B} be a C ∗ superscript 𝐶 C^{*} –algebra. Recall that a pre-Hilbert ℬ ℬ \mathcal{B} –module is a right ℬ ℬ \mathcal{B} –module E 𝐸 E with a sesquilinear inner product ⟨ ∙ , ∙ ⟩ : E × E → ℬ : ∙ ∙ → 𝐸 𝐸 ℬ \langle\bullet,\bullet\rangle\colon E\times E\rightarrow\mathcal{B} satisfying ⟨ x , x ⟩ ≥ 0 𝑥 𝑥 0 \langle x,x\rangle\geq 0 for all x ∈ E 𝑥 𝐸 x\in E (positivity), ⟨ x , y ​ b ⟩ = ⟨ x , y ⟩ ​ b 𝑥 𝑦 𝑏 𝑥 𝑦 𝑏 \langle x,yb\rangle=\langle x,y\rangle b for all x , y ∈ E ; b ∈ ℬ formulae-sequence 𝑥 𝑦 𝐸 𝑏 ℬ x,y\in E;b\in\mathcal{B} (right linearity), and ⟨ x , x ⟩ = 0 ⟹ x = 0 𝑥 𝑥 0 ⟹ 𝑥 0 \langle x,x\rangle=0~{}\Longrightarrow~{}x=0 (definiteness). If definiteness is missing, then E 𝐸 E is a semi-Hilbert ℬ ℬ \mathcal{B} –module . (Properties like ⟨ x , y ⟩ ∗ = ⟨ y , x ⟩ superscript 𝑥 𝑦 𝑦 𝑥 \langle x,y\rangle^{*}=\langle y,x\rangle and ⟨ x ​ b , y ⟩ = b ∗ ​ ⟨ x , y ⟩ 𝑥 𝑏 𝑦 superscript 𝑏 𝑥 𝑦 \langle xb,y\rangle=b^{*}\langle x,y\rangle are automatic.) The most basic property of the inner product in a semi-Hilbert ℬ ℬ \mathcal{B} –module is the following Cauchy-Schwartz inequality ⟨ x , y ⟩ ​ ⟨ y , x ⟩ ≤ ∥ ⟨ y , y ⟩ ∥ ​ ⟨ x , x ⟩ . 𝑥 𝑦 𝑦 𝑥 delimited-∥∥ 𝑦 𝑦 𝑥 𝑥 \langle x,y\rangle\langle y,x\rangle~{}\leq~{}\left\lVert\langle y,y\rangle\right\rVert\langle x,x\rangle. By Cauchy-Schwartz inequality it is possible to quotient out length-zero elements. By Cauchy-Schwartz inequality ∥ x ∥ := ⟨ x , x ⟩ assign delimited-∥∥ 𝑥 𝑥 𝑥 \left\lVert x\right\rVert:=\sqrt{\langle x,x\rangle} defines a norm on the pre-Hilbert module E 𝐸 E . If E 𝐸 E is complete in that norm, then E 𝐸 E is a Hilbert ℬ ℬ \mathcal{B} –module . By Cauchy-Schwartz inequality the operator norm turns the algebra of bounded adjointable operators ℬ a ​ ( E ) superscript ℬ 𝑎 𝐸 \mathscr{B}^{a}(E) on the pre-Hilbert module E 𝐸 E into a pre– C ∗ superscript 𝐶 C^{*} –algebra. Recall that a map a 𝑎 a on E 𝐸 E is adjointable , if it admits an adjoint a ∗ superscript 𝑎 a^{*} such that ⟨ x , a ​ y ⟩ = ⟨ a ∗ ​ x , y ⟩ 𝑥 𝑎 𝑦 superscript 𝑎 𝑥 𝑦 \langle x,ay\rangle=\langle a^{*}x,y\rangle for all x , y ∈ E 𝑥 𝑦 𝐸 x,y\in E . Every adjointable map is closeable. Therefore, by the closed graph theorem , an adjointable map on a Hilbert module is bounded, automatically. In order to speak about product systems we need the (internal) tensor product, and the tensor product is among bimodules or correspondences . If 𝒜 𝒜 \mathcal{A} is another C ∗ superscript 𝐶 C^{*} –algebra, then a correspondence from 𝒜 𝒜 \mathcal{A} to ℬ ℬ \mathcal{B} (or a 𝒜 𝒜 \mathcal{A} – ℬ ℬ \mathcal{B} –correspondence) is a Hilbert ℬ ℬ \mathcal{B} –module with a nondegenerate representation of 𝒜 𝒜 \mathcal{A} by adjointable operators. If 𝒜 = ℬ 𝒜 ℬ \mathcal{A}=\mathcal{B} , then we speak also of a correspondence over ℬ ℬ \mathcal{B} or of a ℬ ℬ \mathcal{B} –correspondence. 1 1 1 The nondegeneracy condition is crucial in all what follows. For the right action of ℬ ℬ \mathcal{B} on a Hilbert ℬ ℬ \mathcal{B} –module it is automatic. (Exercise: Why?) But, there are left actions that act degenerately. However, in that case we will never say 𝒜 𝒜 \mathcal{A} – ℬ ℬ \mathcal{B} –module, but rather speak of a (possibly degenerate) representation of 𝒜 𝒜 \mathcal{A} . The ( internal ) tensor product of a correspondence E 𝐸 E from 𝒜 𝒜 \mathcal{A} to ℬ ℬ \mathcal{B} and a correspondence F 𝐹 F from ℬ ℬ \mathcal{B} to 𝒞 𝒞 \mathcal{C} is the unique correspondence E ⊙ F direct-product 𝐸 𝐹 E\odot F from 𝒜 𝒜 \mathcal{A} to 𝒞 𝒞 \mathcal{C} that is generated by elementary tensors x ⊙ y direct-product 𝑥 𝑦 x\odot y with inner product ⟨ x ⊙ y , x ′ ⊙ y ′ ⟩ = ⟨ y ′ , ⟨ x , x ′ ⟩ ​ y ′ ⟩ direct-product 𝑥 𝑦 direct-product superscript 𝑥 ′ superscript 𝑦 ′ superscript 𝑦 ′ 𝑥 superscript 𝑥 ′ superscript 𝑦 ′ \langle x\odot y,x^{\prime}\odot y^{\prime}\rangle~{}=~{}\bigl{\langle}y^{\prime},\langle x,x^{\prime}\rangle y^{\prime}\bigr{\rangle} (1.1) and the obvious bimodule operation. Uniqueness is, in the sense of a universal property, up to canonical isomorphism. (In two realizations, simply identify the elementary tensors. For a construction take the vector space tensor product E ⊗ F tensor-product 𝐸 𝐹 E\otimes F , define a semiinner product by ( 1.1 ) and divide by the length-zero elements.) The tensor product applies also if E 𝐸 E is just a Hilbert ℬ ℬ \mathcal{B} –module, as every Hilbert ℬ ℬ \mathcal{B} –module E 𝐸 E may be viewed as a correspondence from ℬ a ​ ( E ) superscript ℬ 𝑎 𝐸 \mathscr{B}^{a}(E) to ℬ ℬ \mathcal{B} . This also shows that E ⊙ F direct-product 𝐸 𝐹 E\odot F carries a canonical nondegenerate left action of a ∈ ℬ a ​ ( E ) 𝑎 superscript ℬ 𝑎 𝐸 a\in\mathscr{B}^{a}(E) which we denote by a ⊙ id F direct-product 𝑎 subscript id 𝐹 a\odot\operatorname{\text{\small$\textsf{id}$}}_{F} or, sometimes, simply by a 𝑎 a , too. (Attention! The unital embedding ℬ a ​ ( E ) → ℬ a ​ ( E ) ⊙ id F ⊂ ℬ a ​ ( E ⊙ F ) → superscript ℬ 𝑎 𝐸 direct-product superscript ℬ 𝑎 𝐸 subscript id 𝐹 superscript ℬ 𝑎 direct-product 𝐸 𝐹 \mathscr{B}^{a}(E)\rightarrow\mathscr{B}^{a}(E)\odot\operatorname{\text{\small$\textsf{id}$}}_{F}\subset\mathscr{B}^{a}(E\odot F) need not be faithful.) By ℬ a , b ​ i ​ l ​ ( F ) superscript ℬ 𝑎 𝑏 𝑖 𝑙 𝐹 \mathscr{B}^{a,bil}(F) we denote the space of those elements a ∈ ℬ a ​ ( F ) 𝑎 superscript ℬ 𝑎 𝐹 a\in\mathscr{B}^{a}(F) that are bilinear , that is, which fulfill a ​ ( b ​ y ) = b ​ ( a ​ y ) 𝑎 𝑏 𝑦 𝑏 𝑎 𝑦 a(by)=b(ay) for all b ∈ ℬ , y ∈ F formulae-sequence 𝑏 ℬ 𝑦 𝐹 b\in\mathcal{B},y\in F . There is an embedding ℬ a , b ​ i ​ l ​ ( F ) → id E ⊙ ℬ a , b ​ i ​ l ​ ( F ) ⊂ ℬ a ​ ( E ⊙ F ) → superscript ℬ 𝑎 𝑏 𝑖 𝑙 𝐹 direct-product subscript id 𝐸 superscript ℬ 𝑎 𝑏 𝑖 𝑙 𝐹 superscript ℬ 𝑎 direct-product 𝐸 𝐹 \mathscr{B}^{a,bil}(F)\rightarrow\operatorname{\text{\small$\textsf{id}$}}_{E}\odot\mathscr{B}^{a,bil}(F)\subset\mathscr{B}^{a}(E\odot F) . If E 𝐸 E is full , that is, if the range ideal ℬ E := span ¯ ⁡ ⟨ E , E ⟩ assign subscript ℬ 𝐸 ¯ span 𝐸 𝐸 \mathcal{B}_{E}:=\operatorname{\overline{\operatorname{\text{\small$\textsf{span}$}}}}\langle E,E\rangle in ℬ ℬ \mathcal{B} is ℬ ℬ \mathcal{B} , then one may show that this embedding is an isomorphism onto the relative commutant of ℬ a ​ ( E ) ⊙ id F direct-product superscript ℬ 𝑎 𝐸 subscript id 𝐹 \mathscr{B}^{a}(E)\odot\operatorname{\text{\small$\textsf{id}$}}_{F} in ℬ a ​ ( E ⊙ F ) superscript ℬ 𝑎 direct-product 𝐸 𝐹 \mathscr{B}^{a}(E\odot F) . If ( v , w ) ↦ v ⋅ w maps-to 𝑣 𝑤 ⋅ 𝑣 𝑤 (v,w)\mapsto v\cdot w is bilinear or sesquilinear operation, then V ​ W 𝑉 𝑊 VW is the set { v ⋅ w : v ∈ V , w ∈ W } conditional-set ⋅ 𝑣 𝑤 formulae-sequence 𝑣 𝑉 𝑤 𝑊 \{v\cdot w\colon v\in V,w\in W\} . We do not adopt the convention that V ​ W = span ⁡ V ​ W 𝑉 𝑊 span 𝑉 𝑊 VW=\operatorname{\text{\small$\textsf{span}$}}VW or even V ​ W = span ¯ ⁡ V ​ W 𝑉 𝑊 ¯ span 𝑉 𝑊 VW=\operatorname{\overline{\operatorname{\text{\small$\textsf{span}$}}}}VW .","role":"user"},{"content":"Abstract: The theory of product systems both of Hilbert spaces (Arveson systems) and product systems of Hilbert modules has reached a status where it seems appropriate to rest a moment and to have a look at what is known so far and what are open problems. However, the attempt to give an approximately complete account in view pages is destined to fail already for Arveson systems since Tsirelson, Powers and Liebscher have discovered their powerful methods to construct large classes of examples. In this survey we concentrate on that part of the theory that works also for Hilbert modules. This does not only help to make a selection among the possible topics, but it also helps to shed some new light on the case of Arveson systems. Often, proofs that work for Hilbert modules also lead to simpler proofs in the case of Hilbert spaces. We put emphasis on those aspects that arise from recent results about commutants of von Neumann correspondences, which, in the case of Hilbert spaces, explain the relation between the Arveson system and the Bhat system associated with an E_0--semigroup on B(H).","role":"assistant"}]

Experimental Review on Lepton Universality and Lepton Flavour Violation tests at the B-factories

In recent years, the \babar and Belle experiments have contributed results on tau lepton physics, which improved the experimental picture of lepton universality and of lepton flavor violation searches. Both experiments rely on “B-factories” operating at a centre-of-mass energy of 10.58 GeV on the Υ ​ ( 4 ​ s ) Υ 4 𝑠 \Upsilon(4s) peak, just above the threshold for producing B 𝐵 B -mesons. \babar operates at the PEP-II complex at SLAC, which collides 9 GeV electrons against 3.1 GeV positrons, and has recorded about 420 fb − 1 superscript fb 1 \,\text{fb}^{-1} of data by May 2007. Belle operates at the KEKB B-factory in Japan, which collides 8 GeV electrons against 3.5 GeV positrons, and has recorded about 710 fb − 1 superscript fb 1 \,\text{fb}^{-1} of data by May 2007. The \babar [ 1 ] and Belle [ 2 ] detectors share several similarities and both include a silicon microvertex detector, a drift chamber, a 1.5 T solenoidal superconducting magnet, an electromagnetic calorimeter based on Cesium Iodide crystals, and a segmented muon detector in the magnet return yoke. The two experiments differ in the particle identification strategy: Belle uses an aerogel threshold Cherenkov detector together with time-of-flight and tracker dE/dx, whereas \babar relies on a ring-imaging Cerenkov detector supplemented by the dE/dx in the trackers. With a total now exceeding 1.1 ab -1 of integrated luminosity and a e + ​ e − → τ + ​ τ − → superscript 𝑒 superscript 𝑒 superscript 𝜏 superscript 𝜏 e^{+}e^{-}\to\tau^{+}\tau^{-} cross-section at 10.58 GeV of 0.919 nb [ 3 ] , B-factories recorded in excess of 10 9 superscript 10 9 10^{9} tau pairs, which allow for improving statistics-limited results, like in particular searches for tau lepton flavor violating decays.

Since 1999, the B-factories collaborations BABAR and Belle have accumulated and studied large samples of tau lepton pairs. The experimental results on Lepton Universality checks and Lepton Flavour Violation searches are reported.

[{"content":"Help me to generate the abstract of a survey paper given the title:*Experimental Review on Lepton Universality and Lepton Flavour Violation tests at the B-factories* and the introduction:In recent years, the \babar and Belle experiments have contributed results on tau lepton physics, which improved the experimental picture of lepton universality and of lepton flavor violation searches. Both experiments rely on B-factories operating at a centre-of-mass energy of 10.58 GeV on the Υ ​ ( 4 ​ s ) Υ 4 𝑠 \Upsilon(4s) peak, just above the threshold for producing B 𝐵 B -mesons. \babar operates at the PEP-II complex at SLAC, which collides 9 GeV electrons against 3.1 GeV positrons, and has recorded about 420 fb − 1 superscript fb 1 \,\text{fb}^{-1} of data by May 2007. Belle operates at the KEKB B-factory in Japan, which collides 8 GeV electrons against 3.5 GeV positrons, and has recorded about 710 fb − 1 superscript fb 1 \,\text{fb}^{-1} of data by May 2007. The \babar 1 and Belle 2 detectors share several similarities and both include a silicon microvertex detector, a drift chamber, a 1.5 T solenoidal superconducting magnet, an electromagnetic calorimeter based on Cesium Iodide crystals, and a segmented muon detector in the magnet return yoke. The two experiments differ in the particle identification strategy: Belle uses an aerogel threshold Cherenkov detector together with time-of-flight and tracker dE/dx, whereas \babar relies on a ring-imaging Cerenkov detector supplemented by the dE/dx in the trackers. With a total now exceeding 1.1 ab -1 of integrated luminosity and a e + ​ e − → τ + ​ τ − → superscript 𝑒 superscript 𝑒 superscript 𝜏 superscript 𝜏 e^{+}e^{-}\to\tau^{+}\tau^{-} cross-section at 10.58 GeV of 0.919 nb 3 , B-factories recorded in excess of 10 9 superscript 10 9 10^{9} tau pairs, which allow for improving statistics-limited results, like in particular searches for tau lepton flavor violating decays.","role":"user"},{"content":"Abstract: Since 1999, the B-factories collaborations BABAR and Belle have accumulated and studied large samples of tau lepton pairs. The experimental results on Lepton Universality checks and Lepton Flavour Violation searches are reported.","role":"assistant"}]

Modeling time in computing: A taxonomy and a comparative survey

In many fields of science and engineering, the term dynamics is intrinsically bound to a notion of time. In fact, in classical physics a mathematical model of a dynamical system most often consists of a set of equations that state a relation between a time variable and other quantities characterizing the system, often referred to as system state . In the theory of computation, conversely, the notion of time does not always play a major role. At the root of the theory, a problem is formalized as a function from some input domain to an output range. An algorithm is a process aimed at computing the value of the function; in this process, dynamic aspects are usually abstracted away, since the only concern is the result produced. Timing aspects, however, are quite relevant in computing too, for many reasons; let us recall some of them by adopting a somewhat historical perspective. • First, hardware design leads down to electronic devices where the physical world of circuits comes back into play, for instance when the designer must verify that the sequence of logical gate switches that is necessary to execute an instruction can be completed within a clock’s tick. The time models adopted here are borrowed from physics and electronics, and range from differential equations on continuous time for modeling devices and circuits, to discrete time (coupled with discrete mathematics) for describing logical gates and digital circuits. • When the level of description changes from hardware to software, physical time is progressively disregarded in favor of more “coarse-grained” views of time, where a time unit represents a computational step, possibly in a high-level programming language; or it is even completely abstracted away when adopting a purely functional view of software, as a mapping from some input to the computed output. In this framework, computational complexity theory was developed as a natural complement of computability theory: it was soon apparent that knowing an algorithm to solve a problem is not enough if the execution of such an algorithm takes an unaffordable amount of time. As a consequence, models of abstract machines have been developed or refined so as to measure the time needed for their operations. Then, such an abstract notion of time measure (typically the number of elementary computation steps) could be mapped easily to physical time. • The advent of parallel processing mandated a further investigation of timing issues in the theory of computing. To coordinate appropriately the various concurrent activities, in fact, it is necessary to take into account their temporal evolution. Not by chance the term synchronization derives from the two Greek words σ ​ υ ​ ν 𝜎 𝜐 𝜈 \sigma\upsilon\nu (meaning “together”) and χ ​ ρ ​ o ​ ν ​ o ​ σ 𝜒 𝜌 𝑜 𝜈 𝑜 𝜎 \chi\rho o\nu o\sigma (meaning “time”). • In relatively recent times the advent of novel methods for the design and verification of real-time systems also requires the inclusion of the environment with which the computer interacts in the models under analysis. Therefore the various activities are, in general, not fully synchronized, that is, it is impossible to delay indefinitely one activity while waiting for another one to come alive. Significant classes of systems that possess real-time features are, among others, social organizations (in a broad sense), and distributed and embedded systems. For instance, in a plant control system, the control apparatus must react to the stimuli coming from the plant at a pace that is mandated by the dynamics of the plant. Hence physical time, which was progressively abstracted away, once again plays a prominent role. As a consequence, some type of time modeling is necessary in the theory of computing as well as in any discipline that involves dynamics. Unlike other fields of science and engineering, however, time modeling in computing is far from exhibiting a unitary and comprehensive framework that would be suitable in a general way for most needs of system analysis: this is probably due to the fact that the issue of time modeling arose in different fields, in different circumstances, and was often attacked in a fairly ad hoc manner. In this article we survey various approaches that have been proposed to tackle the issue of time modeling in computing. Rather than pursuing an exhaustive list of formalisms, our main goal is to provide a unifying framework so that the various models can be put in perspective, compared, evaluated, and possibly adapted to the peculiar needs of specific application fields. In this respect, we selected the notations among those that are most prominent in the scientific literature, both as basic research targets and as useful modeling tools in applications. We also aimed at providing suitable “coverage” of the most important features that arise in time modeling. We tried to keep our exposition at a level palatable for the nonspecialist who wishes to gain an overall but not superficial understanding of the issue. Also, although the main goal of time modeling is certainly to use it in the practice of system design, we focus on the conceptual aspects of the problem (what can and cannot be done with a given model; how easy it is to derive properties, etc.) rather than on practical “recipes” of how to apply a formal language in specific projects. The presentation is accompanied by many examples from different domains; most of them are inspired by embedded systems concepts, others, however, show that the same concepts apply as well to a wider class of systems such as biological and social ones. We deliberately excluded from our survey time modeling approaches based on stochastic formalisms. This sector is certainly important and very relevant for several applications, and it has recently received increasing attention from the research community (e.g., [ RKNP04 , DK05 ] ). In fact, most of the formal notations presented in this survey have some variants that include stochastic or probabilistic features. However, including such variants in our presentation would have also required us to present the additional mathematical notions and tools needed to tackle stochastic processes. These are largely different from the notions discussed in the article, which aim at gaining “certainty” (e.g., “the system will not crash under any circumstances”) rather than a “measure of uncertainty” (e.g., “the system will crash with probability 10 − 3 superscript 10 3 10^{-3} ”) as happens with probabilistic approaches. Thus, including stochastic formalisms would have weakened the focus of the article and made it excessively long. The first part of this article introduces an informal reference framework within which the various formalisms can be explained and evaluated. First, Section 2 presents the notion of language, and gives a coarse categorization of formalisms; then, Section 3 proposes a collection of “dimensions” along which the various modeling approaches can be classified. The second part of the article is the actual survey of time modeling formalisms. We do not aim at exhaustiveness; rather, we focus on several relevant formalisms, those that better exemplify the various approaches found in the literature, and analyze them through the dimensions introduced in Section 3 . We complement the exposition, however, with an extensive set of bibliographic references. In the survey, we follow a rather historical ordering: Section 4 summarizes the most traditional ways of taking care of timing aspects in computing, whereas Section 5 is devoted to the more recent proposals, often motivated by the needs of new, critical, real-time applications. Finally, Section 6 contains some concluding remarks.

The increasing relevance of areas such as real-time and embedded systems, pervasive computing, hybrid systems control, and biological and social systems modeling is bringing a growing attention to the temporal aspects of computing, not only in the computer science domain, but also in more traditional fields of engineering. This article surveys various approaches to the formal modeling and analysis of the temporal features of computer-based systems, with a level of detail that is also suitable for nonspecialists. In doing so, it provides a unifying framework, rather than just a comprehensive list of formalisms. The article first lays out some key dimensions along which the various formalisms can be evaluated and compared. Then, a significant sample of formalisms for time modeling in computing are presented and discussed according to these dimensions. The adopted perspective is, to some extent, historical, going from “traditional” models and formalisms to more modern ones.

[{"content":"Help me to generate the abstract of a survey paper given the title:*Modeling time in computing: A taxonomy and a comparative survey* and the introduction:In many fields of science and engineering, the term dynamics is intrinsically bound to a notion of time. In fact, in classical physics a mathematical model of a dynamical system most often consists of a set of equations that state a relation between a time variable and other quantities characterizing the system, often referred to as system state . In the theory of computation, conversely, the notion of time does not always play a major role. At the root of the theory, a problem is formalized as a function from some input domain to an output range. An algorithm is a process aimed at computing the value of the function; in this process, dynamic aspects are usually abstracted away, since the only concern is the result produced. Timing aspects, however, are quite relevant in computing too, for many reasons; let us recall some of them by adopting a somewhat historical perspective. • First, hardware design leads down to electronic devices where the physical world of circuits comes back into play, for instance when the designer must verify that the sequence of logical gate switches that is necessary to execute an instruction can be completed within a clocks tick. The time models adopted here are borrowed from physics and electronics, and range from differential equations on continuous time for modeling devices and circuits, to discrete time (coupled with discrete mathematics) for describing logical gates and digital circuits. • When the level of description changes from hardware to software, physical time is progressively disregarded in favor of more coarse-grained views of time, where a time unit represents a computational step, possibly in a high-level programming language; or it is even completely abstracted away when adopting a purely functional view of software, as a mapping from some input to the computed output. In this framework, computational complexity theory was developed as a natural complement of computability theory: it was soon apparent that knowing an algorithm to solve a problem is not enough if the execution of such an algorithm takes an unaffordable amount of time. As a consequence, models of abstract machines have been developed or refined so as to measure the time needed for their operations. Then, such an abstract notion of time measure (typically the number of elementary computation steps) could be mapped easily to physical time. • The advent of parallel processing mandated a further investigation of timing issues in the theory of computing. To coordinate appropriately the various concurrent activities, in fact, it is necessary to take into account their temporal evolution. Not by chance the term synchronization derives from the two Greek words σ ​ υ ​ ν 𝜎 𝜐 𝜈 \sigma\upsilon\nu (meaning together) and χ ​ ρ ​ o ​ ν ​ o ​ σ 𝜒 𝜌 𝑜 𝜈 𝑜 𝜎 \chi\rho o\nu o\sigma (meaning time). • In relatively recent times the advent of novel methods for the design and verification of real-time systems also requires the inclusion of the environment with which the computer interacts in the models under analysis. Therefore the various activities are, in general, not fully synchronized, that is, it is impossible to delay indefinitely one activity while waiting for another one to come alive. Significant classes of systems that possess real-time features are, among others, social organizations (in a broad sense), and distributed and embedded systems. For instance, in a plant control system, the control apparatus must react to the stimuli coming from the plant at a pace that is mandated by the dynamics of the plant. Hence physical time, which was progressively abstracted away, once again plays a prominent role. As a consequence, some type of time modeling is necessary in the theory of computing as well as in any discipline that involves dynamics. Unlike other fields of science and engineering, however, time modeling in computing is far from exhibiting a unitary and comprehensive framework that would be suitable in a general way for most needs of system analysis: this is probably due to the fact that the issue of time modeling arose in different fields, in different circumstances, and was often attacked in a fairly ad hoc manner. In this article we survey various approaches that have been proposed to tackle the issue of time modeling in computing. Rather than pursuing an exhaustive list of formalisms, our main goal is to provide a unifying framework so that the various models can be put in perspective, compared, evaluated, and possibly adapted to the peculiar needs of specific application fields. In this respect, we selected the notations among those that are most prominent in the scientific literature, both as basic research targets and as useful modeling tools in applications. We also aimed at providing suitable coverage of the most important features that arise in time modeling. We tried to keep our exposition at a level palatable for the nonspecialist who wishes to gain an overall but not superficial understanding of the issue. Also, although the main goal of time modeling is certainly to use it in the practice of system design, we focus on the conceptual aspects of the problem (what can and cannot be done with a given model; how easy it is to derive properties, etc.) rather than on practical recipes of how to apply a formal language in specific projects. The presentation is accompanied by many examples from different domains; most of them are inspired by embedded systems concepts, others, however, show that the same concepts apply as well to a wider class of systems such as biological and social ones. We deliberately excluded from our survey time modeling approaches based on stochastic formalisms. This sector is certainly important and very relevant for several applications, and it has recently received increasing attention from the research community (e.g., RKNP04 , DK05 ). In fact, most of the formal notations presented in this survey have some variants that include stochastic or probabilistic features. However, including such variants in our presentation would have also required us to present the additional mathematical notions and tools needed to tackle stochastic processes. These are largely different from the notions discussed in the article, which aim at gaining certainty (e.g., the system will not crash under any circumstances) rather than a measure of uncertainty (e.g., the system will crash with probability 10 − 3 superscript 10 3 10^{-3} ) as happens with probabilistic approaches. Thus, including stochastic formalisms would have weakened the focus of the article and made it excessively long. The first part of this article introduces an informal reference framework within which the various formalisms can be explained and evaluated. First, Section 2 presents the notion of language, and gives a coarse categorization of formalisms; then, Section 3 proposes a collection of dimensions along which the various modeling approaches can be classified. The second part of the article is the actual survey of time modeling formalisms. We do not aim at exhaustiveness; rather, we focus on several relevant formalisms, those that better exemplify the various approaches found in the literature, and analyze them through the dimensions introduced in Section 3 . We complement the exposition, however, with an extensive set of bibliographic references. In the survey, we follow a rather historical ordering: Section 4 summarizes the most traditional ways of taking care of timing aspects in computing, whereas Section 5 is devoted to the more recent proposals, often motivated by the needs of new, critical, real-time applications. Finally, Section 6 contains some concluding remarks.","role":"user"},{"content":"Abstract: The increasing relevance of areas such as real-time and embedded systems, pervasive computing, hybrid systems control, and biological and social systems modeling is bringing a growing attention to the temporal aspects of computing, not only in the computer science domain, but also in more traditional fields of engineering. This article surveys various approaches to the formal modeling and analysis of the temporal features of computer-based systems, with a level of detail that is also suitable for nonspecialists. In doing so, it provides a unifying framework, rather than just a comprehensive list of formalisms. The article first lays out some key dimensions along which the various formalisms can be evaluated and compared. Then, a significant sample of formalisms for time modeling in computing are presented and discussed according to these dimensions. The adopted perspective is, to some extent, historical, going from traditional models and formalisms to more modern ones.","role":"assistant"}]

Target mass corrections

The scattering of electrons on hadronic targets has historically played an essential role in our understanding of the proton as a composite particle made up of partons: quarks and gluons [ 1 , 2 , 3 , 4 ] . Presently, data from electron and neutrino scattering at large momentum transfers, that is, deeply inelastic scattering (DIS), are used to determine the parton distribution functions (PDFs) which characterize the substructure of hadrons [ 5 , 6 , 7 , 8 , 9 , 10 ] . At lower energies, the resonant components of hadronic structure, and the duality between hadronic and partonic descriptions of interactions, continue to be explored [ 11 , 12 ] . As the precision of the recent lepton–hadron scattering data has improved, it is vital for the theoretical analysis to keep pace. For example, the calculation of the Wilson coefficients has progressed to encompass next-to-leading order (NLO) quantum chromodynamics (QCD) and beyond [ 13 , 14 , 15 , 16 , 17 , 18 , 19 , 20 , 21 , 22 ] . It is important, therefore, to consider all sources of corrections which may contribute at a comparable magnitude, such as electroweak radiative corrections [ 23 , 24 ] , quark mass effects [ 25 , 26 , 27 , 28 , 29 , 30 , 31 , 32 ] , and target mass corrections [ 33 , 34 , 35 , 36 , 37 , 38 , 39 , 40 ] . In this review, we will focus on the problem of target mass corrections (TMCs), which formally are subleading 1 / Q 2 1 superscript 𝑄 2 1/Q^{2} corrections to leading twist structure functions, where Q 2 superscript 𝑄 2 Q^{2} is the squared four-momentum transfer to the hadron. Understanding TMCs is important for several reasons. Their effects are most pronounced at large x 𝑥 x and moderate Q 2 superscript 𝑄 2 Q^{2} , which coincides with the region where parton distribution functions (PDFs) are not very well determined. A reliable extraction of PDFs here therefore demands an accurate description of the TMCs. Furthermore, a reliable interpretation of data on multiparton correlations at low momentum transfer depends on the proper accounting of TMCs. While target mass corrections have a long history, implementing these has not been entirely straightforward, as there exist a number of conventions, prescriptions, and potential scheme choices which can lead to differences in the final numerical results. The target mass corrections to electroweak structure functions were first determined by Georgi and Politzer in 1976 [ 34 ] within the operator product expansion (OPE) at the leading order of QCD. In the same year Barbieri et al. [ 35 , 36 ] rederived the mass corrections to scaling in DIS, including effects arising from non-zero quark masses. These same corrections were later derived from a parton model approach by Ellis, Furmanski and Petronzio [ 37 , 38 ] . Beyond leading order, the NLO QCD corrections to the target mass corrected structure functions were derived by De Rújula, Georgi and Politzer [ 39 ] . Recently, Kretzer and Reno [ 41 , 42 ] reevaluated the TMCs for charged current (CC) and neutral current (NC) μ 𝜇 \mu - and τ 𝜏 \tau - neutrino DIS, including NLO QCD corrections. There is a number of theoretical ingredients necessary to derive the TMCs to hadronic structure functions in lepton–hadron scattering in the context of the OPE, in order to relate them to the quark-parton model (QPM). The OPE method makes use of basic fundamental symmetries to relate the cross section to a reduced matrix element; as such, the OPE takes the hadron mass fully into account. In order to relate the reduced matrix elements to quantities which can be computed in the QPM, one then inherits the associated limitations. For example, the QPM describes the interaction as involving the scattering from a single, free parton (the leading twist contribution, see Sec. 2 ). Multi-parton correlations, formally higher twist, are discarded. Additionally, the QPM also imposes collinear kinematics on the parton involved in the interaction. The potential for the parton momentum to have a non-zero component transverse to the hadron momentum vector is neglected. This effect can omit mass contributions and introduce ambiguities if not addressed carefully. When using the QPM approach, the target mass corrections are taken into account in the following places: • in relating the parton fraction of the hadron’s light-cone momentum (called the Nachtmann variable) to the Bjorken scaling variable x = Q 2 / 2 ​ M ​ ν 𝑥 superscript 𝑄 2 2 𝑀 𝜈 x=Q^{2}/2M\nu , where M 𝑀 M is the hadron mass and ν 𝜈 \nu the energy transfer; • in the mixing between the partonic and hadronic structure functions; • in a collinear expansion, where the TMCs appear from the p T subscript 𝑝 𝑇 p_{T} effects . In the limit of negligible target mass relative to Q 2 superscript 𝑄 2 Q^{2} , the Nachtmann variable reduces to Bjorken- x 𝑥 x . The QPM approach including TMCs has further limitations in the limit as x → 1 → 𝑥 1 x\to 1 . A “threshold problem” arises when trying to enforce that the structure functions vanish in kinematically forbidden regions. In the next section, we review structure functions and the OPE approach. We show the OPE results for the structure functions. Section 3 discusses the relation of the OPE to the parton model. We illustrate the formalism with the example of neutrino charged current scattering in Section 4. Section 5 describes the x → 1 → 𝑥 1 x\rightarrow 1 problem and recent attempts to resolve this theoretical issue. Finally, in Section 6, we make some numerical comparisons using structure functions with and without target mass corrections. Our conclusions are summarized in Section 7. Three appendices detail the notation used in this review, the inclusion of charm mass corrections in neutrino charged current scattering, and a comparison of notation and results with Refs. [ 34 , 35 , 40 ] and [ 42 ] .

With recent advances in the precision of inclusive lepton–nuclear scattering experiments, it has become apparent that comparable improvements are needed in the accuracy of the theoretical analysis tools. In particular, when extracting parton distribution functions in the large-x region, it is crucial to correct the data for effects associated with the nonzero mass of the target. We present here a comprehensive review of these target mass corrections (TMC) to structure functions data, summarizing the relevant formulas for TMCs in electromagnetic and weak processes. We include a full analysis of both hadronic and partonic masses, and trace how these effects appear in the operator product expansion and the factorized parton model formalism, as well as their limitations when applied to data in the x → 1 limit. We evaluate the numerical effects of TMCs on various structure functions, and compare fits to data with and without these corrections.

[{"content":"Help me to generate the abstract of a survey paper given the title:*Target mass corrections* and the introduction:The scattering of electrons on hadronic targets has historically played an essential role in our understanding of the proton as a composite particle made up of partons: quarks and gluons 1 , 2 , 3 , 4 . Presently, data from electron and neutrino scattering at large momentum transfers, that is, deeply inelastic scattering (DIS), are used to determine the parton distribution functions (PDFs) which characterize the substructure of hadrons 5 , 6 , 7 , 8 , 9 , 10 . At lower energies, the resonant components of hadronic structure, and the duality between hadronic and partonic descriptions of interactions, continue to be explored 11 , 12 . As the precision of the recent lepton–hadron scattering data has improved, it is vital for the theoretical analysis to keep pace. For example, the calculation of the Wilson coefficients has progressed to encompass next-to-leading order (NLO) quantum chromodynamics (QCD) and beyond 13 , 14 , 15 , 16 , 17 , 18 , 19 , 20 , 21 , 22 . It is important, therefore, to consider all sources of corrections which may contribute at a comparable magnitude, such as electroweak radiative corrections 23 , 24 , quark mass effects 25 , 26 , 27 , 28 , 29 , 30 , 31 , 32 , and target mass corrections 33 , 34 , 35 , 36 , 37 , 38 , 39 , 40 . In this review, we will focus on the problem of target mass corrections (TMCs), which formally are subleading 1 / Q 2 1 superscript 𝑄 2 1/Q^{2} corrections to leading twist structure functions, where Q 2 superscript 𝑄 2 Q^{2} is the squared four-momentum transfer to the hadron. Understanding TMCs is important for several reasons. Their effects are most pronounced at large x 𝑥 x and moderate Q 2 superscript 𝑄 2 Q^{2} , which coincides with the region where parton distribution functions (PDFs) are not very well determined. A reliable extraction of PDFs here therefore demands an accurate description of the TMCs. Furthermore, a reliable interpretation of data on multiparton correlations at low momentum transfer depends on the proper accounting of TMCs. While target mass corrections have a long history, implementing these has not been entirely straightforward, as there exist a number of conventions, prescriptions, and potential scheme choices which can lead to differences in the final numerical results. The target mass corrections to electroweak structure functions were first determined by Georgi and Politzer in 1976 34 within the operator product expansion (OPE) at the leading order of QCD. In the same year Barbieri et al. 35 , 36 rederived the mass corrections to scaling in DIS, including effects arising from non-zero quark masses. These same corrections were later derived from a parton model approach by Ellis, Furmanski and Petronzio 37 , 38 . Beyond leading order, the NLO QCD corrections to the target mass corrected structure functions were derived by De Rújula, Georgi and Politzer 39 . Recently, Kretzer and Reno 41 , 42 reevaluated the TMCs for charged current (CC) and neutral current (NC) μ 𝜇 \mu - and τ 𝜏 \tau - neutrino DIS, including NLO QCD corrections. There is a number of theoretical ingredients necessary to derive the TMCs to hadronic structure functions in lepton–hadron scattering in the context of the OPE, in order to relate them to the quark-parton model (QPM). The OPE method makes use of basic fundamental symmetries to relate the cross section to a reduced matrix element; as such, the OPE takes the hadron mass fully into account. In order to relate the reduced matrix elements to quantities which can be computed in the QPM, one then inherits the associated limitations. For example, the QPM describes the interaction as involving the scattering from a single, free parton (the leading twist contribution, see Sec. 2 ). Multi-parton correlations, formally higher twist, are discarded. Additionally, the QPM also imposes collinear kinematics on the parton involved in the interaction. The potential for the parton momentum to have a non-zero component transverse to the hadron momentum vector is neglected. This effect can omit mass contributions and introduce ambiguities if not addressed carefully. When using the QPM approach, the target mass corrections are taken into account in the following places: • in relating the parton fraction of the hadrons light-cone momentum (called the Nachtmann variable) to the Bjorken scaling variable x = Q 2 / 2 ​ M ​ ν 𝑥 superscript 𝑄 2 2 𝑀 𝜈 x=Q^{2}/2M\nu , where M 𝑀 M is the hadron mass and ν 𝜈 \nu the energy transfer; • in the mixing between the partonic and hadronic structure functions; • in a collinear expansion, where the TMCs appear from the p T subscript 𝑝 𝑇 p_{T} effects . In the limit of negligible target mass relative to Q 2 superscript 𝑄 2 Q^{2} , the Nachtmann variable reduces to Bjorken- x 𝑥 x . The QPM approach including TMCs has further limitations in the limit as x → 1 → 𝑥 1 x\to 1 . A threshold problem arises when trying to enforce that the structure functions vanish in kinematically forbidden regions. In the next section, we review structure functions and the OPE approach. We show the OPE results for the structure functions. Section 3 discusses the relation of the OPE to the parton model. We illustrate the formalism with the example of neutrino charged current scattering in Section 4. Section 5 describes the x → 1 → 𝑥 1 x\rightarrow 1 problem and recent attempts to resolve this theoretical issue. Finally, in Section 6, we make some numerical comparisons using structure functions with and without target mass corrections. Our conclusions are summarized in Section 7. Three appendices detail the notation used in this review, the inclusion of charm mass corrections in neutrino charged current scattering, and a comparison of notation and results with Refs. 34 , 35 , 40 and 42 .","role":"user"},{"content":"Abstract: With recent advances in the precision of inclusive lepton–nuclear scattering experiments, it has become apparent that comparable improvements are needed in the accuracy of the theoretical analysis tools. In particular, when extracting parton distribution functions in the large-x region, it is crucial to correct the data for effects associated with the nonzero mass of the target. We present here a comprehensive review of these target mass corrections (TMC) to structure functions data, summarizing the relevant formulas for TMCs in electromagnetic and weak processes. We include a full analysis of both hadronic and partonic masses, and trace how these effects appear in the operator product expansion and the factorized parton model formalism, as well as their limitations when applied to data in the x → 1 limit. We evaluate the numerical effects of TMCs on various structure functions, and compare fits to data with and without these corrections.","role":"assistant"}]

Theoretical Review on CP Violation in Rare B decays

I Importance of CP violation As pointed out by Sakharov Sakharov:1967dj in the 60’s, one of the necessary conditions for the observed Universe is CP violation in physical processes. In the standard model (SM) of particle physics, the only source of CP violation is given by the so-called Kobayashi-Maskawa mechanism Kobayashi:1973fv in the quark sector. In weak transitions, the up-type quarks and the down-type quarks are coupled through the 3 × 3 3 3 3\times 3 Cabibbo-Kobayashi-Maskawa (CKM) matrix Cabibbo:1963yz ; Kobayashi:1973fv , which contains a CP-violating phase. Therefore, studying and understanding the origin of such a phase in the SM is crucial to particle physics and cosmology Hou:2008xd . More importantly, it is possible to shed some light on new physics in such studies. Due to its hierarchical structure, the CKM matrix has one useful unitarity condition, which connects its first and third column: V u ​ d ​ V u ​ b ∗ + V c ​ d ​ V c ​ b ∗ + V t ​ d ​ V t ​ b ∗ = 0 . subscript 𝑉 𝑢 𝑑 superscript subscript 𝑉 𝑢 𝑏 subscript 𝑉 𝑐 𝑑 superscript subscript 𝑉 𝑐 𝑏 subscript 𝑉 𝑡 𝑑 superscript subscript 𝑉 𝑡 𝑏 0 \displaystyle V_{ud}V_{ub}^{*}+V_{cd}V_{cb}^{*}+V_{td}V_{tb}^{*}=0~{}. (1) This relation has a special status because it renders on a complex plane a triangle that has all sides about the same size (and so are the angles). An important program of current B 𝐵 B -factories is to use various processes to overconstrain this unitarity triangle (UT). Through such an exercise, we hope not only to measure precisely the sides and angles of the UT but also to obtain hints of physics beyond the SM that provides additional CP-violating sources. The indirect CP violation in the B 𝐵 B system has been first established in the charmonium modes in 2001, and is now measured at a precision better than 5 % percent 5 5\% . Soon after that first measurement, the direct CP violation in the B 𝐵 B system has also been observed in the B 0 → K + ​ π − → superscript 𝐵 0 superscript 𝐾 superscript 𝜋 B^{0}\to K^{+}\pi^{-} decay mode in 2004. This is a result of the interference between color-allowed tree and QCD penguin amplitudes. In the following, we concentrate exclusively on the direct CP asymmetries in rare B 𝐵 B decays.

We discuss several issues related to direct CP violation in rare B meson decays. We review the use of CP asymmetries in extracting information of strong and weak phases, how the experimental data fit into the overall

[{"content":"Help me to generate the abstract of a survey paper given the title:*Theoretical Review on CP Violation in Rare B decays* and the introduction:I Importance of CP violation As pointed out by Sakharov Sakharov:1967dj in the 60s, one of the necessary conditions for the observed Universe is CP violation in physical processes. In the standard model (SM) of particle physics, the only source of CP violation is given by the so-called Kobayashi-Maskawa mechanism Kobayashi:1973fv in the quark sector. In weak transitions, the up-type quarks and the down-type quarks are coupled through the 3 × 3 3 3 3\times 3 Cabibbo-Kobayashi-Maskawa (CKM) matrix Cabibbo:1963yz ; Kobayashi:1973fv , which contains a CP-violating phase. Therefore, studying and understanding the origin of such a phase in the SM is crucial to particle physics and cosmology Hou:2008xd . More importantly, it is possible to shed some light on new physics in such studies. Due to its hierarchical structure, the CKM matrix has one useful unitarity condition, which connects its first and third column: V u ​ d ​ V u ​ b ∗ + V c ​ d ​ V c ​ b ∗ + V t ​ d ​ V t ​ b ∗ = 0 . subscript 𝑉 𝑢 𝑑 superscript subscript 𝑉 𝑢 𝑏 subscript 𝑉 𝑐 𝑑 superscript subscript 𝑉 𝑐 𝑏 subscript 𝑉 𝑡 𝑑 superscript subscript 𝑉 𝑡 𝑏 0 \displaystyle V_{ud}V_{ub}^{*}+V_{cd}V_{cb}^{*}+V_{td}V_{tb}^{*}=0~{}. (1) This relation has a special status because it renders on a complex plane a triangle that has all sides about the same size (and so are the angles). An important program of current B 𝐵 B -factories is to use various processes to overconstrain this unitarity triangle (UT). Through such an exercise, we hope not only to measure precisely the sides and angles of the UT but also to obtain hints of physics beyond the SM that provides additional CP-violating sources. The indirect CP violation in the B 𝐵 B system has been first established in the charmonium modes in 2001, and is now measured at a precision better than 5 % percent 5 5\% . Soon after that first measurement, the direct CP violation in the B 𝐵 B system has also been observed in the B 0 → K + ​ π − → superscript 𝐵 0 superscript 𝐾 superscript 𝜋 B^{0}\to K^{+}\pi^{-} decay mode in 2004. This is a result of the interference between color-allowed tree and QCD penguin amplitudes. In the following, we concentrate exclusively on the direct CP asymmetries in rare B 𝐵 B decays.","role":"user"},{"content":"Abstract: We discuss several issues related to direct CP violation in rare B meson decays. We review the use of CP asymmetries in extracting information of strong and weak phases, how the experimental data fit into the overall","role":"assistant"}]

Adaptive coevolutionary networks: a review

I Introduction Complex networks are ubiquitous in nature. They occur in a large variety of real-world systems ranging from ecology and epidemiology to neuroscience, socio-economics and computer science Albert and Barabasi ( 2002 ); Dorogovtsev and Mendes ( 2003 ); Newman ( 2003 ); Newman et al. ( 2006 ) . While physics has for a long time been concerned with well-mixed systems, lattices and spatially explicit models, the investigation of complex networks has in the recent years received a rapidly increasing amount of attention. In particular, the need to protect or optimize natural networks as well as the goal to create robust and efficient technical nets, prove to be strong incentives for research. Figure 1: (Color Online) In an adaptive network the evolution of the topology depends on the dynamics of the nodes. Thus a feedback loop is created in which a dynamical exchange of information is possible A network consists of a number of network nodes connected by links (see also Box 1). The specific pattern of connections defines the network’s topology . For many applications it is not necessary to capture the topology of a given real world network exactly in a model. Rather, the processes of interest depend in many cases only on certain topological properties Costa et al. ( 2007 ) . The majority of recent studies revolve around two key questions corresponding to two distinct lines of research: what are the values of important topological properties of a network that is evolving in time and, secondly, how does the functioning of the network depend on these properties? The first line of research is concerned with the dynamics of networks . Here, the topology of the network itself is regarded as a dynamical system. It changes in time according to specific, often local, rules. Investigations in this area have revealed that certain evolution rules give rise to peculiar network topologies with special properties. Notable examples include the formation of small world Watts and Strogatz ( 1998 ) and scale free networks Price ( 1965 ); Barabàsi and Albert ( 1999 ) . The second major line of network research focuses on the dynamics on networks . Here, each node of the network represents a dynamical system. The individual systems are coupled according to the network topology. Thus, the topology of the network remains static while the states of the nodes change dynamically. Important processes that are studied within this framework include synchronization of the individual dynamical systems Barahona and Pecora ( 2002 ) and contact processes, such as opinion formation and epidemic spreading May and Lloyd ( 2001 ); Pastor-Satorras and Vespignani ( 2001 ); Kuperman and Abramson ( 2001 ); Newman ( 2002 ); Boguñá et al. ( 2003 ) . These studies have made it clear that certain topological properties have a strong impact on the dynamics. For instance it was shown that vaccination of a fraction of the nodes cannot stop epidemics on a scale free network Pastor-Satorras and Vespignani ( 2001 ); May and Lloyd ( 2001 ) . Until recently, the two lines of network research were pursued almost independently in the physical literature. While there was certainly a strong interaction and cross-fertilization, a given model would either describe the dynamics of a certain network or the dynamics on a certain network. Nevertheless, it is clear that in most real world networks the evolution of the topology is invariably linked to the state of the network and vice versa. Consider for instance a road network. The topology of the network, that is the pattern of roads, influences the dynamic state, i.e. the flow and density of traffic. But, if traffic congestions are common on a given road, it is likely that new roads will be build in order to decrease the load on the congested one. In this way a feedback loop between the state and topology of the network is formed. This feedback loop can give rise to a complicated mutual interaction between a time varying network topology and the nodes’ dynamics. Networks which exhibit such a feedback loop are called coevolutionary or adaptive networks (s. also Fig. 1 ). Based on the successes of the two lines of research mentioned earlier, it is the next logical step to bring these strands back together and to investigate the dynamics of adaptive networks. Indeed, a number of papers on the dynamics of adaptive networks have recently appeared. Since adaptive networks occur over a large variety of scientific disciplines they are currently investigated from many different directions. While the present studies can only be considered as first steps toward a theory of adaptive networks, they already crystallise certain general insights. Despite the thematic diversification, the reported results, considered together, show that certain dynamical phenomena repeatedly appear in adaptive networks: the formation of complex topologies, robust dynamical self-organization, spontaneous emergence of different classes of nodes from an initially inhomogeneous population, and complex mutual dynamics in state and topology. In the following we argue that the mechanisms that give rise to these phenomena arise from the dynamical interplay between state and topology. They are therefore genuine adaptive network effects that cannot be observed in non-adaptive networks. In this review it is our aim to point out that many recent findings reported mainly in the physical literature describe generic dynamical properties of adaptive networks. These findings are therefore of potential importance in many fields of research. In particular we aim to make recent insights accessible to researchers in the biological sciences, where adaptive networks frequently appear and have been studied implicitly for a long time. We start in Sec. II by discussing several examples that illustrate the abundance of adaptive networks in the real world and in applied models. Thereafter we proceed to the core of the review. In Sec. III adaptive Boolean networks are studied to explain how adaptive networks can self-organize towards dynamical criticality. Other, less obvious, but no less intriguing forms of the self-organization are discussed in Sec. IV while we review investigations of adaptive coupled map lattices. In particular, it is shown that a spontaneous ‘division of labour’ can be observed in which the nodes differentiate into separate classes, which play distinct functional roles in the network. Further examples of this functional differentiation of nodes are discussed in Sec. V , which focuses on games on adaptive networks. Finally, in Sec. VI we discuss the dynamics of the spreading of opinions and diseases on social networks, which shows that the adaptive networks can exhibit complex dynamics and can give rise to new phase transitions. We conclude this review in Sec. VII with a summary, synthesis and outlook.

Adaptive networks appear in many biological applications. They combine topological evolution of the network with dynamics in the network nodes. Recently, the dynamics of adaptive networks has been investigated in a number of parallel studies from different fields, ranging from genomics to game theory. Here we review these recent developments and show that they can be viewed from a unique angle. We demonstrate that all these studies are characterized by common themes, most prominently: complex dynamics and robust topological self-organization based on simple local rules.

[{"content":"Help me to generate the abstract of a survey paper given the title:*Adaptive coevolutionary networks: a review* and the introduction:I Introduction Complex networks are ubiquitous in nature. They occur in a large variety of real-world systems ranging from ecology and epidemiology to neuroscience, socio-economics and computer science Albert and Barabasi ( 2002 ); Dorogovtsev and Mendes ( 2003 ); Newman ( 2003 ); Newman et al. ( 2006 ) . While physics has for a long time been concerned with well-mixed systems, lattices and spatially explicit models, the investigation of complex networks has in the recent years received a rapidly increasing amount of attention. In particular, the need to protect or optimize natural networks as well as the goal to create robust and efficient technical nets, prove to be strong incentives for research. Figure 1: (Color Online) In an adaptive network the evolution of the topology depends on the dynamics of the nodes. Thus a feedback loop is created in which a dynamical exchange of information is possible A network consists of a number of network nodes connected by links (see also Box 1). The specific pattern of connections defines the networks topology . For many applications it is not necessary to capture the topology of a given real world network exactly in a model. Rather, the processes of interest depend in many cases only on certain topological properties Costa et al. ( 2007 ) . The majority of recent studies revolve around two key questions corresponding to two distinct lines of research: what are the values of important topological properties of a network that is evolving in time and, secondly, how does the functioning of the network depend on these properties? The first line of research is concerned with the dynamics of networks . Here, the topology of the network itself is regarded as a dynamical system. It changes in time according to specific, often local, rules. Investigations in this area have revealed that certain evolution rules give rise to peculiar network topologies with special properties. Notable examples include the formation of small world Watts and Strogatz ( 1998 ) and scale free networks Price ( 1965 ); Barabàsi and Albert ( 1999 ) . The second major line of network research focuses on the dynamics on networks . Here, each node of the network represents a dynamical system. The individual systems are coupled according to the network topology. Thus, the topology of the network remains static while the states of the nodes change dynamically. Important processes that are studied within this framework include synchronization of the individual dynamical systems Barahona and Pecora ( 2002 ) and contact processes, such as opinion formation and epidemic spreading May and Lloyd ( 2001 ); Pastor-Satorras and Vespignani ( 2001 ); Kuperman and Abramson ( 2001 ); Newman ( 2002 ); Boguñá et al. ( 2003 ) . These studies have made it clear that certain topological properties have a strong impact on the dynamics. For instance it was shown that vaccination of a fraction of the nodes cannot stop epidemics on a scale free network Pastor-Satorras and Vespignani ( 2001 ); May and Lloyd ( 2001 ) . Until recently, the two lines of network research were pursued almost independently in the physical literature. While there was certainly a strong interaction and cross-fertilization, a given model would either describe the dynamics of a certain network or the dynamics on a certain network. Nevertheless, it is clear that in most real world networks the evolution of the topology is invariably linked to the state of the network and vice versa. Consider for instance a road network. The topology of the network, that is the pattern of roads, influences the dynamic state, i.e. the flow and density of traffic. But, if traffic congestions are common on a given road, it is likely that new roads will be build in order to decrease the load on the congested one. In this way a feedback loop between the state and topology of the network is formed. This feedback loop can give rise to a complicated mutual interaction between a time varying network topology and the nodes dynamics. Networks which exhibit such a feedback loop are called coevolutionary or adaptive networks (s. also Fig. 1 ). Based on the successes of the two lines of research mentioned earlier, it is the next logical step to bring these strands back together and to investigate the dynamics of adaptive networks. Indeed, a number of papers on the dynamics of adaptive networks have recently appeared. Since adaptive networks occur over a large variety of scientific disciplines they are currently investigated from many different directions. While the present studies can only be considered as first steps toward a theory of adaptive networks, they already crystallise certain general insights. Despite the thematic diversification, the reported results, considered together, show that certain dynamical phenomena repeatedly appear in adaptive networks: the formation of complex topologies, robust dynamical self-organization, spontaneous emergence of different classes of nodes from an initially inhomogeneous population, and complex mutual dynamics in state and topology. In the following we argue that the mechanisms that give rise to these phenomena arise from the dynamical interplay between state and topology. They are therefore genuine adaptive network effects that cannot be observed in non-adaptive networks. In this review it is our aim to point out that many recent findings reported mainly in the physical literature describe generic dynamical properties of adaptive networks. These findings are therefore of potential importance in many fields of research. In particular we aim to make recent insights accessible to researchers in the biological sciences, where adaptive networks frequently appear and have been studied implicitly for a long time. We start in Sec. II by discussing several examples that illustrate the abundance of adaptive networks in the real world and in applied models. Thereafter we proceed to the core of the review. In Sec. III adaptive Boolean networks are studied to explain how adaptive networks can self-organize towards dynamical criticality. Other, less obvious, but no less intriguing forms of the self-organization are discussed in Sec. IV while we review investigations of adaptive coupled map lattices. In particular, it is shown that a spontaneous division of labour can be observed in which the nodes differentiate into separate classes, which play distinct functional roles in the network. Further examples of this functional differentiation of nodes are discussed in Sec. V , which focuses on games on adaptive networks. Finally, in Sec. VI we discuss the dynamics of the spreading of opinions and diseases on social networks, which shows that the adaptive networks can exhibit complex dynamics and can give rise to new phase transitions. We conclude this review in Sec. VII with a summary, synthesis and outlook.","role":"user"},{"content":"Abstract: Adaptive networks appear in many biological applications. They combine topological evolution of the network with dynamics in the network nodes. Recently, the dynamics of adaptive networks has been investigated in a number of parallel studies from different fields, ranging from genomics to game theory. Here we review these recent developments and show that they can be viewed from a unique angle. We demonstrate that all these studies are characterized by common themes, most prominently: complex dynamics and robust topological self-organization based on simple local rules.","role":"assistant"}]

Area laws for the entanglement entropy - a review

I Introduction In classical physics concepts of entropy quantify the extent to which we are uncertain about the exact state of a physical system at hand or, in other words, the amount of information that is lacking to identify the microstate of a system from all possibilities compatible with the macrostate of the system. If we are not quite sure what microstate of a system to expect, notions of entropy will reflect this lack of knowledge. Randomness, after all, is always and necessarily related to ignorance about the state. Figure 1: A lattice L 𝐿 L with a distinguished set I ⊂ L 𝐼 𝐿 I\subset L (shaded area). Vertices depict the boundary ∂ I 𝐼 \partial I of I 𝐼 I with surface area s ​ ( I ) = | ∂ I | 𝑠 𝐼 𝐼 s(I)=|\partial I| . In quantum mechanics positive entropies may arise even without an objective lack of information. To see this, let us consider a quantum lattice systems (see e.g., Fig. 1) as an example for a quantum many-body system where each of the vertices i 𝑖 i of the lattice L 𝐿 L is associated with an individual quantum system. This quantum many-body system is thought to be in its non-degenerate pure ground state ρ = | ψ ⟩ ​ ⟨ ψ | 𝜌 ket 𝜓 bra 𝜓 \rho=|\psi\rangle\langle\psi| at zero temperature which has vanishing von-Neumann entropy S ​ ( ρ ) = − tr ​ [ ρ ​ log 2 ⁡ ρ ] . 𝑆 𝜌 tr delimited-[] 𝜌 subscript 2 𝜌 S(\rho)=-\text{tr}[\rho\log_{2}\rho]. Let us now distinguish a region of this quantum lattice system, denoting its sites with the set I 𝐼 I and all other sites with O = L \ I 𝑂 \ 𝐿 𝐼 O=L\backslash I . If we consider the reduced state ρ I = tr O ​ [ ρ ] subscript 𝜌 𝐼 subscript tr 𝑂 delimited-[] 𝜌 \rho_{I}=\text{tr}_{O}[\rho] of the sites of the region I 𝐼 I , the state will not be pure in general and will have a non-vanishing von-Neumann entropy S ​ ( ρ I ) 𝑆 subscript 𝜌 𝐼 S(\rho_{I}) . 1 1 1 Of interest are also other entropies, such as the Renyi entropies , S α ​ ( ρ ) = ( 1 − α ) − 1 ​ log 2 ⁡ tr ​ [ ρ α ] subscript 𝑆 𝛼 𝜌 superscript 1 𝛼 1 subscript 2 tr delimited-[] superscript 𝜌 𝛼 S_{\alpha}(\rho)=(1-\alpha)^{-1}\log_{2}\text{tr}[\rho^{\alpha}] with α ≥ 0 𝛼 0 \alpha\geq 0 . For α ↘ 1 ↘ 𝛼 1 \alpha\searrow 1 the usual von-Neumann entropy is recovered. In particular in the context of simulatability, Renyi entropies for arbitrary α 𝛼 \alpha play an important role. In contrast to thermal states this entropy does not originate from a lack of knowledge about the microstate of the system. Even at zero temperature we will encounter a non-zero entropy! This entropy arises because of a very fundamental property of quantum mechanics: Entanglement. This quite intriguing trait of quantum mechanics gives rise to correlations even in situations where the randomness cannot be traced back to a mere lack of knowledge. The mentioned quantity, the entropy of a subregion is called entanglement entropy or geometric entropy and, in quantum information, entropy of entanglement , which represents an operationally defined entanglement measure for pure states (for recent reviews see refs. 125 , 186 ). In the context of quantum field theory, questions of scaling of entanglement entropies in the size of I 𝐼 I have some tradition. Seminal work on the geometric entropy of the free Klein-Gordon field 23 , 207 and subsequent work on conformal field theories 122 , 43 , 226 , 36 , 118 was driven in part by the intriguing suggested connection to the Bekenstein-Hawking black hole entropy 17 , 117 , 18 . In recent years, studies of properties of the entanglement entropy in this sense have enjoyed a revival initiated in refs. 7 , 171 , 172 , 223 . Importantly, this renewed activity is benefitting from the new perspectives and ideas of quantum information theory, and from the realisation of their significance for the understanding of numerical methods and especially their efficiency for describing quantum many-body physics. Quantum information theory also provides novel conceptual and mathematical techniques for determining properties of the geometric entropy analytically. At the heart of these studies are questions like: What role do genuine quantum correlations—entanglement—play in quantum many-body systems? Typically, in such investigations, one abstracts to a large extent from the microscopic specifics of the system: Quite in the spirit of studies of critical phenomena , one thinks less of very detailed properties, but is rather interested in the scaling of the entanglement entropy when the distinguished region grows in size. In fact, for quantum chains, this scaling of entanglement as genuine quantum correlations—a priori very different from the scaling of two-point correlation functions—reflects to a large extent the critical behavior of the quantum many-body system, and shares some relationship to conformal charges. At first sight one might be tempted to think that the entropy of a distinguished region I 𝐼 I , will always possess an extensive character. Such a behavior is referred to as a volume scaling and is observed for thermal states. Intriguingly, for typical ground states, however, this is not at all what one encounters: Instead, one typically finds an area law , or an area law with a small (often logarithmic) correction: This means that if one distinguishes a region, the scaling of the entropy is merely linear in the boundary area of the region. The entanglement entropy is then said to fulfill an area law . It is the purpose of this article to review studies on area laws and the scaling of the entanglement entropy in a non-technical manner. The main four motivations to approach this question (known to the authors) are as follows: • The holographic principle and black hole entropy: The historical motivation to study the entanglement or geometric entropy stems from considerations of black hole physics: It has been suggested in the seminal work of refs. 23 , 207 that the area law of the geometric entropy for a discrete version of a massless free scalar field—then numerically found for an imaginary sphere in a radial symmetry—could be related to the physics of black holes , 118 in particular the Bekenstein-Hawking entropy of a black hole which is proportional to its boundary surface. It has been muted that the holographic principle 29 —the conjecture that the information contained in a volume of space can be represented by a theory which lives in the boundary of that region—could be related to the area law behavior of the entanglement entropy in microscopic theories. • Distribution of quantum correlations in quantum many-body systems: Area laws also say something quite profound on how quantum correlations are distributed in ground states of local quantum many-body systems. Interactions in quantum many-body systems are typically local, which means that systems interact only over a short distance with a finite number of neighbors. The emergence of an area law then provides support for the intuition that short ranged interactions require that quantum correlations between a distinguished region and its exterior are established via its boundary surface. That a strict area law emerges is by no means obvious from the decay of two-point correlators, as we will see. Quantum phase transitions are governed by quantum fluctuations at zero temperature, so it is more than plausible to observe signatures of criticality on the level of entanglement and quantum correlations . This situation is now particularly clear in one-dimensional systems, 7 , 223 , 83 , 134 , 92 , 36 , 147 , 72 , 132 , 138 , 85 , 69 , 73 , 13 , 225 , 146 , 46 , 38 , 111 , 44 , 4 but progress has also been made in higher-dimensional systems, 184 , 55 , 57 , 120 , 31 , 191 , 219 , 238 , 86 , 140 , 90 with rigorous area laws specifically for quasi-free bosonic 184 , 55 , 57 and fermionic 238 , 96 , 56 , 86 systems, as well as in disordered systems 190 . • Complexity of quantum many-body systems and their simulation: One of the key motivations for studying area laws stems from a quite practical context: The numerical simulation of quantum many-body systems. In fact, if there is little entanglement in a ground state of a many-body systems, one might suspect on intuitive grounds that one can describe this ground state with relatively few parameters. More specifically, for one-dimensional systems, one would expect numerical algorithms like the powerful density-matrix renormalization group method 232 , 199 (DMRG) to perform well if the ground state contains a small amount of entanglement. This suspicion can in fact be made rigorous 176 , 215 , 111 , 203 as it turns out that the scaling of entanglement specifies how well a given state can be approximated by a matrix-product state 84 , 199 as generated in DMRG. It is hence not the decay behavior of correlation functions as such that matters here, but in fact the scaling of entanglement. • Topological entanglement entropy: The topological entanglement entropy is an indicator of topological order , 228 , 236 , 165 a new kind of order in quantum many-body systems that cannot be described by local order parameters 140 , 148 , 175 , 108 , 165 . Lattice models having a non-vanishing topological entanglement entropy may be seen as lattice instances of topological quantum field theories. Here a global feature is detected by means of the scaling of geometric entropies. In this Colloquium we do not have sufficient space to give an account of all known derivations of area laws for the entanglement entropy. However, we will try not to merely remain at a superficial level and only state results, but will explain a number of key techniques and arguments. When we label main statements as “theorems” this is done to highlight their special role, to make it easier to follow the line of reasoning. For details of arguments and proofs, often technically involved, we refer the reader to the original work. The reason for the technicality of proofs originates from the type of question that is posed: To distinguish a region of a lattice breaks the translational symmetry of the problem – even in a translationally invariant setting. While numerical studies are sometimes easier to come by, analytical argument can be technically involved, even for quasi-free models. In this article, we discuss the study of entanglement entropy primarily (i) from the viewpoint of quantum information theory, (ii) with an emphasis on rigorous and analytical results, and (iii) the implications on the efficiency of numerical simulation.

Physical interactions in quantum many-body systems are typically local: Individual constituents interact mainly with their few nearest neighbors. This locality of interactions is inherited by a decay of correlation functions, but also reflected by scaling laws of a quite profound quantity: The entanglement entropy of ground states. This entropy of the reduced state of a subregion often merely grows like the boundary area of the subregion, and not like its volume, in sharp contrast with an expected extensive behavior. Such 'area laws' for the entanglement entropy and related quantities have received considerable attention in recent years. They emerge in several seemingly unrelated fields, in the context of black hole physics, quantum information science, and quantum many-body physics where they have important implications on the numerical simulation of lattice models. In this Colloquium we review the current status of area laws in these fields. Center stage is taken by rigorous results on lattice models in one and higher spatial dimensions. The differences and similarities between bosonic and fermionic models are stressed, area laws are related to the velocity of information propagation, and disordered systems, non-equilibrium situations, classical correlation concepts, and topological entanglement entropies are discussed. A significant proportion of the article is devoted to the quantitative connection between the entanglement content of states and the possibility of their efficient numerical simulation. We discuss matrix-product states, higher-dimensional analogues, and states from entanglement renormalization and conclude by highlighting the implications of area laws on quantifying the effective degrees of freedom that need to be considered in simulations.

[{"content":"Help me to generate the abstract of a survey paper given the title:*Area laws for the entanglement entropy - a review* and the introduction:I Introduction In classical physics concepts of entropy quantify the extent to which we are uncertain about the exact state of a physical system at hand or, in other words, the amount of information that is lacking to identify the microstate of a system from all possibilities compatible with the macrostate of the system. If we are not quite sure what microstate of a system to expect, notions of entropy will reflect this lack of knowledge. Randomness, after all, is always and necessarily related to ignorance about the state. Figure 1: A lattice L 𝐿 L with a distinguished set I ⊂ L 𝐼 𝐿 I\subset L (shaded area). Vertices depict the boundary ∂ I 𝐼 \partial I of I 𝐼 I with surface area s ​ ( I ) = | ∂ I | 𝑠 𝐼 𝐼 s(I)=|\partial I| . In quantum mechanics positive entropies may arise even without an objective lack of information. To see this, let us consider a quantum lattice systems (see e.g., Fig. 1) as an example for a quantum many-body system where each of the vertices i 𝑖 i of the lattice L 𝐿 L is associated with an individual quantum system. This quantum many-body system is thought to be in its non-degenerate pure ground state ρ = | ψ ⟩ ​ ⟨ ψ | 𝜌 ket 𝜓 bra 𝜓 \rho=|\psi\rangle\langle\psi| at zero temperature which has vanishing von-Neumann entropy S ​ ( ρ ) = − tr ​ ρ ​ log 2 ⁡ ρ . 𝑆 𝜌 tr delimited- 𝜌 subscript 2 𝜌 S(\rho)=-\text{tr}\rho\log_{2}\rho. Let us now distinguish a region of this quantum lattice system, denoting its sites with the set I 𝐼 I and all other sites with O = L \ I 𝑂 \ 𝐿 𝐼 O=L\backslash I . If we consider the reduced state ρ I = tr O ​ ρ subscript 𝜌 𝐼 subscript tr 𝑂 delimited- 𝜌 \rho_{I}=\text{tr}_{O}\rho of the sites of the region I 𝐼 I , the state will not be pure in general and will have a non-vanishing von-Neumann entropy S ​ ( ρ I ) 𝑆 subscript 𝜌 𝐼 S(\rho_{I}) . 1 1 1 Of interest are also other entropies, such as the Renyi entropies , S α ​ ( ρ ) = ( 1 − α ) − 1 ​ log 2 ⁡ tr ​ ρ α subscript 𝑆 𝛼 𝜌 superscript 1 𝛼 1 subscript 2 tr delimited- superscript 𝜌 𝛼 S_{\alpha}(\rho)=(1-\alpha)^{-1}\log_{2}\text{tr}\rho^{\alpha} with α ≥ 0 𝛼 0 \alpha\geq 0 . For α ↘ 1 ↘ 𝛼 1 \alpha\searrow 1 the usual von-Neumann entropy is recovered. In particular in the context of simulatability, Renyi entropies for arbitrary α 𝛼 \alpha play an important role. In contrast to thermal states this entropy does not originate from a lack of knowledge about the microstate of the system. Even at zero temperature we will encounter a non-zero entropy! This entropy arises because of a very fundamental property of quantum mechanics: Entanglement. This quite intriguing trait of quantum mechanics gives rise to correlations even in situations where the randomness cannot be traced back to a mere lack of knowledge. The mentioned quantity, the entropy of a subregion is called entanglement entropy or geometric entropy and, in quantum information, entropy of entanglement , which represents an operationally defined entanglement measure for pure states (for recent reviews see refs. 125 , 186 ). In the context of quantum field theory, questions of scaling of entanglement entropies in the size of I 𝐼 I have some tradition. Seminal work on the geometric entropy of the free Klein-Gordon field 23 , 207 and subsequent work on conformal field theories 122 , 43 , 226 , 36 , 118 was driven in part by the intriguing suggested connection to the Bekenstein-Hawking black hole entropy 17 , 117 , 18 . In recent years, studies of properties of the entanglement entropy in this sense have enjoyed a revival initiated in refs. 7 , 171 , 172 , 223 . Importantly, this renewed activity is benefitting from the new perspectives and ideas of quantum information theory, and from the realisation of their significance for the understanding of numerical methods and especially their efficiency for describing quantum many-body physics. Quantum information theory also provides novel conceptual and mathematical techniques for determining properties of the geometric entropy analytically. At the heart of these studies are questions like: What role do genuine quantum correlations—entanglement—play in quantum many-body systems? Typically, in such investigations, one abstracts to a large extent from the microscopic specifics of the system: Quite in the spirit of studies of critical phenomena , one thinks less of very detailed properties, but is rather interested in the scaling of the entanglement entropy when the distinguished region grows in size. In fact, for quantum chains, this scaling of entanglement as genuine quantum correlations—a priori very different from the scaling of two-point correlation functions—reflects to a large extent the critical behavior of the quantum many-body system, and shares some relationship to conformal charges. At first sight one might be tempted to think that the entropy of a distinguished region I 𝐼 I , will always possess an extensive character. Such a behavior is referred to as a volume scaling and is observed for thermal states. Intriguingly, for typical ground states, however, this is not at all what one encounters: Instead, one typically finds an area law , or an area law with a small (often logarithmic) correction: This means that if one distinguishes a region, the scaling of the entropy is merely linear in the boundary area of the region. The entanglement entropy is then said to fulfill an area law . It is the purpose of this article to review studies on area laws and the scaling of the entanglement entropy in a non-technical manner. The main four motivations to approach this question (known to the authors) are as follows: • The holographic principle and black hole entropy: The historical motivation to study the entanglement or geometric entropy stems from considerations of black hole physics: It has been suggested in the seminal work of refs. 23 , 207 that the area law of the geometric entropy for a discrete version of a massless free scalar field—then numerically found for an imaginary sphere in a radial symmetry—could be related to the physics of black holes , 118 in particular the Bekenstein-Hawking entropy of a black hole which is proportional to its boundary surface. It has been muted that the holographic principle 29 —the conjecture that the information contained in a volume of space can be represented by a theory which lives in the boundary of that region—could be related to the area law behavior of the entanglement entropy in microscopic theories. • Distribution of quantum correlations in quantum many-body systems: Area laws also say something quite profound on how quantum correlations are distributed in ground states of local quantum many-body systems. Interactions in quantum many-body systems are typically local, which means that systems interact only over a short distance with a finite number of neighbors. The emergence of an area law then provides support for the intuition that short ranged interactions require that quantum correlations between a distinguished region and its exterior are established via its boundary surface. That a strict area law emerges is by no means obvious from the decay of two-point correlators, as we will see. Quantum phase transitions are governed by quantum fluctuations at zero temperature, so it is more than plausible to observe signatures of criticality on the level of entanglement and quantum correlations . This situation is now particularly clear in one-dimensional systems, 7 , 223 , 83 , 134 , 92 , 36 , 147 , 72 , 132 , 138 , 85 , 69 , 73 , 13 , 225 , 146 , 46 , 38 , 111 , 44 , 4 but progress has also been made in higher-dimensional systems, 184 , 55 , 57 , 120 , 31 , 191 , 219 , 238 , 86 , 140 , 90 with rigorous area laws specifically for quasi-free bosonic 184 , 55 , 57 and fermionic 238 , 96 , 56 , 86 systems, as well as in disordered systems 190 . • Complexity of quantum many-body systems and their simulation: One of the key motivations for studying area laws stems from a quite practical context: The numerical simulation of quantum many-body systems. In fact, if there is little entanglement in a ground state of a many-body systems, one might suspect on intuitive grounds that one can describe this ground state with relatively few parameters. More specifically, for one-dimensional systems, one would expect numerical algorithms like the powerful density-matrix renormalization group method 232 , 199 (DMRG) to perform well if the ground state contains a small amount of entanglement. This suspicion can in fact be made rigorous 176 , 215 , 111 , 203 as it turns out that the scaling of entanglement specifies how well a given state can be approximated by a matrix-product state 84 , 199 as generated in DMRG. It is hence not the decay behavior of correlation functions as such that matters here, but in fact the scaling of entanglement. • Topological entanglement entropy: The topological entanglement entropy is an indicator of topological order , 228 , 236 , 165 a new kind of order in quantum many-body systems that cannot be described by local order parameters 140 , 148 , 175 , 108 , 165 . Lattice models having a non-vanishing topological entanglement entropy may be seen as lattice instances of topological quantum field theories. Here a global feature is detected by means of the scaling of geometric entropies. In this Colloquium we do not have sufficient space to give an account of all known derivations of area laws for the entanglement entropy. However, we will try not to merely remain at a superficial level and only state results, but will explain a number of key techniques and arguments. When we label main statements as theorems this is done to highlight their special role, to make it easier to follow the line of reasoning. For details of arguments and proofs, often technically involved, we refer the reader to the original work. The reason for the technicality of proofs originates from the type of question that is posed: To distinguish a region of a lattice breaks the translational symmetry of the problem – even in a translationally invariant setting. While numerical studies are sometimes easier to come by, analytical argument can be technically involved, even for quasi-free models. In this article, we discuss the study of entanglement entropy primarily (i) from the viewpoint of quantum information theory, (ii) with an emphasis on rigorous and analytical results, and (iii) the implications on the efficiency of numerical simulation.","role":"user"},{"content":"Abstract: Physical interactions in quantum many-body systems are typically local: Individual constituents interact mainly with their few nearest neighbors. This locality of interactions is inherited by a decay of correlation functions, but also reflected by scaling laws of a quite profound quantity: The entanglement entropy of ground states. This entropy of the reduced state of a subregion often merely grows like the boundary area of the subregion, and not like its volume, in sharp contrast with an expected extensive behavior. Such 'area laws' for the entanglement entropy and related quantities have received considerable attention in recent years. They emerge in several seemingly unrelated fields, in the context of black hole physics, quantum information science, and quantum many-body physics where they have important implications on the numerical simulation of lattice models. In this Colloquium we review the current status of area laws in these fields. Center stage is taken by rigorous results on lattice models in one and higher spatial dimensions. The differences and similarities between bosonic and fermionic models are stressed, area laws are related to the velocity of information propagation, and disordered systems, non-equilibrium situations, classical correlation concepts, and topological entanglement entropies are discussed. A significant proportion of the article is devoted to the quantitative connection between the entanglement content of states and the possibility of their efficient numerical simulation. We discuss matrix-product states, higher-dimensional analogues, and states from entanglement renormalization and conclude by highlighting the implications of area laws on quantifying the effective degrees of freedom that need to be considered in simulations.","role":"assistant"}]

QUANTIZED DETECTOR NETWORKS: A REVIEW OF RECENT DEVELOPMENTS

and historical perspective The landscape of QM is littered with the debris of various interpretations devised to explain away its strange non-classical properties such as wave-particle duality and quantum interference. There is no space to review any of these attempts here. Usually, they failed because their authors tried to view too much of physical reality in terms of familiar classical concepts, such as particles and waves moving about in a background spacetime. Applied to everyday (large-scale) processes, such a strategy usually works well, as evinced by the success of Newtonian and relativistic mechanics, but when it comes to QM, it leads to various traps waiting for the unwary theorist. A notable example is Schrödinger, who originally thought of his quantum waves as smeared out electronic charge, in contrast with the Born probabilistic interpretation [ 2 , 3 ] generally accepted today. Classical thinking in QM persists to this day in one form or another, ranging from attempts to split electrons [ 4 ] to paradigms such as the Multiverse [ 5 ] and decoherence [ 6 ] , which in their original formulations were based on the assertion that the Schrödinger equation alone suffices to explain all of physics. Such schools of thought view the quantum wavefunction as a fundamental object in its own right. In this article we will emphasize the point that even to talk casually about an “electron wavefunction”, as is commonplace amongst quantum theorists, is to risk applying classical thinking inappropriately to quantum physics. Our view is that a quantum wavefunction is contextual, i.e., without reference to any observer and their apparatus, such a wavefunction is a physically meaningless concept. Our aim in QDN is to eliminate as far as possible concepts which are inessential, potentially misleading, or simply metaphysical (i.e., incapable of verification). Some mathematical concepts such as labstates and Hilbert spaces are used heavily, but in such cases, the motivation for them stems from a desire to avoid mental imagery as much as possible. The most important guiding principle has been to ask what exactly do experimentalists do when they perform quantum physics experiments, and then model the answer to that question according to established quantum mechanical principles.

QDN (quantized detector networks) is a description of quantum processes in which the principal focus is on observers and their apparatus, rather than on states of SUOs (systems under observation). It is a realization of Heisenberg's original instrumentalist approach to quantum physics, and can deal with time dependent apparatus, multiple observers and inter-frame physics. QDN is most naturally expressed in the mathematical language of quantum computation, a language ideally suited to describe quantum experiments as processes of information exchange between observers and their apparatus. Examples in quantum optics are given, showing how the formalism deals with quantum interference, non-locality and entanglement. Particle decays, relativity and non-linearity in quantum mechanics are also discussed.

[{"content":"Help me to generate the abstract of a survey paper given the title:*QUANTIZED DETECTOR NETWORKS: A REVIEW OF RECENT DEVELOPMENTS* and the introduction:and historical perspective The landscape of QM is littered with the debris of various interpretations devised to explain away its strange non-classical properties such as wave-particle duality and quantum interference. There is no space to review any of these attempts here. Usually, they failed because their authors tried to view too much of physical reality in terms of familiar classical concepts, such as particles and waves moving about in a background spacetime. Applied to everyday (large-scale) processes, such a strategy usually works well, as evinced by the success of Newtonian and relativistic mechanics, but when it comes to QM, it leads to various traps waiting for the unwary theorist. A notable example is Schrödinger, who originally thought of his quantum waves as smeared out electronic charge, in contrast with the Born probabilistic interpretation 2 , 3 generally accepted today. Classical thinking in QM persists to this day in one form or another, ranging from attempts to split electrons 4 to paradigms such as the Multiverse 5 and decoherence 6 , which in their original formulations were based on the assertion that the Schrödinger equation alone suffices to explain all of physics. Such schools of thought view the quantum wavefunction as a fundamental object in its own right. In this article we will emphasize the point that even to talk casually about an electron wavefunction, as is commonplace amongst quantum theorists, is to risk applying classical thinking inappropriately to quantum physics. Our view is that a quantum wavefunction is contextual, i.e., without reference to any observer and their apparatus, such a wavefunction is a physically meaningless concept. Our aim in QDN is to eliminate as far as possible concepts which are inessential, potentially misleading, or simply metaphysical (i.e., incapable of verification). Some mathematical concepts such as labstates and Hilbert spaces are used heavily, but in such cases, the motivation for them stems from a desire to avoid mental imagery as much as possible. The most important guiding principle has been to ask what exactly do experimentalists do when they perform quantum physics experiments, and then model the answer to that question according to established quantum mechanical principles.","role":"user"},{"content":"Abstract: QDN (quantized detector networks) is a description of quantum processes in which the principal focus is on observers and their apparatus, rather than on states of SUOs (systems under observation). It is a realization of Heisenberg's original instrumentalist approach to quantum physics, and can deal with time dependent apparatus, multiple observers and inter-frame physics. QDN is most naturally expressed in the mathematical language of quantum computation, a language ideally suited to describe quantum experiments as processes of information exchange between observers and their apparatus. Examples in quantum optics are given, showing how the formalism deals with quantum interference, non-locality and entanglement. Particle decays, relativity and non-linearity in quantum mechanics are also discussed.","role":"assistant"}]

An Overview of the Transmission Capacity of Wireless Networks

I Introduction This paper presents the recently developed framework for the outage probability and transmission capacity [ 1 ] in a one hop wireless ad hoc network. The transmission capacity is defined as the number of successful transmissions taking place in the network per unit area, subject to a constraint on outage probability. In addition to being of general interest, the advantange of transmission capacity – relative to, say, the transport capacity or average sum throughput – lies largely in that it can be exactly derived in some important cases, and tightly bounded in many others, as we shall show. From the expressions and approach given in this paper the exact dependence between system performance (transmission capacity, outage probability) and the possible design choices and network parameters are laid bare. In contrast to the proposed framework, nearly all other work on ad hoc networks must resort to scaling laws or numerical simulations, in which case intuition and/or precision is usually lost. The first goal of this paper is to concisely summarize the new analytical tools (largely drawn from the field of stochastic geometry [ 2 , 3 ] ) that have been developed over numerous papers by the authors and others. Because these techniques have been developed somewhat independently depending on the problem of interest, the system model in § II applied to the baseline model of pathloss attenuation without fading in § III will help newcomers to the area understand the various approaches in context. The second goal is to show how this framework can be used to give crisp insights into wireless network design problems. In the past few years, the transmission capacity approach has been applied to various design problems by a growing group of researchers (see [ 1 , 4 , 5 , 6 , 7 , 8 ] ). Although transmission capacity was originally developed to analyze spread spectrum in ad hoc networks, it has proven to be a metric with considerable breadth of application. Since decentralized wireless networks are generally very difficult to characterize, the intuitive and simple-to-compute qualities of transmission capacity have made it a popular choice for a large number of possible systems, including: i ) i) direct-sequence and frequency-hopping spread spectrum [ 1 , 9 , 4 ] , i i ) ii) interference cancellation [ 10 , 5 ] , i i i ) iii) spectrum sharing in unlicensed, overlaid, and cognitive radio networks [ 11 , 6 , 7 , 12 ] , i v ) iv) scheduling [ 10 ] and power control [ 13 , 14 ] , v ) v) and the use of multiple antennas (which had resisted characterization by other methods) [ 15 , 8 , 16 , 17 , 18 , 19 , 20 , 21 ] . Other researchers have also further studied the basic tradeoffs between outage probability, data rate, and transmission capacity for general networks [ 22 ] . We selectively discuss some of these applications. § IV addresses networks with fading channels, with a focus on Rayleigh (§ IV-B ) and Nakagami (§ IV-C ) fading, scheduling (§ IV-D ), and power control (§ IV-E ). § V addresses the use of multiple antennas, with discussions of diversity (§ V-A ), spatial interference cancellation (§ V-B ), and spatial multiplexing (§ V-C ). The third goal of the paper is to stimulate new efforts to further the tools presented here, both in making them more general and in applying them to new problems. We readily concede that the presented model has some nontrivial shortcomings at present, and we identify those as well as possible avenues forward in § VI .

This paper surveys and unifies a number of recent contributions that have collectively developed a metric for decentralized wireless network analysis known as transmission capacity. Although it is notoriously difficult to derive general end-to-end capacity results for multi-terminal or adhoc networks, the transmission capacity (TC) framework allows for quantification of achievable single-hop rates by focusing on a simplified physical/MAC-layer model. By using stochastic geometry to quantify the multi-user interference in the network, the relationship between the optimal spatial density and success probability of transmissions in the network can be determined, and expressed-often fairly simply-in terms of the key network parameters. The basic model and analytical tools are first discussed and applied to a simple network with path loss only and we present tight upper and lower bounds on transmission capacity (via lower and upper bounds on outage probability). We then introduce random channels (fading/shadowing) and give TC and outage approximations for an arbitrary channel distribution, as well as exact results for the special cases of Rayleigh and Nakagami fading. We then apply these results to show how TC can be used to better understand scheduling, power control, and the deployment of multiple antennas in a decentralized network. The paper closes by discussing shortcomings in the model as well as future research directions.

[{"content":"Help me to generate the abstract of a survey paper given the title:*An Overview of the Transmission Capacity of Wireless Networks* and the introduction:I Introduction This paper presents the recently developed framework for the outage probability and transmission capacity 1 in a one hop wireless ad hoc network. The transmission capacity is defined as the number of successful transmissions taking place in the network per unit area, subject to a constraint on outage probability. In addition to being of general interest, the advantange of transmission capacity – relative to, say, the transport capacity or average sum throughput – lies largely in that it can be exactly derived in some important cases, and tightly bounded in many others, as we shall show. From the expressions and approach given in this paper the exact dependence between system performance (transmission capacity, outage probability) and the possible design choices and network parameters are laid bare. In contrast to the proposed framework, nearly all other work on ad hoc networks must resort to scaling laws or numerical simulations, in which case intuition and/or precision is usually lost. The first goal of this paper is to concisely summarize the new analytical tools (largely drawn from the field of stochastic geometry 2 , 3 ) that have been developed over numerous papers by the authors and others. Because these techniques have been developed somewhat independently depending on the problem of interest, the system model in § II applied to the baseline model of pathloss attenuation without fading in § III will help newcomers to the area understand the various approaches in context. The second goal is to show how this framework can be used to give crisp insights into wireless network design problems. In the past few years, the transmission capacity approach has been applied to various design problems by a growing group of researchers (see 1 , 4 , 5 , 6 , 7 , 8 ). Although transmission capacity was originally developed to analyze spread spectrum in ad hoc networks, it has proven to be a metric with considerable breadth of application. Since decentralized wireless networks are generally very difficult to characterize, the intuitive and simple-to-compute qualities of transmission capacity have made it a popular choice for a large number of possible systems, including: i ) i) direct-sequence and frequency-hopping spread spectrum 1 , 9 , 4 , i i ) ii) interference cancellation 10 , 5 , i i i ) iii) spectrum sharing in unlicensed, overlaid, and cognitive radio networks 11 , 6 , 7 , 12 , i v ) iv) scheduling 10 and power control 13 , 14 , v ) v) and the use of multiple antennas (which had resisted characterization by other methods) 15 , 8 , 16 , 17 , 18 , 19 , 20 , 21 . Other researchers have also further studied the basic tradeoffs between outage probability, data rate, and transmission capacity for general networks 22 . We selectively discuss some of these applications. § IV addresses networks with fading channels, with a focus on Rayleigh (§ IV-B ) and Nakagami (§ IV-C ) fading, scheduling (§ IV-D ), and power control (§ IV-E ). § V addresses the use of multiple antennas, with discussions of diversity (§ V-A ), spatial interference cancellation (§ V-B ), and spatial multiplexing (§ V-C ). The third goal of the paper is to stimulate new efforts to further the tools presented here, both in making them more general and in applying them to new problems. We readily concede that the presented model has some nontrivial shortcomings at present, and we identify those as well as possible avenues forward in § VI .","role":"user"},{"content":"Abstract: This paper surveys and unifies a number of recent contributions that have collectively developed a metric for decentralized wireless network analysis known as transmission capacity. Although it is notoriously difficult to derive general end-to-end capacity results for multi-terminal or adhoc networks, the transmission capacity (TC) framework allows for quantification of achievable single-hop rates by focusing on a simplified physical/MAC-layer model. By using stochastic geometry to quantify the multi-user interference in the network, the relationship between the optimal spatial density and success probability of transmissions in the network can be determined, and expressed-often fairly simply-in terms of the key network parameters. The basic model and analytical tools are first discussed and applied to a simple network with path loss only and we present tight upper and lower bounds on transmission capacity (via lower and upper bounds on outage probability). We then introduce random channels (fading/shadowing) and give TC and outage approximations for an arbitrary channel distribution, as well as exact results for the special cases of Rayleigh and Nakagami fading. We then apply these results to show how TC can be used to better understand scheduling, power control, and the deployment of multiple antennas in a decentralized network. The paper closes by discussing shortcomings in the model as well as future research directions.","role":"assistant"}]

Q-manifolds and Mackenzie theory: an overview

1. Introduction This text is meant to be a brief overview of the topics announced in the title. It does not contain new results (except probably for the remark concerning Q 𝑄 Q -manifold homology, which we wish to elaborate elsewhere). The original exposition of Mackenzie’s constructions is in his papers [ 10 , 21 , 11 , 12 , 13 , 14 , 15 , 16 , 17 , 18 ] . See also the book [ 19 ] . There is a paper [ 20 ] , which is now in the process of publication and which is a substantially reworked version of the earlier paper [ 16 ] . The main point of this text is to give an introduction to the approach to double Lie algebroids based on (graded) Q 𝑄 Q -manifolds. Details can be found in [ 29 ] , where the result (the equivalence between Mackenzie’s double Lie algebroids and a certain class of Q 𝑄 Q -manifolds, see below) was formulated and proved for the first time. Some background information can also be found in [ 27 ] . Double Lie algebroids first appeared as the infinitesimals corresponding to the double Lie groupoids introduced by Mackenzie [ 10 , 11 ] as double objects in the sense of Ehresmann (Lie groupoid objects in the category of Lie groupoids). The abstract notion came about later [ 12 , 13 , 14 ] and turned out to be quite complicated in formulation. (One of the reasons is that a Lie algebroid is not defined diagrammatically, so it is not possible to follow a categorical approach, which works well for the groupoid case.) Although there was absolutely no doubt that this was the ‘right’ notion — one justification was in the appearance of double Lie algebroids as ‘Drinfeld doubles’ of Lie bialgebroids [ 14 ] — their application was somewhat hindered by the complexity of the definition. It had been of a considerable interest for experts to give an alternative simpler description for them. This was achieved in [ 29 ] . This paper is an outcome of my participation in the program on “Poisson sigma models, Lie algebroids, deformations, and higher analogues” at the Erwin Schrödinger Institute in Vienna in August–September 2007, where it was written. It is a pleasant duty to express my gratitude to the Institute for the hospitality and the wonderful atmosphere for research and communications, and to the organizers of the program (Thomas Strobl, Henrique Bursztyn and Harald Grosse) for the invitation. The first announcement of the results of [ 29 ] was made in July 2006 at the XXV Białowieża meeting.

This text is meant to be a brief overview of the topics announced in the title and is based on my talk in Vienna (August/September 2007). It does not contain new results (except probably for a remark concerning Q-manifold homology, which I wish to elaborate elsewhere). "Mackenzie theory" stands for the rich circle of notions that have been put forward by Kirill Mackenzie (solo or in collaboration): double structures such as double Lie groupoids and double Lie algebroids, Lie bialgebroids and their doubles, nontrivial dualities for double and multiple vector bundles, etc. "Q-manifolds" are (super)manifolds with a homological vector field, i.e., a self-commuting odd vector field. They may have an extra Z-grading (called weight) not necessarily linked with the Z_2-grading (parity). I discuss double Lie algebroids (discovered by Mackenzie) and explain how this quite complicated fundamental notion is equivalent to a very simple one if the language of Q-manifolds is used. In particular, it shows how the two seemingly different notions of a "Drinfeld double" of a Lie bialgebroid due to Mackenzie and Roytenberg respectively, turn out to be the same thing if properly understood.

[{"content":"Help me to generate the abstract of a survey paper given the title:*Q-manifolds and Mackenzie theory: an overview* and the introduction:1. Introduction This text is meant to be a brief overview of the topics announced in the title. It does not contain new results (except probably for the remark concerning Q 𝑄 Q -manifold homology, which we wish to elaborate elsewhere). The original exposition of Mackenzies constructions is in his papers 10 , 21 , 11 , 12 , 13 , 14 , 15 , 16 , 17 , 18 . See also the book 19 . There is a paper 20 , which is now in the process of publication and which is a substantially reworked version of the earlier paper 16 . The main point of this text is to give an introduction to the approach to double Lie algebroids based on (graded) Q 𝑄 Q -manifolds. Details can be found in 29 , where the result (the equivalence between Mackenzies double Lie algebroids and a certain class of Q 𝑄 Q -manifolds, see below) was formulated and proved for the first time. Some background information can also be found in 27 . Double Lie algebroids first appeared as the infinitesimals corresponding to the double Lie groupoids introduced by Mackenzie 10 , 11 as double objects in the sense of Ehresmann (Lie groupoid objects in the category of Lie groupoids). The abstract notion came about later 12 , 13 , 14 and turned out to be quite complicated in formulation. (One of the reasons is that a Lie algebroid is not defined diagrammatically, so it is not possible to follow a categorical approach, which works well for the groupoid case.) Although there was absolutely no doubt that this was the right notion — one justification was in the appearance of double Lie algebroids as Drinfeld doubles of Lie bialgebroids 14 — their application was somewhat hindered by the complexity of the definition. It had been of a considerable interest for experts to give an alternative simpler description for them. This was achieved in 29 . This paper is an outcome of my participation in the program on Poisson sigma models, Lie algebroids, deformations, and higher analogues at the Erwin Schrödinger Institute in Vienna in August–September 2007, where it was written. It is a pleasant duty to express my gratitude to the Institute for the hospitality and the wonderful atmosphere for research and communications, and to the organizers of the program (Thomas Strobl, Henrique Bursztyn and Harald Grosse) for the invitation. The first announcement of the results of 29 was made in July 2006 at the XXV Białowieża meeting.","role":"user"},{"content":"Abstract: This text is meant to be a brief overview of the topics announced in the title and is based on my talk in Vienna (August/September 2007). It does not contain new results (except probably for a remark concerning Q-manifold homology, which I wish to elaborate elsewhere). "Mackenzie theory" stands for the rich circle of notions that have been put forward by Kirill Mackenzie (solo or in collaboration): double structures such as double Lie groupoids and double Lie algebroids, Lie bialgebroids and their doubles, nontrivial dualities for double and multiple vector bundles, etc. "Q-manifolds" are (super)manifolds with a homological vector field, i.e., a self-commuting odd vector field. They may have an extra Z-grading (called weight) not necessarily linked with the Z_2-grading (parity). I discuss double Lie algebroids (discovered by Mackenzie) and explain how this quite complicated fundamental notion is equivalent to a very simple one if the language of Q-manifolds is used. In particular, it shows how the two seemingly different notions of a "Drinfeld double" of a Lie bialgebroid due to Mackenzie and Roytenberg respectively, turn out to be the same thing if properly understood.","role":"assistant"}]

Unparticle Phenomenology—A Mini Review

Figure 1: Angular distributions for e − ​ e + → μ − ​ μ + → superscript 𝑒 superscript 𝑒 superscript 𝜇 superscript 𝜇 e^{-}e^{+}\to\mu^{-}\mu^{+} with various scaling dimension d U subscript 𝑑 𝑈 d_{U} for the spin 1 unparticle exchange with L ​ L + R ​ R 𝐿 𝐿 𝑅 𝑅 LL+RR contact terms plus the SM contributions at s = 200 𝑠 200 \sqrt{s}=200 GeV (left panel) and for the spin 2 unparticle exchange plus SM contributions at s 𝑠 \sqrt{s} = 0.5 TeV (right panel). We have set Λ U = 1 subscript Λ 𝑈 1 \Lambda_{U}=1 TeV and λ 1 = λ 2 = 1 subscript 𝜆 1 subscript 𝜆 2 1 \lambda_{1}=\lambda_{2}=1 . See CKY-long for details. The notion of unparticle introduced by Georgi unparticle was based on the hypothesis that there could be an exact scale invariant hidden sector resisted at a high energy scale. A prototype model of such sector is given by the Banks-Zaks theory Banks-Zaks which flows to an infrared fixed point at a lower energy scale Λ U subscript Λ 𝑈 \Lambda_{U} through dimensional transmutation. Below Λ U subscript Λ 𝑈 \Lambda_{U} , unparticle physics emerges and manifest itself as interpolating fields O 𝑂 O of various scaling dimensions and Lorentz structures. One of the interesting feature of unparticle operator is that it has a continuous spectral density ρ ​ ( P 2 ) 𝜌 superscript 𝑃 2 \rho(P^{2}) as a consequence of scale invariance unparticle ρ ​ ( P 2 ) = A d U ​ θ ​ ( P 0 ) ​ θ ​ ( P 2 ) ​ ( P 2 ) d U − 2 𝜌 superscript 𝑃 2 subscript 𝐴 subscript 𝑑 𝑈 𝜃 superscript 𝑃 0 𝜃 superscript 𝑃 2 superscript superscript 𝑃 2 subscript 𝑑 𝑈 2 \rho(P^{2})=A_{d_{U}}\theta(P^{0})\theta(P^{2})(P^{2})^{d_{U}-2} (1) where d U subscript 𝑑 𝑈 d_{U} is the scaling dimension of the unparticle operator with 4-momentum P 𝑃 P and A d U subscript 𝐴 subscript 𝑑 𝑈 A_{d_{U}} is a free normalization factor. Due to their similar kinematical exponent, Georgi unparticle chose A d U subscript 𝐴 subscript 𝑑 𝑈 A_{d_{U}} to be the prefactor of d U subscript 𝑑 𝑈 d_{U} massless particle phase space d ​ LIPS d U = A d U ​ ( ( p 1 + p 2 + ⋯ + p d U ) 2 ) d U − 2 𝑑 subscript LIPS subscript 𝑑 𝑈 subscript 𝐴 subscript 𝑑 𝑈 superscript superscript subscript 𝑝 1 subscript 𝑝 2 ⋯ subscript 𝑝 subscript 𝑑 𝑈 2 subscript 𝑑 𝑈 2 d{\rm LIPS}_{d_{U}}=A_{d_{U}}\left(\left(p_{1}+p_{2}+\cdots+p_{d_{U}}\right)^{2}\right)^{d_{U}-2} (2) where p i 2 = 0 ​ ( i = 1 , … , d U ) superscript subscript 𝑝 𝑖 2 0 𝑖 1 … subscript 𝑑 𝑈 p_{i}^{2}=0\,(i=1,...,d_{U}) . Thus A d U = 16 ​ π 5 / 2 ( 2 ​ π ) 2 ​ d U ​ Γ ​ ( d U + 1 2 ) Γ ​ ( d U − 1 ) ​ Γ ​ ( 2 ​ d U ) . subscript 𝐴 subscript 𝑑 𝑈 16 superscript 𝜋 5 2 superscript 2 𝜋 2 subscript 𝑑 𝑈 Γ subscript 𝑑 𝑈 1 2 Γ subscript 𝑑 𝑈 1 Γ 2 subscript 𝑑 𝑈 A_{d_{U}}=\frac{16\pi^{5/2}}{(2\pi)^{2d_{U}}}\frac{\Gamma(d_{U}+\frac{1}{2})}{\Gamma(d_{U}-1)\Gamma(2d_{U})}\;. (3) As d U subscript 𝑑 𝑈 d_{U} approaches 1, Eq.( 1 ) reduces to the familiar one massless particle phase space. This suggests unparticle behaves like a collection of d U subscript 𝑑 𝑈 d_{U} massless particles. Since d U subscript 𝑑 𝑈 d_{U} can be non-integral, one can now speak of fractional particle. This metaphor draws immediate attention of many physicists worldwide as well as the general public. 1 1 1 One can even find an entry of unparticle in the Wikipedia database. Soon after the introduction of unparticle, its propagator was deduced by using unitarity cuts georgi2 and the spectral decomposition formula CKY-short . For a scalar unparticle, its Feynman propagator is given by Δ F ​ ( P ) = A d U 2 ​ sin ⁡ ( d U ​ π ) ​ 1 ( − P 2 − i ​ 0 + ) 2 − d U . subscript Δ 𝐹 𝑃 subscript 𝐴 subscript 𝑑 𝑈 2 subscript 𝑑 𝑈 𝜋 1 superscript superscript 𝑃 2 𝑖 superscript 0 2 subscript 𝑑 𝑈 \Delta_{F}(P)=\frac{A_{d_{U}}}{2\,\sin\left(d_{U}\pi\right)}\frac{1}{\left(-P^{2}-i\,0^{+}\right)^{2-d_{U}}}\;. (4) The presence of an extra CP-conserving phase factor exp ⁡ ( − i ​ d U ​ π ) 𝑖 subscript 𝑑 𝑈 𝜋 \exp\left(-id_{U}\pi\right) in ( 4 ) for non-integral d U subscript 𝑑 𝑈 d_{U} and time-like P 2 > 0 superscript 𝑃 2 0 P^{2}>0 was pointed out in georgi2 ; CKY-short and the intricate interference effects in phenomenology were also first studied there. Analogous formulas can also be written down for the spin 1 georgi2 ; CKY-short and spin 2 CKY-long unparticle operators, while the fermionic spin 1/2 case was given in fermion-unparticle . Unparticle operators can interact with the Standard Model (SM) fields via exchange of some heavy particles of mass M 𝑀 M . Integrating out such heavy fields induces a series of effective operators describing how unparticle interact with SM fields at low energy unparticle . Some examples of these operators are unparticle ; CKY-short Spin ​ 0 : : Spin 0 absent \displaystyle\mathrm{Spin\;0:} λ 0 ′ ​ 1 Λ U d U − 1 ​ f ¯ ​ f ​ O , λ 0 ′′ ​ 1 Λ U d U − 1 ​ f ¯ ​ i ​ γ 5 ​ f ​ O , superscript subscript 𝜆 0 ′ 1 superscript subscript Λ 𝑈 subscript 𝑑 𝑈 1 ¯ 𝑓 𝑓 𝑂 superscript subscript 𝜆 0 ′′ 1 superscript subscript Λ 𝑈 subscript 𝑑 𝑈 1 ¯ 𝑓 𝑖 superscript 𝛾 5 𝑓 𝑂 \displaystyle\lambda_{0}^{\prime}\frac{1}{\Lambda_{U}^{d_{U}-1}}\overline{f}fO\;,\;\;\lambda_{0}^{\prime\prime}\frac{1}{\Lambda_{U}^{d_{U}-1}}\overline{f}i\gamma^{5}fO\;,\;\; λ 0 ​ 1 Λ U d U ​ G α ​ β ​ G α ​ β ​ O , ⋯ subscript 𝜆 0 1 superscript subscript Λ 𝑈 subscript 𝑑 𝑈 subscript 𝐺 𝛼 𝛽 superscript 𝐺 𝛼 𝛽 𝑂 ⋯ \displaystyle\lambda_{0}\frac{1}{\Lambda_{U}^{d_{U}}}G_{\alpha\beta}G^{\alpha\beta}O\;,\;\cdots Spin ​ 1 : : Spin 1 absent \displaystyle\mathrm{Spin\;1:} λ 1 ​ 1 Λ U d U − 1 ​ f ¯ ​ γ μ ​ f ​ O μ , λ 1 ′ ​ 1 Λ U d U − 1 ​ f ¯ ​ γ μ ​ γ 5 ​ f ​ O μ , ⋯ subscript 𝜆 1 1 superscript subscript Λ 𝑈 subscript 𝑑 𝑈 1 ¯ 𝑓 subscript 𝛾 𝜇 𝑓 superscript 𝑂 𝜇 superscript subscript 𝜆 1 ′ 1 superscript subscript Λ 𝑈 subscript 𝑑 𝑈 1 ¯ 𝑓 subscript 𝛾 𝜇 subscript 𝛾 5 𝑓 superscript 𝑂 𝜇 ⋯ \displaystyle\lambda_{1}\frac{1}{\Lambda_{U}^{d_{U}-1}}\,\overline{f}\gamma_{\mu}f\,O^{\mu}\;,\;\;\lambda_{1}^{\prime}\frac{1}{\Lambda_{U}^{d_{U}-1}}\,\overline{f}\gamma_{\mu}\gamma_{5}f\,O^{\mu}\;,\;\cdots Spin ​ 2 : : Spin 2 absent \displaystyle\mathrm{Spin\;2:} − 1 4 λ 2 ′ 1 Λ U d U ψ ¯ i ( γ μ 𝐃 ν ↔ + γ ν 𝐃 μ ↔ ) ψ O μ ​ ν , \displaystyle-\frac{1}{4}\lambda_{2}^{\prime}\frac{1}{\Lambda_{U}^{d_{U}}}\overline{\psi}\,i\left(\gamma_{\mu}\stackrel{{\scriptstyle\leftrightarrow}}{{{\bf D}}}_{\nu}+\gamma_{\nu}\stackrel{{\scriptstyle\leftrightarrow}}{{{\bf D}}}_{\mu}\right)\psi\,O^{\mu\nu}\;, λ 2 ​ 1 Λ U d U ​ G μ ​ α ​ G ν α ​ O μ ​ ν , ⋯ subscript 𝜆 2 1 superscript subscript Λ 𝑈 subscript 𝑑 𝑈 subscript 𝐺 𝜇 𝛼 superscript subscript 𝐺 𝜈 𝛼 superscript 𝑂 𝜇 𝜈 ⋯ \displaystyle\lambda_{2}\frac{1}{\Lambda_{U}^{d_{U}}}G_{\mu\alpha}G_{\nu}^{\;\alpha}O^{\mu\nu}\;,\;\cdots and so on. Here O , O μ 𝑂 superscript 𝑂 𝜇 O,O^{\mu} and O μ ​ ν superscript 𝑂 𝜇 𝜈 O^{\mu\nu} denote the spin 0, 1 and 2 unparticle operators respectively with d U subscript 𝑑 𝑈 d_{U} its corresponding scaling dimension, 2 2 2 To simplify notation, we do not distinguish the various scaling dimensions of the unparticle operators of different spins. f 𝑓 f is a SM fermion field, G α ​ β superscript 𝐺 𝛼 𝛽 G^{\alpha\beta} is a SM gauge field strength, 𝐃 𝐃 \bf D is a gauge covariant derivative acting on the SM fermion doublet ψ 𝜓 \psi . The λ 𝜆 \lambda -coefficients in front of these operators are effective couplings depend on the short distance physics among the heavy exchange particles with the hidden sector and the SM sector. Thus they are free parameters and a priori of order unity. Besides the above operators, more complete lists of gauge invariant operators have been written down in Chen-He . Antisymmetric rank 2 tensor unparticle operator was discussed in Hur-Ko-Wu . The hypothesis of scale invariance of unparticle physics can be extended further to become conformal. The implication to unparticle physics of unitarity constraints on the scaling dimensions deduced from general conformal field theories Mack was emphasized in Nakayama . The unparticle propagators for this general case have been worked out in GIR . Other theoretical aspects of unparticle were reviewed by Rajaraman Rajaraman-plenary .

We review some collider phenomenology of unparticle physics, including real emissions and virtual exchanges of unparticle. Existing experimental constraints from collider physics as well as astrophysics are briefly discussed.

[{"content":"Help me to generate the abstract of a survey paper given the title:*Unparticle Phenomenology—A Mini Review* and the introduction:Figure 1: Angular distributions for e − ​ e + → μ − ​ μ + → superscript 𝑒 superscript 𝑒 superscript 𝜇 superscript 𝜇 e^{-}e^{+}\to\mu^{-}\mu^{+} with various scaling dimension d U subscript 𝑑 𝑈 d_{U} for the spin 1 unparticle exchange with L ​ L + R ​ R 𝐿 𝐿 𝑅 𝑅 LL+RR contact terms plus the SM contributions at s = 200 𝑠 200 \sqrt{s}=200 GeV (left panel) and for the spin 2 unparticle exchange plus SM contributions at s 𝑠 \sqrt{s} = 0.5 TeV (right panel). We have set Λ U = 1 subscript Λ 𝑈 1 \Lambda_{U}=1 TeV and λ 1 = λ 2 = 1 subscript 𝜆 1 subscript 𝜆 2 1 \lambda_{1}=\lambda_{2}=1 . See CKY-long for details. The notion of unparticle introduced by Georgi unparticle was based on the hypothesis that there could be an exact scale invariant hidden sector resisted at a high energy scale. A prototype model of such sector is given by the Banks-Zaks theory Banks-Zaks which flows to an infrared fixed point at a lower energy scale Λ U subscript Λ 𝑈 \Lambda_{U} through dimensional transmutation. Below Λ U subscript Λ 𝑈 \Lambda_{U} , unparticle physics emerges and manifest itself as interpolating fields O 𝑂 O of various scaling dimensions and Lorentz structures. One of the interesting feature of unparticle operator is that it has a continuous spectral density ρ ​ ( P 2 ) 𝜌 superscript 𝑃 2 \rho(P^{2}) as a consequence of scale invariance unparticle ρ ​ ( P 2 ) = A d U ​ θ ​ ( P 0 ) ​ θ ​ ( P 2 ) ​ ( P 2 ) d U − 2 𝜌 superscript 𝑃 2 subscript 𝐴 subscript 𝑑 𝑈 𝜃 superscript 𝑃 0 𝜃 superscript 𝑃 2 superscript superscript 𝑃 2 subscript 𝑑 𝑈 2 \rho(P^{2})=A_{d_{U}}\theta(P^{0})\theta(P^{2})(P^{2})^{d_{U}-2} (1) where d U subscript 𝑑 𝑈 d_{U} is the scaling dimension of the unparticle operator with 4-momentum P 𝑃 P and A d U subscript 𝐴 subscript 𝑑 𝑈 A_{d_{U}} is a free normalization factor. Due to their similar kinematical exponent, Georgi unparticle chose A d U subscript 𝐴 subscript 𝑑 𝑈 A_{d_{U}} to be the prefactor of d U subscript 𝑑 𝑈 d_{U} massless particle phase space d ​ LIPS d U = A d U ​ ( ( p 1 + p 2 + ⋯ + p d U ) 2 ) d U − 2 𝑑 subscript LIPS subscript 𝑑 𝑈 subscript 𝐴 subscript 𝑑 𝑈 superscript superscript subscript 𝑝 1 subscript 𝑝 2 ⋯ subscript 𝑝 subscript 𝑑 𝑈 2 subscript 𝑑 𝑈 2 d{\rm LIPS}_{d_{U}}=A_{d_{U}}\left(\left(p_{1}+p_{2}+\cdots+p_{d_{U}}\right)^{2}\right)^{d_{U}-2} (2) where p i 2 = 0 ​ ( i = 1 , … , d U ) superscript subscript 𝑝 𝑖 2 0 𝑖 1 … subscript 𝑑 𝑈 p_{i}^{2}=0\,(i=1,...,d_{U}) . Thus A d U = 16 ​ π 5 / 2 ( 2 ​ π ) 2 ​ d U ​ Γ ​ ( d U + 1 2 ) Γ ​ ( d U − 1 ) ​ Γ ​ ( 2 ​ d U ) . subscript 𝐴 subscript 𝑑 𝑈 16 superscript 𝜋 5 2 superscript 2 𝜋 2 subscript 𝑑 𝑈 Γ subscript 𝑑 𝑈 1 2 Γ subscript 𝑑 𝑈 1 Γ 2 subscript 𝑑 𝑈 A_{d_{U}}=\frac{16\pi^{5/2}}{(2\pi)^{2d_{U}}}\frac{\Gamma(d_{U}+\frac{1}{2})}{\Gamma(d_{U}-1)\Gamma(2d_{U})}\;. (3) As d U subscript 𝑑 𝑈 d_{U} approaches 1, Eq.( 1 ) reduces to the familiar one massless particle phase space. This suggests unparticle behaves like a collection of d U subscript 𝑑 𝑈 d_{U} massless particles. Since d U subscript 𝑑 𝑈 d_{U} can be non-integral, one can now speak of fractional particle. This metaphor draws immediate attention of many physicists worldwide as well as the general public. 1 1 1 One can even find an entry of unparticle in the Wikipedia database. Soon after the introduction of unparticle, its propagator was deduced by using unitarity cuts georgi2 and the spectral decomposition formula CKY-short . For a scalar unparticle, its Feynman propagator is given by Δ F ​ ( P ) = A d U 2 ​ sin ⁡ ( d U ​ π ) ​ 1 ( − P 2 − i ​ 0 + ) 2 − d U . subscript Δ 𝐹 𝑃 subscript 𝐴 subscript 𝑑 𝑈 2 subscript 𝑑 𝑈 𝜋 1 superscript superscript 𝑃 2 𝑖 superscript 0 2 subscript 𝑑 𝑈 \Delta_{F}(P)=\frac{A_{d_{U}}}{2\,\sin\left(d_{U}\pi\right)}\frac{1}{\left(-P^{2}-i\,0^{+}\right)^{2-d_{U}}}\;. (4) The presence of an extra CP-conserving phase factor exp ⁡ ( − i ​ d U ​ π ) 𝑖 subscript 𝑑 𝑈 𝜋 \exp\left(-id_{U}\pi\right) in ( 4 ) for non-integral d U subscript 𝑑 𝑈 d_{U} and time-like P 2 > 0 superscript 𝑃 2 0 P^{2}>0 was pointed out in georgi2 ; CKY-short and the intricate interference effects in phenomenology were also first studied there. Analogous formulas can also be written down for the spin 1 georgi2 ; CKY-short and spin 2 CKY-long unparticle operators, while the fermionic spin 1/2 case was given in fermion-unparticle . Unparticle operators can interact with the Standard Model (SM) fields via exchange of some heavy particles of mass M 𝑀 M . Integrating out such heavy fields induces a series of effective operators describing how unparticle interact with SM fields at low energy unparticle . Some examples of these operators are unparticle ; CKY-short Spin ​ 0 : : Spin 0 absent \displaystyle\mathrm{Spin\;0:} λ 0 ′ ​ 1 Λ U d U − 1 ​ f ¯ ​ f ​ O , λ 0 ′′ ​ 1 Λ U d U − 1 ​ f ¯ ​ i ​ γ 5 ​ f ​ O , superscript subscript 𝜆 0 ′ 1 superscript subscript Λ 𝑈 subscript 𝑑 𝑈 1 ¯ 𝑓 𝑓 𝑂 superscript subscript 𝜆 0 ′′ 1 superscript subscript Λ 𝑈 subscript 𝑑 𝑈 1 ¯ 𝑓 𝑖 superscript 𝛾 5 𝑓 𝑂 \displaystyle\lambda_{0}^{\prime}\frac{1}{\Lambda_{U}^{d_{U}-1}}\overline{f}fO\;,\;\;\lambda_{0}^{\prime\prime}\frac{1}{\Lambda_{U}^{d_{U}-1}}\overline{f}i\gamma^{5}fO\;,\;\; λ 0 ​ 1 Λ U d U ​ G α ​ β ​ G α ​ β ​ O , ⋯ subscript 𝜆 0 1 superscript subscript Λ 𝑈 subscript 𝑑 𝑈 subscript 𝐺 𝛼 𝛽 superscript 𝐺 𝛼 𝛽 𝑂 ⋯ \displaystyle\lambda_{0}\frac{1}{\Lambda_{U}^{d_{U}}}G_{\alpha\beta}G^{\alpha\beta}O\;,\;\cdots Spin ​ 1 : : Spin 1 absent \displaystyle\mathrm{Spin\;1:} λ 1 ​ 1 Λ U d U − 1 ​ f ¯ ​ γ μ ​ f ​ O μ , λ 1 ′ ​ 1 Λ U d U − 1 ​ f ¯ ​ γ μ ​ γ 5 ​ f ​ O μ , ⋯ subscript 𝜆 1 1 superscript subscript Λ 𝑈 subscript 𝑑 𝑈 1 ¯ 𝑓 subscript 𝛾 𝜇 𝑓 superscript 𝑂 𝜇 superscript subscript 𝜆 1 ′ 1 superscript subscript Λ 𝑈 subscript 𝑑 𝑈 1 ¯ 𝑓 subscript 𝛾 𝜇 subscript 𝛾 5 𝑓 superscript 𝑂 𝜇 ⋯ \displaystyle\lambda_{1}\frac{1}{\Lambda_{U}^{d_{U}-1}}\,\overline{f}\gamma_{\mu}f\,O^{\mu}\;,\;\;\lambda_{1}^{\prime}\frac{1}{\Lambda_{U}^{d_{U}-1}}\,\overline{f}\gamma_{\mu}\gamma_{5}f\,O^{\mu}\;,\;\cdots Spin ​ 2 : : Spin 2 absent \displaystyle\mathrm{Spin\;2:} − 1 4 λ 2 ′ 1 Λ U d U ψ ¯ i ( γ μ 𝐃 ν ↔ + γ ν 𝐃 μ ↔ ) ψ O μ ​ ν , \displaystyle-\frac{1}{4}\lambda_{2}^{\prime}\frac{1}{\Lambda_{U}^{d_{U}}}\overline{\psi}\,i\left(\gamma_{\mu}\stackrel{{\scriptstyle\leftrightarrow}}{{{\bf D}}}_{\nu}+\gamma_{\nu}\stackrel{{\scriptstyle\leftrightarrow}}{{{\bf D}}}_{\mu}\right)\psi\,O^{\mu\nu}\;, λ 2 ​ 1 Λ U d U ​ G μ ​ α ​ G ν α ​ O μ ​ ν , ⋯ subscript 𝜆 2 1 superscript subscript Λ 𝑈 subscript 𝑑 𝑈 subscript 𝐺 𝜇 𝛼 superscript subscript 𝐺 𝜈 𝛼 superscript 𝑂 𝜇 𝜈 ⋯ \displaystyle\lambda_{2}\frac{1}{\Lambda_{U}^{d_{U}}}G_{\mu\alpha}G_{\nu}^{\;\alpha}O^{\mu\nu}\;,\;\cdots and so on. Here O , O μ 𝑂 superscript 𝑂 𝜇 O,O^{\mu} and O μ ​ ν superscript 𝑂 𝜇 𝜈 O^{\mu\nu} denote the spin 0, 1 and 2 unparticle operators respectively with d U subscript 𝑑 𝑈 d_{U} its corresponding scaling dimension, 2 2 2 To simplify notation, we do not distinguish the various scaling dimensions of the unparticle operators of different spins. f 𝑓 f is a SM fermion field, G α ​ β superscript 𝐺 𝛼 𝛽 G^{\alpha\beta} is a SM gauge field strength, 𝐃 𝐃 \bf D is a gauge covariant derivative acting on the SM fermion doublet ψ 𝜓 \psi . The λ 𝜆 \lambda -coefficients in front of these operators are effective couplings depend on the short distance physics among the heavy exchange particles with the hidden sector and the SM sector. Thus they are free parameters and a priori of order unity. Besides the above operators, more complete lists of gauge invariant operators have been written down in Chen-He . Antisymmetric rank 2 tensor unparticle operator was discussed in Hur-Ko-Wu . The hypothesis of scale invariance of unparticle physics can be extended further to become conformal. The implication to unparticle physics of unitarity constraints on the scaling dimensions deduced from general conformal field theories Mack was emphasized in Nakayama . The unparticle propagators for this general case have been worked out in GIR . Other theoretical aspects of unparticle were reviewed by Rajaraman Rajaraman-plenary .","role":"user"},{"content":"Abstract: We review some collider phenomenology of unparticle physics, including real emissions and virtual exchanges of unparticle. Existing experimental constraints from collider physics as well as astrophysics are briefly discussed.","role":"assistant"}]

A survey of large N continuum phase transitions

1 Large N 𝑁 N QCD in the ’t Hooft limit Large N 𝑁 N gauge theories [ 1 ] are qualitatively similar to QCD with three colors and it is long held hope to solve it analytically for N = ∞ 𝑁 N=\infty . Only planar diagrams contribute in this limit and fermions in the fundamental representation are naturally quenched as long as the number of flavors is finite. Researchers in string theory and gravity also address the problem of large N 𝑁 N gauge theories but they are a long way from solving it analytically [ 2 ] . We will discuss various physical properties of large N 𝑁 N QCD in the ’t Hooft limit. We will start with well known results in d = 2 𝑑 2 d=2 and proceed to a discussion of relatively new results in d = 3 𝑑 3 d=3 and d = 4 𝑑 4 d=4 . The theory will be regulated using the lattice formalism and the lattice bare coupling, b = 1 g 2 ​ N 𝑏 1 superscript 𝑔 2 𝑁 b=\frac{1}{g^{2}N} , will be held fixed as g → 0 → 𝑔 0 g\rightarrow 0 and N → ∞ → 𝑁 N\rightarrow\infty . We will assume that we only have a finite number of fermion flavors and therefore fermions will be naturally quenched as long as we are in the confined phase and there is no chemical potential. The continuum limit corresponds to b → ∞ → 𝑏 b\rightarrow\infty . All our discussion will be on a periodic lattice at a finite physical volume. Specifically, • d = 2 𝑑 2 d=2 : l x , y = L x , y b subscript 𝑙 𝑥 𝑦 subscript 𝐿 𝑥 𝑦 𝑏 l_{x,y}=\frac{L_{x,y}}{\sqrt{b}} . The physical sizes l x ≤ l y subscript 𝑙 𝑥 subscript 𝑙 𝑦 l_{x}\leq l_{y} are kept fixed as L x , y subscript 𝐿 𝑥 𝑦 L_{x,y} and b 𝑏 b are taken to ∞ \infty . • d = 3 𝑑 3 d=3 : l x , y , z = L x , y , z b I subscript 𝑙 𝑥 𝑦 𝑧 subscript 𝐿 𝑥 𝑦 𝑧 subscript 𝑏 𝐼 l_{x,y,z}=\frac{L_{x,y,z}}{b_{I}} ; We will use the tadpole improved coupling [ 3 ] , b I = b ​ e ​ ( b ) subscript 𝑏 𝐼 𝑏 𝑒 𝑏 b_{I}=be(b) where e ​ ( b ) 𝑒 𝑏 e(b) is the average value of the plaquette. The physical sizes l x ≤ l y ≤ l z subscript 𝑙 𝑥 subscript 𝑙 𝑦 subscript 𝑙 𝑧 l_{x}\leq l_{y}\leq l_{z} are kept fixed as L x , y , z subscript 𝐿 𝑥 𝑦 𝑧 L_{x,y,z} and b 𝑏 b are taken to ∞ \infty . • d = 4 𝑑 4 d=4 : l x , y , z , t = L x , y , z , t ​ [ 48 ​ π 2 ​ b I 11 ] 51 121 ​ e − 24 ​ π 2 ​ b I 11 subscript 𝑙 𝑥 𝑦 𝑧 𝑡 subscript 𝐿 𝑥 𝑦 𝑧 𝑡 superscript delimited-[] 48 superscript 𝜋 2 subscript 𝑏 𝐼 11 51 121 superscript 𝑒 24 superscript 𝜋 2 subscript 𝑏 𝐼 11 l_{x,y,z,t}=L_{x,y,z,t}\left[\frac{48\pi^{2}b_{I}}{11}\right]^{\frac{51}{121}}e^{-\frac{24\pi^{2}b_{I}}{11}} . The physical sizes l x ≤ l y ≤ l z ≤ l t subscript 𝑙 𝑥 subscript 𝑙 𝑦 subscript 𝑙 𝑧 subscript 𝑙 𝑡 l_{x}\leq l_{y}\leq l_{z}\leq l_{t} are kept fixed as L x , y , z , t subscript 𝐿 𝑥 𝑦 𝑧 𝑡 L_{x,y,z,t} and b 𝑏 b are taken to ∞ \infty .

The main focus of this talk is the physics of large N QCD on a continuum torus. A cascade of phase transitions associated with the breaking of U(1) symmetries will be discussed. The continuum Wilson loop as a function of its area will be discussed along with its universality properties and the associated double scaling limit. Some recent progress in twisted Eguchi-Kawai is presented. Gauge field topology and $\theta$ vacuua are also discussed in the context of large N gauge theories. Phase transitions in 2D large N principal chiral models are compared with similar transitions in large $N$ gauge theories. Finally, connections to some topics in string theory and gravity are briefly described.

[{"content":"Help me to generate the abstract of a survey paper given the title:*A survey of large N continuum phase transitions* and the introduction:1 Large N 𝑁 N QCD in the t Hooft limit Large N 𝑁 N gauge theories 1 are qualitatively similar to QCD with three colors and it is long held hope to solve it analytically for N = ∞ 𝑁 N=\infty . Only planar diagrams contribute in this limit and fermions in the fundamental representation are naturally quenched as long as the number of flavors is finite. Researchers in string theory and gravity also address the problem of large N 𝑁 N gauge theories but they are a long way from solving it analytically 2 . We will discuss various physical properties of large N 𝑁 N QCD in the t Hooft limit. We will start with well known results in d = 2 𝑑 2 d=2 and proceed to a discussion of relatively new results in d = 3 𝑑 3 d=3 and d = 4 𝑑 4 d=4 . The theory will be regulated using the lattice formalism and the lattice bare coupling, b = 1 g 2 ​ N 𝑏 1 superscript 𝑔 2 𝑁 b=\frac{1}{g^{2}N} , will be held fixed as g → 0 → 𝑔 0 g\rightarrow 0 and N → ∞ → 𝑁 N\rightarrow\infty . We will assume that we only have a finite number of fermion flavors and therefore fermions will be naturally quenched as long as we are in the confined phase and there is no chemical potential. The continuum limit corresponds to b → ∞ → 𝑏 b\rightarrow\infty . All our discussion will be on a periodic lattice at a finite physical volume. Specifically, • d = 2 𝑑 2 d=2 : l x , y = L x , y b subscript 𝑙 𝑥 𝑦 subscript 𝐿 𝑥 𝑦 𝑏 l_{x,y}=\frac{L_{x,y}}{\sqrt{b}} . The physical sizes l x ≤ l y subscript 𝑙 𝑥 subscript 𝑙 𝑦 l_{x}\leq l_{y} are kept fixed as L x , y subscript 𝐿 𝑥 𝑦 L_{x,y} and b 𝑏 b are taken to ∞ \infty . • d = 3 𝑑 3 d=3 : l x , y , z = L x , y , z b I subscript 𝑙 𝑥 𝑦 𝑧 subscript 𝐿 𝑥 𝑦 𝑧 subscript 𝑏 𝐼 l_{x,y,z}=\frac{L_{x,y,z}}{b_{I}} ; We will use the tadpole improved coupling 3 , b I = b ​ e ​ ( b ) subscript 𝑏 𝐼 𝑏 𝑒 𝑏 b_{I}=be(b) where e ​ ( b ) 𝑒 𝑏 e(b) is the average value of the plaquette. The physical sizes l x ≤ l y ≤ l z subscript 𝑙 𝑥 subscript 𝑙 𝑦 subscript 𝑙 𝑧 l_{x}\leq l_{y}\leq l_{z} are kept fixed as L x , y , z subscript 𝐿 𝑥 𝑦 𝑧 L_{x,y,z} and b 𝑏 b are taken to ∞ \infty . • d = 4 𝑑 4 d=4 : l x , y , z , t = L x , y , z , t ​ 48 ​ π 2 ​ b I 11 51 121 ​ e − 24 ​ π 2 ​ b I 11 subscript 𝑙 𝑥 𝑦 𝑧 𝑡 subscript 𝐿 𝑥 𝑦 𝑧 𝑡 superscript delimited- 48 superscript 𝜋 2 subscript 𝑏 𝐼 11 51 121 superscript 𝑒 24 superscript 𝜋 2 subscript 𝑏 𝐼 11 l_{x,y,z,t}=L_{x,y,z,t}\left\frac{48\pi^{2}b_{I}}{11}\right^{\frac{51}{121}}e^{-\frac{24\pi^{2}b_{I}}{11}} . The physical sizes l x ≤ l y ≤ l z ≤ l t subscript 𝑙 𝑥 subscript 𝑙 𝑦 subscript 𝑙 𝑧 subscript 𝑙 𝑡 l_{x}\leq l_{y}\leq l_{z}\leq l_{t} are kept fixed as L x , y , z , t subscript 𝐿 𝑥 𝑦 𝑧 𝑡 L_{x,y,z,t} and b 𝑏 b are taken to ∞ \infty .","role":"user"},{"content":"Abstract: The main focus of this talk is the physics of large N QCD on a continuum torus. A cascade of phase transitions associated with the breaking of U(1) symmetries will be discussed. The continuum Wilson loop as a function of its area will be discussed along with its universality properties and the associated double scaling limit. Some recent progress in twisted Eguchi-Kawai is presented. Gauge field topology and $\theta$ vacuua are also discussed in the context of large N gauge theories. Phase transitions in 2D large N principal chiral models are compared with similar transitions in large $N$ gauge theories. Finally, connections to some topics in string theory and gravity are briefly described.","role":"assistant"}]

Survey on Affine Spheres

1. Introduction Affine spheres were introduced by Ţiţeica in [ 72 , 73 ] , and studied later by Blaschke, Calabi, and Cheng-Yau, among others. These are hypersurfaces in affine ℝ n + 1 superscript ℝ 𝑛 1 \mathbb{R}^{n+1} which are related to real Monge-Ampère equations, to projective structures on manifolds, and to the geometry of Calabi-Yau manifolds. In this survey article, we will outline the theory of affine spheres their relationships to these topics. Affine differential geometry is the study of those differential properties of hypersurfaces of ℝ n + 1 superscript ℝ 𝑛 1 \mathbb{R}^{n+1} which are invariant under all volume-preserving affine transformations. Affine differential geometry is largely traced to Ţiţeica’s papers in 1908-09, although for curves in ℝ 2 superscript ℝ 2 \mathbb{R}^{2} , one of the main invariants, the affine normal, was already introduced by Transon [ 69 ] in 1841. Given a smooth hypersurface H ⊂ ℝ n + 1 𝐻 superscript ℝ 𝑛 1 H\subset\mathbb{R}^{n+1} , the affine normal ξ 𝜉 \xi is an affine-invariant transverse vector field to H 𝐻 H . Define the special affine group as 𝐒𝐀 ​ ( n + 1 , ℝ ) = { Φ : x ↦ A ​ x + b , det A = 1 } . 𝐒𝐀 𝑛 1 ℝ conditional-set Φ formulae-sequence maps-to 𝑥 𝐴 𝑥 𝑏 𝐴 1 \mathbf{SA}(n+1,\mathbb{R})=\{\Phi\!:x\mapsto Ax+b,\,\det A=1\}. The invariance property of the affine normal is then Φ ∗ ​ ξ H ​ ( x ) = ξ Φ ​ ( H ) ​ ( Φ ​ ( x ) ) subscript Φ subscript 𝜉 𝐻 𝑥 subscript 𝜉 Φ 𝐻 Φ 𝑥 \Phi_{*}\xi_{H}(x)=\xi_{\Phi(H)}(\Phi(x)) for any x ∈ H 𝑥 𝐻 x\in H . An improper affine sphere is a hypersurface H 𝐻 H whose affine normals are all parallel, while a proper affine sphere is a hypersurface whose affine normal lines all meet in a point, the center of the affine sphere. By symmetry, a Euclidean sphere must be a proper affine sphere, and affine invariance then shows that all ellipsoids are affine spheres also. More generally, quadric hypersurfaces are the canonical examples of affine spheres. We will mainly focus on the case of convex hypersurfaces in this survey, since in this case the natural invariant metric, the affine metric, is positive definite, and so we can exploit techniques of Riemannian geometry and elliptic PDEs. We assume for a convex hypersurface that the affine normal points to the convex side of the hypersurface. Improper affine sphere are then called parabolic affine spheres , and the primary example is an elliptic paraboloid. For convex hypersurfaces, there are naturally two types of proper affine spheres, depending on whether the affine normal points toward or away from the center. For an elliptic affine sphere , such as an ellipsoid, the affine normals point inward toward the center. Hyperbolic affine spheres have affine normals which point away from the center. One component of a hyperboloid of two sheets is the quadric example of a hyperbolic affine sphere. Some 15 years Ţiţeica’s papers, Blaschke’s book [ 6 ] records much of the early development of affine geometry. Calabi’s papers contain many advances in the theory of affine spheres and related subjects, and Cheng-Yau’s resolution of the structure of hyperbolic affine spheres provides crucial analytic estimates related to Monge-Ampère equations [ 12 , 14 ] . We refer the reader to the books of Nomizu-Sasaki [ 57 ] and Li-Simon-Zhao [ 44 ] for overviews of affine differential geometry. Nomizu-Sasaki [ 57 ] develop the theory in the modern notation of connections, while Li-Simon-Zhao [ 44 ] use Cartan’s moving frame techniques and also provide many analytic details. This survey article focuses on the relationship between the geometry of affine spheres to geometric structures on manifolds. To this end in Section 4 , we describe the semi-linear PDE of Ţiţeica and Wang involving the cubic differential and developing map for affine spheres in ℝ 3 superscript ℝ 3 \mathbb{R}^{3} . Then we outline the relationship, due to Blaschke and Calabi, of affine spheres to real Monge-Ampère equations and the basic duality results related related to the Legendre transform in Section 5 . In Sections 7 - 8 , we discuss Cheng-Yau’s work on hyperbolic affine spheres and invariants of convex cones. In Section 9 , we relate affine spheres to the geometry of affine manifolds, and the conjecture of Strominger-Yau-Zaslow, relating parabolic affine spheres to Calabi-Yau manifolds. In the last two sections, we discuss two generalizations of affine spheres: Affine maximal hypersurfaces are generalizations of parabolic affine spheres which have led to significant progress in the theory of fourth-order elliptic equations in the solution of Chern’s conjecture for affine maximal surfaces in ℝ 3 superscript ℝ 3 \mathbb{R}^{3} by Trudinger-Wang [ 70 ] . We also briefly discuss the affine normal flow, which is the natural parabolic analog of the elliptic PDEs of affine spheres. The selection of topics reflects the author’s interests, and there are many important subjects which lie outside the author’s expertise. We largely do not treat nonconvex affine spheres, and we only mention a few of the recent results classifying affine hypersurfaces with extremal geometric conditions. This article is dedicated to Prof. S.T. Yau, who introduced me to affine differential geometry, for his rich insight and kind encouragement over the years, on the occasion of his 59 th birthday.

We give a survey of the theory of affine spheres, emphasizing the convex cases and relationsships to Monge-Ampere equations and geometric structures on manifolds.

[{"content":"Help me to generate the abstract of a survey paper given the title:*Survey on Affine Spheres* and the introduction:1. Introduction Affine spheres were introduced by Ţiţeica in 72 , 73 , and studied later by Blaschke, Calabi, and Cheng-Yau, among others. These are hypersurfaces in affine ℝ n + 1 superscript ℝ 𝑛 1 \mathbb{R}^{n+1} which are related to real Monge-Ampère equations, to projective structures on manifolds, and to the geometry of Calabi-Yau manifolds. In this survey article, we will outline the theory of affine spheres their relationships to these topics. Affine differential geometry is the study of those differential properties of hypersurfaces of ℝ n + 1 superscript ℝ 𝑛 1 \mathbb{R}^{n+1} which are invariant under all volume-preserving affine transformations. Affine differential geometry is largely traced to Ţiţeicas papers in 1908-09, although for curves in ℝ 2 superscript ℝ 2 \mathbb{R}^{2} , one of the main invariants, the affine normal, was already introduced by Transon 69 in 1841. Given a smooth hypersurface H ⊂ ℝ n + 1 𝐻 superscript ℝ 𝑛 1 H\subset\mathbb{R}^{n+1} , the affine normal ξ 𝜉 \xi is an affine-invariant transverse vector field to H 𝐻 H . Define the special affine group as 𝐒𝐀 ​ ( n + 1 , ℝ ) = { Φ : x ↦ A ​ x + b , det A = 1 } . 𝐒𝐀 𝑛 1 ℝ conditional-set Φ formulae-sequence maps-to 𝑥 𝐴 𝑥 𝑏 𝐴 1 \mathbf{SA}(n+1,\mathbb{R})=\{\Phi\!:x\mapsto Ax+b,\,\det A=1\}. The invariance property of the affine normal is then Φ ∗ ​ ξ H ​ ( x ) = ξ Φ ​ ( H ) ​ ( Φ ​ ( x ) ) subscript Φ subscript 𝜉 𝐻 𝑥 subscript 𝜉 Φ 𝐻 Φ 𝑥 \Phi_{*}\xi_{H}(x)=\xi_{\Phi(H)}(\Phi(x)) for any x ∈ H 𝑥 𝐻 x\in H . An improper affine sphere is a hypersurface H 𝐻 H whose affine normals are all parallel, while a proper affine sphere is a hypersurface whose affine normal lines all meet in a point, the center of the affine sphere. By symmetry, a Euclidean sphere must be a proper affine sphere, and affine invariance then shows that all ellipsoids are affine spheres also. More generally, quadric hypersurfaces are the canonical examples of affine spheres. We will mainly focus on the case of convex hypersurfaces in this survey, since in this case the natural invariant metric, the affine metric, is positive definite, and so we can exploit techniques of Riemannian geometry and elliptic PDEs. We assume for a convex hypersurface that the affine normal points to the convex side of the hypersurface. Improper affine sphere are then called parabolic affine spheres , and the primary example is an elliptic paraboloid. For convex hypersurfaces, there are naturally two types of proper affine spheres, depending on whether the affine normal points toward or away from the center. For an elliptic affine sphere , such as an ellipsoid, the affine normals point inward toward the center. Hyperbolic affine spheres have affine normals which point away from the center. One component of a hyperboloid of two sheets is the quadric example of a hyperbolic affine sphere. Some 15 years Ţiţeicas papers, Blaschkes book 6 records much of the early development of affine geometry. Calabis papers contain many advances in the theory of affine spheres and related subjects, and Cheng-Yaus resolution of the structure of hyperbolic affine spheres provides crucial analytic estimates related to Monge-Ampère equations 12 , 14 . We refer the reader to the books of Nomizu-Sasaki 57 and Li-Simon-Zhao 44 for overviews of affine differential geometry. Nomizu-Sasaki 57 develop the theory in the modern notation of connections, while Li-Simon-Zhao 44 use Cartans moving frame techniques and also provide many analytic details. This survey article focuses on the relationship between the geometry of affine spheres to geometric structures on manifolds. To this end in Section 4 , we describe the semi-linear PDE of Ţiţeica and Wang involving the cubic differential and developing map for affine spheres in ℝ 3 superscript ℝ 3 \mathbb{R}^{3} . Then we outline the relationship, due to Blaschke and Calabi, of affine spheres to real Monge-Ampère equations and the basic duality results related related to the Legendre transform in Section 5 . In Sections 7 - 8 , we discuss Cheng-Yaus work on hyperbolic affine spheres and invariants of convex cones. In Section 9 , we relate affine spheres to the geometry of affine manifolds, and the conjecture of Strominger-Yau-Zaslow, relating parabolic affine spheres to Calabi-Yau manifolds. In the last two sections, we discuss two generalizations of affine spheres: Affine maximal hypersurfaces are generalizations of parabolic affine spheres which have led to significant progress in the theory of fourth-order elliptic equations in the solution of Cherns conjecture for affine maximal surfaces in ℝ 3 superscript ℝ 3 \mathbb{R}^{3} by Trudinger-Wang 70 . We also briefly discuss the affine normal flow, which is the natural parabolic analog of the elliptic PDEs of affine spheres. The selection of topics reflects the authors interests, and there are many important subjects which lie outside the authors expertise. We largely do not treat nonconvex affine spheres, and we only mention a few of the recent results classifying affine hypersurfaces with extremal geometric conditions. This article is dedicated to Prof. S.T. Yau, who introduced me to affine differential geometry, for his rich insight and kind encouragement over the years, on the occasion of his 59 th birthday.","role":"user"},{"content":"Abstract: We give a survey of the theory of affine spheres, emphasizing the convex cases and relationsships to Monge-Ampere equations and geometric structures on manifolds.","role":"assistant"}]

Quantum information processing using nuclear and electron magnetic resonance: review and prospects

I Liquid state NMR I.1 Magnetic interactions In the semi-classical picture, the spin of a nucleus behaves like the dipolar moment of a magnet possessing angular momentum parallel to its magnetic moment. When placed in a constant magnetic field pointing along a certain direction, (customarily defined as the z 𝑧 z direction) the dipolar moment precesses around this axis. The frequency of this precession is called the Larmor frequency and is dependent on the external magnetic field, the nuclear isotope and its chemical environment within the molecule. For quantum information purposes, we are mainly interested in spin-1/2 nuclei (e.g. 1 H, 13 C, 15 N, 19 F, 29 Si and 31 P to name a few). Placed in magnetic fields generated by modern superconducting magnets, different species of nuclei have differences in Larmor frequency on the order of MHz. For example, the Larmor frequency of 1 H is about 500 MHz in a 11.7 Tesla magnet, while that of 13 C is about 125 MHz. Depending on the symmetry of the molecule, two nuclei of the same species can either have the same Larmor frequency, or can have a frequency difference (called chemical shift) ranging from a few Hz to several kHz. Typical liquid state NMR experiments involve an ensemble of around 10 20 superscript 10 20 10^{20} identical molecules dissolved in a solvent whose effect on the nuclear magnetic moments of our molecules can be neglected. When two spins are spatially close, their dipolar moments interact with each other. The strength of this coupling is dependent on the distance between the two spins and their relative orientation with respect to the external magnetic field. In a liquid, the molecules move and rotate around each other on a much shorter time scale than the interactions occurring between them. This causes the intermolecular and intra-molecular dipolar interactions to average to zero on the NMR time scale (i.e. the Larmor period time scale). In solid state NMR, however, dipolar interactions remain and can be controlled as discussed in section II . Within the same molecule, there are still interactions between the spins in the liquid state. If the wavefunctions of bonding electrons overlap spatially with a pair of nuclear spins, the electron mediates an effective interaction between the nuclear spins. This interaction is independent of the external magnetic field and the orientation of the molecule, which inspires its name: scalar coupling (also called indirect spin-spin coupling, or J 𝐽 J -coupling). I.2 The NMR Hamiltonian As mentioned above, in liquid state NMR the intermolecular spin interactions are suppressed. This causes the molecules to be effectively isolated from each other, and therefore a description of the spin dynamics of an ensemble of molecules is well approximated by the spin dynamics of a single molecule. If we consider a molecule containing N 𝑁 N spin-1/2 nuclei in one of the molecules, then the natural Hamiltonian of this system in a large homogeneous magnetic field B → 0 subscript → 𝐵 0 \vec{B}_{0} pointing in the z 𝑧 z direction is given by ℋ n ​ a ​ t subscript ℋ 𝑛 𝑎 𝑡 \displaystyle\mathcal{{H}}_{nat} = \displaystyle= 1 2 ​ ∑ i = 1 N 2 ​ π ​ ν i L ​ σ z i + π 2 ​ ∑ i < j J i ​ j ​ σ z i ​ σ z j 1 2 superscript subscript 𝑖 1 𝑁 2 𝜋 superscript subscript 𝜈 𝑖 𝐿 superscript subscript 𝜎 𝑧 𝑖 𝜋 2 subscript 𝑖 𝑗 subscript 𝐽 𝑖 𝑗 superscript subscript 𝜎 𝑧 𝑖 superscript subscript 𝜎 𝑧 𝑗 \displaystyle\frac{1}{2}\sum_{i=1}^{N}2\pi\nu_{i}^{L}{{\sigma}_{z}^{i}}+\frac{\pi}{2}\sum_{i<j}J_{ij}{{\sigma}_{z}^{i}}{{\sigma}_{z}^{j}} (1) where ν i L = ω i L / 2 ​ π = γ i ​ | B → 0 | superscript subscript 𝜈 𝑖 𝐿 superscript subscript 𝜔 𝑖 𝐿 2 𝜋 subscript 𝛾 𝑖 subscript → 𝐵 0 \nu_{i}^{L}=\omega_{i}^{L}/2\pi=\gamma_{i}|\vec{B}_{0}| is the Larmor frequency of the i t ​ h superscript 𝑖 𝑡 ℎ i^{th} nucleus with gyromagnetic ratio γ i subscript 𝛾 𝑖 \gamma_{i} , J i ​ j subscript 𝐽 𝑖 𝑗 J_{ij} is the coupling strength between nucleus i 𝑖 i and j 𝑗 j and σ z i superscript subscript 𝜎 𝑧 𝑖 {{\sigma}_{z}^{i}} is the z 𝑧 z Pauli matrix of the i th superscript 𝑖 th i^{\textrm{th}} spin. The first term in the Hamiltonian describes the precession of the spins due to their coupling to the external magnetic field, while the second term describes the J 𝐽 J -coupling between pairs of nuclei. This Hamiltonian corresponds to the weak coupling limit, where we assume that the chemical shifts between coupled spins are much greater than their respective couplings, i.e. | ν i L − ν j L | >> J i ​ j / 2 much-greater-than superscript subscript 𝜈 𝑖 𝐿 superscript subscript 𝜈 𝑗 𝐿 subscript 𝐽 𝑖 𝑗 2 |\nu_{i}^{L}-\nu_{j}^{L}|>>J_{ij}/2 . If this approximation is not valid, we need to use the full coupling Hamiltonian σ x i ​ σ x j + σ y i ​ σ y j + σ z i ​ σ z j superscript subscript 𝜎 𝑥 𝑖 superscript subscript 𝜎 𝑥 𝑗 superscript subscript 𝜎 𝑦 𝑖 superscript subscript 𝜎 𝑦 𝑗 superscript subscript 𝜎 𝑧 𝑖 superscript subscript 𝜎 𝑧 𝑗 {{\sigma}_{x}^{i}}{{\sigma}_{x}^{j}}+{{\sigma}_{y}^{i}}{{\sigma}_{y}^{j}}+{{\sigma}_{z}^{i}}{{\sigma}_{z}^{j}} in place of σ z i ​ σ z j superscript subscript 𝜎 𝑧 𝑖 superscript subscript 𝜎 𝑧 𝑗 {{\sigma}_{z}^{i}}{{\sigma}_{z}^{j}} . The exact values of the Hamiltonian parameters are determined by fitting experimental data. I.3 Single-spin control For quantum information processing, we need to be able to perform arbitrary manipulations of a single spin, which is equivalent to arbitrary rotations about any axis. As an example, consider the application of a magnetic field B → 1 subscript → 𝐵 1 \vec{B}_{1} perpendicular to the z 𝑧 z axis which oscillates at the nuclear spin’s Larmor frequency: B → 1 subscript → 𝐵 1 \displaystyle\vec{B}_{1} = \displaystyle= | B → 1 | ​ ( cos ⁡ ( ω r ​ f ​ t ) ​ x → + sin ⁡ ( ω r ​ f ​ t ) ​ y → ) subscript → 𝐵 1 superscript 𝜔 𝑟 𝑓 𝑡 → 𝑥 superscript 𝜔 𝑟 𝑓 𝑡 → 𝑦 \displaystyle|\vec{B}_{1}|\left(\cos(\omega^{rf}t)\,\vec{x}+\sin(\omega^{rf}t)\,\vec{y}\right) (2) where ω r ​ f = 2 ​ π ​ ν r ​ f superscript 𝜔 𝑟 𝑓 2 𝜋 superscript 𝜈 𝑟 𝑓 \omega^{rf}=2\pi\nu^{rf} is the angular frequency of the field. In the rotating frame of the nucleus (i.e. the frame rotating at the same frequency as the spin), B → 1 subscript → 𝐵 1 \vec{B}_{1} will appear as a constant field pointing along its rotating x 𝑥 x axis. The spin will start to precess about this axis. Rotation about any axis in the x ​ y 𝑥 𝑦 xy -plane is possible by adjusting the phase of the B → 1 subscript → 𝐵 1 \vec{B}_{1} field, e.g. ω L ​ t → ω L ​ t + ϕ → superscript 𝜔 𝐿 𝑡 superscript 𝜔 𝐿 𝑡 italic-ϕ \omega^{L}t\to\omega^{L}t+\phi , which will create a rotation around the axis making an angle ϕ italic-ϕ \phi with the x 𝑥 x axis. In the laboratory, such a rotating field can be applied by sending a radio-frequency (RF) pulse of a particular duration and phase to a conducting coil surrounding the sample, calculated according to the rotating wave approximation (see Levitt ( 2001 ) for more details). To better understand this phenomenon from the viewpoint of quantum mechanics, consider the rotating frame picture: suppose the spin is in the state | ψ ​ ( t ) ⟩ ket 𝜓 𝑡 |\psi(t)\rangle , and define the state in the rotating frame of the pulse with angular frequency ω r ​ f superscript 𝜔 𝑟 𝑓 \omega^{rf} as | ψ ⟩ r subscript ket 𝜓 𝑟 \displaystyle|\psi\rangle_{r} = \displaystyle= R z ​ ( − ω r ​ f ​ t ) ​ | ψ ​ ( t ) ⟩ subscript 𝑅 𝑧 superscript 𝜔 𝑟 𝑓 𝑡 ket 𝜓 𝑡 \displaystyle{R}_{z}(-\omega^{rf}t)|\psi(t)\rangle (3) = \displaystyle= R z ​ ( − ω r ​ f ​ t ) ​ e − i ​ t ℏ ​ ℋ n ​ a ​ t ​ | ψ ​ ( 0 ) ⟩ subscript 𝑅 𝑧 superscript 𝜔 𝑟 𝑓 𝑡 superscript 𝑒 𝑖 𝑡 Planck-constant-over-2-pi subscript ℋ 𝑛 𝑎 𝑡 ket 𝜓 0 \displaystyle{R}_{z}(-\omega^{rf}t)e^{-\frac{it}{\hbar}\mathcal{{H}}_{nat}}|\psi(0)\rangle = \displaystyle= e i ℏ ​ σ z ​ ω r ​ f 2 ​ t ​ e − i ​ t ℏ ​ ℋ n ​ a ​ t ​ | ψ ​ ( 0 ) ⟩ superscript 𝑒 𝑖 Planck-constant-over-2-pi subscript 𝜎 𝑧 superscript 𝜔 𝑟 𝑓 2 𝑡 superscript 𝑒 𝑖 𝑡 Planck-constant-over-2-pi subscript ℋ 𝑛 𝑎 𝑡 ket 𝜓 0 \displaystyle e^{\frac{i}{\hbar}{\sigma}_{z}\frac{\omega^{rf}}{2}t}e^{-\frac{it}{\hbar}\mathcal{{H}}_{nat}}|\psi(0)\rangle = \displaystyle= | ψ ​ ( 0 ) ⟩ , for a single spin with ω r ​ f = ω L . ket 𝜓 0 for a single spin with ω r f = ω L \displaystyle|\psi(0)\rangle,\,\,\textrm{for a single spin with $\omega^{rf}=\omega^{L}$}. (4) If we apply a time derivative to equation 3 , it can be shown that the state in the rotating frame | ψ ⟩ r subscript ket 𝜓 𝑟 |\psi\rangle_{r} evolves according to the Schrödinger equation with the new Hamiltonian ℋ r subscript ℋ 𝑟 \displaystyle{\mathcal{{H}}}_{r} = \displaystyle= R z ​ ( − ω r ​ f ​ t ) ​ ℋ n ​ a ​ t ​ R z ​ ( ω r ​ f ​ t ) − ω r ​ f 2 ​ σ z . subscript 𝑅 𝑧 superscript 𝜔 𝑟 𝑓 𝑡 subscript ℋ 𝑛 𝑎 𝑡 subscript 𝑅 𝑧 superscript 𝜔 𝑟 𝑓 𝑡 superscript 𝜔 𝑟 𝑓 2 subscript 𝜎 𝑧 \displaystyle{R}_{z}(-\omega^{rf}t)\mathcal{{H}}_{nat}{R}_{z}(\omega^{rf}t)-\frac{\omega^{rf}}{2}{\sigma}_{z}. (5) When an RF pulse with phase ϕ italic-ϕ \phi is applied to the spin, the laboratory frame Hamiltonian is: ℋ ℋ \displaystyle\mathcal{{H}} = \displaystyle= ω L 2 ​ σ z + ω n ​ u ​ t 2 ​ ( cos ⁡ ( ω r ​ f ​ t + ϕ ) ​ σ x + sin ⁡ ( ω r ​ f ​ t + ϕ ) ​ σ y ) superscript 𝜔 𝐿 2 subscript 𝜎 𝑧 superscript 𝜔 𝑛 𝑢 𝑡 2 superscript 𝜔 𝑟 𝑓 𝑡 italic-ϕ subscript 𝜎 𝑥 superscript 𝜔 𝑟 𝑓 𝑡 italic-ϕ subscript 𝜎 𝑦 \displaystyle\frac{\omega^{L}}{2}{\sigma}_{z}+\frac{\omega^{nut}}{2}\left(\cos{(\omega^{rf}t+\phi)}{\sigma}_{x}+\sin{(\omega^{rf}t+\phi)}{\sigma}_{y}\right) (6) where ω n ​ u ​ t = π ​ γ i ​ | B → 1 | superscript 𝜔 𝑛 𝑢 𝑡 𝜋 subscript 𝛾 𝑖 subscript → 𝐵 1 \omega^{nut}=\pi\gamma_{i}|\vec{B}_{1}| . In the rotating frame this becomes ℋ r subscript ℋ 𝑟 \displaystyle{\mathcal{{H}}}_{r} = \displaystyle= 1 2 ​ ( ω L − ω r ​ f ) ​ σ z + 1 2 ​ ω n ​ u ​ t ​ ( cos ⁡ ϕ ​ σ x + sin ⁡ ϕ ​ σ y ) . 1 2 superscript 𝜔 𝐿 superscript 𝜔 𝑟 𝑓 subscript 𝜎 𝑧 1 2 superscript 𝜔 𝑛 𝑢 𝑡 italic-ϕ subscript 𝜎 𝑥 italic-ϕ subscript 𝜎 𝑦 \displaystyle\frac{1}{2}(\omega^{L}-\omega^{rf}){\sigma}_{z}+\frac{1}{2}\omega^{nut}(\cos\phi\,{\sigma}_{x}+\sin\phi\,{\sigma}_{y}). (7) Thus, if the RF pulse is at the same frequency as the spin, the spin will see a constant field in the x ​ y 𝑥 𝑦 xy plane, and will precess about it. The rotation angle θ 𝜃 \theta is determine by the interval τ 𝜏 \tau during which the RF field is applied, according to θ = ω n ​ u ​ t ​ τ 𝜃 superscript 𝜔 𝑛 𝑢 𝑡 𝜏 \theta=\omega^{nut}\tau . I.4 Adding a second spin It is also possible to independently control two spins with different Larmor frequencies. Applying an RF pulse at the frequency of the first spin, the rotating frame Hamiltonian is given by ℋ ~ n ​ a ​ t subscript ~ ℋ 𝑛 𝑎 𝑡 \displaystyle\widetilde{\mathcal{{H}}}_{nat} = \displaystyle= 1 2 ​ ω 1 n ​ u ​ t ​ σ x 1 + 1 2 ​ ω 2 n ​ u ​ t ​ σ x 2 + 1 2 ​ ( ω 2 L − ω 1 L ) ​ σ z 2 + π 2 ​ J 12 ​ σ z 1 ​ σ z 2 1 2 superscript subscript 𝜔 1 𝑛 𝑢 𝑡 superscript subscript 𝜎 𝑥 1 1 2 superscript subscript 𝜔 2 𝑛 𝑢 𝑡 superscript subscript 𝜎 𝑥 2 1 2 superscript subscript 𝜔 2 𝐿 superscript subscript 𝜔 1 𝐿 superscript subscript 𝜎 𝑧 2 𝜋 2 subscript 𝐽 12 superscript subscript 𝜎 𝑧 1 superscript subscript 𝜎 𝑧 2 \displaystyle\frac{1}{2}\omega_{1}^{nut}{\sigma}_{x}^{1}+\frac{1}{2}\omega_{2}^{nut}{\sigma}_{x}^{2}+\frac{1}{2}(\omega_{2}^{L}-\omega_{1}^{L}){\sigma}_{z}^{2}+\frac{\pi}{2}J_{12}{\sigma}_{z}^{1}{\sigma}_{z}^{2} (8) where we have set ϕ = 0 italic-ϕ 0 \phi=0 for simplicity. While the first spin undergoes a rotation around the x 𝑥 x axis, the second spin experiences a field with an additional non-zero z 𝑧 z component. This is called the off resonance effect. If we consider the case where ω 2 L − ω 2 L >> ω 1 n ​ u ​ t much-greater-than superscript subscript 𝜔 2 𝐿 superscript subscript 𝜔 2 𝐿 superscript subscript 𝜔 1 𝑛 𝑢 𝑡 \omega_{2}^{L}-\omega_{2}^{L}>>\omega_{1}^{nut} then the second spin rotation around the x 𝑥 x axis will average to zero during the time the first spin has completed its rotation. Typically, ω n ​ u ​ t superscript 𝜔 𝑛 𝑢 𝑡 \omega^{nut} is smaller than 1 MHz, so this condition is automatically satisfied if the two nuclei belong to different species. If the spins are of the same species, this condition can also be satisfied if a very low amplitude pulse is used due to the small nutation frequency. In this case, one drawback is that the pulse will necessarily take much longer to achieve the same angle of rotation, and if the two spins have a significant coupling constant J 12 subscript 𝐽 12 J_{12} coupling effects might introduce significant errors and therefore limit our control. Fortunately, there exist well known techniques to address different nuclei of the same species with high precision. The most common technique is to control the spins using shaped pulses. The frequency response to the pulse will depend on the pulse shape (Fourier theorem) and so by applying the pulse with a time varying power we can control the power spectrum of the pulse. For example, if a Gaussian shaped pulse is applied at frequency ω r ​ f superscript 𝜔 𝑟 𝑓 \omega^{rf} , then only spins within a Gaussian distribution of frequencies around ω r ​ f superscript 𝜔 𝑟 𝑓 \omega^{rf} will respond to this RF field. Therefore, if the height and the length of the Gaussian pulse is carefully chosen, one spin can be “addressed”, causing negligible effects to others. This technique permits control of spin pairs with smaller chemical shift differences in shorter periods of time, hence allowing stronger coupling. The length of a Gaussian pulse is proportional to the inverse of the chemical shift between the spins. Therefore, in the limiting case of small chemical shift differences and large J 𝐽 J -coupling values, control of the qubits is more difficult. For most liquid state experiments on a few spins, where chemical shifts are comparatively large and J 𝐽 J -couplings are small, the use of Gaussian pulses is sufficient to achieve very high precision spin rotations. The situation becomes more complicated when there are more homonuclear spins (implying smaller chemical shift differences on average), or stronger coupling like in solid state or liquid crystalenvironments. It is still possible to overcome these drawbacks by considering more complicated pulse shapes and phase modulation. For example, in section I.7 , we will describe how it is possible to find shaped pulses that can implement any desired evolution by simulating the full quantum dynamics. I.5 The controlled-NOT operation In the previous subsection we discussed a method used to independently control different spins. In order to perform quantum computing, we need to achieve universal control and hence be able to have spins interact with each other. A two-qubit gate that is useful for quantum information processing is the controlled-NOT, which acts as | 00 ⟩ → | 00 ⟩ → ket 00 ket 00 \displaystyle|00\rangle\to|00\rangle , | 10 ⟩ → | 11 ⟩ → ket 10 ket 11 \displaystyle|10\rangle\to|11\rangle | 01 ⟩ → | 01 ⟩ → ket 01 ket 01 \displaystyle|01\rangle\to|01\rangle , | 11 ⟩ → | 10 ⟩ . → ket 11 ket 10 \displaystyle|11\rangle\to|10\rangle. (9) The operation must flip the target qubit (second bit) if and only if the first qubit is in the state | 1 ⟩ ket 1 |1\rangle . In NMR, | 0 ⟩ ket 0 |0\rangle and | 1 ⟩ ket 1 |1\rangle are associated with the state of the spin pointing up, | ↑ ⟩ ket ↑ |\uparrow\rangle or pointing down | ↓ ⟩ ket ↓ |\downarrow\rangle respectively. If we look at the Hamiltonian in equation ( 1 ), and consider its effect on spin 2 depending on whether the state of spin 1 is up or down, we obtain an effective Hamiltonian for the second spin: ℋ ↑ 2 = 1 2 ​ ( ω 2 L + π ​ J 12 ) ​ σ z 2 superscript subscript ℋ ↑ 2 1 2 superscript subscript 𝜔 2 𝐿 𝜋 subscript 𝐽 12 superscript subscript 𝜎 𝑧 2 \displaystyle\mathcal{{H}}_{\uparrow}^{2}=\frac{1}{2}(\omega_{2}^{L}+\pi J_{12}){\sigma}_{z}^{2} (10) ℋ ↓ 2 = 1 2 ​ ( ω 2 L − π ​ J 12 ) ​ σ z 2 . superscript subscript ℋ ↓ 2 1 2 superscript subscript 𝜔 2 𝐿 𝜋 subscript 𝐽 12 superscript subscript 𝜎 𝑧 2 \displaystyle\mathcal{{H}}_{\downarrow}^{2}=\frac{1}{2}(\omega_{2}^{L}-\pi J_{12}){\sigma}_{z}^{2}. (11) Therefore, spin 2 will rotate slower or faster depending on the state of spin 1. If the coupling evolves for a time τ = 1 2 ​ J 12 𝜏 1 2 subscript 𝐽 12 \tau=\frac{1}{2J_{12}} , we obtain a controlled-Z rotation of π 2 𝜋 2 \frac{\pi}{2} degrees, which can be transformed into a controlled-NOT by a few single spin pulses applied before and after (see figure 1 for the complete sequence to implement a controlled-NOT). Figure 1: Implementation of a controlled-NOT gate in liquid state NMR. The left circuit is the quantum circuit representation of a controlled-NOT gate with control qubit q 1 and target qubit q 2 . On the right is the NMR implementation of such a gate by combining single qubit rotations and the natural two spin interaction of the system. The single qubit rotation properties are given in the rectangles, e.g. X : 90 : 𝑋 90 X:90 is a short notation for R x ​ ( π 2 ) = e − i ​ π 4 ​ σ x subscript 𝑅 𝑥 𝜋 2 superscript 𝑒 𝑖 𝜋 4 subscript 𝜎 𝑥 R_{x}(\frac{\pi}{2})=e^{-i\frac{\pi}{4}\sigma_{x}} . Notice that the size of the rectangles are not to scale for liquid state NMR, where the RF pulses are usually much shorter than the time to implement a π 2 𝜋 2 \frac{\pi}{2} J-coupling In the previous sections, we demonstrated that it is possible to implement any rotation around an arbitrary axis in the x ​ y 𝑥 𝑦 xy -plane, as well as perform a controlled-NOT gate with two spins. These two conditions are sufficient to perform universal quantum computing. For practical purposes, it is convenient to use only π 2 𝜋 2 \frac{\displaystyle\pi}{\displaystyle 2} and π 𝜋 \pi RF pulses as they are easier to calibrate. Arbitrary angle rotations can still be applied because z 𝑧 z -axis rotations can be obtained by changing the definition of the rotating frame, which is equivalent to changing the phase of subsequent pulses. It can be verified that R x ​ ( π 2 ) ​ R z ​ ( θ ) = R z ​ ( θ ) ​ R n ​ ( π 2 ) subscript 𝑅 𝑥 𝜋 2 subscript 𝑅 𝑧 𝜃 subscript 𝑅 𝑧 𝜃 subscript 𝑅 𝑛 𝜋 2 \displaystyle R_{x}(\frac{\pi}{2})R_{z}(\theta)=R_{z}(\theta)R_{{n}}(\frac{\pi}{2}) (12) where the vector n = cos ⁡ θ ​ x − sin ⁡ θ ​ y 𝑛 𝜃 𝑥 𝜃 𝑦 {n}=\cos{\theta}\,{x}-\sin{\theta}\,{y} . Therefore, since z 𝑧 z rotations commute with the internal Hamiltonian of the system, we may commute all the z 𝑧 z rotations to the end of the pulse sequence and compensate for any remaining z 𝑧 z rotation during the post-processing of the data. Moreover, the overall z 𝑧 z -rotation takes no time and is far more precise when using this procedure because the RF phase has a higher accuracy than the RF amplitude in modern NMR spectrometers. I.6 Refocusing and control techniques In an NMR system, spins constantly couple to each other, and we must “turn off” these couplings on demand to implement generic quantum gates. For example, consider a three spin system in which we wish to implement a controlled-NOT between the first and second qubits. As mentioned above, a π 2 𝜋 2 \frac{\pi}{2} J 𝐽 J -coupling between spins 1 and 2 is needed, which is accomplished by allowing the system to evolve under the natural Hamiltonian for a time τ = 1 2 ​ J 12 𝜏 1 2 subscript 𝐽 12 \tau=\frac{1}{2J_{12}} . During this time spin 3 will also couple to spins 1 and 2, giving an unwanted evolution. However, if we apply a π 𝜋 \pi pulse on the third spin half-way through the free evolution (at time τ 2 𝜏 2 \frac{\tau}{2} ), this spin will effectively decouple from the other two spins and, upon an extra π 𝜋 \pi pulse at the end of the evolution (at time τ 𝜏 \tau ), it will be returned to its initial state (this pulse sequence is shown in Figure 2 ). Considering only the interaqction term of the intramolecular component of the Hamiltonian, we can write the evolution of the system as: U ^ ​ ( t ) ^ 𝑈 𝑡 \displaystyle\hat{U}(t) = \displaystyle= R x † 3 ​ ( π ) ​ e − i ​ ℋ ​ τ / 2 ​ R x 3 ​ ( π ) ​ e − i ​ ℋ ​ τ / 2 subscript superscript 𝑅 † absent 3 𝑥 𝜋 superscript 𝑒 𝑖 ℋ 𝜏 2 subscript superscript 𝑅 3 𝑥 𝜋 superscript 𝑒 𝑖 ℋ 𝜏 2 \displaystyle R^{\dagger 3}_{x}(\pi)e^{-i\mathcal{{H}}\tau/2}R^{3}_{x}(\pi)e^{-i\mathcal{{H}}\tau/2} (13) = \displaystyle= ( i ​ σ x 3 ) ​ e − i π ​ τ 4 ( J 12 σ z 1 σ z 2 + J 13 σ z 1 σ z 3 + J 23 σ z 2 σ z 3 ​ ( − i ​ σ x 3 ) ​ e − i ​ π ​ τ 4 ​ ( J 12 ​ σ z 1 ​ σ z 2 + J 13 ​ σ z 1 ​ σ z 3 + J 23 ​ σ z 2 ​ σ z 3 ) \displaystyle(i{\sigma}_{x}^{3})e^{-i\frac{\pi\tau}{4}(J_{12}{{\sigma}_{z}^{1}}{{\sigma}_{z}^{2}}+J_{13}{{\sigma}_{z}^{1}}{{\sigma}_{z}^{3}}+J_{23}{{\sigma}_{z}^{2}}{{\sigma}_{z}^{3}}}(-i{\sigma}_{x}^{3})e^{-i\frac{\pi\tau}{4}(J_{12}{{\sigma}_{z}^{1}}{{\sigma}_{z}^{2}}+J_{13}{{\sigma}_{z}^{1}}{{\sigma}_{z}^{3}}+J_{23}{{\sigma}_{z}^{2}}{{\sigma}_{z}^{3}})} = \displaystyle= e − i ​ τ 2 ​ ( J 12 ​ σ z 1 ​ σ z 2 − J 13 ​ σ z 1 ​ σ z 3 − J 23 ​ σ z 2 ​ σ z 3 ) ​ e − i ​ τ 2 ​ ( J 12 ​ σ z 1 ​ σ z 2 + J 13 ​ σ z 1 ​ σ z 3 + J 23 ​ σ z 2 ​ σ z 3 ) superscript 𝑒 𝑖 𝜏 2 subscript 𝐽 12 superscript subscript 𝜎 𝑧 1 superscript subscript 𝜎 𝑧 2 subscript 𝐽 13 superscript subscript 𝜎 𝑧 1 superscript subscript 𝜎 𝑧 3 subscript 𝐽 23 superscript subscript 𝜎 𝑧 2 superscript subscript 𝜎 𝑧 3 superscript 𝑒 𝑖 𝜏 2 subscript 𝐽 12 superscript subscript 𝜎 𝑧 1 superscript subscript 𝜎 𝑧 2 subscript 𝐽 13 superscript subscript 𝜎 𝑧 1 superscript subscript 𝜎 𝑧 3 subscript 𝐽 23 superscript subscript 𝜎 𝑧 2 superscript subscript 𝜎 𝑧 3 \displaystyle e^{-i\frac{\tau}{2}(J_{12}{{\sigma}_{z}^{1}}{{\sigma}_{z}^{2}}-J_{13}{{\sigma}_{z}^{1}}{{\sigma}_{z}^{3}}-J_{23}{{\sigma}_{z}^{2}}{{\sigma}_{z}^{3}})}e^{-i\frac{\tau}{2}(J_{12}{{\sigma}_{z}^{1}}{{\sigma}_{z}^{2}}+J_{13}{{\sigma}_{z}^{1}}{{\sigma}_{z}^{3}}+J_{23}{{\sigma}_{z}^{2}}{{\sigma}_{z}^{3}})} = \displaystyle= e − i ​ π ​ τ 2 ​ J 12 ​ σ z 1 ​ σ z 2 , superscript 𝑒 𝑖 𝜋 𝜏 2 subscript 𝐽 12 superscript subscript 𝜎 𝑧 1 superscript subscript 𝜎 𝑧 2 \displaystyle e^{-i\frac{\pi\tau}{2}J_{12}{{\sigma}_{z}^{1}}{{\sigma}_{z}^{2}}}, where R x 3 ​ ( π ) subscript superscript 𝑅 3 𝑥 𝜋 R^{3}_{x}(\pi) is the operator of a π 𝜋 \pi -pulse about the x 𝑥 x -axis on spin 3. This is called a refocusing scheme and can be readily generalized to any number of coupled spins, i.e. a π 𝜋 \pi pulse on spin i 𝑖 i will effectively decouple it from all the other spins. This scheme can also be efficiently generalized to decouple m 𝑚 m spins from each other and from the N − m 𝑁 𝑚 N-m remaining spins. In practice, the situation is more complex. For example, one must keep track of the Zeeman evolution of all the spins (which is called phase tracking). This evolution can be taken into account by changing the phase of subsequent pulses according to the relation given in equation 12 . Figure 2: In NMR, the coupling between spins is always active. It is possible to refocus 2-qubit interactions using special pulse sequences. An example is given above, where halfway through a period τ 𝜏 \tau , a π 𝜋 \pi pulse on one of the nuclei is implemented that reverses the direction of its spin. Note that when we have a coupling of the form σ z ^ ( 1 ) ​ σ z ^ ( 2 ) superscript ^ subscript 𝜎 𝑧 1 superscript ^ subscript 𝜎 𝑧 2 \hat{\sigma_{z}}^{(1)}\hat{\sigma_{z}}^{(2)} the effect of the pulse can be thought of as reversing the sign of the coupling, and thus allows to cancel the interaction that occurred during the first τ / 2 𝜏 2 \tau/2 period. This pulse sequence effectively decouples the third qubit from the system. while leaving the coupling between q 1 and q 2 unchanged. This situation is mathematically treated in equation ( 13 ) For systems involving up to a few spins, pulse phases and decoupling sequences are derived by hand, but for molecules containing greater number of spins, these calculations become tedious and computer assisted techniques are used Bowdrey et al. ( 2005 ) . Efficient classical algorithms can be implemented that optimize pulse sequences with respect to phase and residual coupling errors. A major source of pulse errors are off resonance and coupling errors. It is possible to estimate and compensate for these errors by evaluating the first-order coupling and phase errors generated by a pulse. This is done by assuming that the real pulse can be decomposed as the ideal pulse preceded and followed by phase and coupling errors (see figure 3 ). Since the error terms all commute with each other, they can be estimated using pairwise spin simulations, which requires reasonable computational resources, i.e. is efficient as we scale the number of nuclei. With small J 𝐽 J -couplings and short pulses it is reasonable to expect error rates below a fraction of a percent for each pulse. Figure 3: A selective pulse designed to implement a single-qubit rotation in an N 𝑁 N -spin system will, in general, also affect the other spins. This can be studied in small systems by simulating the full quantum dynamics to obtain the unitary U s ​ i ​ m subscript 𝑈 𝑠 𝑖 𝑚 U_{sim} . The unwanted evolution of the non-target qubits is represented by the broken line on the left figure. If the pulse is carefully designed so that its implementation is very close to the ideal desired unitary U i ​ d subscript 𝑈 𝑖 𝑑 U_{id} , the error can be efficiently estimated to first order by phase errors (represented by θ i subscript 𝜃 𝑖 \theta_{i} and ϕ i subscript italic-ϕ 𝑖 \phi_{i} ) and coupling errors occurring before and after the pulse. Once the errors generated by each pulse are known they can be taken into account and corrected for by optimizing the durations of the free evolution periods and the timing of the refocusing pulses. Such an algorithm can also perform phase tracking and modify the pulse phases accordingly. Very high gate fidelities have been demonstrated using this efficient pairwise simulation technique. I.7 Optimal control for strongly coupled spins In some cases, spins are so strongly coupled that the approximation of σ z i ​ σ z j superscript subscript 𝜎 𝑧 𝑖 superscript subscript 𝜎 𝑧 𝑗 {{\sigma}_{z}^{i}}{{\sigma}_{z}^{j}} couplings used above breaks down. In those cases another technique can be used: strongly modulating pulses designed using numerical optimal control techniques Fortunato et al. ( 2002 ); Khaneja et al. ( 2005 ) . For systems containing about less than ten qubits, we can find extremely high fidelity and robust control by applying optimal control principles. Just as classical optimal control theory can tell how to best steer a rocket, quantum optimal control gives the tools to best steer a quantum system to a desired unitary gate. Quantum optimal control has been used for some time in the context of driving chemical reactions with shaped laser pulses Rabitz et al. ( 2000 ) . There, the goal is to maximize the transfer from a known initial state to a known final state. In the context of quantum computing, we do not necessarily know what the input state will be, and so we must find unitary gates which will work correctly for any input state. The Hamiltonian at any point in time can be written down as ℋ t ​ o ​ t ​ ( t ) = ℋ n ​ a ​ t + ℋ r ​ f ​ ( t ) , subscript ℋ 𝑡 𝑜 𝑡 𝑡 subscript ℋ 𝑛 𝑎 𝑡 subscript ℋ 𝑟 𝑓 𝑡 \mathcal{{H}}_{tot}(t)=\mathcal{{H}}_{nat}+\mathcal{{H}}_{rf}(t), (14) where, ℋ n ​ a ​ t subscript ℋ 𝑛 𝑎 𝑡 \mathcal{{H}}_{nat} is the time-independent, natural (or drift) Hamiltonian, and ℋ r ​ f ​ ( t ) subscript ℋ 𝑟 𝑓 𝑡 \mathcal{{H}}_{rf}(t) represents the controllable time-dependent RF fields which we can use to drive the evolution of the system. The task is to find the sequence of control fields that will produce the correct unitary evolution. For reasonably complicated systems analytical solutions are beyond reach; however, using a numerical search we can find control sequences with fidelities as high as 0.999999 0.999999 0.999999 . The control fields are discretized at a suitable rate (see figure 4 ) and a random guess for the control sequence is chosen. We can simulate the evolution of the system under this sequence and obtain the simulated unitary U s ​ i ​ m subscript 𝑈 𝑠 𝑖 𝑚 U_{sim} . We can then compare this to our goal unitary U g ​ o ​ a ​ l subscript 𝑈 𝑔 𝑜 𝑎 𝑙 U_{goal} through a fitness function, Φ = | T ​ r ​ ( U s ​ i ​ m † ​ U g ​ o ​ a ​ l ) | 2 Φ superscript 𝑇 𝑟 superscript subscript 𝑈 𝑠 𝑖 𝑚 † subscript 𝑈 𝑔 𝑜 𝑎 𝑙 2 \Phi=\left|Tr\left(U_{sim}^{\dagger}U_{goal}\right)\right|^{2} (15) which is equivalent to the state overlap fidelity averaged over all input states Fortunato et al. ( 2002 ) . We can then use any optimization routine to search for the highest Φ Φ \Phi . The first step in this direction for NMR quantum information processing was the scheme of Fortunato et al. Fortunato et al. ( 2002 ) who, by limiting the form of the control fields to a small number of constant amplitude, phase and frequency periods, were able to find high fidelity control sequences through a simplex search. Many more control periods can be considered and the numerical search substantially sped up through the use of standard optimal control techniques to obtain information about the gradient of the fitness function at each point. This is the GRadient Ascent Pulse Engineering or GRAPE algorithm introduced by Khaneja et al. Khaneja et al. ( 2005 ) . From the simulation information we can calculate approximate gradients of the fitness function with respect to the control amplitude at each timestep (see Figure 4 ). With this gradient information we can update the control fields by moving along the steepest ascent direction and then repeat the procedure. Convergence to a global maximum is of course not guaranteed as this hill climbing algorithm is a local search. However, empirically we find we can achieve sufficiently high fidelities from such local maxima. Convergence of the algorithm can be further improved by using non-linear conjugate gradients Bryson and Ho ( 1975 ) . The control sequences drive the system through a complicated and non-intuitive path and small errors in our modeling of the system’s Hamiltonian might lead to a low fidelity pulse. Fortunately, the control sequences can also be made robust against static inhomogeneities or uncertainties such as field inhomogenties and amplitude miscalibration of pulses. Robustness to both these effects for a particular pulse is shown in figure 6 b. Figure 4: An example section of a GRAPE pulse. The bars show the control amplitude which is constant for each time step. The arrows show the derivatives from the GRAPE algorithm at each point which tell us how to update the pulse for the next iteration (adapted from Khaneja et al. ( 2005 ) ). Recent work on a 12-qubit liquid state system that compared this method with the more traditional approach outlined in previous sections can be found in Negrevergne et al. ( 2006 ) . Not surprisingly, the control using strongly modulating pulses was more precise but at an exponential cost in searching for the pulse sequences. The scheme described above can be generalized if we can break the molecule into strongly coupled subsystems which are themselves weakly coupled, and thus it is possible to perform the decomposition described in previous sections between subsystems using pairwise simulation of subsystems. Gates to implement rotations within the subsystem will however require numerical optimization methods such as GRAPE. The details of this hybrid method are presently under investigation. The ability to control liquid state nuclear spin systems has allowed implementation of a variety of benchmarking experiments and algorithms on small qubit registers. For example, we have developed sufficient control to implement quantum error correcting protocols Cory et al. ( 1998 ); Laforest et al. ( 2007 ); Cory et al. ( 2000 ); Knill et al. ( 2001 ); Boulant et al. ( 2004 ) , the simulation of quantum systems Chen et al. ( 2004 ); Negrevergne et al. ( 2004 ); Havel et al. ( 2001 ); Tseng et al. ( 1999 ); Somaroo et al. ( 1999 ) and the benchmark 12 qubit quantum processor Negrevergne et al. ( 2006 ) . An extensive list of these and other experiments can be found at http://arxiv.org/.

This paper describes recent progress using nuclear magnetic resonance (NMR) as a platform for implementing quantum information processing (QIP) tasks. The basic ideas of NMR QIP are detailed, examining the successes and limitations of liquid and solid state experiments. Finally, a future direction for implementing quantum processors is suggested,utilizing both nuclear and electron spin degrees of freedom.

[{"content":"Help me to generate the abstract of a survey paper given the title:*Quantum information processing using nuclear and electron magnetic resonance: review and prospects* and the introduction:I Liquid state NMR I.1 Magnetic interactions In the semi-classical picture, the spin of a nucleus behaves like the dipolar moment of a magnet possessing angular momentum parallel to its magnetic moment. When placed in a constant magnetic field pointing along a certain direction, (customarily defined as the z 𝑧 z direction) the dipolar moment precesses around this axis. The frequency of this precession is called the Larmor frequency and is dependent on the external magnetic field, the nuclear isotope and its chemical environment within the molecule. For quantum information purposes, we are mainly interested in spin-1/2 nuclei (e.g. 1 H, 13 C, 15 N, 19 F, 29 Si and 31 P to name a few). Placed in magnetic fields generated by modern superconducting magnets, different species of nuclei have differences in Larmor frequency on the order of MHz. For example, the Larmor frequency of 1 H is about 500 MHz in a 11.7 Tesla magnet, while that of 13 C is about 125 MHz. Depending on the symmetry of the molecule, two nuclei of the same species can either have the same Larmor frequency, or can have a frequency difference (called chemical shift) ranging from a few Hz to several kHz. Typical liquid state NMR experiments involve an ensemble of around 10 20 superscript 10 20 10^{20} identical molecules dissolved in a solvent whose effect on the nuclear magnetic moments of our molecules can be neglected. When two spins are spatially close, their dipolar moments interact with each other. The strength of this coupling is dependent on the distance between the two spins and their relative orientation with respect to the external magnetic field. In a liquid, the molecules move and rotate around each other on a much shorter time scale than the interactions occurring between them. This causes the intermolecular and intra-molecular dipolar interactions to average to zero on the NMR time scale (i.e. the Larmor period time scale). In solid state NMR, however, dipolar interactions remain and can be controlled as discussed in section II . Within the same molecule, there are still interactions between the spins in the liquid state. If the wavefunctions of bonding electrons overlap spatially with a pair of nuclear spins, the electron mediates an effective interaction between the nuclear spins. This interaction is independent of the external magnetic field and the orientation of the molecule, which inspires its name: scalar coupling (also called indirect spin-spin coupling, or J 𝐽 J -coupling). I.2 The NMR Hamiltonian As mentioned above, in liquid state NMR the intermolecular spin interactions are suppressed. This causes the molecules to be effectively isolated from each other, and therefore a description of the spin dynamics of an ensemble of molecules is well approximated by the spin dynamics of a single molecule. If we consider a molecule containing N 𝑁 N spin-1/2 nuclei in one of the molecules, then the natural Hamiltonian of this system in a large homogeneous magnetic field B → 0 subscript → 𝐵 0 \vec{B}_{0} pointing in the z 𝑧 z direction is given by ℋ n ​ a ​ t subscript ℋ 𝑛 𝑎 𝑡 \displaystyle\mathcal{{H}}_{nat} = \displaystyle= 1 2 ​ ∑ i = 1 N 2 ​ π ​ ν i L ​ σ z i + π 2 ​ ∑ i < j J i ​ j ​ σ z i ​ σ z j 1 2 superscript subscript 𝑖 1 𝑁 2 𝜋 superscript subscript 𝜈 𝑖 𝐿 superscript subscript 𝜎 𝑧 𝑖 𝜋 2 subscript 𝑖 𝑗 subscript 𝐽 𝑖 𝑗 superscript subscript 𝜎 𝑧 𝑖 superscript subscript 𝜎 𝑧 𝑗 \displaystyle\frac{1}{2}\sum_{i=1}^{N}2\pi\nu_{i}^{L}{{\sigma}_{z}^{i}}+\frac{\pi}{2}\sum_{i<j}J_{ij}{{\sigma}_{z}^{i}}{{\sigma}_{z}^{j}} (1) where ν i L = ω i L / 2 ​ π = γ i ​ | B → 0 | superscript subscript 𝜈 𝑖 𝐿 superscript subscript 𝜔 𝑖 𝐿 2 𝜋 subscript 𝛾 𝑖 subscript → 𝐵 0 \nu_{i}^{L}=\omega_{i}^{L}/2\pi=\gamma_{i}|\vec{B}_{0}| is the Larmor frequency of the i t ​ h superscript 𝑖 𝑡 ℎ i^{th} nucleus with gyromagnetic ratio γ i subscript 𝛾 𝑖 \gamma_{i} , J i ​ j subscript 𝐽 𝑖 𝑗 J_{ij} is the coupling strength between nucleus i 𝑖 i and j 𝑗 j and σ z i superscript subscript 𝜎 𝑧 𝑖 {{\sigma}_{z}^{i}} is the z 𝑧 z Pauli matrix of the i th superscript 𝑖 th i^{\textrm{th}} spin. The first term in the Hamiltonian describes the precession of the spins due to their coupling to the external magnetic field, while the second term describes the J 𝐽 J -coupling between pairs of nuclei. This Hamiltonian corresponds to the weak coupling limit, where we assume that the chemical shifts between coupled spins are much greater than their respective couplings, i.e. | ν i L − ν j L | >> J i ​ j / 2 much-greater-than superscript subscript 𝜈 𝑖 𝐿 superscript subscript 𝜈 𝑗 𝐿 subscript 𝐽 𝑖 𝑗 2 |\nu_{i}^{L}-\nu_{j}^{L}|>>J_{ij}/2 . If this approximation is not valid, we need to use the full coupling Hamiltonian σ x i ​ σ x j + σ y i ​ σ y j + σ z i ​ σ z j superscript subscript 𝜎 𝑥 𝑖 superscript subscript 𝜎 𝑥 𝑗 superscript subscript 𝜎 𝑦 𝑖 superscript subscript 𝜎 𝑦 𝑗 superscript subscript 𝜎 𝑧 𝑖 superscript subscript 𝜎 𝑧 𝑗 {{\sigma}_{x}^{i}}{{\sigma}_{x}^{j}}+{{\sigma}_{y}^{i}}{{\sigma}_{y}^{j}}+{{\sigma}_{z}^{i}}{{\sigma}_{z}^{j}} in place of σ z i ​ σ z j superscript subscript 𝜎 𝑧 𝑖 superscript subscript 𝜎 𝑧 𝑗 {{\sigma}_{z}^{i}}{{\sigma}_{z}^{j}} . The exact values of the Hamiltonian parameters are determined by fitting experimental data. I.3 Single-spin control For quantum information processing, we need to be able to perform arbitrary manipulations of a single spin, which is equivalent to arbitrary rotations about any axis. As an example, consider the application of a magnetic field B → 1 subscript → 𝐵 1 \vec{B}_{1} perpendicular to the z 𝑧 z axis which oscillates at the nuclear spins Larmor frequency: B → 1 subscript → 𝐵 1 \displaystyle\vec{B}_{1} = \displaystyle= | B → 1 | ​ ( cos ⁡ ( ω r ​ f ​ t ) ​ x → + sin ⁡ ( ω r ​ f ​ t ) ​ y → ) subscript → 𝐵 1 superscript 𝜔 𝑟 𝑓 𝑡 → 𝑥 superscript 𝜔 𝑟 𝑓 𝑡 → 𝑦 \displaystyle|\vec{B}_{1}|\left(\cos(\omega^{rf}t)\,\vec{x}+\sin(\omega^{rf}t)\,\vec{y}\right) (2) where ω r ​ f = 2 ​ π ​ ν r ​ f superscript 𝜔 𝑟 𝑓 2 𝜋 superscript 𝜈 𝑟 𝑓 \omega^{rf}=2\pi\nu^{rf} is the angular frequency of the field. In the rotating frame of the nucleus (i.e. the frame rotating at the same frequency as the spin), B → 1 subscript → 𝐵 1 \vec{B}_{1} will appear as a constant field pointing along its rotating x 𝑥 x axis. The spin will start to precess about this axis. Rotation about any axis in the x ​ y 𝑥 𝑦 xy -plane is possible by adjusting the phase of the B → 1 subscript → 𝐵 1 \vec{B}_{1} field, e.g. ω L ​ t → ω L ​ t + ϕ → superscript 𝜔 𝐿 𝑡 superscript 𝜔 𝐿 𝑡 italic-ϕ \omega^{L}t\to\omega^{L}t+\phi , which will create a rotation around the axis making an angle ϕ italic-ϕ \phi with the x 𝑥 x axis. In the laboratory, such a rotating field can be applied by sending a radio-frequency (RF) pulse of a particular duration and phase to a conducting coil surrounding the sample, calculated according to the rotating wave approximation (see Levitt ( 2001 ) for more details). To better understand this phenomenon from the viewpoint of quantum mechanics, consider the rotating frame picture: suppose the spin is in the state | ψ ​ ( t ) ⟩ ket 𝜓 𝑡 |\psi(t)\rangle , and define the state in the rotating frame of the pulse with angular frequency ω r ​ f superscript 𝜔 𝑟 𝑓 \omega^{rf} as | ψ ⟩ r subscript ket 𝜓 𝑟 \displaystyle|\psi\rangle_{r} = \displaystyle= R z ​ ( − ω r ​ f ​ t ) ​ | ψ ​ ( t ) ⟩ subscript 𝑅 𝑧 superscript 𝜔 𝑟 𝑓 𝑡 ket 𝜓 𝑡 \displaystyle{R}_{z}(-\omega^{rf}t)|\psi(t)\rangle (3) = \displaystyle= R z ​ ( − ω r ​ f ​ t ) ​ e − i ​ t ℏ ​ ℋ n ​ a ​ t ​ | ψ ​ ( 0 ) ⟩ subscript 𝑅 𝑧 superscript 𝜔 𝑟 𝑓 𝑡 superscript 𝑒 𝑖 𝑡 Planck-constant-over-2-pi subscript ℋ 𝑛 𝑎 𝑡 ket 𝜓 0 \displaystyle{R}_{z}(-\omega^{rf}t)e^{-\frac{it}{\hbar}\mathcal{{H}}_{nat}}|\psi(0)\rangle = \displaystyle= e i ℏ ​ σ z ​ ω r ​ f 2 ​ t ​ e − i ​ t ℏ ​ ℋ n ​ a ​ t ​ | ψ ​ ( 0 ) ⟩ superscript 𝑒 𝑖 Planck-constant-over-2-pi subscript 𝜎 𝑧 superscript 𝜔 𝑟 𝑓 2 𝑡 superscript 𝑒 𝑖 𝑡 Planck-constant-over-2-pi subscript ℋ 𝑛 𝑎 𝑡 ket 𝜓 0 \displaystyle e^{\frac{i}{\hbar}{\sigma}_{z}\frac{\omega^{rf}}{2}t}e^{-\frac{it}{\hbar}\mathcal{{H}}_{nat}}|\psi(0)\rangle = \displaystyle= | ψ ​ ( 0 ) ⟩ , for a single spin with ω r ​ f = ω L . ket 𝜓 0 for a single spin with ω r f = ω L \displaystyle|\psi(0)\rangle,\,\,\textrm{for a single spin with $\omega^{rf}=\omega^{L}$}. (4) If we apply a time derivative to equation 3 , it can be shown that the state in the rotating frame | ψ ⟩ r subscript ket 𝜓 𝑟 |\psi\rangle_{r} evolves according to the Schrödinger equation with the new Hamiltonian ℋ r subscript ℋ 𝑟 \displaystyle{\mathcal{{H}}}_{r} = \displaystyle= R z ​ ( − ω r ​ f ​ t ) ​ ℋ n ​ a ​ t ​ R z ​ ( ω r ​ f ​ t ) − ω r ​ f 2 ​ σ z . subscript 𝑅 𝑧 superscript 𝜔 𝑟 𝑓 𝑡 subscript ℋ 𝑛 𝑎 𝑡 subscript 𝑅 𝑧 superscript 𝜔 𝑟 𝑓 𝑡 superscript 𝜔 𝑟 𝑓 2 subscript 𝜎 𝑧 \displaystyle{R}_{z}(-\omega^{rf}t)\mathcal{{H}}_{nat}{R}_{z}(\omega^{rf}t)-\frac{\omega^{rf}}{2}{\sigma}_{z}. (5) When an RF pulse with phase ϕ italic-ϕ \phi is applied to the spin, the laboratory frame Hamiltonian is: ℋ ℋ \displaystyle\mathcal{{H}} = \displaystyle= ω L 2 ​ σ z + ω n ​ u ​ t 2 ​ ( cos ⁡ ( ω r ​ f ​ t + ϕ ) ​ σ x + sin ⁡ ( ω r ​ f ​ t + ϕ ) ​ σ y ) superscript 𝜔 𝐿 2 subscript 𝜎 𝑧 superscript 𝜔 𝑛 𝑢 𝑡 2 superscript 𝜔 𝑟 𝑓 𝑡 italic-ϕ subscript 𝜎 𝑥 superscript 𝜔 𝑟 𝑓 𝑡 italic-ϕ subscript 𝜎 𝑦 \displaystyle\frac{\omega^{L}}{2}{\sigma}_{z}+\frac{\omega^{nut}}{2}\left(\cos{(\omega^{rf}t+\phi)}{\sigma}_{x}+\sin{(\omega^{rf}t+\phi)}{\sigma}_{y}\right) (6) where ω n ​ u ​ t = π ​ γ i ​ | B → 1 | superscript 𝜔 𝑛 𝑢 𝑡 𝜋 subscript 𝛾 𝑖 subscript → 𝐵 1 \omega^{nut}=\pi\gamma_{i}|\vec{B}_{1}| . In the rotating frame this becomes ℋ r subscript ℋ 𝑟 \displaystyle{\mathcal{{H}}}_{r} = \displaystyle= 1 2 ​ ( ω L − ω r ​ f ) ​ σ z + 1 2 ​ ω n ​ u ​ t ​ ( cos ⁡ ϕ ​ σ x + sin ⁡ ϕ ​ σ y ) . 1 2 superscript 𝜔 𝐿 superscript 𝜔 𝑟 𝑓 subscript 𝜎 𝑧 1 2 superscript 𝜔 𝑛 𝑢 𝑡 italic-ϕ subscript 𝜎 𝑥 italic-ϕ subscript 𝜎 𝑦 \displaystyle\frac{1}{2}(\omega^{L}-\omega^{rf}){\sigma}_{z}+\frac{1}{2}\omega^{nut}(\cos\phi\,{\sigma}_{x}+\sin\phi\,{\sigma}_{y}). (7) Thus, if the RF pulse is at the same frequency as the spin, the spin will see a constant field in the x ​ y 𝑥 𝑦 xy plane, and will precess about it. The rotation angle θ 𝜃 \theta is determine by the interval τ 𝜏 \tau during which the RF field is applied, according to θ = ω n ​ u ​ t ​ τ 𝜃 superscript 𝜔 𝑛 𝑢 𝑡 𝜏 \theta=\omega^{nut}\tau . I.4 Adding a second spin It is also possible to independently control two spins with different Larmor frequencies. Applying an RF pulse at the frequency of the first spin, the rotating frame Hamiltonian is given by ℋ ~ n ​ a ​ t subscript ~ ℋ 𝑛 𝑎 𝑡 \displaystyle\widetilde{\mathcal{{H}}}_{nat} = \displaystyle= 1 2 ​ ω 1 n ​ u ​ t ​ σ x 1 + 1 2 ​ ω 2 n ​ u ​ t ​ σ x 2 + 1 2 ​ ( ω 2 L − ω 1 L ) ​ σ z 2 + π 2 ​ J 12 ​ σ z 1 ​ σ z 2 1 2 superscript subscript 𝜔 1 𝑛 𝑢 𝑡 superscript subscript 𝜎 𝑥 1 1 2 superscript subscript 𝜔 2 𝑛 𝑢 𝑡 superscript subscript 𝜎 𝑥 2 1 2 superscript subscript 𝜔 2 𝐿 superscript subscript 𝜔 1 𝐿 superscript subscript 𝜎 𝑧 2 𝜋 2 subscript 𝐽 12 superscript subscript 𝜎 𝑧 1 superscript subscript 𝜎 𝑧 2 \displaystyle\frac{1}{2}\omega_{1}^{nut}{\sigma}_{x}^{1}+\frac{1}{2}\omega_{2}^{nut}{\sigma}_{x}^{2}+\frac{1}{2}(\omega_{2}^{L}-\omega_{1}^{L}){\sigma}_{z}^{2}+\frac{\pi}{2}J_{12}{\sigma}_{z}^{1}{\sigma}_{z}^{2} (8) where we have set ϕ = 0 italic-ϕ 0 \phi=0 for simplicity. While the first spin undergoes a rotation around the x 𝑥 x axis, the second spin experiences a field with an additional non-zero z 𝑧 z component. This is called the off resonance effect. If we consider the case where ω 2 L − ω 2 L >> ω 1 n ​ u ​ t much-greater-than superscript subscript 𝜔 2 𝐿 superscript subscript 𝜔 2 𝐿 superscript subscript 𝜔 1 𝑛 𝑢 𝑡 \omega_{2}^{L}-\omega_{2}^{L}>>\omega_{1}^{nut} then the second spin rotation around the x 𝑥 x axis will average to zero during the time the first spin has completed its rotation. Typically, ω n ​ u ​ t superscript 𝜔 𝑛 𝑢 𝑡 \omega^{nut} is smaller than 1 MHz, so this condition is automatically satisfied if the two nuclei belong to different species. If the spins are of the same species, this condition can also be satisfied if a very low amplitude pulse is used due to the small nutation frequency. In this case, one drawback is that the pulse will necessarily take much longer to achieve the same angle of rotation, and if the two spins have a significant coupling constant J 12 subscript 𝐽 12 J_{12} coupling effects might introduce significant errors and therefore limit our control. Fortunately, there exist well known techniques to address different nuclei of the same species with high precision. The most common technique is to control the spins using shaped pulses. The frequency response to the pulse will depend on the pulse shape (Fourier theorem) and so by applying the pulse with a time varying power we can control the power spectrum of the pulse. For example, if a Gaussian shaped pulse is applied at frequency ω r ​ f superscript 𝜔 𝑟 𝑓 \omega^{rf} , then only spins within a Gaussian distribution of frequencies around ω r ​ f superscript 𝜔 𝑟 𝑓 \omega^{rf} will respond to this RF field. Therefore, if the height and the length of the Gaussian pulse is carefully chosen, one spin can be addressed, causing negligible effects to others. This technique permits control of spin pairs with smaller chemical shift differences in shorter periods of time, hence allowing stronger coupling. The length of a Gaussian pulse is proportional to the inverse of the chemical shift between the spins. Therefore, in the limiting case of small chemical shift differences and large J 𝐽 J -coupling values, control of the qubits is more difficult. For most liquid state experiments on a few spins, where chemical shifts are comparatively large and J 𝐽 J -couplings are small, the use of Gaussian pulses is sufficient to achieve very high precision spin rotations. The situation becomes more complicated when there are more homonuclear spins (implying smaller chemical shift differences on average), or stronger coupling like in solid state or liquid crystalenvironments. It is still possible to overcome these drawbacks by considering more complicated pulse shapes and phase modulation. For example, in section I.7 , we will describe how it is possible to find shaped pulses that can implement any desired evolution by simulating the full quantum dynamics. I.5 The controlled-NOT operation In the previous subsection we discussed a method used to independently control different spins. In order to perform quantum computing, we need to achieve universal control and hence be able to have spins interact with each other. A two-qubit gate that is useful for quantum information processing is the controlled-NOT, which acts as | 00 ⟩ → | 00 ⟩ → ket 00 ket 00 \displaystyle|00\rangle\to|00\rangle , | 10 ⟩ → | 11 ⟩ → ket 10 ket 11 \displaystyle|10\rangle\to|11\rangle | 01 ⟩ → | 01 ⟩ → ket 01 ket 01 \displaystyle|01\rangle\to|01\rangle , | 11 ⟩ → | 10 ⟩ . → ket 11 ket 10 \displaystyle|11\rangle\to|10\rangle. (9) The operation must flip the target qubit (second bit) if and only if the first qubit is in the state | 1 ⟩ ket 1 |1\rangle . In NMR, | 0 ⟩ ket 0 |0\rangle and | 1 ⟩ ket 1 |1\rangle are associated with the state of the spin pointing up, | ↑ ⟩ ket ↑ |\uparrow\rangle or pointing down | ↓ ⟩ ket ↓ |\downarrow\rangle respectively. If we look at the Hamiltonian in equation ( 1 ), and consider its effect on spin 2 depending on whether the state of spin 1 is up or down, we obtain an effective Hamiltonian for the second spin: ℋ ↑ 2 = 1 2 ​ ( ω 2 L + π ​ J 12 ) ​ σ z 2 superscript subscript ℋ ↑ 2 1 2 superscript subscript 𝜔 2 𝐿 𝜋 subscript 𝐽 12 superscript subscript 𝜎 𝑧 2 \displaystyle\mathcal{{H}}_{\uparrow}^{2}=\frac{1}{2}(\omega_{2}^{L}+\pi J_{12}){\sigma}_{z}^{2} (10) ℋ ↓ 2 = 1 2 ​ ( ω 2 L − π ​ J 12 ) ​ σ z 2 . superscript subscript ℋ ↓ 2 1 2 superscript subscript 𝜔 2 𝐿 𝜋 subscript 𝐽 12 superscript subscript 𝜎 𝑧 2 \displaystyle\mathcal{{H}}_{\downarrow}^{2}=\frac{1}{2}(\omega_{2}^{L}-\pi J_{12}){\sigma}_{z}^{2}. (11) Therefore, spin 2 will rotate slower or faster depending on the state of spin 1. If the coupling evolves for a time τ = 1 2 ​ J 12 𝜏 1 2 subscript 𝐽 12 \tau=\frac{1}{2J_{12}} , we obtain a controlled-Z rotation of π 2 𝜋 2 \frac{\pi}{2} degrees, which can be transformed into a controlled-NOT by a few single spin pulses applied before and after (see figure 1 for the complete sequence to implement a controlled-NOT). Figure 1: Implementation of a controlled-NOT gate in liquid state NMR. The left circuit is the quantum circuit representation of a controlled-NOT gate with control qubit q 1 and target qubit q 2 . On the right is the NMR implementation of such a gate by combining single qubit rotations and the natural two spin interaction of the system. The single qubit rotation properties are given in the rectangles, e.g. X : 90 : 𝑋 90 X:90 is a short notation for R x ​ ( π 2 ) = e − i ​ π 4 ​ σ x subscript 𝑅 𝑥 𝜋 2 superscript 𝑒 𝑖 𝜋 4 subscript 𝜎 𝑥 R_{x}(\frac{\pi}{2})=e^{-i\frac{\pi}{4}\sigma_{x}} . Notice that the size of the rectangles are not to scale for liquid state NMR, where the RF pulses are usually much shorter than the time to implement a π 2 𝜋 2 \frac{\pi}{2} J-coupling In the previous sections, we demonstrated that it is possible to implement any rotation around an arbitrary axis in the x ​ y 𝑥 𝑦 xy -plane, as well as perform a controlled-NOT gate with two spins. These two conditions are sufficient to perform universal quantum computing. For practical purposes, it is convenient to use only π 2 𝜋 2 \frac{\displaystyle\pi}{\displaystyle 2} and π 𝜋 \pi RF pulses as they are easier to calibrate. Arbitrary angle rotations can still be applied because z 𝑧 z -axis rotations can be obtained by changing the definition of the rotating frame, which is equivalent to changing the phase of subsequent pulses. It can be verified that R x ​ ( π 2 ) ​ R z ​ ( θ ) = R z ​ ( θ ) ​ R n ​ ( π 2 ) subscript 𝑅 𝑥 𝜋 2 subscript 𝑅 𝑧 𝜃 subscript 𝑅 𝑧 𝜃 subscript 𝑅 𝑛 𝜋 2 \displaystyle R_{x}(\frac{\pi}{2})R_{z}(\theta)=R_{z}(\theta)R_{{n}}(\frac{\pi}{2}) (12) where the vector n = cos ⁡ θ ​ x − sin ⁡ θ ​ y 𝑛 𝜃 𝑥 𝜃 𝑦 {n}=\cos{\theta}\,{x}-\sin{\theta}\,{y} . Therefore, since z 𝑧 z rotations commute with the internal Hamiltonian of the system, we may commute all the z 𝑧 z rotations to the end of the pulse sequence and compensate for any remaining z 𝑧 z rotation during the post-processing of the data. Moreover, the overall z 𝑧 z -rotation takes no time and is far more precise when using this procedure because the RF phase has a higher accuracy than the RF amplitude in modern NMR spectrometers. I.6 Refocusing and control techniques In an NMR system, spins constantly couple to each other, and we must turn off these couplings on demand to implement generic quantum gates. For example, consider a three spin system in which we wish to implement a controlled-NOT between the first and second qubits. As mentioned above, a π 2 𝜋 2 \frac{\pi}{2} J 𝐽 J -coupling between spins 1 and 2 is needed, which is accomplished by allowing the system to evolve under the natural Hamiltonian for a time τ = 1 2 ​ J 12 𝜏 1 2 subscript 𝐽 12 \tau=\frac{1}{2J_{12}} . During this time spin 3 will also couple to spins 1 and 2, giving an unwanted evolution. However, if we apply a π 𝜋 \pi pulse on the third spin half-way through the free evolution (at time τ 2 𝜏 2 \frac{\tau}{2} ), this spin will effectively decouple from the other two spins and, upon an extra π 𝜋 \pi pulse at the end of the evolution (at time τ 𝜏 \tau ), it will be returned to its initial state (this pulse sequence is shown in Figure 2 ). Considering only the interaqction term of the intramolecular component of the Hamiltonian, we can write the evolution of the system as: U ^ ​ ( t ) ^ 𝑈 𝑡 \displaystyle\hat{U}(t) = \displaystyle= R x † 3 ​ ( π ) ​ e − i ​ ℋ ​ τ / 2 ​ R x 3 ​ ( π ) ​ e − i ​ ℋ ​ τ / 2 subscript superscript 𝑅 † absent 3 𝑥 𝜋 superscript 𝑒 𝑖 ℋ 𝜏 2 subscript superscript 𝑅 3 𝑥 𝜋 superscript 𝑒 𝑖 ℋ 𝜏 2 \displaystyle R^{\dagger 3}_{x}(\pi)e^{-i\mathcal{{H}}\tau/2}R^{3}_{x}(\pi)e^{-i\mathcal{{H}}\tau/2} (13) = \displaystyle= ( i ​ σ x 3 ) ​ e − i π ​ τ 4 ( J 12 σ z 1 σ z 2 + J 13 σ z 1 σ z 3 + J 23 σ z 2 σ z 3 ​ ( − i ​ σ x 3 ) ​ e − i ​ π ​ τ 4 ​ ( J 12 ​ σ z 1 ​ σ z 2 + J 13 ​ σ z 1 ​ σ z 3 + J 23 ​ σ z 2 ​ σ z 3 ) \displaystyle(i{\sigma}_{x}^{3})e^{-i\frac{\pi\tau}{4}(J_{12}{{\sigma}_{z}^{1}}{{\sigma}_{z}^{2}}+J_{13}{{\sigma}_{z}^{1}}{{\sigma}_{z}^{3}}+J_{23}{{\sigma}_{z}^{2}}{{\sigma}_{z}^{3}}}(-i{\sigma}_{x}^{3})e^{-i\frac{\pi\tau}{4}(J_{12}{{\sigma}_{z}^{1}}{{\sigma}_{z}^{2}}+J_{13}{{\sigma}_{z}^{1}}{{\sigma}_{z}^{3}}+J_{23}{{\sigma}_{z}^{2}}{{\sigma}_{z}^{3}})} = \displaystyle= e − i ​ τ 2 ​ ( J 12 ​ σ z 1 ​ σ z 2 − J 13 ​ σ z 1 ​ σ z 3 − J 23 ​ σ z 2 ​ σ z 3 ) ​ e − i ​ τ 2 ​ ( J 12 ​ σ z 1 ​ σ z 2 + J 13 ​ σ z 1 ​ σ z 3 + J 23 ​ σ z 2 ​ σ z 3 ) superscript 𝑒 𝑖 𝜏 2 subscript 𝐽 12 superscript subscript 𝜎 𝑧 1 superscript subscript 𝜎 𝑧 2 subscript 𝐽 13 superscript subscript 𝜎 𝑧 1 superscript subscript 𝜎 𝑧 3 subscript 𝐽 23 superscript subscript 𝜎 𝑧 2 superscript subscript 𝜎 𝑧 3 superscript 𝑒 𝑖 𝜏 2 subscript 𝐽 12 superscript subscript 𝜎 𝑧 1 superscript subscript 𝜎 𝑧 2 subscript 𝐽 13 superscript subscript 𝜎 𝑧 1 superscript subscript 𝜎 𝑧 3 subscript 𝐽 23 superscript subscript 𝜎 𝑧 2 superscript subscript 𝜎 𝑧 3 \displaystyle e^{-i\frac{\tau}{2}(J_{12}{{\sigma}_{z}^{1}}{{\sigma}_{z}^{2}}-J_{13}{{\sigma}_{z}^{1}}{{\sigma}_{z}^{3}}-J_{23}{{\sigma}_{z}^{2}}{{\sigma}_{z}^{3}})}e^{-i\frac{\tau}{2}(J_{12}{{\sigma}_{z}^{1}}{{\sigma}_{z}^{2}}+J_{13}{{\sigma}_{z}^{1}}{{\sigma}_{z}^{3}}+J_{23}{{\sigma}_{z}^{2}}{{\sigma}_{z}^{3}})} = \displaystyle= e − i ​ π ​ τ 2 ​ J 12 ​ σ z 1 ​ σ z 2 , superscript 𝑒 𝑖 𝜋 𝜏 2 subscript 𝐽 12 superscript subscript 𝜎 𝑧 1 superscript subscript 𝜎 𝑧 2 \displaystyle e^{-i\frac{\pi\tau}{2}J_{12}{{\sigma}_{z}^{1}}{{\sigma}_{z}^{2}}}, where R x 3 ​ ( π ) subscript superscript 𝑅 3 𝑥 𝜋 R^{3}_{x}(\pi) is the operator of a π 𝜋 \pi -pulse about the x 𝑥 x -axis on spin 3. This is called a refocusing scheme and can be readily generalized to any number of coupled spins, i.e. a π 𝜋 \pi pulse on spin i 𝑖 i will effectively decouple it from all the other spins. This scheme can also be efficiently generalized to decouple m 𝑚 m spins from each other and from the N − m 𝑁 𝑚 N-m remaining spins. In practice, the situation is more complex. For example, one must keep track of the Zeeman evolution of all the spins (which is called phase tracking). This evolution can be taken into account by changing the phase of subsequent pulses according to the relation given in equation 12 . Figure 2: In NMR, the coupling between spins is always active. It is possible to refocus 2-qubit interactions using special pulse sequences. An example is given above, where halfway through a period τ 𝜏 \tau , a π 𝜋 \pi pulse on one of the nuclei is implemented that reverses the direction of its spin. Note that when we have a coupling of the form σ z ^ ( 1 ) ​ σ z ^ ( 2 ) superscript ^ subscript 𝜎 𝑧 1 superscript ^ subscript 𝜎 𝑧 2 \hat{\sigma_{z}}^{(1)}\hat{\sigma_{z}}^{(2)} the effect of the pulse can be thought of as reversing the sign of the coupling, and thus allows to cancel the interaction that occurred during the first τ / 2 𝜏 2 \tau/2 period. This pulse sequence effectively decouples the third qubit from the system. while leaving the coupling between q 1 and q 2 unchanged. This situation is mathematically treated in equation ( 13 ) For systems involving up to a few spins, pulse phases and decoupling sequences are derived by hand, but for molecules containing greater number of spins, these calculations become tedious and computer assisted techniques are used Bowdrey et al. ( 2005 ) . Efficient classical algorithms can be implemented that optimize pulse sequences with respect to phase and residual coupling errors. A major source of pulse errors are off resonance and coupling errors. It is possible to estimate and compensate for these errors by evaluating the first-order coupling and phase errors generated by a pulse. This is done by assuming that the real pulse can be decomposed as the ideal pulse preceded and followed by phase and coupling errors (see figure 3 ). Since the error terms all commute with each other, they can be estimated using pairwise spin simulations, which requires reasonable computational resources, i.e. is efficient as we scale the number of nuclei. With small J 𝐽 J -couplings and short pulses it is reasonable to expect error rates below a fraction of a percent for each pulse. Figure 3: A selective pulse designed to implement a single-qubit rotation in an N 𝑁 N -spin system will, in general, also affect the other spins. This can be studied in small systems by simulating the full quantum dynamics to obtain the unitary U s ​ i ​ m subscript 𝑈 𝑠 𝑖 𝑚 U_{sim} . The unwanted evolution of the non-target qubits is represented by the broken line on the left figure. If the pulse is carefully designed so that its implementation is very close to the ideal desired unitary U i ​ d subscript 𝑈 𝑖 𝑑 U_{id} , the error can be efficiently estimated to first order by phase errors (represented by θ i subscript 𝜃 𝑖 \theta_{i} and ϕ i subscript italic-ϕ 𝑖 \phi_{i} ) and coupling errors occurring before and after the pulse. Once the errors generated by each pulse are known they can be taken into account and corrected for by optimizing the durations of the free evolution periods and the timing of the refocusing pulses. Such an algorithm can also perform phase tracking and modify the pulse phases accordingly. Very high gate fidelities have been demonstrated using this efficient pairwise simulation technique. I.7 Optimal control for strongly coupled spins In some cases, spins are so strongly coupled that the approximation of σ z i ​ σ z j superscript subscript 𝜎 𝑧 𝑖 superscript subscript 𝜎 𝑧 𝑗 {{\sigma}_{z}^{i}}{{\sigma}_{z}^{j}} couplings used above breaks down. In those cases another technique can be used: strongly modulating pulses designed using numerical optimal control techniques Fortunato et al. ( 2002 ); Khaneja et al. ( 2005 ) . For systems containing about less than ten qubits, we can find extremely high fidelity and robust control by applying optimal control principles. Just as classical optimal control theory can tell how to best steer a rocket, quantum optimal control gives the tools to best steer a quantum system to a desired unitary gate. Quantum optimal control has been used for some time in the context of driving chemical reactions with shaped laser pulses Rabitz et al. ( 2000 ) . There, the goal is to maximize the transfer from a known initial state to a known final state. In the context of quantum computing, we do not necessarily know what the input state will be, and so we must find unitary gates which will work correctly for any input state. The Hamiltonian at any point in time can be written down as ℋ t ​ o ​ t ​ ( t ) = ℋ n ​ a ​ t + ℋ r ​ f ​ ( t ) , subscript ℋ 𝑡 𝑜 𝑡 𝑡 subscript ℋ 𝑛 𝑎 𝑡 subscript ℋ 𝑟 𝑓 𝑡 \mathcal{{H}}_{tot}(t)=\mathcal{{H}}_{nat}+\mathcal{{H}}_{rf}(t), (14) where, ℋ n ​ a ​ t subscript ℋ 𝑛 𝑎 𝑡 \mathcal{{H}}_{nat} is the time-independent, natural (or drift) Hamiltonian, and ℋ r ​ f ​ ( t ) subscript ℋ 𝑟 𝑓 𝑡 \mathcal{{H}}_{rf}(t) represents the controllable time-dependent RF fields which we can use to drive the evolution of the system. The task is to find the sequence of control fields that will produce the correct unitary evolution. For reasonably complicated systems analytical solutions are beyond reach; however, using a numerical search we can find control sequences with fidelities as high as 0.999999 0.999999 0.999999 . The control fields are discretized at a suitable rate (see figure 4 ) and a random guess for the control sequence is chosen. We can simulate the evolution of the system under this sequence and obtain the simulated unitary U s ​ i ​ m subscript 𝑈 𝑠 𝑖 𝑚 U_{sim} . We can then compare this to our goal unitary U g ​ o ​ a ​ l subscript 𝑈 𝑔 𝑜 𝑎 𝑙 U_{goal} through a fitness function, Φ = | T ​ r ​ ( U s ​ i ​ m † ​ U g ​ o ​ a ​ l ) | 2 Φ superscript 𝑇 𝑟 superscript subscript 𝑈 𝑠 𝑖 𝑚 † subscript 𝑈 𝑔 𝑜 𝑎 𝑙 2 \Phi=\left|Tr\left(U_{sim}^{\dagger}U_{goal}\right)\right|^{2} (15) which is equivalent to the state overlap fidelity averaged over all input states Fortunato et al. ( 2002 ) . We can then use any optimization routine to search for the highest Φ Φ \Phi . The first step in this direction for NMR quantum information processing was the scheme of Fortunato et al. Fortunato et al. ( 2002 ) who, by limiting the form of the control fields to a small number of constant amplitude, phase and frequency periods, were able to find high fidelity control sequences through a simplex search. Many more control periods can be considered and the numerical search substantially sped up through the use of standard optimal control techniques to obtain information about the gradient of the fitness function at each point. This is the GRadient Ascent Pulse Engineering or GRAPE algorithm introduced by Khaneja et al. Khaneja et al. ( 2005 ) . From the simulation information we can calculate approximate gradients of the fitness function with respect to the control amplitude at each timestep (see Figure 4 ). With this gradient information we can update the control fields by moving along the steepest ascent direction and then repeat the procedure. Convergence to a global maximum is of course not guaranteed as this hill climbing algorithm is a local search. However, empirically we find we can achieve sufficiently high fidelities from such local maxima. Convergence of the algorithm can be further improved by using non-linear conjugate gradients Bryson and Ho ( 1975 ) . The control sequences drive the system through a complicated and non-intuitive path and small errors in our modeling of the systems Hamiltonian might lead to a low fidelity pulse. Fortunately, the control sequences can also be made robust against static inhomogeneities or uncertainties such as field inhomogenties and amplitude miscalibration of pulses. Robustness to both these effects for a particular pulse is shown in figure 6 b. Figure 4: An example section of a GRAPE pulse. The bars show the control amplitude which is constant for each time step. The arrows show the derivatives from the GRAPE algorithm at each point which tell us how to update the pulse for the next iteration (adapted from Khaneja et al. ( 2005 ) ). Recent work on a 12-qubit liquid state system that compared this method with the more traditional approach outlined in previous sections can be found in Negrevergne et al. ( 2006 ) . Not surprisingly, the control using strongly modulating pulses was more precise but at an exponential cost in searching for the pulse sequences. The scheme described above can be generalized if we can break the molecule into strongly coupled subsystems which are themselves weakly coupled, and thus it is possible to perform the decomposition described in previous sections between subsystems using pairwise simulation of subsystems. Gates to implement rotations within the subsystem will however require numerical optimization methods such as GRAPE. The details of this hybrid method are presently under investigation. The ability to control liquid state nuclear spin systems has allowed implementation of a variety of benchmarking experiments and algorithms on small qubit registers. For example, we have developed sufficient control to implement quantum error correcting protocols Cory et al. ( 1998 ); Laforest et al. ( 2007 ); Cory et al. ( 2000 ); Knill et al. ( 2001 ); Boulant et al. ( 2004 ) , the simulation of quantum systems Chen et al. ( 2004 ); Negrevergne et al. ( 2004 ); Havel et al. ( 2001 ); Tseng et al. ( 1999 ); Somaroo et al. ( 1999 ) and the benchmark 12 qubit quantum processor Negrevergne et al. ( 2006 ) . An extensive list of these and other experiments can be found at http://arxiv.org/.","role":"user"},{"content":"Abstract: This paper describes recent progress using nuclear magnetic resonance (NMR) as a platform for implementing quantum information processing (QIP) tasks. The basic ideas of NMR QIP are detailed, examining the successes and limitations of liquid and solid state experiments. Finally, a future direction for implementing quantum processors is suggested,utilizing both nuclear and electron spin degrees of freedom.","role":"assistant"}]

On the field intersection problem of generic polynomials : a survey (Algebraic Number Theory and Related Topics 2007)

Let G 𝐺 G be a finite group, k 𝑘 k a field of characteristic ≠ 2 absent 2 \neq 2 , M 𝑀 M a field containing k 𝑘 k with # ​ M = ∞ # 𝑀 \#M=\infty , and k ​ ( 𝐭 ) 𝑘 𝐭 k(\mathbf{t}) the rational function field over k 𝑘 k with n 𝑛 n indeterminates 𝐭 = ( t 1 , … , t n ) 𝐭 subscript 𝑡 1 … subscript 𝑡 𝑛 \mathbf{t}=(t_{1},\ldots,t_{n}) . Our main interest in this note is a k 𝑘 k -generic polynomial for G 𝐺 G (cf. [ DeM83 ] , [ Kem01 ] , [ JLY02 ] ). Definition . A polynomial f 𝐭 ​ ( X ) ∈ k ​ ( 𝐭 ) ​ [ X ] subscript 𝑓 𝐭 𝑋 𝑘 𝐭 delimited-[] 𝑋 f_{\mathbf{t}}(X)\in k(\mathbf{t})[X] is called k 𝑘 k -generic for G 𝐺 G if it has the following property: the Galois group of f 𝐭 ​ ( X ) subscript 𝑓 𝐭 𝑋 f_{\mathbf{t}}(X) over k ​ ( 𝐭 ) 𝑘 𝐭 k(\mathbf{t}) is isomorphic to G 𝐺 G and every G 𝐺 G -Galois extension L / M 𝐿 𝑀 L/M over an arbitrary infinite field M ⊃ k 𝑘 𝑀 M\supset k can be obtained as L = Spl M ​ f 𝐚 ​ ( X ) 𝐿 subscript Spl 𝑀 subscript 𝑓 𝐚 𝑋 L=\mathrm{Spl}_{M}f_{\mathbf{a}}(X) , the splitting field of f 𝐚 ​ ( X ) subscript 𝑓 𝐚 𝑋 f_{\mathbf{a}}(X) over M 𝑀 M , for some 𝐚 = ( a 1 , … , a n ) ∈ M n 𝐚 subscript 𝑎 1 … subscript 𝑎 𝑛 superscript 𝑀 𝑛 \mathbf{a}=(a_{1},\ldots,a_{n})\in M^{n} . Let f 𝐭 G ​ ( X ) ∈ k ​ ( 𝐭 ) ​ [ X ] superscript subscript 𝑓 𝐭 𝐺 𝑋 𝑘 𝐭 delimited-[] 𝑋 f_{\mathbf{t}}^{G}(X)\in k(\mathbf{t})[X] be a k 𝑘 k -generic polynomial for G 𝐺 G . Examples of k 𝑘 k -generic polynomials for G 𝐺 G are known for various pairs of ( k , G ) 𝑘 𝐺 (k,G) (for example, see [ Kem94 ] , [ KM00 ] , [ JLY02 ] , [ Rik04 ] ). Since a k 𝑘 k -generic polynomial f 𝐭 G ​ ( X ) superscript subscript 𝑓 𝐭 𝐺 𝑋 f_{\mathbf{t}}^{G}(X) for G 𝐺 G covers all G 𝐺 G -Galois extensions over M ⊃ k 𝑘 𝑀 M\supset k by specializing parameters, it is natural to ask the following problem: Field isomorphism problem of a generic polynomial For a field M ⊃ k 𝑘 𝑀 M\supset k and 𝐚 , 𝐛 ∈ M n 𝐚 𝐛 superscript 𝑀 𝑛 \mathbf{a},\mathbf{b}\in M^{n} , determine whether Spl M ​ f 𝐚 G ​ ( X ) subscript Spl 𝑀 superscript subscript 𝑓 𝐚 𝐺 𝑋 \mathrm{Spl}_{M}f_{\mathbf{a}}^{G}(X) and Spl M ​ f 𝐛 G ​ ( X ) subscript Spl 𝑀 superscript subscript 𝑓 𝐛 𝐺 𝑋 \mathrm{Spl}_{M}f_{\mathbf{b}}^{G}(X) are isomorphic over M 𝑀 M or not. It would be desired to give an answer to the problem within the base field M 𝑀 M by using the data 𝐚 , 𝐛 ∈ M n 𝐚 𝐛 superscript 𝑀 𝑛 \mathbf{a},\mathbf{b}\in M^{n} . Throughout this paper, we assume that f 𝐚 G ​ ( X ) superscript subscript 𝑓 𝐚 𝐺 𝑋 f_{\mathbf{a}}^{G}(X) is separable for 𝐚 ∈ M n 𝐚 superscript 𝑀 𝑛 \mathbf{a}\in M^{n} . Let S n subscript 𝑆 𝑛 S_{n} (resp. D n subscript 𝐷 𝑛 D_{n} , C n subscript 𝐶 𝑛 C_{n} ) be the symmetric (resp. the dihedral, the cyclic) group of degree n 𝑛 n . We take k 𝑘 k -generic polynomials f t C 3 ​ ( X ) superscript subscript 𝑓 𝑡 subscript 𝐶 3 𝑋 \displaystyle f_{t}^{C_{3}}(X) := X 3 − t ​ X 2 − ( t + 3 ) ​ X − 1 ∈ k ​ ( t ) ​ [ X ] , assign absent superscript 𝑋 3 𝑡 superscript 𝑋 2 𝑡 3 𝑋 1 𝑘 𝑡 delimited-[] 𝑋 \displaystyle:=X^{3}-tX^{2}-(t+3)X-1\,\in k(t)[X], f t S 3 ​ ( X ) superscript subscript 𝑓 𝑡 subscript 𝑆 3 𝑋 \displaystyle f_{t}^{S_{3}}(X) := X 3 + t ​ X + t ∈ k ​ ( t ) ​ [ X ] , assign absent superscript 𝑋 3 𝑡 𝑋 𝑡 𝑘 𝑡 delimited-[] 𝑋 \displaystyle:=X^{3}+tX+t\,\in k(t)[X], f s , t D 4 ​ ( X ) superscript subscript 𝑓 𝑠 𝑡 subscript 𝐷 4 𝑋 \displaystyle f_{s,t}^{D_{4}}(X) := X 4 + s ​ X 2 + t ∈ k ​ ( s , t ) ​ [ X ] assign absent superscript 𝑋 4 𝑠 superscript 𝑋 2 𝑡 𝑘 𝑠 𝑡 delimited-[] 𝑋 \displaystyle:=X^{4}+sX^{2}+t\,\in k(s,t)[X] for G = C 3 𝐺 subscript 𝐶 3 G=C_{3} , S 3 subscript 𝑆 3 S_{3} , D 4 subscript 𝐷 4 D_{4} , respectively. By using formal Tschirnhausen transformation, we showed the following theorem which is an analogue to the results of Morton [ Mor94 ] and Chapman [ Cha96 ] . Theorem 1.1 ( [ Mor94 ] , [ Cha96 ] , [ HM ] ) For m , n ∈ M 𝑚 𝑛 𝑀 m,n\in M , the splitting fields of f m C 3 ​ ( X ) superscript subscript 𝑓 𝑚 subscript 𝐶 3 𝑋 f_{m}^{C_{3}}(X) and of f n C 3 ​ ( X ) superscript subscript 𝑓 𝑛 subscript 𝐶 3 𝑋 f_{n}^{C_{3}}(X) over M 𝑀 M coincide if and only if there exists z ∈ M 𝑧 𝑀 z\in M such that either n = m ​ ( z 3 − 3 ​ z − 1 ) − 9 ​ z ​ ( z + 1 ) m ​ z ​ ( z + 1 ) + z 3 + 3 ​ z 2 − 1 𝑜𝑟 n = − m ​ ( z 3 + 3 ​ z 2 − 1 ) + 3 ​ ( z 3 − 3 ​ z − 1 ) m ​ z ​ ( z + 1 ) + z 3 + 3 ​ z 2 − 1 . formulae-sequence 𝑛 𝑚 superscript 𝑧 3 3 𝑧 1 9 𝑧 𝑧 1 𝑚 𝑧 𝑧 1 superscript 𝑧 3 3 superscript 𝑧 2 1 𝑜𝑟 𝑛 𝑚 superscript 𝑧 3 3 superscript 𝑧 2 1 3 superscript 𝑧 3 3 𝑧 1 𝑚 𝑧 𝑧 1 superscript 𝑧 3 3 superscript 𝑧 2 1 n\,=\,\frac{m(z^{3}-3z-1)-9z(z+1)}{mz(z+1)+z^{3}+3z^{2}-1}\ \ \mathit{or}\ \ n\,=\,-\frac{m(z^{3}+3z^{2}-1)+3(z^{3}-3z-1)}{mz(z+1)+z^{3}+3z^{2}-1}. We also have analogues to the above theorem for two non-abelian groups S 3 subscript 𝑆 3 S_{3} and D 4 subscript 𝐷 4 D_{4} via formal Tschirnhausen transformation. Theorem 1.2 ( [ HM07 ] ) Assume that char k ≠ 3 𝑘 3 k\neq 3 . For a , b ∈ M 𝑎 𝑏 𝑀 a,b\in M with a ≠ b 𝑎 𝑏 a\neq b , the splitting fields of f a S 3 ​ ( X ) superscript subscript 𝑓 𝑎 subscript 𝑆 3 𝑋 f_{a}^{S_{3}}(X) and of f b S 3 ​ ( X ) superscript subscript 𝑓 𝑏 subscript 𝑆 3 𝑋 f_{b}^{S_{3}}(X) over M 𝑀 M coincide if and only if there exists u ∈ M 𝑢 𝑀 u\in M such that b = a ​ ( u 2 + 9 ​ u − 3 ​ a ) 3 ( u 3 − 2 ​ a ​ u 2 − 9 ​ a ​ u − 2 ​ a 2 − 27 ​ a ) 2 . 𝑏 𝑎 superscript superscript 𝑢 2 9 𝑢 3 𝑎 3 superscript superscript 𝑢 3 2 𝑎 superscript 𝑢 2 9 𝑎 𝑢 2 superscript 𝑎 2 27 𝑎 2 b=\frac{a(u^{2}+9u-3a)^{3}}{(u^{3}-2au^{2}-9au-2a^{2}-27a)^{2}}. Theorem 1.3 ( [ HM-2 ] ) For a , b ∈ M 𝑎 𝑏 𝑀 a,b\in M , we assume that Gal ​ ( f a , b D 4 / M ) = D 4 Gal superscript subscript 𝑓 𝑎 𝑏 subscript 𝐷 4 𝑀 subscript 𝐷 4 \mathrm{Gal}(f_{a,b}^{D_{4}}/M)=D_{4} . Then for a , b , a ′ , b ′ ∈ M 𝑎 𝑏 superscript 𝑎 ′ superscript 𝑏 ′ 𝑀 a,b,a^{\prime},b^{\prime}\in M , the splitting fields of f a , b D 4 ​ ( X ) superscript subscript 𝑓 𝑎 𝑏 subscript 𝐷 4 𝑋 f_{a,b}^{D_{4}}(X) and of f a ′ , b ′ D 4 ​ ( X ) superscript subscript 𝑓 superscript 𝑎 ′ superscript 𝑏 ′ subscript 𝐷 4 𝑋 f_{a^{\prime},b^{\prime}}^{D_{4}}(X) over M 𝑀 M coincide if and only if there exist p , q ∈ M 𝑝 𝑞 𝑀 p,q\in M such that either ( i ) a ′ i superscript 𝑎 ′ \displaystyle{\rm(i)}\quad a^{\prime} = a ​ p 2 − 4 ​ b ​ p ​ q + a ​ b ​ q 2 , b ′ = b ​ ( p 2 − a ​ p ​ q + b ​ q 2 ) 2 𝑜𝑟 formulae-sequence absent 𝑎 superscript 𝑝 2 4 𝑏 𝑝 𝑞 𝑎 𝑏 superscript 𝑞 2 superscript 𝑏 ′ 𝑏 superscript superscript 𝑝 2 𝑎 𝑝 𝑞 𝑏 superscript 𝑞 2 2 𝑜𝑟 \displaystyle=ap^{2}-4bpq+abq^{2},\quad b^{\prime}=b(p^{2}-apq+bq^{2})^{2}\ \ \mathit{or} ( ii ) a ′ ii superscript 𝑎 ′ \displaystyle{\rm(ii)}\quad a^{\prime} = 2 ​ ( a ​ p 2 − 4 ​ b ​ p ​ q + a ​ b ​ q 2 ) , b ′ = ( a 2 − 4 ​ b ) ​ ( p 2 − b ​ q 2 ) 2 . formulae-sequence absent 2 𝑎 superscript 𝑝 2 4 𝑏 𝑝 𝑞 𝑎 𝑏 superscript 𝑞 2 superscript 𝑏 ′ superscript 𝑎 2 4 𝑏 superscript superscript 𝑝 2 𝑏 superscript 𝑞 2 2 \displaystyle=2(ap^{2}-4bpq+abq^{2}),\quad b^{\prime}=(a^{2}-4b)(p^{2}-bq^{2})^{2}. By applying Hilbert’s irreducibility theorem (cf. for example [ JLY02 , Chapter 3] ) and Siegel’s theorem for curves of genus 0 0 (cf. for example [ Lan78 , Theorem 6.1] ) to the theorems above respectively, we get the following corollaries: Corollary 1.4 Let f 𝐚 G ​ ( X ) = f m C 3 ​ ( X ) superscript subscript 𝑓 𝐚 𝐺 𝑋 superscript subscript 𝑓 𝑚 subscript 𝐶 3 𝑋 f_{\mathbf{a}}^{G}(X)=f_{m}^{C_{3}}(X) ( ( ( resp. f a S 3 ​ ( X ) superscript subscript 𝑓 𝑎 subscript 𝑆 3 𝑋 f_{a}^{S_{3}}(X) , f a , b D 4 ( X ) ) f_{a,b}^{D_{4}}(X)) be as above in Theorem 1.1 ( ( ( resp. Theorem 1.2 , Theorem 1.3 ) ) ) with given 𝐚 ∈ M n 𝐚 superscript 𝑀 𝑛 \mathbf{a}\in M^{n} , and suppose that M ⊃ k 𝑘 𝑀 M\supset k is Hilbertian ( ( ( e.g. a number field ) ) ) . Then there exist infinitely many 𝐛 ∈ M n 𝐛 superscript 𝑀 𝑛 \mathbf{b}\in M^{n} such that Spl M ​ f 𝐚 G ​ ( X ) = Spl M ​ f 𝐛 G ​ ( X ) subscript Spl 𝑀 superscript subscript 𝑓 𝐚 𝐺 𝑋 subscript Spl 𝑀 superscript subscript 𝑓 𝐛 𝐺 𝑋 \mathrm{Spl}_{M}f_{\mathbf{a}}^{G}(X)=\mathrm{Spl}_{M}f_{\mathbf{b}}^{G}(X) . Corollary 1.5 Let M 𝑀 M be a number field and 𝒪 M subscript 𝒪 𝑀 \mathcal{O}_{M} the ring of integers in M 𝑀 M . For f a G ​ ( X ) = f m C 3 ​ ( X ) superscript subscript 𝑓 𝑎 𝐺 𝑋 superscript subscript 𝑓 𝑚 subscript 𝐶 3 𝑋 f_{a}^{G}(X)=f_{m}^{C_{3}}(X) ( ( ( resp. f a S 3 ( X ) ) f_{a}^{S_{3}}(X)) as above in Theorem 1.1 ( ( ( resp. Theorem 1.2 ) ) ) with a given integer a ∈ 𝒪 M 𝑎 subscript 𝒪 𝑀 a\in\mathcal{O}_{M} , there exist only finitely many integers b ∈ 𝒪 M 𝑏 subscript 𝒪 𝑀 b\in\mathcal{O}_{M} such that Spl M ​ f a G ​ ( X ) = Spl M ​ f b G ​ ( X ) subscript Spl 𝑀 superscript subscript 𝑓 𝑎 𝐺 𝑋 subscript Spl 𝑀 superscript subscript 𝑓 𝑏 𝐺 𝑋 \mathrm{Spl}_{M}f_{a}^{G}(X)=\mathrm{Spl}_{M}f_{b}^{G}(X) . Indeed integers b ∈ 𝒪 M 𝑏 subscript 𝒪 𝑀 b\in\mathcal{O}_{M} as in Corollary 1.5 are derived from some integer solutions of (finitely many) cubic Thue equations a ​ X ​ Y ​ ( X + Y ) + X 3 + 3 ​ X ​ Y 2 − Y 3 = λ 𝑎 𝑋 𝑌 𝑋 𝑌 superscript 𝑋 3 3 𝑋 superscript 𝑌 2 superscript 𝑌 3 𝜆 aXY(X+Y)+X^{3}+3XY^{2}-Y^{3}=\lambda (resp. X 3 − 2 ​ a ​ X 2 ​ Y − 9 ​ a ​ X ​ Y 2 − 2 ​ a ​ Y 3 − 27 ​ a ​ Y 3 = λ superscript 𝑋 3 2 𝑎 superscript 𝑋 2 𝑌 9 𝑎 𝑋 superscript 𝑌 2 2 𝑎 superscript 𝑌 3 27 𝑎 superscript 𝑌 3 𝜆 X^{3}-2aX^{2}Y-9aXY^{2}-2aY^{3}-27aY^{3}=\lambda ) which are curves of genus 1 1 1 (see also the proof of [ Lan78 , Theorem 6.1] ). Kemper [ Kem01 ] , furthermore, showed that for a subgroup H 𝐻 H of G 𝐺 G every H 𝐻 H -Galois extension over M 𝑀 M is also given by a specialization f 𝐚 G ​ ( X ) superscript subscript 𝑓 𝐚 𝐺 𝑋 f_{\mathbf{a}}^{G}(X) , 𝐚 ∈ M n 𝐚 superscript 𝑀 𝑛 \mathbf{a}\in M^{n} , of a generic polynomial f 𝐭 G ​ ( X ) superscript subscript 𝑓 𝐭 𝐺 𝑋 f_{\mathbf{t}}^{G}(X) for G 𝐺 G . Hence a problem naturally arises. Field intersection problem of a generic polynomial For a field M ⊃ k 𝑘 𝑀 M\supset k and 𝐚 , 𝐛 ∈ M n 𝐚 𝐛 superscript 𝑀 𝑛 \mathbf{a},\mathbf{b}\in M^{n} , determine the intersection of Spl M ​ f 𝐚 G ​ ( X ) subscript Spl 𝑀 superscript subscript 𝑓 𝐚 𝐺 𝑋 \mathrm{Spl}_{M}f_{\mathbf{a}}^{G}(X) and Spl M ​ f 𝐛 G ​ ( X ) subscript Spl 𝑀 superscript subscript 𝑓 𝐛 𝐺 𝑋 \mathrm{Spl}_{M}f_{\mathbf{b}}^{G}(X) . Clearly if we get an answer to the field intersection problem of a k 𝑘 k -generic polynomial, we also obtain an answer to the corresponding field isomorphism problem. The aim of this note is to survey a method to give an answer to the isomorphism problem and the intersection problem of k 𝑘 k -generic polynomials via formal Tschirnhausen transformation and multi-resolvent polynomials. In Section 2 , we review known results about resolvent polynomials. In Section 3 , we recall a formal Tschirnhausen transformation which is given in [ HM ] . In Section 4 , we give a general method to solve the intersection problem of k 𝑘 k -generic polynomials. In Section 5 , we obtain an explicit answer to the problems in the cubic case. We give a proof of Theorem 1.2 as a special case of the intersection problem of f t S 3 ​ ( X ) = X 3 + t ​ X + t superscript subscript 𝑓 𝑡 subscript 𝑆 3 𝑋 superscript 𝑋 3 𝑡 𝑋 𝑡 f_{t}^{S_{3}}(X)=X^{3}+tX+t via formal Tschirnhausen transformation. In Section 6 , we take the k 𝑘 k -generic polynomial f s , t D 5 ​ ( X ) superscript subscript 𝑓 𝑠 𝑡 subscript 𝐷 5 𝑋 \displaystyle f_{s,t}^{D_{5}}(X) = X 5 + ( t − 3 ) ​ X 4 + ( s − t + 3 ) ​ X 3 + ( t 2 − t − 2 ​ s − 1 ) ​ X 2 + s ​ X + t ∈ k ​ ( s , t ) ​ [ X ] absent superscript 𝑋 5 𝑡 3 superscript 𝑋 4 𝑠 𝑡 3 superscript 𝑋 3 superscript 𝑡 2 𝑡 2 𝑠 1 superscript 𝑋 2 𝑠 𝑋 𝑡 𝑘 𝑠 𝑡 delimited-[] 𝑋 \displaystyle=X^{5}+(t-3)X^{4}+(s-t+3)X^{3}+(t^{2}-t-2s-1)X^{2}+sX+t\in k(s,t)[X] for D 5 subscript 𝐷 5 D_{5} which is called Brumer’s quintic (cf. [ JLY02 ] ). Based on the general result, we illustrate how to give an answer to the problem for f s , t D 5 ​ ( X ) superscript subscript 𝑓 𝑠 𝑡 subscript 𝐷 5 𝑋 f_{s,t}^{D_{5}}(X) by multi-resolvent polynomials. We also give some numerical examples.

Let k be a eld of characteristic 2. We survey a general method of the eld intersection problem of generic polynomials via formal Tschirnhausen transformation. We announce some of our recent results of cubic, quartic and quintic cases the details of which are to appear elsewhere. In this note, we give an explicit answer to the problem in the cases of cubic and dihedral quintic by using multi-resolvent polynomials.

[{"content":"Help me to generate the abstract of a survey paper given the title:*On the field intersection problem of generic polynomials : a survey (Algebraic Number Theory and Related Topics 2007)* and the introduction:Let G 𝐺 G be a finite group, k 𝑘 k a field of characteristic ≠ 2 absent 2 \neq 2 , M 𝑀 M a field containing k 𝑘 k with # ​ M = ∞ # 𝑀 \#M=\infty , and k ​ ( 𝐭 ) 𝑘 𝐭 k(\mathbf{t}) the rational function field over k 𝑘 k with n 𝑛 n indeterminates 𝐭 = ( t 1 , … , t n ) 𝐭 subscript 𝑡 1 … subscript 𝑡 𝑛 \mathbf{t}=(t_{1},\ldots,t_{n}) . Our main interest in this note is a k 𝑘 k -generic polynomial for G 𝐺 G (cf. DeM83 , Kem01 , JLY02 ). Definition . A polynomial f 𝐭 ​ ( X ) ∈ k ​ ( 𝐭 ) ​ X subscript 𝑓 𝐭 𝑋 𝑘 𝐭 delimited- 𝑋 f_{\mathbf{t}}(X)\in k(\mathbf{t})X is called k 𝑘 k -generic for G 𝐺 G if it has the following property: the Galois group of f 𝐭 ​ ( X ) subscript 𝑓 𝐭 𝑋 f_{\mathbf{t}}(X) over k ​ ( 𝐭 ) 𝑘 𝐭 k(\mathbf{t}) is isomorphic to G 𝐺 G and every G 𝐺 G -Galois extension L / M 𝐿 𝑀 L/M over an arbitrary infinite field M ⊃ k 𝑘 𝑀 M\supset k can be obtained as L = Spl M ​ f 𝐚 ​ ( X ) 𝐿 subscript Spl 𝑀 subscript 𝑓 𝐚 𝑋 L=\mathrm{Spl}_{M}f_{\mathbf{a}}(X) , the splitting field of f 𝐚 ​ ( X ) subscript 𝑓 𝐚 𝑋 f_{\mathbf{a}}(X) over M 𝑀 M , for some 𝐚 = ( a 1 , … , a n ) ∈ M n 𝐚 subscript 𝑎 1 … subscript 𝑎 𝑛 superscript 𝑀 𝑛 \mathbf{a}=(a_{1},\ldots,a_{n})\in M^{n} . Let f 𝐭 G ​ ( X ) ∈ k ​ ( 𝐭 ) ​ X superscript subscript 𝑓 𝐭 𝐺 𝑋 𝑘 𝐭 delimited- 𝑋 f_{\mathbf{t}}^{G}(X)\in k(\mathbf{t})X be a k 𝑘 k -generic polynomial for G 𝐺 G . Examples of k 𝑘 k -generic polynomials for G 𝐺 G are known for various pairs of ( k , G ) 𝑘 𝐺 (k,G) (for example, see Kem94 , KM00 , JLY02 , Rik04 ). Since a k 𝑘 k -generic polynomial f 𝐭 G ​ ( X ) superscript subscript 𝑓 𝐭 𝐺 𝑋 f_{\mathbf{t}}^{G}(X) for G 𝐺 G covers all G 𝐺 G -Galois extensions over M ⊃ k 𝑘 𝑀 M\supset k by specializing parameters, it is natural to ask the following problem: Field isomorphism problem of a generic polynomial For a field M ⊃ k 𝑘 𝑀 M\supset k and 𝐚 , 𝐛 ∈ M n 𝐚 𝐛 superscript 𝑀 𝑛 \mathbf{a},\mathbf{b}\in M^{n} , determine whether Spl M ​ f 𝐚 G ​ ( X ) subscript Spl 𝑀 superscript subscript 𝑓 𝐚 𝐺 𝑋 \mathrm{Spl}_{M}f_{\mathbf{a}}^{G}(X) and Spl M ​ f 𝐛 G ​ ( X ) subscript Spl 𝑀 superscript subscript 𝑓 𝐛 𝐺 𝑋 \mathrm{Spl}_{M}f_{\mathbf{b}}^{G}(X) are isomorphic over M 𝑀 M or not. It would be desired to give an answer to the problem within the base field M 𝑀 M by using the data 𝐚 , 𝐛 ∈ M n 𝐚 𝐛 superscript 𝑀 𝑛 \mathbf{a},\mathbf{b}\in M^{n} . Throughout this paper, we assume that f 𝐚 G ​ ( X ) superscript subscript 𝑓 𝐚 𝐺 𝑋 f_{\mathbf{a}}^{G}(X) is separable for 𝐚 ∈ M n 𝐚 superscript 𝑀 𝑛 \mathbf{a}\in M^{n} . Let S n subscript 𝑆 𝑛 S_{n} (resp. D n subscript 𝐷 𝑛 D_{n} , C n subscript 𝐶 𝑛 C_{n} ) be the symmetric (resp. the dihedral, the cyclic) group of degree n 𝑛 n . We take k 𝑘 k -generic polynomials f t C 3 ​ ( X ) superscript subscript 𝑓 𝑡 subscript 𝐶 3 𝑋 \displaystyle f_{t}^{C_{3}}(X) := X 3 − t ​ X 2 − ( t + 3 ) ​ X − 1 ∈ k ​ ( t ) ​ X , assign absent superscript 𝑋 3 𝑡 superscript 𝑋 2 𝑡 3 𝑋 1 𝑘 𝑡 delimited- 𝑋 \displaystyle:=X^{3}-tX^{2}-(t+3)X-1\,\in k(t)X, f t S 3 ​ ( X ) superscript subscript 𝑓 𝑡 subscript 𝑆 3 𝑋 \displaystyle f_{t}^{S_{3}}(X) := X 3 + t ​ X + t ∈ k ​ ( t ) ​ X , assign absent superscript 𝑋 3 𝑡 𝑋 𝑡 𝑘 𝑡 delimited- 𝑋 \displaystyle:=X^{3}+tX+t\,\in k(t)X, f s , t D 4 ​ ( X ) superscript subscript 𝑓 𝑠 𝑡 subscript 𝐷 4 𝑋 \displaystyle f_{s,t}^{D_{4}}(X) := X 4 + s ​ X 2 + t ∈ k ​ ( s , t ) ​ X assign absent superscript 𝑋 4 𝑠 superscript 𝑋 2 𝑡 𝑘 𝑠 𝑡 delimited- 𝑋 \displaystyle:=X^{4}+sX^{2}+t\,\in k(s,t)X for G = C 3 𝐺 subscript 𝐶 3 G=C_{3} , S 3 subscript 𝑆 3 S_{3} , D 4 subscript 𝐷 4 D_{4} , respectively. By using formal Tschirnhausen transformation, we showed the following theorem which is an analogue to the results of Morton Mor94 and Chapman Cha96 . Theorem 1.1 ( Mor94 , Cha96 , HM ) For m , n ∈ M 𝑚 𝑛 𝑀 m,n\in M , the splitting fields of f m C 3 ​ ( X ) superscript subscript 𝑓 𝑚 subscript 𝐶 3 𝑋 f_{m}^{C_{3}}(X) and of f n C 3 ​ ( X ) superscript subscript 𝑓 𝑛 subscript 𝐶 3 𝑋 f_{n}^{C_{3}}(X) over M 𝑀 M coincide if and only if there exists z ∈ M 𝑧 𝑀 z\in M such that either n = m ​ ( z 3 − 3 ​ z − 1 ) − 9 ​ z ​ ( z + 1 ) m ​ z ​ ( z + 1 ) + z 3 + 3 ​ z 2 − 1 𝑜𝑟 n = − m ​ ( z 3 + 3 ​ z 2 − 1 ) + 3 ​ ( z 3 − 3 ​ z − 1 ) m ​ z ​ ( z + 1 ) + z 3 + 3 ​ z 2 − 1 . formulae-sequence 𝑛 𝑚 superscript 𝑧 3 3 𝑧 1 9 𝑧 𝑧 1 𝑚 𝑧 𝑧 1 superscript 𝑧 3 3 superscript 𝑧 2 1 𝑜𝑟 𝑛 𝑚 superscript 𝑧 3 3 superscript 𝑧 2 1 3 superscript 𝑧 3 3 𝑧 1 𝑚 𝑧 𝑧 1 superscript 𝑧 3 3 superscript 𝑧 2 1 n\,=\,\frac{m(z^{3}-3z-1)-9z(z+1)}{mz(z+1)+z^{3}+3z^{2}-1}\ \ \mathit{or}\ \ n\,=\,-\frac{m(z^{3}+3z^{2}-1)+3(z^{3}-3z-1)}{mz(z+1)+z^{3}+3z^{2}-1}. We also have analogues to the above theorem for two non-abelian groups S 3 subscript 𝑆 3 S_{3} and D 4 subscript 𝐷 4 D_{4} via formal Tschirnhausen transformation. Theorem 1.2 ( HM07 ) Assume that char k ≠ 3 𝑘 3 k\neq 3 . For a , b ∈ M 𝑎 𝑏 𝑀 a,b\in M with a ≠ b 𝑎 𝑏 a\neq b , the splitting fields of f a S 3 ​ ( X ) superscript subscript 𝑓 𝑎 subscript 𝑆 3 𝑋 f_{a}^{S_{3}}(X) and of f b S 3 ​ ( X ) superscript subscript 𝑓 𝑏 subscript 𝑆 3 𝑋 f_{b}^{S_{3}}(X) over M 𝑀 M coincide if and only if there exists u ∈ M 𝑢 𝑀 u\in M such that b = a ​ ( u 2 + 9 ​ u − 3 ​ a ) 3 ( u 3 − 2 ​ a ​ u 2 − 9 ​ a ​ u − 2 ​ a 2 − 27 ​ a ) 2 . 𝑏 𝑎 superscript superscript 𝑢 2 9 𝑢 3 𝑎 3 superscript superscript 𝑢 3 2 𝑎 superscript 𝑢 2 9 𝑎 𝑢 2 superscript 𝑎 2 27 𝑎 2 b=\frac{a(u^{2}+9u-3a)^{3}}{(u^{3}-2au^{2}-9au-2a^{2}-27a)^{2}}. Theorem 1.3 ( HM-2 ) For a , b ∈ M 𝑎 𝑏 𝑀 a,b\in M , we assume that Gal ​ ( f a , b D 4 / M ) = D 4 Gal superscript subscript 𝑓 𝑎 𝑏 subscript 𝐷 4 𝑀 subscript 𝐷 4 \mathrm{Gal}(f_{a,b}^{D_{4}}/M)=D_{4} . Then for a , b , a ′ , b ′ ∈ M 𝑎 𝑏 superscript 𝑎 ′ superscript 𝑏 ′ 𝑀 a,b,a^{\prime},b^{\prime}\in M , the splitting fields of f a , b D 4 ​ ( X ) superscript subscript 𝑓 𝑎 𝑏 subscript 𝐷 4 𝑋 f_{a,b}^{D_{4}}(X) and of f a ′ , b ′ D 4 ​ ( X ) superscript subscript 𝑓 superscript 𝑎 ′ superscript 𝑏 ′ subscript 𝐷 4 𝑋 f_{a^{\prime},b^{\prime}}^{D_{4}}(X) over M 𝑀 M coincide if and only if there exist p , q ∈ M 𝑝 𝑞 𝑀 p,q\in M such that either ( i ) a ′ i superscript 𝑎 ′ \displaystyle{\rm(i)}\quad a^{\prime} = a ​ p 2 − 4 ​ b ​ p ​ q + a ​ b ​ q 2 , b ′ = b ​ ( p 2 − a ​ p ​ q + b ​ q 2 ) 2 𝑜𝑟 formulae-sequence absent 𝑎 superscript 𝑝 2 4 𝑏 𝑝 𝑞 𝑎 𝑏 superscript 𝑞 2 superscript 𝑏 ′ 𝑏 superscript superscript 𝑝 2 𝑎 𝑝 𝑞 𝑏 superscript 𝑞 2 2 𝑜𝑟 \displaystyle=ap^{2}-4bpq+abq^{2},\quad b^{\prime}=b(p^{2}-apq+bq^{2})^{2}\ \ \mathit{or} ( ii ) a ′ ii superscript 𝑎 ′ \displaystyle{\rm(ii)}\quad a^{\prime} = 2 ​ ( a ​ p 2 − 4 ​ b ​ p ​ q + a ​ b ​ q 2 ) , b ′ = ( a 2 − 4 ​ b ) ​ ( p 2 − b ​ q 2 ) 2 . formulae-sequence absent 2 𝑎 superscript 𝑝 2 4 𝑏 𝑝 𝑞 𝑎 𝑏 superscript 𝑞 2 superscript 𝑏 ′ superscript 𝑎 2 4 𝑏 superscript superscript 𝑝 2 𝑏 superscript 𝑞 2 2 \displaystyle=2(ap^{2}-4bpq+abq^{2}),\quad b^{\prime}=(a^{2}-4b)(p^{2}-bq^{2})^{2}. By applying Hilberts irreducibility theorem (cf. for example JLY02 , Chapter 3 ) and Siegels theorem for curves of genus 0 0 (cf. for example Lan78 , Theorem 6.1 ) to the theorems above respectively, we get the following corollaries: Corollary 1.4 Let f 𝐚 G ​ ( X ) = f m C 3 ​ ( X ) superscript subscript 𝑓 𝐚 𝐺 𝑋 superscript subscript 𝑓 𝑚 subscript 𝐶 3 𝑋 f_{\mathbf{a}}^{G}(X)=f_{m}^{C_{3}}(X) ( ( ( resp. f a S 3 ​ ( X ) superscript subscript 𝑓 𝑎 subscript 𝑆 3 𝑋 f_{a}^{S_{3}}(X) , f a , b D 4 ( X ) ) f_{a,b}^{D_{4}}(X)) be as above in Theorem 1.1 ( ( ( resp. Theorem 1.2 , Theorem 1.3 ) ) ) with given 𝐚 ∈ M n 𝐚 superscript 𝑀 𝑛 \mathbf{a}\in M^{n} , and suppose that M ⊃ k 𝑘 𝑀 M\supset k is Hilbertian ( ( ( e.g. a number field ) ) ) . Then there exist infinitely many 𝐛 ∈ M n 𝐛 superscript 𝑀 𝑛 \mathbf{b}\in M^{n} such that Spl M ​ f 𝐚 G ​ ( X ) = Spl M ​ f 𝐛 G ​ ( X ) subscript Spl 𝑀 superscript subscript 𝑓 𝐚 𝐺 𝑋 subscript Spl 𝑀 superscript subscript 𝑓 𝐛 𝐺 𝑋 \mathrm{Spl}_{M}f_{\mathbf{a}}^{G}(X)=\mathrm{Spl}_{M}f_{\mathbf{b}}^{G}(X) . Corollary 1.5 Let M 𝑀 M be a number field and 𝒪 M subscript 𝒪 𝑀 \mathcal{O}_{M} the ring of integers in M 𝑀 M . For f a G ​ ( X ) = f m C 3 ​ ( X ) superscript subscript 𝑓 𝑎 𝐺 𝑋 superscript subscript 𝑓 𝑚 subscript 𝐶 3 𝑋 f_{a}^{G}(X)=f_{m}^{C_{3}}(X) ( ( ( resp. f a S 3 ( X ) ) f_{a}^{S_{3}}(X)) as above in Theorem 1.1 ( ( ( resp. Theorem 1.2 ) ) ) with a given integer a ∈ 𝒪 M 𝑎 subscript 𝒪 𝑀 a\in\mathcal{O}_{M} , there exist only finitely many integers b ∈ 𝒪 M 𝑏 subscript 𝒪 𝑀 b\in\mathcal{O}_{M} such that Spl M ​ f a G ​ ( X ) = Spl M ​ f b G ​ ( X ) subscript Spl 𝑀 superscript subscript 𝑓 𝑎 𝐺 𝑋 subscript Spl 𝑀 superscript subscript 𝑓 𝑏 𝐺 𝑋 \mathrm{Spl}_{M}f_{a}^{G}(X)=\mathrm{Spl}_{M}f_{b}^{G}(X) . Indeed integers b ∈ 𝒪 M 𝑏 subscript 𝒪 𝑀 b\in\mathcal{O}_{M} as in Corollary 1.5 are derived from some integer solutions of (finitely many) cubic Thue equations a ​ X ​ Y ​ ( X + Y ) + X 3 + 3 ​ X ​ Y 2 − Y 3 = λ 𝑎 𝑋 𝑌 𝑋 𝑌 superscript 𝑋 3 3 𝑋 superscript 𝑌 2 superscript 𝑌 3 𝜆 aXY(X+Y)+X^{3}+3XY^{2}-Y^{3}=\lambda (resp. X 3 − 2 ​ a ​ X 2 ​ Y − 9 ​ a ​ X ​ Y 2 − 2 ​ a ​ Y 3 − 27 ​ a ​ Y 3 = λ superscript 𝑋 3 2 𝑎 superscript 𝑋 2 𝑌 9 𝑎 𝑋 superscript 𝑌 2 2 𝑎 superscript 𝑌 3 27 𝑎 superscript 𝑌 3 𝜆 X^{3}-2aX^{2}Y-9aXY^{2}-2aY^{3}-27aY^{3}=\lambda ) which are curves of genus 1 1 1 (see also the proof of Lan78 , Theorem 6.1 ). Kemper Kem01 , furthermore, showed that for a subgroup H 𝐻 H of G 𝐺 G every H 𝐻 H -Galois extension over M 𝑀 M is also given by a specialization f 𝐚 G ​ ( X ) superscript subscript 𝑓 𝐚 𝐺 𝑋 f_{\mathbf{a}}^{G}(X) , 𝐚 ∈ M n 𝐚 superscript 𝑀 𝑛 \mathbf{a}\in M^{n} , of a generic polynomial f 𝐭 G ​ ( X ) superscript subscript 𝑓 𝐭 𝐺 𝑋 f_{\mathbf{t}}^{G}(X) for G 𝐺 G . Hence a problem naturally arises. Field intersection problem of a generic polynomial For a field M ⊃ k 𝑘 𝑀 M\supset k and 𝐚 , 𝐛 ∈ M n 𝐚 𝐛 superscript 𝑀 𝑛 \mathbf{a},\mathbf{b}\in M^{n} , determine the intersection of Spl M ​ f 𝐚 G ​ ( X ) subscript Spl 𝑀 superscript subscript 𝑓 𝐚 𝐺 𝑋 \mathrm{Spl}_{M}f_{\mathbf{a}}^{G}(X) and Spl M ​ f 𝐛 G ​ ( X ) subscript Spl 𝑀 superscript subscript 𝑓 𝐛 𝐺 𝑋 \mathrm{Spl}_{M}f_{\mathbf{b}}^{G}(X) . Clearly if we get an answer to the field intersection problem of a k 𝑘 k -generic polynomial, we also obtain an answer to the corresponding field isomorphism problem. The aim of this note is to survey a method to give an answer to the isomorphism problem and the intersection problem of k 𝑘 k -generic polynomials via formal Tschirnhausen transformation and multi-resolvent polynomials. In Section 2 , we review known results about resolvent polynomials. In Section 3 , we recall a formal Tschirnhausen transformation which is given in HM . In Section 4 , we give a general method to solve the intersection problem of k 𝑘 k -generic polynomials. In Section 5 , we obtain an explicit answer to the problems in the cubic case. We give a proof of Theorem 1.2 as a special case of the intersection problem of f t S 3 ​ ( X ) = X 3 + t ​ X + t superscript subscript 𝑓 𝑡 subscript 𝑆 3 𝑋 superscript 𝑋 3 𝑡 𝑋 𝑡 f_{t}^{S_{3}}(X)=X^{3}+tX+t via formal Tschirnhausen transformation. In Section 6 , we take the k 𝑘 k -generic polynomial f s , t D 5 ​ ( X ) superscript subscript 𝑓 𝑠 𝑡 subscript 𝐷 5 𝑋 \displaystyle f_{s,t}^{D_{5}}(X) = X 5 + ( t − 3 ) ​ X 4 + ( s − t + 3 ) ​ X 3 + ( t 2 − t − 2 ​ s − 1 ) ​ X 2 + s ​ X + t ∈ k ​ ( s , t ) ​ X absent superscript 𝑋 5 𝑡 3 superscript 𝑋 4 𝑠 𝑡 3 superscript 𝑋 3 superscript 𝑡 2 𝑡 2 𝑠 1 superscript 𝑋 2 𝑠 𝑋 𝑡 𝑘 𝑠 𝑡 delimited- 𝑋 \displaystyle=X^{5}+(t-3)X^{4}+(s-t+3)X^{3}+(t^{2}-t-2s-1)X^{2}+sX+t\in k(s,t)X for D 5 subscript 𝐷 5 D_{5} which is called Brumers quintic (cf. JLY02 ). Based on the general result, we illustrate how to give an answer to the problem for f s , t D 5 ​ ( X ) superscript subscript 𝑓 𝑠 𝑡 subscript 𝐷 5 𝑋 f_{s,t}^{D_{5}}(X) by multi-resolvent polynomials. We also give some numerical examples.","role":"user"},{"content":"Abstract: Let k be a eld of characteristic 2. We survey a general method of the eld intersection problem of generic polynomials via formal Tschirnhausen transformation. We announce some of our recent results of cubic, quartic and quintic cases the details of which are to appear elsewhere. In this note, we give an explicit answer to the problem in the cases of cubic and dihedral quintic by using multi-resolvent polynomials.","role":"assistant"}]

Analog Chaos-based Secure Communications and Cryptanalysis: A Brief Survey

I Introduction Since the late 1980s, chaos-based cryptography has attracted more and more attention from researchers in many different areas. It has been found that chaotic systems and cryptosystems share many similar properties. For instance, chaotic systems are sensitive to the initial conditions, which corresponds to the diffusion property of good cryptosystems (for a comparison of chaos and cryptography, see Table 1 in [ 1 ] ). Basically, there are two major types of chaos-based cryptosystems: analog chaos-based secure communication systems and digital chaos-based ciphers, which are designed employing completely different principles. Almost all analog chaos-based secure communication systems are designed based on the technique for chaos synchronization, which was first discovered in the 1980s and then well developed in the 1990s [ 2 ] . The establishment of chaos synchronization between two remote chaotic systems actually means that some information has successfully been transmitted from one end to the other. This fact naturally leads to the foundation of a chaos-based communication system. Then, by keeping some part of the involved chaotic systems secret, a third party not knowing the secret key will not be able to reconstruct the information transmitted. Thus, a chaos-based secure communication system is created. Following this basic idea, a large number of analog chaos-based secure communication systems have been proposed since the 1990s. Meanwhile, related cryptanalytic work has also been developed to evaluate performance (mainly the security) of various analog chaos-based secure communication systems. Though a number of surveys have been published to introduce progress in this area, they become relatively obsolete due to the rapid growth of new research work in recent years. The purpose of this paper is to give a brief survey of analog chaos-based secure communications and related cryptanalytic work, especially focusing on latest work reported since the year 2000. This paper is organized as follows. In the next section we first introduce some preliminary knowledge about the underlying chaos synchronization technique. Then, we classify most early chaos-based secure communication systems into three basic types. Next, different kinds of cryptanalysis are discussed with some concrete examples. Finally, we enumerate some new countermeasures that have been proposed to resist known attacks. A few concluding remarks are given at the end of the paper to express our opinion on future trends in this area.

A large number of analog chaos-based secure communication systems have been proposed since the early 1990s exploiting the technique of chaos synchronization. A brief survey of these chaos-based cryptosystems and of related cryptanalytic results is given. Some recently proposed countermeasures against known attacks are also introduced.

[{"content":"Help me to generate the abstract of a survey paper given the title:*Analog Chaos-based Secure Communications and Cryptanalysis: A Brief Survey* and the introduction:I Introduction Since the late 1980s, chaos-based cryptography has attracted more and more attention from researchers in many different areas. It has been found that chaotic systems and cryptosystems share many similar properties. For instance, chaotic systems are sensitive to the initial conditions, which corresponds to the diffusion property of good cryptosystems (for a comparison of chaos and cryptography, see Table 1 in 1 ). Basically, there are two major types of chaos-based cryptosystems: analog chaos-based secure communication systems and digital chaos-based ciphers, which are designed employing completely different principles. Almost all analog chaos-based secure communication systems are designed based on the technique for chaos synchronization, which was first discovered in the 1980s and then well developed in the 1990s 2 . The establishment of chaos synchronization between two remote chaotic systems actually means that some information has successfully been transmitted from one end to the other. This fact naturally leads to the foundation of a chaos-based communication system. Then, by keeping some part of the involved chaotic systems secret, a third party not knowing the secret key will not be able to reconstruct the information transmitted. Thus, a chaos-based secure communication system is created. Following this basic idea, a large number of analog chaos-based secure communication systems have been proposed since the 1990s. Meanwhile, related cryptanalytic work has also been developed to evaluate performance (mainly the security) of various analog chaos-based secure communication systems. Though a number of surveys have been published to introduce progress in this area, they become relatively obsolete due to the rapid growth of new research work in recent years. The purpose of this paper is to give a brief survey of analog chaos-based secure communications and related cryptanalytic work, especially focusing on latest work reported since the year 2000. This paper is organized as follows. In the next section we first introduce some preliminary knowledge about the underlying chaos synchronization technique. Then, we classify most early chaos-based secure communication systems into three basic types. Next, different kinds of cryptanalysis are discussed with some concrete examples. Finally, we enumerate some new countermeasures that have been proposed to resist known attacks. A few concluding remarks are given at the end of the paper to express our opinion on future trends in this area.","role":"user"},{"content":"Abstract: A large number of analog chaos-based secure communication systems have been proposed since the early 1990s exploiting the technique of chaos synchronization. A brief survey of these chaos-based cryptosystems and of related cryptanalytic results is given. Some recently proposed countermeasures against known attacks are also introduced.","role":"assistant"}]

Leaky Quantum Graphs: A Review

1. Introduction In this paper we are going to review results concerning a class of “different” quantum graph models. With this aim in mind, it would be natural to start by recalling briefly the standard quantum graphs, their description, properties, and numerous applications. In this volume, however, this would be clearly to bring owls to Athens 1 1 1 Or, depending on your taste, coal to Newcastle, firewood to the forest, etc. As usual, one can also refer to the Bard: to throw a perfume on the violet (The Life and Death of King John). and we refrain from doing that referring to the other articles in these proceedings, or to [ BCFK06 ] as another rich bibliography source. To motivate a need to look for alternative description of graph-like structures, let us observe that — despite its mathematical simplicity, beauty, and versatility — the standard quantum-graph model has also some drawbacks. In our opinion, the following two are the most important: • the presence of ad hoc parameters in the boundary conditions which describe how the wave functions are matched at the graphs vertices • the fact that particles are strictly confined to graph edges. While this is often a reasonable approximation when dealing, say, with electrons in semiconductor quantum wires, such a model neglects quantum tunneling which can play role once such wires are placed close to each other. One consequence is that in such a description, with the graph embedded in ℝ ν superscript ℝ 𝜈 \mathbb{R}^{\nu} , spectral properties reflect the topology while the graph geometry enters only through the edge lengths, their shapes being irrelevant A way to deal with the first problem is to regard a quantum graph as an idealization of a more “realistic” system without such ambiguities; a natural candidate for this role are various “fat graphs”. Limits of such objects when the edge width squeezes to zero were studied extensively, first in the easier Neumann-type case [ FW93 , KZ01 , RS01 , Sa01 , Ku02 , KZ03 , EP05 , Po06 , EP07 ] and quite recently also in the situation with Dirichlet boundaries [ Po05 , MV07 , CE07 , Gr07 ] . These results give a partial answer to the first question 2 2 2 An alternative approach is to keep the graph fixed and to approximate the vertex coupling through suitably scaled families of regular or singular interactions – see [ Ex96 , CE04 , ETu07 ] . while the second problem remains. Here we are going to discuss a class of quantum graph models which are free of both difficulties; the price we pay is — similarly as for the fat graphs mentioned above — that instead of ordinary differential equations we have to deal with a PDE problem. The idea is to preserve the whole Euclidean space as the configuration space and to suppose that the particle is kept in the vicinity of the graph Γ ⊂ ℝ ν Γ superscript ℝ 𝜈 \Gamma\subset\mathbb{R}^{\nu} by an attractive singular interaction. Formally such a Hamiltonian expresses as (1.1) − Δ − α ​ ( x ) ​ δ ​ ( x − Γ ) Δ 𝛼 𝑥 𝛿 𝑥 Γ -\Delta-\alpha(x)\delta(x-\Gamma)\, with α ​ ( x ) > 0 𝛼 𝑥 0 \alpha(x)>0\, ; we will consider mostly the situation where the attraction is position independent, α ​ ( x ) = α > 0 𝛼 𝑥 𝛼 0 \alpha(x)=\alpha>0 . Before proceeding to a definition of such singular Schrödinger operators and discussion of their properties, let us make a few remarks. First of all, it is clear that there is no ambiguity related to the graph vertices once Γ Γ \Gamma and α 𝛼 \alpha are given. It is equally obvious that the confinement in this model takes place at negative energies only. The particle now “lives” in the whole space and can be found even at large distances from Γ Γ \Gamma , although with a small probability, because the complement ℝ ν ∖ Γ superscript ℝ 𝜈 Γ \mathbb{R}^{\nu}\setminus\Gamma is the classically forbidden region. The presence of the tunneling is the reason why we dub such systems as leaky quantum graphs . Schrödinger operators with interactions supported by curves and other manifolds of a lower dimension were studied already in the early nineties [ BT92 ] , and even earlier in examples with a particular symmetry [ AGS87 , Sha88 ] . A more systematic investigation motivated by the above considerations was undertaken in a last few years; it is the aim of this review to describe its results. One should stress, however, that such mathematical structures can be studied also from other points of view. A prominent example comes from studies of high contrast optical systems 3 3 3 Another situation where one arrives at a leaky-graph-type model arises when one deals with contact interactions of several one-dimensional particles [ Du08 ] . used to model photonic crystals — see, e.g. [ FKu96 , KK98 ] — which in a suitable approximation yield an analogue of the spectral problem for the operator ( 1.1 ); the two differ only by the physical interpretation, the roles of the coupling and spectral parameters being switched. A derivation of leaky-graph models in this context was given in the paper [ FKu98 ] , see also the review [ Ku01 ] and recall that the corresponding operators can be cast also in a pseudo-differential form [ FKu98 , PP04 ] . The material we are going to review is relatively extensive. We will take care, of course, to explain properly all the notions and the results. On the other hand, proofs will be mostly sketched. However, we will always give references to original papers where the particular complete argument can be found. Let us finally remark that the subject reviewed here cannot be regarded as closed, on the contrary, there are many open questions. We devote to them the closing section, and the author of this survey can only hope that his reader will take this problem list as a challenge and a program which will keep him or her busy for some time.

The aim of this review is to provide an overview of a recent work concerning ``leaky'' quantum graphs described by Hamiltonians given formally by the expression $-\Delta -\alpha \delta (x-\Gamma)$ with a singular attractive interaction supported by a graph-like set in $\mathbb{R}^\nu,\: \nu=2,3$. We will explain how such singular Schr\"odinger operators can be properly defined for different codimensions of $\Gamma$. Furthermore, we are going to discuss their properties, in particular, the way in which the geometry of $\Gamma$ influences their spectra and the scattering, strong-coupling asymptotic behavior, and a discrete counterpart to leaky-graph Hamiltonians using point interactions. The subject cannot be regarded as closed at present, and we will add a list of open problems hoping that the reader will take some of them as a challenge.

[{"content":"Help me to generate the abstract of a survey paper given the title:*Leaky Quantum Graphs: A Review* and the introduction:1. Introduction In this paper we are going to review results concerning a class of different quantum graph models. With this aim in mind, it would be natural to start by recalling briefly the standard quantum graphs, their description, properties, and numerous applications. In this volume, however, this would be clearly to bring owls to Athens 1 1 1 Or, depending on your taste, coal to Newcastle, firewood to the forest, etc. As usual, one can also refer to the Bard: to throw a perfume on the violet (The Life and Death of King John). and we refrain from doing that referring to the other articles in these proceedings, or to BCFK06 as another rich bibliography source. To motivate a need to look for alternative description of graph-like structures, let us observe that — despite its mathematical simplicity, beauty, and versatility — the standard quantum-graph model has also some drawbacks. In our opinion, the following two are the most important: • the presence of ad hoc parameters in the boundary conditions which describe how the wave functions are matched at the graphs vertices • the fact that particles are strictly confined to graph edges. While this is often a reasonable approximation when dealing, say, with electrons in semiconductor quantum wires, such a model neglects quantum tunneling which can play role once such wires are placed close to each other. One consequence is that in such a description, with the graph embedded in ℝ ν superscript ℝ 𝜈 \mathbb{R}^{\nu} , spectral properties reflect the topology while the graph geometry enters only through the edge lengths, their shapes being irrelevant A way to deal with the first problem is to regard a quantum graph as an idealization of a more realistic system without such ambiguities; a natural candidate for this role are various fat graphs. Limits of such objects when the edge width squeezes to zero were studied extensively, first in the easier Neumann-type case FW93 , KZ01 , RS01 , Sa01 , Ku02 , KZ03 , EP05 , Po06 , EP07 and quite recently also in the situation with Dirichlet boundaries Po05 , MV07 , CE07 , Gr07 . These results give a partial answer to the first question 2 2 2 An alternative approach is to keep the graph fixed and to approximate the vertex coupling through suitably scaled families of regular or singular interactions – see Ex96 , CE04 , ETu07 . while the second problem remains. Here we are going to discuss a class of quantum graph models which are free of both difficulties; the price we pay is — similarly as for the fat graphs mentioned above — that instead of ordinary differential equations we have to deal with a PDE problem. The idea is to preserve the whole Euclidean space as the configuration space and to suppose that the particle is kept in the vicinity of the graph Γ ⊂ ℝ ν Γ superscript ℝ 𝜈 \Gamma\subset\mathbb{R}^{\nu} by an attractive singular interaction. Formally such a Hamiltonian expresses as (1.1) − Δ − α ​ ( x ) ​ δ ​ ( x − Γ ) Δ 𝛼 𝑥 𝛿 𝑥 Γ -\Delta-\alpha(x)\delta(x-\Gamma)\, with α ​ ( x ) > 0 𝛼 𝑥 0 \alpha(x)>0\, ; we will consider mostly the situation where the attraction is position independent, α ​ ( x ) = α > 0 𝛼 𝑥 𝛼 0 \alpha(x)=\alpha>0 . Before proceeding to a definition of such singular Schrödinger operators and discussion of their properties, let us make a few remarks. First of all, it is clear that there is no ambiguity related to the graph vertices once Γ Γ \Gamma and α 𝛼 \alpha are given. It is equally obvious that the confinement in this model takes place at negative energies only. The particle now lives in the whole space and can be found even at large distances from Γ Γ \Gamma , although with a small probability, because the complement ℝ ν ∖ Γ superscript ℝ 𝜈 Γ \mathbb{R}^{\nu}\setminus\Gamma is the classically forbidden region. The presence of the tunneling is the reason why we dub such systems as leaky quantum graphs . Schrödinger operators with interactions supported by curves and other manifolds of a lower dimension were studied already in the early nineties BT92 , and even earlier in examples with a particular symmetry AGS87 , Sha88 . A more systematic investigation motivated by the above considerations was undertaken in a last few years; it is the aim of this review to describe its results. One should stress, however, that such mathematical structures can be studied also from other points of view. A prominent example comes from studies of high contrast optical systems 3 3 3 Another situation where one arrives at a leaky-graph-type model arises when one deals with contact interactions of several one-dimensional particles Du08 . used to model photonic crystals — see, e.g. FKu96 , KK98 — which in a suitable approximation yield an analogue of the spectral problem for the operator ( 1.1 ); the two differ only by the physical interpretation, the roles of the coupling and spectral parameters being switched. A derivation of leaky-graph models in this context was given in the paper FKu98 , see also the review Ku01 and recall that the corresponding operators can be cast also in a pseudo-differential form FKu98 , PP04 . The material we are going to review is relatively extensive. We will take care, of course, to explain properly all the notions and the results. On the other hand, proofs will be mostly sketched. However, we will always give references to original papers where the particular complete argument can be found. Let us finally remark that the subject reviewed here cannot be regarded as closed, on the contrary, there are many open questions. We devote to them the closing section, and the author of this survey can only hope that his reader will take this problem list as a challenge and a program which will keep him or her busy for some time.","role":"user"},{"content":"Abstract: The aim of this review is to provide an overview of a recent work concerning ``leaky'' quantum graphs described by Hamiltonians given formally by the expression $-\Delta -\alpha \delta (x-\Gamma)$ with a singular attractive interaction supported by a graph-like set in $\mathbb{R}^\nu,\: \nu=2,3$. We will explain how such singular Schr\"odinger operators can be properly defined for different codimensions of $\Gamma$. Furthermore, we are going to discuss their properties, in particular, the way in which the geometry of $\Gamma$ influences their spectra and the scattering, strong-coupling asymptotic behavior, and a discrete counterpart to leaky-graph Hamiltonians using point interactions. The subject cannot be regarded as closed at present, and we will add a list of open problems hoping that the reader will take some of them as a challenge.","role":"assistant"}]

An overview of abelian varieties in homotopy theory

The goal of this paper is to provide an overview of joint work with Behrens on topological automorphic forms [ 8 ] . The ultimate hope is to introduce a somewhat broad audience of topologists to this subject matter connecting modern homotopy theory, algebraic geometry, and number theory. Through an investigation of properties of Chern classes, Quillen discovered a connection between stable homotopy theory and 1 1 1 -dimensional formal group laws [ 41 ] . After almost 40 years, the impacts of this connection are still being felt. The stratification of formal group laws in finite characteristic gives rise to the chromatic filtration in stable homotopy theory [ 42 ] , and has definite calculational consequences. The nilpotence and periodicity phenomena in stable homotopy groups of spheres arise from a deep investigation of this connection [ 13 ] . Formal group laws have at least one other major manifestation: the study of abelian varieties. The examination of this connection led to elliptic cohomology theories and topological modular forms, or tmf tmf {\rm tmf} [ 25 ] . One of the main results in this theory is the construction of a spectrum tmf tmf {\rm tmf} , a structured ring object in the stable homotopy category. The homotopy groups of tmf tmf {\rm tmf} are, up to finite kernel and cokernel, the ring of integral modular forms [ 10 ] via a natural comparison map. The spectrum tmf tmf {\rm tmf} is often viewed as a “universal” elliptic cohomology theory corresponding to the moduli of elliptic curves. Unfortunately, the major involved parties have not yet published a full exposition of this theory. The near-future reader is urged to consult [ 5 ] , as well as seek out some of the unpublished literature and reading lists on topological modular forms if more background study is desired. Algebraic topology is explicitly tied to 1 1 1 -dimensional formal group laws, and so the formal group laws of higher-dimensional abelian varieties (and larger possible “height” invariants of those) are initially not connected to topology. The goal of [ 8 ] was to create generalizations of the theory of topological modular forms, through certain moduli of abelian varieties with extra data specifying 1 1 1 -dimensional summands of their formal group laws. The author doubts that it is possible to cover all of this background to any degree of detail within the confines of a paper of reasonable size, even restricting to those subjects that are of interest from a topological point of view. In addition, there are existing (and better) sources for this material. Therefore, our presentation of this material is informal, and we will try to list references for those who find some subject of interest to them. We assume a basic understanding of stable homotopy theory, and an inevitable aspect of the theory is that we require more and more of the language of algebraic geometry as we proceed. A rough outline of the topics covered follows. In sections 2 and 3 we begin with some background on the connection between the theory of complex bordism and formal group laws. We next discuss in section 4 the basic theories of Hopf algebroids and stacks, and the relation between stack cohomology and the Adams-Novikov spectral sequence in section 5 . We then discuss the problem of realizing formal group law data by spectra, such as is achieved by the Landweber exact functor theorem and the Goerss-Hopkins-Miller theorem, in section 6 . Examples of multiplicative group laws are discussed in section 7 , and the theories of elliptic cohomology and topological modular forms in sections 8 and 9 . We then discuss the possibility of moving forward from these known examples in section 10 , by discussing some of the geometry of the moduli of formal groups and height invariants. The generalization of the Goerss-Hopkins-Miller theorem due to Lurie, without which the subject of topological automorphic forms would be pure speculation, is introduced in section 11 . We view it as our point of entry: given this theorem, what kinds of new structures in homotopy theory can we produce? The answer, in the form of various moduli of higher-dimensional abelian varieties, appears in section 12 . Though the definitions of these moduli are lifted almost directly from the study of automorphic forms, we attempt in sections 13 , 14 , and 15 to indicate why this data is natural to require in order produce moduli satisfying the hypotheses of Lurie’s theorem. In section 16 , we try to indicate why some initial choices are made the way they are. One of the applications in mind has been the construction of finite resolutions of the K ​ ( n ) 𝐾 𝑛 K(n) -local sphere. Henn has given finite length algebraic resolutions allowing computation of the cohomology of the Morava stabilizer group in terms of the cohomology of finite subgroups [ 22 ] . Goerss-Henn-Mahowald-Rezk [ 17 ] and Behrens [ 6 ] gave analogous constructions of the K ​ ( 2 ) 𝐾 2 K(2) -local sphere at the prime 3 3 3 out of a finite number of spectra of the form E 2 h ​ G superscript subscript 𝐸 2 ℎ 𝐺 E_{2}^{hG} , where E 2 subscript 𝐸 2 E_{2} is a Lubin-Tate spectrum and G 𝐺 G is a finite subgroup of the Morava stabilizer group. The hope is that these constructions will generalize to other primes and higher height by considering diagrams of abelian varieties and isogenies. None of the (correct) material in this paper is new.

This is an expository article on the theory of formal group laws in homotopy theory, with the goal of leading to the connection with higher-dimensional abelian varieties and automorphic forms. These are roughly based on a talk at the conference "New Topological Contexts for Galois Theory and Algebraic Geometry."

[{"content":"Help me to generate the abstract of a survey paper given the title:*An overview of abelian varieties in homotopy theory* and the introduction:The goal of this paper is to provide an overview of joint work with Behrens on topological automorphic forms 8 . The ultimate hope is to introduce a somewhat broad audience of topologists to this subject matter connecting modern homotopy theory, algebraic geometry, and number theory. Through an investigation of properties of Chern classes, Quillen discovered a connection between stable homotopy theory and 1 1 1 -dimensional formal group laws 41 . After almost 40 years, the impacts of this connection are still being felt. The stratification of formal group laws in finite characteristic gives rise to the chromatic filtration in stable homotopy theory 42 , and has definite calculational consequences. The nilpotence and periodicity phenomena in stable homotopy groups of spheres arise from a deep investigation of this connection 13 . Formal group laws have at least one other major manifestation: the study of abelian varieties. The examination of this connection led to elliptic cohomology theories and topological modular forms, or tmf tmf {\rm tmf} 25 . One of the main results in this theory is the construction of a spectrum tmf tmf {\rm tmf} , a structured ring object in the stable homotopy category. The homotopy groups of tmf tmf {\rm tmf} are, up to finite kernel and cokernel, the ring of integral modular forms 10 via a natural comparison map. The spectrum tmf tmf {\rm tmf} is often viewed as a universal elliptic cohomology theory corresponding to the moduli of elliptic curves. Unfortunately, the major involved parties have not yet published a full exposition of this theory. The near-future reader is urged to consult 5 , as well as seek out some of the unpublished literature and reading lists on topological modular forms if more background study is desired. Algebraic topology is explicitly tied to 1 1 1 -dimensional formal group laws, and so the formal group laws of higher-dimensional abelian varieties (and larger possible height invariants of those) are initially not connected to topology. The goal of 8 was to create generalizations of the theory of topological modular forms, through certain moduli of abelian varieties with extra data specifying 1 1 1 -dimensional summands of their formal group laws. The author doubts that it is possible to cover all of this background to any degree of detail within the confines of a paper of reasonable size, even restricting to those subjects that are of interest from a topological point of view. In addition, there are existing (and better) sources for this material. Therefore, our presentation of this material is informal, and we will try to list references for those who find some subject of interest to them. We assume a basic understanding of stable homotopy theory, and an inevitable aspect of the theory is that we require more and more of the language of algebraic geometry as we proceed. A rough outline of the topics covered follows. In sections 2 and 3 we begin with some background on the connection between the theory of complex bordism and formal group laws. We next discuss in section 4 the basic theories of Hopf algebroids and stacks, and the relation between stack cohomology and the Adams-Novikov spectral sequence in section 5 . We then discuss the problem of realizing formal group law data by spectra, such as is achieved by the Landweber exact functor theorem and the Goerss-Hopkins-Miller theorem, in section 6 . Examples of multiplicative group laws are discussed in section 7 , and the theories of elliptic cohomology and topological modular forms in sections 8 and 9 . We then discuss the possibility of moving forward from these known examples in section 10 , by discussing some of the geometry of the moduli of formal groups and height invariants. The generalization of the Goerss-Hopkins-Miller theorem due to Lurie, without which the subject of topological automorphic forms would be pure speculation, is introduced in section 11 . We view it as our point of entry: given this theorem, what kinds of new structures in homotopy theory can we produce? The answer, in the form of various moduli of higher-dimensional abelian varieties, appears in section 12 . Though the definitions of these moduli are lifted almost directly from the study of automorphic forms, we attempt in sections 13 , 14 , and 15 to indicate why this data is natural to require in order produce moduli satisfying the hypotheses of Luries theorem. In section 16 , we try to indicate why some initial choices are made the way they are. One of the applications in mind has been the construction of finite resolutions of the K ​ ( n ) 𝐾 𝑛 K(n) -local sphere. Henn has given finite length algebraic resolutions allowing computation of the cohomology of the Morava stabilizer group in terms of the cohomology of finite subgroups 22 . Goerss-Henn-Mahowald-Rezk 17 and Behrens 6 gave analogous constructions of the K ​ ( 2 ) 𝐾 2 K(2) -local sphere at the prime 3 3 3 out of a finite number of spectra of the form E 2 h ​ G superscript subscript 𝐸 2 ℎ 𝐺 E_{2}^{hG} , where E 2 subscript 𝐸 2 E_{2} is a Lubin-Tate spectrum and G 𝐺 G is a finite subgroup of the Morava stabilizer group. The hope is that these constructions will generalize to other primes and higher height by considering diagrams of abelian varieties and isogenies. None of the (correct) material in this paper is new.","role":"user"},{"content":"Abstract: This is an expository article on the theory of formal group laws in homotopy theory, with the goal of leading to the connection with higher-dimensional abelian varieties and automorphic forms. These are roughly based on a talk at the conference "New Topological Contexts for Galois Theory and Algebraic Geometry."","role":"assistant"}]

Stability, Singularities and Mass Thresholds in Child Universe Production — a concise survey including some recent results and prospects —

The questions about the origin, formation and development of our universe are among the most challenging in physics, since they involve difficult theoretical problems and, at the same time, are not subject to a direct experimental approach. Fortunately, in the last years, theoretical investigations have found valuable allies in the increasing amount of data, which is coming from continuously refined observational techniques. This has given the possibility to aid our theoretical understanding not only with a much more clear picture of the universe, especially at the earliest stage of its evolution, but also with the results of computational simulations, which have a consistent realization of these data as a target of more and more refined models. From the theoretical point of view, after the development of the inflationary scenario [ 38 , 54 , 3 , 55 , 56 , 37 , 53 , 67 ] , a primary goal has been a successful consistent description of the earliest instants of life of our universe; this problem became tightly bound with developing ideas of vacuum and vacuum decay [ 26 , 27 , 24 ] about 30 years ago, when the interplay of these processes with gravitation was studied [ 28 , 51 ] . These studies go under the name of vacuum bubble dynamics : in the case of false vacuum their coupling with gravity was firstly analyzed by Sato et al. [ 64 , 47 , 62 , 57 , 63 , 48 ] ; interesting developments soon followed with the work of Blau et al. [ 20 ] and Berezin et al. [ 18 , 19 ] . A very interesting aspect of false vacuum bubble dynamics is that it can give rise to the formation of a child universe 1 1 1 The terminology child universe is here favored over the one baby universe used elsewhere. . Child universe formation is the process in which a new universe (the child one) emerges from an existing one (which we will call the parent universe) in such a way that the structure of the parent universe is preserved. Since the definition of a universe implicitly assumes that it can expand to a sufficiently large size and live for a sufficiently long time (so that in it structures similar to the one that we observe in our universe can be formed) child universe formation (as defined above) may seem impossible. This is, instead, not the case in a general relativistic framework, which allows a rich enough structure in the causal structure of spacetime to satisfy both the above conditions. Diverse realizations of the child universe creation process are possible and we will provide a short review of some of them below: for concreteness, our attention will concentrate on the model originally developed in [ 20 ] as well as later refinements of it. In this context, the importance of [ 20 ] resides in the fact that, for the first time, geodesically complete coordinate systems were used to describe the universe formation process. This, apparently purely technical, point has instead direct advantages in the physical interpretation of the model, since it emphasizes the crucial role played by wormholes. As we will also repeat below, if we consider a small bubble of false vacuum which expands to a very large size (the baby universe) the energy density inside the bubble is higher than the energy density outside it; the mechanical force is then directed from the parent space toward the baby universe. On one side of the wormhole this is the direction of decreasing radius (so that in this case there can be no expansion). But if the solution can evolve on the other side of the wormhole, the opposite situation is realized, i.e. the mechanical force pushes in the direction of increasing radius. Thus, the false vacuum bubble can (and will) expand; from the point of view of the parent spacetime, this process is taking place on the other side of the wormhole throat. Thus the growth of the child universe takes place without affecting in any way the evolution of the parent universe, which witnesses at most the formation of a black hole 2 2 2 These ideas can be generalized to situations in which the black hole is created as a result of gravitational collapse (see, for instance, [ 14 , 29 ] and references therein; more references can also be found in the additional contribution of S. Ansoldi to the proceedings of this conference). . The child universe, in fact, grows creating its own space . After introducing in a very concise way some necessary background formalism, with the intent of fixing notations and conventions (section 2 ), in section 3 we present a (non exhaustive) review of some of the above ideas. In particular, in three separate subsections we revisit some features of child universe formation: in subsection 3.1 we analyze the occurrence of singularities and the problems with the tunnelling process that can be used for singularity avoidance in semiclassical, minisuperspace models; in subsection 3.2 we review the stabilization of the initial (i.e. pre -tunnelling) configuration, both from the classical and semiclassical standpoint; in subsection 3.3 we discuss the presence of a critical mass threshold for child universe formation as well as early proposals to reduce it or trade it with other properties of spacetime and/or its matter content. Section 4 follows, where we describe a recent realization of child universe production free of a mass threshold; in this model child universe production can, surprisingly enough, take place out of almost empty space , a fact that is quite suggestive, especially in the perspective of unsuppressed transplanckian child universe production, which has also been recently discussed. We conclude the paper in section 5 , with a synopsis and a concise discussion/remark, stressing again what is the main idea that we have revisited in the rest of the paper.

We present a review of selected topics concerning the creation and evolution of child universes, together with a concise account of some recent progress in the field.

[{"content":"Help me to generate the abstract of a survey paper given the title:*Stability, Singularities and Mass Thresholds in Child Universe Production — a concise survey including some recent results and prospects —* and the introduction:The questions about the origin, formation and development of our universe are among the most challenging in physics, since they involve difficult theoretical problems and, at the same time, are not subject to a direct experimental approach. Fortunately, in the last years, theoretical investigations have found valuable allies in the increasing amount of data, which is coming from continuously refined observational techniques. This has given the possibility to aid our theoretical understanding not only with a much more clear picture of the universe, especially at the earliest stage of its evolution, but also with the results of computational simulations, which have a consistent realization of these data as a target of more and more refined models. From the theoretical point of view, after the development of the inflationary scenario 38 , 54 , 3 , 55 , 56 , 37 , 53 , 67 , a primary goal has been a successful consistent description of the earliest instants of life of our universe; this problem became tightly bound with developing ideas of vacuum and vacuum decay 26 , 27 , 24 about 30 years ago, when the interplay of these processes with gravitation was studied 28 , 51 . These studies go under the name of vacuum bubble dynamics : in the case of false vacuum their coupling with gravity was firstly analyzed by Sato et al. 64 , 47 , 62 , 57 , 63 , 48 ; interesting developments soon followed with the work of Blau et al. 20 and Berezin et al. 18 , 19 . A very interesting aspect of false vacuum bubble dynamics is that it can give rise to the formation of a child universe 1 1 1 The terminology child universe is here favored over the one baby universe used elsewhere. . Child universe formation is the process in which a new universe (the child one) emerges from an existing one (which we will call the parent universe) in such a way that the structure of the parent universe is preserved. Since the definition of a universe implicitly assumes that it can expand to a sufficiently large size and live for a sufficiently long time (so that in it structures similar to the one that we observe in our universe can be formed) child universe formation (as defined above) may seem impossible. This is, instead, not the case in a general relativistic framework, which allows a rich enough structure in the causal structure of spacetime to satisfy both the above conditions. Diverse realizations of the child universe creation process are possible and we will provide a short review of some of them below: for concreteness, our attention will concentrate on the model originally developed in 20 as well as later refinements of it. In this context, the importance of 20 resides in the fact that, for the first time, geodesically complete coordinate systems were used to describe the universe formation process. This, apparently purely technical, point has instead direct advantages in the physical interpretation of the model, since it emphasizes the crucial role played by wormholes. As we will also repeat below, if we consider a small bubble of false vacuum which expands to a very large size (the baby universe) the energy density inside the bubble is higher than the energy density outside it; the mechanical force is then directed from the parent space toward the baby universe. On one side of the wormhole this is the direction of decreasing radius (so that in this case there can be no expansion). But if the solution can evolve on the other side of the wormhole, the opposite situation is realized, i.e. the mechanical force pushes in the direction of increasing radius. Thus, the false vacuum bubble can (and will) expand; from the point of view of the parent spacetime, this process is taking place on the other side of the wormhole throat. Thus the growth of the child universe takes place without affecting in any way the evolution of the parent universe, which witnesses at most the formation of a black hole 2 2 2 These ideas can be generalized to situations in which the black hole is created as a result of gravitational collapse (see, for instance, 14 , 29 and references therein; more references can also be found in the additional contribution of S. Ansoldi to the proceedings of this conference). . The child universe, in fact, grows creating its own space . After introducing in a very concise way some necessary background formalism, with the intent of fixing notations and conventions (section 2 ), in section 3 we present a (non exhaustive) review of some of the above ideas. In particular, in three separate subsections we revisit some features of child universe formation: in subsection 3.1 we analyze the occurrence of singularities and the problems with the tunnelling process that can be used for singularity avoidance in semiclassical, minisuperspace models; in subsection 3.2 we review the stabilization of the initial (i.e. pre -tunnelling) configuration, both from the classical and semiclassical standpoint; in subsection 3.3 we discuss the presence of a critical mass threshold for child universe formation as well as early proposals to reduce it or trade it with other properties of spacetime and/or its matter content. Section 4 follows, where we describe a recent realization of child universe production free of a mass threshold; in this model child universe production can, surprisingly enough, take place out of almost empty space , a fact that is quite suggestive, especially in the perspective of unsuppressed transplanckian child universe production, which has also been recently discussed. We conclude the paper in section 5 , with a synopsis and a concise discussion/remark, stressing again what is the main idea that we have revisited in the rest of the paper.","role":"user"},{"content":"Abstract: We present a review of selected topics concerning the creation and evolution of child universes, together with a concise account of some recent progress in the field.","role":"assistant"}]

Scaling limits of two‐dimensional percolation: an overview

Percolation as a mathematical theory was introduced by Broadbent and Hammersley [ 11 , 12 ] to model the spread of a gas or a fluid through a porous medium. To mimic the randomness of the medium, they declared the edges of the d 𝑑 d -dimensional cubic lattice independently open (to the passage of the gas or fluid) with probability p 𝑝 p or closed with probability 1 − p 1 𝑝 1-p . Since then, many variants of this simple model have been studied, attracting the interest of both mathematicians and physicists. Mathematicians are interested in percolation because of its deceiving simplicity which hides difficult and elegant results. From the point of view of physicists, percolation is maybe the simplest statistical mechanical model undergoing a continuous phase transition as the value of the parameter p 𝑝 p is varied, with all the standard features typical of critical phenomena (scaling laws, conformal invariance, universality). On the applied side, percolation has been used to model the spread of a disease, a fire or a rumor, the displacement of oil by water, the behavior of random electrical circuits, and more recently the connectivity properties of communication networks. We will concentrate on a version of the model in which each vertex of the triangular lattice, identified by duality with the corresponding face of the hexagonal lattice (see Figure 1 ), is independently colored black with probability p 𝑝 p or white otherwise. Questions regarding the geometry of this random coloring (for instance, whether there exists a path on black sites connecting the opposite edges of a given rectangle) can be expressed in terms of the behavior of clusters (i.e., maximal connected monochromatic subsets of the lattice) or of the boundaries between them (we will sometimes call such boundaries or portions of them percolation interfaces ). It is well known (see, e.g., [ 25 , 23 , 9 ] ) that in this model if p > 1 / 2 𝑝 1 2 p>1/2 (respectively, p < 1 / 2 𝑝 1 2 p<1/2 ) there is an infinite black (resp., white) cluster, while for p = 1 / 2 𝑝 1 2 p=1/2 there is no infinite cluster of either color. The latter value is the critical threshold p c subscript 𝑝 𝑐 p_{c} of the model, at which the percolation phase transition occurs. As testified by [ 25 , 23 ] , we have had for some time a good understanding of the subcritical ( p < p c 𝑝 subscript 𝑝 𝑐 p<p_{c} ) and supercritical ( p > p c 𝑝 subscript 𝑝 𝑐 p>p_{c} ) phases. As for the critical behavior ( p 𝑝 p equal to or approaching p c subscript 𝑝 𝑐 p_{c} ), despite some important achievements (see, in particular, [ 26 ] and [ 25 , 23 ] as general references), a complete and rigorous understanding seemed out of reach for any two-dimensional percolation model until the introduction of the Stochastic Loewner Evolution (SLE) by Oded Schramm [ 37 ] and the proof of conformal invariance by Stanislav Smirnov [ 40 , 41 ] . The percolation phase transition is a purely geometric transition that physicists have successfully studied with the methods of continuous phase transitions, or critical phenomena. In the theory of critical phenomena it is usually assumed that a physical system near a continuous phase transition is characterized by a single length scale (the correlation length ) in terms of which all other lengths should be measured. When combined with the experimental observation that the correlation length diverges at the phase transition, this simple but strong assumption, known as the scaling hypothesis, leads to the belief that at criticality the system has no characteristic length, and is therefore invariant under scale transformations. This implies that all thermodynamic functions at criticality are homogeneous functions, and predicts the appearance of power laws. It also suggests that for models of critical systems realized on a lattice, one can attempt to take a continuum scaling limit in which the mesh of the lattice is sent to zero while focus is kept on “macroscopic” observables that capture the large scale behavior. In the limit, the discrete model should converge to a continuum one that encodes the large scale properties of the original model, containing at the same time more symmetry. In many cases, this allows to derive extra insight by combining methods of discrete mathematics with considerations inspired by the continuum limit picture. The simplest example of such a continuum random model is Brownian motion, which is the scaling limit of the simple random walk. In general, though, the complexity of the discrete model makes it impossible to even guess the nature of the scaling limit, unless some additional feature can be shown to hold, which can be used to pin down properties of the continuum limit. Two-dimensional critical systems belong to the class of models for which this can be done, and the additional feature is conformal invariance , as predicted by physicists since the early seventies [ 33 , 34 ] . The connection between the scaling limit of critical percolation interfaces (i.e., boundaries between clusters of different colors) and SLE has led to tremendous progress in recent years, not only providing a rigorous derivation of many of the results previously obtained by physicists, but also deepening our geometric understanding of critical percolation, and critical phenomena in general. The main power of SLE stems from the fact that it allows to compute different quantities; for example, percolation crossing probabilities and various percolation critical exponents. In general, relating the scaling limit of a critical lattice model to SLE allows for a rigorous determination of some aspects of the large scale behavior of the lattice model. It also provides deeper insight into geometric aspects that are not easily accessible with the methods developed by physicists to study critical phenomena. For mathematicians, the biggest advantage of SLE over those methods lies maybe in its mathematical rigor. However, many physicists working on critical phenomena have promptly recognized the importance of SLE and added it to their toolbox.

We present a review of the recent progress on percolation scaling limits in two dimensions. In particular, we consider the convergence of critical crossing probabilities to Cardy's formula and of the critical exploration path to chordal stochastic Loewner evolution (SLE6), the full scaling limit of critical cluster boundaries, and near‐critical scaling limits.

[{"content":"Help me to generate the abstract of a survey paper given the title:*Scaling limits of two‐dimensional percolation: an overview* and the introduction:Percolation as a mathematical theory was introduced by Broadbent and Hammersley 11 , 12 to model the spread of a gas or a fluid through a porous medium. To mimic the randomness of the medium, they declared the edges of the d 𝑑 d -dimensional cubic lattice independently open (to the passage of the gas or fluid) with probability p 𝑝 p or closed with probability 1 − p 1 𝑝 1-p . Since then, many variants of this simple model have been studied, attracting the interest of both mathematicians and physicists. Mathematicians are interested in percolation because of its deceiving simplicity which hides difficult and elegant results. From the point of view of physicists, percolation is maybe the simplest statistical mechanical model undergoing a continuous phase transition as the value of the parameter p 𝑝 p is varied, with all the standard features typical of critical phenomena (scaling laws, conformal invariance, universality). On the applied side, percolation has been used to model the spread of a disease, a fire or a rumor, the displacement of oil by water, the behavior of random electrical circuits, and more recently the connectivity properties of communication networks. We will concentrate on a version of the model in which each vertex of the triangular lattice, identified by duality with the corresponding face of the hexagonal lattice (see Figure 1 ), is independently colored black with probability p 𝑝 p or white otherwise. Questions regarding the geometry of this random coloring (for instance, whether there exists a path on black sites connecting the opposite edges of a given rectangle) can be expressed in terms of the behavior of clusters (i.e., maximal connected monochromatic subsets of the lattice) or of the boundaries between them (we will sometimes call such boundaries or portions of them percolation interfaces ). It is well known (see, e.g., 25 , 23 , 9 ) that in this model if p > 1 / 2 𝑝 1 2 p>1/2 (respectively, p < 1 / 2 𝑝 1 2 p<1/2 ) there is an infinite black (resp., white) cluster, while for p = 1 / 2 𝑝 1 2 p=1/2 there is no infinite cluster of either color. The latter value is the critical threshold p c subscript 𝑝 𝑐 p_{c} of the model, at which the percolation phase transition occurs. As testified by 25 , 23 , we have had for some time a good understanding of the subcritical ( p < p c 𝑝 subscript 𝑝 𝑐 p<p_{c} ) and supercritical ( p > p c 𝑝 subscript 𝑝 𝑐 p>p_{c} ) phases. As for the critical behavior ( p 𝑝 p equal to or approaching p c subscript 𝑝 𝑐 p_{c} ), despite some important achievements (see, in particular, 26 and 25 , 23 as general references), a complete and rigorous understanding seemed out of reach for any two-dimensional percolation model until the introduction of the Stochastic Loewner Evolution (SLE) by Oded Schramm 37 and the proof of conformal invariance by Stanislav Smirnov 40 , 41 . The percolation phase transition is a purely geometric transition that physicists have successfully studied with the methods of continuous phase transitions, or critical phenomena. In the theory of critical phenomena it is usually assumed that a physical system near a continuous phase transition is characterized by a single length scale (the correlation length ) in terms of which all other lengths should be measured. When combined with the experimental observation that the correlation length diverges at the phase transition, this simple but strong assumption, known as the scaling hypothesis, leads to the belief that at criticality the system has no characteristic length, and is therefore invariant under scale transformations. This implies that all thermodynamic functions at criticality are homogeneous functions, and predicts the appearance of power laws. It also suggests that for models of critical systems realized on a lattice, one can attempt to take a continuum scaling limit in which the mesh of the lattice is sent to zero while focus is kept on macroscopic observables that capture the large scale behavior. In the limit, the discrete model should converge to a continuum one that encodes the large scale properties of the original model, containing at the same time more symmetry. In many cases, this allows to derive extra insight by combining methods of discrete mathematics with considerations inspired by the continuum limit picture. The simplest example of such a continuum random model is Brownian motion, which is the scaling limit of the simple random walk. In general, though, the complexity of the discrete model makes it impossible to even guess the nature of the scaling limit, unless some additional feature can be shown to hold, which can be used to pin down properties of the continuum limit. Two-dimensional critical systems belong to the class of models for which this can be done, and the additional feature is conformal invariance , as predicted by physicists since the early seventies 33 , 34 . The connection between the scaling limit of critical percolation interfaces (i.e., boundaries between clusters of different colors) and SLE has led to tremendous progress in recent years, not only providing a rigorous derivation of many of the results previously obtained by physicists, but also deepening our geometric understanding of critical percolation, and critical phenomena in general. The main power of SLE stems from the fact that it allows to compute different quantities; for example, percolation crossing probabilities and various percolation critical exponents. In general, relating the scaling limit of a critical lattice model to SLE allows for a rigorous determination of some aspects of the large scale behavior of the lattice model. It also provides deeper insight into geometric aspects that are not easily accessible with the methods developed by physicists to study critical phenomena. For mathematicians, the biggest advantage of SLE over those methods lies maybe in its mathematical rigor. However, many physicists working on critical phenomena have promptly recognized the importance of SLE and added it to their toolbox.","role":"user"},{"content":"Abstract: We present a review of the recent progress on percolation scaling limits in two dimensions. In particular, we consider the convergence of critical crossing probabilities to Cardy's formula and of the critical exploration path to chordal stochastic Loewner evolution (SLE6), the full scaling limit of critical cluster boundaries, and near‐critical scaling limits.","role":"assistant"}]

Analyzing and modeling real-world phenomena with complex networks: a survey of applications

The many achievements of physics over the last few centuries have been based on reductionist approaches, whereby the system of interest is reduced to a small, isolated portion of the world, with full control of the parameters involved ( e.g. , temperature, pressure, electric field). An interesting instance of reductionism, which is seldom realized, is the modeling of non-linear phenomena with linear models by restricting the parameters and variables in terms of a linear approximation. In establishing the structure of matter with the quantum theory in the first few decades of the 20th century, for example, reductionism was key to reaching quantitative treatment of the properties of atoms, molecules and then sophisticated structures such as crystalline solids. Indeed, deciphering the structure of matter was decisive for many developments – not only in physics but also in chemistry, materials science and more recently in biology [ 1 ] . Nevertheless, with reductionist approaches only limited classes of real-world systems may be treated, for the complexity inherent in naturally-occurring phenomena cannot be embedded in the theoretical analysis. There is now a trend in science to extend the scientific method to become more integrationist and deal explicitly with non-linear approaches. The impressive evolution of the field of complex networks fits perfectly within such a scientific framework. Its origins can be traced back to Leonhard Euler’s solution of the Königsberg bridges problem ( e.g. , [ 2 ] ), after which the theory of graphs has been useful for theoretical physics, economy, sociology and biology. However, most of such studies focused on static graphs, i.e. graphs whose structure remained fixed. Important developments on dynamic networks were addressed by Erdős and Rényi [ 3 ] , among others, particularly for the so-called random networks, including the model now known as Erdős and Rényi — ER [ 4 , 5 , 6 ] . This type of network is characterized by the feature that in a network with N 𝑁 N initially isolated nodes, new connections are progressively established with uniform probability between any pair of nodes. Such networks are well described in terms of their average degree, implying they have a relatively simple structure. Despite the formalism and comprehensiveness of the theoretical results obtained by Erdős and collaborators, random networks ultimately proved not to be good models for natural structures and phenomena. Indeed, heterogeneous structuring, not the relative uniformity and simplicity of ER networks, is the rule in Nature. Therefore, it was mainly thanks to the efforts of sociologists along the last decades ( e.g. , [ 7 , 8 ] ) that graph theory started to be systematically applied to represent and model natural phenomena, more specifically social relations. These efforts were mainly related to the concept of the small-world phenomena in networks, which are characterized by small average shortest path lengths between pairs of nodes and relatively high clustering coefficients. Interestingly, the small-world property turned out to be ubiquitous. The next decisive development in graph applications took place quite recently, including Faloutsos et al. characterization of the Internet power law organization [ 9 ] and the identification of such a kind of connectivity in the WWW [ 10 ] , giving rise to the scale-free paradigm [ 11 ] . Subsequent investigations showed that many natural and human-made networks also exhibited scale-free organization, including protein-protein interaction networks [ 12 ] , domain interaction networks [ 13 ] , metabolic networks [ 14 ] , food webs [ 15 ] , networks of collaborators [ 16 ] , networks of airports [ 17 ] and roads [ 18 ] . The success of complex networks is therefore to a large extent a consequence of their natural suitability to represent virtually any discrete system. Moreover, the organization and evolution of such networks, as well as dynamical processes on them [ 19 , 20 ] , involve non-linear models and effects. The connectivity of networks is ultimately decisive in constraining and defining many aspects of systems dynamics. The key importance of this principle has been highlighted in many comprehensive surveys [ 21 , 22 , 23 , 19 , 24 ] . For instance, the behavior of biological neuronal networks, one of the greatest remaining scientific challenges, is largely defined by connectivity ( e.g. , [ 25 , 26 , 27 ] ). Because of its virtually unlimited generality for representing connectivity in the most diverse real systems in an integrative way, complex networks are promising for integration and unification of several aspects of modern science, including the inter-relationships between structure and dynamics [ 28 , 29 ] . Such a potential has been confirmed with a diversity of applications for complex networks, encompassing areas such as ecology, genetics, epidemiology, physics, the Internet and WWW, computing, etc. In fact, applications of complex networks are redefining the scientific method through incorporation of dynamic and multidisciplinary aspects of statistical physics and computer science. This survey is aimed at a comprehensive review of the myriad of applications of complex networks, discussing how they have been applied to real data to obtain useful insights. In order to ensure a coherent, integrated presentation of the related works, the survey has been organized according to main areas and subareas. The survey starts with a section describing the basic concepts related to complex networks theory, such as measurements.

The success of new scientific areas can be assessed by their potential in contributing to new theoretical approaches and in applications to real-world problems. Complex networks have fared extremely well in both of these aspects, with their sound theoretical basis being developed over the years and with a variety of applications. In this survey, we analyze the applications of complex networks to real-world problems and data, with emphasis in representation, analysis and modeling. A diversity of phenomena are surveyed, which may be classified into no less than 11 areas, providing a clear indication of the impact of the field of complex networks.

[{"content":"Help me to generate the abstract of a survey paper given the title:*Analyzing and modeling real-world phenomena with complex networks: a survey of applications* and the introduction:The many achievements of physics over the last few centuries have been based on reductionist approaches, whereby the system of interest is reduced to a small, isolated portion of the world, with full control of the parameters involved ( e.g. , temperature, pressure, electric field). An interesting instance of reductionism, which is seldom realized, is the modeling of non-linear phenomena with linear models by restricting the parameters and variables in terms of a linear approximation. In establishing the structure of matter with the quantum theory in the first few decades of the 20th century, for example, reductionism was key to reaching quantitative treatment of the properties of atoms, molecules and then sophisticated structures such as crystalline solids. Indeed, deciphering the structure of matter was decisive for many developments – not only in physics but also in chemistry, materials science and more recently in biology 1 . Nevertheless, with reductionist approaches only limited classes of real-world systems may be treated, for the complexity inherent in naturally-occurring phenomena cannot be embedded in the theoretical analysis. There is now a trend in science to extend the scientific method to become more integrationist and deal explicitly with non-linear approaches. The impressive evolution of the field of complex networks fits perfectly within such a scientific framework. Its origins can be traced back to Leonhard Eulers solution of the Königsberg bridges problem ( e.g. , 2 ), after which the theory of graphs has been useful for theoretical physics, economy, sociology and biology. However, most of such studies focused on static graphs, i.e. graphs whose structure remained fixed. Important developments on dynamic networks were addressed by Erdős and Rényi 3 , among others, particularly for the so-called random networks, including the model now known as Erdős and Rényi — ER 4 , 5 , 6 . This type of network is characterized by the feature that in a network with N 𝑁 N initially isolated nodes, new connections are progressively established with uniform probability between any pair of nodes. Such networks are well described in terms of their average degree, implying they have a relatively simple structure. Despite the formalism and comprehensiveness of the theoretical results obtained by Erdős and collaborators, random networks ultimately proved not to be good models for natural structures and phenomena. Indeed, heterogeneous structuring, not the relative uniformity and simplicity of ER networks, is the rule in Nature. Therefore, it was mainly thanks to the efforts of sociologists along the last decades ( e.g. , 7 , 8 ) that graph theory started to be systematically applied to represent and model natural phenomena, more specifically social relations. These efforts were mainly related to the concept of the small-world phenomena in networks, which are characterized by small average shortest path lengths between pairs of nodes and relatively high clustering coefficients. Interestingly, the small-world property turned out to be ubiquitous. The next decisive development in graph applications took place quite recently, including Faloutsos et al. characterization of the Internet power law organization 9 and the identification of such a kind of connectivity in the WWW 10 , giving rise to the scale-free paradigm 11 . Subsequent investigations showed that many natural and human-made networks also exhibited scale-free organization, including protein-protein interaction networks 12 , domain interaction networks 13 , metabolic networks 14 , food webs 15 , networks of collaborators 16 , networks of airports 17 and roads 18 . The success of complex networks is therefore to a large extent a consequence of their natural suitability to represent virtually any discrete system. Moreover, the organization and evolution of such networks, as well as dynamical processes on them 19 , 20 , involve non-linear models and effects. The connectivity of networks is ultimately decisive in constraining and defining many aspects of systems dynamics. The key importance of this principle has been highlighted in many comprehensive surveys 21 , 22 , 23 , 19 , 24 . For instance, the behavior of biological neuronal networks, one of the greatest remaining scientific challenges, is largely defined by connectivity ( e.g. , 25 , 26 , 27 ). Because of its virtually unlimited generality for representing connectivity in the most diverse real systems in an integrative way, complex networks are promising for integration and unification of several aspects of modern science, including the inter-relationships between structure and dynamics 28 , 29 . Such a potential has been confirmed with a diversity of applications for complex networks, encompassing areas such as ecology, genetics, epidemiology, physics, the Internet and WWW, computing, etc. In fact, applications of complex networks are redefining the scientific method through incorporation of dynamic and multidisciplinary aspects of statistical physics and computer science. This survey is aimed at a comprehensive review of the myriad of applications of complex networks, discussing how they have been applied to real data to obtain useful insights. In order to ensure a coherent, integrated presentation of the related works, the survey has been organized according to main areas and subareas. The survey starts with a section describing the basic concepts related to complex networks theory, such as measurements.","role":"user"},{"content":"Abstract: The success of new scientific areas can be assessed by their potential in contributing to new theoretical approaches and in applications to real-world problems. Complex networks have fared extremely well in both of these aspects, with their sound theoretical basis being developed over the years and with a variety of applications. In this survey, we analyze the applications of complex networks to real-world problems and data, with emphasis in representation, analysis and modeling. A diversity of phenomena are surveyed, which may be classified into no less than 11 areas, providing a clear indication of the impact of the field of complex networks.","role":"assistant"}]

An Experimental Review of Charmonium

I INTRODUCTION After more than 30 years, the charmonium system continues to provide an important laboratory for the study of the strong force. This brief review reflects the wide scope of strong force physics that is accessible in the charmonium system. The physics impact of the results presented here ranges from tests of lattice QCD calculations to our understanding of bound gluonic states (glueballs) to an evaluation of models of the spin-spin component of the q ​ q ¯ 𝑞 ¯ 𝑞 q\overline{q} potential to our ability to calculate strong radiative and relativistic corrections. In small increments, the charmonium system has, over time, given us a variety of opportunities to add to our global understanding of the strong force 1 1 1 For a comprehensive review of the range of theoretical ideas experimentally accessible in the charmonium system, refer to QWGYellow or BESPhysics ; also refer to the references cited in the experimental papers described below. . The charmonium system can be accessed experimentally in a number of complementary ways. This review covers recent results obtained by three different methods. (1) The CLEO-c experiment at Cornell University collects data at an e + ​ e − superscript 𝑒 superscript 𝑒 e^{+}e^{-} collider (CESR) that has a center of mass energy in the charmonium region. CLEO-c has recently collected a total and final sample of 27 million ψ ​ ( 2 ​ S ) 𝜓 2 𝑆 \psi(2S) decays. Many other charmonium states can be reached and studied through transitions of the ψ ​ ( 2 ​ S ) 𝜓 2 𝑆 \psi(2S) . (2,3) The Belle (at KEKB) and BaBar (at SLAC) experiments run at e + ​ e − superscript 𝑒 superscript 𝑒 e^{+}e^{-} colliders with center of mass energies in the bottomonium region. At these higher energies, among other possibilities, one can (2) study the collisions of two photons radiated from the e + ​ e − superscript 𝑒 superscript 𝑒 e^{+}e^{-} , where the center of mass energy of the two photons is in the charmonium region; or one can (3) study charmonium states produced in B 𝐵 B decays. Results from BES-II are not covered in this review. It should be noted, however, that BES-III is on the horizon and will soon increase, substantially, the world sample of J / ψ ​ ( 1 ​ S ) 𝐽 𝜓 1 𝑆 J/\psi(1S) and ψ ​ ( 2 ​ S ) 𝜓 2 𝑆 \psi(2S) decays.

This review briefly outlines recent experimental results from the charmonium system. These include new measurements of the M1 radiative transition rates $B(\psi(1S,2S)\to\gamma\eta_c(1S))$; new insights into the mass of the $\eta_c(1S)$; the observation of $J/\psi(1S)$ decays to three photons; new measurements of the two-photon widths of the $\chi_{cJ}(1P)$; the observation of radiative decays of the $\chi_{cj}(1P)$ to light quarks; a precise measurement of the $h_c(1P)$ mass; and an update on properties of the $\eta_c(2S)$. Each result adds piecemeal to our understanding of the strong force.

[{"content":"Help me to generate the abstract of a survey paper given the title:*An Experimental Review of Charmonium* and the introduction:I INTRODUCTION After more than 30 years, the charmonium system continues to provide an important laboratory for the study of the strong force. This brief review reflects the wide scope of strong force physics that is accessible in the charmonium system. The physics impact of the results presented here ranges from tests of lattice QCD calculations to our understanding of bound gluonic states (glueballs) to an evaluation of models of the spin-spin component of the q ​ q ¯ 𝑞 ¯ 𝑞 q\overline{q} potential to our ability to calculate strong radiative and relativistic corrections. In small increments, the charmonium system has, over time, given us a variety of opportunities to add to our global understanding of the strong force 1 1 1 For a comprehensive review of the range of theoretical ideas experimentally accessible in the charmonium system, refer to QWGYellow or BESPhysics ; also refer to the references cited in the experimental papers described below. . The charmonium system can be accessed experimentally in a number of complementary ways. This review covers recent results obtained by three different methods. (1) The CLEO-c experiment at Cornell University collects data at an e + ​ e − superscript 𝑒 superscript 𝑒 e^{+}e^{-} collider (CESR) that has a center of mass energy in the charmonium region. CLEO-c has recently collected a total and final sample of 27 million ψ ​ ( 2 ​ S ) 𝜓 2 𝑆 \psi(2S) decays. Many other charmonium states can be reached and studied through transitions of the ψ ​ ( 2 ​ S ) 𝜓 2 𝑆 \psi(2S) . (2,3) The Belle (at KEKB) and BaBar (at SLAC) experiments run at e + ​ e − superscript 𝑒 superscript 𝑒 e^{+}e^{-} colliders with center of mass energies in the bottomonium region. At these higher energies, among other possibilities, one can (2) study the collisions of two photons radiated from the e + ​ e − superscript 𝑒 superscript 𝑒 e^{+}e^{-} , where the center of mass energy of the two photons is in the charmonium region; or one can (3) study charmonium states produced in B 𝐵 B decays. Results from BES-II are not covered in this review. It should be noted, however, that BES-III is on the horizon and will soon increase, substantially, the world sample of J / ψ ​ ( 1 ​ S ) 𝐽 𝜓 1 𝑆 J/\psi(1S) and ψ ​ ( 2 ​ S ) 𝜓 2 𝑆 \psi(2S) decays.","role":"user"},{"content":"Abstract: This review briefly outlines recent experimental results from the charmonium system. These include new measurements of the M1 radiative transition rates $B(\psi(1S,2S)\to\gamma\eta_c(1S))$; new insights into the mass of the $\eta_c(1S)$; the observation of $J/\psi(1S)$ decays to three photons; new measurements of the two-photon widths of the $\chi_{cJ}(1P)$; the observation of radiative decays of the $\chi_{cj}(1P)$ to light quarks; a precise measurement of the $h_c(1P)$ mass; and an update on properties of the $\eta_c(2S)$. Each result adds piecemeal to our understanding of the strong force.","role":"assistant"}]

An Overview of Hopf Algebras of Trees and Their Actions on Functions

There is growing interest in Hopf algebras defined from trees, permutations, and other combinatorial structures. In this paper, we provide an expository account of some of the Hopf algebras that can be defined using trees, labeled trees, ordered trees and heap ordered trees. We also describe some actions of these Hopf algebras on algebra of functions. We assume that the reader is familiar with the basic definitions and properties of Hopf algebras, as covered, for example, in [ 14 ] . This paper is based in part on [ 9 ] .

We provide an expository account of some of the Hopf algebras that can be defined using trees, labeled trees, ordered trees and heap ordered trees. We also describe some actions of these Hopf algebras on algebra of functions.

[{"content":"Help me to generate the abstract of a survey paper given the title:*An Overview of Hopf Algebras of Trees and Their Actions on Functions* and the introduction:There is growing interest in Hopf algebras defined from trees, permutations, and other combinatorial structures. In this paper, we provide an expository account of some of the Hopf algebras that can be defined using trees, labeled trees, ordered trees and heap ordered trees. We also describe some actions of these Hopf algebras on algebra of functions. We assume that the reader is familiar with the basic definitions and properties of Hopf algebras, as covered, for example, in 14 . This paper is based in part on 9 .","role":"user"},{"content":"Abstract: We provide an expository account of some of the Hopf algebras that can be defined using trees, labeled trees, ordered trees and heap ordered trees. We also describe some actions of these Hopf algebras on algebra of functions.","role":"assistant"}]

Loops in the Hamiltonian group: a survey

1. Introduction Let ( M , ω ) 𝑀 𝜔 (M,{\omega}) be a symplectic manifold that is closed, i.e. compact and without boundary. We denote its group of diffeomorphisms by Diff := Diff ​ M assign Diff Diff 𝑀 {\rm Diff}:={\rm Diff}\,M and by Symp := Symp ​ ( M , ω ) assign Symp Symp 𝑀 𝜔 {\rm Symp}:={\rm Symp}(M,{\omega}) the subgroup of symplectomorphisms, i.e. diffeomorphisms that preserve the symplectic form. Its identity component Symp 0 := Symp 0 ​ ( M , ω ) assign subscript Symp 0 subscript Symp 0 𝑀 𝜔 {\rm Symp}_{0}:={\rm Symp}_{0}(M,{\omega}) has an important normal subgroup Ham := Ham ​ ( M , ω ) assign Ham Ham 𝑀 𝜔 {\rm Ham}:={\rm Ham}(M,{\omega}) consisting of all symplectomorphisms ϕ ∈ Symp 0 italic-ϕ subscript Symp 0 \phi\in{\rm Symp}_{0} with zero flux, or equivalently, of all time 1 1 1 maps ϕ 1 H superscript subscript italic-ϕ 1 𝐻 \phi_{1}^{H} of Hamiltonian flows ϕ t H superscript subscript italic-ϕ 𝑡 𝐻 \phi_{t}^{H} , where H : M × S 1 → ℝ : 𝐻 → 𝑀 superscript 𝑆 1 ℝ H:M\times S^{1}\to{\mathbb{R}} is a (smooth) time dependent function. Basic information about these groups may be found in [ 40 , 41 , 54 ] and the survey articles [ 35 , 36 ] . Note that Ham = Symp 0 Ham subscript Symp 0 {\rm Ham}={\rm Symp}_{0} when H 1 ​ ( M ; ℝ ) = 0 superscript 𝐻 1 𝑀 ℝ 0 H^{1}(M;{\mathbb{R}})=0 . This survey is mostly concerned with questions about based Hamiltonian loops, i.e. smooth paths { ϕ t } 0 ≤ t ≤ 1 subscript subscript italic-ϕ 𝑡 0 𝑡 1 \{\phi_{t}\}_{0\leq t\leq 1} in the Hamiltonian group Ham Ham {\rm Ham} for which ϕ 1 = ϕ 0 = i ​ d subscript italic-ϕ 1 subscript italic-ϕ 0 𝑖 𝑑 \phi_{1}=\phi_{0}=id . Each such loop is the flow of some time dependent Hamiltonian H t , subscript 𝐻 𝑡 H_{t}, and may be reparametrized so that H 0 = H 1 subscript 𝐻 0 subscript 𝐻 1 H_{0}=H_{1} and the induced map H : M × S 1 → ℝ : 𝐻 → 𝑀 superscript 𝑆 1 ℝ H:M\times S^{1}\to{\mathbb{R}} is smooth. If this generating Hamiltonian is time independent then its flow ϕ t H , t ∈ S 1 subscript superscript italic-ϕ 𝐻 𝑡 𝑡 superscript 𝑆 1 \phi^{H}_{t},t\in S^{1} , is a subgroup of Ham Ham {\rm Ham} isomorphic to S 1 superscript 𝑆 1 S^{1} . (The function H : M → ℝ : 𝐻 → 𝑀 ℝ H:M\to{\mathbb{R}} is then called the moment map .) Although these loops are the easiest to understand, there are still many unsolved questions about them. Our first group of questions concerns the structure of π 1 ​ ( Symp ) subscript 𝜋 1 Symp \pi_{1}({\rm Symp}) and π 1 ​ ( Ham ) subscript 𝜋 1 Ham \pi_{1}({\rm Ham}) . Note that when dim M = 2 dimension 𝑀 2 \dim\,M=2 , Moser’s homotopy argument implies that the symplectomorphism group of M 𝑀 M is homotopy equivalent to its group of orientation preserving diffeomorphisms. Thus the homotopy type of the groups Symp Symp {\rm Symp} and Ham Ham {\rm Ham} are known. Ham Ham {\rm Ham} is contractible unless M = S 2 𝑀 superscript 𝑆 2 M=S^{2} , in which case both it and Symp 0 subscript Symp 0 {\rm Symp}_{0} are homotopic to SO ​ ( 3 ) SO 3 {\rm SO}(3) ; while in higher genus Symp 0 subscript Symp 0 {\rm Symp}_{0} is contractible except in the case of the torus, when it is homotopic to the torus ℝ 2 / ℤ 2 superscript ℝ 2 superscript ℤ 2 {\mathbb{R}}^{2}/{\mathbb{Z}}^{2} . The group π 0 ​ ( Symp ) subscript 𝜋 0 Symp \pi_{0}({\rm Symp}) is the well-known mapping class group. Thus the questions listed below are not interesting in this case. The case when dim M = 4 dimension 𝑀 4 \dim M=4 is also moderately well understood. In particular, every 4 4 4 -manifold with a Hamiltonian S 1 superscript 𝑆 1 S^{1} action is the blow up of a rational or ruled 4 4 4 -manifold. (This was first proved by Audin [ 4 ] and Ahara–Hattori [ 3 ] ; see also Karshon [ 19 ] .) Moreover the homotopy type of Ham Ham {\rm Ham} is understood when M = ℂ ​ P 2 𝑀 ℂ superscript 𝑃 2 M={\mathbb{C}}P^{2} or S 2 × S 2 superscript 𝑆 2 superscript 𝑆 2 S^{2}\times S^{2} or a one point blow up of such; see for example Gromov [ 15 ] , Abreu–McDuff [ 2 ] , Abreu–Granja–Kitchloo [ 1 ] and Lalonde–Pinsonnault [ 28 ] . For work on nonHamiltonian S 1 superscript 𝑆 1 S^{1} actions in dimensions 4 4 4 and above see Bouyakoub [ 6 ] , Duistermaat–Pelayo [ 7 ] and Pelayo [ 50 ] . There is information on smooth circle actions in 4 4 4 -dimensions, see Fintushel [ 11 ] and Baldridge [ 5 ] . But almost nothing is known about the diffeomorphism group of a manifold of dimension ≥ 4 absent 4 \geq 4 ; for example it is not known whether Diff ​ ℂ ​ P 2 Diff ℂ superscript 𝑃 2 {\rm Diff}\,{\mathbb{C}}P^{2} is homotopy equivalent to the projective unitary group P ​ U ​ ( 3 ) 𝑃 𝑈 3 PU(3) as is Symp ​ ( ℂ ​ P 2 ) Symp ℂ superscript 𝑃 2 {\rm Symp}({\mathbb{C}}P^{2}) . On the other hand it follows from [ 2 ] that Ham ​ ( S 2 × S 2 , ω ) Ham superscript 𝑆 2 superscript 𝑆 2 𝜔 {\rm Ham}(S^{2}\times S^{2},{\omega}) is not homotopy equivalent to Diff ​ ( S 2 × S 2 ) Diff superscript 𝑆 2 superscript 𝑆 2 {\rm Diff}(S^{2}\times S^{2}) for any symplectic form [ ω ] delimited-[] 𝜔 [{\omega}] . Question 1.1 . When does a circle subgroup γ 𝛾 {\gamma} of Symp Symp {\rm Symp} represent a nonzero element in π 1 ​ ( Symp ) subscript 𝜋 1 Symp \pi_{1}({\rm Symp}) , or even one of infinite order? (Entov–Polterovich [ 9 ] call circles of infinite order incompressible .) The first problem here is to decide when a loop is Hamiltonian, i.e. is in the kernel of the Flux homomorphism. Recall from [ 40 ] that Flux is defined on the universal cover Symp ~ 0 subscript ~ Symp 0 \widetilde{\rm Symp}_{0} of Symp 0 subscript Symp 0 {\rm Symp}_{0} by (1.1) Flux : Symp ~ 0 → H 1 ​ ( M ; ℝ ) , ϕ ~ ↦ ∫ 0 1 [ ω ​ ( ϕ ˙ t , ⋅ ) ] ​ 𝑑 t , : Flux formulae-sequence → subscript ~ Symp 0 superscript 𝐻 1 𝑀 ℝ maps-to ~ italic-ϕ superscript subscript 0 1 delimited-[] 𝜔 subscript ˙ italic-ϕ 𝑡 ⋅ differential-d 𝑡 {\rm Flux}:\widetilde{\rm Symp}_{0}\to H^{1}(M;{\mathbb{R}}),\quad{\widetilde{\phi}}\mapsto\int_{0}^{1}[{\omega}(\dot{\phi}_{t},\cdot)]\,dt, where ϕ ~ = ( ϕ 1 , { ϕ t } ) ∈ Symp ~ 0 ~ italic-ϕ subscript italic-ϕ 1 subscript italic-ϕ 𝑡 subscript ~ Symp 0 {\widetilde{\phi}}=(\phi_{1},\{\phi_{t}\})\in\widetilde{\rm Symp}_{0} . The (symplectic) Flux group Γ ω subscript Γ 𝜔 {\Gamma}_{\omega} is defined to be the image of π 1 ​ ( Symp ) subscript 𝜋 1 Symp \pi_{1}({\rm Symp}) under Flux, so that there is an induced homomorphism Flux : π 1 ​ ( Symp ) → H 1 ​ ( M ; ℝ ) / Γ ω : Flux → subscript 𝜋 1 Symp superscript 𝐻 1 𝑀 ℝ subscript Γ 𝜔 {\rm Flux}:\pi_{1}({\rm Symp})\to H^{1}(M;{\mathbb{R}})/{\Gamma}_{\omega} with kernel Ham Ham {\rm Ham} . Ono [ 48 ] recently proved that Γ ω subscript Γ 𝜔 {\Gamma}_{\omega} is discrete. Unfortunately there seem to be no good techniques for understanding when Flux ​ ( γ ) Flux 𝛾 {\rm Flux}({\gamma}) is trivial. Since Hamiltonian S 1 superscript 𝑆 1 S^{1} actions always have fixed points at the critical points of the moment map H : M → ℝ : 𝐻 → 𝑀 ℝ H:M\to{\mathbb{R}} , a first guess might be that every symplectic action with fixed points is Hamiltonian. However McDuff [ 31 ] shows that this is not the case except in dimension 4 4 4 . In fact, the following basic problem is still unsolved in dimensions > 4 absent 4 >4 . Question 1.2 . Suppose that S 1 superscript 𝑆 1 S^{1} acts symplectically on the closed symplectic manifold ( M , ω ) 𝑀 𝜔 (M,{\omega}) with a finite but nonzero number of fixed points. Is the action Hamiltonian? If the action is semifree (i.e. the stabilizer of a point is either the identity or the whole group) Tolman–Weitsman [ 60 ] show that the answer to Question 1.2 is affirmative by computing various equivariant cohomology classes. Some other information on this question has been obtained by Feldman [ 10 ] and Pelayo–Tolman [ 51 ] . One might hope to extend the Tolman–Weitsman result to semifree actions with more general conditions on the fixed point components, for example that they are simply connected; note that these cannot be arbitrary because of the example in [ 31 ] of a semifree but nonHamiltonian action on a 6 6 6 -manifold with fixed point sets that are 2 2 2 -tori. In the current discussion we will largely ignore this problem, for the most part considering only Hamiltonian loops and their images in π 1 ​ ( Ham ) subscript 𝜋 1 Ham \pi_{1}({\rm Ham}) . Question 1.3 . To what extent are π 1 ​ ( Ham ) subscript 𝜋 1 Ham \pi_{1}({\rm Ham}) and π 1 ​ ( Symp ) subscript 𝜋 1 Symp \pi_{1}({\rm Symp}) generated by symplectic S 1 superscript 𝑆 1 S^{1} actions? This question is a measure of our ignorance. S 1 superscript 𝑆 1 S^{1} actions do generate π 1 ​ ( Symp ) subscript 𝜋 1 Symp \pi_{1}({\rm Symp}) in very special cases such as ℂ ​ P 2 ℂ superscript 𝑃 2 {\mathbb{C}}P^{2} or ( S 2 × S 2 , p ​ r 1 ∗ ​ ( σ ) + p ​ r 2 ∗ ​ ( σ ) ) superscript 𝑆 2 superscript 𝑆 2 𝑝 superscript subscript 𝑟 1 𝜎 𝑝 superscript subscript 𝑟 2 𝜎 \bigl{(}S^{2}\times S^{2},pr_{1}^{*}({\sigma})+pr_{2}^{*}({\sigma})\bigr{)} (note that the factors have equal area). Indeed in these cases Symp Symp {\rm Symp} itself is known to have the homotopy type of a compact Lie group (see [ 35 ] ). However, as we see below, this does not hold in general. Question 1.4 . What can one say about the relation between π 1 ​ ( Ham ) subscript 𝜋 1 Ham \pi_{1}({\rm Ham}) , π 1 ​ ( Symp ) subscript 𝜋 1 Symp \pi_{1}({\rm Symp}) and π 1 ​ ( Diff ) subscript 𝜋 1 Diff \pi_{1}({\rm Diff}) ? For example, under what circumstances is the map π 1 ​ ( Symp ) → π 1 ​ ( Diff ) → subscript 𝜋 1 Symp subscript 𝜋 1 Diff \pi_{1}({\rm Symp})\to\pi_{1}({\rm Diff}) injective or surjective? The symplectic Flux group Γ ω subscript Γ 𝜔 {\Gamma}_{\omega} is the quotient of π 1 ​ ( Symp ) subscript 𝜋 1 Symp \pi_{1}({\rm Symp}) by π 1 ​ ( Ham ) subscript 𝜋 1 Ham \pi_{1}({\rm Ham}) and hence precisely measures their difference. By Kȩdra–Kotschick–Morita [ 22 ] , this group vanishes in many cases. Much of their paper in fact applies to the volume 1 1 1 The volume flux is defined by equation ( 1.1 ), but ω 𝜔 {\omega} should be understood as a volume form and the homomorphism takes values in H m − 1 ​ ( M ) superscript 𝐻 𝑚 1 𝑀 H^{m-1}(M) , where m := dim M assign 𝑚 dimension 𝑀 m:=\dim M . Accordingly, Γ vol subscript Γ vol {\Gamma}_{{\rm vol}} is the image of π 1 ​ ( Diff vol ) subscript 𝜋 1 subscript Diff vol \pi_{1}({\rm Diff}_{{\rm vol}}) under the flux. flux group Γ vol subscript Γ vol {\Gamma}_{{\rm vol}} , which is in principle of a more topological nature than Γ ω subscript Γ 𝜔 {\Gamma}_{\omega} ; it would be interesting to find conditions for the vanishing of Γ ω subscript Γ 𝜔 {\Gamma}_{\omega} that involve symplectic geometry at a deeper level. In this note, we begin by describing some classical methods for exploring the above questions, the first based on properties of the action functional 𝒜 H subscript 𝒜 𝐻 {\mathcal{A}}_{H} and the second using volume. These methods give rather good information in the following cases: ∙ ∙ \bullet Question 1.1 for toric manifolds (see Corollary 2.4 below) ∙ ∙ \bullet Questions 1.3 and 1.4 for pointwise blow ups M ~ ~ 𝑀 {\widetilde{M}} of arbitrary symplectic manifolds M 𝑀 M (see Proposition 2.7 and its corollaries). If M 𝑀 M is noncompact and one considers the group Ham c := Ham c ​ M assign superscript Ham 𝑐 superscript Ham 𝑐 𝑀 {\rm Ham}^{c}:={\rm Ham}^{c}\,M of compactly supported Hamiltonian symplectomorphisms of M 𝑀 M , then there is another classical homomorphism called the Calabi homomorphism : Cal : π 1 ​ ( Ham c ) → ℝ , γ ↦ 1 n ! ​ ∫ 0 1 ( ∫ M H t ​ ω n ) ​ 𝑑 t , : Cal formulae-sequence → subscript 𝜋 1 superscript Ham 𝑐 ℝ maps-to 𝛾 1 𝑛 superscript subscript 0 1 subscript 𝑀 subscript 𝐻 𝑡 superscript 𝜔 𝑛 differential-d 𝑡 {\rm Cal}:\pi_{1}({\rm Ham}^{c})\to{\mathbb{R}},\quad{\gamma}\mapsto\frac{1}{n!}\int_{0}^{1}\Bigl{(}\int_{M}H_{t}{\omega}^{n}\bigr{)}dt, where H t subscript 𝐻 𝑡 H_{t} is the generating Hamiltonian for γ 𝛾 {\gamma} , normalized to have compact support. We explain briefly in Lemma 3.9 why ∙ ∙ \bullet the Calabi homomorphism need not vanish. As we shall see, this question, though classical in origin, is very closely related to questions about the Seidel representation in quantum homology. One might wonder if it is possible to get better information about the above questions by using more modern (i.e. quantum) techniques. In fact, Question 1.1 first arose in McDuff–Slimowitz [ 42 ] , a paper that uses Floer theoretic techniques to study paths in Ham Ham {\rm Ham} that are geodesic with respect to the Hofer norm. 2 2 2 This is defined in § 3.2 below. An unexpected consequence of the ideas developed there is that semifree Hamiltonian circle actions do not contract in Ham Ham {\rm Ham} , though they might have finite order (for example, a rotation of S 2 superscript 𝑆 2 S^{2} .) The main tool that has proved useful in this context is an extension of the action homomorphism due to Seidel [ 58 ] , that is called the Seidel representation ; see § 3.1 . This homomorphism assigns to every γ ∈ π 1 ​ ( Ham ) 𝛾 subscript 𝜋 1 Ham {\gamma}\in\pi_{1}({\rm Ham}) a unit (i.e. invertible element) 𝒮 ​ ( γ ) 𝒮 𝛾 {\mathcal{S}}({\gamma}) in the (small) quantum homology Q ​ H ∗ ​ ( M ) 𝑄 subscript 𝐻 𝑀 QH_{*}(M) . Corollary 3.2 gives some more results on the above questions obtained using 𝒮 𝒮 {\mathcal{S}} . Because 𝒮 𝒮 {\mathcal{S}} is usually very hard to calculate, the classical methods often work better in specific examples. Nevertheless, 𝒮 𝒮 {\mathcal{S}} is a key tool in other contexts. One very interesting question is the following. Note that in two dimensions, the only symplectic manifold with a Hamiltonian S 1 superscript 𝑆 1 S^{1} action is S 2 superscript 𝑆 2 S^{2} , while T 2 superscript 𝑇 2 T^{2} has nonHamiltonian actions and higher genus surfaces Σ g subscript Σ 𝑔 {\Sigma}_{g} have none. Question 1.5 . Is there a meaningful extension of the classification of Riemann surfaces into spheres, tori and higher genus to higher dimensional symplectic manifolds? If so, is any aspect of it reflected in the properties of π 1 ​ ( Ham ) subscript 𝜋 1 Ham \pi_{1}({\rm Ham}) ? This would be an analog of minimal model theory in algebraic geometry. A first step, accomplished by Ruan and his coworkers Hu and T.-J. Li [ 56 , 18 ] , is to understand what it means for two symplectic manifolds to be birationally equivalent. Their results imply that a reasonable class of manifolds to take as the analog of spheres are the symplectically uniruled manifolds. These are the manifolds for which there is a nonzero genus zero Gromov–Witten invariant ⟨ a 1 , a 2 , … , a k ⟩ β M {\bigl{\langle}}a_{1},a_{2},\dots,a_{k}{\bigl{\rangle}}^{M}_{\beta} (for some k ≥ 1 , 𝑘 1 k\geq 1, a i ∈ H ∗ ​ ( M ) subscript 𝑎 𝑖 subscript 𝐻 𝑀 a_{i}\in H_{*}(M) and β ∈ H 2 ​ ( M ) 𝛽 subscript 𝐻 2 𝑀 {\beta}\in H_{2}(M) ) with one of the constraints a i subscript 𝑎 𝑖 a_{i} equal to a point. This class includes all projective manifolds that are uniruled in the sense of algebraic geometry. In this case ω ​ ( β ) ≠ 0 𝜔 𝛽 0 {\omega}({\beta})\neq 0 and c 1 ​ ( β ) ≠ 0 subscript 𝑐 1 𝛽 0 c_{1}({\beta})\neq 0 . At the other extreme are the symplectically aspherical manifolds for which the restriction ω | π 2 ​ ( M ) evaluated-at 𝜔 subscript 𝜋 2 𝑀 {\omega}|_{\pi_{2}(M)} of [ ω ] delimited-[] 𝜔 [{\omega}] to π 2 ​ ( M ) subscript 𝜋 2 𝑀 \pi_{2}(M) vanishes, and possibly also (depending on the author) the restriction c 1 | π 2 ​ ( M ) evaluated-at subscript 𝑐 1 subscript 𝜋 2 𝑀 c_{1}|_{\pi_{2}(M)} of the first Chern class c 1 subscript 𝑐 1 c_{1} of ( M , ω ) 𝑀 𝜔 (M,{\omega}) . These manifolds have no J 𝐽 J -holomorphic curves at all, and hence all nontrivial (i.e. β ≠ 0 𝛽 0 {\beta}\neq 0 ) Gromov–Witten invariants vanish. To a first approximation, one can characterize symplectically uniruled manifolds in terms of their quantum homology Q ​ H ∗ ​ ( M ) 𝑄 subscript 𝐻 𝑀 QH_{*}(M) in the following way. If ( M , ω ) 𝑀 𝜔 (M,{\omega}) is not uniruled then all invertible elements in Q ​ H 2 ​ n ​ ( M ) 𝑄 subscript 𝐻 2 𝑛 𝑀 QH_{2n}(M) have the form 1 ​ l ⊗ λ + x tensor-product 1 l 𝜆 𝑥 {{\mathchoice{\rm 1\mskip-4.0mul}{\rm 1\mskip-4.0mul}{\rm 1\mskip-4.5mul}{\rm 1\mskip-5.0mul}}}\otimes{\lambda}+x where λ 𝜆 {\lambda} is invertible in the coefficient field Λ Λ {\Lambda} and x ∈ H < 2 ​ n ​ ( M ) ⊗ Λ 𝑥 tensor-product subscript 𝐻 absent 2 𝑛 𝑀 Λ x\in H_{<2n}(M)\otimes{\Lambda} . (This is nearly an iff statement and can be improved to such: cf. the appendix to [ 38 ] . Here 1 ​ l 1 l {{\mathchoice{\rm 1\mskip-4.0mul}{\rm 1\mskip-4.0mul}{\rm 1\mskip-4.5mul}{\rm 1\mskip-5.0mul}}} denotes the fundamental class [ M ] ∈ Q ​ H ∗ ​ ( M ) delimited-[] 𝑀 𝑄 subscript 𝐻 𝑀 [M]\in QH_{*}(M) ; it is the identity element in Q ​ H ∗ ​ ( M ) 𝑄 subscript 𝐻 𝑀 QH_{*}(M) . The notation is explained in more detail below.) Therefore another version of the previous question is: Question 1.6 . To what extent is the structure of the quantum homology ring Q ​ H ∗ ​ ( M ) 𝑄 subscript 𝐻 𝑀 QH_{*}(M) reflected in the algebraic/topological/geometric structure of Ham Ham {\rm Ham} ? For example, according to Polterovich [ 55 ] , if ω | π 2 ​ ( M ) = 0 evaluated-at 𝜔 subscript 𝜋 2 𝑀 0 {\omega}|_{\pi_{2}(M)}=0 , Ham Ham {\rm Ham} contains no elements of finite order except for i ​ d 𝑖 𝑑 id . Does this property continue to hold if the condition on ω 𝜔 {\omega} is weakened to the vanishing of all Gromov–Witten invariants? Since S 1 superscript 𝑆 1 S^{1} -manifolds certainly support nontrivial symplectomorphisms of finite order (such as a half turn), it is natural to ask whether these manifolds have nontrivial Gromov–Witten invariants. Since the Seidel element 𝒮 ​ ( γ ) 𝒮 𝛾 {\mathcal{S}}({\gamma}) is a unit in quantum homology one might also expect to see traces of the uniruled/non-uniruled dichotomy in its properties. This was the guiding idea in my recent proof [ 38 ] that every closed symplectic manifold that supports a Hamiltonian S 1 superscript 𝑆 1 S^{1} action is uniruled . The argument in [ 38 ] applies more generally to manifolds with Hamiltonian loops that are nondegenerate Hofer geodesics. This opens up many interesting questions of a more dynamical flavor. Here we shall discuss the following basic (but unrelated) problem, which is still open in many cases. Question 1.7 . Does Ham Ham {\rm Ham} have infinite diameter with respect to the Hofer metric? One expects the answer to be positive always. However the proofs for spheres (due to Polterovich [ 53 ] ) and other Riemann surfaces (due to Lalonde–McDuff [ 25 ] ) are very different. In fact, as noted by Ostrover [ 49 ] one can use the spectral invariants of Schwarz [ 57 ] and Oh [ 46 , 47 ] (see also Usher [ 61 ] ) to show that the universal cover Ham ~ ~ Ham {\widetilde{\rm Ham}} of Ham Ham {\rm Ham} always has infinite diameter with respect to the induced (pseudo)metric. Therefore the question becomes: when does this result transfer down to Ham Ham {\rm Ham} ? Schwarz shows in [ 57 ] that this happens when ω 𝜔 {\omega} and c 1 subscript 𝑐 1 c_{1} both vanish on π 2 ​ ( M ) subscript 𝜋 2 𝑀 \pi_{2}(M) . I recently extended his work in [ 39 ] , showing that the asymptotic spectral invariants descend to Ham Ham {\rm Ham} if, for example, all nontrivial genus zero Gromov–Witten invariants vanish and rank ​ H 2 ​ ( M ; ℝ ) > 1 rank subscript 𝐻 2 𝑀 ℝ 1 {\rm rank\,}H_{2}(M;{\mathbb{R}})>1 . As we explain in § 3.2 below, Schwarz’s argument hinges on the properties of the Seidel elements 𝒮 ​ ( γ ) 𝒮 𝛾 {\mathcal{S}}({\gamma}) of γ ∈ π 1 ​ ( Ham ) 𝛾 subscript 𝜋 1 Ham {\gamma}\in\pi_{1}({\rm Ham}) . Here we shall sketch a different extension of his result. In particular we show: Ham Ham {\rm Ham} has infinite diameter when M 𝑀 M is a “small” one point blow up of ℂ ​ P 2 ℂ superscript 𝑃 2 {\mathbb{C}}P^{2} . This manifold M 𝑀 M is of course uniruled (and the spectral invariants do not descend). To my knowledge, it is not yet known whether Ham Ham {\rm Ham} has infinite diameter for all one point blow ups of ℂ ​ P 2 ℂ superscript 𝑃 2 {\mathbb{C}}P^{2} , though it does for ℂ ​ P 2 ℂ superscript 𝑃 2 {\mathbb{C}}P^{2} itself (and indeed for any ℂ ​ P n ℂ superscript 𝑃 𝑛 {\mathbb{C}}P^{n} ) by the results of Entov–Polterovich [ 8 ] . For further results on this problem see McDuff [ 39 ] .

This note describes some recent results about the homotopy properties of Hamiltonian loops in various manifolds, including toric manifolds and one point blow ups. We describe conditions under which a circle action does not contract in the Hamiltonian group, and construct an example of a loop $\ga$ of diffeomorphisms of a symplectic manifold M with the property that none of the loops smoothly isotopic to $\ga$ preserve any symplectic form on M. We also discuss some new conditions under which the Hamiltonian group has infinite Hofer diameter. Some of the methods used are classical (Weinstein's action homomorphism and volume calculations), while others use quantum methods (the Seidel representation and spectral invariants).

[{"content":"Help me to generate the abstract of a survey paper given the title:*Loops in the Hamiltonian group: a survey* and the introduction:1. Introduction Let ( M , ω ) 𝑀 𝜔 (M,{\omega}) be a symplectic manifold that is closed, i.e. compact and without boundary. We denote its group of diffeomorphisms by Diff := Diff ​ M assign Diff Diff 𝑀 {\rm Diff}:={\rm Diff}\,M and by Symp := Symp ​ ( M , ω ) assign Symp Symp 𝑀 𝜔 {\rm Symp}:={\rm Symp}(M,{\omega}) the subgroup of symplectomorphisms, i.e. diffeomorphisms that preserve the symplectic form. Its identity component Symp 0 := Symp 0 ​ ( M , ω ) assign subscript Symp 0 subscript Symp 0 𝑀 𝜔 {\rm Symp}_{0}:={\rm Symp}_{0}(M,{\omega}) has an important normal subgroup Ham := Ham ​ ( M , ω ) assign Ham Ham 𝑀 𝜔 {\rm Ham}:={\rm Ham}(M,{\omega}) consisting of all symplectomorphisms ϕ ∈ Symp 0 italic-ϕ subscript Symp 0 \phi\in{\rm Symp}_{0} with zero flux, or equivalently, of all time 1 1 1 maps ϕ 1 H superscript subscript italic-ϕ 1 𝐻 \phi_{1}^{H} of Hamiltonian flows ϕ t H superscript subscript italic-ϕ 𝑡 𝐻 \phi_{t}^{H} , where H : M × S 1 → ℝ : 𝐻 → 𝑀 superscript 𝑆 1 ℝ H:M\times S^{1}\to{\mathbb{R}} is a (smooth) time dependent function. Basic information about these groups may be found in 40 , 41 , 54 and the survey articles 35 , 36 . Note that Ham = Symp 0 Ham subscript Symp 0 {\rm Ham}={\rm Symp}_{0} when H 1 ​ ( M ; ℝ ) = 0 superscript 𝐻 1 𝑀 ℝ 0 H^{1}(M;{\mathbb{R}})=0 . This survey is mostly concerned with questions about based Hamiltonian loops, i.e. smooth paths { ϕ t } 0 ≤ t ≤ 1 subscript subscript italic-ϕ 𝑡 0 𝑡 1 \{\phi_{t}\}_{0\leq t\leq 1} in the Hamiltonian group Ham Ham {\rm Ham} for which ϕ 1 = ϕ 0 = i ​ d subscript italic-ϕ 1 subscript italic-ϕ 0 𝑖 𝑑 \phi_{1}=\phi_{0}=id . Each such loop is the flow of some time dependent Hamiltonian H t , subscript 𝐻 𝑡 H_{t}, and may be reparametrized so that H 0 = H 1 subscript 𝐻 0 subscript 𝐻 1 H_{0}=H_{1} and the induced map H : M × S 1 → ℝ : 𝐻 → 𝑀 superscript 𝑆 1 ℝ H:M\times S^{1}\to{\mathbb{R}} is smooth. If this generating Hamiltonian is time independent then its flow ϕ t H , t ∈ S 1 subscript superscript italic-ϕ 𝐻 𝑡 𝑡 superscript 𝑆 1 \phi^{H}_{t},t\in S^{1} , is a subgroup of Ham Ham {\rm Ham} isomorphic to S 1 superscript 𝑆 1 S^{1} . (The function H : M → ℝ : 𝐻 → 𝑀 ℝ H:M\to{\mathbb{R}} is then called the moment map .) Although these loops are the easiest to understand, there are still many unsolved questions about them. Our first group of questions concerns the structure of π 1 ​ ( Symp ) subscript 𝜋 1 Symp \pi_{1}({\rm Symp}) and π 1 ​ ( Ham ) subscript 𝜋 1 Ham \pi_{1}({\rm Ham}) . Note that when dim M = 2 dimension 𝑀 2 \dim\,M=2 , Mosers homotopy argument implies that the symplectomorphism group of M 𝑀 M is homotopy equivalent to its group of orientation preserving diffeomorphisms. Thus the homotopy type of the groups Symp Symp {\rm Symp} and Ham Ham {\rm Ham} are known. Ham Ham {\rm Ham} is contractible unless M = S 2 𝑀 superscript 𝑆 2 M=S^{2} , in which case both it and Symp 0 subscript Symp 0 {\rm Symp}_{0} are homotopic to SO ​ ( 3 ) SO 3 {\rm SO}(3) ; while in higher genus Symp 0 subscript Symp 0 {\rm Symp}_{0} is contractible except in the case of the torus, when it is homotopic to the torus ℝ 2 / ℤ 2 superscript ℝ 2 superscript ℤ 2 {\mathbb{R}}^{2}/{\mathbb{Z}}^{2} . The group π 0 ​ ( Symp ) subscript 𝜋 0 Symp \pi_{0}({\rm Symp}) is the well-known mapping class group. Thus the questions listed below are not interesting in this case. The case when dim M = 4 dimension 𝑀 4 \dim M=4 is also moderately well understood. In particular, every 4 4 4 -manifold with a Hamiltonian S 1 superscript 𝑆 1 S^{1} action is the blow up of a rational or ruled 4 4 4 -manifold. (This was first proved by Audin 4 and Ahara–Hattori 3 ; see also Karshon 19 .) Moreover the homotopy type of Ham Ham {\rm Ham} is understood when M = ℂ ​ P 2 𝑀 ℂ superscript 𝑃 2 M={\mathbb{C}}P^{2} or S 2 × S 2 superscript 𝑆 2 superscript 𝑆 2 S^{2}\times S^{2} or a one point blow up of such; see for example Gromov 15 , Abreu–McDuff 2 , Abreu–Granja–Kitchloo 1 and Lalonde–Pinsonnault 28 . For work on nonHamiltonian S 1 superscript 𝑆 1 S^{1} actions in dimensions 4 4 4 and above see Bouyakoub 6 , Duistermaat–Pelayo 7 and Pelayo 50 . There is information on smooth circle actions in 4 4 4 -dimensions, see Fintushel 11 and Baldridge 5 . But almost nothing is known about the diffeomorphism group of a manifold of dimension ≥ 4 absent 4 \geq 4 ; for example it is not known whether Diff ​ ℂ ​ P 2 Diff ℂ superscript 𝑃 2 {\rm Diff}\,{\mathbb{C}}P^{2} is homotopy equivalent to the projective unitary group P ​ U ​ ( 3 ) 𝑃 𝑈 3 PU(3) as is Symp ​ ( ℂ ​ P 2 ) Symp ℂ superscript 𝑃 2 {\rm Symp}({\mathbb{C}}P^{2}) . On the other hand it follows from 2 that Ham ​ ( S 2 × S 2 , ω ) Ham superscript 𝑆 2 superscript 𝑆 2 𝜔 {\rm Ham}(S^{2}\times S^{2},{\omega}) is not homotopy equivalent to Diff ​ ( S 2 × S 2 ) Diff superscript 𝑆 2 superscript 𝑆 2 {\rm Diff}(S^{2}\times S^{2}) for any symplectic form ω delimited- 𝜔 {\omega} . Question 1.1 . When does a circle subgroup γ 𝛾 {\gamma} of Symp Symp {\rm Symp} represent a nonzero element in π 1 ​ ( Symp ) subscript 𝜋 1 Symp \pi_{1}({\rm Symp}) , or even one of infinite order? (Entov–Polterovich 9 call circles of infinite order incompressible .) The first problem here is to decide when a loop is Hamiltonian, i.e. is in the kernel of the Flux homomorphism. Recall from 40 that Flux is defined on the universal cover Symp ~ 0 subscript ~ Symp 0 \widetilde{\rm Symp}_{0} of Symp 0 subscript Symp 0 {\rm Symp}_{0} by (1.1) Flux : Symp ~ 0 → H 1 ​ ( M ; ℝ ) , ϕ ~ ↦ ∫ 0 1 ω ​ ( ϕ ˙ t , ⋅ ) ​ 𝑑 t , : Flux formulae-sequence → subscript ~ Symp 0 superscript 𝐻 1 𝑀 ℝ maps-to ~ italic-ϕ superscript subscript 0 1 delimited- 𝜔 subscript ˙ italic-ϕ 𝑡 ⋅ differential-d 𝑡 {\rm Flux}:\widetilde{\rm Symp}_{0}\to H^{1}(M;{\mathbb{R}}),\quad{\widetilde{\phi}}\mapsto\int_{0}^{1}{\omega}(\dot{\phi}_{t},\cdot)\,dt, where ϕ ~ = ( ϕ 1 , { ϕ t } ) ∈ Symp ~ 0 ~ italic-ϕ subscript italic-ϕ 1 subscript italic-ϕ 𝑡 subscript ~ Symp 0 {\widetilde{\phi}}=(\phi_{1},\{\phi_{t}\})\in\widetilde{\rm Symp}_{0} . The (symplectic) Flux group Γ ω subscript Γ 𝜔 {\Gamma}_{\omega} is defined to be the image of π 1 ​ ( Symp ) subscript 𝜋 1 Symp \pi_{1}({\rm Symp}) under Flux, so that there is an induced homomorphism Flux : π 1 ​ ( Symp ) → H 1 ​ ( M ; ℝ ) / Γ ω : Flux → subscript 𝜋 1 Symp superscript 𝐻 1 𝑀 ℝ subscript Γ 𝜔 {\rm Flux}:\pi_{1}({\rm Symp})\to H^{1}(M;{\mathbb{R}})/{\Gamma}_{\omega} with kernel Ham Ham {\rm Ham} . Ono 48 recently proved that Γ ω subscript Γ 𝜔 {\Gamma}_{\omega} is discrete. Unfortunately there seem to be no good techniques for understanding when Flux ​ ( γ ) Flux 𝛾 {\rm Flux}({\gamma}) is trivial. Since Hamiltonian S 1 superscript 𝑆 1 S^{1} actions always have fixed points at the critical points of the moment map H : M → ℝ : 𝐻 → 𝑀 ℝ H:M\to{\mathbb{R}} , a first guess might be that every symplectic action with fixed points is Hamiltonian. However McDuff 31 shows that this is not the case except in dimension 4 4 4 . In fact, the following basic problem is still unsolved in dimensions > 4 absent 4 >4 . Question 1.2 . Suppose that S 1 superscript 𝑆 1 S^{1} acts symplectically on the closed symplectic manifold ( M , ω ) 𝑀 𝜔 (M,{\omega}) with a finite but nonzero number of fixed points. Is the action Hamiltonian? If the action is semifree (i.e. the stabilizer of a point is either the identity or the whole group) Tolman–Weitsman 60 show that the answer to Question 1.2 is affirmative by computing various equivariant cohomology classes. Some other information on this question has been obtained by Feldman 10 and Pelayo–Tolman 51 . One might hope to extend the Tolman–Weitsman result to semifree actions with more general conditions on the fixed point components, for example that they are simply connected; note that these cannot be arbitrary because of the example in 31 of a semifree but nonHamiltonian action on a 6 6 6 -manifold with fixed point sets that are 2 2 2 -tori. In the current discussion we will largely ignore this problem, for the most part considering only Hamiltonian loops and their images in π 1 ​ ( Ham ) subscript 𝜋 1 Ham \pi_{1}({\rm Ham}) . Question 1.3 . To what extent are π 1 ​ ( Ham ) subscript 𝜋 1 Ham \pi_{1}({\rm Ham}) and π 1 ​ ( Symp ) subscript 𝜋 1 Symp \pi_{1}({\rm Symp}) generated by symplectic S 1 superscript 𝑆 1 S^{1} actions? This question is a measure of our ignorance. S 1 superscript 𝑆 1 S^{1} actions do generate π 1 ​ ( Symp ) subscript 𝜋 1 Symp \pi_{1}({\rm Symp}) in very special cases such as ℂ ​ P 2 ℂ superscript 𝑃 2 {\mathbb{C}}P^{2} or ( S 2 × S 2 , p ​ r 1 ∗ ​ ( σ ) + p ​ r 2 ∗ ​ ( σ ) ) superscript 𝑆 2 superscript 𝑆 2 𝑝 superscript subscript 𝑟 1 𝜎 𝑝 superscript subscript 𝑟 2 𝜎 \bigl{(}S^{2}\times S^{2},pr_{1}^{*}({\sigma})+pr_{2}^{*}({\sigma})\bigr{)} (note that the factors have equal area). Indeed in these cases Symp Symp {\rm Symp} itself is known to have the homotopy type of a compact Lie group (see 35 ). However, as we see below, this does not hold in general. Question 1.4 . What can one say about the relation between π 1 ​ ( Ham ) subscript 𝜋 1 Ham \pi_{1}({\rm Ham}) , π 1 ​ ( Symp ) subscript 𝜋 1 Symp \pi_{1}({\rm Symp}) and π 1 ​ ( Diff ) subscript 𝜋 1 Diff \pi_{1}({\rm Diff}) ? For example, under what circumstances is the map π 1 ​ ( Symp ) → π 1 ​ ( Diff ) → subscript 𝜋 1 Symp subscript 𝜋 1 Diff \pi_{1}({\rm Symp})\to\pi_{1}({\rm Diff}) injective or surjective? The symplectic Flux group Γ ω subscript Γ 𝜔 {\Gamma}_{\omega} is the quotient of π 1 ​ ( Symp ) subscript 𝜋 1 Symp \pi_{1}({\rm Symp}) by π 1 ​ ( Ham ) subscript 𝜋 1 Ham \pi_{1}({\rm Ham}) and hence precisely measures their difference. By Kȩdra–Kotschick–Morita 22 , this group vanishes in many cases. Much of their paper in fact applies to the volume 1 1 1 The volume flux is defined by equation ( 1.1 ), but ω 𝜔 {\omega} should be understood as a volume form and the homomorphism takes values in H m − 1 ​ ( M ) superscript 𝐻 𝑚 1 𝑀 H^{m-1}(M) , where m := dim M assign 𝑚 dimension 𝑀 m:=\dim M . Accordingly, Γ vol subscript Γ vol {\Gamma}_{{\rm vol}} is the image of π 1 ​ ( Diff vol ) subscript 𝜋 1 subscript Diff vol \pi_{1}({\rm Diff}_{{\rm vol}}) under the flux. flux group Γ vol subscript Γ vol {\Gamma}_{{\rm vol}} , which is in principle of a more topological nature than Γ ω subscript Γ 𝜔 {\Gamma}_{\omega} ; it would be interesting to find conditions for the vanishing of Γ ω subscript Γ 𝜔 {\Gamma}_{\omega} that involve symplectic geometry at a deeper level. In this note, we begin by describing some classical methods for exploring the above questions, the first based on properties of the action functional 𝒜 H subscript 𝒜 𝐻 {\mathcal{A}}_{H} and the second using volume. These methods give rather good information in the following cases: ∙ ∙ \bullet Question 1.1 for toric manifolds (see Corollary 2.4 below) ∙ ∙ \bullet Questions 1.3 and 1.4 for pointwise blow ups M ~ ~ 𝑀 {\widetilde{M}} of arbitrary symplectic manifolds M 𝑀 M (see Proposition 2.7 and its corollaries). If M 𝑀 M is noncompact and one considers the group Ham c := Ham c ​ M assign superscript Ham 𝑐 superscript Ham 𝑐 𝑀 {\rm Ham}^{c}:={\rm Ham}^{c}\,M of compactly supported Hamiltonian symplectomorphisms of M 𝑀 M , then there is another classical homomorphism called the Calabi homomorphism : Cal : π 1 ​ ( Ham c ) → ℝ , γ ↦ 1 n ! ​ ∫ 0 1 ( ∫ M H t ​ ω n ) ​ 𝑑 t , : Cal formulae-sequence → subscript 𝜋 1 superscript Ham 𝑐 ℝ maps-to 𝛾 1 𝑛 superscript subscript 0 1 subscript 𝑀 subscript 𝐻 𝑡 superscript 𝜔 𝑛 differential-d 𝑡 {\rm Cal}:\pi_{1}({\rm Ham}^{c})\to{\mathbb{R}},\quad{\gamma}\mapsto\frac{1}{n!}\int_{0}^{1}\Bigl{(}\int_{M}H_{t}{\omega}^{n}\bigr{)}dt, where H t subscript 𝐻 𝑡 H_{t} is the generating Hamiltonian for γ 𝛾 {\gamma} , normalized to have compact support. We explain briefly in Lemma 3.9 why ∙ ∙ \bullet the Calabi homomorphism need not vanish. As we shall see, this question, though classical in origin, is very closely related to questions about the Seidel representation in quantum homology. One might wonder if it is possible to get better information about the above questions by using more modern (i.e. quantum) techniques. In fact, Question 1.1 first arose in McDuff–Slimowitz 42 , a paper that uses Floer theoretic techniques to study paths in Ham Ham {\rm Ham} that are geodesic with respect to the Hofer norm. 2 2 2 This is defined in § 3.2 below. An unexpected consequence of the ideas developed there is that semifree Hamiltonian circle actions do not contract in Ham Ham {\rm Ham} , though they might have finite order (for example, a rotation of S 2 superscript 𝑆 2 S^{2} .) The main tool that has proved useful in this context is an extension of the action homomorphism due to Seidel 58 , that is called the Seidel representation ; see § 3.1 . This homomorphism assigns to every γ ∈ π 1 ​ ( Ham ) 𝛾 subscript 𝜋 1 Ham {\gamma}\in\pi_{1}({\rm Ham}) a unit (i.e. invertible element) 𝒮 ​ ( γ ) 𝒮 𝛾 {\mathcal{S}}({\gamma}) in the (small) quantum homology Q ​ H ∗ ​ ( M ) 𝑄 subscript 𝐻 𝑀 QH_{*}(M) . Corollary 3.2 gives some more results on the above questions obtained using 𝒮 𝒮 {\mathcal{S}} . Because 𝒮 𝒮 {\mathcal{S}} is usually very hard to calculate, the classical methods often work better in specific examples. Nevertheless, 𝒮 𝒮 {\mathcal{S}} is a key tool in other contexts. One very interesting question is the following. Note that in two dimensions, the only symplectic manifold with a Hamiltonian S 1 superscript 𝑆 1 S^{1} action is S 2 superscript 𝑆 2 S^{2} , while T 2 superscript 𝑇 2 T^{2} has nonHamiltonian actions and higher genus surfaces Σ g subscript Σ 𝑔 {\Sigma}_{g} have none. Question 1.5 . Is there a meaningful extension of the classification of Riemann surfaces into spheres, tori and higher genus to higher dimensional symplectic manifolds? If so, is any aspect of it reflected in the properties of π 1 ​ ( Ham ) subscript 𝜋 1 Ham \pi_{1}({\rm Ham}) ? This would be an analog of minimal model theory in algebraic geometry. A first step, accomplished by Ruan and his coworkers Hu and T.-J. Li 56 , 18 , is to understand what it means for two symplectic manifolds to be birationally equivalent. Their results imply that a reasonable class of manifolds to take as the analog of spheres are the symplectically uniruled manifolds. These are the manifolds for which there is a nonzero genus zero Gromov–Witten invariant ⟨ a 1 , a 2 , … , a k ⟩ β M {\bigl{\langle}}a_{1},a_{2},\dots,a_{k}{\bigl{\rangle}}^{M}_{\beta} (for some k ≥ 1 , 𝑘 1 k\geq 1, a i ∈ H ∗ ​ ( M ) subscript 𝑎 𝑖 subscript 𝐻 𝑀 a_{i}\in H_{*}(M) and β ∈ H 2 ​ ( M ) 𝛽 subscript 𝐻 2 𝑀 {\beta}\in H_{2}(M) ) with one of the constraints a i subscript 𝑎 𝑖 a_{i} equal to a point. This class includes all projective manifolds that are uniruled in the sense of algebraic geometry. In this case ω ​ ( β ) ≠ 0 𝜔 𝛽 0 {\omega}({\beta})\neq 0 and c 1 ​ ( β ) ≠ 0 subscript 𝑐 1 𝛽 0 c_{1}({\beta})\neq 0 . At the other extreme are the symplectically aspherical manifolds for which the restriction ω | π 2 ​ ( M ) evaluated-at 𝜔 subscript 𝜋 2 𝑀 {\omega}|_{\pi_{2}(M)} of ω delimited- 𝜔 {\omega} to π 2 ​ ( M ) subscript 𝜋 2 𝑀 \pi_{2}(M) vanishes, and possibly also (depending on the author) the restriction c 1 | π 2 ​ ( M ) evaluated-at subscript 𝑐 1 subscript 𝜋 2 𝑀 c_{1}|_{\pi_{2}(M)} of the first Chern class c 1 subscript 𝑐 1 c_{1} of ( M , ω ) 𝑀 𝜔 (M,{\omega}) . These manifolds have no J 𝐽 J -holomorphic curves at all, and hence all nontrivial (i.e. β ≠ 0 𝛽 0 {\beta}\neq 0 ) Gromov–Witten invariants vanish. To a first approximation, one can characterize symplectically uniruled manifolds in terms of their quantum homology Q ​ H ∗ ​ ( M ) 𝑄 subscript 𝐻 𝑀 QH_{*}(M) in the following way. If ( M , ω ) 𝑀 𝜔 (M,{\omega}) is not uniruled then all invertible elements in Q ​ H 2 ​ n ​ ( M ) 𝑄 subscript 𝐻 2 𝑛 𝑀 QH_{2n}(M) have the form 1 ​ l ⊗ λ + x tensor-product 1 l 𝜆 𝑥 {{\mathchoice{\rm 1\mskip-4.0mul}{\rm 1\mskip-4.0mul}{\rm 1\mskip-4.5mul}{\rm 1\mskip-5.0mul}}}\otimes{\lambda}+x where λ 𝜆 {\lambda} is invertible in the coefficient field Λ Λ {\Lambda} and x ∈ H < 2 ​ n ​ ( M ) ⊗ Λ 𝑥 tensor-product subscript 𝐻 absent 2 𝑛 𝑀 Λ x\in H_{<2n}(M)\otimes{\Lambda} . (This is nearly an iff statement and can be improved to such: cf. the appendix to 38 . Here 1 ​ l 1 l {{\mathchoice{\rm 1\mskip-4.0mul}{\rm 1\mskip-4.0mul}{\rm 1\mskip-4.5mul}{\rm 1\mskip-5.0mul}}} denotes the fundamental class M ∈ Q ​ H ∗ ​ ( M ) delimited- 𝑀 𝑄 subscript 𝐻 𝑀 M\in QH_{*}(M) ; it is the identity element in Q ​ H ∗ ​ ( M ) 𝑄 subscript 𝐻 𝑀 QH_{*}(M) . The notation is explained in more detail below.) Therefore another version of the previous question is: Question 1.6 . To what extent is the structure of the quantum homology ring Q ​ H ∗ ​ ( M ) 𝑄 subscript 𝐻 𝑀 QH_{*}(M) reflected in the algebraic/topological/geometric structure of Ham Ham {\rm Ham} ? For example, according to Polterovich 55 , if ω | π 2 ​ ( M ) = 0 evaluated-at 𝜔 subscript 𝜋 2 𝑀 0 {\omega}|_{\pi_{2}(M)}=0 , Ham Ham {\rm Ham} contains no elements of finite order except for i ​ d 𝑖 𝑑 id . Does this property continue to hold if the condition on ω 𝜔 {\omega} is weakened to the vanishing of all Gromov–Witten invariants? Since S 1 superscript 𝑆 1 S^{1} -manifolds certainly support nontrivial symplectomorphisms of finite order (such as a half turn), it is natural to ask whether these manifolds have nontrivial Gromov–Witten invariants. Since the Seidel element 𝒮 ​ ( γ ) 𝒮 𝛾 {\mathcal{S}}({\gamma}) is a unit in quantum homology one might also expect to see traces of the uniruled/non-uniruled dichotomy in its properties. This was the guiding idea in my recent proof 38 that every closed symplectic manifold that supports a Hamiltonian S 1 superscript 𝑆 1 S^{1} action is uniruled . The argument in 38 applies more generally to manifolds with Hamiltonian loops that are nondegenerate Hofer geodesics. This opens up many interesting questions of a more dynamical flavor. Here we shall discuss the following basic (but unrelated) problem, which is still open in many cases. Question 1.7 . Does Ham Ham {\rm Ham} have infinite diameter with respect to the Hofer metric? One expects the answer to be positive always. However the proofs for spheres (due to Polterovich 53 ) and other Riemann surfaces (due to Lalonde–McDuff 25 ) are very different. In fact, as noted by Ostrover 49 one can use the spectral invariants of Schwarz 57 and Oh 46 , 47 (see also Usher 61 ) to show that the universal cover Ham ~ ~ Ham {\widetilde{\rm Ham}} of Ham Ham {\rm Ham} always has infinite diameter with respect to the induced (pseudo)metric. Therefore the question becomes: when does this result transfer down to Ham Ham {\rm Ham} ? Schwarz shows in 57 that this happens when ω 𝜔 {\omega} and c 1 subscript 𝑐 1 c_{1} both vanish on π 2 ​ ( M ) subscript 𝜋 2 𝑀 \pi_{2}(M) . I recently extended his work in 39 , showing that the asymptotic spectral invariants descend to Ham Ham {\rm Ham} if, for example, all nontrivial genus zero Gromov–Witten invariants vanish and rank ​ H 2 ​ ( M ; ℝ ) > 1 rank subscript 𝐻 2 𝑀 ℝ 1 {\rm rank\,}H_{2}(M;{\mathbb{R}})>1 . As we explain in § 3.2 below, Schwarzs argument hinges on the properties of the Seidel elements 𝒮 ​ ( γ ) 𝒮 𝛾 {\mathcal{S}}({\gamma}) of γ ∈ π 1 ​ ( Ham ) 𝛾 subscript 𝜋 1 Ham {\gamma}\in\pi_{1}({\rm Ham}) . Here we shall sketch a different extension of his result. In particular we show: Ham Ham {\rm Ham} has infinite diameter when M 𝑀 M is a small one point blow up of ℂ ​ P 2 ℂ superscript 𝑃 2 {\mathbb{C}}P^{2} . This manifold M 𝑀 M is of course uniruled (and the spectral invariants do not descend). To my knowledge, it is not yet known whether Ham Ham {\rm Ham} has infinite diameter for all one point blow ups of ℂ ​ P 2 ℂ superscript 𝑃 2 {\mathbb{C}}P^{2} , though it does for ℂ ​ P 2 ℂ superscript 𝑃 2 {\mathbb{C}}P^{2} itself (and indeed for any ℂ ​ P n ℂ superscript 𝑃 𝑛 {\mathbb{C}}P^{n} ) by the results of Entov–Polterovich 8 . For further results on this problem see McDuff 39 .","role":"user"},{"content":"Abstract: This note describes some recent results about the homotopy properties of Hamiltonian loops in various manifolds, including toric manifolds and one point blow ups. We describe conditions under which a circle action does not contract in the Hamiltonian group, and construct an example of a loop $\ga$ of diffeomorphisms of a symplectic manifold M with the property that none of the loops smoothly isotopic to $\ga$ preserve any symplectic form on M. We also discuss some new conditions under which the Hamiltonian group has infinite Hofer diameter. Some of the methods used are classical (Weinstein's action homomorphism and volume calculations), while others use quantum methods (the Seidel representation and spectral invariants).","role":"assistant"}]

Top quark theory review for the Tevatron, LHC, and ILC

I THE MANY FACETS OF TOP QUARK PHYSICS The study of top-quark properties and dynamics provides a unique window to the mechanism of electroweak symmetry breaking (EWSB). The large mass of the top quark suggests that it plays a special role in EWSB, and that new physics connected to EWSB may be found first through precision studies of top-quark observables. Deviations of experimental measurements from the SM predictions, including electroweak (EW) and QCD corrections, could indicate new non-standard top production or decay mechanisms. Since the top quark immediately decays before it hadronizes or flips its spin, it provides an excellent testing ground for perturbative QCD. Moreover, information about spin correlation and polarization, imprinted by the production process, is preserved, and can be measured in angular distributions of top-decay products, providing another way to search for deviations from the SM expectation. The precise measurement of the top-quark mass ( m t subscript 𝑚 𝑡 m_{t} ) allows for improved bounds on the mass of the Standard Model (SM) Higgs boson ( M H subscript 𝑀 𝐻 M_{H} ), which is presently constrained to be smaller than 185 GeV ( 95 % percent 95 95\% C.L.) Alcaraz:2007ri . Measuring precisely the properties of the top quark and studying its dynamics therefore is an important goal at the Tevatron Run II, LHC and ILC. To fully exploit the potential of these colliders for precision top-quark physics, it is crucial that predictions for top-quark observables are under theoretical control and include higher-order corrections within the SM and beyond. In the following, I will briefly describe some theory aspects of top-quark physics relevant to the Tevatron, LHC and ILC. For detailed reviews of theoretical and experimental results please see, e.g., Refs. Bernreuther:2008ju ; Kehoe:2007px ; Gerber:2007xk ; Abe:2001nqa ; Beneke:2000hk (as well as presentations in this session).

I will briefly review the status of higher-order calculations for top-quark observables, comment on the need for improvements, discuss some of the recent theoretical advances, and present a few examples to highlight the role of top-quark observables in searches for signals of physics beyond the Standard Model.

[{"content":"Help me to generate the abstract of a survey paper given the title:*Top quark theory review for the Tevatron, LHC, and ILC* and the introduction:I THE MANY FACETS OF TOP QUARK PHYSICS The study of top-quark properties and dynamics provides a unique window to the mechanism of electroweak symmetry breaking (EWSB). The large mass of the top quark suggests that it plays a special role in EWSB, and that new physics connected to EWSB may be found first through precision studies of top-quark observables. Deviations of experimental measurements from the SM predictions, including electroweak (EW) and QCD corrections, could indicate new non-standard top production or decay mechanisms. Since the top quark immediately decays before it hadronizes or flips its spin, it provides an excellent testing ground for perturbative QCD. Moreover, information about spin correlation and polarization, imprinted by the production process, is preserved, and can be measured in angular distributions of top-decay products, providing another way to search for deviations from the SM expectation. The precise measurement of the top-quark mass ( m t subscript 𝑚 𝑡 m_{t} ) allows for improved bounds on the mass of the Standard Model (SM) Higgs boson ( M H subscript 𝑀 𝐻 M_{H} ), which is presently constrained to be smaller than 185 GeV ( 95 % percent 95 95\% C.L.) Alcaraz:2007ri . Measuring precisely the properties of the top quark and studying its dynamics therefore is an important goal at the Tevatron Run II, LHC and ILC. To fully exploit the potential of these colliders for precision top-quark physics, it is crucial that predictions for top-quark observables are under theoretical control and include higher-order corrections within the SM and beyond. In the following, I will briefly describe some theory aspects of top-quark physics relevant to the Tevatron, LHC and ILC. For detailed reviews of theoretical and experimental results please see, e.g., Refs. Bernreuther:2008ju ; Kehoe:2007px ; Gerber:2007xk ; Abe:2001nqa ; Beneke:2000hk (as well as presentations in this session).","role":"user"},{"content":"Abstract: I will briefly review the status of higher-order calculations for top-quark observables, comment on the need for improvements, discuss some of the recent theoretical advances, and present a few examples to highlight the role of top-quark observables in searches for signals of physics beyond the Standard Model.","role":"assistant"}]

Survey on recent invariants on classical knot theory

1. Link diagrams and Reidemeister moves The classical knot theory studies the position of a circle (knots) or of several circles (links) in S 3 superscript 𝑆 3 S^{3} (or ℝ 3 superscript ℝ 3 \mathbb{R}^{3} ). We say that two links L 1 subscript 𝐿 1 L_{1} and L 2 subscript 𝐿 2 L_{2} in S 3 superscript 𝑆 3 S^{3} are isotopic, written L 1 ≈ L 2 subscript 𝐿 1 subscript 𝐿 2 L_{1}\approx L_{2} , if there exists an isotopy F : S 3 × I ⟶ S 3 × I : 𝐹 ⟶ superscript 𝑆 3 𝐼 superscript 𝑆 3 𝐼 F:S^{3}\times I\longrightarrow S^{3}\times I such that F 0 = Id subscript 𝐹 0 Id F_{0}=\mathrm{Id} and F 1 ​ ( L 1 ) = L 2 subscript 𝐹 1 subscript 𝐿 1 subscript 𝐿 2 F_{1}(L_{1})=L_{2} . If the links L 1 subscript 𝐿 1 L_{1} and L 2 subscript 𝐿 2 L_{2} are oriented, we assume additionally that F 1 subscript 𝐹 1 F_{1} preserves orientations of the links. We work all the time in the PL category; smooth category could be considered equally well. S 3 = ℝ 3 ∪ ∞ superscript 𝑆 3 superscript ℝ 3 S^{3}=\mathbb{R}^{3}\cup\infty and we can always assume that the link omits ∞ \infty . It is not difficult to show that two links are isotopic in ℝ 3 superscript ℝ 3 \mathbb{R}^{3} if and only if they are isotopic in S 3 superscript 𝑆 3 S^{3} . Links (up to isotopy) can be represented by their diagrams on the plane. Namely, let p : ℝ 3 ⟶ ℝ 2 : 𝑝 ⟶ superscript ℝ 3 superscript ℝ 2 p:\mathbb{R}^{3}\longrightarrow\mathbb{R}^{2} be a projection and L ⊂ ℝ 3 𝐿 superscript ℝ 3 L\subset\mathbb{R}^{3} a link. A point P ∈ p ​ ( L ) ⊂ ℝ 2 𝑃 𝑝 𝐿 superscript ℝ 2 P\in p(L)\subset\mathbb{R}^{2} whose preimage, p − 1 ​ ( P ) superscript 𝑝 1 𝑃 p^{-1}(P) , contains more than one point is called a multiple point. A projection p 𝑝 p is called regular if (1) There are only finitely many multiple points, and all multiple points are double points (called crossings), and (2) P / L : L → ℝ 2 : 𝑃 𝐿 → 𝐿 superscript ℝ 2 P/L:L\to\mathbb{R}^{2} is the general position projection (in some triangulation of ( ℝ 3 , L ) superscript ℝ 3 𝐿 (\mathbb{R}^{3},L) and ℝ 2 superscript ℝ 2 \mathbb{R}^{2} , in which P / L 𝑃 𝐿 P/L is simplicial, no double points of L 𝐿 L are vertices). If, for a given regular projection of a link, all over-crossings (bridges) at every crossing are marked, then the link can be reconstructed from the projection. The projection of the link with just described additional information is called the diagram of the link. We call two diagrams equivalent (in oriented or unoriented category) if they describe isotopic links. The following theorem of Reidemeister allows us to work entirely with diagrams. Theorem 1.0.1 . Two link diagrams are equivalent if and only if they are connected by a finite sequence of Reidemeister moves, Ω i ± 1 subscript superscript Ω plus-or-minus 1 𝑖 \Omega^{\pm 1}_{i} ( i = 1 , 2 , 3 ) 𝑖 1 2 3 (i=1,2,3) see Figure 1.1 . Figure 1.1.

The survey we are presenting is over 22 years old but it has still some ideas which where never published (except in Polish). This survey is the base of the third Chapter of my book: KNOTS: From combinatorics of knot diagrams to combinatorial topology based on knots, which is still in preparation (but compare this http URL). The purpose of this survey is to present a new combinatorial method of constructing invariants of isotopy classes of tame links. The period of time between the spring of 1984 and the summer of 1985 was full of discoveries which revolutionized the knot theory and will have a deep impact on some other branches of mathematics. It started by the discovery of Jones of the new polynomial invariant of links (in May 1984), and the last big step (which will be described in this survey) has been made by Kauffman in August 1985 when Kauffman applied his method which allowed him to unify almost all previous work. This survey is far from being complete, even if we limit ourselves to the purely combinatorial methods and to the period May, 1984 -- September, 1985.

[{"content":"Help me to generate the abstract of a survey paper given the title:*Survey on recent invariants on classical knot theory* and the introduction:1. Link diagrams and Reidemeister moves The classical knot theory studies the position of a circle (knots) or of several circles (links) in S 3 superscript 𝑆 3 S^{3} (or ℝ 3 superscript ℝ 3 \mathbb{R}^{3} ). We say that two links L 1 subscript 𝐿 1 L_{1} and L 2 subscript 𝐿 2 L_{2} in S 3 superscript 𝑆 3 S^{3} are isotopic, written L 1 ≈ L 2 subscript 𝐿 1 subscript 𝐿 2 L_{1}\approx L_{2} , if there exists an isotopy F : S 3 × I ⟶ S 3 × I : 𝐹 ⟶ superscript 𝑆 3 𝐼 superscript 𝑆 3 𝐼 F:S^{3}\times I\longrightarrow S^{3}\times I such that F 0 = Id subscript 𝐹 0 Id F_{0}=\mathrm{Id} and F 1 ​ ( L 1 ) = L 2 subscript 𝐹 1 subscript 𝐿 1 subscript 𝐿 2 F_{1}(L_{1})=L_{2} . If the links L 1 subscript 𝐿 1 L_{1} and L 2 subscript 𝐿 2 L_{2} are oriented, we assume additionally that F 1 subscript 𝐹 1 F_{1} preserves orientations of the links. We work all the time in the PL category; smooth category could be considered equally well. S 3 = ℝ 3 ∪ ∞ superscript 𝑆 3 superscript ℝ 3 S^{3}=\mathbb{R}^{3}\cup\infty and we can always assume that the link omits ∞ \infty . It is not difficult to show that two links are isotopic in ℝ 3 superscript ℝ 3 \mathbb{R}^{3} if and only if they are isotopic in S 3 superscript 𝑆 3 S^{3} . Links (up to isotopy) can be represented by their diagrams on the plane. Namely, let p : ℝ 3 ⟶ ℝ 2 : 𝑝 ⟶ superscript ℝ 3 superscript ℝ 2 p:\mathbb{R}^{3}\longrightarrow\mathbb{R}^{2} be a projection and L ⊂ ℝ 3 𝐿 superscript ℝ 3 L\subset\mathbb{R}^{3} a link. A point P ∈ p ​ ( L ) ⊂ ℝ 2 𝑃 𝑝 𝐿 superscript ℝ 2 P\in p(L)\subset\mathbb{R}^{2} whose preimage, p − 1 ​ ( P ) superscript 𝑝 1 𝑃 p^{-1}(P) , contains more than one point is called a multiple point. A projection p 𝑝 p is called regular if (1) There are only finitely many multiple points, and all multiple points are double points (called crossings), and (2) P / L : L → ℝ 2 : 𝑃 𝐿 → 𝐿 superscript ℝ 2 P/L:L\to\mathbb{R}^{2} is the general position projection (in some triangulation of ( ℝ 3 , L ) superscript ℝ 3 𝐿 (\mathbb{R}^{3},L) and ℝ 2 superscript ℝ 2 \mathbb{R}^{2} , in which P / L 𝑃 𝐿 P/L is simplicial, no double points of L 𝐿 L are vertices). If, for a given regular projection of a link, all over-crossings (bridges) at every crossing are marked, then the link can be reconstructed from the projection. The projection of the link with just described additional information is called the diagram of the link. We call two diagrams equivalent (in oriented or unoriented category) if they describe isotopic links. The following theorem of Reidemeister allows us to work entirely with diagrams. Theorem 1.0.1 . Two link diagrams are equivalent if and only if they are connected by a finite sequence of Reidemeister moves, Ω i ± 1 subscript superscript Ω plus-or-minus 1 𝑖 \Omega^{\pm 1}_{i} ( i = 1 , 2 , 3 ) 𝑖 1 2 3 (i=1,2,3) see Figure 1.1 . Figure 1.1.","role":"user"},{"content":"Abstract: The survey we are presenting is over 22 years old but it has still some ideas which where never published (except in Polish). This survey is the base of the third Chapter of my book: KNOTS: From combinatorics of knot diagrams to combinatorial topology based on knots, which is still in preparation (but compare this http URL). The purpose of this survey is to present a new combinatorial method of constructing invariants of isotopy classes of tame links. The period of time between the spring of 1984 and the summer of 1985 was full of discoveries which revolutionized the knot theory and will have a deep impact on some other branches of mathematics. It started by the discovery of Jones of the new polynomial invariant of links (in May 1984), and the last big step (which will be described in this survey) has been made by Kauffman in August 1985 when Kauffman applied his method which allowed him to unify almost all previous work. This survey is far from being complete, even if we limit ourselves to the purely combinatorial methods and to the period May, 1984 -- September, 1985.","role":"assistant"}]

A survey of Einstein metrics on 4-manifolds

1. Introduction. The Einstein equations (1.1) R ​ i ​ c g = λ ​ g , λ ∈ ℝ , formulae-sequence 𝑅 𝑖 subscript 𝑐 𝑔 𝜆 𝑔 𝜆 ℝ Ric_{g}=\lambda g,\ \ \lambda\in{\mathbb{R}}, for a Riemannian metric g 𝑔 g are the simplest and most natural set of equations for a metric on a given compact manifold M 𝑀 M . Historically these equations arose in the context of Einstein’s general theory of relativity, where the metric g 𝑔 g is of Lorentzian signature. However, over the past several decades there has also been much mathematical interest in Einstein metrics of Riemannian signature on compact manifolds, especially in low dimensions, and in particular in relation to the topology of the underlying manifold. A strong motivation for this comes from the understanding developed in dimension 2 and more recently in dimension 3. To explain this, there is a complete classification of compact oriented 2-manifolds by the Euler characteristic, originally obtained by purely topological methods through work of Möbius, Dehn, Heegard and Rado. This classification was later reproved via the Poincaré-Koebe uniformization theorem for surfaces, i.e. any compact oriented surface carries a metric of constant curvature. The structure of such metrics then easily gives the full list of possible topological types of such surfaces, (and much more). It has long been a goal of mathematicians to prove a similar classification of compact oriented 3-manifolds. Thurston [ 71 ] realized that the key to this should be in studying the possible geometric stuctures on 3-manifolds; the most important such structures are again the constant curvature, i.e. Einstein, metrics. However, in contrast to surfaces, most 3-manifolds do not admit an Einstein metric ( 1.1 ); instead one had a simple, and conjecturally complete, list of well understood obstructions. A general 3-manifold should decompose into a collection of domains, each of which carries a natural geometry, the most important geometry being that of Einstein metrics. The recent solution of the Thurston geometrization conjecture by Perelman [ 61 ] - [ 63 ] and Hamilton [ 36 ] - [ 37 ] has accomplished this goal of completely classifying all 3-manifolds. This has been obtained by understanding how to obtain solutions to the Einstein equations via the parabolic analogue of ( 1.1 ), namely the Ricci flow. Ideally, one would like to carry out a similar program in dimension 4. However, (as with the passage from 2 to 3 dimensions), the world of 4-manifolds is much more complicated than lower dimensions. One encounters a vast variety of exotic smooth structures, there are severe complications in understanding the fundamental group, and so on. Moreover, there are no canonical local models of Einstein metrics, and even when such metrics exist, the tie of the global geometry of these metrics with the underlying topology remains currently poorly understood. In fact, as discussed eloquently by Gromov in [ 34 ] , the ”dream” of trying to understand 4 4 4 -manifolds via Einstein or other canonical metrics may well be impossible to realize. On the other hand, one should keep in mind that the deepest understanding to date of smooth 4 4 4 -manifolds comes via the geometry of connections or gauge fields, in the theories developed by Yang-Mills, Donaldson and Seiberg-Witten. How far such theories can be carried over to metrics, (the gravitational field), remains to be seen. This paper is an introductory survey of basic results to date on the existence, uniqueness, and structure of moduli spaces of Einstein metrics on 4-manifolds. Many interesting topics have been omitted or presented only briefly, due partly to the limits of the author’s knowledge and taste, and partly to keep the article at a reasonable length. For this reason, we have excluded all discussion of Einstein metrics in higher dimensions. Also, any area of research is only as vital as the interest of significant open questions and problems. Accordingly, we present a number of such open problems throughout the paper, some of which are well-known and others less so.

We survey some aspects of the current state of research on Einstein metrics on compact 4-manifolds. A number of open problems are presented and discussed.

[{"content":"Help me to generate the abstract of a survey paper given the title:*A survey of Einstein metrics on 4-manifolds* and the introduction:1. Introduction. The Einstein equations (1.1) R ​ i ​ c g = λ ​ g , λ ∈ ℝ , formulae-sequence 𝑅 𝑖 subscript 𝑐 𝑔 𝜆 𝑔 𝜆 ℝ Ric_{g}=\lambda g,\ \ \lambda\in{\mathbb{R}}, for a Riemannian metric g 𝑔 g are the simplest and most natural set of equations for a metric on a given compact manifold M 𝑀 M . Historically these equations arose in the context of Einsteins general theory of relativity, where the metric g 𝑔 g is of Lorentzian signature. However, over the past several decades there has also been much mathematical interest in Einstein metrics of Riemannian signature on compact manifolds, especially in low dimensions, and in particular in relation to the topology of the underlying manifold. A strong motivation for this comes from the understanding developed in dimension 2 and more recently in dimension 3. To explain this, there is a complete classification of compact oriented 2-manifolds by the Euler characteristic, originally obtained by purely topological methods through work of Möbius, Dehn, Heegard and Rado. This classification was later reproved via the Poincaré-Koebe uniformization theorem for surfaces, i.e. any compact oriented surface carries a metric of constant curvature. The structure of such metrics then easily gives the full list of possible topological types of such surfaces, (and much more). It has long been a goal of mathematicians to prove a similar classification of compact oriented 3-manifolds. Thurston 71 realized that the key to this should be in studying the possible geometric stuctures on 3-manifolds; the most important such structures are again the constant curvature, i.e. Einstein, metrics. However, in contrast to surfaces, most 3-manifolds do not admit an Einstein metric ( 1.1 ); instead one had a simple, and conjecturally complete, list of well understood obstructions. A general 3-manifold should decompose into a collection of domains, each of which carries a natural geometry, the most important geometry being that of Einstein metrics. The recent solution of the Thurston geometrization conjecture by Perelman 61 - 63 and Hamilton 36 - 37 has accomplished this goal of completely classifying all 3-manifolds. This has been obtained by understanding how to obtain solutions to the Einstein equations via the parabolic analogue of ( 1.1 ), namely the Ricci flow. Ideally, one would like to carry out a similar program in dimension 4. However, (as with the passage from 2 to 3 dimensions), the world of 4-manifolds is much more complicated than lower dimensions. One encounters a vast variety of exotic smooth structures, there are severe complications in understanding the fundamental group, and so on. Moreover, there are no canonical local models of Einstein metrics, and even when such metrics exist, the tie of the global geometry of these metrics with the underlying topology remains currently poorly understood. In fact, as discussed eloquently by Gromov in 34 , the dream of trying to understand 4 4 4 -manifolds via Einstein or other canonical metrics may well be impossible to realize. On the other hand, one should keep in mind that the deepest understanding to date of smooth 4 4 4 -manifolds comes via the geometry of connections or gauge fields, in the theories developed by Yang-Mills, Donaldson and Seiberg-Witten. How far such theories can be carried over to metrics, (the gravitational field), remains to be seen. This paper is an introductory survey of basic results to date on the existence, uniqueness, and structure of moduli spaces of Einstein metrics on 4-manifolds. Many interesting topics have been omitted or presented only briefly, due partly to the limits of the authors knowledge and taste, and partly to keep the article at a reasonable length. For this reason, we have excluded all discussion of Einstein metrics in higher dimensions. Also, any area of research is only as vital as the interest of significant open questions and problems. Accordingly, we present a number of such open problems throughout the paper, some of which are well-known and others less so.","role":"user"},{"content":"Abstract: We survey some aspects of the current state of research on Einstein metrics on compact 4-manifolds. A number of open problems are presented and discussed.","role":"assistant"}]

Review of Properties of the Top Quark from Measurements at the Tevatron

The existence of a third and most massive generation of fundamental fermions was unveiled in 1975 with the discovery of the τ 𝜏 \tau lepton at SLAC-LBL [ 1 ] . In 1977, the discovery of the bottom quark [ 2 ] at Fermilab extended the knowledge of a third generation into the quark sector and immediately raised the question of the existence of the top quark as the weak isospin partner of the bottom quark. To remain self consistent, the standard model (SM) of elementary particle physics required the existence of the top quark, and electroweak precision measurements offered increasingly precise predictions of properties such as its mass. The top quark’s large mass prevented its discovery for almost two decades, but by 1994 it was indirectly constrained to be 178 ± plus-or-minus \pm 11 − 19 + 18 subscript superscript absent 18 19 {}^{+18}_{-19} GeV/c 2 [ 3 ] . After mounting experimental evidence [ 4 , 5 , 6 , 7 , 8 , 9 ] , the top quark ( t 𝑡 t ) was finally discovered in 1995 at Fermilab by the CDF and D0 collaborations [ 10 , 11 ] in the mass range predicted by the standard model. The completion of the quark sector once again demonstrated the enormous predictive power of the SM. By now, the mass of the top quark is measured to be 172.4 ± plus-or-minus \pm 1.2 GeV/c 2 [ 12 ] , marking the most precisely measured quark mass and the most massive fundamental particle known to date. The consequent lifetime of the top quark in the SM of ≈ 5 ⋅ 10 − 25 absent ⋅ 5 superscript 10 25 \approx 5\cdot 10^{-25} s is extremely short, suggesting that it decays before hadronizing. This makes it the only quark that does not form bound states, allowing the study of an essentially bare quark with properties such as spin undisturbed by hadronization [ 13 ] . The measurement of top quark pair ( t ​ t ¯ 𝑡 ¯ 𝑡 t\overline{t} ) production probes our understanding of the strong interaction and predictions from perturbative QCD, while the decay of top quarks and the production of single top quarks reflect the electroweak interaction. Measuring other properties of the top quark, such as its electric charge, the helicity of the W 𝑊 W boson in t → W ​ b → 𝑡 𝑊 𝑏 t\to Wb decay, the branching fraction ℬ ​ ( t → W ​ b ) ℬ → 𝑡 𝑊 𝑏 {\cal B}(t\rightarrow Wb) , etc. , and comparing these with predictions of the SM is a very powerful tool in searching for new physics beyond the standard model. The top quark can also be used to constrain the mass range of the last yet to be observed particle of the standard model, the Higgs boson, because their masses and the mass of the W 𝑊 W boson are linked through radiative corrections [ 14 ] . The Higgs boson is a manifestation of the Higgs mechanism [ 15 , 16 , 17 , 18 , 19 , 20 ] , implemented in the standard model to provide the needed breaking of the electroweak symmetry to which the top quark may be intimately connected because of its large mass. Because of its fairly recent discovery, the top quark’s properties have not yet been explored with the same scrutiny as those of the lighter quarks. However, in the ongoing data taking at Fermilab’s Tevatron proton-antiproton collider, an integrated luminosity of more than 4 fb -1 has already been recorded by each of the collider experiments CDF and D0, corresponding to an increase of about a factor seventy relative to the data that was available for the discovery of the top quark. The new data can be used to refine previous measurements to higher precision that starts to become limited by systematic rather than statistical uncertainties. In addition, measurements that have never been performed become feasible, such as the first evidence for electroweak production of single top quarks and the consequent first direct measurement of the CKM matrix element | V t ​ b | subscript 𝑉 𝑡 𝑏 |V_{tb}| , recently published by D0 [ 21 , 22 ] and CDF [ 23 ] . This article is intended to provide an overview of the current status of the top quark physics program pursued at the Tevatron. Results available until the LHC startup in September 2008 have been included, utilizing samples of data of up to 2.8 fb -1 in integrated luminosity. Previous reviews of the top quark are available in Refs. [ 24 , 25 , 26 , 27 , 28 ] . The outline of this article is as follows: The second chapter provides a brief introduction to the standard model, with emphasis on the special role played by the top quark. Chapter 3 describes production and decay modes for top quarks in the framework of the standard model. Chapter 4 outlines the experimental setup used for the measurements described in the following sections. Chapter 5 presents studies of the production of top quarks, including measurements of cross section that form the basis for other measurements of top quark characteristics. Chapter 6 elaborates on the different results for top quark decay properties, followed in Chapter 7 by a discussion of measurements of fundamental attributes of the top quark, such as its charge and mass. The final chapter (8) contains a brief summary of the achievements to date.

This review summarizes the program in the physics of the top quark being pursued at Fermilab's Tevatron proton–antiproton collider at a center-of-mass energy of 1.96 TeV. More than a decade after the discovery of the top quark at the two collider detectors CDF and D0, the Tevatron has been the only accelerator to produce top quarks and to study them directly. The Tevatron's increased luminosity and center-of-mass energy offer the possibility to scrutinize the properties of this heaviest fundamental particle through new measurements that were not feasible before, such as the first evidence for electroweak production of top quarks and the resulting direct constraints on the involved couplings. Better measurements of top quark properties provide more stringent tests of predictions from the SM of elementary particle physics. In particular, the improvement in measurements of the mass of the top quark, with the latest uncertainty of 0.7% marking the most precisely measured quark mass to date, further constrains the prediction of the mass of the still to be discovered Higgs boson.

[{"content":"Help me to generate the abstract of a survey paper given the title:*Review of Properties of the Top Quark from Measurements at the Tevatron* and the introduction:The existence of a third and most massive generation of fundamental fermions was unveiled in 1975 with the discovery of the τ 𝜏 \tau lepton at SLAC-LBL 1 . In 1977, the discovery of the bottom quark 2 at Fermilab extended the knowledge of a third generation into the quark sector and immediately raised the question of the existence of the top quark as the weak isospin partner of the bottom quark. To remain self consistent, the standard model (SM) of elementary particle physics required the existence of the top quark, and electroweak precision measurements offered increasingly precise predictions of properties such as its mass. The top quarks large mass prevented its discovery for almost two decades, but by 1994 it was indirectly constrained to be 178 ± plus-or-minus \pm 11 − 19 + 18 subscript superscript absent 18 19 {}^{+18}_{-19} GeV/c 2 3 . After mounting experimental evidence 4 , 5 , 6 , 7 , 8 , 9 , the top quark ( t 𝑡 t ) was finally discovered in 1995 at Fermilab by the CDF and D0 collaborations 10 , 11 in the mass range predicted by the standard model. The completion of the quark sector once again demonstrated the enormous predictive power of the SM. By now, the mass of the top quark is measured to be 172.4 ± plus-or-minus \pm 1.2 GeV/c 2 12 , marking the most precisely measured quark mass and the most massive fundamental particle known to date. The consequent lifetime of the top quark in the SM of ≈ 5 ⋅ 10 − 25 absent ⋅ 5 superscript 10 25 \approx 5\cdot 10^{-25} s is extremely short, suggesting that it decays before hadronizing. This makes it the only quark that does not form bound states, allowing the study of an essentially bare quark with properties such as spin undisturbed by hadronization 13 . The measurement of top quark pair ( t ​ t ¯ 𝑡 ¯ 𝑡 t\overline{t} ) production probes our understanding of the strong interaction and predictions from perturbative QCD, while the decay of top quarks and the production of single top quarks reflect the electroweak interaction. Measuring other properties of the top quark, such as its electric charge, the helicity of the W 𝑊 W boson in t → W ​ b → 𝑡 𝑊 𝑏 t\to Wb decay, the branching fraction ℬ ​ ( t → W ​ b ) ℬ → 𝑡 𝑊 𝑏 {\cal B}(t\rightarrow Wb) , etc. , and comparing these with predictions of the SM is a very powerful tool in searching for new physics beyond the standard model. The top quark can also be used to constrain the mass range of the last yet to be observed particle of the standard model, the Higgs boson, because their masses and the mass of the W 𝑊 W boson are linked through radiative corrections 14 . The Higgs boson is a manifestation of the Higgs mechanism 15 , 16 , 17 , 18 , 19 , 20 , implemented in the standard model to provide the needed breaking of the electroweak symmetry to which the top quark may be intimately connected because of its large mass. Because of its fairly recent discovery, the top quarks properties have not yet been explored with the same scrutiny as those of the lighter quarks. However, in the ongoing data taking at Fermilabs Tevatron proton-antiproton collider, an integrated luminosity of more than 4 fb -1 has already been recorded by each of the collider experiments CDF and D0, corresponding to an increase of about a factor seventy relative to the data that was available for the discovery of the top quark. The new data can be used to refine previous measurements to higher precision that starts to become limited by systematic rather than statistical uncertainties. In addition, measurements that have never been performed become feasible, such as the first evidence for electroweak production of single top quarks and the consequent first direct measurement of the CKM matrix element | V t ​ b | subscript 𝑉 𝑡 𝑏 |V_{tb}| , recently published by D0 21 , 22 and CDF 23 . This article is intended to provide an overview of the current status of the top quark physics program pursued at the Tevatron. Results available until the LHC startup in September 2008 have been included, utilizing samples of data of up to 2.8 fb -1 in integrated luminosity. Previous reviews of the top quark are available in Refs. 24 , 25 , 26 , 27 , 28 . The outline of this article is as follows: The second chapter provides a brief introduction to the standard model, with emphasis on the special role played by the top quark. Chapter 3 describes production and decay modes for top quarks in the framework of the standard model. Chapter 4 outlines the experimental setup used for the measurements described in the following sections. Chapter 5 presents studies of the production of top quarks, including measurements of cross section that form the basis for other measurements of top quark characteristics. Chapter 6 elaborates on the different results for top quark decay properties, followed in Chapter 7 by a discussion of measurements of fundamental attributes of the top quark, such as its charge and mass. The final chapter (8) contains a brief summary of the achievements to date.","role":"user"},{"content":"Abstract: This review summarizes the program in the physics of the top quark being pursued at Fermilab's Tevatron proton–antiproton collider at a center-of-mass energy of 1.96 TeV. More than a decade after the discovery of the top quark at the two collider detectors CDF and D0, the Tevatron has been the only accelerator to produce top quarks and to study them directly. The Tevatron's increased luminosity and center-of-mass energy offer the possibility to scrutinize the properties of this heaviest fundamental particle through new measurements that were not feasible before, such as the first evidence for electroweak production of top quarks and the resulting direct constraints on the involved couplings. Better measurements of top quark properties provide more stringent tests of predictions from the SM of elementary particle physics. In particular, the improvement in measurements of the mass of the top quark, with the latest uncertainty of 0.7% marking the most precisely measured quark mass to date, further constrains the prediction of the mass of the still to be discovered Higgs boson.","role":"assistant"}]

Non-Invasive Glucose Monitoring Techniques: A Review and current trends

I Introduction Diabetes mellitus is a medical condition in which the body does not adequately produce the quantity or quality of insulin needed to maintain a normal circulating blood glucose. Insulin is a hormone that enables glucose (sugar) to enter the body’s cells to be used for energy. Two types of diabetes are common. Type - I is also known as Insulin Dependent Diabetes Mellitus (IDDM) and accounts for 5-10% of all cases. Type - II or Non-Insulin Dependent Diabetes Mellitus (NIDDM) occurs in 90-95% of the diabetic population. IDDM occurs in childhood. It requires insulin doses to maintain life, in addition to healthy practices. Frequent self-monitoring of blood glucose is crucial for effective treatment and reduction of the morbidity and mortality of diabetes. Unmonitored diabetes can lead to severe complications over time, including blindness, kidney failure, heart failure, and peripheral neuropathy associated with limb pain, poor circulation, gangrene and subsequent amputation Davidson . These complications are largely due to poor glucose control. The Diabetes Care and Complications Trial (DCCT) demonstrate that more frequent monitoring of blood glucose and insulin levels could prevent many of the long term complications of diabetes Auxter . However, the conventional blood (finger stick) glucose testing & monitoring are painful, inconvenient due to disruption of daily life. Also, it causes fear of hyoglycemia resulting from tighter glucose control and may be difficult to perform in long term diabetic patients due to calluses on the fingers and poor circulation. A glucose measurement with following qualities, i) non-invasive, ii) non-contact, iii) fast measurement capability, iv) painless measurements, v) convenience for glucose monitoring and vi) cost effective which could provide adequate control and greatly reduce the complications seen in these patients. At present, the simplest and less painful method for glucose measurements are done by pricking a finger and extracting a drop of blood ( 50 ​ μ ​ l / d ​ l 50 𝜇 𝑙 𝑑 𝑙 50\mu l/dl ) which is applied to a test strip composed of chemicals sensitive to the glucose in the blood sample. An optical meter (glucometer) is used to analyze the blood sample and gives a numerical glucose reading. This paper refers the widely used monitoring techniques of blood glucose monitoring. A summary of the techniques discussed are given as a chart in figure 1. Figure 1: Methods of glucose measurement techniques. I.1 Glucose in human body D-glucose is a molecule with the chemical formula C 6 H 12 O 6 . in human body, food is converted into sugar and provides energy to all tissues and organs through blood circulation. In terms of its chemical composition, human blood sugar consists of D-glucose that exists mainly in the water base of blood plasma. The daily variation of glucose concentration in the human body is in the range of 60 - 160 mg/dl Auses . Arterial and capillary blood taken from the fingertip have an identical glucose content, while the glucose level of venous blood is lower than the corresponding arterial value (1 - 17 mg/dl in healthy subjects and up to 30 mg/dl in diabetic patients). Besides, blood glucose also exists in other biofluids such as intracellular fluid, interstitial fluid, humour, saliva, sweet and urine. Researchers have established that, in the steady state condition, the glucose level in the intracellular and interstitial fluid is identical with the concentration of glucose in the blood. It is also known that the glucose level in humour correlates strongly with the glucose content of blood, while the glucose level in saliva, sweet and urine does not. It is the basic energy source for cellular metabolism. Glucose concentration varies between different anatomic regions and in different parts of the blood circulation. Figure 2: Anomers of D-glucose in an aqueous solution Glucose permeates red blood cells (RBCs) via passive diffusion, which is supported by 55-kD-glycoprotein Eggins . RBCs work as a buffer in blood to control the plasma glucose concentration Shults . It is suspected that D-glucose does not need a transporter to move between blood plasma and interstitial fluid (ISF). This process is thought to be driven by diffusion gradients. When the glucose concentration of plasma increases, the initial response is water movement from ISF to plasma; then the glucose diffuses into the ISF, where it is used as an energy source by cells. Because of these gradients, the glucose concentration in tissues is not a constant. The glucose concentration is higher in arterial blood than in venous blood, and the venous glucose concentration depends on the arterial blood flow rate and the rate of glucose uptake by the tissue Eggins . Unlike between blood plasma and ISF, there are many glucose transporters through the cell membrane. The transport process of glucose is called passive-mediated or facilitated diffusion. In many organelles these transporters are stimulated by insulin. The distribution of the transporters varies between organelles. The transport rate of glucose into cells is possibly limited by glucose transporters or by glucose phosphorylation Shults . In muscle cells, the glucose is phosphorylated rapidly after the intake into the cell Shults . Although many body fluids and tissues have been studied for non-invasive glucose sensing, in the best case scenario, the blood glucose concentration would be measured directly from the blood vessels. To make a diagnosis of Diabetes Mellitus, the vein plasma glucose should be measured Fischer . I.2 Glucose oxidation methods Traditional glucose determination is based on the glucose oxidation reaction, catalyzed by glucose oxidase (GOD). It may be described by glucose + O 2 + H 2 ​ O ​ G O D → ​ H 2 ​ O 2 + gluconic acid \hbox{glucose}+O_{2}+H_{2}O\;\;\underrightarrow{\;\;\;\;GOD\;\;\;\;}\;\;\;H_{2}O_{2}+\hbox{gluconic acid} (1) Glucose oxidase catalyzes the oxidation of β 𝛽 \beta -D-glucose to D-gluconic acid and hydrogen peroxide. It is highly specific for β 𝛽 \beta -D-glucose and does not act on α 𝛼 \alpha -D-glucose. Its major use is in the determination of free glucose in body fluids. Although specific for β 𝛽 \beta -D-glucose, glucose oxidase can be used to measure the total amount of glucose. This is because, following the consumption of β 𝛽 \beta -glucose, α 𝛼 \alpha -glucose at equilibrium is converted to the β 𝛽 \beta -form by mutarotation. The consumed oxygen or the ensuing production of gluconic acid or hydrogen peroxide (H 2 O 2 ) is in direct proportion to the glucose content. The glucose oxidase method is characterized by high sensitivity, accuracy and reliability. Traditional electrochemical methods such as potentiometry or amperometry can be used to determine the glucose content during the glucose oxidation reaction. Very recently, the results obtained on the glycemia measurement by an indirect method are presented. amar The measurement method uses a sensor to electromagnetic coupling based on loss currents. This sensor allows the detection of the blood s glucose level through the variation of the sensor impedance that is depend on the dielectric parameters of blood, in particular the conductivity. The in vitro results presented a standard deviation of 2.1 mV, an average error of 1.8 mV, and a maximum variation of 3.6 mV equivalents to a maximum error of ± plus-or-minus \pm 197 mg dl -1 on the concentration of D + glucose compared to the line of tendency.

Diabetes mellitus is a complex group of syndromes that have in common a disturbance in the body's use of glucose, resulting in an elevated blood sugar. Once detected, sugar diabetes can be controlled by an appropriate regimen that should include diet therapy, a weight reduction program for those persons who are overweight, a program of exercise and insulin injections or oral drugs to lower blood glucose. Blood glucose monitoring by the patient and the physician is an important aspect in the control of the devastating complications (heart disease, blindness, kidney failure or amputations) due to the disease. Intensive therapy and frequent glucose testing has numerous benefits. With ever improving advances in diagnostic technology, the race for the next generation of bloodless, painless, accurate glucose instruments has begun. In this paper, we reviewed various methods, techniques and approaches successfully demonstrated for measuring or monitoring blood glucose. Invasive, minimally invasive and noninvasive techniques available in literature are summarised.

[{"content":"Help me to generate the abstract of a survey paper given the title:*Non-Invasive Glucose Monitoring Techniques: A Review and current trends* and the introduction:I Introduction Diabetes mellitus is a medical condition in which the body does not adequately produce the quantity or quality of insulin needed to maintain a normal circulating blood glucose. Insulin is a hormone that enables glucose (sugar) to enter the bodys cells to be used for energy. Two types of diabetes are common. Type - I is also known as Insulin Dependent Diabetes Mellitus (IDDM) and accounts for 5-10% of all cases. Type - II or Non-Insulin Dependent Diabetes Mellitus (NIDDM) occurs in 90-95% of the diabetic population. IDDM occurs in childhood. It requires insulin doses to maintain life, in addition to healthy practices. Frequent self-monitoring of blood glucose is crucial for effective treatment and reduction of the morbidity and mortality of diabetes. Unmonitored diabetes can lead to severe complications over time, including blindness, kidney failure, heart failure, and peripheral neuropathy associated with limb pain, poor circulation, gangrene and subsequent amputation Davidson . These complications are largely due to poor glucose control. The Diabetes Care and Complications Trial (DCCT) demonstrate that more frequent monitoring of blood glucose and insulin levels could prevent many of the long term complications of diabetes Auxter . However, the conventional blood (finger stick) glucose testing & monitoring are painful, inconvenient due to disruption of daily life. Also, it causes fear of hyoglycemia resulting from tighter glucose control and may be difficult to perform in long term diabetic patients due to calluses on the fingers and poor circulation. A glucose measurement with following qualities, i) non-invasive, ii) non-contact, iii) fast measurement capability, iv) painless measurements, v) convenience for glucose monitoring and vi) cost effective which could provide adequate control and greatly reduce the complications seen in these patients. At present, the simplest and less painful method for glucose measurements are done by pricking a finger and extracting a drop of blood ( 50 ​ μ ​ l / d ​ l 50 𝜇 𝑙 𝑑 𝑙 50\mu l/dl ) which is applied to a test strip composed of chemicals sensitive to the glucose in the blood sample. An optical meter (glucometer) is used to analyze the blood sample and gives a numerical glucose reading. This paper refers the widely used monitoring techniques of blood glucose monitoring. A summary of the techniques discussed are given as a chart in figure 1. Figure 1: Methods of glucose measurement techniques. I.1 Glucose in human body D-glucose is a molecule with the chemical formula C 6 H 12 O 6 . in human body, food is converted into sugar and provides energy to all tissues and organs through blood circulation. In terms of its chemical composition, human blood sugar consists of D-glucose that exists mainly in the water base of blood plasma. The daily variation of glucose concentration in the human body is in the range of 60 - 160 mg/dl Auses . Arterial and capillary blood taken from the fingertip have an identical glucose content, while the glucose level of venous blood is lower than the corresponding arterial value (1 - 17 mg/dl in healthy subjects and up to 30 mg/dl in diabetic patients). Besides, blood glucose also exists in other biofluids such as intracellular fluid, interstitial fluid, humour, saliva, sweet and urine. Researchers have established that, in the steady state condition, the glucose level in the intracellular and interstitial fluid is identical with the concentration of glucose in the blood. It is also known that the glucose level in humour correlates strongly with the glucose content of blood, while the glucose level in saliva, sweet and urine does not. It is the basic energy source for cellular metabolism. Glucose concentration varies between different anatomic regions and in different parts of the blood circulation. Figure 2: Anomers of D-glucose in an aqueous solution Glucose permeates red blood cells (RBCs) via passive diffusion, which is supported by 55-kD-glycoprotein Eggins . RBCs work as a buffer in blood to control the plasma glucose concentration Shults . It is suspected that D-glucose does not need a transporter to move between blood plasma and interstitial fluid (ISF). This process is thought to be driven by diffusion gradients. When the glucose concentration of plasma increases, the initial response is water movement from ISF to plasma; then the glucose diffuses into the ISF, where it is used as an energy source by cells. Because of these gradients, the glucose concentration in tissues is not a constant. The glucose concentration is higher in arterial blood than in venous blood, and the venous glucose concentration depends on the arterial blood flow rate and the rate of glucose uptake by the tissue Eggins . Unlike between blood plasma and ISF, there are many glucose transporters through the cell membrane. The transport process of glucose is called passive-mediated or facilitated diffusion. In many organelles these transporters are stimulated by insulin. The distribution of the transporters varies between organelles. The transport rate of glucose into cells is possibly limited by glucose transporters or by glucose phosphorylation Shults . In muscle cells, the glucose is phosphorylated rapidly after the intake into the cell Shults . Although many body fluids and tissues have been studied for non-invasive glucose sensing, in the best case scenario, the blood glucose concentration would be measured directly from the blood vessels. To make a diagnosis of Diabetes Mellitus, the vein plasma glucose should be measured Fischer . I.2 Glucose oxidation methods Traditional glucose determination is based on the glucose oxidation reaction, catalyzed by glucose oxidase (GOD). It may be described by glucose + O 2 + H 2 ​ O ​ G O D → ​ H 2 ​ O 2 + gluconic acid \hbox{glucose}+O_{2}+H_{2}O\;\;\underrightarrow{\;\;\;\;GOD\;\;\;\;}\;\;\;H_{2}O_{2}+\hbox{gluconic acid} (1) Glucose oxidase catalyzes the oxidation of β 𝛽 \beta -D-glucose to D-gluconic acid and hydrogen peroxide. It is highly specific for β 𝛽 \beta -D-glucose and does not act on α 𝛼 \alpha -D-glucose. Its major use is in the determination of free glucose in body fluids. Although specific for β 𝛽 \beta -D-glucose, glucose oxidase can be used to measure the total amount of glucose. This is because, following the consumption of β 𝛽 \beta -glucose, α 𝛼 \alpha -glucose at equilibrium is converted to the β 𝛽 \beta -form by mutarotation. The consumed oxygen or the ensuing production of gluconic acid or hydrogen peroxide (H 2 O 2 ) is in direct proportion to the glucose content. The glucose oxidase method is characterized by high sensitivity, accuracy and reliability. Traditional electrochemical methods such as potentiometry or amperometry can be used to determine the glucose content during the glucose oxidation reaction. Very recently, the results obtained on the glycemia measurement by an indirect method are presented. amar The measurement method uses a sensor to electromagnetic coupling based on loss currents. This sensor allows the detection of the blood s glucose level through the variation of the sensor impedance that is depend on the dielectric parameters of blood, in particular the conductivity. The in vitro results presented a standard deviation of 2.1 mV, an average error of 1.8 mV, and a maximum variation of 3.6 mV equivalents to a maximum error of ± plus-or-minus \pm 197 mg dl -1 on the concentration of D + glucose compared to the line of tendency.","role":"user"},{"content":"Abstract: Diabetes mellitus is a complex group of syndromes that have in common a disturbance in the body's use of glucose, resulting in an elevated blood sugar. Once detected, sugar diabetes can be controlled by an appropriate regimen that should include diet therapy, a weight reduction program for those persons who are overweight, a program of exercise and insulin injections or oral drugs to lower blood glucose. Blood glucose monitoring by the patient and the physician is an important aspect in the control of the devastating complications (heart disease, blindness, kidney failure or amputations) due to the disease. Intensive therapy and frequent glucose testing has numerous benefits. With ever improving advances in diagnostic technology, the race for the next generation of bloodless, painless, accurate glucose instruments has begun. In this paper, we reviewed various methods, techniques and approaches successfully demonstrated for measuring or monitoring blood glucose. Invasive, minimally invasive and noninvasive techniques available in literature are summarised.","role":"assistant"}]

A survey of combinatorial aspects in the topology of complex hyperplane arrangements

1. Introduction By a hyperplane arrangement 𝒜 𝒜 {\mathcal{A}} we understand in what follows either a collection { H 1 , … , H n } subscript 𝐻 1 … subscript 𝐻 𝑛 \{H_{1},\dots,H_{n}\} of codimension one subspaces in a finite dimensional vector space V 𝑉 V or the subspace of the ambient space V 𝑉 V given by their union. We consider here only arrangements in complex affine spaces. The starting point in the study of complex hyperplane arrangements was the braid arrangement, 𝒜 = { H i ​ j = Ker ⁡ ( z i − z j ) | 1 ≤ i ≠ j ≤ n } ⊂ ℂ n 𝒜 conditional-set subscript 𝐻 𝑖 𝑗 Ker subscript 𝑧 𝑖 subscript 𝑧 𝑗 1 𝑖 𝑗 𝑛 superscript ℂ 𝑛 {\mathcal{A}}=\{H_{ij}=\operatorname{Ker}(z_{i}-z_{j})\;|\;1\leq i\neq j\leq n\}\subset\mathbb{C}^{n} . The results obtained for the complement M 𝒜 = { z ∈ ℂ n | z i ≠ z j ​ ∀ i ≠ j } subscript 𝑀 𝒜 conditional-set 𝑧 superscript ℂ 𝑛 subscript 𝑧 𝑖 subscript 𝑧 𝑗 for-all 𝑖 𝑗 M_{{\mathcal{A}}}=\{z\in\mathbb{C}^{n}\;|\;z_{i}\neq z_{j}\;\forall i\neq j\} – factorization of the Poincaré polynomial and the presentation of the cohomology algebra in terms of generators and relations (Arnold [ 2 ] ), or the K ​ ( π , 1 ) 𝐾 𝜋 1 K(\pi,1) property (Fadell-Neuwirth [ 16 ] ) – initiated typical directions of investigation in arrangement theory. Resolving a conjecture of Arnold, Brieskorn showed that the cohomology ring of the complement, M 𝒜 = V ∖ ⋃ i = 1 n H i subscript 𝑀 𝒜 𝑉 superscript subscript 𝑖 1 𝑛 subscript 𝐻 𝑖 M_{{\mathcal{A}}}=V\setminus\bigcup_{i=1}^{n}H_{i} , of an arbitrary complex arrangement 𝒜 𝒜 {\mathcal{A}} is generated by the classes of the closed 1 1 1 -forms 1 2 ​ π ​ − 1 ​ ( d ​ α H α H ) 1 2 𝜋 1 𝑑 subscript 𝛼 𝐻 subscript 𝛼 𝐻 \frac{1}{2\pi\sqrt{-1}}(\frac{d\alpha_{H}}{\alpha_{H}}) , where α H = 0 subscript 𝛼 𝐻 0 \alpha_{H}=0 are the defining equations for the hyperplanes H ∈ 𝒜 𝐻 𝒜 H\in{\mathcal{A}} . An emblematic result for the field is the theorem of Orlik and Solomon [ 39 ] , that proves an isomorphism between the cohomology ring of the complement, H ∗ ​ ( M 𝒜 ) superscript 𝐻 subscript 𝑀 𝒜 H^{*}(M_{\mathcal{A}}) , and the Orlik-Solomon algebra of the arrangement, A ∗ ​ ( 𝒜 ) superscript 𝐴 𝒜 A^{*}({\mathcal{A}}) , which is determined by the lattice ℒ ​ ( 𝒜 ) = { ⋂ H ∈ ℬ H | ℬ ⊂ 𝒜 } ℒ 𝒜 conditional-set subscript 𝐻 ℬ 𝐻 ℬ 𝒜 \mathcal{L}({\mathcal{A}})=\{\bigcap_{H\in{{\mathcal{B}}}}H\;|\;{\mathcal{B}}\subset{\mathcal{A}}\} , ordered by reverse inclusion. We say that a topological invariant (property) of the complement M 𝒜 subscript 𝑀 𝒜 M_{{\mathcal{A}}} is combinatorial if it depends only on the intersection lattice ℒ ​ ( 𝒜 ) ℒ 𝒜 \mathcal{L}({\mathcal{A}}) . We have chosen to survey here several natural, interesting topological facets of the combinatorial determination problem (based on personal taste, and with no intention of exhausting the subject). Furthermore, ample existing expository work covers various aspects of the field of hyperplane arrangements, see for instance [ 21 ] , [ 22 ] , [ 17 ] , [ 50 ] , [ 54 ] . A brief discussion of our main topics follows. Projection onto the first coordinates, ℂ n → ℂ n − 1 → superscript ℂ 𝑛 superscript ℂ 𝑛 1 \mathbb{C}^{n}\to\mathbb{C}^{n-1} , restricts to a topological fibration between the corresponding braid complements, with fiber ℂ ∖ { n − 1 ​ points } ℂ 𝑛 1 points \mathbb{C}\setminus\{n-1\;\text{points}\} . Arrangements whose complement may be obtained from the simplest arrangement, { 0 } ⊂ ℂ 0 ℂ \{0\}\subset\mathbb{C} , by iterating a similar fibration construction, were introduced by Falk-Randell [ 20 ] and called fiber-type . A remarkable result (due to Terao [ 51 ] and described in Section 2 ) says that this is equivalent to ℒ ​ ( 𝒜 ) ℒ 𝒜 \mathcal{L}({\mathcal{A}}) being supersolvable , in the sense of Stanley [ 48 ] . In Section 3 , we examine the converse of the Orlik-Solomon theorem: the cohomology ring H ∗ ​ ( M 𝒜 ) superscript 𝐻 subscript 𝑀 𝒜 H^{*}(M_{\mathcal{A}}) , together with its natural 1 1 1 -marking (in the sense of Choudary-Dimca-Papadima [ 5 ] ), determines the lattice ℒ ​ ( 𝒜 ) ℒ 𝒜 \mathcal{L}({\mathcal{A}}) . Section 4 treats the homotopy groups, { π i ​ ( M 𝒜 ) } i ≥ 1 subscript subscript 𝜋 𝑖 subscript 𝑀 𝒜 𝑖 1 \{\pi_{i}(M_{{\mathcal{A}}})\}_{i\geq 1} . As shown by Rybnikov [ 44 ] , the fundamental group π 1 ​ ( M 𝒜 ) subscript 𝜋 1 subscript 𝑀 𝒜 \pi_{1}(M_{{\mathcal{A}}}) is not combinatorial (see also [ 3 ] , [ 4 ] ). Nevertheless, its rational associated graded Lie algebra , gr ∗ ⁡ ( π 1 ​ ( M 𝒜 ) ) ⊗ ℚ tensor-product subscript gr subscript 𝜋 1 subscript 𝑀 𝒜 ℚ \operatorname{gr}_{*}(\pi_{1}(M_{{\mathcal{A}}}))\otimes\mathbb{Q} , is combinatorial, by a result of Kohno [ 29 ] . The nilpotence of π 1 ​ ( M 𝒜 ) subscript 𝜋 1 subscript 𝑀 𝒜 \pi_{1}(M_{{\mathcal{A}}}) is equivalent to the property of ℒ ​ ( 𝒜 ) ℒ 𝒜 \mathcal{L}({\mathcal{A}}) of being in general position in codimension 2 2 2 ; this fact is the first step in the proof of a famous conjecture of Zariski. The K ​ ( π , 1 ) 𝐾 𝜋 1 K(\pi,1) property of M 𝒜 subscript 𝑀 𝒜 M_{{\mathcal{A}}} means by definition the vanishing of the higher homotopy groups π i ​ ( M 𝒜 ) subscript 𝜋 𝑖 subscript 𝑀 𝒜 \pi_{i}(M_{{\mathcal{A}}}) , for i > 1 𝑖 1 i>1 . It is easy to see that fiber-type implies K ​ ( π , 1 ) 𝐾 𝜋 1 K(\pi,1) , but the converse is not true, as shown by an example of Falk [ 18 ] . A basic result of Hattori [ 25 ] implies that the K ​ ( π , 1 ) 𝐾 𝜋 1 K(\pi,1) property does not hold, if the lattice ℒ ​ ( 𝒜 ) ℒ 𝒜 \mathcal{L}({\mathcal{A}}) is generic . Jambu-Papadima introduced in [ 26 ] the (combinatorially defined) hypersolvable class of arrangements, which (strictly) contains both the fiber-type and the generic arrangements. They also proved that the K ​ ( π , 1 ) 𝐾 𝜋 1 K(\pi,1) property is combinatorial, within the hypersolvable class. In general, this question is open. Section 5 is devoted to the hypersolvable class. For the split solvable subclass, we highlight a remarkable perfect match, namely the fact that the lattice ℒ ​ ( 𝒜 ) ℒ 𝒜 \mathcal{L}({\mathcal{A}}) and the homeomorphism type of the complement M 𝒜 subscript 𝑀 𝒜 M_{{\mathcal{A}}} determine each other [ 5 ] . Rank 1 1 1 complex locally constant sheaves (alias local systems ) on M 𝒜 subscript 𝑀 𝒜 M_{{\mathcal{A}}} may also be viewed as characters ρ 𝜌 \rho belonging to the character torus, 𝕋 π 1 ​ ( M 𝒜 ) = ( ℂ ∗ ) n subscript 𝕋 subscript 𝜋 1 subscript 𝑀 𝒜 superscript superscript ℂ 𝑛 \mathbb{T}_{\pi_{1}(M_{{\mathcal{A}}})}=(\mathbb{C}^{*})^{n} . Combinatorial determinacy of the twisted cohomology H ∗ ​ ( M 𝒜 , ℂ ρ ) superscript 𝐻 subscript 𝑀 𝒜 subscript ℂ 𝜌 H^{*}(M_{{\mathcal{A}}},{}_{\rho}\mathbb{C}) is a major open problem in the field. We review in Section 6 several known facts in this direction. The Milnor fiber of a degree n 𝑛 n homogeneous polynomial f ∈ ℂ ​ [ x 1 , … , x l ] 𝑓 ℂ subscript 𝑥 1 … subscript 𝑥 𝑙 f\in\mathbb{C}[x_{1},\dots,x_{l}] is an important object in singularity theory, introduced by Milnor in [ 37 ] . It is defined by F f := f − 1 ​ ( 1 ) assign subscript 𝐹 𝑓 superscript 𝑓 1 1 F_{f}:=f^{-1}(1) , and comes endowed with a natural automorphism, induced by multiplication in ℂ l superscript ℂ 𝑙 \mathbb{C}^{l} by a primitive n 𝑛 n -root of unity. The induced action on H ∗ ​ ( F f , ℚ ) subscript 𝐻 subscript 𝐹 𝑓 ℚ H_{*}(F_{f},\mathbb{Q}) is called the algebraic monodromy . Its study is a very active area of research in singularity theory. In arrangement theory, the Milnor fiber of 𝒜 𝒜 {\mathcal{A}} , F 𝒜 subscript 𝐹 𝒜 F_{{\mathcal{A}}} , is associated to the defining polynomial of 𝒜 𝒜 {\mathcal{A}} , Q ​ ( 𝒜 ) := Π H ∈ 𝒜 ​ α H assign 𝑄 𝒜 subscript Π 𝐻 𝒜 subscript 𝛼 𝐻 Q({\mathcal{A}}):=\Pi_{H\in{\mathcal{A}}}\alpha_{H} . A long-standing open problem in this context is the combinatorial determination of the algebraic monodromy action on H 1 ​ ( F 𝒜 , ℚ ) subscript 𝐻 1 subscript 𝐹 𝒜 ℚ H_{1}(F_{{\mathcal{A}}},\mathbb{Q}) . We address this question in Section 7 . Following [ 36 ] , we present an affirmative answer, in terms of combinatorial objects defined in positive characteristic, for subarrangements of type B 𝐵 B Coxeter arrangements, parametrized by graphs . We also emphasize a general relation between algebraic monodromy and twisted cohomology, which is key to the proof.

We survey interactions between the topology and the combinatorics of complex hyperplane arrangements. Without claiming to be exhaustive, we examine in this setting combinatorial aspects of fundamental groups, associated graded Lie algebras, higher homotopy groups, cohomology rings, twisted homology with rank 1 complex coefficients, and Milnor fibers.

[{"content":"Help me to generate the abstract of a survey paper given the title:*A survey of combinatorial aspects in the topology of complex hyperplane arrangements* and the introduction:1. Introduction By a hyperplane arrangement 𝒜 𝒜 {\mathcal{A}} we understand in what follows either a collection { H 1 , … , H n } subscript 𝐻 1 … subscript 𝐻 𝑛 \{H_{1},\dots,H_{n}\} of codimension one subspaces in a finite dimensional vector space V 𝑉 V or the subspace of the ambient space V 𝑉 V given by their union. We consider here only arrangements in complex affine spaces. The starting point in the study of complex hyperplane arrangements was the braid arrangement, 𝒜 = { H i ​ j = Ker ⁡ ( z i − z j ) | 1 ≤ i ≠ j ≤ n } ⊂ ℂ n 𝒜 conditional-set subscript 𝐻 𝑖 𝑗 Ker subscript 𝑧 𝑖 subscript 𝑧 𝑗 1 𝑖 𝑗 𝑛 superscript ℂ 𝑛 {\mathcal{A}}=\{H_{ij}=\operatorname{Ker}(z_{i}-z_{j})\;|\;1\leq i\neq j\leq n\}\subset\mathbb{C}^{n} . The results obtained for the complement M 𝒜 = { z ∈ ℂ n | z i ≠ z j ​ ∀ i ≠ j } subscript 𝑀 𝒜 conditional-set 𝑧 superscript ℂ 𝑛 subscript 𝑧 𝑖 subscript 𝑧 𝑗 for-all 𝑖 𝑗 M_{{\mathcal{A}}}=\{z\in\mathbb{C}^{n}\;|\;z_{i}\neq z_{j}\;\forall i\neq j\} – factorization of the Poincaré polynomial and the presentation of the cohomology algebra in terms of generators and relations (Arnold 2 ), or the K ​ ( π , 1 ) 𝐾 𝜋 1 K(\pi,1) property (Fadell-Neuwirth 16 ) – initiated typical directions of investigation in arrangement theory. Resolving a conjecture of Arnold, Brieskorn showed that the cohomology ring of the complement, M 𝒜 = V ∖ ⋃ i = 1 n H i subscript 𝑀 𝒜 𝑉 superscript subscript 𝑖 1 𝑛 subscript 𝐻 𝑖 M_{{\mathcal{A}}}=V\setminus\bigcup_{i=1}^{n}H_{i} , of an arbitrary complex arrangement 𝒜 𝒜 {\mathcal{A}} is generated by the classes of the closed 1 1 1 -forms 1 2 ​ π ​ − 1 ​ ( d ​ α H α H ) 1 2 𝜋 1 𝑑 subscript 𝛼 𝐻 subscript 𝛼 𝐻 \frac{1}{2\pi\sqrt{-1}}(\frac{d\alpha_{H}}{\alpha_{H}}) , where α H = 0 subscript 𝛼 𝐻 0 \alpha_{H}=0 are the defining equations for the hyperplanes H ∈ 𝒜 𝐻 𝒜 H\in{\mathcal{A}} . An emblematic result for the field is the theorem of Orlik and Solomon 39 , that proves an isomorphism between the cohomology ring of the complement, H ∗ ​ ( M 𝒜 ) superscript 𝐻 subscript 𝑀 𝒜 H^{*}(M_{\mathcal{A}}) , and the Orlik-Solomon algebra of the arrangement, A ∗ ​ ( 𝒜 ) superscript 𝐴 𝒜 A^{*}({\mathcal{A}}) , which is determined by the lattice ℒ ​ ( 𝒜 ) = { ⋂ H ∈ ℬ H | ℬ ⊂ 𝒜 } ℒ 𝒜 conditional-set subscript 𝐻 ℬ 𝐻 ℬ 𝒜 \mathcal{L}({\mathcal{A}})=\{\bigcap_{H\in{{\mathcal{B}}}}H\;|\;{\mathcal{B}}\subset{\mathcal{A}}\} , ordered by reverse inclusion. We say that a topological invariant (property) of the complement M 𝒜 subscript 𝑀 𝒜 M_{{\mathcal{A}}} is combinatorial if it depends only on the intersection lattice ℒ ​ ( 𝒜 ) ℒ 𝒜 \mathcal{L}({\mathcal{A}}) . We have chosen to survey here several natural, interesting topological facets of the combinatorial determination problem (based on personal taste, and with no intention of exhausting the subject). Furthermore, ample existing expository work covers various aspects of the field of hyperplane arrangements, see for instance 21 , 22 , 17 , 50 , 54 . A brief discussion of our main topics follows. Projection onto the first coordinates, ℂ n → ℂ n − 1 → superscript ℂ 𝑛 superscript ℂ 𝑛 1 \mathbb{C}^{n}\to\mathbb{C}^{n-1} , restricts to a topological fibration between the corresponding braid complements, with fiber ℂ ∖ { n − 1 ​ points } ℂ 𝑛 1 points \mathbb{C}\setminus\{n-1\;\text{points}\} . Arrangements whose complement may be obtained from the simplest arrangement, { 0 } ⊂ ℂ 0 ℂ \{0\}\subset\mathbb{C} , by iterating a similar fibration construction, were introduced by Falk-Randell 20 and called fiber-type . A remarkable result (due to Terao 51 and described in Section 2 ) says that this is equivalent to ℒ ​ ( 𝒜 ) ℒ 𝒜 \mathcal{L}({\mathcal{A}}) being supersolvable , in the sense of Stanley 48 . In Section 3 , we examine the converse of the Orlik-Solomon theorem: the cohomology ring H ∗ ​ ( M 𝒜 ) superscript 𝐻 subscript 𝑀 𝒜 H^{*}(M_{\mathcal{A}}) , together with its natural 1 1 1 -marking (in the sense of Choudary-Dimca-Papadima 5 ), determines the lattice ℒ ​ ( 𝒜 ) ℒ 𝒜 \mathcal{L}({\mathcal{A}}) . Section 4 treats the homotopy groups, { π i ​ ( M 𝒜 ) } i ≥ 1 subscript subscript 𝜋 𝑖 subscript 𝑀 𝒜 𝑖 1 \{\pi_{i}(M_{{\mathcal{A}}})\}_{i\geq 1} . As shown by Rybnikov 44 , the fundamental group π 1 ​ ( M 𝒜 ) subscript 𝜋 1 subscript 𝑀 𝒜 \pi_{1}(M_{{\mathcal{A}}}) is not combinatorial (see also 3 , 4 ). Nevertheless, its rational associated graded Lie algebra , gr ∗ ⁡ ( π 1 ​ ( M 𝒜 ) ) ⊗ ℚ tensor-product subscript gr subscript 𝜋 1 subscript 𝑀 𝒜 ℚ \operatorname{gr}_{*}(\pi_{1}(M_{{\mathcal{A}}}))\otimes\mathbb{Q} , is combinatorial, by a result of Kohno 29 . The nilpotence of π 1 ​ ( M 𝒜 ) subscript 𝜋 1 subscript 𝑀 𝒜 \pi_{1}(M_{{\mathcal{A}}}) is equivalent to the property of ℒ ​ ( 𝒜 ) ℒ 𝒜 \mathcal{L}({\mathcal{A}}) of being in general position in codimension 2 2 2 ; this fact is the first step in the proof of a famous conjecture of Zariski. The K ​ ( π , 1 ) 𝐾 𝜋 1 K(\pi,1) property of M 𝒜 subscript 𝑀 𝒜 M_{{\mathcal{A}}} means by definition the vanishing of the higher homotopy groups π i ​ ( M 𝒜 ) subscript 𝜋 𝑖 subscript 𝑀 𝒜 \pi_{i}(M_{{\mathcal{A}}}) , for i > 1 𝑖 1 i>1 . It is easy to see that fiber-type implies K ​ ( π , 1 ) 𝐾 𝜋 1 K(\pi,1) , but the converse is not true, as shown by an example of Falk 18 . A basic result of Hattori 25 implies that the K ​ ( π , 1 ) 𝐾 𝜋 1 K(\pi,1) property does not hold, if the lattice ℒ ​ ( 𝒜 ) ℒ 𝒜 \mathcal{L}({\mathcal{A}}) is generic . Jambu-Papadima introduced in 26 the (combinatorially defined) hypersolvable class of arrangements, which (strictly) contains both the fiber-type and the generic arrangements. They also proved that the K ​ ( π , 1 ) 𝐾 𝜋 1 K(\pi,1) property is combinatorial, within the hypersolvable class. In general, this question is open. Section 5 is devoted to the hypersolvable class. For the split solvable subclass, we highlight a remarkable perfect match, namely the fact that the lattice ℒ ​ ( 𝒜 ) ℒ 𝒜 \mathcal{L}({\mathcal{A}}) and the homeomorphism type of the complement M 𝒜 subscript 𝑀 𝒜 M_{{\mathcal{A}}} determine each other 5 . Rank 1 1 1 complex locally constant sheaves (alias local systems ) on M 𝒜 subscript 𝑀 𝒜 M_{{\mathcal{A}}} may also be viewed as characters ρ 𝜌 \rho belonging to the character torus, 𝕋 π 1 ​ ( M 𝒜 ) = ( ℂ ∗ ) n subscript 𝕋 subscript 𝜋 1 subscript 𝑀 𝒜 superscript superscript ℂ 𝑛 \mathbb{T}_{\pi_{1}(M_{{\mathcal{A}}})}=(\mathbb{C}^{*})^{n} . Combinatorial determinacy of the twisted cohomology H ∗ ​ ( M 𝒜 , ℂ ρ ) superscript 𝐻 subscript 𝑀 𝒜 subscript ℂ 𝜌 H^{*}(M_{{\mathcal{A}}},{}_{\rho}\mathbb{C}) is a major open problem in the field. We review in Section 6 several known facts in this direction. The Milnor fiber of a degree n 𝑛 n homogeneous polynomial f ∈ ℂ ​ x 1 , … , x l 𝑓 ℂ subscript 𝑥 1 … subscript 𝑥 𝑙 f\in\mathbb{C}x_{1},\dots,x_{l} is an important object in singularity theory, introduced by Milnor in 37 . It is defined by F f := f − 1 ​ ( 1 ) assign subscript 𝐹 𝑓 superscript 𝑓 1 1 F_{f}:=f^{-1}(1) , and comes endowed with a natural automorphism, induced by multiplication in ℂ l superscript ℂ 𝑙 \mathbb{C}^{l} by a primitive n 𝑛 n -root of unity. The induced action on H ∗ ​ ( F f , ℚ ) subscript 𝐻 subscript 𝐹 𝑓 ℚ H_{*}(F_{f},\mathbb{Q}) is called the algebraic monodromy . Its study is a very active area of research in singularity theory. In arrangement theory, the Milnor fiber of 𝒜 𝒜 {\mathcal{A}} , F 𝒜 subscript 𝐹 𝒜 F_{{\mathcal{A}}} , is associated to the defining polynomial of 𝒜 𝒜 {\mathcal{A}} , Q ​ ( 𝒜 ) := Π H ∈ 𝒜 ​ α H assign 𝑄 𝒜 subscript Π 𝐻 𝒜 subscript 𝛼 𝐻 Q({\mathcal{A}}):=\Pi_{H\in{\mathcal{A}}}\alpha_{H} . A long-standing open problem in this context is the combinatorial determination of the algebraic monodromy action on H 1 ​ ( F 𝒜 , ℚ ) subscript 𝐻 1 subscript 𝐹 𝒜 ℚ H_{1}(F_{{\mathcal{A}}},\mathbb{Q}) . We address this question in Section 7 . Following 36 , we present an affirmative answer, in terms of combinatorial objects defined in positive characteristic, for subarrangements of type B 𝐵 B Coxeter arrangements, parametrized by graphs . We also emphasize a general relation between algebraic monodromy and twisted cohomology, which is key to the proof.","role":"user"},{"content":"Abstract: We survey interactions between the topology and the combinatorics of complex hyperplane arrangements. Without claiming to be exhaustive, we examine in this setting combinatorial aspects of fundamental groups, associated graded Lie algebras, higher homotopy groups, cohomology rings, twisted homology with rank 1 complex coefficients, and Milnor fibers.","role":"assistant"}]

Permutation Patterns: A survey of simple permutations

An interval of a permutation π 𝜋 \pi corresponds to a set of contiguous indices I = [ a , b ] 𝐼 𝑎 𝑏 I=[a,b] such that the set of values π ​ ( I ) = { π ​ ( i ) : i ∈ I } 𝜋 𝐼 conditional-set 𝜋 𝑖 𝑖 𝐼 \pi(I)=\{\pi(i):i\in I\} is also contiguous. Every permutation of length n 𝑛 n has intervals of lengths 0 0 , 1 1 1 and n 𝑛 n . If a permutation π 𝜋 \pi has no other intervals, then π 𝜋 \pi is said to be simple . For example, the permutation π = 28146357 𝜋 28146357 \pi=28146357 is not simple as witnessed by the non-trivial interval 4635 4635 4635 ( = π ​ ( 4 ) ​ π ​ ( 5 ) ​ π ​ ( 6 ) ​ π ​ ( 7 ) absent 𝜋 4 𝜋 5 𝜋 6 𝜋 7 =\pi(4)\pi(5)\pi(6)\pi(7) ), while σ = 51742683 𝜎 51742683 \sigma=51742683 is simple. 1 1 1 Simplicity may be defined for all relational structures; such structures have variously been called prime or indecomposable . While intervals of permutations have applications in biomathematics, particularly to genetic algorithms and the matching of gene sequences (see Corteel, Louchard, and Pemantle [ 21 ] for extensive references), simple permutations form the “building blocks” of permutation classes and have thus received intensive study in recent years. We will see in Section 3 the various ways in which simplicity plays a role in the study of permutation classes, but we begin this short survey by introducing the substitution decomposition in Subsection 1.1 , and thence by reviewing the structural and enumerative results of simple permutations themselves in Section 2 . The rest of this subsection will cover several basic definitions that we will require. Two finite sequences of the same length, α = a 1 ​ a 2 ​ ⋯ ​ a n 𝛼 subscript 𝑎 1 subscript 𝑎 2 ⋯ subscript 𝑎 𝑛 \alpha=a_{1}a_{2}\cdots a_{n} and β = b 1 ​ b 2 ​ ⋯ ​ b n 𝛽 subscript 𝑏 1 subscript 𝑏 2 ⋯ subscript 𝑏 𝑛 \beta=b_{1}b_{2}\cdots b_{n} , are said to be order isomorphic if, for all i , j 𝑖 𝑗 i,j , we have a i < a j subscript 𝑎 𝑖 subscript 𝑎 𝑗 a_{i}<a_{j} if and only if b i < b j subscript 𝑏 𝑖 subscript 𝑏 𝑗 b_{i}<b_{j} . As such, each sequence of distinct real numbers is order isomorphic to a unique permutation. Similarly, any given subsequence (or pattern ) of a permutation π 𝜋 \pi is order isomorphic to a smaller permutation, σ 𝜎 \sigma say, and such a subsequence is called a copy of σ 𝜎 \sigma in π 𝜋 \pi . We may also say that π 𝜋 \pi contains σ 𝜎 \sigma (or, in some texts, π 𝜋 \pi involves σ 𝜎 \sigma ) and write σ ≤ π 𝜎 𝜋 \sigma\leq\pi . If, on the other hand, π 𝜋 \pi does not contain a copy of some given σ 𝜎 \sigma , then π 𝜋 \pi is said to avoid σ 𝜎 \sigma . For example, π = 918572346 𝜋 918572346 \pi=918572346 contains 51342 51342 51342 because of the subsequence 91572 91572 91572 ( = π ​ ( 1 ) ​ π ​ ( 2 ) ​ π ​ ( 4 ) ​ π ​ ( 5 ) ​ π ​ ( 6 ) absent 𝜋 1 𝜋 2 𝜋 4 𝜋 5 𝜋 6 =\pi(1)\pi(2)\pi(4)\pi(5)\pi(6) ), but avoids 3142 3142 3142 . It will often be useful to view permutations and order isomorphism graphically. Two sets S 𝑆 S and T 𝑇 T of points in the plane are said to be order isomorphic if the axes for the set S 𝑆 S can be stretched and shrunk in some manner to map the points of S 𝑆 S bijectively onto the points of T 𝑇 T , i.e., if there are strictly increasing functions f , g : ℝ → ℝ : 𝑓 𝑔 → ℝ ℝ f,g:\mathbb{R}\rightarrow\mathbb{R} such that { ( f ​ ( s 1 ) , g ​ ( s 2 ) ) : ( s 1 , s 2 ) ∈ S } = T conditional-set 𝑓 subscript 𝑠 1 𝑔 subscript 𝑠 2 subscript 𝑠 1 subscript 𝑠 2 𝑆 𝑇 \{(f(s_{1}),g(s_{2})):(s_{1},s_{2})\in S\}=T . Note that this forms an equivalence relation since the inverse of a strictly increasing function is also strictly increasing. The plot of the permutation π 𝜋 \pi is then the point set { ( i , π ​ ( i ) ) } 𝑖 𝜋 𝑖 \{(i,\pi(i))\} , and every finite point set in the plane in which no two points share a coordinate (often called a generic or noncorectilinear set) is order isomorphic to the plot of a unique permutation (see Figure 1 for an example). Note that, with a slight abuse of notation, we will say that a point set is order isomorphic to a permutation. \psaxes [dy=10,Dy=1,dx=10,Dx=1,tickstyle=bottom,showorigin=false,labels=none](0,0)(90,90) Figure 1: The plot of the permutation π = 934826715 𝜋 934826715 \pi=934826715 . The pattern containment order forms a partial order on the set of all permutations. Downsets of permutations under this order are called permutation classes . 2 2 2 In the past, permutation classes have also been called closed classes or pattern classes . In other words, if 𝒞 𝒞 \mathcal{C} is a permutation class and π ∈ 𝒞 𝜋 𝒞 \pi\in\mathcal{C} , then for any permutation σ 𝜎 \sigma with σ ≤ π 𝜎 𝜋 \sigma\leq\pi we have σ ∈ 𝒞 𝜎 𝒞 \sigma\in\mathcal{C} . A given permutation class is often described in terms of its minimal avoidance set, or basis . More formally, the basis B 𝐵 B of a permutation class 𝒞 𝒞 \mathcal{C} is the smallest set for which 𝒞 = { π ∣ β ≰ π ​ for all ​ β ∈ B } 𝒞 conditional-set 𝜋 not-less-than-or-equals 𝛽 𝜋 for all 𝛽 𝐵 \mathcal{C}=\{\pi\mid\beta\not\leq\pi\textrm{ for all }\beta\in B\} . For a permutation class 𝒞 𝒞 \mathcal{C} , we denote by 𝒞 n subscript 𝒞 𝑛 \mathcal{C}_{n} the set 𝒞 ∩ S n 𝒞 subscript 𝑆 𝑛 \mathcal{C}\cap S_{n} , i.e. the permutations in 𝒞 𝒞 \mathcal{C} of length n 𝑛 n , and we refer to f ​ ( x ) = ∑ | 𝒞 n | ​ x n 𝑓 𝑥 subscript 𝒞 𝑛 superscript 𝑥 𝑛 f(x)=\sum|\mathcal{C}_{n}|x^{n} as the generating function for 𝒞 𝒞 \mathcal{C} . Analogues of pattern containment exist for other relational structures; sets of structures closed under taking induced substructures are known as hereditary properties . Hereditary properties of graphs have received considerable attention (see Bollobás [ 11 ] for a survey of some older results), while more recently attention has been given to hereditary properties of a variety of structures including tournaments, ordered graphs and posets (see, for example, Balogh et al. [ 6 , 8 , 7 ] , and Bollobás’s recent survey [ 12 ] ). 1.1 Substitution Decomposition The simple permutations form the elemental building blocks upon which all other permutations are constructed by means of the substitution decomposition. 3 3 3 This decomposition is also called the modular decomposition, disjunctive decomposition or X 𝑋 X -join in other contexts. Analogues of this decomposition exist for every relational structure, and it has frequently arisen in a wide variety of perspectives, ranging from game theory to combinatorial optimization — for references see Möhring [ 35 ] or Möhring and Radermacher [ 36 ] . Its first appearance seems to be in a 1953 talk by Fraïssé (though only the abstract of this talk [ 25 ] survives). It did not appear in an article until Gallai [ 26 ] (for an English translation, see [ 27 ] ), who applied them particularly to the study of transitive orientations of graphs. Given a permutation σ 𝜎 \sigma of length m 𝑚 m and nonempty permutations α 1 , … , α m subscript 𝛼 1 … subscript 𝛼 𝑚 \alpha_{1},\dots,\alpha_{m} , the inflation of σ 𝜎 \sigma by α 1 , … , α m subscript 𝛼 1 … subscript 𝛼 𝑚 \alpha_{1},\dots,\alpha_{m} — denoted σ ​ [ α 1 , … , α m ] 𝜎 subscript 𝛼 1 … subscript 𝛼 𝑚 \sigma[\alpha_{1},\dots,\alpha_{m}] — is the permutation obtained by replacing each entry σ ​ ( i ) 𝜎 𝑖 \sigma(i) by an interval that is order isomorphic to α i subscript 𝛼 𝑖 \alpha_{i} . For example, 2413 ​ [ 1 , 132 , 321 , 12 ] = 479832156 2413 1 132 321 12 479832156 2413[1,132,321,12]=479832156 . Conversely, a deflation of π 𝜋 \pi is any expression of π 𝜋 \pi as an inflation π = σ ​ [ π 1 , π 2 , … , π m ] 𝜋 𝜎 subscript 𝜋 1 subscript 𝜋 2 … subscript 𝜋 𝑚 \pi=\sigma[\pi_{1},\pi_{2},\ldots,\pi_{m}] , and we will call σ 𝜎 \sigma a skeleton of π 𝜋 \pi . We then have the substitution decomposition of permutations: Proposition 1.1 (Albert and Atkinson [ 1 ] ) . Every permutation may be written as the inflation of a unique simple permutation. Moreover, if π 𝜋 \pi can be written as σ ​ [ α 1 , … , α m ] 𝜎 subscript 𝛼 1 … subscript 𝛼 𝑚 \sigma[\alpha_{1},\dots,\alpha_{m}] where σ 𝜎 \sigma is simple and m ≥ 4 𝑚 4 m\geq 4 , then the α i subscript 𝛼 𝑖 \alpha_{i} ’s are unique. Non-unique cases arise when a permutation can be written as an inflation of either 12 12 12 or 21 21 21 , and to recover uniqueness we may choose a particular decomposition in a variety of ways. The one we will use is as follows. Proposition 1.2 (Albert and Atkinson [ 1 ] ) . If π 𝜋 \pi is an inflation of 12 12 12 , then there is a unique sum indecomposable α 1 subscript 𝛼 1 \alpha_{1} such that π = 12 ​ [ α 1 , α 2 ] 𝜋 12 subscript 𝛼 1 subscript 𝛼 2 \pi=12[\alpha_{1},\alpha_{2}] for some α 2 subscript 𝛼 2 \alpha_{2} , which is itself unique. The same holds with 12 12 12 replaced by 21 21 21 and “sum” replaced by “skew”. The substitution decomposition tree for a permutation is obtained by recursively decomposing until we are left only with inflations of simple permutations by singletons. For example, the permutation π = 452398167 𝜋 452398167 \pi=452398167 is decomposed as 452398167 452398167 \displaystyle 452398167 = \displaystyle= 2413 ​ [ 3412 , 21 , 1 , 12 ] 2413 3412 21 1 12 \displaystyle 2413[3412,21,1,12] = \displaystyle= 2413 ​ [ 21 ​ [ 12 , 12 ] , 21 ​ [ 1 , 1 ] , 1 , 12 ​ [ 1 , 1 ] ] 2413 21 12 12 21 1 1 1 12 1 1 \displaystyle 2413[21[12,12],21[1,1],1,12[1,1]] = \displaystyle= 2413 ​ [ 21 ​ [ 12 ​ [ 1 , 1 ] , 12 ​ [ 1 , 1 ] ] , 21 ​ [ 1 , 1 ] , 1 , 12 ​ [ 1 , 1 ] ] 2413 21 12 1 1 12 1 1 21 1 1 1 12 1 1 \displaystyle 2413[21[12[1,1],12[1,1]],21[1,1],1,12[1,1]] and its substitution decomposition tree is given in Figure 2 . \pstree [nodesep=2pt,levelsep=5ex] \TR 452398167 452398167 452398167 \pstree \TR 4523 4523 4523 \pstree \TR 45 45 45 \TR 4 4 4 \TR 5 5 5 \pstree \TR 23 23 23 \TR 2 2 2 \TR 3 3 3 \pstree \TR 98 98 98 \TR 9 9 9 \TR 8 8 8 \TR 1 1 1 \pstree \TR 67 67 67 \TR 6 6 6 \TR 7 7 7 Figure 2: The substitution decomposition tree of π = 452398167 𝜋 452398167 \pi=452398167 . Computation in Linear Time. The substitution decomposition is most frequently used in solving algorithmic problems, and consequently much attention has been given to its computation in optimal time. 4 4 4 In particular, graph decomposition has received significant attention, with the first O ​ ( | V | + | E | ) 𝑂 𝑉 𝐸 O(|V|+|E|) algorithms appearing in 1994 by McConnell and Spinrad [ 33 ] and Cournier and Habib [ 22 ] . By its connection to the intervals of a permutation, a first approach to compute the substitution decomposition might be simply to compute all the intervals of our given permutation. Since there may be as many as N = n ​ ( n − 1 ) / 2 𝑁 𝑛 𝑛 1 2 N=n(n-1)/2 such intervals in a permutation of length n 𝑛 n , listing these will not yield a linear O ​ ( n ) 𝑂 𝑛 O(n) algorithm for the substitution decomposition. However, this computation has received significant attention through its connections with biomathematics, with an O ​ ( n + N ) 𝑂 𝑛 𝑁 O(n+N) time algorithm being given by Bergeron, Chauve, Montgolfier and Raffinot [ 10 ] . 5 5 5 In fact, Bergeron et al show how to compute the “common intervals” — a generalisation of our notion of interval applied to sets of permutations. The first algorithm to compute the substitution decomposition of a permutation in linear time was given by Uno and Yagiura [ 41 ] , while Bergeron et al have since given a simpler algorithm. A strong interval of a permutation π 𝜋 \pi is an interval I 𝐼 I for which every other interval J 𝐽 J satisfies one of J ⊆ I 𝐽 𝐼 J\subseteq I , I ⊆ J 𝐼 𝐽 I\subseteq J or I ∩ J = ∅ 𝐼 𝐽 I\cap J=\emptyset . For example, given π = 234615 𝜋 234615 \pi=234615 , the interval 234 234 234 ( = π ​ ( 1 ) ​ π ​ ( 2 ) ​ π ​ ( 3 ) absent 𝜋 1 𝜋 2 𝜋 3 =\pi(1)\pi(2)\pi(3) ) is a strong interval, but 23 23 23 is not, because it has non-trivial intersection with 34 34 34 . A permutation can have at most 2 ​ n − 1 2 𝑛 1 2n-1 strong intervals (note that the n 𝑛 n singletons and the whole permutation are all strong intervals), and Bergeron et al give an optimal O ​ ( n ) 𝑂 𝑛 O(n) algorithm to list them all. The substitution decomposition tree of the permutation follows immediately. It is worth noticing that this algorithm does not give the simple skeletons for each internal node of the decomposition tree — indeed, there are currently no linear time algorithms to do this. It is, however, straightforward to compute the label of any particular node in linear time, e.g. by finding a representative symbol for each strong interval lying below the node, and then computing the permutation order isomorphic to this sequence of representatives.

We survey the known results about simple permutations. In particular, we present a number of recent enumerative and structural results pertaining to simple permutations, and show how simple permutations play an important role in the study of permutation classes. We demonstrate how classes containing only finitely many simple permutations satisfy a number of special properties relating to enumeration, partial well-order and the property of being finitely based.

[{"content":"Help me to generate the abstract of a survey paper given the title:*Permutation Patterns: A survey of simple permutations* and the introduction:An interval of a permutation π 𝜋 \pi corresponds to a set of contiguous indices I = a , b 𝐼 𝑎 𝑏 I=a,b such that the set of values π ​ ( I ) = { π ​ ( i ) : i ∈ I } 𝜋 𝐼 conditional-set 𝜋 𝑖 𝑖 𝐼 \pi(I)=\{\pi(i):i\in I\} is also contiguous. Every permutation of length n 𝑛 n has intervals of lengths 0 0 , 1 1 1 and n 𝑛 n . If a permutation π 𝜋 \pi has no other intervals, then π 𝜋 \pi is said to be simple . For example, the permutation π = 28146357 𝜋 28146357 \pi=28146357 is not simple as witnessed by the non-trivial interval 4635 4635 4635 ( = π ​ ( 4 ) ​ π ​ ( 5 ) ​ π ​ ( 6 ) ​ π ​ ( 7 ) absent 𝜋 4 𝜋 5 𝜋 6 𝜋 7 =\pi(4)\pi(5)\pi(6)\pi(7) ), while σ = 51742683 𝜎 51742683 \sigma=51742683 is simple. 1 1 1 Simplicity may be defined for all relational structures; such structures have variously been called prime or indecomposable . While intervals of permutations have applications in biomathematics, particularly to genetic algorithms and the matching of gene sequences (see Corteel, Louchard, and Pemantle 21 for extensive references), simple permutations form the building blocks of permutation classes and have thus received intensive study in recent years. We will see in Section 3 the various ways in which simplicity plays a role in the study of permutation classes, but we begin this short survey by introducing the substitution decomposition in Subsection 1.1 , and thence by reviewing the structural and enumerative results of simple permutations themselves in Section 2 . The rest of this subsection will cover several basic definitions that we will require. Two finite sequences of the same length, α = a 1 ​ a 2 ​ ⋯ ​ a n 𝛼 subscript 𝑎 1 subscript 𝑎 2 ⋯ subscript 𝑎 𝑛 \alpha=a_{1}a_{2}\cdots a_{n} and β = b 1 ​ b 2 ​ ⋯ ​ b n 𝛽 subscript 𝑏 1 subscript 𝑏 2 ⋯ subscript 𝑏 𝑛 \beta=b_{1}b_{2}\cdots b_{n} , are said to be order isomorphic if, for all i , j 𝑖 𝑗 i,j , we have a i < a j subscript 𝑎 𝑖 subscript 𝑎 𝑗 a_{i}<a_{j} if and only if b i < b j subscript 𝑏 𝑖 subscript 𝑏 𝑗 b_{i}<b_{j} . As such, each sequence of distinct real numbers is order isomorphic to a unique permutation. Similarly, any given subsequence (or pattern ) of a permutation π 𝜋 \pi is order isomorphic to a smaller permutation, σ 𝜎 \sigma say, and such a subsequence is called a copy of σ 𝜎 \sigma in π 𝜋 \pi . We may also say that π 𝜋 \pi contains σ 𝜎 \sigma (or, in some texts, π 𝜋 \pi involves σ 𝜎 \sigma ) and write σ ≤ π 𝜎 𝜋 \sigma\leq\pi . If, on the other hand, π 𝜋 \pi does not contain a copy of some given σ 𝜎 \sigma , then π 𝜋 \pi is said to avoid σ 𝜎 \sigma . For example, π = 918572346 𝜋 918572346 \pi=918572346 contains 51342 51342 51342 because of the subsequence 91572 91572 91572 ( = π ​ ( 1 ) ​ π ​ ( 2 ) ​ π ​ ( 4 ) ​ π ​ ( 5 ) ​ π ​ ( 6 ) absent 𝜋 1 𝜋 2 𝜋 4 𝜋 5 𝜋 6 =\pi(1)\pi(2)\pi(4)\pi(5)\pi(6) ), but avoids 3142 3142 3142 . It will often be useful to view permutations and order isomorphism graphically. Two sets S 𝑆 S and T 𝑇 T of points in the plane are said to be order isomorphic if the axes for the set S 𝑆 S can be stretched and shrunk in some manner to map the points of S 𝑆 S bijectively onto the points of T 𝑇 T , i.e., if there are strictly increasing functions f , g : ℝ → ℝ : 𝑓 𝑔 → ℝ ℝ f,g:\mathbb{R}\rightarrow\mathbb{R} such that { ( f ​ ( s 1 ) , g ​ ( s 2 ) ) : ( s 1 , s 2 ) ∈ S } = T conditional-set 𝑓 subscript 𝑠 1 𝑔 subscript 𝑠 2 subscript 𝑠 1 subscript 𝑠 2 𝑆 𝑇 \{(f(s_{1}),g(s_{2})):(s_{1},s_{2})\in S\}=T . Note that this forms an equivalence relation since the inverse of a strictly increasing function is also strictly increasing. The plot of the permutation π 𝜋 \pi is then the point set { ( i , π ​ ( i ) ) } 𝑖 𝜋 𝑖 \{(i,\pi(i))\} , and every finite point set in the plane in which no two points share a coordinate (often called a generic or noncorectilinear set) is order isomorphic to the plot of a unique permutation (see Figure 1 for an example). Note that, with a slight abuse of notation, we will say that a point set is order isomorphic to a permutation. \psaxes dy=10,Dy=1,dx=10,Dx=1,tickstyle=bottom,showorigin=false,labels=none(0,0)(90,90) Figure 1: The plot of the permutation π = 934826715 𝜋 934826715 \pi=934826715 . The pattern containment order forms a partial order on the set of all permutations. Downsets of permutations under this order are called permutation classes . 2 2 2 In the past, permutation classes have also been called closed classes or pattern classes . In other words, if 𝒞 𝒞 \mathcal{C} is a permutation class and π ∈ 𝒞 𝜋 𝒞 \pi\in\mathcal{C} , then for any permutation σ 𝜎 \sigma with σ ≤ π 𝜎 𝜋 \sigma\leq\pi we have σ ∈ 𝒞 𝜎 𝒞 \sigma\in\mathcal{C} . A given permutation class is often described in terms of its minimal avoidance set, or basis . More formally, the basis B 𝐵 B of a permutation class 𝒞 𝒞 \mathcal{C} is the smallest set for which 𝒞 = { π ∣ β ≰ π ​ for all ​ β ∈ B } 𝒞 conditional-set 𝜋 not-less-than-or-equals 𝛽 𝜋 for all 𝛽 𝐵 \mathcal{C}=\{\pi\mid\beta\not\leq\pi\textrm{ for all }\beta\in B\} . For a permutation class 𝒞 𝒞 \mathcal{C} , we denote by 𝒞 n subscript 𝒞 𝑛 \mathcal{C}_{n} the set 𝒞 ∩ S n 𝒞 subscript 𝑆 𝑛 \mathcal{C}\cap S_{n} , i.e. the permutations in 𝒞 𝒞 \mathcal{C} of length n 𝑛 n , and we refer to f ​ ( x ) = ∑ | 𝒞 n | ​ x n 𝑓 𝑥 subscript 𝒞 𝑛 superscript 𝑥 𝑛 f(x)=\sum|\mathcal{C}_{n}|x^{n} as the generating function for 𝒞 𝒞 \mathcal{C} . Analogues of pattern containment exist for other relational structures; sets of structures closed under taking induced substructures are known as hereditary properties . Hereditary properties of graphs have received considerable attention (see Bollobás 11 for a survey of some older results), while more recently attention has been given to hereditary properties of a variety of structures including tournaments, ordered graphs and posets (see, for example, Balogh et al. 6 , 8 , 7 , and Bollobáss recent survey 12 ). 1.1 Substitution Decomposition The simple permutations form the elemental building blocks upon which all other permutations are constructed by means of the substitution decomposition. 3 3 3 This decomposition is also called the modular decomposition, disjunctive decomposition or X 𝑋 X -join in other contexts. Analogues of this decomposition exist for every relational structure, and it has frequently arisen in a wide variety of perspectives, ranging from game theory to combinatorial optimization — for references see Möhring 35 or Möhring and Radermacher 36 . Its first appearance seems to be in a 1953 talk by Fraïssé (though only the abstract of this talk 25 survives). It did not appear in an article until Gallai 26 (for an English translation, see 27 ), who applied them particularly to the study of transitive orientations of graphs. Given a permutation σ 𝜎 \sigma of length m 𝑚 m and nonempty permutations α 1 , … , α m subscript 𝛼 1 … subscript 𝛼 𝑚 \alpha_{1},\dots,\alpha_{m} , the inflation of σ 𝜎 \sigma by α 1 , … , α m subscript 𝛼 1 … subscript 𝛼 𝑚 \alpha_{1},\dots,\alpha_{m} — denoted σ ​ α 1 , … , α m 𝜎 subscript 𝛼 1 … subscript 𝛼 𝑚 \sigma\alpha_{1},\dots,\alpha_{m} — is the permutation obtained by replacing each entry σ ​ ( i ) 𝜎 𝑖 \sigma(i) by an interval that is order isomorphic to α i subscript 𝛼 𝑖 \alpha_{i} . For example, 2413 ​ 1 , 132 , 321 , 12 = 479832156 2413 1 132 321 12 479832156 24131,132,321,12=479832156 . Conversely, a deflation of π 𝜋 \pi is any expression of π 𝜋 \pi as an inflation π = σ ​ π 1 , π 2 , … , π m 𝜋 𝜎 subscript 𝜋 1 subscript 𝜋 2 … subscript 𝜋 𝑚 \pi=\sigma\pi_{1},\pi_{2},\ldots,\pi_{m} , and we will call σ 𝜎 \sigma a skeleton of π 𝜋 \pi . We then have the substitution decomposition of permutations: Proposition 1.1 (Albert and Atkinson 1 ) . Every permutation may be written as the inflation of a unique simple permutation. Moreover, if π 𝜋 \pi can be written as σ ​ α 1 , … , α m 𝜎 subscript 𝛼 1 … subscript 𝛼 𝑚 \sigma\alpha_{1},\dots,\alpha_{m} where σ 𝜎 \sigma is simple and m ≥ 4 𝑚 4 m\geq 4 , then the α i subscript 𝛼 𝑖 \alpha_{i} s are unique. Non-unique cases arise when a permutation can be written as an inflation of either 12 12 12 or 21 21 21 , and to recover uniqueness we may choose a particular decomposition in a variety of ways. The one we will use is as follows. Proposition 1.2 (Albert and Atkinson 1 ) . If π 𝜋 \pi is an inflation of 12 12 12 , then there is a unique sum indecomposable α 1 subscript 𝛼 1 \alpha_{1} such that π = 12 ​ α 1 , α 2 𝜋 12 subscript 𝛼 1 subscript 𝛼 2 \pi=12\alpha_{1},\alpha_{2} for some α 2 subscript 𝛼 2 \alpha_{2} , which is itself unique. The same holds with 12 12 12 replaced by 21 21 21 and sum replaced by skew. The substitution decomposition tree for a permutation is obtained by recursively decomposing until we are left only with inflations of simple permutations by singletons. For example, the permutation π = 452398167 𝜋 452398167 \pi=452398167 is decomposed as 452398167 452398167 \displaystyle 452398167 = \displaystyle= 2413 ​ 3412 , 21 , 1 , 12 2413 3412 21 1 12 \displaystyle 24133412,21,1,12 = \displaystyle= 2413 ​ 21 ​ 12 , 12 , 21 ​ 1 , 1 , 1 , 12 ​ 1 , 1 2413 21 12 12 21 1 1 1 12 1 1 \displaystyle 24132112,12,211,1,1,121,1 = \displaystyle= 2413 ​ 21 ​ 12 ​ 1 , 1 , 12 ​ 1 , 1 , 21 ​ 1 , 1 , 1 , 12 ​ 1 , 1 2413 21 12 1 1 12 1 1 21 1 1 1 12 1 1 \displaystyle 241321121,1,121,1,211,1,1,121,1 and its substitution decomposition tree is given in Figure 2 . \pstree nodesep=2pt,levelsep=5ex \TR 452398167 452398167 452398167 \pstree \TR 4523 4523 4523 \pstree \TR 45 45 45 \TR 4 4 4 \TR 5 5 5 \pstree \TR 23 23 23 \TR 2 2 2 \TR 3 3 3 \pstree \TR 98 98 98 \TR 9 9 9 \TR 8 8 8 \TR 1 1 1 \pstree \TR 67 67 67 \TR 6 6 6 \TR 7 7 7 Figure 2: The substitution decomposition tree of π = 452398167 𝜋 452398167 \pi=452398167 . Computation in Linear Time. The substitution decomposition is most frequently used in solving algorithmic problems, and consequently much attention has been given to its computation in optimal time. 4 4 4 In particular, graph decomposition has received significant attention, with the first O ​ ( | V | + | E | ) 𝑂 𝑉 𝐸 O(|V|+|E|) algorithms appearing in 1994 by McConnell and Spinrad 33 and Cournier and Habib 22 . By its connection to the intervals of a permutation, a first approach to compute the substitution decomposition might be simply to compute all the intervals of our given permutation. Since there may be as many as N = n ​ ( n − 1 ) / 2 𝑁 𝑛 𝑛 1 2 N=n(n-1)/2 such intervals in a permutation of length n 𝑛 n , listing these will not yield a linear O ​ ( n ) 𝑂 𝑛 O(n) algorithm for the substitution decomposition. However, this computation has received significant attention through its connections with biomathematics, with an O ​ ( n + N ) 𝑂 𝑛 𝑁 O(n+N) time algorithm being given by Bergeron, Chauve, Montgolfier and Raffinot 10 . 5 5 5 In fact, Bergeron et al show how to compute the common intervals — a generalisation of our notion of interval applied to sets of permutations. The first algorithm to compute the substitution decomposition of a permutation in linear time was given by Uno and Yagiura 41 , while Bergeron et al have since given a simpler algorithm. A strong interval of a permutation π 𝜋 \pi is an interval I 𝐼 I for which every other interval J 𝐽 J satisfies one of J ⊆ I 𝐽 𝐼 J\subseteq I , I ⊆ J 𝐼 𝐽 I\subseteq J or I ∩ J = ∅ 𝐼 𝐽 I\cap J=\emptyset . For example, given π = 234615 𝜋 234615 \pi=234615 , the interval 234 234 234 ( = π ​ ( 1 ) ​ π ​ ( 2 ) ​ π ​ ( 3 ) absent 𝜋 1 𝜋 2 𝜋 3 =\pi(1)\pi(2)\pi(3) ) is a strong interval, but 23 23 23 is not, because it has non-trivial intersection with 34 34 34 . A permutation can have at most 2 ​ n − 1 2 𝑛 1 2n-1 strong intervals (note that the n 𝑛 n singletons and the whole permutation are all strong intervals), and Bergeron et al give an optimal O ​ ( n ) 𝑂 𝑛 O(n) algorithm to list them all. The substitution decomposition tree of the permutation follows immediately. It is worth noticing that this algorithm does not give the simple skeletons for each internal node of the decomposition tree — indeed, there are currently no linear time algorithms to do this. It is, however, straightforward to compute the label of any particular node in linear time, e.g. by finding a representative symbol for each strong interval lying below the node, and then computing the permutation order isomorphic to this sequence of representatives.","role":"user"},{"content":"Abstract: We survey the known results about simple permutations. In particular, we present a number of recent enumerative and structural results pertaining to simple permutations, and show how simple permutations play an important role in the study of permutation classes. We demonstrate how classes containing only finitely many simple permutations satisfy a number of special properties relating to enumeration, partial well-order and the property of being finitely based.","role":"assistant"}]

An overview of arithmetic motivic integration

1. Introduction The aim of these notes is to provide an elementary introduction to some aspects of the theory of arithmetic motivic integration, as well as a brief guide to the extensive literature on the subject. The idea of motivic integration was introduced by M. Kontsevich in 1995. It was quickly developed by J. Denef and F. Loeser in a series of papers [ DL ] , [ DL.McKay ] , [ DL.Igusa ] , and by others. This theory, whose applications are mostly in algebraic geometry over algebraically closed fields, now is often referred to as “geometric motivic integration”, to distinguish it from the so-called arithmetic motivic integration that specifies to integration over p 𝑝 p -adic fields. The theory of arithmetic motivic integration first appeared in the 1999 paper by J. Denef and F. Loeser [ DL.arithm ] . The articles [ Tom.intro ] and [ DL.congr ] together provide an excellent exposition of this work. In 2004, R. Cluckers and F. Loeser developed a different and very effective approach to motivic integration (both geometric and arithmetic) [ CL ] . Even though there is an expository version [ CL.expo ] , this theory seems to be not yet well-known. This note is intended in part to be a companion with examples to [ CL ] . The aim is not just to describe what motivic integration achieves, but to give some clues as to how it works. We have stayed very close to the work of Cluckers and Loeser in the main part of this exposition. In fact, much of these notes is a direct quotation, most frequently from the articles [ CL ] , [ CL.expo ] , and also [ DL.congr ] , and [ DL.arithm ] . Even though we try to give precise references all the time, some quotes from these sources might not always be acknowledged since they are so ubiquitous. Some ideas, especially in the appendices, are clearly borrowed from [ Tom.intro ] . The secondary goal was to collect references to many sources on motivic integration, and to provide some information on the relationship and logical interconnections between them. This is done in Appendix 1 (Section 7 ). Our ultimate hope is that the reader would be able to start using motivic integration instead of p 𝑝 p -adic integration, if there is any advantage in doing integration independently of p 𝑝 p at the cost of losing a finite number of primes. Acknowledgment. The first author thanks T.C. Hales for introducing her to the subject; Jonathan Korman – for many hours of discussions, and Raf Cluckers – for explaining his work on several occasions. We have learned a lot of what appears in these notes at the joint University of Toronto-McMaster University seminar on motivic integration in 2004-2005, and thank Elliot Lawes, Jonathan Korman and Alfred Dolich for their lectures. The contributions of the second author are limited to sections 1-5. Finally, the first author thanks the organizers and participants of the mini-courses on motivic integration at the University of Utah and at the Fields Institute Workshop at the University of Ottawa, where most of this material was presented, and the editors of this volume for multiple suggestions and corrections.

This is an attempt at an elementary exposition, with examples, of the theory of motivic integration developed by R. Cluckers and F. Loeser, with the view towards applications in representation theory of p-adic groups.

[{"content":"Help me to generate the abstract of a survey paper given the title:*An overview of arithmetic motivic integration* and the introduction:1. Introduction The aim of these notes is to provide an elementary introduction to some aspects of the theory of arithmetic motivic integration, as well as a brief guide to the extensive literature on the subject. The idea of motivic integration was introduced by M. Kontsevich in 1995. It was quickly developed by J. Denef and F. Loeser in a series of papers DL , DL.McKay , DL.Igusa , and by others. This theory, whose applications are mostly in algebraic geometry over algebraically closed fields, now is often referred to as geometric motivic integration, to distinguish it from the so-called arithmetic motivic integration that specifies to integration over p 𝑝 p -adic fields. The theory of arithmetic motivic integration first appeared in the 1999 paper by J. Denef and F. Loeser DL.arithm . The articles Tom.intro and DL.congr together provide an excellent exposition of this work. In 2004, R. Cluckers and F. Loeser developed a different and very effective approach to motivic integration (both geometric and arithmetic) CL . Even though there is an expository version CL.expo , this theory seems to be not yet well-known. This note is intended in part to be a companion with examples to CL . The aim is not just to describe what motivic integration achieves, but to give some clues as to how it works. We have stayed very close to the work of Cluckers and Loeser in the main part of this exposition. In fact, much of these notes is a direct quotation, most frequently from the articles CL , CL.expo , and also DL.congr , and DL.arithm . Even though we try to give precise references all the time, some quotes from these sources might not always be acknowledged since they are so ubiquitous. Some ideas, especially in the appendices, are clearly borrowed from Tom.intro . The secondary goal was to collect references to many sources on motivic integration, and to provide some information on the relationship and logical interconnections between them. This is done in Appendix 1 (Section 7 ). Our ultimate hope is that the reader would be able to start using motivic integration instead of p 𝑝 p -adic integration, if there is any advantage in doing integration independently of p 𝑝 p at the cost of losing a finite number of primes. Acknowledgment. The first author thanks T.C. Hales for introducing her to the subject; Jonathan Korman – for many hours of discussions, and Raf Cluckers – for explaining his work on several occasions. We have learned a lot of what appears in these notes at the joint University of Toronto-McMaster University seminar on motivic integration in 2004-2005, and thank Elliot Lawes, Jonathan Korman and Alfred Dolich for their lectures. The contributions of the second author are limited to sections 1-5. Finally, the first author thanks the organizers and participants of the mini-courses on motivic integration at the University of Utah and at the Fields Institute Workshop at the University of Ottawa, where most of this material was presented, and the editors of this volume for multiple suggestions and corrections.","role":"user"},{"content":"Abstract: This is an attempt at an elementary exposition, with examples, of the theory of motivic integration developed by R. Cluckers and F. Loeser, with the view towards applications in representation theory of p-adic groups.","role":"assistant"}]

Episturmian words: a survey

1.1 From Sturmian to episturmian Most renowned amongst the branches of combinatorics on words is the theory of infinite binary sequences called Sturmian words , which are fascinating in many respects, having been studied from combinatorial, algebraic, and geometric points of view. Their beautiful properties are related to many fields such as Number Theory, Geometry, Symbolic Dynamical Systems, Theoretical Physics, and Theoretical Computer Science (see [ 7 , 83 , 96 ] for recent surveys). Since the seminal works of Morse and Hedlund [ 91 ] , Sturmian words have been shown to admit numerous equivalent definitions and characterizations. For instance, it is well known that an infinite word 𝐰 𝐰 \mathbf{w} over { a , b } 𝑎 𝑏 \{a,b\} is Sturmian if and only if 𝐰 𝐰 \mathbf{w} is aperiodic and balanced : for any two factors u 𝑢 u , v 𝑣 v of 𝐰 𝐰 \mathbf{w} of the same length, the number of a 𝑎 a ’s in each of u 𝑢 u and v 𝑣 v differs by at most 1 1 1 . Sturmian words are also characterized by their factor complexity function (which counts the number of distinct factors of each length): they have exactly n + 1 𝑛 1 n+1 distinct factors of length n 𝑛 n for each n 𝑛 n . In this sense, Sturmian words are precisely the aperiodic infinite words of minimal factor complexity since, as is well known, an infinite word is ultimately periodic if and only if it has less than n + 1 𝑛 1 n+1 factors of length n 𝑛 n for some n 𝑛 n (see [ 37 ] ). Many interesting properties of Sturmian words can be attributed to their low complexity, which induces certain regularities in such words without, however, making them periodic. Sturmian words can also be geometrically realized as cutting sequences by considering the sequence of ‘cuts’ in an integer grid made by a line of irrational slope (see for instance [ 38 , 13 ] ). They also provide a symbolic coding of the orbit of a point on a circle with respect to a rotation by an irrational number (see [ 91 , 4 ] ). All of the above characteristic properties of Sturmian words lead to natural generalizations on arbitrary finite alphabets. In one direction, the balance property naturally extends to an alphabet with more than two letters (e.g., see [ 68 , 110 , 115 ] ) as does the following generalized balance property that also characterizes Sturmian words (see [ 49 , 1 ] ): the difference between the number of occurrences of a word u 𝑢 u in any pair of factors of the same length is at most 1 1 1 . In another direction, we could consider relaxing the minimality condition for the factor complexity p ​ ( n ) 𝑝 𝑛 p(n) . For example, quasi-Sturmian words are infinite words for which there exist two positive integers N 𝑁 N and c 𝑐 c such that n + 1 ≤ p ​ ( n ) ≤ n + c 𝑛 1 𝑝 𝑛 𝑛 𝑐 n+1\leq p(n)\leq n+c for all n ≥ N 𝑛 𝑁 n\geq N . This generalization was introduced in [ 5 ] when studying the transcendence of certain continued fraction expansions. See also [ 31 , 36 , 66 , 105 ] for similar extensions of Sturmian words with respect to factor complexity. From the geometric point of view, cutting sequences naturally generalize to trajectories in the hypercube billiard (e.g., see [ 25 ] ), and codings of rotational orbits carry over to codings of interval exchange transformations (e.g., see [ 18 ] ). Two other very interesting natural generalizations of Sturmian words are Arnoux-Rauzy sequences [ 12 , 97 ] and episturmian words [ 43 , 73 ] , which we will now define. From the factor complexity of Sturmian words, it immediately follows that any Sturmian word is over a 2 2 2 -letter alphabet and has exactly one left special factor of each length. A factor u 𝑢 u of a finite or infinite word w 𝑤 w is said to be left special (resp. right special ) in w 𝑤 w if there exists at least two distinct letters a 𝑎 a , b 𝑏 b such that a ​ u 𝑎 𝑢 au and b ​ u 𝑏 𝑢 bu (resp. u ​ a 𝑢 𝑎 ua , u ​ b 𝑢 𝑏 ub ) are factors of w 𝑤 w . Extending the left special property of Sturmian words, a recurrent infinite word 𝐰 𝐰 \mathbf{w} over a finite alphabet 𝒜 𝒜 \mathcal{A} is said to be an Arnoux-Rauzy sequence (or a strict episturmian word ) if it has exactly one left special factor and one right special factor of each length, and for every left (resp. right) special factor u 𝑢 u of 𝐰 𝐰 \mathbf{w} , x ​ u 𝑥 𝑢 xu (resp. u ​ x 𝑢 𝑥 ux ) is a factor of 𝐰 𝐰 \mathbf{w} for all letters x ∈ 𝒜 𝑥 𝒜 x\in\mathcal{A} . A noteable property that is shared by Sturmian words and Arnouxy-Rauzy sequences is their closure under reversal, i.e., if u 𝑢 u is a factor of such a word, then its reversal is also a factor. This nice property inspired Droubay, Justin, and Pirillo’s generalization of Sturmian words in [ 43 ] : an infinite word is episturmian if it is closed under reversal and has at most one left special factor of each length. Sturmian, Arnoux-Rauzy, and episturmian words all have standard (or characteristic ) elements, which are those having all of their left special factors as prefixes. Within these families of words, standard words are good representatives in the sense that an infinite word belongs to one such family if and only if it has the same set of factors as some standard word in that family. From the definitions, it is clear that the family of Arnoux-Rauzy sequences is a particular subclass of the family of episturmian words. More precisely, episturmian words are composed of the Arnoux-Rauzy sequences, images of the Arnoux-Rauzy sequences by episturmian morphisms , and certain periodic infinite words (see Section 5 ). In the 2 2 2 -letter case, Arnoux-Rauzy sequences are exactly the Sturmian words whereas episturmian words include all recurrent balanced words, i.e., periodic balanced words and Sturmian words. The study of episturmian words and Arnoux-Rauzy sequences has enjoyed a great deal of popularity in recent times, owing mostly to the many properties that they share with Sturmian words. In this paper we survey the purely combinatorial work on episturmian words, beginning with their definition and basic properties in Section 2 . Then, in Section 3 , we recall episturmian morphisms which allow for a deeper study of episturmian words. In particular, any episturmian word is the image of another episturmian word by some so-called pure episturmian morphism . Even more, any episturmian word can be infinitely decomposed over the set of pure episturmian morphisms. This last property allows an episturmian word to be defined by one of its morphic decompositions or, equivalently, by a certain directive word , which is an infinite sequence of rules for decomposing the given episturmian word by morphisms. In Section 4 we consider notions such as shifts , spins , and block-equivalence in connection with directive words, which allow us to study when two different spinned infinite words direct the same episturmian word. We also consider periodic and purely morphic episturmian words. In Section 5 , our discussion briefly turns to Arnoux-Rauzy sequences and finite episturmian words . Following this, we study in Section 6 some properties of factors of episturmian words (and Arnoux-Rauzy sequences), including factor complexity, palindromes, fractional powers, frequencies, and return words. Lastly, we consider more recent work involving lexicographic order and the balance property (including Fraenkel’s conjecture ). 1.2 Notation & terminology We assume the reader is familiar with combinatorics on words and morphisms (e.g., see [ 82 , 83 ] ). In this section, we recall some basic definitions and properties relating to episturmian words which are needed throughout the paper. For the most part, we follow the notation and terminology of [ 43 , 73 , 75 , 62 ] . Let 𝒜 𝒜 \mathcal{A} denote a finite alphabet , i.e., a non-empty finite set of symbols called letters . A finite word over 𝒜 𝒜 \mathcal{A} is a finite sequence of letters from 𝒜 𝒜 \mathcal{A} . The empty word ε 𝜀 \varepsilon is the empty sequence. Under the operation of concatenation, the set 𝒜 ∗ superscript 𝒜 \mathcal{A}^{*} of all finite words over 𝒜 𝒜 \mathcal{A} is a free monoid with identity element ε 𝜀 \varepsilon and set of generators 𝒜 𝒜 \mathcal{A} . The set of non-empty words over 𝒜 𝒜 \mathcal{A} is the free semigroup 𝒜 + := 𝒜 ∗ ∖ { ε } assign superscript 𝒜 superscript 𝒜 𝜀 \mathcal{A}^{+}:=\mathcal{A}^{*}\setminus\{\varepsilon\} . A right-infinite (resp. left-infinite , bi-infinite ) word over 𝒜 𝒜 \mathcal{A} is a sequence indexed by ℕ + superscript ℕ \mathbb{N}^{+} (resp. ℤ ∖ ℕ + ℤ superscript ℕ \mathbb{Z}\setminus\mathbb{N}^{+} , ℤ ℤ \mathbb{Z} ) with values in 𝒜 𝒜 \mathcal{A} . For instance, a left-infinite word is represented by 𝐮 = ⋯ ​ b − 2 ​ b − 1 ​ b 0 𝐮 ⋯ subscript 𝑏 2 subscript 𝑏 1 subscript 𝑏 0 \mathbf{u}=\cdots b_{-2}b_{-1}b_{0} and a right-infinite word by 𝐯 = b 1 ​ b 2 ​ b 3 ​ ⋯ 𝐯 subscript 𝑏 1 subscript 𝑏 2 subscript 𝑏 3 ⋯ \mathbf{v}=b_{1}b_{2}b_{3}\cdots where b i ∈ 𝒜 subscript 𝑏 𝑖 𝒜 b_{i}\in\mathcal{A} . The concatenation of 𝐮 𝐮 \mathbf{u} and 𝐯 𝐯 \mathbf{v} gives the bi-infinite word 𝐮 . 𝐯 = ⋯ ​ b − 2 ​ b − 1 ​ b 0 . b 1 ​ b 2 ​ b 3 ​ ⋯ formulae-sequence 𝐮 𝐯 ⋯ subscript 𝑏 2 subscript 𝑏 1 subscript 𝑏 0 subscript 𝑏 1 subscript 𝑏 2 subscript 𝑏 3 ⋯ \mathbf{u}.\mathbf{v}=\cdots b_{-2}b_{-1}b_{0}.b_{1}b_{2}b_{3}\cdots with a dot written between b 0 subscript 𝑏 0 b_{0} and b 1 subscript 𝑏 1 b_{1} to avoid ambiguity. For easier reading, infinite words are hereafter typically typed in boldface to distinguish them from finite words. The shift map T T \mathrm{T} is defined for bi-infinite words 𝐛 = ( b i ) i ∈ ℤ 𝐛 subscript subscript 𝑏 𝑖 𝑖 ℤ \mathbf{b}=(b_{i})_{i\in\mathbb{Z}} by T ​ ( 𝐛 ) = ( b i + 1 ) i ∈ ℤ T 𝐛 subscript subscript 𝑏 𝑖 1 𝑖 ℤ \mathrm{T}(\mathbf{b})=(b_{i+1})_{i\in\mathbb{Z}} and its k 𝑘 k -th iteration is denoted by T k superscript T 𝑘 \mathrm{T}^{k} . This extends to right-infinite words for k ≥ 0 𝑘 0 k\geq 0 and left-infinite words for k ≤ 0 𝑘 0 k\leq 0 . For finite words w ∈ 𝒜 ∗ 𝑤 superscript 𝒜 w\in\mathcal{A}^{*} , the shift map T T \mathrm{T} acts circularly, i.e., if w = x ​ v 𝑤 𝑥 𝑣 w=xv where x ∈ 𝒜 𝑥 𝒜 x\in\mathcal{A} , then T ​ ( w ) = v ​ x T 𝑤 𝑣 𝑥 \mathrm{T}(w)=vx . The set of all right-infinite words over 𝒜 𝒜 \mathcal{A} is denoted by 𝒜 ω superscript 𝒜 𝜔 \mathcal{A}^{\omega} , and we define 𝒜 ∞ := 𝒜 ∗ ∪ 𝒜 ω assign superscript 𝒜 superscript 𝒜 superscript 𝒜 𝜔 \mathcal{A}^{\infty}:=\mathcal{A}^{*}\cup\mathcal{A}^{\omega} . An ultimately periodic right-infinite word can be written as u ​ v ω = u ​ v ​ v ​ v ​ ⋯ 𝑢 superscript 𝑣 𝜔 𝑢 𝑣 𝑣 𝑣 ⋯ uv^{\omega}=uvvv\cdots , for some u 𝑢 u , v ∈ 𝒜 ∗ 𝑣 superscript 𝒜 v\in\mathcal{A}^{*} , v ≠ ε 𝑣 𝜀 v\neq\varepsilon . If u = ε 𝑢 𝜀 u=\varepsilon , then such a word is periodic . A right-infinite word that is not ultimately periodic is said to be aperiodic . Given a finite word w = x 1 ​ x 2 ​ ⋯ ​ x m ∈ 𝒜 ∗ 𝑤 subscript 𝑥 1 subscript 𝑥 2 ⋯ subscript 𝑥 𝑚 superscript 𝒜 w=x_{1}x_{2}\cdots x_{m}\in\mathcal{A}^{*} with each x i ∈ 𝒜 subscript 𝑥 𝑖 𝒜 x_{i}\in\mathcal{A} , the length of w 𝑤 w , denoted by | w | 𝑤 |w| , is equal to m 𝑚 m . By convention, the empty word ε 𝜀 \varepsilon is the unique word of length 0 0 . The number of occurrences of a letter a 𝑎 a in w 𝑤 w is denoted by | w | a subscript 𝑤 𝑎 |w|_{a} . If | w | a = 0 subscript 𝑤 𝑎 0 |w|_{a}=0 , then w 𝑤 w is said to be a 𝑎 a -free . The reversal w ~ ~ 𝑤 \widetilde{w} of w 𝑤 w is its mirror image: w ~ = x m ​ x m − 1 ​ ⋯ ​ x 1 ~ 𝑤 subscript 𝑥 𝑚 subscript 𝑥 𝑚 1 ⋯ subscript 𝑥 1 \widetilde{w}=x_{m}x_{m-1}\cdots x_{1} , and if w = w ~ 𝑤 ~ 𝑤 w=\widetilde{w} , then w 𝑤 w is called a palindrome . The reversal operator naturally extends to bi-infinite words; that is, the reversal of the bi-infinite word 𝐛 = 𝐥 . 𝐫 formulae-sequence 𝐛 𝐥 𝐫 \mathbf{b}=\mathbf{l}.\mathbf{r} , with 𝐥 𝐥 \mathbf{l} left-infinite and 𝐫 𝐫 \mathbf{r} right-infinite, is given by 𝐛 ~ = 𝐫 ~ . 𝐥 ~ formulae-sequence ~ 𝐛 ~ 𝐫 ~ 𝐥 \widetilde{\mathbf{b}}=\widetilde{\mathbf{r}}.\widetilde{\mathbf{l}} . A finite word w 𝑤 w is a factor of a finite or infinite word z 𝑧 z if z = u ​ w ​ v 𝑧 𝑢 𝑤 𝑣 z=uwv for some words u 𝑢 u , v 𝑣 v (which are finite or infinite depending on z 𝑧 z ). In the special case u = ε 𝑢 𝜀 u=\varepsilon (resp. v = ε 𝑣 𝜀 v=\varepsilon ), we call w 𝑤 w a prefix (resp. suffix ) of z 𝑧 z . We use the notation p − 1 ​ w superscript 𝑝 1 𝑤 p^{-1}w (resp. w ​ s − 1 𝑤 superscript 𝑠 1 ws^{-1} ) to indicate the removal of a prefix p 𝑝 p (resp. suffix s 𝑠 s ) of a finite word w 𝑤 w . Note that a prefix or suffix u 𝑢 u of a finite word w 𝑤 w is said to be proper if u ≠ w 𝑢 𝑤 u\neq w . A factor u 𝑢 u of a finite or infinite word w 𝑤 w is right (resp. left ) special if u ​ a 𝑢 𝑎 ua , u ​ b 𝑢 𝑏 ub (resp. a ​ u 𝑎 𝑢 au , b ​ u 𝑏 𝑢 bu ) are factors of w 𝑤 w for some letters a 𝑎 a , b ∈ 𝒜 𝑏 𝒜 b\in\mathcal{A} , a ≠ b 𝑎 𝑏 a\neq b . For any finite or infinite word w 𝑤 w , F ​ ( w ) 𝐹 𝑤 F(w) denotes the set of all its factors. Moreover, the alphabet of w 𝑤 w is Alph ( w ) := F ​ ( w ) ∩ 𝒜 assign 𝑤 𝐹 𝑤 𝒜 (w):=F(w)\cap\mathcal{A} and, if w 𝑤 w is infinite, we denote by Ult ( w ) 𝑤 (w) the set of all letters occurring infinitely often in w 𝑤 w . Any two infinite words 𝐱 𝐱 \mathbf{x} , 𝐲 𝐲 \mathbf{y} are said to be factor-equivalent if F ​ ( 𝐱 ) = F ​ ( 𝐲 ) 𝐹 𝐱 𝐹 𝐲 F(\mathbf{x})=F(\mathbf{y}) , i.e., if 𝐱 𝐱 \mathbf{x} and 𝐲 𝐲 \mathbf{y} have the same set of factors. A factor of an infinite word 𝐱 𝐱 \mathbf{x} is recurrent in 𝐱 𝐱 \mathbf{x} if it occurs infinitely often in 𝐱 𝐱 \mathbf{x} , and 𝐱 𝐱 \mathbf{x} itself is said to be recurrent if all of its factors are recurrent in it. For a bi-infinite word to be recurrent, any factor must occur infinitely often to the left and to the right. An infinite word is said to be uniformly recurrent if any factor occurs infinitely many times in it with bounded gaps [ 37 ] . A morphism φ 𝜑 \varphi on 𝒜 𝒜 \mathcal{A} is a map from 𝒜 ∗ superscript 𝒜 \mathcal{A}^{*} to 𝒜 ∗ superscript 𝒜 \mathcal{A}^{*} such that φ ​ ( u ​ v ) = φ ​ ( u ) ​ φ ​ ( v ) 𝜑 𝑢 𝑣 𝜑 𝑢 𝜑 𝑣 \varphi(uv)=\varphi(u)\varphi(v) for any words u 𝑢 u , v 𝑣 v over 𝒜 𝒜 \mathcal{A} . A morphism on 𝒜 𝒜 \mathcal{A} is entirely defined by the images of letters in 𝒜 𝒜 \mathcal{A} . All morphisms considered in this paper will be non-erasing : the image of any non-empty word is never empty. Hence the action of a morphism φ 𝜑 \varphi on 𝒜 ∗ superscript 𝒜 \mathcal{A}^{*} can be naturally extended to infinite words; that is, if 𝐱 = x 1 ​ x 2 ​ x 3 ​ ⋯ ∈ 𝒜 ω 𝐱 subscript 𝑥 1 subscript 𝑥 2 subscript 𝑥 3 ⋯ superscript 𝒜 𝜔 \mathbf{x}=x_{1}x_{2}x_{3}\cdots\in\mathcal{A}^{\omega} , then f ​ ( 𝐱 ) = f ​ ( x 1 ) ​ f ​ ( x 2 ) ​ f ​ ( x 3 ) ​ ⋯ 𝑓 𝐱 𝑓 subscript 𝑥 1 𝑓 subscript 𝑥 2 𝑓 subscript 𝑥 3 ⋯ f(\mathbf{x})=f(x_{1})f(x_{2})f(x_{3})\cdots . An infinite word 𝐱 𝐱 \mathbf{x} can therefore be a fixed point of a morphism φ 𝜑 \varphi , i.e., φ ​ ( 𝐱 ) = 𝐱 𝜑 𝐱 𝐱 \varphi(\mathbf{x})=\mathbf{x} . If φ 𝜑 \varphi is a (non-erasing) morphism such that φ ​ ( a ) = a ​ w 𝜑 𝑎 𝑎 𝑤 \varphi(a)=aw for some letter a ∈ 𝒜 𝑎 𝒜 a\in\mathcal{A} and w ∈ 𝒜 + 𝑤 superscript 𝒜 w\in\mathcal{A}^{+} , then φ n ​ ( a ) superscript 𝜑 𝑛 𝑎 \varphi^{n}(a) is a proper prefix of the word φ n + 1 ​ ( a ) superscript 𝜑 𝑛 1 𝑎 \varphi^{n+1}(a) for each n ∈ ℕ 𝑛 ℕ n\in\mathbb{N} , and the limit of the sequence ( φ n ​ ( a ) ) n ≥ 0 subscript superscript 𝜑 𝑛 𝑎 𝑛 0 (\varphi^{n}(a))_{n\geq 0} is the unique infinite word: 𝐰 = lim n → ∞ ​ φ n ​ ( a ) = φ ω ​ ( a ) ( = a ​ w ​ φ ​ ( w ) ​ φ 2 ​ ( w ) ​ φ 3 ​ ( w ) ​ ⋯ ) . 𝐰 → 𝑛 superscript 𝜑 𝑛 𝑎 annotated superscript 𝜑 𝜔 𝑎 absent 𝑎 𝑤 𝜑 𝑤 superscript 𝜑 2 𝑤 superscript 𝜑 3 𝑤 ⋯ \mathbf{w}=\underset{n\rightarrow\infty}{\lim}\varphi^{n}(a)=\varphi^{\omega}(a)~{}(=aw\varphi(w)\varphi^{2}(w)\varphi^{3}(w)\cdots). Clearly, 𝐰 𝐰 \mathbf{w} is a fixed point of φ 𝜑 \varphi and we say that 𝐰 𝐰 \mathbf{w} is generated by φ 𝜑 \varphi . Furthermore, an infinite word generated by a morphism is said to be purely morphic . In what follows, we will denote the composition of morphisms by juxtaposition as for concatenation of words.

In this paper, we survey the rich theory of infinite episturmian words which generalize to any finite alphabet, in a rather resembling way, the well-known family of Sturmian words on two letters. After recalling definitions and basic properties, we consider episturmian morphisms that allow for a deeper study of these words. Some properties of factors are described, including factor complexity, palindromes, fractional powers, frequencies, and return words. We also consider lexicographical properties of episturmian words, as well as their connection to the balance property, and related notions such as finite episturmian words, Arnoux-Rauzy sequences, and "episkew words" that generalize the skew words of Morse and Hedlund.

[{"content":"Help me to generate the abstract of a survey paper given the title:*Episturmian words: a survey* and the introduction:1.1 From Sturmian to episturmian Most renowned amongst the branches of combinatorics on words is the theory of infinite binary sequences called Sturmian words , which are fascinating in many respects, having been studied from combinatorial, algebraic, and geometric points of view. Their beautiful properties are related to many fields such as Number Theory, Geometry, Symbolic Dynamical Systems, Theoretical Physics, and Theoretical Computer Science (see 7 , 83 , 96 for recent surveys). Since the seminal works of Morse and Hedlund 91 , Sturmian words have been shown to admit numerous equivalent definitions and characterizations. For instance, it is well known that an infinite word 𝐰 𝐰 \mathbf{w} over { a , b } 𝑎 𝑏 \{a,b\} is Sturmian if and only if 𝐰 𝐰 \mathbf{w} is aperiodic and balanced : for any two factors u 𝑢 u , v 𝑣 v of 𝐰 𝐰 \mathbf{w} of the same length, the number of a 𝑎 a s in each of u 𝑢 u and v 𝑣 v differs by at most 1 1 1 . Sturmian words are also characterized by their factor complexity function (which counts the number of distinct factors of each length): they have exactly n + 1 𝑛 1 n+1 distinct factors of length n 𝑛 n for each n 𝑛 n . In this sense, Sturmian words are precisely the aperiodic infinite words of minimal factor complexity since, as is well known, an infinite word is ultimately periodic if and only if it has less than n + 1 𝑛 1 n+1 factors of length n 𝑛 n for some n 𝑛 n (see 37 ). Many interesting properties of Sturmian words can be attributed to their low complexity, which induces certain regularities in such words without, however, making them periodic. Sturmian words can also be geometrically realized as cutting sequences by considering the sequence of cuts in an integer grid made by a line of irrational slope (see for instance 38 , 13 ). They also provide a symbolic coding of the orbit of a point on a circle with respect to a rotation by an irrational number (see 91 , 4 ). All of the above characteristic properties of Sturmian words lead to natural generalizations on arbitrary finite alphabets. In one direction, the balance property naturally extends to an alphabet with more than two letters (e.g., see 68 , 110 , 115 ) as does the following generalized balance property that also characterizes Sturmian words (see 49 , 1 ): the difference between the number of occurrences of a word u 𝑢 u in any pair of factors of the same length is at most 1 1 1 . In another direction, we could consider relaxing the minimality condition for the factor complexity p ​ ( n ) 𝑝 𝑛 p(n) . For example, quasi-Sturmian words are infinite words for which there exist two positive integers N 𝑁 N and c 𝑐 c such that n + 1 ≤ p ​ ( n ) ≤ n + c 𝑛 1 𝑝 𝑛 𝑛 𝑐 n+1\leq p(n)\leq n+c for all n ≥ N 𝑛 𝑁 n\geq N . This generalization was introduced in 5 when studying the transcendence of certain continued fraction expansions. See also 31 , 36 , 66 , 105 for similar extensions of Sturmian words with respect to factor complexity. From the geometric point of view, cutting sequences naturally generalize to trajectories in the hypercube billiard (e.g., see 25 ), and codings of rotational orbits carry over to codings of interval exchange transformations (e.g., see 18 ). Two other very interesting natural generalizations of Sturmian words are Arnoux-Rauzy sequences 12 , 97 and episturmian words 43 , 73 , which we will now define. From the factor complexity of Sturmian words, it immediately follows that any Sturmian word is over a 2 2 2 -letter alphabet and has exactly one left special factor of each length. A factor u 𝑢 u of a finite or infinite word w 𝑤 w is said to be left special (resp. right special ) in w 𝑤 w if there exists at least two distinct letters a 𝑎 a , b 𝑏 b such that a ​ u 𝑎 𝑢 au and b ​ u 𝑏 𝑢 bu (resp. u ​ a 𝑢 𝑎 ua , u ​ b 𝑢 𝑏 ub ) are factors of w 𝑤 w . Extending the left special property of Sturmian words, a recurrent infinite word 𝐰 𝐰 \mathbf{w} over a finite alphabet 𝒜 𝒜 \mathcal{A} is said to be an Arnoux-Rauzy sequence (or a strict episturmian word ) if it has exactly one left special factor and one right special factor of each length, and for every left (resp. right) special factor u 𝑢 u of 𝐰 𝐰 \mathbf{w} , x ​ u 𝑥 𝑢 xu (resp. u ​ x 𝑢 𝑥 ux ) is a factor of 𝐰 𝐰 \mathbf{w} for all letters x ∈ 𝒜 𝑥 𝒜 x\in\mathcal{A} . A noteable property that is shared by Sturmian words and Arnouxy-Rauzy sequences is their closure under reversal, i.e., if u 𝑢 u is a factor of such a word, then its reversal is also a factor. This nice property inspired Droubay, Justin, and Pirillos generalization of Sturmian words in 43 : an infinite word is episturmian if it is closed under reversal and has at most one left special factor of each length. Sturmian, Arnoux-Rauzy, and episturmian words all have standard (or characteristic ) elements, which are those having all of their left special factors as prefixes. Within these families of words, standard words are good representatives in the sense that an infinite word belongs to one such family if and only if it has the same set of factors as some standard word in that family. From the definitions, it is clear that the family of Arnoux-Rauzy sequences is a particular subclass of the family of episturmian words. More precisely, episturmian words are composed of the Arnoux-Rauzy sequences, images of the Arnoux-Rauzy sequences by episturmian morphisms , and certain periodic infinite words (see Section 5 ). In the 2 2 2 -letter case, Arnoux-Rauzy sequences are exactly the Sturmian words whereas episturmian words include all recurrent balanced words, i.e., periodic balanced words and Sturmian words. The study of episturmian words and Arnoux-Rauzy sequences has enjoyed a great deal of popularity in recent times, owing mostly to the many properties that they share with Sturmian words. In this paper we survey the purely combinatorial work on episturmian words, beginning with their definition and basic properties in Section 2 . Then, in Section 3 , we recall episturmian morphisms which allow for a deeper study of episturmian words. In particular, any episturmian word is the image of another episturmian word by some so-called pure episturmian morphism . Even more, any episturmian word can be infinitely decomposed over the set of pure episturmian morphisms. This last property allows an episturmian word to be defined by one of its morphic decompositions or, equivalently, by a certain directive word , which is an infinite sequence of rules for decomposing the given episturmian word by morphisms. In Section 4 we consider notions such as shifts , spins , and block-equivalence in connection with directive words, which allow us to study when two different spinned infinite words direct the same episturmian word. We also consider periodic and purely morphic episturmian words. In Section 5 , our discussion briefly turns to Arnoux-Rauzy sequences and finite episturmian words . Following this, we study in Section 6 some properties of factors of episturmian words (and Arnoux-Rauzy sequences), including factor complexity, palindromes, fractional powers, frequencies, and return words. Lastly, we consider more recent work involving lexicographic order and the balance property (including Fraenkels conjecture ). 1.2 Notation & terminology We assume the reader is familiar with combinatorics on words and morphisms (e.g., see 82 , 83 ). In this section, we recall some basic definitions and properties relating to episturmian words which are needed throughout the paper. For the most part, we follow the notation and terminology of 43 , 73 , 75 , 62 . Let 𝒜 𝒜 \mathcal{A} denote a finite alphabet , i.e., a non-empty finite set of symbols called letters . A finite word over 𝒜 𝒜 \mathcal{A} is a finite sequence of letters from 𝒜 𝒜 \mathcal{A} . The empty word ε 𝜀 \varepsilon is the empty sequence. Under the operation of concatenation, the set 𝒜 ∗ superscript 𝒜 \mathcal{A}^{*} of all finite words over 𝒜 𝒜 \mathcal{A} is a free monoid with identity element ε 𝜀 \varepsilon and set of generators 𝒜 𝒜 \mathcal{A} . The set of non-empty words over 𝒜 𝒜 \mathcal{A} is the free semigroup 𝒜 + := 𝒜 ∗ ∖ { ε } assign superscript 𝒜 superscript 𝒜 𝜀 \mathcal{A}^{+}:=\mathcal{A}^{*}\setminus\{\varepsilon\} . A right-infinite (resp. left-infinite , bi-infinite ) word over 𝒜 𝒜 \mathcal{A} is a sequence indexed by ℕ + superscript ℕ \mathbb{N}^{+} (resp. ℤ ∖ ℕ + ℤ superscript ℕ \mathbb{Z}\setminus\mathbb{N}^{+} , ℤ ℤ \mathbb{Z} ) with values in 𝒜 𝒜 \mathcal{A} . For instance, a left-infinite word is represented by 𝐮 = ⋯ ​ b − 2 ​ b − 1 ​ b 0 𝐮 ⋯ subscript 𝑏 2 subscript 𝑏 1 subscript 𝑏 0 \mathbf{u}=\cdots b_{-2}b_{-1}b_{0} and a right-infinite word by 𝐯 = b 1 ​ b 2 ​ b 3 ​ ⋯ 𝐯 subscript 𝑏 1 subscript 𝑏 2 subscript 𝑏 3 ⋯ \mathbf{v}=b_{1}b_{2}b_{3}\cdots where b i ∈ 𝒜 subscript 𝑏 𝑖 𝒜 b_{i}\in\mathcal{A} . The concatenation of 𝐮 𝐮 \mathbf{u} and 𝐯 𝐯 \mathbf{v} gives the bi-infinite word 𝐮 . 𝐯 = ⋯ ​ b − 2 ​ b − 1 ​ b 0 . b 1 ​ b 2 ​ b 3 ​ ⋯ formulae-sequence 𝐮 𝐯 ⋯ subscript 𝑏 2 subscript 𝑏 1 subscript 𝑏 0 subscript 𝑏 1 subscript 𝑏 2 subscript 𝑏 3 ⋯ \mathbf{u}.\mathbf{v}=\cdots b_{-2}b_{-1}b_{0}.b_{1}b_{2}b_{3}\cdots with a dot written between b 0 subscript 𝑏 0 b_{0} and b 1 subscript 𝑏 1 b_{1} to avoid ambiguity. For easier reading, infinite words are hereafter typically typed in boldface to distinguish them from finite words. The shift map T T \mathrm{T} is defined for bi-infinite words 𝐛 = ( b i ) i ∈ ℤ 𝐛 subscript subscript 𝑏 𝑖 𝑖 ℤ \mathbf{b}=(b_{i})_{i\in\mathbb{Z}} by T ​ ( 𝐛 ) = ( b i + 1 ) i ∈ ℤ T 𝐛 subscript subscript 𝑏 𝑖 1 𝑖 ℤ \mathrm{T}(\mathbf{b})=(b_{i+1})_{i\in\mathbb{Z}} and its k 𝑘 k -th iteration is denoted by T k superscript T 𝑘 \mathrm{T}^{k} . This extends to right-infinite words for k ≥ 0 𝑘 0 k\geq 0 and left-infinite words for k ≤ 0 𝑘 0 k\leq 0 . For finite words w ∈ 𝒜 ∗ 𝑤 superscript 𝒜 w\in\mathcal{A}^{*} , the shift map T T \mathrm{T} acts circularly, i.e., if w = x ​ v 𝑤 𝑥 𝑣 w=xv where x ∈ 𝒜 𝑥 𝒜 x\in\mathcal{A} , then T ​ ( w ) = v ​ x T 𝑤 𝑣 𝑥 \mathrm{T}(w)=vx . The set of all right-infinite words over 𝒜 𝒜 \mathcal{A} is denoted by 𝒜 ω superscript 𝒜 𝜔 \mathcal{A}^{\omega} , and we define 𝒜 ∞ := 𝒜 ∗ ∪ 𝒜 ω assign superscript 𝒜 superscript 𝒜 superscript 𝒜 𝜔 \mathcal{A}^{\infty}:=\mathcal{A}^{*}\cup\mathcal{A}^{\omega} . An ultimately periodic right-infinite word can be written as u ​ v ω = u ​ v ​ v ​ v ​ ⋯ 𝑢 superscript 𝑣 𝜔 𝑢 𝑣 𝑣 𝑣 ⋯ uv^{\omega}=uvvv\cdots , for some u 𝑢 u , v ∈ 𝒜 ∗ 𝑣 superscript 𝒜 v\in\mathcal{A}^{*} , v ≠ ε 𝑣 𝜀 v\neq\varepsilon . If u = ε 𝑢 𝜀 u=\varepsilon , then such a word is periodic . A right-infinite word that is not ultimately periodic is said to be aperiodic . Given a finite word w = x 1 ​ x 2 ​ ⋯ ​ x m ∈ 𝒜 ∗ 𝑤 subscript 𝑥 1 subscript 𝑥 2 ⋯ subscript 𝑥 𝑚 superscript 𝒜 w=x_{1}x_{2}\cdots x_{m}\in\mathcal{A}^{*} with each x i ∈ 𝒜 subscript 𝑥 𝑖 𝒜 x_{i}\in\mathcal{A} , the length of w 𝑤 w , denoted by | w | 𝑤 |w| , is equal to m 𝑚 m . By convention, the empty word ε 𝜀 \varepsilon is the unique word of length 0 0 . The number of occurrences of a letter a 𝑎 a in w 𝑤 w is denoted by | w | a subscript 𝑤 𝑎 |w|_{a} . If | w | a = 0 subscript 𝑤 𝑎 0 |w|_{a}=0 , then w 𝑤 w is said to be a 𝑎 a -free . The reversal w ~ ~ 𝑤 \widetilde{w} of w 𝑤 w is its mirror image: w ~ = x m ​ x m − 1 ​ ⋯ ​ x 1 ~ 𝑤 subscript 𝑥 𝑚 subscript 𝑥 𝑚 1 ⋯ subscript 𝑥 1 \widetilde{w}=x_{m}x_{m-1}\cdots x_{1} , and if w = w ~ 𝑤 ~ 𝑤 w=\widetilde{w} , then w 𝑤 w is called a palindrome . The reversal operator naturally extends to bi-infinite words; that is, the reversal of the bi-infinite word 𝐛 = 𝐥 . 𝐫 formulae-sequence 𝐛 𝐥 𝐫 \mathbf{b}=\mathbf{l}.\mathbf{r} , with 𝐥 𝐥 \mathbf{l} left-infinite and 𝐫 𝐫 \mathbf{r} right-infinite, is given by 𝐛 ~ = 𝐫 ~ . 𝐥 ~ formulae-sequence ~ 𝐛 ~ 𝐫 ~ 𝐥 \widetilde{\mathbf{b}}=\widetilde{\mathbf{r}}.\widetilde{\mathbf{l}} . A finite word w 𝑤 w is a factor of a finite or infinite word z 𝑧 z if z = u ​ w ​ v 𝑧 𝑢 𝑤 𝑣 z=uwv for some words u 𝑢 u , v 𝑣 v (which are finite or infinite depending on z 𝑧 z ). In the special case u = ε 𝑢 𝜀 u=\varepsilon (resp. v = ε 𝑣 𝜀 v=\varepsilon ), we call w 𝑤 w a prefix (resp. suffix ) of z 𝑧 z . We use the notation p − 1 ​ w superscript 𝑝 1 𝑤 p^{-1}w (resp. w ​ s − 1 𝑤 superscript 𝑠 1 ws^{-1} ) to indicate the removal of a prefix p 𝑝 p (resp. suffix s 𝑠 s ) of a finite word w 𝑤 w . Note that a prefix or suffix u 𝑢 u of a finite word w 𝑤 w is said to be proper if u ≠ w 𝑢 𝑤 u\neq w . A factor u 𝑢 u of a finite or infinite word w 𝑤 w is right (resp. left ) special if u ​ a 𝑢 𝑎 ua , u ​ b 𝑢 𝑏 ub (resp. a ​ u 𝑎 𝑢 au , b ​ u 𝑏 𝑢 bu ) are factors of w 𝑤 w for some letters a 𝑎 a , b ∈ 𝒜 𝑏 𝒜 b\in\mathcal{A} , a ≠ b 𝑎 𝑏 a\neq b . For any finite or infinite word w 𝑤 w , F ​ ( w ) 𝐹 𝑤 F(w) denotes the set of all its factors. Moreover, the alphabet of w 𝑤 w is Alph ( w ) := F ​ ( w ) ∩ 𝒜 assign 𝑤 𝐹 𝑤 𝒜 (w):=F(w)\cap\mathcal{A} and, if w 𝑤 w is infinite, we denote by Ult ( w ) 𝑤 (w) the set of all letters occurring infinitely often in w 𝑤 w . Any two infinite words 𝐱 𝐱 \mathbf{x} , 𝐲 𝐲 \mathbf{y} are said to be factor-equivalent if F ​ ( 𝐱 ) = F ​ ( 𝐲 ) 𝐹 𝐱 𝐹 𝐲 F(\mathbf{x})=F(\mathbf{y}) , i.e., if 𝐱 𝐱 \mathbf{x} and 𝐲 𝐲 \mathbf{y} have the same set of factors. A factor of an infinite word 𝐱 𝐱 \mathbf{x} is recurrent in 𝐱 𝐱 \mathbf{x} if it occurs infinitely often in 𝐱 𝐱 \mathbf{x} , and 𝐱 𝐱 \mathbf{x} itself is said to be recurrent if all of its factors are recurrent in it. For a bi-infinite word to be recurrent, any factor must occur infinitely often to the left and to the right. An infinite word is said to be uniformly recurrent if any factor occurs infinitely many times in it with bounded gaps 37 . A morphism φ 𝜑 \varphi on 𝒜 𝒜 \mathcal{A} is a map from 𝒜 ∗ superscript 𝒜 \mathcal{A}^{*} to 𝒜 ∗ superscript 𝒜 \mathcal{A}^{*} such that φ ​ ( u ​ v ) = φ ​ ( u ) ​ φ ​ ( v ) 𝜑 𝑢 𝑣 𝜑 𝑢 𝜑 𝑣 \varphi(uv)=\varphi(u)\varphi(v) for any words u 𝑢 u , v 𝑣 v over 𝒜 𝒜 \mathcal{A} . A morphism on 𝒜 𝒜 \mathcal{A} is entirely defined by the images of letters in 𝒜 𝒜 \mathcal{A} . All morphisms considered in this paper will be non-erasing : the image of any non-empty word is never empty. Hence the action of a morphism φ 𝜑 \varphi on 𝒜 ∗ superscript 𝒜 \mathcal{A}^{*} can be naturally extended to infinite words; that is, if 𝐱 = x 1 ​ x 2 ​ x 3 ​ ⋯ ∈ 𝒜 ω 𝐱 subscript 𝑥 1 subscript 𝑥 2 subscript 𝑥 3 ⋯ superscript 𝒜 𝜔 \mathbf{x}=x_{1}x_{2}x_{3}\cdots\in\mathcal{A}^{\omega} , then f ​ ( 𝐱 ) = f ​ ( x 1 ) ​ f ​ ( x 2 ) ​ f ​ ( x 3 ) ​ ⋯ 𝑓 𝐱 𝑓 subscript 𝑥 1 𝑓 subscript 𝑥 2 𝑓 subscript 𝑥 3 ⋯ f(\mathbf{x})=f(x_{1})f(x_{2})f(x_{3})\cdots . An infinite word 𝐱 𝐱 \mathbf{x} can therefore be a fixed point of a morphism φ 𝜑 \varphi , i.e., φ ​ ( 𝐱 ) = 𝐱 𝜑 𝐱 𝐱 \varphi(\mathbf{x})=\mathbf{x} . If φ 𝜑 \varphi is a (non-erasing) morphism such that φ ​ ( a ) = a ​ w 𝜑 𝑎 𝑎 𝑤 \varphi(a)=aw for some letter a ∈ 𝒜 𝑎 𝒜 a\in\mathcal{A} and w ∈ 𝒜 + 𝑤 superscript 𝒜 w\in\mathcal{A}^{+} , then φ n ​ ( a ) superscript 𝜑 𝑛 𝑎 \varphi^{n}(a) is a proper prefix of the word φ n + 1 ​ ( a ) superscript 𝜑 𝑛 1 𝑎 \varphi^{n+1}(a) for each n ∈ ℕ 𝑛 ℕ n\in\mathbb{N} , and the limit of the sequence ( φ n ​ ( a ) ) n ≥ 0 subscript superscript 𝜑 𝑛 𝑎 𝑛 0 (\varphi^{n}(a))_{n\geq 0} is the unique infinite word: 𝐰 = lim n → ∞ ​ φ n ​ ( a ) = φ ω ​ ( a ) ( = a ​ w ​ φ ​ ( w ) ​ φ 2 ​ ( w ) ​ φ 3 ​ ( w ) ​ ⋯ ) . 𝐰 → 𝑛 superscript 𝜑 𝑛 𝑎 annotated superscript 𝜑 𝜔 𝑎 absent 𝑎 𝑤 𝜑 𝑤 superscript 𝜑 2 𝑤 superscript 𝜑 3 𝑤 ⋯ \mathbf{w}=\underset{n\rightarrow\infty}{\lim}\varphi^{n}(a)=\varphi^{\omega}(a)~{}(=aw\varphi(w)\varphi^{2}(w)\varphi^{3}(w)\cdots). Clearly, 𝐰 𝐰 \mathbf{w} is a fixed point of φ 𝜑 \varphi and we say that 𝐰 𝐰 \mathbf{w} is generated by φ 𝜑 \varphi . Furthermore, an infinite word generated by a morphism is said to be purely morphic . In what follows, we will denote the composition of morphisms by juxtaposition as for concatenation of words.","role":"user"},{"content":"Abstract: In this paper, we survey the rich theory of infinite episturmian words which generalize to any finite alphabet, in a rather resembling way, the well-known family of Sturmian words on two letters. After recalling definitions and basic properties, we consider episturmian morphisms that allow for a deeper study of these words. Some properties of factors are described, including factor complexity, palindromes, fractional powers, frequencies, and return words. We also consider lexicographical properties of episturmian words, as well as their connection to the balance property, and related notions such as finite episturmian words, Arnoux-Rauzy sequences, and "episkew words" that generalize the skew words of Morse and Hedlund.","role":"assistant"}]

The Charm Renaissance: $D^-$ Physics - A Selective Review

In recent years, no doubt because of the success of the B 𝐵 B -physics programme at BaBar and Belle, charm has been considered the poor relation of heavy-quark physics. This is now changing: while B 𝐵 B -physics remains the biggest show in town, it is no longer the only show. As will be argued, measurements in the D 𝐷 D -sector have a vital, although indirect, role to play in pinning down the value of certain critical parameters of flavour-physics. Furthermore, the charm system provides a powerful laboratory in its own right to search for contributions from non-Standard Model (SM) processes. Three main reasons can be identified which explain why charm physics is once more, quite correctly, being perceived as an important and exciting discipline: 1. Precision CKM Tests The success of the B 𝐵 B -factories and the Tevatron has meant that CKM unitarity triangle tests are achieving successively higher levels of precision. This progress will continue with the LHCb experiment at CERN. Although the CKM elements being studied are those accessible in B 𝐵 B -decays, charm turns out to be a vital ingredient in the programme. 2. Charm Mixing and its Legacy The discovery of D 0 − D 0 ¯ superscript 𝐷 0 ¯ superscript 𝐷 0 D^{0}-\overline{D^{0}} oscillations has been the most exciting event of the past couple of years in high energy physics. The higher than expected rate is (arguably) intriguing in its own right, and points the way forward to searches for CP violation (CPV) in the charm sector. 3. Recent Discoveries in Spectroscopy The discovery of several missing charmonium states, and a number of unexpected and possibly exotic resonances (the X 𝑋 X , Y 𝑌 Y and Z 𝑍 Z ) has rekindled interest in the c ​ c ¯ 𝑐 ¯ 𝑐 c\overline{c} system as a laboratory for studying QCD. In this review we focus on the first two topics. Useful discussion of the third item may be found in [ 1 ] .

In recent years, no doubt because of the success of the B-physics programme at BaBar and Belle, charm has been considered the poor relation of heavy-quark physics. This is now changing: while B-physics remains the biggest show in town, it is no longer the only show. As will be argued, measurements in the D-sector have a vital, although indirect, role to play in pinning down the value of certain critical parameters of flavour-physics. Furthermore, the charm system provides a powerful laboratory in its own right to search for contributions from non-Standard Model (SM) processes. Three main reasons can be identified which explain why charm physics is once more, quite correctly, being perceived as an important and exciting discipline:

[{"content":"Help me to generate the abstract of a survey paper given the title:*The Charm Renaissance: $D^-$ Physics - A Selective Review* and the introduction:In recent years, no doubt because of the success of the B 𝐵 B -physics programme at BaBar and Belle, charm has been considered the poor relation of heavy-quark physics. This is now changing: while B 𝐵 B -physics remains the biggest show in town, it is no longer the only show. As will be argued, measurements in the D 𝐷 D -sector have a vital, although indirect, role to play in pinning down the value of certain critical parameters of flavour-physics. Furthermore, the charm system provides a powerful laboratory in its own right to search for contributions from non-Standard Model (SM) processes. Three main reasons can be identified which explain why charm physics is once more, quite correctly, being perceived as an important and exciting discipline: 1. Precision CKM Tests The success of the B 𝐵 B -factories and the Tevatron has meant that CKM unitarity triangle tests are achieving successively higher levels of precision. This progress will continue with the LHCb experiment at CERN. Although the CKM elements being studied are those accessible in B 𝐵 B -decays, charm turns out to be a vital ingredient in the programme. 2. Charm Mixing and its Legacy The discovery of D 0 − D 0 ¯ superscript 𝐷 0 ¯ superscript 𝐷 0 D^{0}-\overline{D^{0}} oscillations has been the most exciting event of the past couple of years in high energy physics. The higher than expected rate is (arguably) intriguing in its own right, and points the way forward to searches for CP violation (CPV) in the charm sector. 3. Recent Discoveries in Spectroscopy The discovery of several missing charmonium states, and a number of unexpected and possibly exotic resonances (the X 𝑋 X , Y 𝑌 Y and Z 𝑍 Z ) has rekindled interest in the c ​ c ¯ 𝑐 ¯ 𝑐 c\overline{c} system as a laboratory for studying QCD. In this review we focus on the first two topics. Useful discussion of the third item may be found in 1 .","role":"user"},{"content":"Abstract: In recent years, no doubt because of the success of the B-physics programme at BaBar and Belle, charm has been considered the poor relation of heavy-quark physics. This is now changing: while B-physics remains the biggest show in town, it is no longer the only show. As will be argued, measurements in the D-sector have a vital, although indirect, role to play in pinning down the value of certain critical parameters of flavour-physics. Furthermore, the charm system provides a powerful laboratory in its own right to search for contributions from non-Standard Model (SM) processes. Three main reasons can be identified which explain why charm physics is once more, quite correctly, being perceived as an important and exciting discipline:","role":"assistant"}]

Generalized permutation patterns - a short survey

1. Introduction Patterns in permutations have been studied sporadically, often implicitly, for over a century, but in the last two decades this area has grown explosively, with several hundred published papers. As seems to be the case with most things in enumerative combinatorics, some instances of permutation patterns can be found already in MacMahon’s classical book from 1915, Combinatory Analysis [ 48 ] . In the seminal paper Restricted permutations of Simion and Schmidt [ 56 ] from 1985 the systematic study of permutation patterns was launched, and it now seems clear that this field will continue growing for a long time to come, due to its plethora of problems that range from the easy to the seemingly impossible, with a rich middle ground of challenging but solvable problems. Most important, perhaps, for the future growth of the subject, is the wealth of connections to other branches of combinatorics, other fields of mathematics, and to other disciplines such as computer science and physics. Whereas an occurrence of a classical pattern p 𝑝 p in a permutation π 𝜋 \pi is simply a subsequence of π 𝜋 \pi whose letters are in the same relative order (of size) as those in p 𝑝 p , in an occurrence of a generalized pattern , some letters of that subsequence may be required to be adjacent in the permutation. For example, the classical pattern 1 ​ ​ 2 ​ ​ 3 ​ ​ 4 1 2 3 4 1\makebox[4.73611pt]{\rule[2.49721pt]{3.05696pt}{0.6458pt}}2\makebox[4.73611pt]{\rule[2.49721pt]{3.05696pt}{0.6458pt}}3\makebox[4.73611pt]{\rule[2.49721pt]{3.05696pt}{0.6458pt}}4 simply corresponds to an increasing subsequence of length four, whereas an occurrence of the generalized pattern 1 ​ ​ 23 ​ ​ 4 1 23 4 1\makebox[4.73611pt]{\rule[2.49721pt]{3.05696pt}{0.6458pt}}23\makebox[4.73611pt]{\rule[2.49721pt]{3.05696pt}{0.6458pt}}4 would require the middle two letters of that sequence to be adjacent in π 𝜋 \pi , due to the absence of a dash between 2 and 3. Thus, the permutation 23145 23145 23145 contains 1 ​ ​ 2 ​ ​ 3 ​ ​ 4 1 2 3 4 1\makebox[4.73611pt]{\rule[2.49721pt]{3.05696pt}{0.6458pt}}2\makebox[4.73611pt]{\rule[2.49721pt]{3.05696pt}{0.6458pt}}3\makebox[4.73611pt]{\rule[2.49721pt]{3.05696pt}{0.6458pt}}4 but not 1 ​ ​ 23 ​ ​ 4 1 23 4 1\makebox[4.73611pt]{\rule[2.49721pt]{3.05696pt}{0.6458pt}}23\makebox[4.73611pt]{\rule[2.49721pt]{3.05696pt}{0.6458pt}}4 . Note that for the classical patterns, our notation differs from the usual one, since the dashes we have between every pair of adjacent letters in a classical pattern are usually omitted when only classical patterns are being considered. It is well known that the number of permutations of length n 𝑛 n avoiding any one classical pattern of length 3 is the n 𝑛 n -th Catalan number, which counts a myriad different combinatorial objects. There are many other results in this direction, relating pattern avoiding permutations to various other combinatorial structures, either via bijections, or by such classes of permutations being equinumerous to the structures in question without there being a known bijection. Counting permutations according to the number of occurrences of generalized patterns one comes up with a vast array of known sequences, such as the Euler numbers, Stirling numbers of both kinds, Motzkin numbers, Entringer numbers, Schröder numbers, Fibonacci numbers, Pell numbers and many more. Also, one often finds lesser known sequences that are nevertheless related to known structures, such as directed animals, planar maps, permutation tableaux, various kinds of trees and involutions in S n subscript 𝑆 𝑛 S_{n} , to name a few. Thus, generalized patterns provide a significant addition to the already sizable flora of classical patterns and their connections to other combinatorial structures. In fact, due to their great diversity, the non-classical generalized patterns are likely to provide richer connections to other combinatorial structures than the classical ones do. Supporting this is the fact that the recently proved Stanley-Wilf conjecture—which gives a strong bound for the growth rate of the number of permutations of length n 𝑛 n avoiding a classical pattern—does not hold for some generalized patterns. This paper is organized as follows: In Section 2 we introduce definitions and in Section 3 we mention implicit appearances of generalized patterns in the literature. In Sections 4 and 5 we survey what is known about the avoidance of generalized patterns of length three and four, respectively. In Section 6 we give some examples where generalized patterns have shown up in very different contexts, establishing connections to various other combinatorial structures, some of which seem quite deep. Section 7 lists several instances of so-called barred patterns that turn out to be equivalent to generalized patterns and Section 8 deals with asymptotics for avoidance of generalized patterns. Finally, in Section 9 , we mention some further generalizations of the generalized patterns.

An occurrence of a classical pattern p in a permutation π is a subsequence of π whose letters are in the same relative order (of size) as those in p. In an occurrence of a generalized pattern, some letters of that subsequence may be required to be adjacent in the permutation. Subsets of permutations characterized by the avoidance—or the prescribed number of occurrences— of generalized patterns exhibit connections to an enormous variety of other combinatorial structures, some of them apparently deep. We give a short overview of the state of the art for generalized patterns.

[{"content":"Help me to generate the abstract of a survey paper given the title:*Generalized permutation patterns - a short survey* and the introduction:1. Introduction Patterns in permutations have been studied sporadically, often implicitly, for over a century, but in the last two decades this area has grown explosively, with several hundred published papers. As seems to be the case with most things in enumerative combinatorics, some instances of permutation patterns can be found already in MacMahons classical book from 1915, Combinatory Analysis 48 . In the seminal paper Restricted permutations of Simion and Schmidt 56 from 1985 the systematic study of permutation patterns was launched, and it now seems clear that this field will continue growing for a long time to come, due to its plethora of problems that range from the easy to the seemingly impossible, with a rich middle ground of challenging but solvable problems. Most important, perhaps, for the future growth of the subject, is the wealth of connections to other branches of combinatorics, other fields of mathematics, and to other disciplines such as computer science and physics. Whereas an occurrence of a classical pattern p 𝑝 p in a permutation π 𝜋 \pi is simply a subsequence of π 𝜋 \pi whose letters are in the same relative order (of size) as those in p 𝑝 p , in an occurrence of a generalized pattern , some letters of that subsequence may be required to be adjacent in the permutation. For example, the classical pattern 1 ​ ​ 2 ​ ​ 3 ​ ​ 4 1 2 3 4 1\makebox4.73611pt{\rule2.49721pt{3.05696pt}{0.6458pt}}2\makebox4.73611pt{\rule2.49721pt{3.05696pt}{0.6458pt}}3\makebox4.73611pt{\rule2.49721pt{3.05696pt}{0.6458pt}}4 simply corresponds to an increasing subsequence of length four, whereas an occurrence of the generalized pattern 1 ​ ​ 23 ​ ​ 4 1 23 4 1\makebox4.73611pt{\rule2.49721pt{3.05696pt}{0.6458pt}}23\makebox4.73611pt{\rule2.49721pt{3.05696pt}{0.6458pt}}4 would require the middle two letters of that sequence to be adjacent in π 𝜋 \pi , due to the absence of a dash between 2 and 3. Thus, the permutation 23145 23145 23145 contains 1 ​ ​ 2 ​ ​ 3 ​ ​ 4 1 2 3 4 1\makebox4.73611pt{\rule2.49721pt{3.05696pt}{0.6458pt}}2\makebox4.73611pt{\rule2.49721pt{3.05696pt}{0.6458pt}}3\makebox4.73611pt{\rule2.49721pt{3.05696pt}{0.6458pt}}4 but not 1 ​ ​ 23 ​ ​ 4 1 23 4 1\makebox4.73611pt{\rule2.49721pt{3.05696pt}{0.6458pt}}23\makebox4.73611pt{\rule2.49721pt{3.05696pt}{0.6458pt}}4 . Note that for the classical patterns, our notation differs from the usual one, since the dashes we have between every pair of adjacent letters in a classical pattern are usually omitted when only classical patterns are being considered. It is well known that the number of permutations of length n 𝑛 n avoiding any one classical pattern of length 3 is the n 𝑛 n -th Catalan number, which counts a myriad different combinatorial objects. There are many other results in this direction, relating pattern avoiding permutations to various other combinatorial structures, either via bijections, or by such classes of permutations being equinumerous to the structures in question without there being a known bijection. Counting permutations according to the number of occurrences of generalized patterns one comes up with a vast array of known sequences, such as the Euler numbers, Stirling numbers of both kinds, Motzkin numbers, Entringer numbers, Schröder numbers, Fibonacci numbers, Pell numbers and many more. Also, one often finds lesser known sequences that are nevertheless related to known structures, such as directed animals, planar maps, permutation tableaux, various kinds of trees and involutions in S n subscript 𝑆 𝑛 S_{n} , to name a few. Thus, generalized patterns provide a significant addition to the already sizable flora of classical patterns and their connections to other combinatorial structures. In fact, due to their great diversity, the non-classical generalized patterns are likely to provide richer connections to other combinatorial structures than the classical ones do. Supporting this is the fact that the recently proved Stanley-Wilf conjecture—which gives a strong bound for the growth rate of the number of permutations of length n 𝑛 n avoiding a classical pattern—does not hold for some generalized patterns. This paper is organized as follows: In Section 2 we introduce definitions and in Section 3 we mention implicit appearances of generalized patterns in the literature. In Sections 4 and 5 we survey what is known about the avoidance of generalized patterns of length three and four, respectively. In Section 6 we give some examples where generalized patterns have shown up in very different contexts, establishing connections to various other combinatorial structures, some of which seem quite deep. Section 7 lists several instances of so-called barred patterns that turn out to be equivalent to generalized patterns and Section 8 deals with asymptotics for avoidance of generalized patterns. Finally, in Section 9 , we mention some further generalizations of the generalized patterns.","role":"user"},{"content":"Abstract: An occurrence of a classical pattern p in a permutation π is a subsequence of π whose letters are in the same relative order (of size) as those in p. In an occurrence of a generalized pattern, some letters of that subsequence may be required to be adjacent in the permutation. Subsets of permutations characterized by the avoidance—or the prescribed number of occurrences— of generalized patterns exhibit connections to an enormous variety of other combinatorial structures, some of them apparently deep. We give a short overview of the state of the art for generalized patterns.","role":"assistant"}]

Critical review of K − ppn bound states

I Introduction The possibility of having deeply bound K − superscript 𝐾 K^{-} states in nuclei is drawing much attention both theoretically and experimentally. The starting point to face this problem is obviously the understanding of the elementary K ¯ ​ N ¯ 𝐾 𝑁 \bar{K}N interaction, and lots of efforts have been devoted to this topic, mostly using unitary extensions of chiral perturbation theory weise ; angels ; Oller:2000fj ; Oset:2001cn ; jido ; Garcia-Recio:2002td ; Garcia-Recio:2003ks ; Hyodo:2002pk ; GarciaRecio:2005hy ; Hyodo:2007jq . The recent determination of the K − ​ p superscript 𝐾 𝑝 K^{-}p scattering length from the study of K − ​ p superscript 𝐾 𝑝 K^{-}p atoms in DEAR at DA Φ Φ \Phi NE Beer:2005qi has stimulated a revival of the interest on this issue and several studies have already incorporated chiral Lagrangians of higher order Borasoy:2005ie ; Oller:2005ig ; Oller:2006jw ; borasoy in addition to the lowest order one used in angels ; Oller:2000fj ; Oset:2001cn . Much work has also been done along these lines in order to study the interaction of kaons with nuclei, deducing K − superscript 𝐾 K^{-} nucleus optical potentials with a moderate attraction of about 50 MeV at normal nuclear matter density lutz ; angelsself ; schaffner ; galself ; Tolos:2006ny . The selfconsistency of the calculation is an important requirement for the construction of the potential, due to the presence of the Λ ​ ( 1405 ) Λ 1405 \Lambda(1405) resonance below threshold, and is responsible for a fast transition from a repulsive potential in the t ​ ρ 𝑡 𝜌 t\rho approximation at very low densities to an attraction at the densities felt by measured kaonic atom states. This “shallow” theoretical potential was shown to reproduce satisfactorily the data on shifts and widths of kaonic atoms okumura . However, reduced chi-squared values were obtained from phenomenological fits to kaonic atoms which favored strongly attractive potentials of the order of − 200 200 -200 MeV at the center of the nucleus friedman-gal . A combination of theory and phenomenology was attempted in Ref. baca , where an excellent fit to the full set of kaonic atom data was found with a potential that deviated at most by 20% from the theoretical one of angelsself . This potential also generated deeply bound K − superscript 𝐾 K^{-} nuclear states having a width of the order of 100 MeV, much bigger than the corresponding binding energy. The bound states would then overlap among themselves and with the continuum and, consequently, would not show up as narrow peaks in an experiment. Other phenomenological potentials of sizable attraction (with potential depths around 100–200 MeV at ρ 0 subscript 𝜌 0 \rho_{0} ) that could in principle accommodate deeply bound states, have been discussed in gal1 ; gal2 ; gal3 ; gal4 ; gal ; muto ; amigo1 ; amigo2 . In these latter works a relativistic mean field approach is followed, introducing σ 𝜎 \sigma and ω 𝜔 \omega fields which couple to kaons and nucleons to obtain the K ¯ ¯ 𝐾 \bar{K} nucleus optical potential. Less attractive potentials within this framework are also found in Zhong:2004wa ; Zhong:2006hd ; Dang:2007ai . A new look at these relativistic mean field potentials from the perspective of the microscopic chiral unitary approach is presented in Torres:2007rz . The opposite extreme has been represented by some highly attractive phenomenological potentials with about 600 MeV strength in the center of the nucleus akaishi ; akainew . These potentials, leading to compressed nuclear matter of ten times nuclear matter density, met criticisms from toki and more recently from Hyodo:2007jq . The work of toki met criticism from akanuc concerning the ”narrow peak” predicted in toki , but actually the width of the peak was not calculated in toki . It was calculated in hyper showing that it was not narrow and indeed agrees with the revised experiment of the KEK work as we shall discuss below. The criticisms of akanuc were rebutted in hyper and more recently in npangels . Predictions of deeply bound K − superscript 𝐾 K^{-} states for few nucleon systems have been first done in akaishi ; akainew . More recently, Faddeev-type calculations were made for the K ¯ ​ N ​ N ¯ 𝐾 𝑁 𝑁 \bar{K}NN system using phenomenological input in Shevchenko:2006xy ; Shevchenko:2007zz and a leading-order chiral interaction in Ikeda:2007nz . Both studies found a K − ​ p ​ p superscript 𝐾 𝑝 𝑝 K^{-}pp quasibound state above the π ​ Σ ​ N 𝜋 Σ 𝑁 \pi\Sigma N threshold with a relatively large width. A variational approach with phenomenological local potentials has also been applied in Yamazaki:2007cs to study the K ¯ ​ N ​ N ¯ 𝐾 𝑁 𝑁 \bar{K}NN system, leading to a bound state at about 50 MeV below the K ¯ ​ N ​ N ¯ 𝐾 𝑁 𝑁 \bar{K}NN threshold. A more recent variational calculation Weise:2007rd ; Dote:2007rk emphasizes the important role of the repulsive N ​ N 𝑁 𝑁 NN interaction at short distances and obtains preliminary results having smaller bindings and larger widths than those found by the other earlier approaches. On the experimental side the situation is still at a very early stage. Initial hopes that a peak seen in the ( K s ​ t ​ o ​ p − , p ) subscript superscript 𝐾 𝑠 𝑡 𝑜 𝑝 𝑝 (K^{-}_{stop},p) reaction on 4 He Suzuki:2004ep could be a signal of a K − superscript 𝐾 K^{-} bound in the trinucleon with a binding of 195 MeV gradually lost a support. First, an alternative explanation of the peak was presented in toki , showing that a peak with the strength claimed in the experiment was coming from K − superscript 𝐾 K^{-} absorption on a pair of nucleons going to p ​ Σ 𝑝 Σ p\Sigma , leaving the other two nucleons as spectators. This hypothesis led to the prediction that such a peak should be seen in other light or medium nuclei where it should be narrower and weaker as the nuclear size increases. This was confirmed with the finding of such a peak in the ( K s ​ t ​ o ​ p − , p ) subscript superscript 𝐾 𝑠 𝑡 𝑜 𝑝 𝑝 (K^{-}_{stop},p) reaction on 6 Li, which already fades away in 12 C nuclei at FINUDA agnellonuc . In toki the K − superscript 𝐾 K^{-} absorption was described as taking place from (np) pairs of the Fermi sea. In agnellonuc the same explanation was given for the peaks suggesting that the (np) pairs would be correlated in ”quasi”-deuteron clusters. The final development in this discussion has come from a new experiment of the KEK reaction of Suzuki:2004ep reported in Sato:2007sb where, performing a more precise measurement, which subsaned deficiencies in the efficiency corrections, the relatively narrow peak seen in Suzuki:2004ep disappears and only a broad bump remains around the region where the peak was initially claimed. The position and width of this broad bump are in agreement with the estimations done in hyper ; npangels based on the kaon absorption mechanism of toki . The second source of initial hope came from the experiment of the FINUDA collaboration Agnello:2005qj , where a peak seen in the invariant mass distribution of Λ ​ p Λ 𝑝 \Lambda p following K − superscript 𝐾 K^{-} absorption in a mixture of light nuclei was interpreted as evidence for a K − ​ p ​ p superscript 𝐾 𝑝 𝑝 K^{-}pp bound state, with 115 MeV binding and 67 MeV width. However, it was shown in Magas:2006fn ; Kpp that the peak seen could be interpreted in terms of K − superscript 𝐾 K^{-} absorption on a pair of nucleons leading to a Λ ​ p Λ 𝑝 \Lambda p pair, followed by the rescattering of the p 𝑝 p or the Λ Λ \Lambda on the remnant nucleus. More recently, a new experiment of the FINUDA collaboration :2007ph found a peak on the invariant mass of Λ ​ d Λ 𝑑 \Lambda d following the absorption of a K − superscript 𝐾 K^{-} on 6 Li, which was interpreted as a signature for a bound K ¯ ​ N ​ N ​ N ¯ 𝐾 𝑁 𝑁 𝑁 \bar{K}NNN state with 58 MeV binding and 37 MeV width. These results are puzzling, since the bound state of the K ¯ ¯ 𝐾 {\bar{K}} in the three nucleon system has significantly smaller values for the binding and width than those claimed for the bound state of the K ¯ ¯ 𝐾 {\bar{K}} in the two nucleon system Agnello:2005qj . These unexpected results require serious thoughts but no discussion was done in :2007ph . About the same time as the FINUDA experiment :2007ph a similar experiment was performed at KEK Suzuki:2007kn , looking also at the Λ ​ d Λ 𝑑 \Lambda d invariant mass following K − superscript 𝐾 K^{-} absorption but on a 4 He target. The authors of this latter work do not share the conclusions of :2007ph concerning the association of the peak to a K ¯ ¯ 𝐾 \bar{K} bound state, and claim instead that the peak could be a signature of three body absorption. In the present work we perform detailed calculations of K − superscript 𝐾 K^{-} absorption from three nucleons in 6 Li and show that all features observed in the experiment of :2007ph can be well interpreted in the picture of three body kaon absorption, as suggested in Suzuki:2007kn , with the rest of the nucleons acting as spectators.

We make a thorough study of the process of three-body kaon absorption in nuclei, in connection with a recent FINUDA experiment which claims the existence of a deeply bound kaonic state from the observation of a peak in the {lambda}d invariant mass distribution following K{sup -} absorption on {sup 6}Li. We show that the peak is naturally explained in terms of K{sup -} absorption from three nucleons leaving the rest as spectators. We can also reproduce all the other observables measured in the same experiment and used to support the hypothesis of the deeply bound kaon state. Our study also reveals interesting aspects of kaon absorption in nuclei, a process that must be understood in order to make progress in the search for K{sup -} deeply bound states in nuclei.

[{"content":"Help me to generate the abstract of a survey paper given the title:*Critical review of K − ppn bound states* and the introduction:I Introduction The possibility of having deeply bound K − superscript 𝐾 K^{-} states in nuclei is drawing much attention both theoretically and experimentally. The starting point to face this problem is obviously the understanding of the elementary K ¯ ​ N ¯ 𝐾 𝑁 \bar{K}N interaction, and lots of efforts have been devoted to this topic, mostly using unitary extensions of chiral perturbation theory weise ; angels ; Oller:2000fj ; Oset:2001cn ; jido ; Garcia-Recio:2002td ; Garcia-Recio:2003ks ; Hyodo:2002pk ; GarciaRecio:2005hy ; Hyodo:2007jq . The recent determination of the K − ​ p superscript 𝐾 𝑝 K^{-}p scattering length from the study of K − ​ p superscript 𝐾 𝑝 K^{-}p atoms in DEAR at DA Φ Φ \Phi NE Beer:2005qi has stimulated a revival of the interest on this issue and several studies have already incorporated chiral Lagrangians of higher order Borasoy:2005ie ; Oller:2005ig ; Oller:2006jw ; borasoy in addition to the lowest order one used in angels ; Oller:2000fj ; Oset:2001cn . Much work has also been done along these lines in order to study the interaction of kaons with nuclei, deducing K − superscript 𝐾 K^{-} nucleus optical potentials with a moderate attraction of about 50 MeV at normal nuclear matter density lutz ; angelsself ; schaffner ; galself ; Tolos:2006ny . The selfconsistency of the calculation is an important requirement for the construction of the potential, due to the presence of the Λ ​ ( 1405 ) Λ 1405 \Lambda(1405) resonance below threshold, and is responsible for a fast transition from a repulsive potential in the t ​ ρ 𝑡 𝜌 t\rho approximation at very low densities to an attraction at the densities felt by measured kaonic atom states. This shallow theoretical potential was shown to reproduce satisfactorily the data on shifts and widths of kaonic atoms okumura . However, reduced chi-squared values were obtained from phenomenological fits to kaonic atoms which favored strongly attractive potentials of the order of − 200 200 -200 MeV at the center of the nucleus friedman-gal . A combination of theory and phenomenology was attempted in Ref. baca , where an excellent fit to the full set of kaonic atom data was found with a potential that deviated at most by 20% from the theoretical one of angelsself . This potential also generated deeply bound K − superscript 𝐾 K^{-} nuclear states having a width of the order of 100 MeV, much bigger than the corresponding binding energy. The bound states would then overlap among themselves and with the continuum and, consequently, would not show up as narrow peaks in an experiment. Other phenomenological potentials of sizable attraction (with potential depths around 100–200 MeV at ρ 0 subscript 𝜌 0 \rho_{0} ) that could in principle accommodate deeply bound states, have been discussed in gal1 ; gal2 ; gal3 ; gal4 ; gal ; muto ; amigo1 ; amigo2 . In these latter works a relativistic mean field approach is followed, introducing σ 𝜎 \sigma and ω 𝜔 \omega fields which couple to kaons and nucleons to obtain the K ¯ ¯ 𝐾 \bar{K} nucleus optical potential. Less attractive potentials within this framework are also found in Zhong:2004wa ; Zhong:2006hd ; Dang:2007ai . A new look at these relativistic mean field potentials from the perspective of the microscopic chiral unitary approach is presented in Torres:2007rz . The opposite extreme has been represented by some highly attractive phenomenological potentials with about 600 MeV strength in the center of the nucleus akaishi ; akainew . These potentials, leading to compressed nuclear matter of ten times nuclear matter density, met criticisms from toki and more recently from Hyodo:2007jq . The work of toki met criticism from akanuc concerning the narrow peak predicted in toki , but actually the width of the peak was not calculated in toki . It was calculated in hyper showing that it was not narrow and indeed agrees with the revised experiment of the KEK work as we shall discuss below. The criticisms of akanuc were rebutted in hyper and more recently in npangels . Predictions of deeply bound K − superscript 𝐾 K^{-} states for few nucleon systems have been first done in akaishi ; akainew . More recently, Faddeev-type calculations were made for the K ¯ ​ N ​ N ¯ 𝐾 𝑁 𝑁 \bar{K}NN system using phenomenological input in Shevchenko:2006xy ; Shevchenko:2007zz and a leading-order chiral interaction in Ikeda:2007nz . Both studies found a K − ​ p ​ p superscript 𝐾 𝑝 𝑝 K^{-}pp quasibound state above the π ​ Σ ​ N 𝜋 Σ 𝑁 \pi\Sigma N threshold with a relatively large width. A variational approach with phenomenological local potentials has also been applied in Yamazaki:2007cs to study the K ¯ ​ N ​ N ¯ 𝐾 𝑁 𝑁 \bar{K}NN system, leading to a bound state at about 50 MeV below the K ¯ ​ N ​ N ¯ 𝐾 𝑁 𝑁 \bar{K}NN threshold. A more recent variational calculation Weise:2007rd ; Dote:2007rk emphasizes the important role of the repulsive N ​ N 𝑁 𝑁 NN interaction at short distances and obtains preliminary results having smaller bindings and larger widths than those found by the other earlier approaches. On the experimental side the situation is still at a very early stage. Initial hopes that a peak seen in the ( K s ​ t ​ o ​ p − , p ) subscript superscript 𝐾 𝑠 𝑡 𝑜 𝑝 𝑝 (K^{-}_{stop},p) reaction on 4 He Suzuki:2004ep could be a signal of a K − superscript 𝐾 K^{-} bound in the trinucleon with a binding of 195 MeV gradually lost a support. First, an alternative explanation of the peak was presented in toki , showing that a peak with the strength claimed in the experiment was coming from K − superscript 𝐾 K^{-} absorption on a pair of nucleons going to p ​ Σ 𝑝 Σ p\Sigma , leaving the other two nucleons as spectators. This hypothesis led to the prediction that such a peak should be seen in other light or medium nuclei where it should be narrower and weaker as the nuclear size increases. This was confirmed with the finding of such a peak in the ( K s ​ t ​ o ​ p − , p ) subscript superscript 𝐾 𝑠 𝑡 𝑜 𝑝 𝑝 (K^{-}_{stop},p) reaction on 6 Li, which already fades away in 12 C nuclei at FINUDA agnellonuc . In toki the K − superscript 𝐾 K^{-} absorption was described as taking place from (np) pairs of the Fermi sea. In agnellonuc the same explanation was given for the peaks suggesting that the (np) pairs would be correlated in quasi-deuteron clusters. The final development in this discussion has come from a new experiment of the KEK reaction of Suzuki:2004ep reported in Sato:2007sb where, performing a more precise measurement, which subsaned deficiencies in the efficiency corrections, the relatively narrow peak seen in Suzuki:2004ep disappears and only a broad bump remains around the region where the peak was initially claimed. The position and width of this broad bump are in agreement with the estimations done in hyper ; npangels based on the kaon absorption mechanism of toki . The second source of initial hope came from the experiment of the FINUDA collaboration Agnello:2005qj , where a peak seen in the invariant mass distribution of Λ ​ p Λ 𝑝 \Lambda p following K − superscript 𝐾 K^{-} absorption in a mixture of light nuclei was interpreted as evidence for a K − ​ p ​ p superscript 𝐾 𝑝 𝑝 K^{-}pp bound state, with 115 MeV binding and 67 MeV width. However, it was shown in Magas:2006fn ; Kpp that the peak seen could be interpreted in terms of K − superscript 𝐾 K^{-} absorption on a pair of nucleons leading to a Λ ​ p Λ 𝑝 \Lambda p pair, followed by the rescattering of the p 𝑝 p or the Λ Λ \Lambda on the remnant nucleus. More recently, a new experiment of the FINUDA collaboration :2007ph found a peak on the invariant mass of Λ ​ d Λ 𝑑 \Lambda d following the absorption of a K − superscript 𝐾 K^{-} on 6 Li, which was interpreted as a signature for a bound K ¯ ​ N ​ N ​ N ¯ 𝐾 𝑁 𝑁 𝑁 \bar{K}NNN state with 58 MeV binding and 37 MeV width. These results are puzzling, since the bound state of the K ¯ ¯ 𝐾 {\bar{K}} in the three nucleon system has significantly smaller values for the binding and width than those claimed for the bound state of the K ¯ ¯ 𝐾 {\bar{K}} in the two nucleon system Agnello:2005qj . These unexpected results require serious thoughts but no discussion was done in :2007ph . About the same time as the FINUDA experiment :2007ph a similar experiment was performed at KEK Suzuki:2007kn , looking also at the Λ ​ d Λ 𝑑 \Lambda d invariant mass following K − superscript 𝐾 K^{-} absorption but on a 4 He target. The authors of this latter work do not share the conclusions of :2007ph concerning the association of the peak to a K ¯ ¯ 𝐾 \bar{K} bound state, and claim instead that the peak could be a signature of three body absorption. In the present work we perform detailed calculations of K − superscript 𝐾 K^{-} absorption from three nucleons in 6 Li and show that all features observed in the experiment of :2007ph can be well interpreted in the picture of three body kaon absorption, as suggested in Suzuki:2007kn , with the rest of the nucleons acting as spectators.","role":"user"},{"content":"Abstract: We make a thorough study of the process of three-body kaon absorption in nuclei, in connection with a recent FINUDA experiment which claims the existence of a deeply bound kaonic state from the observation of a peak in the {lambda}d invariant mass distribution following K{sup -} absorption on {sup 6}Li. We show that the peak is naturally explained in terms of K{sup -} absorption from three nucleons leaving the rest as spectators. We can also reproduce all the other observables measured in the same experiment and used to support the hypothesis of the deeply bound kaon state. Our study also reveals interesting aspects of kaon absorption in nuclei, a process that must be understood in order to make progress in the search for K{sup -} deeply bound states in nuclei.","role":"assistant"}]

Solid ^4He and the Supersolid Phase: from Theoretical Speculation to the Discovery of a New State of Matter? : A Review of the Past and Present Status of Research( New Frontiers of Quantum Fluids and Solids)

Should we see atoms flowing without resistance in and out of a solid made of the same atoms, this would certainly be counterintuitive because the same entities have to simultaneously give the rigidity intrinsic to a solid and provide the moving particles. Note the difference between this system and the case of a superconductor, where the flowing charges are the electrons and the solidity of the system is given by the ions. Still, this possibility is what should happen in a supersolid, a state of matter conjectured to exist almost forty years ago [ 1 , 2 ] , and solid 4 He appeared to be the most likely candidate. A plethora of counterintuitive behaviours have their origin in quantum mechanics and in the indistinguishability of identical particles, and supersolidity also has its roots in quantum mechanics. In the case of identical particles with integer spin, such as 4 He, which has zero total spin, the extensive occupation of a single quantum state, i.e., the phenomenon of Bose-Einstein condensation (BEC), is at the heart of (in the sense that BEC is proven to imply [ 3 ] ) superfluidity. BEC has been recognized in the superfluid phase of liquid 4 He, and more recently it has been directly observed in the metastable phases of low-density ultracold trapped atomic systems. Typical characteristics of the superflow in such phases [ 4 ] are nonclassical rotational inertia (NCRI) and also the absence of dissipation for flow velocities under the critical velocity as well as the presence of vortices with a velocity flow field with quantized circulation in units of h / m ℎ 𝑚 h/m , where h ℎ h is Planck’s constant and m 𝑚 m is the mass of the particles. As already mentioned, a solid phase with such superflow properties seems paradoxical at first sight. However, from a theoretical point of view, nothing prohibits such an occurrence. In fact, model quantum systems exist in which the existence of a supersolid phase has been proved (an example inherent to the topic of this review is presented in ref. \citeonline reatto) or found (for quantum lattice models see, for example, refs. \citeonline lattice1,lattice2,lattice3). A supersolid phase would correspond to a phase in which two kinds of order are simultaneously present: crystalline long-range order and off-diagonal long-range order [ 9 ] (ODLRO). Thus, in such a state, both order in real space (crystalline order) and order in momentum space (corresponding to the off-diagonal order) should be simultaneously present. These two different kinds of order originate from two different broken symmetries: the translational symmetry, as observed in any crystalline solid (directly detectable in the Bragg scattering), and the gauge symmetry, whose breakage allows for the phase coherence throughout the system. Our discussion is devoted to solid 4 He, in which the translational symmetry is spontaneously broken and only the interatomic interaction plays a role in determining the reference lattice. We are not going to discuss the cases of adsorbed phases of helium, of bulk systems of different species such as hydrogen, of quantum gases in optical lattices, or of lattice models. Note that lattice models are qualitatively different from the case of 4 He, because in such models the Hamiltonian is invariant only for lattice translations so that the spatial broken symmetry refers to a discrete group, not to a continuous group such as in a quantum solid. In this review we have not tried to produce an exhaustive report of the research contributions to this (presently hot) topic, which are rapidly growing in number. We suggest that the reader also refers to other reviews [ 10 , 11 , 12 ] to obtain a more comprehensive view of what has been achieved and what remains to be achieved in the theoretical and experimental investigation of a possible supersolid 4 He phase. After this brief introduction, the review is structured as follows: in § ​ 2 § 2 \S 2 we will review the main theoretical discussions and some experimental studies that appeared before the torsional oscillator (TO) “revolution” [ 13 , 14 ] ; in § ​ 3 § 3 \S 3 we will discuss the main experimental results in the last four years; section 4 is devoted to a discussion of the theoretical contributions (we will discuss mostly quantum simulation results) stimulated by the new experimental findings, whereas § ​ 5 § 5 \S 5 contains a discussion of what remains to be investigated and clariied on the basis of the present status of the research in this field.

The possibility of a supersolid state of matter, i.e., a crystalline solid exhibiting superfluid properties, first appeared in theoretical studies about forty years ago. After a long period of little interest due to the lack of experimental evidence, it has attracted strong experimental and theoretical attention in the last few years since Kim and Chan (Penn State, U.S.A.) reported evidence for nonclassical rotational inertia effects, a typical signature of superfluidity, in samples of solid 4 He. Since this “first observation”, other experimental groups have observed such effects in the response to the rotation of samples of crystalline helium, and it has become clear that the response of the solid is extremely sensitive to growth conditions, annealing processes, and 3 He impurities. A peak in the specific heat in the same range of temperatures has been reported as well as anomalies in the elastic behaviour of solid 4 He with a strong resemblance to the phenomena revealed by torsional oscillator experime...

[{"content":"Help me to generate the abstract of a survey paper given the title:*Solid ^4He and the Supersolid Phase: from Theoretical Speculation to the Discovery of a New State of Matter? : A Review of the Past and Present Status of Research( New Frontiers of Quantum Fluids and Solids)* and the introduction:Should we see atoms flowing without resistance in and out of a solid made of the same atoms, this would certainly be counterintuitive because the same entities have to simultaneously give the rigidity intrinsic to a solid and provide the moving particles. Note the difference between this system and the case of a superconductor, where the flowing charges are the electrons and the solidity of the system is given by the ions. Still, this possibility is what should happen in a supersolid, a state of matter conjectured to exist almost forty years ago 1 , 2 , and solid 4 He appeared to be the most likely candidate. A plethora of counterintuitive behaviours have their origin in quantum mechanics and in the indistinguishability of identical particles, and supersolidity also has its roots in quantum mechanics. In the case of identical particles with integer spin, such as 4 He, which has zero total spin, the extensive occupation of a single quantum state, i.e., the phenomenon of Bose-Einstein condensation (BEC), is at the heart of (in the sense that BEC is proven to imply 3 ) superfluidity. BEC has been recognized in the superfluid phase of liquid 4 He, and more recently it has been directly observed in the metastable phases of low-density ultracold trapped atomic systems. Typical characteristics of the superflow in such phases 4 are nonclassical rotational inertia (NCRI) and also the absence of dissipation for flow velocities under the critical velocity as well as the presence of vortices with a velocity flow field with quantized circulation in units of h / m ℎ 𝑚 h/m , where h ℎ h is Plancks constant and m 𝑚 m is the mass of the particles. As already mentioned, a solid phase with such superflow properties seems paradoxical at first sight. However, from a theoretical point of view, nothing prohibits such an occurrence. In fact, model quantum systems exist in which the existence of a supersolid phase has been proved (an example inherent to the topic of this review is presented in ref. \citeonline reatto) or found (for quantum lattice models see, for example, refs. \citeonline lattice1,lattice2,lattice3). A supersolid phase would correspond to a phase in which two kinds of order are simultaneously present: crystalline long-range order and off-diagonal long-range order 9 (ODLRO). Thus, in such a state, both order in real space (crystalline order) and order in momentum space (corresponding to the off-diagonal order) should be simultaneously present. These two different kinds of order originate from two different broken symmetries: the translational symmetry, as observed in any crystalline solid (directly detectable in the Bragg scattering), and the gauge symmetry, whose breakage allows for the phase coherence throughout the system. Our discussion is devoted to solid 4 He, in which the translational symmetry is spontaneously broken and only the interatomic interaction plays a role in determining the reference lattice. We are not going to discuss the cases of adsorbed phases of helium, of bulk systems of different species such as hydrogen, of quantum gases in optical lattices, or of lattice models. Note that lattice models are qualitatively different from the case of 4 He, because in such models the Hamiltonian is invariant only for lattice translations so that the spatial broken symmetry refers to a discrete group, not to a continuous group such as in a quantum solid. In this review we have not tried to produce an exhaustive report of the research contributions to this (presently hot) topic, which are rapidly growing in number. We suggest that the reader also refers to other reviews 10 , 11 , 12 to obtain a more comprehensive view of what has been achieved and what remains to be achieved in the theoretical and experimental investigation of a possible supersolid 4 He phase. After this brief introduction, the review is structured as follows: in § ​ 2 § 2 \S 2 we will review the main theoretical discussions and some experimental studies that appeared before the torsional oscillator (TO) revolution 13 , 14 ; in § ​ 3 § 3 \S 3 we will discuss the main experimental results in the last four years; section 4 is devoted to a discussion of the theoretical contributions (we will discuss mostly quantum simulation results) stimulated by the new experimental findings, whereas § ​ 5 § 5 \S 5 contains a discussion of what remains to be investigated and clariied on the basis of the present status of the research in this field.","role":"user"},{"content":"Abstract: The possibility of a supersolid state of matter, i.e., a crystalline solid exhibiting superfluid properties, first appeared in theoretical studies about forty years ago. After a long period of little interest due to the lack of experimental evidence, it has attracted strong experimental and theoretical attention in the last few years since Kim and Chan (Penn State, U.S.A.) reported evidence for nonclassical rotational inertia effects, a typical signature of superfluidity, in samples of solid 4 He. Since this first observation, other experimental groups have observed such effects in the response to the rotation of samples of crystalline helium, and it has become clear that the response of the solid is extremely sensitive to growth conditions, annealing processes, and 3 He impurities. A peak in the specific heat in the same range of temperatures has been reported as well as anomalies in the elastic behaviour of solid 4 He with a strong resemblance to the phenomena revealed by torsional oscillator experime...","role":"assistant"}]

Time-fractional derivatives in relaxation processes: a tutorial survey

1 Definitions and properties For a sufficiently well-behaved function f ​ ( t ) 𝑓 𝑡 f(t) (with t ∈ 𝐑 + 𝑡 superscript 𝐑 t\in{\bf R}^{+} ) we may define the derivative of a positive non-integer order in two different senses, that we refer here as to Riemann-Liouville (R-L) derivative and Caputo (C) derivative, respectively. Both derivatives are related to the so-called Riemann-Liouville fractional integral. For any α > 0 𝛼 0 \alpha>0 this fractional integral is defined as J t α ​ f ​ ( t ) := 1 Γ ​ ( α ) ​ ∫ 0 t ( t − τ ) α − 1 ​ f ​ ( τ ) ​ 𝑑 τ , assign superscript subscript 𝐽 𝑡 𝛼 𝑓 𝑡 1 Γ 𝛼 superscript subscript 0 𝑡 superscript 𝑡 𝜏 𝛼 1 𝑓 𝜏 differential-d 𝜏 J_{t}^{\alpha}\,f(t):={1\over{\Gamma(\alpha)}}\,\int_{0}^{t}\!(t-\tau)^{\alpha-1}\,f(\tau)\,d\tau\,, ( 1.1 ) 1.1 where Γ ​ ( α ) := ∫ 0 ∞ e − u ​ u α − 1 ​ 𝑑 u assign Γ 𝛼 superscript subscript 0 superscript e 𝑢 superscript 𝑢 𝛼 1 differential-d 𝑢 {\displaystyle\Gamma(\alpha):=\int_{0}^{\infty}{\rm e}^{-u}\,u^{\alpha-1}\,du} denotes the Gamma function. For existence of the integral (1) it is sufficient that the function f ​ ( t ) 𝑓 𝑡 f(t) is locally integrable in 𝐑 + superscript 𝐑 {\bf R}^{+} and for t → 0 → 𝑡 0 t\to 0 behaves like O ​ ( t − ν ) 𝑂 superscript 𝑡 𝜈 O(t^{-\nu}) with a number ν < α 𝜈 𝛼 \nu<\alpha . For completion we define J t 0 = I superscript subscript 𝐽 𝑡 0 𝐼 J_{t}^{0}=I (Identity operator). We recall the semigroup property J t α ​ J t β = J t β ​ J t α = J t α + β , α , β ≥ 0 . formulae-sequence superscript subscript 𝐽 𝑡 𝛼 superscript subscript 𝐽 𝑡 𝛽 superscript subscript 𝐽 𝑡 𝛽 superscript subscript 𝐽 𝑡 𝛼 superscript subscript 𝐽 𝑡 𝛼 𝛽 𝛼 𝛽 0 J_{t}^{\alpha}\,J_{t}^{\beta}=\,J_{t}^{\beta}\,J_{t}^{\alpha}=J_{t}^{\alpha+\beta}\,,\quad\alpha,\beta\geq 0\,. ( 1.2 ) 1.2 Furthermore we note that for α ≥ 0 𝛼 0 \alpha\geq 0 J t α ​ t γ = Γ ​ ( γ + 1 ) Γ ​ ( γ + 1 + α ) ​ t γ + α , γ > − 1 . formulae-sequence superscript subscript 𝐽 𝑡 𝛼 superscript 𝑡 𝛾 Γ 𝛾 1 Γ 𝛾 1 𝛼 superscript 𝑡 𝛾 𝛼 𝛾 1 J_{t}^{\alpha}\,t^{\gamma}={\Gamma(\gamma+1)\over\Gamma(\gamma+1+\alpha)}\,t^{\gamma+\alpha}\,,\quad\gamma>-1\,. ( 1.3 ) 1.3 The fractional derivative of order μ > 0 𝜇 0 \mu>0 in the Riemann-Liouville sense is defined as the operator D t μ superscript subscript 𝐷 𝑡 𝜇 D_{t}^{\mu} which is the left inverse of the Riemann-Liouville integral of order μ 𝜇 \mu (in analogy with the ordinary derivative), that is D t μ ​ J t μ = I , μ > 0 . formulae-sequence superscript subscript 𝐷 𝑡 𝜇 superscript subscript 𝐽 𝑡 𝜇 𝐼 𝜇 0 D_{t}^{\mu}\,J_{t}^{\mu}=I\,,\quad\mu>0\,. ( 1.4 ) 1.4 If m 𝑚 m denotes the positive integer such that m − 1 < μ ≤ m , 𝑚 1 𝜇 𝑚 m-1<\mu\leq m\,, we recognize from Eqs. (1.2) and (1.4): D t μ ​ f ​ ( t ) := D t m ​ J t m − μ ​ f ​ ( t ) , m − 1 < μ ≤ m . formulae-sequence assign superscript subscript 𝐷 𝑡 𝜇 𝑓 𝑡 superscript subscript 𝐷 𝑡 𝑚 superscript subscript 𝐽 𝑡 𝑚 𝜇 𝑓 𝑡 𝑚 1 𝜇 𝑚 D_{t}^{\mu}\,f(t):=D_{t}^{m}\,J_{t}^{m-\mu}\,f(t)\,,\quad m-1<\mu\leq m\,. ( 1.5 ) 1.5 In fact, using the semigroup property (1.2), we have D t μ ​ J t μ = D t m ​ J t m − μ ​ J t μ = D t m ​ J t m = I . superscript subscript 𝐷 𝑡 𝜇 superscript subscript 𝐽 𝑡 𝜇 superscript subscript 𝐷 𝑡 𝑚 superscript subscript 𝐽 𝑡 𝑚 𝜇 superscript subscript 𝐽 𝑡 𝜇 superscript subscript 𝐷 𝑡 𝑚 superscript subscript 𝐽 𝑡 𝑚 𝐼 D_{t}^{\mu}\,J_{t}^{\mu}=D_{t}^{m}\,J_{t}^{m-\mu}\,J_{t}^{\mu}=D_{t}^{m}\,J_{t}^{m}=I\,. Thus (1.5) implies D t μ ​ f ​ ( t ) = { d m d ​ t m ​ [ 1 Γ ​ ( m − μ ) ​ ∫ 0 t f ​ ( τ ) ​ d ​ τ ( t − τ ) μ + 1 − m ] , m − 1 < μ < m ; d m d ​ t m ​ f ​ ( t ) , μ = m . \,\!\!\!\!D_{t}^{\mu}\,f(t)=\cases{{\displaystyle{d^{m}\over dt^{m}}}\left[{\displaystyle{1\over{\Gamma(m-\mu)}}\int_{0}^{t}\!{f(\tau)\,d\tau\over(t-\tau)^{\mu+1-m}}}\right],&$m-1<\mu<m;$\cr\cr{\displaystyle{d^{m}\over dt^{m}}f(t)},&$\mu=m.$} ( 1.5 ′ ) superscript 1.5 ′ For completion we define D t 0 = I . superscript subscript 𝐷 𝑡 0 𝐼 D_{t}^{0}=I\,. On the other hand, the fractional derivative of order μ 𝜇 \mu in the Caputo sense is defined as the operator D t μ ∗ subscript superscript subscript 𝐷 𝑡 𝜇 \,{}_{*}D_{t}^{\mu} such that D t μ ∗ ​ f ​ ( t ) := J t m − μ ​ D t m ​ f ​ ( t ) , m − 1 < μ ≤ m . formulae-sequence assign subscript superscript subscript 𝐷 𝑡 𝜇 𝑓 𝑡 superscript subscript 𝐽 𝑡 𝑚 𝜇 superscript subscript 𝐷 𝑡 𝑚 𝑓 𝑡 𝑚 1 𝜇 𝑚 {}_{*}D_{t}^{\mu}\,f(t):=\,J_{t}^{m-\mu}\,D_{t}^{m}\,f(t)\,,\quad m-1<\mu\leq m\,. ( 1.6 ) 1.6 This implies D t μ ∗ ​ f ​ ( t ) = { 1 Γ ​ ( m − μ ) ​ ∫ 0 t f ( m ) ​ ( τ ) ​ d ​ τ ( t − τ ) μ + 1 − m , m − 1 < μ < m ; d m d ​ t m ​ f ​ ( t ) , μ = m . \,{}_{*}D_{t}^{\mu}\,f(t)=\cases{{\displaystyle{1\over{\Gamma(m-\mu)}}}\,{\displaystyle\int_{0}^{t}\!{\displaystyle{f^{(m)}(\tau)\,d\tau\over(t-\tau)^{\mu+1-m}}}}\,,&$m-1<\mu<m\,;$\cr\cr{\displaystyle{d^{m}\over dt^{m}}f(t)}\,,&$\mu=m\,.$} ( 1.6 ′ ) superscript 1.6 ′ Thus, when the order is not integer the two fractional derivatives differ in that the standard derivative of order m 𝑚 m does not generally commute with the fractional integral. Of course the Caputo derivative (1.6 ′ ) needs higher regularity conditions of f ​ ( t ) 𝑓 𝑡 f(t) than the Riemann-Liouville derivative (1.5 ′ ). We point out that the Caputo fractional derivative satisfies the relevant property of being zero when applied to a constant, and, in general, to any power function of non-negative integer degree less than m , 𝑚 m\,, if its order μ 𝜇 \mu is such that m − 1 < μ ≤ m . 𝑚 1 𝜇 𝑚 m-1<\mu\leq m\,. Furthermore we note for μ ≥ 0 𝜇 0 \mu\geq 0 : D t μ ​ t γ = Γ ​ ( γ + 1 ) Γ ​ ( γ + 1 − μ ) ​ t γ − μ , γ > − 1 . formulae-sequence superscript subscript 𝐷 𝑡 𝜇 superscript 𝑡 𝛾 Γ 𝛾 1 Γ 𝛾 1 𝜇 superscript 𝑡 𝛾 𝜇 𝛾 1 D_{t}^{\mu}\,t^{\gamma}={\Gamma(\gamma+1)\over\Gamma(\gamma+1-\mu)}\,t^{\gamma-\mu}\,,\quad\gamma>-1\,. ( 1.7 ) 1.7 It is instructive to compare Eqs. (1.3), (1.7). In [ 58 ] we have shown the essential relationships between the two fractional derivatives for the same non-integer order D t μ ∗ ​ f ​ ( t ) = { D t μ ​ [ f ​ ( t ) − ∑ k = 0 m − 1 f ( k ) ​ ( 0 + ) ​ t k k ! ] , D t μ ​ f ​ ( t ) − ∑ k = 0 m − 1 f ( k ) ​ ( 0 + ) ​ t k − μ Γ ​ ( k − μ + 1 ) , m − 1 < μ < m . formulae-sequence subscript superscript subscript 𝐷 𝑡 𝜇 𝑓 𝑡 cases superscript subscript 𝐷 𝑡 𝜇 delimited-[] 𝑓 𝑡 superscript subscript 𝑘 0 𝑚 1 superscript 𝑓 𝑘 superscript 0 superscript 𝑡 𝑘 𝑘 otherwise otherwise otherwise superscript subscript 𝐷 𝑡 𝜇 𝑓 𝑡 superscript subscript 𝑘 0 𝑚 1 superscript 𝑓 𝑘 superscript 0 superscript 𝑡 𝑘 𝜇 Γ 𝑘 𝜇 1 otherwise 𝑚 1 𝜇 𝑚 {}_{*}D_{t}^{\mu}\,f(t)=\cases{{\displaystyle\,D_{t}^{\mu}\left[f(t)-\sum_{k=0}^{m-1}f^{(k)}(0^{+})\,{t^{k}\over k!}\right]},\cr\cr{\displaystyle D_{t}^{\mu}\,f(t)-\sum_{k=0}^{m-1}{f^{(k)}(0^{+})\,t^{k-\mu}\over\Gamma(k-\mu+1)}},\cr}\quad m-1<\mu<m\,. ( 1.8 ) 1.8 In particular we have from (1.6 ′ ) and (1.8) D t μ ∗ ​ f ​ ( t ) = 1 Γ ​ ( 1 − μ ) ​ ∫ 0 t f ( 1 ) ​ ( τ ) ( t − τ ) μ ​ 𝑑 τ = D t μ ​ [ f ​ ( t ) − f ​ ( 0 + ) ] = D t μ ​ f ​ ( t ) − f ​ ( 0 + ) ​ t − μ Γ ​ ( 1 − μ ) , ​ 0 < μ < 1 . missing-subexpression subscript superscript subscript 𝐷 𝑡 𝜇 𝑓 𝑡 1 Γ 1 𝜇 superscript subscript 0 𝑡 superscript 𝑓 1 𝜏 superscript 𝑡 𝜏 𝜇 differential-d 𝜏 missing-subexpression absent superscript subscript 𝐷 𝑡 𝜇 delimited-[] 𝑓 𝑡 𝑓 superscript 0 superscript subscript 𝐷 𝑡 𝜇 𝑓 𝑡 𝑓 superscript 0 superscript 𝑡 𝜇 Γ 1 𝜇 0 𝜇 1 \begin{array}[]{ll}&\,{}_{*}D_{t}^{\mu}f(t)={\displaystyle{1\over{\Gamma(1-\mu)}}\,\int_{0}^{t}\!{f^{(1)}(\tau)\over(t-\tau)^{\mu}}\,d\tau}\\ &={\displaystyle\,D_{t}^{\mu}\,\left[f(t)-f(0^{+})\right]}={\displaystyle\,D_{t}^{\mu}\,f(t)-f(0^{+})\,{t^{-\mu}\over\Gamma(1-\mu)}}\,,\end{array}\;0<\mu<1\,. ( 1.9 ) 1.9 The Caputo fractional derivative represents a sort of regularization in the time origin for the Riemann-Liouville fractional derivative. We note that for its existence all the limiting values f ( k ) ​ ( 0 + ) := lim t → 0 + D t k ​ f ​ ( t ) assign superscript 𝑓 𝑘 superscript 0 subscript → 𝑡 superscript 0 superscript subscript 𝐷 𝑡 𝑘 𝑓 𝑡 f^{(k)}(0^{+}):={\displaystyle\lim_{t\to 0^{+}}D_{t}^{k}f(t)} are required to be finite for k = 0 , 1 , 2 , … , m − 1 𝑘 0 1 2 … 𝑚 1 k=0,1,2,\dots,m-1 . In the special case f ( k ) ​ ( 0 + ) = 0 superscript 𝑓 𝑘 superscript 0 0 f^{(k)}(0^{+})=0 for k = 0 , 1 , m − 1 𝑘 0 1 𝑚 1 k=0,1,m-1 , the two fractional derivatives coincide. We observe the different behaviour of the two fractional derivatives at the end points of the interval ( m − 1 , m ) 𝑚 1 𝑚 (m-1,m)\, namely when the order is any positive integer, as it can be noted from their definitions (1.5), (1.6). In fact, whereas for μ → m − → 𝜇 superscript 𝑚 \mu\to m^{-} both derivatives reduce to D t m superscript subscript 𝐷 𝑡 𝑚 D_{t}^{m} , as stated in Eqs. (1.5 ′ ), (1.6 ′ ), due to the fact that the operator J t 0 = I superscript subscript 𝐽 𝑡 0 𝐼 J_{t}^{0}\,=\,I commutes with D t m superscript subscript 𝐷 𝑡 𝑚 D_{t}^{m} , for μ → ( m − 1 ) + → 𝜇 superscript 𝑚 1 \mu\to(m-1)^{+} we have μ → ( m − 1 ) + : { D t μ ​ f ​ ( t ) → D t m ​ J t 1 ​ f ​ ( t ) = D t ( m − 1 ) ​ f ​ ( t ) = f ( m − 1 ​ ( t ) , D t μ ∗ ​ f ​ ( t ) → J t 1 ​ D t m ​ f ​ ( t ) = f ( m − 1 ) ​ ( t ) − f ( m − 1 ) ​ ( 0 + ) . \!\!\mu\to(m-1)^{+}:\cases{{\displaystyle D_{t}^{\mu}f(t)}\to{\displaystyle\,\,D_{t}^{m}\,J_{t}^{1}\,f(t)=D_{t}^{(m-1)}\,f(t)=f^{(m-1}(t)},\cr\cr{\,{}_{*}D_{t}^{\mu}f(t)}\to{\displaystyle J_{t}^{1}\,D_{t}^{m}\,f(t)=f^{(m-1)}(t)-f^{(m-1)}(0^{+})}.} ( 1.10 ) 1.10 As a consequence, roughly speaking, we can say that D t μ superscript subscript 𝐷 𝑡 𝜇 D_{t}^{\mu} is, with respect to its order μ , 𝜇 \mu\,, an operator continuous at any positive integer, whereas D t μ ∗ subscript superscript subscript 𝐷 𝑡 𝜇 \,{}_{*}D_{t}^{\mu} is an operator only left-continuous. The above behaviours have induced us to keep for the Riemann-Liouville derivative the same symbolic notation as for the standard derivative of integer order, while for the Caputo derivative to decorate the corresponding symbol with subscript ∗ * . We also note, with m − 1 < μ ≤ m , 𝑚 1 𝜇 𝑚 m-1<\mu\leq m\,, and c j subscript 𝑐 𝑗 c_{j} arbitrary constants, D t μ ​ f ​ ( t ) = D t μ ​ g ​ ( t ) ⟺ f ​ ( t ) = g ​ ( t ) + ∑ j = 1 m c j ​ t μ − j , ⟺ superscript subscript 𝐷 𝑡 𝜇 𝑓 𝑡 superscript subscript 𝐷 𝑡 𝜇 𝑔 𝑡 𝑓 𝑡 𝑔 𝑡 superscript subscript 𝑗 1 𝑚 subscript 𝑐 𝑗 superscript 𝑡 𝜇 𝑗 \hskip-14.22636ptD_{t}^{\mu}\,f(t)\,=\,D_{t}^{\mu}\,g(t)\,\Longleftrightarrow\,f(t)=g(t)+\sum_{j=1}^{m}c_{j}\,t^{\mu-j}\,, ( 1.11 ) 1.11 D t μ ∗ ​ f ​ ( t ) = ∗ D t μ ​ g ​ ( t ) ⟺ f ​ ( t ) = g ​ ( t ) + ∑ j = 1 m c j ​ t m − j . ⟺ subscript subscript superscript subscript 𝐷 𝑡 𝜇 𝑓 𝑡 superscript subscript 𝐷 𝑡 𝜇 𝑔 𝑡 𝑓 𝑡 𝑔 𝑡 superscript subscript 𝑗 1 𝑚 subscript 𝑐 𝑗 superscript 𝑡 𝑚 𝑗 {}_{*}D_{t}^{\mu}\,f(t)\,=\,_{*}D_{t}^{\mu}\,g(t)\,\Longleftrightarrow\,f(t)=g(t)+\sum_{j=1}^{m}c_{j}\,t^{m-j}\,. ( 1.12 ) 1.12 Furthermore, we observe that in case of a non-integer order for both fractional derivatives the semigroup property (of the standard derivative for integer order) does not hold for both fractional derivatives when the order is not integer. We point out the major utility of the Caputo fractional derivative in treating initial-value problems for physical and engineering applications where initial conditions are usually expressed in terms of integer-order derivatives. This can be easily seen using the Laplace transformation 1 1 1 The Laplace transform of a well-behaved function f ​ ( t ) 𝑓 𝑡 f(t) is defined as f ~ ( s ) = ℒ { f ( t ) ; s } : = ∫ 0 ∞ e − s ​ t f ( t ) d t , ℜ ( s ) > a f . \widetilde{f}(s)={\mathcal{L}}\left\{f(t);s\right\}:={\displaystyle\int_{0}^{\infty}}\!{\rm e}^{\displaystyle\,-st}\,f(t)\,dt\,,\quad\Re\,(s)>a_{f}\,. We recall that under suitable conditions the Laplace transform of the m 𝑚 m -derivative of f ​ ( t ) 𝑓 𝑡 f(t) is given by ℒ ​ { D t m ​ f ​ ( t ) ; s } = s m ​ f ~ ​ ( s ) − ∑ k = 0 m − 1 s m − 1 − k ​ f ( k ) ​ ( 0 + ) , f ( k ) ​ ( 0 + ) := lim t → 0 + D t k ​ f ​ ( t ) . formulae-sequence ℒ superscript subscript 𝐷 𝑡 𝑚 𝑓 𝑡 𝑠 superscript 𝑠 𝑚 ~ 𝑓 𝑠 superscript subscript 𝑘 0 𝑚 1 superscript 𝑠 𝑚 1 𝑘 superscript 𝑓 𝑘 superscript 0 assign superscript 𝑓 𝑘 superscript 0 subscript → 𝑡 superscript 0 superscript subscript 𝐷 𝑡 𝑘 𝑓 𝑡 {\mathcal{L}}\left\{D_{t}^{m}\,f(t);s\right\}=s^{m}\,\widetilde{f}(s)-\sum_{k=0}^{m-1}s^{m-1-k}\,f^{(k)}(0^{+})\,,\quad f^{(k)}(0^{+}):=\lim_{t\to 0^{+}}D_{t}^{k}f(t)\,. For the Caputo derivative of order μ 𝜇 \mu with m − 1 < μ ≤ m 𝑚 1 𝜇 𝑚 m-1<\mu\leq m we have ℒ ​ { D t μ ∗ ​ f ​ ( t ) ; s } = s μ ​ f ~ ​ ( s ) − ∑ k = 0 m − 1 s μ − 1 − k ​ f ( k ) ​ ( 0 + ) , f ( k ) ​ ( 0 + ) := lim t → 0 + D t k ​ f ​ ( t ) . missing-subexpression ℒ subscript superscript subscript 𝐷 𝑡 𝜇 𝑓 𝑡 𝑠 superscript 𝑠 𝜇 ~ 𝑓 𝑠 superscript subscript 𝑘 0 𝑚 1 superscript 𝑠 𝜇 1 𝑘 superscript 𝑓 𝑘 superscript 0 missing-subexpression missing-subexpression missing-subexpression assign superscript 𝑓 𝑘 superscript 0 subscript → 𝑡 superscript 0 superscript subscript 𝐷 𝑡 𝑘 𝑓 𝑡 \begin{array}[]{ll}&{\displaystyle{\mathcal{L}}\left\{{}_{*}D_{t}^{\mu}\,f(t);s\right\}}={\displaystyle s^{\mu}\,\widetilde{f}(s)-\sum_{k=0}^{m-1}s^{\mu-1-k}\,f^{(k)}(0^{+})}\,,\\ \\ &f^{(k)}(0^{+}):={\displaystyle\lim_{t\to 0^{+}}D_{t}^{k}f(t)\,.}\end{array} ( 1.13 ) 1.13 The corresponding rule for the Riemann-Liouville derivative of order μ 𝜇 \mu is ℒ ​ { D t μ ​ f ​ ( t ) ; s } = s μ ​ f ~ ​ ( s ) − ∑ k = 0 m − 1 s m − 1 − k ​ g ( k ) ​ ( 0 + ) , g ( k ) ​ ( 0 + ) := lim t → 0 + D t k ​ g ​ ( t ) , where g ​ ( t ) := J t ( m − μ ) ​ f ​ ( t ) . missing-subexpression ℒ superscript subscript 𝐷 𝑡 𝜇 𝑓 𝑡 𝑠 superscript 𝑠 𝜇 ~ 𝑓 𝑠 superscript subscript 𝑘 0 𝑚 1 superscript 𝑠 𝑚 1 𝑘 superscript 𝑔 𝑘 superscript 0 missing-subexpression missing-subexpression missing-subexpression formulae-sequence assign superscript 𝑔 𝑘 superscript 0 subscript → 𝑡 superscript 0 superscript subscript 𝐷 𝑡 𝑘 𝑔 𝑡 where assign 𝑔 𝑡 superscript subscript 𝐽 𝑡 𝑚 𝜇 𝑓 𝑡 \begin{array}[]{ll}&{\displaystyle{\mathcal{L}}\left\{D_{t}^{\mu}\,f(t);s\right\}}={\displaystyle s^{\mu}\,\widetilde{f}(s)-\sum_{k=0}^{m-1}s^{m-1-k}\,g^{(k)}(0^{+})}\,,\\ \\ &g^{(k)}(0^{+}):={\displaystyle\lim_{t\to 0^{+}}D_{t}^{k}g(t)\,,}\quad\hbox{where}\quad g(t):=J_{t}^{(m-\mu)}\,f(t)\,.\end{array} ( 1.14 ) 1.14 Thus it is more cumbersome to use the rule (1.14) than (1.13). The rule (1.14) requires initial values concerning an extra function g ​ ( t ) 𝑔 𝑡 g(t) related to the given f ​ ( t ) 𝑓 𝑡 f(t) through a fractional integral. However, when all the limiting values f ( k ) ​ ( 0 + ) superscript 𝑓 𝑘 superscript 0 f^{(k)}(0^{+}) for k = 0 , 1 , 2 , … 𝑘 0 1 2 … k=0,1,2,\dots are finite and the order is not integer, we can prove that the corresponding g ( k ) ​ ( 0 + ) superscript 𝑔 𝑘 superscript 0 g^{(k)}(0^{+}) vanish so that formula (1.14) simplifies into ℒ ​ { D t μ ​ f ​ ( t ) ; s } = s μ ​ f ~ ​ ( s ) , m − 1 < μ < m . formulae-sequence ℒ superscript subscript 𝐷 𝑡 𝜇 𝑓 𝑡 𝑠 superscript 𝑠 𝜇 ~ 𝑓 𝑠 𝑚 1 𝜇 𝑚 {\mathcal{L}}\left\{D_{t}^{\mu}\,f(t);s\right\}=s^{\mu}\,\widetilde{f}(s)\,,\quad m-1<\mu<m\,. ( 1.15 ) 1.15 For this proof it is sufficient to apply the Laplace transform to the second equation (8), by recalling that ℒ ​ { t α ; s } = Γ ​ ( α + 1 ) / s α + 1 ℒ superscript 𝑡 𝛼 𝑠 Γ 𝛼 1 superscript 𝑠 𝛼 1 {\mathcal{L}}\left\{t^{\alpha};s\right\}=\Gamma(\alpha+1)/s^{\alpha+1}\, for α > − 1 𝛼 1 \alpha>-1 , and then to compare (1.13) with (1.14). For fractional differentiation on the positive semi-axis we recall another definition for the fractional derivative recently introduced by Hilfer, see [ 68 ] and [ 116 ] , which interpolates the previous definitions (1.5) and (1.6). Like the two derivatives previously discussed, it is related to a Riemann-Liouville integral. In our notation it reads D t μ , ν := J t ν ​ ( 1 − μ ) ​ D t 1 ​ J t ( 1 − ν ) ​ ( 1 − μ ) , 0 < μ < 1 , 0 ≤ ν ≤ 1 . formulae-sequence formulae-sequence assign superscript subscript 𝐷 𝑡 𝜇 𝜈 superscript subscript 𝐽 𝑡 𝜈 1 𝜇 superscript subscript 𝐷 𝑡 1 superscript subscript 𝐽 𝑡 1 𝜈 1 𝜇 0 𝜇 1 0 𝜈 1 D_{t}^{\mu,\nu}:=J_{t}^{\nu(1-\mu)}\,D_{t}^{1}\,J_{t}^{(1-\nu)(1-\mu)}\,,\quad 0<\mu<1\,,\quad 0\leq\nu\leq 1\,. ( 1.16 ) 1.16 We can refer it to as the Hilfer (H) fractional derivative of order μ 𝜇 \mu and type ν 𝜈 \nu . The Riemann-Liouville derivative corresponds to the type ν = 0 𝜈 0 \nu=0 whereas that Caputo derivative to the type ν = 1 𝜈 1 \nu=1 . We have here not discussed the Beyer-Kempfle approach investigated and used in several papers by Beyer and Kempfle et al.: this approach is appropriate for causal processes not starting at a finite instant of time, see e.g. [ 9 , 69 ] . They define the time-fractional derivative on the whole real line as a pseudo-differential operator via its Fourier symbol. The interested reader is referred to the above mentioned papers and references therein. For further reading on the theory and applications of fractional integrals and derivatives (more generally of fractional calculus) we may recommend e.g. our CISM Lecture Notes [ 56 , 58 , 81 ] , the review papers [ 86 , 87 , 117 ] , and the books [ 89 , 71 , 67 , 70 , 78 , 95 , 109 , 119 , 120 ] with references therein.

The aim of this tutorial survey is to revisit the basic theory of relaxation processes governed by linear differential equations of fractional order. The fractional derivatives are intended both in the Rieamann-Liouville sense and in the Caputo sense. After giving a necessary outline of the classical theory of linear viscoelasticity, we contrast these two types of fractional derivatives in their ability to take into account initial conditions in the constitutive equations of fractional order. We also provide historical notes on the origins of the Caputo derivative and on the use of fractional calculus in viscoelasticity.

[{"content":"Help me to generate the abstract of a survey paper given the title:*Time-fractional derivatives in relaxation processes: a tutorial survey* and the introduction:1 Definitions and properties For a sufficiently well-behaved function f ​ ( t ) 𝑓 𝑡 f(t) (with t ∈ 𝐑 + 𝑡 superscript 𝐑 t\in{\bf R}^{+} ) we may define the derivative of a positive non-integer order in two different senses, that we refer here as to Riemann-Liouville (R-L) derivative and Caputo (C) derivative, respectively. Both derivatives are related to the so-called Riemann-Liouville fractional integral. For any α > 0 𝛼 0 \alpha>0 this fractional integral is defined as J t α ​ f ​ ( t ) := 1 Γ ​ ( α ) ​ ∫ 0 t ( t − τ ) α − 1 ​ f ​ ( τ ) ​ 𝑑 τ , assign superscript subscript 𝐽 𝑡 𝛼 𝑓 𝑡 1 Γ 𝛼 superscript subscript 0 𝑡 superscript 𝑡 𝜏 𝛼 1 𝑓 𝜏 differential-d 𝜏 J_{t}^{\alpha}\,f(t):={1\over{\Gamma(\alpha)}}\,\int_{0}^{t}\!(t-\tau)^{\alpha-1}\,f(\tau)\,d\tau\,, ( 1.1 ) 1.1 where Γ ​ ( α ) := ∫ 0 ∞ e − u ​ u α − 1 ​ 𝑑 u assign Γ 𝛼 superscript subscript 0 superscript e 𝑢 superscript 𝑢 𝛼 1 differential-d 𝑢 {\displaystyle\Gamma(\alpha):=\int_{0}^{\infty}{\rm e}^{-u}\,u^{\alpha-1}\,du} denotes the Gamma function. For existence of the integral (1) it is sufficient that the function f ​ ( t ) 𝑓 𝑡 f(t) is locally integrable in 𝐑 + superscript 𝐑 {\bf R}^{+} and for t → 0 → 𝑡 0 t\to 0 behaves like O ​ ( t − ν ) 𝑂 superscript 𝑡 𝜈 O(t^{-\nu}) with a number ν < α 𝜈 𝛼 \nu<\alpha . For completion we define J t 0 = I superscript subscript 𝐽 𝑡 0 𝐼 J_{t}^{0}=I (Identity operator). We recall the semigroup property J t α ​ J t β = J t β ​ J t α = J t α + β , α , β ≥ 0 . formulae-sequence superscript subscript 𝐽 𝑡 𝛼 superscript subscript 𝐽 𝑡 𝛽 superscript subscript 𝐽 𝑡 𝛽 superscript subscript 𝐽 𝑡 𝛼 superscript subscript 𝐽 𝑡 𝛼 𝛽 𝛼 𝛽 0 J_{t}^{\alpha}\,J_{t}^{\beta}=\,J_{t}^{\beta}\,J_{t}^{\alpha}=J_{t}^{\alpha+\beta}\,,\quad\alpha,\beta\geq 0\,. ( 1.2 ) 1.2 Furthermore we note that for α ≥ 0 𝛼 0 \alpha\geq 0 J t α ​ t γ = Γ ​ ( γ + 1 ) Γ ​ ( γ + 1 + α ) ​ t γ + α , γ > − 1 . formulae-sequence superscript subscript 𝐽 𝑡 𝛼 superscript 𝑡 𝛾 Γ 𝛾 1 Γ 𝛾 1 𝛼 superscript 𝑡 𝛾 𝛼 𝛾 1 J_{t}^{\alpha}\,t^{\gamma}={\Gamma(\gamma+1)\over\Gamma(\gamma+1+\alpha)}\,t^{\gamma+\alpha}\,,\quad\gamma>-1\,. ( 1.3 ) 1.3 The fractional derivative of order μ > 0 𝜇 0 \mu>0 in the Riemann-Liouville sense is defined as the operator D t μ superscript subscript 𝐷 𝑡 𝜇 D_{t}^{\mu} which is the left inverse of the Riemann-Liouville integral of order μ 𝜇 \mu (in analogy with the ordinary derivative), that is D t μ ​ J t μ = I , μ > 0 . formulae-sequence superscript subscript 𝐷 𝑡 𝜇 superscript subscript 𝐽 𝑡 𝜇 𝐼 𝜇 0 D_{t}^{\mu}\,J_{t}^{\mu}=I\,,\quad\mu>0\,. ( 1.4 ) 1.4 If m 𝑚 m denotes the positive integer such that m − 1 < μ ≤ m , 𝑚 1 𝜇 𝑚 m-1<\mu\leq m\,, we recognize from Eqs. (1.2) and (1.4): D t μ ​ f ​ ( t ) := D t m ​ J t m − μ ​ f ​ ( t ) , m − 1 < μ ≤ m . formulae-sequence assign superscript subscript 𝐷 𝑡 𝜇 𝑓 𝑡 superscript subscript 𝐷 𝑡 𝑚 superscript subscript 𝐽 𝑡 𝑚 𝜇 𝑓 𝑡 𝑚 1 𝜇 𝑚 D_{t}^{\mu}\,f(t):=D_{t}^{m}\,J_{t}^{m-\mu}\,f(t)\,,\quad m-1<\mu\leq m\,. ( 1.5 ) 1.5 In fact, using the semigroup property (1.2), we have D t μ ​ J t μ = D t m ​ J t m − μ ​ J t μ = D t m ​ J t m = I . superscript subscript 𝐷 𝑡 𝜇 superscript subscript 𝐽 𝑡 𝜇 superscript subscript 𝐷 𝑡 𝑚 superscript subscript 𝐽 𝑡 𝑚 𝜇 superscript subscript 𝐽 𝑡 𝜇 superscript subscript 𝐷 𝑡 𝑚 superscript subscript 𝐽 𝑡 𝑚 𝐼 D_{t}^{\mu}\,J_{t}^{\mu}=D_{t}^{m}\,J_{t}^{m-\mu}\,J_{t}^{\mu}=D_{t}^{m}\,J_{t}^{m}=I\,. Thus (1.5) implies D t μ ​ f ​ ( t ) = { d m d ​ t m ​ 1 Γ ​ ( m − μ ) ​ ∫ 0 t f ​ ( τ ) ​ d ​ τ ( t − τ ) μ + 1 − m , m − 1 < μ < m ; d m d ​ t m ​ f ​ ( t ) , μ = m . \,\!\!\!\!D_{t}^{\mu}\,f(t)=\cases{{\displaystyle{d^{m}\over dt^{m}}}\left{\displaystyle{1\over{\Gamma(m-\mu)}}\int_{0}^{t}\!{f(\tau)\,d\tau\over(t-\tau)^{\mu+1-m}}}\right,&$m-1<\mu<m;$\cr\cr{\displaystyle{d^{m}\over dt^{m}}f(t)},&$\mu=m.$} ( 1.5 ′ ) superscript 1.5 ′ For completion we define D t 0 = I . superscript subscript 𝐷 𝑡 0 𝐼 D_{t}^{0}=I\,. On the other hand, the fractional derivative of order μ 𝜇 \mu in the Caputo sense is defined as the operator D t μ ∗ subscript superscript subscript 𝐷 𝑡 𝜇 \,{}_{*}D_{t}^{\mu} such that D t μ ∗ ​ f ​ ( t ) := J t m − μ ​ D t m ​ f ​ ( t ) , m − 1 < μ ≤ m . formulae-sequence assign subscript superscript subscript 𝐷 𝑡 𝜇 𝑓 𝑡 superscript subscript 𝐽 𝑡 𝑚 𝜇 superscript subscript 𝐷 𝑡 𝑚 𝑓 𝑡 𝑚 1 𝜇 𝑚 {}_{*}D_{t}^{\mu}\,f(t):=\,J_{t}^{m-\mu}\,D_{t}^{m}\,f(t)\,,\quad m-1<\mu\leq m\,. ( 1.6 ) 1.6 This implies D t μ ∗ ​ f ​ ( t ) = { 1 Γ ​ ( m − μ ) ​ ∫ 0 t f ( m ) ​ ( τ ) ​ d ​ τ ( t − τ ) μ + 1 − m , m − 1 < μ < m ; d m d ​ t m ​ f ​ ( t ) , μ = m . \,{}_{*}D_{t}^{\mu}\,f(t)=\cases{{\displaystyle{1\over{\Gamma(m-\mu)}}}\,{\displaystyle\int_{0}^{t}\!{\displaystyle{f^{(m)}(\tau)\,d\tau\over(t-\tau)^{\mu+1-m}}}}\,,&$m-1<\mu<m\,;$\cr\cr{\displaystyle{d^{m}\over dt^{m}}f(t)}\,,&$\mu=m\,.$} ( 1.6 ′ ) superscript 1.6 ′ Thus, when the order is not integer the two fractional derivatives differ in that the standard derivative of order m 𝑚 m does not generally commute with the fractional integral. Of course the Caputo derivative (1.6 ′ ) needs higher regularity conditions of f ​ ( t ) 𝑓 𝑡 f(t) than the Riemann-Liouville derivative (1.5 ′ ). We point out that the Caputo fractional derivative satisfies the relevant property of being zero when applied to a constant, and, in general, to any power function of non-negative integer degree less than m , 𝑚 m\,, if its order μ 𝜇 \mu is such that m − 1 < μ ≤ m . 𝑚 1 𝜇 𝑚 m-1<\mu\leq m\,. Furthermore we note for μ ≥ 0 𝜇 0 \mu\geq 0 : D t μ ​ t γ = Γ ​ ( γ + 1 ) Γ ​ ( γ + 1 − μ ) ​ t γ − μ , γ > − 1 . formulae-sequence superscript subscript 𝐷 𝑡 𝜇 superscript 𝑡 𝛾 Γ 𝛾 1 Γ 𝛾 1 𝜇 superscript 𝑡 𝛾 𝜇 𝛾 1 D_{t}^{\mu}\,t^{\gamma}={\Gamma(\gamma+1)\over\Gamma(\gamma+1-\mu)}\,t^{\gamma-\mu}\,,\quad\gamma>-1\,. ( 1.7 ) 1.7 It is instructive to compare Eqs. (1.3), (1.7). In 58 we have shown the essential relationships between the two fractional derivatives for the same non-integer order D t μ ∗ ​ f ​ ( t ) = { D t μ ​ f ​ ( t ) − ∑ k = 0 m − 1 f ( k ) ​ ( 0 + ) ​ t k k ! , D t μ ​ f ​ ( t ) − ∑ k = 0 m − 1 f ( k ) ​ ( 0 + ) ​ t k − μ Γ ​ ( k − μ + 1 ) , m − 1 < μ < m . formulae-sequence subscript superscript subscript 𝐷 𝑡 𝜇 𝑓 𝑡 cases superscript subscript 𝐷 𝑡 𝜇 delimited- 𝑓 𝑡 superscript subscript 𝑘 0 𝑚 1 superscript 𝑓 𝑘 superscript 0 superscript 𝑡 𝑘 𝑘 otherwise otherwise otherwise superscript subscript 𝐷 𝑡 𝜇 𝑓 𝑡 superscript subscript 𝑘 0 𝑚 1 superscript 𝑓 𝑘 superscript 0 superscript 𝑡 𝑘 𝜇 Γ 𝑘 𝜇 1 otherwise 𝑚 1 𝜇 𝑚 {}_{*}D_{t}^{\mu}\,f(t)=\cases{{\displaystyle\,D_{t}^{\mu}\leftf(t)-\sum_{k=0}^{m-1}f^{(k)}(0^{+})\,{t^{k}\over k!}\right},\cr\cr{\displaystyle D_{t}^{\mu}\,f(t)-\sum_{k=0}^{m-1}{f^{(k)}(0^{+})\,t^{k-\mu}\over\Gamma(k-\mu+1)}},\cr}\quad m-1<\mu<m\,. ( 1.8 ) 1.8 In particular we have from (1.6 ′ ) and (1.8) D t μ ∗ ​ f ​ ( t ) = 1 Γ ​ ( 1 − μ ) ​ ∫ 0 t f ( 1 ) ​ ( τ ) ( t − τ ) μ ​ 𝑑 τ = D t μ ​ f ​ ( t ) − f ​ ( 0 + ) = D t μ ​ f ​ ( t ) − f ​ ( 0 + ) ​ t − μ Γ ​ ( 1 − μ ) , ​ 0 < μ < 1 . missing-subexpression subscript superscript subscript 𝐷 𝑡 𝜇 𝑓 𝑡 1 Γ 1 𝜇 superscript subscript 0 𝑡 superscript 𝑓 1 𝜏 superscript 𝑡 𝜏 𝜇 differential-d 𝜏 missing-subexpression absent superscript subscript 𝐷 𝑡 𝜇 delimited- 𝑓 𝑡 𝑓 superscript 0 superscript subscript 𝐷 𝑡 𝜇 𝑓 𝑡 𝑓 superscript 0 superscript 𝑡 𝜇 Γ 1 𝜇 0 𝜇 1 \begin{array}{ll}&\,{}_{*}D_{t}^{\mu}f(t)={\displaystyle{1\over{\Gamma(1-\mu)}}\,\int_{0}^{t}\!{f^{(1)}(\tau)\over(t-\tau)^{\mu}}\,d\tau}\\ &={\displaystyle\,D_{t}^{\mu}\,\leftf(t)-f(0^{+})\right}={\displaystyle\,D_{t}^{\mu}\,f(t)-f(0^{+})\,{t^{-\mu}\over\Gamma(1-\mu)}}\,,\end{array}\;0<\mu<1\,. ( 1.9 ) 1.9 The Caputo fractional derivative represents a sort of regularization in the time origin for the Riemann-Liouville fractional derivative. We note that for its existence all the limiting values f ( k ) ​ ( 0 + ) := lim t → 0 + D t k ​ f ​ ( t ) assign superscript 𝑓 𝑘 superscript 0 subscript → 𝑡 superscript 0 superscript subscript 𝐷 𝑡 𝑘 𝑓 𝑡 f^{(k)}(0^{+}):={\displaystyle\lim_{t\to 0^{+}}D_{t}^{k}f(t)} are required to be finite for k = 0 , 1 , 2 , … , m − 1 𝑘 0 1 2 … 𝑚 1 k=0,1,2,\dots,m-1 . In the special case f ( k ) ​ ( 0 + ) = 0 superscript 𝑓 𝑘 superscript 0 0 f^{(k)}(0^{+})=0 for k = 0 , 1 , m − 1 𝑘 0 1 𝑚 1 k=0,1,m-1 , the two fractional derivatives coincide. We observe the different behaviour of the two fractional derivatives at the end points of the interval ( m − 1 , m ) 𝑚 1 𝑚 (m-1,m)\, namely when the order is any positive integer, as it can be noted from their definitions (1.5), (1.6). In fact, whereas for μ → m − → 𝜇 superscript 𝑚 \mu\to m^{-} both derivatives reduce to D t m superscript subscript 𝐷 𝑡 𝑚 D_{t}^{m} , as stated in Eqs. (1.5 ′ ), (1.6 ′ ), due to the fact that the operator J t 0 = I superscript subscript 𝐽 𝑡 0 𝐼 J_{t}^{0}\,=\,I commutes with D t m superscript subscript 𝐷 𝑡 𝑚 D_{t}^{m} , for μ → ( m − 1 ) + → 𝜇 superscript 𝑚 1 \mu\to(m-1)^{+} we have μ → ( m − 1 ) + : { D t μ ​ f ​ ( t ) → D t m ​ J t 1 ​ f ​ ( t ) = D t ( m − 1 ) ​ f ​ ( t ) = f ( m − 1 ​ ( t ) , D t μ ∗ ​ f ​ ( t ) → J t 1 ​ D t m ​ f ​ ( t ) = f ( m − 1 ) ​ ( t ) − f ( m − 1 ) ​ ( 0 + ) . \!\!\mu\to(m-1)^{+}:\cases{{\displaystyle D_{t}^{\mu}f(t)}\to{\displaystyle\,\,D_{t}^{m}\,J_{t}^{1}\,f(t)=D_{t}^{(m-1)}\,f(t)=f^{(m-1}(t)},\cr\cr{\,{}_{*}D_{t}^{\mu}f(t)}\to{\displaystyle J_{t}^{1}\,D_{t}^{m}\,f(t)=f^{(m-1)}(t)-f^{(m-1)}(0^{+})}.} ( 1.10 ) 1.10 As a consequence, roughly speaking, we can say that D t μ superscript subscript 𝐷 𝑡 𝜇 D_{t}^{\mu} is, with respect to its order μ , 𝜇 \mu\,, an operator continuous at any positive integer, whereas D t μ ∗ subscript superscript subscript 𝐷 𝑡 𝜇 \,{}_{*}D_{t}^{\mu} is an operator only left-continuous. The above behaviours have induced us to keep for the Riemann-Liouville derivative the same symbolic notation as for the standard derivative of integer order, while for the Caputo derivative to decorate the corresponding symbol with subscript ∗ * . We also note, with m − 1 < μ ≤ m , 𝑚 1 𝜇 𝑚 m-1<\mu\leq m\,, and c j subscript 𝑐 𝑗 c_{j} arbitrary constants, D t μ ​ f ​ ( t ) = D t μ ​ g ​ ( t ) ⟺ f ​ ( t ) = g ​ ( t ) + ∑ j = 1 m c j ​ t μ − j , ⟺ superscript subscript 𝐷 𝑡 𝜇 𝑓 𝑡 superscript subscript 𝐷 𝑡 𝜇 𝑔 𝑡 𝑓 𝑡 𝑔 𝑡 superscript subscript 𝑗 1 𝑚 subscript 𝑐 𝑗 superscript 𝑡 𝜇 𝑗 \hskip-14.22636ptD_{t}^{\mu}\,f(t)\,=\,D_{t}^{\mu}\,g(t)\,\Longleftrightarrow\,f(t)=g(t)+\sum_{j=1}^{m}c_{j}\,t^{\mu-j}\,, ( 1.11 ) 1.11 D t μ ∗ ​ f ​ ( t ) = ∗ D t μ ​ g ​ ( t ) ⟺ f ​ ( t ) = g ​ ( t ) + ∑ j = 1 m c j ​ t m − j . ⟺ subscript subscript superscript subscript 𝐷 𝑡 𝜇 𝑓 𝑡 superscript subscript 𝐷 𝑡 𝜇 𝑔 𝑡 𝑓 𝑡 𝑔 𝑡 superscript subscript 𝑗 1 𝑚 subscript 𝑐 𝑗 superscript 𝑡 𝑚 𝑗 {}_{*}D_{t}^{\mu}\,f(t)\,=\,_{*}D_{t}^{\mu}\,g(t)\,\Longleftrightarrow\,f(t)=g(t)+\sum_{j=1}^{m}c_{j}\,t^{m-j}\,. ( 1.12 ) 1.12 Furthermore, we observe that in case of a non-integer order for both fractional derivatives the semigroup property (of the standard derivative for integer order) does not hold for both fractional derivatives when the order is not integer. We point out the major utility of the Caputo fractional derivative in treating initial-value problems for physical and engineering applications where initial conditions are usually expressed in terms of integer-order derivatives. This can be easily seen using the Laplace transformation 1 1 1 The Laplace transform of a well-behaved function f ​ ( t ) 𝑓 𝑡 f(t) is defined as f ~ ( s ) = ℒ { f ( t ) ; s } : = ∫ 0 ∞ e − s ​ t f ( t ) d t , ℜ ( s ) > a f . \widetilde{f}(s)={\mathcal{L}}\left\{f(t);s\right\}:={\displaystyle\int_{0}^{\infty}}\!{\rm e}^{\displaystyle\,-st}\,f(t)\,dt\,,\quad\Re\,(s)>a_{f}\,. We recall that under suitable conditions the Laplace transform of the m 𝑚 m -derivative of f ​ ( t ) 𝑓 𝑡 f(t) is given by ℒ ​ { D t m ​ f ​ ( t ) ; s } = s m ​ f ~ ​ ( s ) − ∑ k = 0 m − 1 s m − 1 − k ​ f ( k ) ​ ( 0 + ) , f ( k ) ​ ( 0 + ) := lim t → 0 + D t k ​ f ​ ( t ) . formulae-sequence ℒ superscript subscript 𝐷 𝑡 𝑚 𝑓 𝑡 𝑠 superscript 𝑠 𝑚 ~ 𝑓 𝑠 superscript subscript 𝑘 0 𝑚 1 superscript 𝑠 𝑚 1 𝑘 superscript 𝑓 𝑘 superscript 0 assign superscript 𝑓 𝑘 superscript 0 subscript → 𝑡 superscript 0 superscript subscript 𝐷 𝑡 𝑘 𝑓 𝑡 {\mathcal{L}}\left\{D_{t}^{m}\,f(t);s\right\}=s^{m}\,\widetilde{f}(s)-\sum_{k=0}^{m-1}s^{m-1-k}\,f^{(k)}(0^{+})\,,\quad f^{(k)}(0^{+}):=\lim_{t\to 0^{+}}D_{t}^{k}f(t)\,. For the Caputo derivative of order μ 𝜇 \mu with m − 1 < μ ≤ m 𝑚 1 𝜇 𝑚 m-1<\mu\leq m we have ℒ ​ { D t μ ∗ ​ f ​ ( t ) ; s } = s μ ​ f ~ ​ ( s ) − ∑ k = 0 m − 1 s μ − 1 − k ​ f ( k ) ​ ( 0 + ) , f ( k ) ​ ( 0 + ) := lim t → 0 + D t k ​ f ​ ( t ) . missing-subexpression ℒ subscript superscript subscript 𝐷 𝑡 𝜇 𝑓 𝑡 𝑠 superscript 𝑠 𝜇 ~ 𝑓 𝑠 superscript subscript 𝑘 0 𝑚 1 superscript 𝑠 𝜇 1 𝑘 superscript 𝑓 𝑘 superscript 0 missing-subexpression missing-subexpression missing-subexpression assign superscript 𝑓 𝑘 superscript 0 subscript → 𝑡 superscript 0 superscript subscript 𝐷 𝑡 𝑘 𝑓 𝑡 \begin{array}{ll}&{\displaystyle{\mathcal{L}}\left\{{}_{*}D_{t}^{\mu}\,f(t);s\right\}}={\displaystyle s^{\mu}\,\widetilde{f}(s)-\sum_{k=0}^{m-1}s^{\mu-1-k}\,f^{(k)}(0^{+})}\,,\\ \\ &f^{(k)}(0^{+}):={\displaystyle\lim_{t\to 0^{+}}D_{t}^{k}f(t)\,.}\end{array} ( 1.13 ) 1.13 The corresponding rule for the Riemann-Liouville derivative of order μ 𝜇 \mu is ℒ ​ { D t μ ​ f ​ ( t ) ; s } = s μ ​ f ~ ​ ( s ) − ∑ k = 0 m − 1 s m − 1 − k ​ g ( k ) ​ ( 0 + ) , g ( k ) ​ ( 0 + ) := lim t → 0 + D t k ​ g ​ ( t ) , where g ​ ( t ) := J t ( m − μ ) ​ f ​ ( t ) . missing-subexpression ℒ superscript subscript 𝐷 𝑡 𝜇 𝑓 𝑡 𝑠 superscript 𝑠 𝜇 ~ 𝑓 𝑠 superscript subscript 𝑘 0 𝑚 1 superscript 𝑠 𝑚 1 𝑘 superscript 𝑔 𝑘 superscript 0 missing-subexpression missing-subexpression missing-subexpression formulae-sequence assign superscript 𝑔 𝑘 superscript 0 subscript → 𝑡 superscript 0 superscript subscript 𝐷 𝑡 𝑘 𝑔 𝑡 where assign 𝑔 𝑡 superscript subscript 𝐽 𝑡 𝑚 𝜇 𝑓 𝑡 \begin{array}{ll}&{\displaystyle{\mathcal{L}}\left\{D_{t}^{\mu}\,f(t);s\right\}}={\displaystyle s^{\mu}\,\widetilde{f}(s)-\sum_{k=0}^{m-1}s^{m-1-k}\,g^{(k)}(0^{+})}\,,\\ \\ &g^{(k)}(0^{+}):={\displaystyle\lim_{t\to 0^{+}}D_{t}^{k}g(t)\,,}\quad\hbox{where}\quad g(t):=J_{t}^{(m-\mu)}\,f(t)\,.\end{array} ( 1.14 ) 1.14 Thus it is more cumbersome to use the rule (1.14) than (1.13). The rule (1.14) requires initial values concerning an extra function g ​ ( t ) 𝑔 𝑡 g(t) related to the given f ​ ( t ) 𝑓 𝑡 f(t) through a fractional integral. However, when all the limiting values f ( k ) ​ ( 0 + ) superscript 𝑓 𝑘 superscript 0 f^{(k)}(0^{+}) for k = 0 , 1 , 2 , … 𝑘 0 1 2 … k=0,1,2,\dots are finite and the order is not integer, we can prove that the corresponding g ( k ) ​ ( 0 + ) superscript 𝑔 𝑘 superscript 0 g^{(k)}(0^{+}) vanish so that formula (1.14) simplifies into ℒ ​ { D t μ ​ f ​ ( t ) ; s } = s μ ​ f ~ ​ ( s ) , m − 1 < μ < m . formulae-sequence ℒ superscript subscript 𝐷 𝑡 𝜇 𝑓 𝑡 𝑠 superscript 𝑠 𝜇 ~ 𝑓 𝑠 𝑚 1 𝜇 𝑚 {\mathcal{L}}\left\{D_{t}^{\mu}\,f(t);s\right\}=s^{\mu}\,\widetilde{f}(s)\,,\quad m-1<\mu<m\,. ( 1.15 ) 1.15 For this proof it is sufficient to apply the Laplace transform to the second equation (8), by recalling that ℒ ​ { t α ; s } = Γ ​ ( α + 1 ) / s α + 1 ℒ superscript 𝑡 𝛼 𝑠 Γ 𝛼 1 superscript 𝑠 𝛼 1 {\mathcal{L}}\left\{t^{\alpha};s\right\}=\Gamma(\alpha+1)/s^{\alpha+1}\, for α > − 1 𝛼 1 \alpha>-1 , and then to compare (1.13) with (1.14). For fractional differentiation on the positive semi-axis we recall another definition for the fractional derivative recently introduced by Hilfer, see 68 and 116 , which interpolates the previous definitions (1.5) and (1.6). Like the two derivatives previously discussed, it is related to a Riemann-Liouville integral. In our notation it reads D t μ , ν := J t ν ​ ( 1 − μ ) ​ D t 1 ​ J t ( 1 − ν ) ​ ( 1 − μ ) , 0 < μ < 1 , 0 ≤ ν ≤ 1 . formulae-sequence formulae-sequence assign superscript subscript 𝐷 𝑡 𝜇 𝜈 superscript subscript 𝐽 𝑡 𝜈 1 𝜇 superscript subscript 𝐷 𝑡 1 superscript subscript 𝐽 𝑡 1 𝜈 1 𝜇 0 𝜇 1 0 𝜈 1 D_{t}^{\mu,\nu}:=J_{t}^{\nu(1-\mu)}\,D_{t}^{1}\,J_{t}^{(1-\nu)(1-\mu)}\,,\quad 0<\mu<1\,,\quad 0\leq\nu\leq 1\,. ( 1.16 ) 1.16 We can refer it to as the Hilfer (H) fractional derivative of order μ 𝜇 \mu and type ν 𝜈 \nu . The Riemann-Liouville derivative corresponds to the type ν = 0 𝜈 0 \nu=0 whereas that Caputo derivative to the type ν = 1 𝜈 1 \nu=1 . We have here not discussed the Beyer-Kempfle approach investigated and used in several papers by Beyer and Kempfle et al.: this approach is appropriate for causal processes not starting at a finite instant of time, see e.g. 9 , 69 . They define the time-fractional derivative on the whole real line as a pseudo-differential operator via its Fourier symbol. The interested reader is referred to the above mentioned papers and references therein. For further reading on the theory and applications of fractional integrals and derivatives (more generally of fractional calculus) we may recommend e.g. our CISM Lecture Notes 56 , 58 , 81 , the review papers 86 , 87 , 117 , and the books 89 , 71 , 67 , 70 , 78 , 95 , 109 , 119 , 120 with references therein.","role":"user"},{"content":"Abstract: The aim of this tutorial survey is to revisit the basic theory of relaxation processes governed by linear differential equations of fractional order. The fractional derivatives are intended both in the Rieamann-Liouville sense and in the Caputo sense. After giving a necessary outline of the classical theory of linear viscoelasticity, we contrast these two types of fractional derivatives in their ability to take into account initial conditions in the constitutive equations of fractional order. We also provide historical notes on the origins of the Caputo derivative and on the use of fractional calculus in viscoelasticity.","role":"assistant"}]

Analytic QCD - a Short Review

I Introduction Perturbative QCD calculations involve coupling a ​ ( Q 2 ) ≡ α s ​ ( Q 2 ) / π 𝑎 superscript 𝑄 2 subscript 𝛼 𝑠 superscript 𝑄 2 𝜋 a(Q^{2})\equiv\alpha_{s}(Q^{2})/\pi which has Landau singularities (poles, cuts) on the space-like semiaxis 0 ≤ Q 2 ≤ Λ 2 0 superscript 𝑄 2 superscript Λ 2 0\leq Q^{2}\leq\Lambda^{2} ( q 2 ≡ − Q 2 superscript 𝑞 2 superscript 𝑄 2 q^{2}\equiv-Q^{2} ). These lead to Landau singularities for the evaluated space-like observables 𝒟 ​ ( Q 2 ) 𝒟 superscript 𝑄 2 {\cal D}(Q^{2}) at low Q 2 ∼ < Λ 2 superscript similar-to superscript 𝑄 2 superscript Λ 2 Q^{2}\stackrel{{\scriptstyle<}}{{\sim}}\Lambda^{2} . The existence of such singularities is in contradiction with the general principles of the local quantum field theories BSh . Further, lattice simulations ls confirm that such singularities are not present in a ​ ( Q 2 ) 𝑎 superscript 𝑄 2 a(Q^{2}) . An analytized coupling 𝒜 1 ​ ( Q 2 ) subscript 𝒜 1 superscript 𝑄 2 \mathcal{A}_{1}(Q^{2}) , which agrees with the perturbative a ​ ( Q 2 ) 𝑎 superscript 𝑄 2 a(Q^{2}) at Q 2 → ∞ → superscript 𝑄 2 {Q^{2}\to\infty} and is analytic in the Euclidean part of the Q 2 superscript 𝑄 2 Q^{2} -plane ( Q 2 ​ ϵ ​ 𝒞 superscript 𝑄 2 italic-ϵ 𝒞 Q^{2}\ \epsilon\ {\cal C} , Q 2 ≰ 0 not-less-than-or-equals superscript 𝑄 2 0 Q^{2}\not\leq 0 ), addresses this problem, and has been constructed by Shirkov and Solovtsov about ten years ago ShS . Several other analytic QCD (anQCD) models for 𝒜 1 ​ ( Q 2 ) subscript 𝒜 1 superscript 𝑄 2 {\mathcal{A}_{1}(Q^{2})} can be constructed, possibly satisfying certain additional constraints at low and/or at high Q 2 superscript 𝑄 2 Q^{2} . Another problem is the analytization of higher power terms a n ↦ 𝒜 n maps-to superscript 𝑎 𝑛 subscript 𝒜 𝑛 {a^{n}}\mapsto\mathcal{A}_{n} in the truncated perturbation series (TPS) for 𝒟 ​ ( Q 2 ) 𝒟 superscript 𝑄 2 {\cal D}(Q^{2}) . Also here, several possibilities appear. Application of the Operator Product Expansion (OPE) approach, in the ITEP sense, to inclusive space-like observables appears to make sense only in a restricted class of such anQCD models. This is a short and incomplete review of the activity in the area; relatively large space is given to the work of the review’s authors. For an earlier and more extensive review, see e. g. Ref. Prosperi:2006hx . Section II contains general aspects of analytization of the Euclidean coupling a ​ ( Q 2 ) ↦ 𝒜 1 ​ ( Q 2 ) maps-to 𝑎 superscript 𝑄 2 subscript 𝒜 1 superscript 𝑄 2 {a(Q^{2})\mapsto\mathcal{A}_{1}(Q^{2})} , and the definition of the time-like (Minkowskian) coupling 𝔄 1 ​ ( s ) subscript 𝔄 1 𝑠 {\mathfrak{A}_{1}(s)} . Further, in Sec. II we review the minimal analytization (MA) procedure developed by Shirkov and Solovtsov ShS , and a variant thereof developed by Nesterenko Neste2000 . In Sec. III we present various approaches of going beyond the MA procedure, i.e., various models for 𝔄 1 ​ ( s ) subscript 𝔄 1 𝑠 {\mathfrak{A}_{1}(s)} , and thus for 𝒜 1 ​ ( Q 2 ) subscript 𝒜 1 superscript 𝑄 2 {\mathcal{A}_{1}(Q^{2})} Alekseev:2005he ; Srivastava:2001ts ; Webber:1998um ; Neste2005 ; Cvetic:2006mk ; Cvetic:2006gc . In Sec. IV , analytization procedures for the higher powers a n ​ ( Q 2 ) ↦ 𝒜 n ​ ( Q 2 ) maps-to superscript 𝑎 𝑛 superscript 𝑄 2 subscript 𝒜 𝑛 superscript 𝑄 2 {a^{n}(Q^{2})\mapsto\mathcal{A}_{n}(Q^{2})} in MA model are presented Milton:1997mi ; Milton:2000fi ; Sh , and an alternative approach which is applicable to any model of analytic 𝒜 1 ​ ( Q 2 ) subscript 𝒜 1 superscript 𝑄 2 {\mathcal{A}_{1}(Q^{2})} Cvetic:2006mk ; Cvetic:2006gc is presented. In Sec. V , an analytization of noninteger powers a ν ​ ( Q 2 ) superscript 𝑎 𝜈 superscript 𝑄 2 a^{\nu}(Q^{2}) is outlined Bakulev . In Sec. VI , methods of evaluations of space-like and of inclusive time-like observables in models with analytic 𝒜 1 ​ ( Q 2 ) subscript 𝒜 1 superscript 𝑄 2 {\mathcal{A}_{1}(Q^{2})} are described, and some numerical results are presented for semihadronic τ 𝜏 \tau decay rate ratio r τ subscript 𝑟 𝜏 {r_{\tau}} , Adler function d v ​ ( Q 2 ) subscript 𝑑 𝑣 superscript 𝑄 2 d_{v}(Q^{2}) and Bjorken polarized sum rule (BjPSR) d b ​ ( Q 2 ) subscript 𝑑 𝑏 superscript 𝑄 2 {d_{b}(Q^{2})} Milton:1997mi ; Milton:2000fi ; Sh ; Shirkov:2006gv ; Cvetic:2006mk ; Cvetic:2006gc . In Sec. VII , two sets of models are presented Raczka ; Cvetic:2007ad whose analytic couplings 𝒜 1 ​ ( Q 2 ) subscript 𝒜 1 superscript 𝑄 2 {\mathcal{A}_{1}(Q^{2})} preserve the OPE-ITEP philosophy, i.e., at high Q 2 superscript 𝑄 2 Q^{2} they fulfill: | 𝒜 1 ​ ( Q 2 ) − a ​ ( Q 2 ) | < ( Λ 2 / Q 2 ) k subscript 𝒜 1 superscript 𝑄 2 𝑎 superscript 𝑄 2 superscript superscript Λ 2 superscript 𝑄 2 𝑘 {|\mathcal{A}_{1}(Q^{2})-a(Q^{2})|<(\Lambda^{2}/Q^{2})^{k}} for any k ​ ϵ ​ 𝒩 𝑘 italic-ϵ 𝒩 k\ \epsilon\ {\cal N} . Section VIII contains a summary of the presented themes.

Analytic versions of QCD are those whose coupling αs(Q2) does not have the unphysical Landau singularities on the space-like axis (-q2= Q2 > 0). The coupling is analytic in the entire complex plane except the time-like axis (Q2 < 0). Such couplings are thus suitable for application of perturbative methods down to energies of order GeV. We present a short review of the activity in the area which started with a seminal paper of Shirkov and Solovtsov ten years ago. Several models for analytic QCD coupling are presented. Strengths and weaknesses of some of these models are pointed out. Further, for such analytic couplings, constructions of the corresponding higher order analytic couplings (the analogs of the higher powers of the perturbative coupling) are outlined, and an approach based on the renormalization group considerations is singled out. Methods of evaluation of the leading-twist part of space-like observables in such analytic frameworks are described. Such methods are applicable also to the inclusive time-like observables. Two analytic models are outlined which respect the ITEP Operator Product Expansion philosophy, and thus allow for an evaluation of higher-twist contributions to observables.

[{"content":"Help me to generate the abstract of a survey paper given the title:*Analytic QCD - a Short Review* and the introduction:I Introduction Perturbative QCD calculations involve coupling a ​ ( Q 2 ) ≡ α s ​ ( Q 2 ) / π 𝑎 superscript 𝑄 2 subscript 𝛼 𝑠 superscript 𝑄 2 𝜋 a(Q^{2})\equiv\alpha_{s}(Q^{2})/\pi which has Landau singularities (poles, cuts) on the space-like semiaxis 0 ≤ Q 2 ≤ Λ 2 0 superscript 𝑄 2 superscript Λ 2 0\leq Q^{2}\leq\Lambda^{2} ( q 2 ≡ − Q 2 superscript 𝑞 2 superscript 𝑄 2 q^{2}\equiv-Q^{2} ). These lead to Landau singularities for the evaluated space-like observables 𝒟 ​ ( Q 2 ) 𝒟 superscript 𝑄 2 {\cal D}(Q^{2}) at low Q 2 ∼ < Λ 2 superscript similar-to superscript 𝑄 2 superscript Λ 2 Q^{2}\stackrel{{\scriptstyle<}}{{\sim}}\Lambda^{2} . The existence of such singularities is in contradiction with the general principles of the local quantum field theories BSh . Further, lattice simulations ls confirm that such singularities are not present in a ​ ( Q 2 ) 𝑎 superscript 𝑄 2 a(Q^{2}) . An analytized coupling 𝒜 1 ​ ( Q 2 ) subscript 𝒜 1 superscript 𝑄 2 \mathcal{A}_{1}(Q^{2}) , which agrees with the perturbative a ​ ( Q 2 ) 𝑎 superscript 𝑄 2 a(Q^{2}) at Q 2 → ∞ → superscript 𝑄 2 {Q^{2}\to\infty} and is analytic in the Euclidean part of the Q 2 superscript 𝑄 2 Q^{2} -plane ( Q 2 ​ ϵ ​ 𝒞 superscript 𝑄 2 italic-ϵ 𝒞 Q^{2}\ \epsilon\ {\cal C} , Q 2 ≰ 0 not-less-than-or-equals superscript 𝑄 2 0 Q^{2}\not\leq 0 ), addresses this problem, and has been constructed by Shirkov and Solovtsov about ten years ago ShS . Several other analytic QCD (anQCD) models for 𝒜 1 ​ ( Q 2 ) subscript 𝒜 1 superscript 𝑄 2 {\mathcal{A}_{1}(Q^{2})} can be constructed, possibly satisfying certain additional constraints at low and/or at high Q 2 superscript 𝑄 2 Q^{2} . Another problem is the analytization of higher power terms a n ↦ 𝒜 n maps-to superscript 𝑎 𝑛 subscript 𝒜 𝑛 {a^{n}}\mapsto\mathcal{A}_{n} in the truncated perturbation series (TPS) for 𝒟 ​ ( Q 2 ) 𝒟 superscript 𝑄 2 {\cal D}(Q^{2}) . Also here, several possibilities appear. Application of the Operator Product Expansion (OPE) approach, in the ITEP sense, to inclusive space-like observables appears to make sense only in a restricted class of such anQCD models. This is a short and incomplete review of the activity in the area; relatively large space is given to the work of the reviews authors. For an earlier and more extensive review, see e. g. Ref. Prosperi:2006hx . Section II contains general aspects of analytization of the Euclidean coupling a ​ ( Q 2 ) ↦ 𝒜 1 ​ ( Q 2 ) maps-to 𝑎 superscript 𝑄 2 subscript 𝒜 1 superscript 𝑄 2 {a(Q^{2})\mapsto\mathcal{A}_{1}(Q^{2})} , and the definition of the time-like (Minkowskian) coupling 𝔄 1 ​ ( s ) subscript 𝔄 1 𝑠 {\mathfrak{A}_{1}(s)} . Further, in Sec. II we review the minimal analytization (MA) procedure developed by Shirkov and Solovtsov ShS , and a variant thereof developed by Nesterenko Neste2000 . In Sec. III we present various approaches of going beyond the MA procedure, i.e., various models for 𝔄 1 ​ ( s ) subscript 𝔄 1 𝑠 {\mathfrak{A}_{1}(s)} , and thus for 𝒜 1 ​ ( Q 2 ) subscript 𝒜 1 superscript 𝑄 2 {\mathcal{A}_{1}(Q^{2})} Alekseev:2005he ; Srivastava:2001ts ; Webber:1998um ; Neste2005 ; Cvetic:2006mk ; Cvetic:2006gc . In Sec. IV , analytization procedures for the higher powers a n ​ ( Q 2 ) ↦ 𝒜 n ​ ( Q 2 ) maps-to superscript 𝑎 𝑛 superscript 𝑄 2 subscript 𝒜 𝑛 superscript 𝑄 2 {a^{n}(Q^{2})\mapsto\mathcal{A}_{n}(Q^{2})} in MA model are presented Milton:1997mi ; Milton:2000fi ; Sh , and an alternative approach which is applicable to any model of analytic 𝒜 1 ​ ( Q 2 ) subscript 𝒜 1 superscript 𝑄 2 {\mathcal{A}_{1}(Q^{2})} Cvetic:2006mk ; Cvetic:2006gc is presented. In Sec. V , an analytization of noninteger powers a ν ​ ( Q 2 ) superscript 𝑎 𝜈 superscript 𝑄 2 a^{\nu}(Q^{2}) is outlined Bakulev . In Sec. VI , methods of evaluations of space-like and of inclusive time-like observables in models with analytic 𝒜 1 ​ ( Q 2 ) subscript 𝒜 1 superscript 𝑄 2 {\mathcal{A}_{1}(Q^{2})} are described, and some numerical results are presented for semihadronic τ 𝜏 \tau decay rate ratio r τ subscript 𝑟 𝜏 {r_{\tau}} , Adler function d v ​ ( Q 2 ) subscript 𝑑 𝑣 superscript 𝑄 2 d_{v}(Q^{2}) and Bjorken polarized sum rule (BjPSR) d b ​ ( Q 2 ) subscript 𝑑 𝑏 superscript 𝑄 2 {d_{b}(Q^{2})} Milton:1997mi ; Milton:2000fi ; Sh ; Shirkov:2006gv ; Cvetic:2006mk ; Cvetic:2006gc . In Sec. VII , two sets of models are presented Raczka ; Cvetic:2007ad whose analytic couplings 𝒜 1 ​ ( Q 2 ) subscript 𝒜 1 superscript 𝑄 2 {\mathcal{A}_{1}(Q^{2})} preserve the OPE-ITEP philosophy, i.e., at high Q 2 superscript 𝑄 2 Q^{2} they fulfill: | 𝒜 1 ​ ( Q 2 ) − a ​ ( Q 2 ) | < ( Λ 2 / Q 2 ) k subscript 𝒜 1 superscript 𝑄 2 𝑎 superscript 𝑄 2 superscript superscript Λ 2 superscript 𝑄 2 𝑘 {|\mathcal{A}_{1}(Q^{2})-a(Q^{2})|<(\Lambda^{2}/Q^{2})^{k}} for any k ​ ϵ ​ 𝒩 𝑘 italic-ϵ 𝒩 k\ \epsilon\ {\cal N} . Section VIII contains a summary of the presented themes.","role":"user"},{"content":"Abstract: Analytic versions of QCD are those whose coupling αs(Q2) does not have the unphysical Landau singularities on the space-like axis (-q2= Q2 > 0). The coupling is analytic in the entire complex plane except the time-like axis (Q2 < 0). Such couplings are thus suitable for application of perturbative methods down to energies of order GeV. We present a short review of the activity in the area which started with a seminal paper of Shirkov and Solovtsov ten years ago. Several models for analytic QCD coupling are presented. Strengths and weaknesses of some of these models are pointed out. Further, for such analytic couplings, constructions of the corresponding higher order analytic couplings (the analogs of the higher powers of the perturbative coupling) are outlined, and an approach based on the renormalization group considerations is singled out. Methods of evaluation of the leading-twist part of space-like observables in such analytic frameworks are described. Such methods are applicable also to the inclusive time-like observables. Two analytic models are outlined which respect the ITEP Operator Product Expansion philosophy, and thus allow for an evaluation of higher-twist contributions to observables.","role":"assistant"}]

Interpretations of Quantum Mechanics: a critical survey

I Quantum Theory: brief Overview Can we explain what the world is through a fundamental physical theory? This question corresponds to the historic disagreement among scientists and epistemologists concerning how to regard physical theories to which people commonly refer as the realist/antirealist debate. The position of the antirealist is the one according to which we should not believe that physics reveals to us something about reality but rather we should be satisfied with physics to be, for example, just empirically adequate. In contrast, the realist is strongly inclined to say not only that physics tells us about reality, but also that it is our only way to actually do metaphysics. In few words, the question is: is there an ontology? We are interested to show through a logical pathway the existence of a possible ontology in Nature. The abstract mathematical structure of the Lorentz transformations was deduced through simple physical principles. Thanks to the existence of these physical principles we do not have a significant debate on the interpretation of the theory of special relativity. The formulation of Quantum Mechanics (QM), on to the contrary, is based on a number of rather abstract axioms without a clear motivation for their existence. The problem about quantum mechanics does not lie on its effectivity, but on its interpretation. Any attempt to interpret quantum mechanics tries to provide a definite meaning to issues such as realism, completeness, local realism and determinism. Despite its success, the absence of elementary physical principles has determined a broad discussion about the interpretation of the theory. For this reason, and not only, Bell called the ordinary QM with the abbreviation FAPP (for all practical purposes). The standard interpretation of quantum mechanics, attempts, as much as possible, to give an ontological model of physical systems using the concept of the quantum state. However, the interpretation does not fully succeed in giving such a model, for this reason one solution to this problem is to abandon any attempt at an ontological model and to put quantum mechanics on a purely epistemological footing ( the context of informational approaches). We believe that a possible ontological model arises by the application of formalism of quantum mechanics to the entire universe (including observers). We will start next sections presenting, first, the basic formalism and postulates of QM, and then overviewing some relevant historical interpretations of QM.

This brief survey analyzes the epistemological implications about the role of observer in the interpretations of Quantum Mechanics. As we know, the goal of most interpretations of quantum mechanics is to avoid the apparent intrusion of the observer into the measurement process. In the same time, there are implicit and hidden assumptions about his role. In fact, most interpretations taking as ontic level one of these fundamental concepts as information, physical law and matter bring us to new problematical questions. We think, that no interpretation of the quantum theory can avoid this intrusion until we do not clarify the nature of observer.

[{"content":"Help me to generate the abstract of a survey paper given the title:*Interpretations of Quantum Mechanics: a critical survey* and the introduction:I Quantum Theory: brief Overview Can we explain what the world is through a fundamental physical theory? This question corresponds to the historic disagreement among scientists and epistemologists concerning how to regard physical theories to which people commonly refer as the realist/antirealist debate. The position of the antirealist is the one according to which we should not believe that physics reveals to us something about reality but rather we should be satisfied with physics to be, for example, just empirically adequate. In contrast, the realist is strongly inclined to say not only that physics tells us about reality, but also that it is our only way to actually do metaphysics. In few words, the question is: is there an ontology? We are interested to show through a logical pathway the existence of a possible ontology in Nature. The abstract mathematical structure of the Lorentz transformations was deduced through simple physical principles. Thanks to the existence of these physical principles we do not have a significant debate on the interpretation of the theory of special relativity. The formulation of Quantum Mechanics (QM), on to the contrary, is based on a number of rather abstract axioms without a clear motivation for their existence. The problem about quantum mechanics does not lie on its effectivity, but on its interpretation. Any attempt to interpret quantum mechanics tries to provide a definite meaning to issues such as realism, completeness, local realism and determinism. Despite its success, the absence of elementary physical principles has determined a broad discussion about the interpretation of the theory. For this reason, and not only, Bell called the ordinary QM with the abbreviation FAPP (for all practical purposes). The standard interpretation of quantum mechanics, attempts, as much as possible, to give an ontological model of physical systems using the concept of the quantum state. However, the interpretation does not fully succeed in giving such a model, for this reason one solution to this problem is to abandon any attempt at an ontological model and to put quantum mechanics on a purely epistemological footing ( the context of informational approaches). We believe that a possible ontological model arises by the application of formalism of quantum mechanics to the entire universe (including observers). We will start next sections presenting, first, the basic formalism and postulates of QM, and then overviewing some relevant historical interpretations of QM.","role":"user"},{"content":"Abstract: This brief survey analyzes the epistemological implications about the role of observer in the interpretations of Quantum Mechanics. As we know, the goal of most interpretations of quantum mechanics is to avoid the apparent intrusion of the observer into the measurement process. In the same time, there are implicit and hidden assumptions about his role. In fact, most interpretations taking as ontic level one of these fundamental concepts as information, physical law and matter bring us to new problematical questions. We think, that no interpretation of the quantum theory can avoid this intrusion until we do not clarify the nature of observer.","role":"assistant"}]

Review of Applications of YFS-Style Resummation in Quantum Field Theory via Monte Carlo Methods

1 Preface It is with great pleasure that I present this review of the application of YFS-style [ 1 ] exact, amplitude based resummation via Monte Carlo methods on the occasion of the 60 t ​ h superscript 60 𝑡 ℎ 60^{th} birthday of Prof. S. Jadach, my friend and collaborator since 1985. In the review, we intend to highlight some of the many pioneering contributions which Prof. Jadach has made to this important subject. We are all grateful to him for all that he has taught us about the subject.

We review the application of exact, amplitude-based, YFS-syle resummation in quantum field theory via Monte Carlo methods.

[{"content":"Help me to generate the abstract of a survey paper given the title:*Review of Applications of YFS-Style Resummation in Quantum Field Theory via Monte Carlo Methods* and the introduction:1 Preface It is with great pleasure that I present this review of the application of YFS-style 1 exact, amplitude based resummation via Monte Carlo methods on the occasion of the 60 t ​ h superscript 60 𝑡 ℎ 60^{th} birthday of Prof. S. Jadach, my friend and collaborator since 1985. In the review, we intend to highlight some of the many pioneering contributions which Prof. Jadach has made to this important subject. We are all grateful to him for all that he has taught us about the subject.","role":"user"},{"content":"Abstract: We review the application of exact, amplitude-based, YFS-syle resummation in quantum field theory via Monte Carlo methods.","role":"assistant"}]

Information Resources in High-Energy Physics: Surveying the Present Landscape and Charting the Future Course

High-Energy Physics (HEP), also known as Particle Physics, has a long record of innovation in scholarly communication. Half a century ago, theoretical physicists and experimental collaborations mailed to their peers hundreds, even thousands, of copies of their manuscripts. This occurred at the time of submission to peer-reviewed journals, whose speed in disseminating scientific information was deemed to be insufficient for the speed at which the field was evolving [ 1 ] . This practice led to the creation of the first electronic catalog for gray literature, later evolving into a catalog of the entire subject literature: the SPIRES database [ 2 ] . In the last two decades, crucial innovation in scholarly communication emerged from the HEP community, ranging from the invention of the world-wide web at CERN [ 3 ] to the inception of arXiv, the first and archetypal repository [ 4 ] . The onset of the web gave SPIRES the honor to be the first web server in America and the first database on the web [ 5 ] . More recently the HEP community inspired the development of Invenio, one of the first open-source digital library software packages [ 6 ] , currently used for repositories in many fields. Thanks to this suite of user-driven innovations, HEP scholars have used a variety of dedicated, field-specific “information resources”. For many decades these have been run by large research institutions as a natural evolution of more conventional library services. At their inception, these resources often provided unique services, or were tailored specifically to the needs of the HEP community. Many of these services still exist and still provide information that cannot be obtained in any other way. For many years now almost all journal literature has been electronically available, the entire web is readily searchable, and commercial online databases provide metadata about all scientific literature. In addition, online services are changing more and more rapidly as new tools are developed and new ways of interacting with users evolve. In light of this fast-changing world, it is important to assess the usage by HEP researchers of HEP-specific information resources. Such a study serves two purposes: within the field, it informs on the need for such community-based resources and their real role in the present internet landscape, inspiring their future evolution; globally, it provides an in-depth case study of the impact of discipline-based information resources, as opposed to institution-based information resources or cross-cutting (commercial) information platforms. This information is particularly relevant in light of recent worldwide moves towards self-archiving of research results at the institutional or disciplinary level, and the need to effectively incorporate these resources in the research workflow. A survey of HEP scholars was designed and deployed in order to provide a unique insight into their information needs and the way their research workflow includes information discovery and retrieval. Its results are presented in this Article. The Article first describes the current landscape of HEP information resources (Section 2), then presents the survey methodology and the demographics of the respondents (Section 3). Two sets of results are presented and discussed: the information resources preferred by HEP researchers (Section 4) and their appreciation of the relative importance of possible features of information resources (Section 5). The survey also provides additional information on user requirements for the future of information resources (Section 6). After the conclusions of the study (Section 7), an Appendix presents some of the most inspiring free-text answers charting the future of information provision in this field.

Access to previous results is of paramount importance in the scientific process. Recent progress in information management focuses on building e-infrastructures for the optimization of the research workflow, through both policy-driven and user-pulled dynamics. For decades, High-Energy Physics (HEP) has pioneered innovative solutions in the field of information management and dissemination. In light of a transforming information environment, it is important to assess the current usage of information resources by researchers and HEP provides a unique test-bed for this assessment. A survey of about 10% of practitioners in the field reveals usage trends and information needs. Community-based services, such as the pioneering arXiv and SPIRES systems, largely answer the need of the scientists, with a limited but increasing fraction of younger users relying on Google. Commercial services offered by publishers or database vendors are essentially unused in the field. The survey offers an insight into the most important features that users require to optimize their research workflow. These results inform the future evolution of information management in HEP and, as these researchers are traditionally ``early adopters'' of innovation in scholarly communication, can inspire developments of disciplinary repositories serving other communities.

[{"content":"Help me to generate the abstract of a survey paper given the title:*Information Resources in High-Energy Physics: Surveying the Present Landscape and Charting the Future Course* and the introduction:High-Energy Physics (HEP), also known as Particle Physics, has a long record of innovation in scholarly communication. Half a century ago, theoretical physicists and experimental collaborations mailed to their peers hundreds, even thousands, of copies of their manuscripts. This occurred at the time of submission to peer-reviewed journals, whose speed in disseminating scientific information was deemed to be insufficient for the speed at which the field was evolving 1 . This practice led to the creation of the first electronic catalog for gray literature, later evolving into a catalog of the entire subject literature: the SPIRES database 2 . In the last two decades, crucial innovation in scholarly communication emerged from the HEP community, ranging from the invention of the world-wide web at CERN 3 to the inception of arXiv, the first and archetypal repository 4 . The onset of the web gave SPIRES the honor to be the first web server in America and the first database on the web 5 . More recently the HEP community inspired the development of Invenio, one of the first open-source digital library software packages 6 , currently used for repositories in many fields. Thanks to this suite of user-driven innovations, HEP scholars have used a variety of dedicated, field-specific information resources. For many decades these have been run by large research institutions as a natural evolution of more conventional library services. At their inception, these resources often provided unique services, or were tailored specifically to the needs of the HEP community. Many of these services still exist and still provide information that cannot be obtained in any other way. For many years now almost all journal literature has been electronically available, the entire web is readily searchable, and commercial online databases provide metadata about all scientific literature. In addition, online services are changing more and more rapidly as new tools are developed and new ways of interacting with users evolve. In light of this fast-changing world, it is important to assess the usage by HEP researchers of HEP-specific information resources. Such a study serves two purposes: within the field, it informs on the need for such community-based resources and their real role in the present internet landscape, inspiring their future evolution; globally, it provides an in-depth case study of the impact of discipline-based information resources, as opposed to institution-based information resources or cross-cutting (commercial) information platforms. This information is particularly relevant in light of recent worldwide moves towards self-archiving of research results at the institutional or disciplinary level, and the need to effectively incorporate these resources in the research workflow. A survey of HEP scholars was designed and deployed in order to provide a unique insight into their information needs and the way their research workflow includes information discovery and retrieval. Its results are presented in this Article. The Article first describes the current landscape of HEP information resources (Section 2), then presents the survey methodology and the demographics of the respondents (Section 3). Two sets of results are presented and discussed: the information resources preferred by HEP researchers (Section 4) and their appreciation of the relative importance of possible features of information resources (Section 5). The survey also provides additional information on user requirements for the future of information resources (Section 6). After the conclusions of the study (Section 7), an Appendix presents some of the most inspiring free-text answers charting the future of information provision in this field.","role":"user"},{"content":"Abstract: Access to previous results is of paramount importance in the scientific process. Recent progress in information management focuses on building e-infrastructures for the optimization of the research workflow, through both policy-driven and user-pulled dynamics. For decades, High-Energy Physics (HEP) has pioneered innovative solutions in the field of information management and dissemination. In light of a transforming information environment, it is important to assess the current usage of information resources by researchers and HEP provides a unique test-bed for this assessment. A survey of about 10% of practitioners in the field reveals usage trends and information needs. Community-based services, such as the pioneering arXiv and SPIRES systems, largely answer the need of the scientists, with a limited but increasing fraction of younger users relying on Google. Commercial services offered by publishers or database vendors are essentially unused in the field. The survey offers an insight into the most important features that users require to optimize their research workflow. These results inform the future evolution of information management in HEP and, as these researchers are traditionally ``early adopters'' of innovation in scholarly communication, can inspire developments of disciplinary repositories serving other communities.","role":"assistant"}]

Superallowed 0+ → 0+ nuclear β decays: A new survey with precision tests of the conserved vector current hypothesis and the standard model

I Introduction Precise measurements of the beta decay between nuclear analog states of spin, J π = 0 + superscript 𝐽 𝜋 superscript 0 J^{\pi}=0^{+} , and isospin, T = 1 𝑇 1 T=1 , provide demanding and fundamental tests of the properties of the electroweak interaction. Collectively, these transitions can sensitively probe the conservation of the vector weak current, set tight limits on the presence of scalar or right-hand currents and, by providing the most precise value for V u ​ d subscript 𝑉 𝑢 𝑑 V_{ud} , the up-down quark-mixing element of the Cabibbo-Kobayashi-Maskawa (CKM) matrix, they can contribute to the most demanding available test of the unitarity of that matrix, a property which is fundamental to the electroweak standard model. We have published five previous surveys of 0 + superscript 0 0^{+} → 0 + → absent superscript 0 \rightarrow 0^{+} superallowed transitions TH73 ; HT75 ; Ko84 ; HT90 ; HT05 , the first having appeared 35 years ago and the most recent, four years ago. In each, we published a complete survey of all relevant nuclear data that pertained to these superallowed transitions and used the results to set limits on the weak-interaction parameters that were important at the time. A particularly noteable outcome of our analysis four years ago HT05 was that the sum of squares of the top-row elements of the CKM matrix – the test of CKM unitarity – remained ambiguous, with the possibility of a significant shortfall in the unitarity sum. Since our last survey closed in November 2004, there has been a great deal of activity in this field prompted at least in part by the tantalizing possibility that new physics could be revealed by a failure in CKM unitarity. New measurements relating to 0 + superscript 0 0^{+} → 0 + → absent superscript 0 \rightarrow 0^{+} superallowed transitions have appeared in 27 publications, an addition of 20% to the papers accumulated up to 2004. Many of these measurements were of unprecedented precision so they did not merely add more of the same: they palpably improved the results, in some cases by tightening their error bars and, in others, by changing their central values. Penning-trap measurements of decay energies, which only became possible after 2004, have been especially effective in this regard. In addition to new measurements, there have also been important improvements to the small theoretical corrections that must be applied to the data in order to extract V u ​ d subscript 𝑉 𝑢 𝑑 V_{ud} and the values of other weak-interaction parameters. In the past four years, the radiative Ma06 and isospin symmetry-breaking corrections TH08 have both been subjected to major re-evaluations, which have undoubtedly improved their values and, in the former case, has reduced the uncertainty by a factor of two. In parallel with these developments, there has also been considerable activity in the determination of V u ​ s subscript 𝑉 𝑢 𝑠 V_{us} , the other matrix element that plays a role in the top-row unitarity test of the CKM matrix. (The third element in the top row, V u ​ b subscript 𝑉 𝑢 𝑏 V_{ub} , is very small and contributes a negligible 0.001% to the unitarity sum.) As with the work related to V u ​ d subscript 𝑉 𝑢 𝑑 V_{ud} , this activity has encompassed new experiments – precise measurements of kaon branching ratios – as well as improved theoretical corrections. However, in contrast with V u ​ d subscript 𝑉 𝑢 𝑑 V_{ud} , not only the uncertainty of V u ​ s subscript 𝑉 𝑢 𝑠 V_{us} but also its central value have been considerably changed by this recent work (see Fl08 for an up-to-date overview of V u ​ s subscript 𝑉 𝑢 𝑠 V_{us} ). Overall, the recent improvements have been numerous enough and their impact on the unitarity test significant enough that this is an opportune time to produce a new and updated survey of the nuclear data used to establish V u ​ d subscript 𝑉 𝑢 𝑑 V_{ud} . We incorporate data on a total of 20 superallowed transitions and have continued the practice we began in 1984 Ko84 of updating all original data to take account of the most modern calibration standards. In addition to including the improved correction terms already referred to, we have also upgraded our calculation of the statistical rate function f 𝑓 f to include provision for excitation of the daughter atom, and we have included a more extensive treatment of possible systematic uncertainties associated with the isospin symmetry-breaking corrections. Superallowed 0 + superscript 0 0^{+} → 0 + → absent superscript 0 \rightarrow 0^{+} β 𝛽 \beta decay between T = 1 𝑇 1 T=1 analog states depends uniquely on the vector part of the weak interaction and, according to the conserved vector current (CVC) hypothesis, its experimental f ​ t 𝑓 𝑡 ft value should be related to the vector coupling constant, a fundamental constant which is the same for all such transitions. In practice, the expression for f ​ t 𝑓 𝑡 ft includes several small ( ∼ similar-to \sim 1%) correction terms. It is convenient to combine some of these terms with the f ​ t 𝑓 𝑡 ft value and define a “corrected” ℱ ​ t ℱ 𝑡 {\cal F}t value. Thus, we write HT05 ℱ ​ t ≡ f ​ t ​ ( 1 + δ R ′ ) ​ ( 1 + δ N ​ S − δ C ) = K 2 ​ G V 2 ​ ( 1 + Δ R V ) , ℱ 𝑡 𝑓 𝑡 1 superscript subscript 𝛿 𝑅 ′ 1 subscript 𝛿 𝑁 𝑆 subscript 𝛿 𝐶 𝐾 2 superscript subscript 𝐺 V 2 1 superscript subscript Δ R V {\cal F}t\equiv ft(1+\delta_{R}^{\prime})(1+\delta_{NS}-\delta_{C})=\frac{K}{2G_{\mbox{\tiny V}}^{2}(1+\Delta_{\mbox{\tiny R}}^{\mbox{\tiny V}})}~{}, (1) where K / ( ℏ ​ c ) 6 = 2 ​ π 3 ​ ℏ ​ ln ⁡ 2 / ( m e ​ c 2 ) 5 = 8120.2787 ​ ( 11 ) × 10 − 10 𝐾 superscript Planck-constant-over-2-pi 𝑐 6 2 superscript 𝜋 3 Planck-constant-over-2-pi 2 superscript subscript 𝑚 𝑒 superscript 𝑐 2 5 8120.2787 11 superscript 10 10 K/(\hbar c)^{6}=2\pi^{3}\hbar\ln 2/(m_{e}c^{2})^{5}=8120.2787(11)\times 10^{-10} GeV -4 s, G V subscript 𝐺 V G_{\mbox{\tiny V}} is the vector coupling constant for semi-leptonic weak interactions, δ C subscript 𝛿 𝐶 \delta_{C} is the isospin-symmetry-breaking correction and Δ R V superscript subscript Δ R V \Delta_{\mbox{\tiny R}}^{\mbox{\tiny V}} is the transition-independent part of the radiative correction. The terms δ R ′ superscript subscript 𝛿 𝑅 ′ \delta_{R}^{\prime} and δ N ​ S subscript 𝛿 𝑁 𝑆 \delta_{NS} comprise the transition-dependent part of the radiative correction, the former being a function only of the electron’s energy and the Z 𝑍 Z of the daughter nucleus, while the latter, like δ C subscript 𝛿 𝐶 \delta_{C} , depends in its evaluation on the details of nuclear structure. From this equation, it can be seen that each measured transition establishes an individual value for G V subscript 𝐺 V G_{\mbox{\tiny V}} and, if the CVC assertion is correct that G V subscript 𝐺 V G_{\mbox{\tiny V}} is not renormalized in the nuclear medium, all such values – and all the ℱ ​ t ℱ 𝑡 {\cal F}t values themselves – should be identical within uncertainties, regardless of the specific nuclei involved. Our procedure in this paper is to examine all experimental data related to 20 superallowed transitions, comprising all those that have been well studied, together with others that are now coming under scrutiny after becoming accessible to precision measurement. The methods used in data evaluation are presented in Sec. II . The calculations and corrections required to extract ℱ ​ t ℱ 𝑡 {\cal F}t values from these data are described and applied in Sec. III ; in the same section, we use the resulting ℱ ​ t ℱ 𝑡 {\cal F}t values to test CVC. Finally, in Sec. IV we explore the impact of these results on a number of weak-interaction issues: CKM unitarity as well as the possible existence of scalar interactions, right-hand currents and extra Z 𝑍 Z bosons. This is much the same pattern as we followed in our last review HT05 so we will not describe the formalism again in detail, referring the reader instead to that earlier work.

A new critical survey is presented of all half-life, decay-energy, and branching-ratio measurements related to 20 superallowed ${0}^{+}\ensuremath{\rightarrow}{0}^{+}\ensuremath{\beta}$ decays. Compared with our last review, there are numerous improvements: First, we have added 27 recently published measurements and eliminated 9 references, either because they have been superseded by much more precise modern results or because there are now reasons to consider them fatally flawed; of particular importance, the new data include a number of high-precision Penning-trap measurements of decay energies. Second, we have used the recently improved isospin symmetry-breaking corrections, which were motivated by these new Penning-trap results. Third, our calculation of the statistical rate function $f$ now accounts for possible excitation in the daughter atom, a small effect but one that merits inclusion at the present level of experimental precision. Finally, we have re-examined the systematic uncertainty associated with the isospin symmetry-breaking corrections by evaluating the radial-overlap correction using Hartree-Fock radial wave functions and comparing the results with our earlier calculations, which used Saxon-Woods wave functions; the provision for systematic uncertainty has been changed as a consequence. The new ``corrected'' $\mathcal{F}t$ values are impressively constant and their average, when combined with the muon lifetime, yields the up-down quark-mixing element of the Cabibbo-Kobayashi-Maskawa (CKM) matrix, ${V}_{\mathit{ud}}=0.97425\ifmmode\pm\else\textpm\fi{}0.00022$. The unitarity test on the top row of the matrix becomes $|{V}_{\mathit{ud}}|{}^{2}+|{V}_{\mathit{us}}|{}^{2}+|{V}_{\mathit{ub}}|{}^{2}=0.99995\ifmmode\pm\else\textpm\fi{}0.00061$. Both ${V}_{\mathit{ud}}$ and the unitarity sum have significantly reduced uncertainties compared with our previous survey, although the new value of ${V}_{\mathit{ud}}$ is statistically consistent with the old one. From these data we also set limits on the possible existence of scalar interactions, right-hand currents, and extra $Z$ bosons. Finally, we discuss the priorities for future theoretical and experimental work with the goal of making the CKM unitarity test even more definitive.

[{"content":"Help me to generate the abstract of a survey paper given the title:*Superallowed 0+ → 0+ nuclear β decays: A new survey with precision tests of the conserved vector current hypothesis and the standard model* and the introduction:I Introduction Precise measurements of the beta decay between nuclear analog states of spin, J π = 0 + superscript 𝐽 𝜋 superscript 0 J^{\pi}=0^{+} , and isospin, T = 1 𝑇 1 T=1 , provide demanding and fundamental tests of the properties of the electroweak interaction. Collectively, these transitions can sensitively probe the conservation of the vector weak current, set tight limits on the presence of scalar or right-hand currents and, by providing the most precise value for V u ​ d subscript 𝑉 𝑢 𝑑 V_{ud} , the up-down quark-mixing element of the Cabibbo-Kobayashi-Maskawa (CKM) matrix, they can contribute to the most demanding available test of the unitarity of that matrix, a property which is fundamental to the electroweak standard model. We have published five previous surveys of 0 + superscript 0 0^{+} → 0 + → absent superscript 0 \rightarrow 0^{+} superallowed transitions TH73 ; HT75 ; Ko84 ; HT90 ; HT05 , the first having appeared 35 years ago and the most recent, four years ago. In each, we published a complete survey of all relevant nuclear data that pertained to these superallowed transitions and used the results to set limits on the weak-interaction parameters that were important at the time. A particularly noteable outcome of our analysis four years ago HT05 was that the sum of squares of the top-row elements of the CKM matrix – the test of CKM unitarity – remained ambiguous, with the possibility of a significant shortfall in the unitarity sum. Since our last survey closed in November 2004, there has been a great deal of activity in this field prompted at least in part by the tantalizing possibility that new physics could be revealed by a failure in CKM unitarity. New measurements relating to 0 + superscript 0 0^{+} → 0 + → absent superscript 0 \rightarrow 0^{+} superallowed transitions have appeared in 27 publications, an addition of 20% to the papers accumulated up to 2004. Many of these measurements were of unprecedented precision so they did not merely add more of the same: they palpably improved the results, in some cases by tightening their error bars and, in others, by changing their central values. Penning-trap measurements of decay energies, which only became possible after 2004, have been especially effective in this regard. In addition to new measurements, there have also been important improvements to the small theoretical corrections that must be applied to the data in order to extract V u ​ d subscript 𝑉 𝑢 𝑑 V_{ud} and the values of other weak-interaction parameters. In the past four years, the radiative Ma06 and isospin symmetry-breaking corrections TH08 have both been subjected to major re-evaluations, which have undoubtedly improved their values and, in the former case, has reduced the uncertainty by a factor of two. In parallel with these developments, there has also been considerable activity in the determination of V u ​ s subscript 𝑉 𝑢 𝑠 V_{us} , the other matrix element that plays a role in the top-row unitarity test of the CKM matrix. (The third element in the top row, V u ​ b subscript 𝑉 𝑢 𝑏 V_{ub} , is very small and contributes a negligible 0.001% to the unitarity sum.) As with the work related to V u ​ d subscript 𝑉 𝑢 𝑑 V_{ud} , this activity has encompassed new experiments – precise measurements of kaon branching ratios – as well as improved theoretical corrections. However, in contrast with V u ​ d subscript 𝑉 𝑢 𝑑 V_{ud} , not only the uncertainty of V u ​ s subscript 𝑉 𝑢 𝑠 V_{us} but also its central value have been considerably changed by this recent work (see Fl08 for an up-to-date overview of V u ​ s subscript 𝑉 𝑢 𝑠 V_{us} ). Overall, the recent improvements have been numerous enough and their impact on the unitarity test significant enough that this is an opportune time to produce a new and updated survey of the nuclear data used to establish V u ​ d subscript 𝑉 𝑢 𝑑 V_{ud} . We incorporate data on a total of 20 superallowed transitions and have continued the practice we began in 1984 Ko84 of updating all original data to take account of the most modern calibration standards. In addition to including the improved correction terms already referred to, we have also upgraded our calculation of the statistical rate function f 𝑓 f to include provision for excitation of the daughter atom, and we have included a more extensive treatment of possible systematic uncertainties associated with the isospin symmetry-breaking corrections. Superallowed 0 + superscript 0 0^{+} → 0 + → absent superscript 0 \rightarrow 0^{+} β 𝛽 \beta decay between T = 1 𝑇 1 T=1 analog states depends uniquely on the vector part of the weak interaction and, according to the conserved vector current (CVC) hypothesis, its experimental f ​ t 𝑓 𝑡 ft value should be related to the vector coupling constant, a fundamental constant which is the same for all such transitions. In practice, the expression for f ​ t 𝑓 𝑡 ft includes several small ( ∼ similar-to \sim 1%) correction terms. It is convenient to combine some of these terms with the f ​ t 𝑓 𝑡 ft value and define a corrected ℱ ​ t ℱ 𝑡 {\cal F}t value. Thus, we write HT05 ℱ ​ t ≡ f ​ t ​ ( 1 + δ R ′ ) ​ ( 1 + δ N ​ S − δ C ) = K 2 ​ G V 2 ​ ( 1 + Δ R V ) , ℱ 𝑡 𝑓 𝑡 1 superscript subscript 𝛿 𝑅 ′ 1 subscript 𝛿 𝑁 𝑆 subscript 𝛿 𝐶 𝐾 2 superscript subscript 𝐺 V 2 1 superscript subscript Δ R V {\cal F}t\equiv ft(1+\delta_{R}^{\prime})(1+\delta_{NS}-\delta_{C})=\frac{K}{2G_{\mbox{\tiny V}}^{2}(1+\Delta_{\mbox{\tiny R}}^{\mbox{\tiny V}})}~{}, (1) where K / ( ℏ ​ c ) 6 = 2 ​ π 3 ​ ℏ ​ ln ⁡ 2 / ( m e ​ c 2 ) 5 = 8120.2787 ​ ( 11 ) × 10 − 10 𝐾 superscript Planck-constant-over-2-pi 𝑐 6 2 superscript 𝜋 3 Planck-constant-over-2-pi 2 superscript subscript 𝑚 𝑒 superscript 𝑐 2 5 8120.2787 11 superscript 10 10 K/(\hbar c)^{6}=2\pi^{3}\hbar\ln 2/(m_{e}c^{2})^{5}=8120.2787(11)\times 10^{-10} GeV -4 s, G V subscript 𝐺 V G_{\mbox{\tiny V}} is the vector coupling constant for semi-leptonic weak interactions, δ C subscript 𝛿 𝐶 \delta_{C} is the isospin-symmetry-breaking correction and Δ R V superscript subscript Δ R V \Delta_{\mbox{\tiny R}}^{\mbox{\tiny V}} is the transition-independent part of the radiative correction. The terms δ R ′ superscript subscript 𝛿 𝑅 ′ \delta_{R}^{\prime} and δ N ​ S subscript 𝛿 𝑁 𝑆 \delta_{NS} comprise the transition-dependent part of the radiative correction, the former being a function only of the electrons energy and the Z 𝑍 Z of the daughter nucleus, while the latter, like δ C subscript 𝛿 𝐶 \delta_{C} , depends in its evaluation on the details of nuclear structure. From this equation, it can be seen that each measured transition establishes an individual value for G V subscript 𝐺 V G_{\mbox{\tiny V}} and, if the CVC assertion is correct that G V subscript 𝐺 V G_{\mbox{\tiny V}} is not renormalized in the nuclear medium, all such values – and all the ℱ ​ t ℱ 𝑡 {\cal F}t values themselves – should be identical within uncertainties, regardless of the specific nuclei involved. Our procedure in this paper is to examine all experimental data related to 20 superallowed transitions, comprising all those that have been well studied, together with others that are now coming under scrutiny after becoming accessible to precision measurement. The methods used in data evaluation are presented in Sec. II . The calculations and corrections required to extract ℱ ​ t ℱ 𝑡 {\cal F}t values from these data are described and applied in Sec. III ; in the same section, we use the resulting ℱ ​ t ℱ 𝑡 {\cal F}t values to test CVC. Finally, in Sec. IV we explore the impact of these results on a number of weak-interaction issues: CKM unitarity as well as the possible existence of scalar interactions, right-hand currents and extra Z 𝑍 Z bosons. This is much the same pattern as we followed in our last review HT05 so we will not describe the formalism again in detail, referring the reader instead to that earlier work.","role":"user"},{"content":"Abstract: A new critical survey is presented of all half-life, decay-energy, and branching-ratio measurements related to 20 superallowed ${0}^{+}\ensuremath{\rightarrow}{0}^{+}\ensuremath{\beta}$ decays. Compared with our last review, there are numerous improvements: First, we have added 27 recently published measurements and eliminated 9 references, either because they have been superseded by much more precise modern results or because there are now reasons to consider them fatally flawed; of particular importance, the new data include a number of high-precision Penning-trap measurements of decay energies. Second, we have used the recently improved isospin symmetry-breaking corrections, which were motivated by these new Penning-trap results. Third, our calculation of the statistical rate function $f$ now accounts for possible excitation in the daughter atom, a small effect but one that merits inclusion at the present level of experimental precision. Finally, we have re-examined the systematic uncertainty associated with the isospin symmetry-breaking corrections by evaluating the radial-overlap correction using Hartree-Fock radial wave functions and comparing the results with our earlier calculations, which used Saxon-Woods wave functions; the provision for systematic uncertainty has been changed as a consequence. The new ``corrected'' $\mathcal{F}t$ values are impressively constant and their average, when combined with the muon lifetime, yields the up-down quark-mixing element of the Cabibbo-Kobayashi-Maskawa (CKM) matrix, ${V}_{\mathit{ud}}=0.97425\ifmmode\pm\else\textpm\fi{}0.00022$. The unitarity test on the top row of the matrix becomes $|{V}_{\mathit{ud}}|{}^{2}+|{V}_{\mathit{us}}|{}^{2}+|{V}_{\mathit{ub}}|{}^{2}=0.99995\ifmmode\pm\else\textpm\fi{}0.00061$. Both ${V}_{\mathit{ud}}$ and the unitarity sum have significantly reduced uncertainties compared with our previous survey, although the new value of ${V}_{\mathit{ud}}$ is statistically consistent with the old one. From these data we also set limits on the possible existence of scalar interactions, right-hand currents, and extra $Z$ bosons. Finally, we discuss the priorities for future theoretical and experimental work with the goal of making the CKM unitarity test even more definitive.","role":"assistant"}]

Some results on non-self-adjoint operators, a survey

1.1 Some background For self-adjoint and more generally normal operators on some complex Hilbert space ℋ ℋ {\cal H} we have a nice theory, including the spectral theorem and a nice estimate on the norm of the resolvent: ‖ ( z − P ) − 1 ‖ ≤ ( dist ​ ( z , σ ​ ( P ) ) ) − 1 , σ ​ ( P ) = the spectrum of ​ P . formulae-sequence norm superscript 𝑧 𝑃 1 superscript dist 𝑧 𝜎 𝑃 1 𝜎 𝑃 the spectrum of 𝑃 \|(z-P)^{-1}\|\leq({\rm dist\,}(z,\sigma(P)))^{-1},\quad\sigma(P)=\hbox{\, the spectrum of }P. (1.1) This has a consequence for the corresponding evolution problem: If σ ​ ( P ) ⊂ { z ∈ 𝐂 ; ℜ ⁡ z ≥ λ 0 } 𝜎 𝑃 formulae-sequence 𝑧 𝐂 𝑧 subscript 𝜆 0 \sigma(P)\subset\{z\in{\bf C};\Re z\geq\lambda_{0}\} , then ‖ e − t ​ P ‖ ≤ e − λ 0 ​ t , t ≥ 0 . formulae-sequence norm superscript 𝑒 𝑡 𝑃 superscript 𝑒 subscript 𝜆 0 𝑡 𝑡 0 \|e^{-tP}\|\leq e^{-\lambda_{0}t},\ t\geq 0. (1.2) However, non-normal operators appear frequently: Scattering poles, Convection-diffusion problems, Kramers-Fokker-Planck equation, damped wave equations, linearized operators in fluid dynamics. Then typically, ‖ ( z − P ) − 1 ‖ norm superscript 𝑧 𝑃 1 \|(z-P)^{-1}\| may be very large even when z 𝑧 z is far from the spectrum. This implies mathematical difficulties: – When studying the distribution of eigenvalues, – When studying functions of the operator, like e − t ​ P superscript 𝑒 𝑡 𝑃 e^{-tP} and its norm. It also implies numerical difficulties like: – Eigenvalue instability. There are (in the author’s opinion) two ways out: – Change the Hilbert space norm to make the operators look more normal. (Complex scaling methods.) – Recognize that the region of the z 𝑧 z -plane where ‖ ( z − P ) − 1 ‖ norm superscript 𝑧 𝑃 1 \|(z-P)^{-1}\| is large, has its own interest. (Pseudospectrum.) The option to choose depends on the problem. • In some problems, like those related to scattering poles, there is no obvious choice of Hilbert space and we are free to make the most natural one. This option is particularly natural when considering a differential operator with analytic coefficients. • In other problems the canonical Hilbert space is L 2 superscript 𝐿 2 L^{2} and we are at most allowed to change the norm into an equivalent one. Here the notion of pseudospectrum is likely to be important. Let P : ℋ → ℋ : 𝑃 → ℋ ℋ P:{\cal H}\to{\cal H} be closed, densely defined, ℋ ℋ {\cal H} a complex Hilbert space and let ρ ​ ( P ) = 𝐂 ∖ σ ​ ( P ) 𝜌 𝑃 𝐂 𝜎 𝑃 \rho(P)={\bf C}\setminus\sigma(P) denote the resolvent set. The notion of pseudospectrum is important in numerical analysis and we refer to L.N. Trefethen [ 52 ] , Trefethen–M. Embree [ 54 ] and further references given there. Thanks to works of E.B. Davies [ 5 ] , [ 7 ] , M. Zworski [ 57 ] and others it has become popular in the non-self-adjoint spectral theory of differential operators. Definition 1.1 Let ϵ > 0 italic-ϵ 0 \epsilon>0 . The ϵ italic-ϵ \epsilon -pseudospectrum of P 𝑃 P is σ ϵ ​ ( P ) := σ ​ ( P ) ∪ { z ∈ ρ ​ ( P ) ; ‖ ( z − P ) − 1 ‖ > 1 / ϵ } . assign subscript 𝜎 italic-ϵ 𝑃 𝜎 𝑃 formulae-sequence 𝑧 𝜌 𝑃 norm superscript 𝑧 𝑃 1 1 italic-ϵ \sigma_{\epsilon}(P):=\sigma(P)\cup\{z\in\rho(P);\,\|(z-P)^{-1}\|>1/\epsilon\}. Unlike the spectrum, the pseudospectrum will in general change when we change the norm on ℋ ℋ {\cal H} . Moreover, it can be characterized as a set of spectral instability as follows from the following version of a theorem of Roch-Silberman [ 43 ] : Theorem 1.2 σ ϵ ​ ( P ) = ⋃ Q ∈ ℒ ​ ( ℋ , ℋ ) ‖ Q ‖ < ϵ σ ​ ( P + Q ) . subscript 𝜎 italic-ϵ 𝑃 subscript FRACOP 𝑄 ℒ ℋ ℋ norm 𝑄 italic-ϵ 𝜎 𝑃 𝑄 \sigma_{\epsilon}(P)=\bigcup_{Q\in{\cal L}({\cal H},{\cal H})\atop\|Q\|<\epsilon}\sigma(P+Q). In his survey [ 52 ] L.N. Trefethen discusses some linearized operators from fluid dynamics: • Orr-Sommerfeld equation ( Orzag, Reddy, Schmid, Hennigson ). • Plane Poiseuille flow ( L.N and A.N Trefethen, Schmid ). • Pipe Poiseuille flow ( L.N and A.N Trefethen, Reddy, Driscoll ), and to what extent stability can be predicted from the sudy of the spectrum of these non-self-adjoint operators: Eigenvalue analysis alone leads in some cases to the prediction of stability for Reynolds numbers R < 5772 𝑅 5772 R<5772 . Experimentally however, we have stability only for R < 1000 𝑅 1000 R<1000 . The rough explanation of this is that the ϵ italic-ϵ \epsilon -pseudospectrum (for a suitable ϵ italic-ϵ \epsilon ) crosses the imaginary axis before the spectrum does, when R 𝑅 R increases. Then ‖ e − t ​ P ‖ norm superscript 𝑒 𝑡 𝑃 \|e^{-tP}\| will grow fast for a limited time even though the growth for very large times is determined by the spectrum. However, since P 𝑃 P appears as a linearization of a non-linear problem, that suffices to cause instability. In the case of differential operators the pseudospectral phenomenon is very general and related to classical works in PDE on local solvability and non-hypoellipticity. E.B. Davies [ 5 ] studied the non-self-adjoint semiclasscial Schrödinger operator with a smooth (complex-valued) potential in dimension 1 and showed under “generic” assumptions that one can construct quasimodes with the spectral parameter varying in an open complex set, containing points that are possibly very far from the spectrum (as can be verified in the case of the complex harmonic oscillator). M. Zworski [ 57 ] observed that this is essentially a rediscovery of an old result of Hörmander [ 32 , 33 ] , and was able to generalize considerably Davies’ result by adapting the one of Hörmander to the semi-classical case. With N. Dencker and M.Zworski [ 9 ] we also gave a direct proof and a corresponding adaptation of old results of Sato-Kawai-Kashiwara [ 44 ] to the analytic case: Theorem 1.3 ( [ 57 ] , [ 9 ] ) Let P ​ ( x , h ​ D x ) = ∑ | α | ≤ m a α ​ ( x ) ​ ( h ​ D x ) α , D x = ∂ ∂ x formulae-sequence 𝑃 𝑥 ℎ subscript 𝐷 𝑥 subscript 𝛼 𝑚 subscript 𝑎 𝛼 𝑥 superscript ℎ subscript 𝐷 𝑥 𝛼 subscript 𝐷 𝑥 𝑥 P(x,hD_{x})=\sum_{|\alpha|\leq m}a_{\alpha}(x)(hD_{x})^{\alpha},\ D_{x}={\partial\over\partial x} (1.3) have smooth coefficients in the open set Ω ⊂ 𝐑 n Ω superscript 𝐑 𝑛 \Omega\subset{\bf R}^{n} . Put p ​ ( x , ξ ) = ∑ | α | ≤ m a α ​ ( x ) ​ ξ α 𝑝 𝑥 𝜉 subscript 𝛼 𝑚 subscript 𝑎 𝛼 𝑥 superscript 𝜉 𝛼 p(x,\xi)=\sum_{|\alpha|\leq m}a_{\alpha}(x)\xi^{\alpha} . Assume z = p ​ ( x 0 , ξ 0 ) 𝑧 𝑝 subscript 𝑥 0 subscript 𝜉 0 z=p(x_{0},\xi_{0}) with the Poisson bracket 1 i ​ { p , p ¯ } ​ ( x 0 , ξ 0 ) > 0 1 𝑖 𝑝 ¯ 𝑝 subscript 𝑥 0 subscript 𝜉 0 0 {1\over i}\{p,\overline{p}\}(x_{0},\xi_{0})>0 . Then ∃ \exists u = u h ∈ C 0 ∞ ​ ( Ω ) 𝑢 subscript 𝑢 ℎ superscript subscript 𝐶 0 Ω u=u_{h}\in C_{0}^{\infty}(\Omega) , with ‖ u ‖ = 1 norm 𝑢 1 \|u\|=1 , ‖ ( P − z ) ​ u ‖ = 𝒪 ​ ( h ∞ ) norm 𝑃 𝑧 𝑢 𝒪 superscript ℎ \|(P-z)u\|={\cal O}(h^{\infty}) , when h → 0 → ℎ 0 h\to 0 . Analytic case: Can replace ” h ∞ superscript ℎ h^{\infty} ” by ” e − 1 / C ​ h superscript 𝑒 1 𝐶 ℎ e^{-1/Ch} ”. Here, we have used standard multi-index notation: ξ α = ξ 1 α 1 ⋅ … ⋅ ξ n α n superscript 𝜉 𝛼 ⋅ superscript subscript 𝜉 1 subscript 𝛼 1 … superscript subscript 𝜉 𝑛 subscript 𝛼 𝑛 \xi^{\alpha}=\xi_{1}^{\alpha_{1}}\cdot...\cdot\xi_{n}^{\alpha_{n}} , | α | = α 1 + … + α n 𝛼 subscript 𝛼 1 … subscript 𝛼 𝑛 |\alpha|=\alpha_{1}+...+\alpha_{n} , and the norms ∥ ⋅ ∥ \|\cdot\| are the ones of L 2 superscript 𝐿 2 L^{2} or ℓ 2 superscript ℓ 2 \ell^{2} if nothing else is indicated. This result was subsequently generalized by K. Pravda-Starov, [ 42 ] . Notice that this implies that when the theorem applies and if the resolvent ( P − z ) − 1 superscript 𝑃 𝑧 1 (P-z)^{-1} exists then its norm is greater than any negative power of h ℎ h when h → 0 → ℎ 0 h\to 0 . Example 1.4 Let P = − h 2 ​ Δ + V ​ ( x ) 𝑃 superscript ℎ 2 Δ 𝑉 𝑥 P=-h^{2}\Delta+V(x) , p ​ ( x , ξ ) = ξ 2 + V ​ ( x ) 𝑝 𝑥 𝜉 superscript 𝜉 2 𝑉 𝑥 p(x,\xi)=\xi^{2}+V(x) , ξ 2 = ξ 1 2 + … + ξ n 2 superscript 𝜉 2 superscript subscript 𝜉 1 2 … superscript subscript 𝜉 𝑛 2 \xi^{2}=\xi_{1}^{2}+...+\xi_{n}^{2} . Then 1 i ​ { p , p ¯ } = − 4 ​ ξ ⋅ ℑ ⁡ V ′ ​ ( x ) 1 𝑖 𝑝 ¯ 𝑝 ⋅ 4 𝜉 superscript 𝑉 ′ 𝑥 {1\over i}\{p,\overline{p}\}=-4\xi\cdot\Im V^{\prime}(x) . “Generically”, if z = p ​ ( x , ξ ) 𝑧 𝑝 𝑥 𝜉 z=p(x,\xi) , then { p , p ¯ } ​ ( x , ξ ) ≠ 0 𝑝 ¯ 𝑝 𝑥 𝜉 0 \{p,\overline{p}\}(x,\xi)\neq 0 and one can show quite generally that if this happens then there is also another point ( y , η ) 𝑦 𝜂 (y,\eta) with p ​ ( y , η ) = z 𝑝 𝑦 𝜂 𝑧 p(y,\eta)=z such that { p , p ¯ } ​ ( x , ξ ) 𝑝 ¯ 𝑝 𝑥 𝜉 \{p,\overline{p}\}(x,\xi) takes the opposite sign. This justifies the following simplified terminology in the semi-classical limit: The semi-classical pseudospectrum of P 𝑃 P is the range ℛ ​ ( p ) ℛ 𝑝 {\cal R}(p) of p 𝑝 p . In [ 9 ] we also showed under suitable assumptions (inspired from scattering theory and from the theory of sub-elliptic operators), that there is no spectrum near the boundary of the semi-classical pseudospectrum and that we may have quite a good control of the norm of the resolvent there. Generalizations to the case of systems were given by Dencker [ 8 ] . 1.2 The topics of this survey We will discuss three subjects involving non-self-adjoint differential and pseudodifferential operators, We will always wok in the semi-classical limit, which means that our operators are of the form P ​ ( x , h ​ D x ; h ) 𝑃 𝑥 ℎ subscript 𝐷 𝑥 ℎ P(x,hD_{x};h) , where P 𝑃 P is a suitable symbol and 0 < h ≪ 1 0 ℎ much-less-than 1 0<h\ll 1 . It is quite clear however that some of our results will also apply to non-semiclassical situations in the limit of large eigenvalues. The subjects are: • The Kramers-Fokker-Planck operator, • Bohr-Sommerfeld rules in dimension 2, • Weyl asymptotics for non-self-adjoint operators with small random perturbations. and most of the works discussed are the results of collaborations with A. Melin, M. Hitrik, F. Hérau, C.Stolk, S. Vũ Ngọc and M. Hager. In the first two topics we exploit the possibility of changing the Hilbert space norm by introducing exponential weights on phase space. In the case of the Kramers-Fokker-Planck operator, we make no analyticity assumptions and the phase-space weights are correspondingly quite weak. In this case however our operator is a differential one, so we are allowed to apply strong exponential weights depending only on the base variables, and this is important when studying small exponential corrections of the eigenvalues via the so called tunnel effect. For the Bohr-Sommerfeld rules, we make analyticity assumptions that allow stronger phase space weights. In both cases the effect of the exponential weights is to make the operator under consideration more normal. In the third topic, we do not use any deformations of the given Hilbert space, but exponential weights play an important role at another level, namely to count zeros of holomorphic functions with exponential growth. The pseudospectrum will not be discussed explicitly below. In the Kramers-Fokker-Planck case, the problems are located near the boundary of the semi-classical pseudospectrum, and it turns out that we have a very nice control of the resolvent there. In the 2 dimensional Bohr-Sommerfeld rules, we have stronger exponential weights, reflecting stronger pseudospectral phenomena. Finally in the subject of Weyl asymptotics, we often have strong pseudospectral behaviour for the unperturbed operator. From the proofs it appears that the random perturbations will weaken the pseudospectral behaviour and this might have very interesting consequences for the associated evolution problems. This is still very much an open problem.

This text is a survey of recent results obtained by the author and collaborators on different problems for non-self-adjoint operators. The topics are: Kramers-Fokker-Planck type operators, spectral asymptotics in two dimensions and Weyl asymptotics for the eigenvalues of non-self-adjoint operators with small random perturbations. In the introduction we also review the notion of pseudo-spectrum and its relation to non-self-adjoint spectral problems.

[{"content":"Help me to generate the abstract of a survey paper given the title:*Some results on non-self-adjoint operators, a survey* and the introduction:1.1 Some background For self-adjoint and more generally normal operators on some complex Hilbert space ℋ ℋ {\cal H} we have a nice theory, including the spectral theorem and a nice estimate on the norm of the resolvent: ‖ ( z − P ) − 1 ‖ ≤ ( dist ​ ( z , σ ​ ( P ) ) ) − 1 , σ ​ ( P ) = the spectrum of ​ P . formulae-sequence norm superscript 𝑧 𝑃 1 superscript dist 𝑧 𝜎 𝑃 1 𝜎 𝑃 the spectrum of 𝑃 \|(z-P)^{-1}\|\leq({\rm dist\,}(z,\sigma(P)))^{-1},\quad\sigma(P)=\hbox{\, the spectrum of }P. (1.1) This has a consequence for the corresponding evolution problem: If σ ​ ( P ) ⊂ { z ∈ 𝐂 ; ℜ ⁡ z ≥ λ 0 } 𝜎 𝑃 formulae-sequence 𝑧 𝐂 𝑧 subscript 𝜆 0 \sigma(P)\subset\{z\in{\bf C};\Re z\geq\lambda_{0}\} , then ‖ e − t ​ P ‖ ≤ e − λ 0 ​ t , t ≥ 0 . formulae-sequence norm superscript 𝑒 𝑡 𝑃 superscript 𝑒 subscript 𝜆 0 𝑡 𝑡 0 \|e^{-tP}\|\leq e^{-\lambda_{0}t},\ t\geq 0. (1.2) However, non-normal operators appear frequently: Scattering poles, Convection-diffusion problems, Kramers-Fokker-Planck equation, damped wave equations, linearized operators in fluid dynamics. Then typically, ‖ ( z − P ) − 1 ‖ norm superscript 𝑧 𝑃 1 \|(z-P)^{-1}\| may be very large even when z 𝑧 z is far from the spectrum. This implies mathematical difficulties: – When studying the distribution of eigenvalues, – When studying functions of the operator, like e − t ​ P superscript 𝑒 𝑡 𝑃 e^{-tP} and its norm. It also implies numerical difficulties like: – Eigenvalue instability. There are (in the authors opinion) two ways out: – Change the Hilbert space norm to make the operators look more normal. (Complex scaling methods.) – Recognize that the region of the z 𝑧 z -plane where ‖ ( z − P ) − 1 ‖ norm superscript 𝑧 𝑃 1 \|(z-P)^{-1}\| is large, has its own interest. (Pseudospectrum.) The option to choose depends on the problem. • In some problems, like those related to scattering poles, there is no obvious choice of Hilbert space and we are free to make the most natural one. This option is particularly natural when considering a differential operator with analytic coefficients. • In other problems the canonical Hilbert space is L 2 superscript 𝐿 2 L^{2} and we are at most allowed to change the norm into an equivalent one. Here the notion of pseudospectrum is likely to be important. Let P : ℋ → ℋ : 𝑃 → ℋ ℋ P:{\cal H}\to{\cal H} be closed, densely defined, ℋ ℋ {\cal H} a complex Hilbert space and let ρ ​ ( P ) = 𝐂 ∖ σ ​ ( P ) 𝜌 𝑃 𝐂 𝜎 𝑃 \rho(P)={\bf C}\setminus\sigma(P) denote the resolvent set. The notion of pseudospectrum is important in numerical analysis and we refer to L.N. Trefethen 52 , Trefethen–M. Embree 54 and further references given there. Thanks to works of E.B. Davies 5 , 7 , M. Zworski 57 and others it has become popular in the non-self-adjoint spectral theory of differential operators. Definition 1.1 Let ϵ > 0 italic-ϵ 0 \epsilon>0 . The ϵ italic-ϵ \epsilon -pseudospectrum of P 𝑃 P is σ ϵ ​ ( P ) := σ ​ ( P ) ∪ { z ∈ ρ ​ ( P ) ; ‖ ( z − P ) − 1 ‖ > 1 / ϵ } . assign subscript 𝜎 italic-ϵ 𝑃 𝜎 𝑃 formulae-sequence 𝑧 𝜌 𝑃 norm superscript 𝑧 𝑃 1 1 italic-ϵ \sigma_{\epsilon}(P):=\sigma(P)\cup\{z\in\rho(P);\,\|(z-P)^{-1}\|>1/\epsilon\}. Unlike the spectrum, the pseudospectrum will in general change when we change the norm on ℋ ℋ {\cal H} . Moreover, it can be characterized as a set of spectral instability as follows from the following version of a theorem of Roch-Silberman 43 : Theorem 1.2 σ ϵ ​ ( P ) = ⋃ Q ∈ ℒ ​ ( ℋ , ℋ ) ‖ Q ‖ < ϵ σ ​ ( P + Q ) . subscript 𝜎 italic-ϵ 𝑃 subscript FRACOP 𝑄 ℒ ℋ ℋ norm 𝑄 italic-ϵ 𝜎 𝑃 𝑄 \sigma_{\epsilon}(P)=\bigcup_{Q\in{\cal L}({\cal H},{\cal H})\atop\|Q\|<\epsilon}\sigma(P+Q). In his survey 52 L.N. Trefethen discusses some linearized operators from fluid dynamics: • Orr-Sommerfeld equation ( Orzag, Reddy, Schmid, Hennigson ). • Plane Poiseuille flow ( L.N and A.N Trefethen, Schmid ). • Pipe Poiseuille flow ( L.N and A.N Trefethen, Reddy, Driscoll ), and to what extent stability can be predicted from the sudy of the spectrum of these non-self-adjoint operators: Eigenvalue analysis alone leads in some cases to the prediction of stability for Reynolds numbers R < 5772 𝑅 5772 R<5772 . Experimentally however, we have stability only for R < 1000 𝑅 1000 R<1000 . The rough explanation of this is that the ϵ italic-ϵ \epsilon -pseudospectrum (for a suitable ϵ italic-ϵ \epsilon ) crosses the imaginary axis before the spectrum does, when R 𝑅 R increases. Then ‖ e − t ​ P ‖ norm superscript 𝑒 𝑡 𝑃 \|e^{-tP}\| will grow fast for a limited time even though the growth for very large times is determined by the spectrum. However, since P 𝑃 P appears as a linearization of a non-linear problem, that suffices to cause instability. In the case of differential operators the pseudospectral phenomenon is very general and related to classical works in PDE on local solvability and non-hypoellipticity. E.B. Davies 5 studied the non-self-adjoint semiclasscial Schrödinger operator with a smooth (complex-valued) potential in dimension 1 and showed under generic assumptions that one can construct quasimodes with the spectral parameter varying in an open complex set, containing points that are possibly very far from the spectrum (as can be verified in the case of the complex harmonic oscillator). M. Zworski 57 observed that this is essentially a rediscovery of an old result of Hörmander 32 , 33 , and was able to generalize considerably Davies result by adapting the one of Hörmander to the semi-classical case. With N. Dencker and M.Zworski 9 we also gave a direct proof and a corresponding adaptation of old results of Sato-Kawai-Kashiwara 44 to the analytic case: Theorem 1.3 ( 57 , 9 ) Let P ​ ( x , h ​ D x ) = ∑ | α | ≤ m a α ​ ( x ) ​ ( h ​ D x ) α , D x = ∂ ∂ x formulae-sequence 𝑃 𝑥 ℎ subscript 𝐷 𝑥 subscript 𝛼 𝑚 subscript 𝑎 𝛼 𝑥 superscript ℎ subscript 𝐷 𝑥 𝛼 subscript 𝐷 𝑥 𝑥 P(x,hD_{x})=\sum_{|\alpha|\leq m}a_{\alpha}(x)(hD_{x})^{\alpha},\ D_{x}={\partial\over\partial x} (1.3) have smooth coefficients in the open set Ω ⊂ 𝐑 n Ω superscript 𝐑 𝑛 \Omega\subset{\bf R}^{n} . Put p ​ ( x , ξ ) = ∑ | α | ≤ m a α ​ ( x ) ​ ξ α 𝑝 𝑥 𝜉 subscript 𝛼 𝑚 subscript 𝑎 𝛼 𝑥 superscript 𝜉 𝛼 p(x,\xi)=\sum_{|\alpha|\leq m}a_{\alpha}(x)\xi^{\alpha} . Assume z = p ​ ( x 0 , ξ 0 ) 𝑧 𝑝 subscript 𝑥 0 subscript 𝜉 0 z=p(x_{0},\xi_{0}) with the Poisson bracket 1 i ​ { p , p ¯ } ​ ( x 0 , ξ 0 ) > 0 1 𝑖 𝑝 ¯ 𝑝 subscript 𝑥 0 subscript 𝜉 0 0 {1\over i}\{p,\overline{p}\}(x_{0},\xi_{0})>0 . Then ∃ \exists u = u h ∈ C 0 ∞ ​ ( Ω ) 𝑢 subscript 𝑢 ℎ superscript subscript 𝐶 0 Ω u=u_{h}\in C_{0}^{\infty}(\Omega) , with ‖ u ‖ = 1 norm 𝑢 1 \|u\|=1 , ‖ ( P − z ) ​ u ‖ = 𝒪 ​ ( h ∞ ) norm 𝑃 𝑧 𝑢 𝒪 superscript ℎ \|(P-z)u\|={\cal O}(h^{\infty}) , when h → 0 → ℎ 0 h\to 0 . Analytic case: Can replace h ∞ superscript ℎ h^{\infty} by e − 1 / C ​ h superscript 𝑒 1 𝐶 ℎ e^{-1/Ch} . Here, we have used standard multi-index notation: ξ α = ξ 1 α 1 ⋅ … ⋅ ξ n α n superscript 𝜉 𝛼 ⋅ superscript subscript 𝜉 1 subscript 𝛼 1 … superscript subscript 𝜉 𝑛 subscript 𝛼 𝑛 \xi^{\alpha}=\xi_{1}^{\alpha_{1}}\cdot...\cdot\xi_{n}^{\alpha_{n}} , | α | = α 1 + … + α n 𝛼 subscript 𝛼 1 … subscript 𝛼 𝑛 |\alpha|=\alpha_{1}+...+\alpha_{n} , and the norms ∥ ⋅ ∥ \|\cdot\| are the ones of L 2 superscript 𝐿 2 L^{2} or ℓ 2 superscript ℓ 2 \ell^{2} if nothing else is indicated. This result was subsequently generalized by K. Pravda-Starov, 42 . Notice that this implies that when the theorem applies and if the resolvent ( P − z ) − 1 superscript 𝑃 𝑧 1 (P-z)^{-1} exists then its norm is greater than any negative power of h ℎ h when h → 0 → ℎ 0 h\to 0 . Example 1.4 Let P = − h 2 ​ Δ + V ​ ( x ) 𝑃 superscript ℎ 2 Δ 𝑉 𝑥 P=-h^{2}\Delta+V(x) , p ​ ( x , ξ ) = ξ 2 + V ​ ( x ) 𝑝 𝑥 𝜉 superscript 𝜉 2 𝑉 𝑥 p(x,\xi)=\xi^{2}+V(x) , ξ 2 = ξ 1 2 + … + ξ n 2 superscript 𝜉 2 superscript subscript 𝜉 1 2 … superscript subscript 𝜉 𝑛 2 \xi^{2}=\xi_{1}^{2}+...+\xi_{n}^{2} . Then 1 i ​ { p , p ¯ } = − 4 ​ ξ ⋅ ℑ ⁡ V ′ ​ ( x ) 1 𝑖 𝑝 ¯ 𝑝 ⋅ 4 𝜉 superscript 𝑉 ′ 𝑥 {1\over i}\{p,\overline{p}\}=-4\xi\cdot\Im V^{\prime}(x) . Generically, if z = p ​ ( x , ξ ) 𝑧 𝑝 𝑥 𝜉 z=p(x,\xi) , then { p , p ¯ } ​ ( x , ξ ) ≠ 0 𝑝 ¯ 𝑝 𝑥 𝜉 0 \{p,\overline{p}\}(x,\xi)\neq 0 and one can show quite generally that if this happens then there is also another point ( y , η ) 𝑦 𝜂 (y,\eta) with p ​ ( y , η ) = z 𝑝 𝑦 𝜂 𝑧 p(y,\eta)=z such that { p , p ¯ } ​ ( x , ξ ) 𝑝 ¯ 𝑝 𝑥 𝜉 \{p,\overline{p}\}(x,\xi) takes the opposite sign. This justifies the following simplified terminology in the semi-classical limit: The semi-classical pseudospectrum of P 𝑃 P is the range ℛ ​ ( p ) ℛ 𝑝 {\cal R}(p) of p 𝑝 p . In 9 we also showed under suitable assumptions (inspired from scattering theory and from the theory of sub-elliptic operators), that there is no spectrum near the boundary of the semi-classical pseudospectrum and that we may have quite a good control of the norm of the resolvent there. Generalizations to the case of systems were given by Dencker 8 . 1.2 The topics of this survey We will discuss three subjects involving non-self-adjoint differential and pseudodifferential operators, We will always wok in the semi-classical limit, which means that our operators are of the form P ​ ( x , h ​ D x ; h ) 𝑃 𝑥 ℎ subscript 𝐷 𝑥 ℎ P(x,hD_{x};h) , where P 𝑃 P is a suitable symbol and 0 < h ≪ 1 0 ℎ much-less-than 1 0<h\ll 1 . It is quite clear however that some of our results will also apply to non-semiclassical situations in the limit of large eigenvalues. The subjects are: • The Kramers-Fokker-Planck operator, • Bohr-Sommerfeld rules in dimension 2, • Weyl asymptotics for non-self-adjoint operators with small random perturbations. and most of the works discussed are the results of collaborations with A. Melin, M. Hitrik, F. Hérau, C.Stolk, S. Vũ Ngọc and M. Hager. In the first two topics we exploit the possibility of changing the Hilbert space norm by introducing exponential weights on phase space. In the case of the Kramers-Fokker-Planck operator, we make no analyticity assumptions and the phase-space weights are correspondingly quite weak. In this case however our operator is a differential one, so we are allowed to apply strong exponential weights depending only on the base variables, and this is important when studying small exponential corrections of the eigenvalues via the so called tunnel effect. For the Bohr-Sommerfeld rules, we make analyticity assumptions that allow stronger phase space weights. In both cases the effect of the exponential weights is to make the operator under consideration more normal. In the third topic, we do not use any deformations of the given Hilbert space, but exponential weights play an important role at another level, namely to count zeros of holomorphic functions with exponential growth. The pseudospectrum will not be discussed explicitly below. In the Kramers-Fokker-Planck case, the problems are located near the boundary of the semi-classical pseudospectrum, and it turns out that we have a very nice control of the resolvent there. In the 2 dimensional Bohr-Sommerfeld rules, we have stronger exponential weights, reflecting stronger pseudospectral phenomena. Finally in the subject of Weyl asymptotics, we often have strong pseudospectral behaviour for the unperturbed operator. From the proofs it appears that the random perturbations will weaken the pseudospectral behaviour and this might have very interesting consequences for the associated evolution problems. This is still very much an open problem.","role":"user"},{"content":"Abstract: This text is a survey of recent results obtained by the author and collaborators on different problems for non-self-adjoint operators. The topics are: Kramers-Fokker-Planck type operators, spectral asymptotics in two dimensions and Weyl asymptotics for the eigenvalues of non-self-adjoint operators with small random perturbations. In the introduction we also review the notion of pseudo-spectrum and its relation to non-self-adjoint spectral problems.","role":"assistant"}]

Surveying the Phenomenology of General Gauge Mediation

Gauge mediation is a predictive and flavor blind communication mechanism for SUSY breaking [ 1 ] . The earliest and simplest implementation of minimal gauge mediation involved communication of SUSY breaking from the hidden sector through a single set of vector like messengers charged under standard model gauge groups. The messengers couple to a hidden sector field ⟨ X ⟩ = x + θ 2 ​ F x delimited-⟨⟩ 𝑋 𝑥 superscript 𝜃 2 subscript 𝐹 𝑥 \langle X\rangle=x+\theta^{2}F_{x} and acquire a supersymmetric and nonsupersymmetric mass term, W = λ ​ X ​ ϕ ​ ϕ ~ → λ ​ ⟨ x ⟩ ​ ϕ ​ ϕ ~ + λ ​ F x ​ ϕ ​ ϕ ~ 𝑊 𝜆 𝑋 italic-ϕ ~ italic-ϕ → 𝜆 delimited-⟨⟩ 𝑥 italic-ϕ ~ italic-ϕ 𝜆 subscript 𝐹 𝑥 italic-ϕ ~ italic-ϕ W=\lambda X\phi\tilde{\phi}\rightarrow\lambda\langle x\rangle\phi\tilde{\phi}+\lambda F_{x}\phi\tilde{\phi} (1.1) Defining Λ ≡ F x / x Λ subscript 𝐹 𝑥 𝑥 \Lambda\equiv F_{x}/x we see that one loop gaugino masses are generated M λ ​ i ∼ α i 4 ​ π ​ Λ similar-to subscript 𝑀 𝜆 𝑖 subscript 𝛼 𝑖 4 𝜋 Λ M_{\lambda i}\sim\frac{\alpha_{i}}{4\pi}\Lambda (1.2) in addition to two loop scalar masses m ~ 2 = 2 ​ Λ 2 ​ [ C 3 ​ ( α 3 4 ​ π ) 2 + C 2 ​ ( α 2 4 ​ π ) 2 + 5 3 ​ ( Y 2 ) 2 ​ ( α 1 4 ​ π ) 2 ] . superscript ~ 𝑚 2 2 superscript Λ 2 delimited-[] subscript 𝐶 3 superscript subscript 𝛼 3 4 𝜋 2 subscript 𝐶 2 superscript subscript 𝛼 2 4 𝜋 2 5 3 superscript 𝑌 2 2 superscript subscript 𝛼 1 4 𝜋 2 \widetilde{m}^{2}={2\Lambda^{2}}\left[C_{3}\left({\alpha_{3}\over 4\pi}\right)^{2}+C_{2}\left({\alpha_{2}\over 4\pi}\right)^{2}+{5\over 3}{\left(Y\over 2\right)^{2}}\left({\alpha_{1}\over 4\pi}\right)^{2}\right]. (1.3) Minimal gauge mediation predicts that all scalar and gaugino mass terms are related by a single mass scale. Sparticles get masses proportional to powers of their gauge couplings, therefore, there is a large mass hierarchy in the spectrum placing sparticles charged under QCD far above the others. A large mass hierarchy in the SUSY spectrum is not just a prediction of gauge mediation but also results from the other leading candidates for SUSY communication, anomaly mediation and mSUGRA [ 2 ] . Recent bounds on quickly decaying NLSPs place a strong limit on Λ Λ \Lambda , the mass scale of minimal gauge mediation, at 91 TeV [ 3 ] . For non-prompt NLSP decay lower bounds come from charged sparticle masses and indirect constraints such as the inclusive tri-lepton signal. In both cases a heavy and thus tuned spectrum results, but this applies especially to the case of prompt NLSP decay. As experiments push the spectrum of minimal gauge mediation higher, non-minimal models become more phenomenologically (and for many philosophically) attractive. Simple extensions to MGM have been scattered throughout the literature, see for example [ 4 ] [ 5 ] . Recently [ 6 ] have proposed a simple framework for counting allowed parameters in general gauge mediated schemes. The general framework allows for at most six mass parameters to determine the low energy spectrum. There are now a variety of viable models implementing GGM in a more systematic way, however a detailed analysis of the parameter space of non-minimal SUSY models has not been done. Though constraints on minimal frameworks like MSUGRA are well understood, in non-minimal scenarios it isn’t even clear exactly what bounds on sparticle masses are - for example see recent work on gluinos [ 7 ] or look through PDG to see how many analyses are solely MSUGRA based [ 8 ] . For a given low energy spectrum in the GGM framework, there may be several ways to complete the model in the hidden sector. Model building in general gauge mediation may be accomplished with weakly coupled renormalizable theories [ 9 ] [ 10 ] , or in frameworks with non-perturbative dynamics [ 11 ] [ 12 ] . GGM models may be direct, with messengers participating in SUSY breaking, or indirect. In fact the general framework of GM does not require messengers at all. The goal of this work is not to build specific UV completions to models, or to insist on a specific fix in the Higgs sector. Rather it is to look at the low energy predictions and pick out interesting regions of parameter space where the phenomenology is far from standard and/or the tuning is significantly less then the minimal case. That said, many implementations of GGM result in split SUSY like spectra, where scalars are much heavier than gauginos and tuning is quite bad. This may be the case for example if Poppitz-Trivedi type mass terms are large. Such terms add to scalar masses only as they preserve R symmetry and are accounted for by non-vanishing messenger supertrace [ 13 ] . Such terms occur generically in certain classes of GGM models. This work will not address models with large mass splitting between scalars and gauginos, rather it will attempt to focus on theories that predict light spectra which look different from both the MGM and split SUSY spectra. The focus here will be limited the predictions of weakly coupled renormalizible hidden sectors with and R symmetry [ 10 ] . Such models avoid split spectra and rely on multiple hidden sector spurions and different numbers of messenger multiplets in various representations of SU(5) to change the number of low energy parameters. For example, consider the case of N sets of messengers in a 5 5 5 , 5 ¯ ¯ 5 \overline{5} representation of SU(5). These messengers may couple to multiple scalar fields which get vevs and F terms. The messenger couplings are λ ℓ a ​ i ​ X a ​ ℓ ¯ i ​ ℓ i + λ q a ​ i ​ X a ​ q ¯ i ​ q i superscript subscript 𝜆 ℓ 𝑎 𝑖 subscript 𝑋 𝑎 subscript ¯ ℓ 𝑖 subscript ℓ 𝑖 superscript subscript 𝜆 𝑞 𝑎 𝑖 subscript 𝑋 𝑎 subscript ¯ 𝑞 𝑖 subscript 𝑞 𝑖 \lambda_{\ell}^{ai}X_{a}\bar{\ell}_{i}\ell_{i}+\lambda_{q}^{ai}X_{a}\bar{q}_{i}q_{i} (1.4) where the index i counts the N messengers and the index a counts the hidden sector fields. There are then four parameters which determine the low energy spectrum, Λ q i = λ q a ​ i ​ F a λ q b ​ i ​ x b ; Λ ℓ i = λ ℓ a ​ i ​ F a λ ℓ b ​ i ​ x b . formulae-sequence superscript subscript Λ 𝑞 𝑖 superscript subscript 𝜆 𝑞 𝑎 𝑖 subscript 𝐹 𝑎 superscript subscript 𝜆 𝑞 𝑏 𝑖 subscript 𝑥 𝑏 superscript subscript Λ ℓ 𝑖 superscript subscript 𝜆 ℓ 𝑎 𝑖 subscript 𝐹 𝑎 superscript subscript 𝜆 ℓ 𝑏 𝑖 subscript 𝑥 𝑏 \Lambda_{q}^{i}={\lambda_{q}^{ai}F_{a}\over\lambda_{q}^{bi}x_{b}};\Lambda_{\ell}^{i}={\lambda_{\ell}^{ai}F_{a}\over\lambda_{\ell}^{bi}x_{b}}. (1.5) Counting this way we see that one set of 5 5 5 , 5 ¯ ¯ 5 \overline{5} leads to two parameters, one set of 10 10 10 , 10 ¯ ¯ 10 \overline{10} yields three and so on. Scanning over the complete space of gauge mediated parameters is complicated. However, the possible low energy phenomenologies become very nonstandard for cases with just two or three parameters. This work will focus on such cases, with the hope that these scans will sample much of the interesting low energy phenomenology of the full six parameter space. What follow in section 2 is an overview of parameter counting for weakly coupled GGM models. Section 3 is a catalogue of the most relevant constraints on GGM parameter space. The remaining sections map out the viable parameter space for different multi-parameter GM scenarios; section 4 deals with 2 parameter GM, section 5 with 3 parameters. Section 6 concludes.

I explore the phenomenology, constraints and tuning for several weakly coupled implementations of multi-parameter gauge mediation and compare to minimal gauge mediation. The low energy spectra are distinct from that of minimal gauge mediation, a wide range of NLSPs is found and spectra are significantly compressed thus tunings may be generically reduced to a part in 10 to a part in 20.

[{"content":"Help me to generate the abstract of a survey paper given the title:*Surveying the Phenomenology of General Gauge Mediation* and the introduction:Gauge mediation is a predictive and flavor blind communication mechanism for SUSY breaking 1 . The earliest and simplest implementation of minimal gauge mediation involved communication of SUSY breaking from the hidden sector through a single set of vector like messengers charged under standard model gauge groups. The messengers couple to a hidden sector field ⟨ X ⟩ = x + θ 2 ​ F x delimited-⟨⟩ 𝑋 𝑥 superscript 𝜃 2 subscript 𝐹 𝑥 \langle X\rangle=x+\theta^{2}F_{x} and acquire a supersymmetric and nonsupersymmetric mass term, W = λ ​ X ​ ϕ ​ ϕ ~ → λ ​ ⟨ x ⟩ ​ ϕ ​ ϕ ~ + λ ​ F x ​ ϕ ​ ϕ ~ 𝑊 𝜆 𝑋 italic-ϕ ~ italic-ϕ → 𝜆 delimited-⟨⟩ 𝑥 italic-ϕ ~ italic-ϕ 𝜆 subscript 𝐹 𝑥 italic-ϕ ~ italic-ϕ W=\lambda X\phi\tilde{\phi}\rightarrow\lambda\langle x\rangle\phi\tilde{\phi}+\lambda F_{x}\phi\tilde{\phi} (1.1) Defining Λ ≡ F x / x Λ subscript 𝐹 𝑥 𝑥 \Lambda\equiv F_{x}/x we see that one loop gaugino masses are generated M λ ​ i ∼ α i 4 ​ π ​ Λ similar-to subscript 𝑀 𝜆 𝑖 subscript 𝛼 𝑖 4 𝜋 Λ M_{\lambda i}\sim\frac{\alpha_{i}}{4\pi}\Lambda (1.2) in addition to two loop scalar masses m ~ 2 = 2 ​ Λ 2 ​ C 3 ​ ( α 3 4 ​ π ) 2 + C 2 ​ ( α 2 4 ​ π ) 2 + 5 3 ​ ( Y 2 ) 2 ​ ( α 1 4 ​ π ) 2 . superscript ~ 𝑚 2 2 superscript Λ 2 delimited- subscript 𝐶 3 superscript subscript 𝛼 3 4 𝜋 2 subscript 𝐶 2 superscript subscript 𝛼 2 4 𝜋 2 5 3 superscript 𝑌 2 2 superscript subscript 𝛼 1 4 𝜋 2 \widetilde{m}^{2}={2\Lambda^{2}}\leftC_{3}\left({\alpha_{3}\over 4\pi}\right)^{2}+C_{2}\left({\alpha_{2}\over 4\pi}\right)^{2}+{5\over 3}{\left(Y\over 2\right)^{2}}\left({\alpha_{1}\over 4\pi}\right)^{2}\right. (1.3) Minimal gauge mediation predicts that all scalar and gaugino mass terms are related by a single mass scale. Sparticles get masses proportional to powers of their gauge couplings, therefore, there is a large mass hierarchy in the spectrum placing sparticles charged under QCD far above the others. A large mass hierarchy in the SUSY spectrum is not just a prediction of gauge mediation but also results from the other leading candidates for SUSY communication, anomaly mediation and mSUGRA 2 . Recent bounds on quickly decaying NLSPs place a strong limit on Λ Λ \Lambda , the mass scale of minimal gauge mediation, at 91 TeV 3 . For non-prompt NLSP decay lower bounds come from charged sparticle masses and indirect constraints such as the inclusive tri-lepton signal. In both cases a heavy and thus tuned spectrum results, but this applies especially to the case of prompt NLSP decay. As experiments push the spectrum of minimal gauge mediation higher, non-minimal models become more phenomenologically (and for many philosophically) attractive. Simple extensions to MGM have been scattered throughout the literature, see for example 4 5 . Recently 6 have proposed a simple framework for counting allowed parameters in general gauge mediated schemes. The general framework allows for at most six mass parameters to determine the low energy spectrum. There are now a variety of viable models implementing GGM in a more systematic way, however a detailed analysis of the parameter space of non-minimal SUSY models has not been done. Though constraints on minimal frameworks like MSUGRA are well understood, in non-minimal scenarios it isnt even clear exactly what bounds on sparticle masses are - for example see recent work on gluinos 7 or look through PDG to see how many analyses are solely MSUGRA based 8 . For a given low energy spectrum in the GGM framework, there may be several ways to complete the model in the hidden sector. Model building in general gauge mediation may be accomplished with weakly coupled renormalizable theories 9 10 , or in frameworks with non-perturbative dynamics 11 12 . GGM models may be direct, with messengers participating in SUSY breaking, or indirect. In fact the general framework of GM does not require messengers at all. The goal of this work is not to build specific UV completions to models, or to insist on a specific fix in the Higgs sector. Rather it is to look at the low energy predictions and pick out interesting regions of parameter space where the phenomenology is far from standard and/or the tuning is significantly less then the minimal case. That said, many implementations of GGM result in split SUSY like spectra, where scalars are much heavier than gauginos and tuning is quite bad. This may be the case for example if Poppitz-Trivedi type mass terms are large. Such terms add to scalar masses only as they preserve R symmetry and are accounted for by non-vanishing messenger supertrace 13 . Such terms occur generically in certain classes of GGM models. This work will not address models with large mass splitting between scalars and gauginos, rather it will attempt to focus on theories that predict light spectra which look different from both the MGM and split SUSY spectra. The focus here will be limited the predictions of weakly coupled renormalizible hidden sectors with and R symmetry 10 . Such models avoid split spectra and rely on multiple hidden sector spurions and different numbers of messenger multiplets in various representations of SU(5) to change the number of low energy parameters. For example, consider the case of N sets of messengers in a 5 5 5 , 5 ¯ ¯ 5 \overline{5} representation of SU(5). These messengers may couple to multiple scalar fields which get vevs and F terms. The messenger couplings are λ ℓ a ​ i ​ X a ​ ℓ ¯ i ​ ℓ i + λ q a ​ i ​ X a ​ q ¯ i ​ q i superscript subscript 𝜆 ℓ 𝑎 𝑖 subscript 𝑋 𝑎 subscript ¯ ℓ 𝑖 subscript ℓ 𝑖 superscript subscript 𝜆 𝑞 𝑎 𝑖 subscript 𝑋 𝑎 subscript ¯ 𝑞 𝑖 subscript 𝑞 𝑖 \lambda_{\ell}^{ai}X_{a}\bar{\ell}_{i}\ell_{i}+\lambda_{q}^{ai}X_{a}\bar{q}_{i}q_{i} (1.4) where the index i counts the N messengers and the index a counts the hidden sector fields. There are then four parameters which determine the low energy spectrum, Λ q i = λ q a ​ i ​ F a λ q b ​ i ​ x b ; Λ ℓ i = λ ℓ a ​ i ​ F a λ ℓ b ​ i ​ x b . formulae-sequence superscript subscript Λ 𝑞 𝑖 superscript subscript 𝜆 𝑞 𝑎 𝑖 subscript 𝐹 𝑎 superscript subscript 𝜆 𝑞 𝑏 𝑖 subscript 𝑥 𝑏 superscript subscript Λ ℓ 𝑖 superscript subscript 𝜆 ℓ 𝑎 𝑖 subscript 𝐹 𝑎 superscript subscript 𝜆 ℓ 𝑏 𝑖 subscript 𝑥 𝑏 \Lambda_{q}^{i}={\lambda_{q}^{ai}F_{a}\over\lambda_{q}^{bi}x_{b}};\Lambda_{\ell}^{i}={\lambda_{\ell}^{ai}F_{a}\over\lambda_{\ell}^{bi}x_{b}}. (1.5) Counting this way we see that one set of 5 5 5 , 5 ¯ ¯ 5 \overline{5} leads to two parameters, one set of 10 10 10 , 10 ¯ ¯ 10 \overline{10} yields three and so on. Scanning over the complete space of gauge mediated parameters is complicated. However, the possible low energy phenomenologies become very nonstandard for cases with just two or three parameters. This work will focus on such cases, with the hope that these scans will sample much of the interesting low energy phenomenology of the full six parameter space. What follow in section 2 is an overview of parameter counting for weakly coupled GGM models. Section 3 is a catalogue of the most relevant constraints on GGM parameter space. The remaining sections map out the viable parameter space for different multi-parameter GM scenarios; section 4 deals with 2 parameter GM, section 5 with 3 parameters. Section 6 concludes.","role":"user"},{"content":"Abstract: I explore the phenomenology, constraints and tuning for several weakly coupled implementations of multi-parameter gauge mediation and compare to minimal gauge mediation. The low energy spectra are distinct from that of minimal gauge mediation, a wide range of NLSPs is found and spectra are significantly compressed thus tunings may be generically reduced to a part in 10 to a part in 20.","role":"assistant"}]

Predictions for the LHC: an overview

In this paper I present an overview of the predictions for Pb-Pb collisions at s NN = 5.5 subscript 𝑠 NN 5.5 \sqrt{s_{\rm NN}}=5.5 TeV at the Large Hadron Collider, mainly based on the material presented during the workshop ’Heavy Ion Collisions at the LHC - Last Call for Predictions’, held in the frame of the CERN Theory Institute from May 14th to June 10th 2007 [ 1 ] . Such compilation (see also [ 2 ] , and [ 3 ] for the analogous one for RHIC) should be useful for: (a) distinguishing pre- from post-dictions; (b) assuming that a model tested at RHIC (and eventually SPS) energies can be extrapolated to the LHC, the huge lever arm in energy provides very strong constraints; and (c) providing a frozen image of our present understanding of ultra-relativistic heavy ion collisions. I have classified the predictions in: bulk properties (Section 2): multiplicities (see pre-RHIC predictions in [ 4 ] ), azimuthal asymmetries, hadronic flavor observables and correlations at low transverse momentum; and hard and electromagnetic probes (Section 3): high transverse momentum observables and jets, quarkonium and heavy quarks, and leptonic probes and photons, extensively documented in [ 5 , 6 , 7 , 8 ] . Unless otherwise stated, the predictions presented here can be found in [ 1 ] and will be referenced by the name of their first author. Finally I draw some conclusions.

I present an overview of predictions for the heavy ion program at the Large Hadron Collider. It is mainly based on the material presented during the workshop 'Heavy Ion Collisions at the LHC—Last Call for Predictions', held in the frame of the CERN Theory Institute from May 14th to June 10th, 2007. Predictions on both bulk properties and hard probes are reviewed.

[{"content":"Help me to generate the abstract of a survey paper given the title:*Predictions for the LHC: an overview* and the introduction:In this paper I present an overview of the predictions for Pb-Pb collisions at s NN = 5.5 subscript 𝑠 NN 5.5 \sqrt{s_{\rm NN}}=5.5 TeV at the Large Hadron Collider, mainly based on the material presented during the workshop Heavy Ion Collisions at the LHC - Last Call for Predictions, held in the frame of the CERN Theory Institute from May 14th to June 10th 2007 1 . Such compilation (see also 2 , and 3 for the analogous one for RHIC) should be useful for: (a) distinguishing pre- from post-dictions; (b) assuming that a model tested at RHIC (and eventually SPS) energies can be extrapolated to the LHC, the huge lever arm in energy provides very strong constraints; and (c) providing a frozen image of our present understanding of ultra-relativistic heavy ion collisions. I have classified the predictions in: bulk properties (Section 2): multiplicities (see pre-RHIC predictions in 4 ), azimuthal asymmetries, hadronic flavor observables and correlations at low transverse momentum; and hard and electromagnetic probes (Section 3): high transverse momentum observables and jets, quarkonium and heavy quarks, and leptonic probes and photons, extensively documented in 5 , 6 , 7 , 8 . Unless otherwise stated, the predictions presented here can be found in 1 and will be referenced by the name of their first author. Finally I draw some conclusions.","role":"user"},{"content":"Abstract: I present an overview of predictions for the heavy ion program at the Large Hadron Collider. It is mainly based on the material presented during the workshop 'Heavy Ion Collisions at the LHC—Last Call for Predictions', held in the frame of the CERN Theory Institute from May 14th to June 10th, 2007. Predictions on both bulk properties and hard probes are reviewed.","role":"assistant"}]

Singularity resolution in loop quantum cosmology: A brief overview

I Issue of the Beginning and the End Over the history of mankind, cosmological paradigms have evolved in interesting ways. It is illuminating to begin with a long range historical perspective by recalling paradigms that seemed obvious and most natural for centuries only to be superseded by radical shifts. Treatise on Time, the Beginning and the End date back at least twenty five centuries. Does the flow of time have an objective, universal meaning beyond human perception? Or, is it fundamentally only a convenient, and perhaps merely psychological, notion? Did the physical universe have a finite beginning or has it been evolving eternally? Leading thinkers across cultures meditated on these issues and arrived at definite but strikingly different answers. For example, in the sixth century BCE, Gautama Buddha taught that ‘a period of time’ is a purely conventional notion, time and space exist only in relation to our experience, and the universe is eternal. In the Christian thought, by contrast, the universe had a finite beginning and there was debate whether time represents ‘movement’ of bodies or if it flows only in the soul. In the fourth century CE, St. Augustine held that time itself started with the world. Founding fathers of modern Science from Galileo to Newton continued to accept that God created the universe. Nonetheless, their work led to a radical change of paradigm. Before Newton, boundaries between the absolute and the relative, the true and the apparent and the mathematical and the common were blurry. Newton rescued time from the psychological and the material world and made it objective and absolute. It now ran uniformly from the infinite past to the infinite future. This paradigm became a dogma over centuries. Philosophers often used it to argue that the universe itself had to be eternal. For, as Immanuel Kant emphasized, otherwise one could ask “what was there before?” General relativity toppled this Newtonian paradigm in one fell swoop. Now the gravitational field is encoded in space-time geometry. Since geometry is a dynamical, physical entity, it is now perfectly feasible for the universe to have had a finite beginning —the big-bang— at which not only matter but space-time itself is born. If space is compact, matter as well as space-time end in the big-crunch singularity. In this respect, general relativity took us back to St. Augustine’s paradigm but in a detailed, specific and mathematically precise form. In semi-popular articles and radio shows, relativists now like to emphasize that the question “what was there before?” is rendered meaningless because the notions of ‘before’ requires a pre-existing space-time geometry. We now have a new paradigm, a new dogma: In the Beginning there was the Big Bang. However, the very fusion of gravity with geometry now gives rise to a new tension. In Newtonian (or Minkowskian) physics, a given physical field could become singular at a space-time point. This generally implied that the field could not be unambiguously evolved to the future of that point. However, this singularity had no effect on the global arena. Since the space-time geometry is unaffected by matter, it remains intact. Other fields could be evolved indefinitely. Trouble was limited to the one field which became ill behaved. However, because gravity is geometry in general relativity, when the gravitational field becomes singular, the continuum tares and the space-time itself ends. There is no more an arena for other fields to live in. All of physics, as we know it, comes to an abrupt halt. Physical observables associated with both matter and geometry simply diverge signalling a fundamental flaw in our description of Nature. This is the new quandary. When faced with deep quandaries, one has to carefully analyze the reasoning that led to the impasse. Typically the reasoning is flawed, possibly for subtle reasons. In the present case the culprit is the premise that general relativity —with its representation of space-time as a smooth continuum— provides an accurate description of Nature arbitrarily close to the singularity. For, general relativity completely ignores quantum physics and over the last century we have learned that quantum effects become important in the physics of the small. They should in fact be dominant in parts of the universe where matter densities become enormous. Thus the occurrence of the big bang and other singularities are predictions of general relativity, precisely in a regime where it is inapplicable! Classical physics of general relativity does come to a halt at the big-bang and the big crunch. But this is not an indication of what really happens because use of general relativity near singularities is an extrapolation which has no physical justification whatsoever. We need a theory that incorporates not only the dynamical nature of geometry but also the ramifications of quantum physics. We need a quantum theory of gravity, a new paradigm. These considerations suggest that singularities of general relativity are perhaps the most promising gates to physics beyond Einstein. They provide a fertile conceptual and technical ground in our search of the new paradigm. Consider some of the deepest conceptual questions we face today: the issue of the Beginning and the end End, the arrow of time, and the puzzle of black hole information loss. Their resolutions hinge on the true nature of singularities. In my view, considerable amount of contemporary confusion about such questions arises from our explicit or implicit insistence that singularities of general relativity are true boundaries of space-time; that we can trust causal structure all the way to these singularities; that notions such as event horizons are absolute even though changes in the metric in a Planck scale neighborhood of the singularity can move event horizons dramatically or even make them disappear altogether ph . Over the last 2-3 years several classically singular space-times have been investigated in detail through the lens of loop quantum gravity (LQG). This is a non-perturbative approach to the unification of general relativity and quantum physics in which one takes Einstein’s encoding of gravity into geometry seriously and elevates it to the quantum level alrev ; crbook ; ttbook . One is thus led to build quantum gravity using quantum Riemannian geometry almmt ; rs ; al5 ; alvol . Both geometry and matter are dynamical and described quantum mechanically from the start. In particular, then, there is no background space-time. The kinematical structure of the theory has been firmly established for some years now. There are also several interesting and concrete proposals for dynamics (see, in particular alrev ; crbook ; ttbook ; spinfoam-rev ). However, in my view there is still considerable ambiguity and none of the proposals is fully satisfactory. Nonetheless, over the last 2-3 years, considerable progress could be made by restricting oneself to subcases where detailed and explicit analysis is possible. These ‘mini’ and ‘midi’ superspaces are well adapted to analyze the deep conceptual tensions discussed above. For, they encompass the most interesting of classically singular space-times —Friedman-Lemaître-Robertson-Walker (FLRW) universes with the big bang singularity and black holes with the Schwarzschild-type singularity— and analyze them in detail using symmetry reduced versions of loop quantum gravity. In all cases studied so far, classical singularities are naturally resolved and the quantum space-time is vastly larger than what general relativity had us believe. As a result, there is a new paradigm to analyze the old questions. In my talk, I focused on cosmological singularities. The material I covered is discussed in greater detail in the original papers mb1 ; abl ; aps1 ; aps2 ; aps3 ; apsv ; aps4 ; acs ; cs and in more comprehensive reviews mb-rev ; aa-badhonef . Discussion of black hole singularities and the issue of information loss can be found either in the original papers ab1 ; ab2 ; bv ; atv or in a comprehensive, recent review addressed to non-experts aa-mink . Here, I will confine myself to a sketch of the singularity resolution in loop quantum cosmology (LQC). Section II will provide a conceptual setting, section III will summarize the main results and section IV will present the outlook.

A brief summary of loop quantum cosmology of homogeneous isotropic models is presented for non-experts with emphasis on the origin of and subtleties associated with the resolution of big bang and big crunch singularities. For further details, and a discussion of some frequently asked questions, the reader is directed to more comprehensive recent reviews.1

[{"content":"Help me to generate the abstract of a survey paper given the title:*Singularity resolution in loop quantum cosmology: A brief overview* and the introduction:I Issue of the Beginning and the End Over the history of mankind, cosmological paradigms have evolved in interesting ways. It is illuminating to begin with a long range historical perspective by recalling paradigms that seemed obvious and most natural for centuries only to be superseded by radical shifts. Treatise on Time, the Beginning and the End date back at least twenty five centuries. Does the flow of time have an objective, universal meaning beyond human perception? Or, is it fundamentally only a convenient, and perhaps merely psychological, notion? Did the physical universe have a finite beginning or has it been evolving eternally? Leading thinkers across cultures meditated on these issues and arrived at definite but strikingly different answers. For example, in the sixth century BCE, Gautama Buddha taught that a period of time is a purely conventional notion, time and space exist only in relation to our experience, and the universe is eternal. In the Christian thought, by contrast, the universe had a finite beginning and there was debate whether time represents movement of bodies or if it flows only in the soul. In the fourth century CE, St. Augustine held that time itself started with the world. Founding fathers of modern Science from Galileo to Newton continued to accept that God created the universe. Nonetheless, their work led to a radical change of paradigm. Before Newton, boundaries between the absolute and the relative, the true and the apparent and the mathematical and the common were blurry. Newton rescued time from the psychological and the material world and made it objective and absolute. It now ran uniformly from the infinite past to the infinite future. This paradigm became a dogma over centuries. Philosophers often used it to argue that the universe itself had to be eternal. For, as Immanuel Kant emphasized, otherwise one could ask what was there before? General relativity toppled this Newtonian paradigm in one fell swoop. Now the gravitational field is encoded in space-time geometry. Since geometry is a dynamical, physical entity, it is now perfectly feasible for the universe to have had a finite beginning —the big-bang— at which not only matter but space-time itself is born. If space is compact, matter as well as space-time end in the big-crunch singularity. In this respect, general relativity took us back to St. Augustines paradigm but in a detailed, specific and mathematically precise form. In semi-popular articles and radio shows, relativists now like to emphasize that the question what was there before? is rendered meaningless because the notions of before requires a pre-existing space-time geometry. We now have a new paradigm, a new dogma: In the Beginning there was the Big Bang. However, the very fusion of gravity with geometry now gives rise to a new tension. In Newtonian (or Minkowskian) physics, a given physical field could become singular at a space-time point. This generally implied that the field could not be unambiguously evolved to the future of that point. However, this singularity had no effect on the global arena. Since the space-time geometry is unaffected by matter, it remains intact. Other fields could be evolved indefinitely. Trouble was limited to the one field which became ill behaved. However, because gravity is geometry in general relativity, when the gravitational field becomes singular, the continuum tares and the space-time itself ends. There is no more an arena for other fields to live in. All of physics, as we know it, comes to an abrupt halt. Physical observables associated with both matter and geometry simply diverge signalling a fundamental flaw in our description of Nature. This is the new quandary. When faced with deep quandaries, one has to carefully analyze the reasoning that led to the impasse. Typically the reasoning is flawed, possibly for subtle reasons. In the present case the culprit is the premise that general relativity —with its representation of space-time as a smooth continuum— provides an accurate description of Nature arbitrarily close to the singularity. For, general relativity completely ignores quantum physics and over the last century we have learned that quantum effects become important in the physics of the small. They should in fact be dominant in parts of the universe where matter densities become enormous. Thus the occurrence of the big bang and other singularities are predictions of general relativity, precisely in a regime where it is inapplicable! Classical physics of general relativity does come to a halt at the big-bang and the big crunch. But this is not an indication of what really happens because use of general relativity near singularities is an extrapolation which has no physical justification whatsoever. We need a theory that incorporates not only the dynamical nature of geometry but also the ramifications of quantum physics. We need a quantum theory of gravity, a new paradigm. These considerations suggest that singularities of general relativity are perhaps the most promising gates to physics beyond Einstein. They provide a fertile conceptual and technical ground in our search of the new paradigm. Consider some of the deepest conceptual questions we face today: the issue of the Beginning and the end End, the arrow of time, and the puzzle of black hole information loss. Their resolutions hinge on the true nature of singularities. In my view, considerable amount of contemporary confusion about such questions arises from our explicit or implicit insistence that singularities of general relativity are true boundaries of space-time; that we can trust causal structure all the way to these singularities; that notions such as event horizons are absolute even though changes in the metric in a Planck scale neighborhood of the singularity can move event horizons dramatically or even make them disappear altogether ph . Over the last 2-3 years several classically singular space-times have been investigated in detail through the lens of loop quantum gravity (LQG). This is a non-perturbative approach to the unification of general relativity and quantum physics in which one takes Einsteins encoding of gravity into geometry seriously and elevates it to the quantum level alrev ; crbook ; ttbook . One is thus led to build quantum gravity using quantum Riemannian geometry almmt ; rs ; al5 ; alvol . Both geometry and matter are dynamical and described quantum mechanically from the start. In particular, then, there is no background space-time. The kinematical structure of the theory has been firmly established for some years now. There are also several interesting and concrete proposals for dynamics (see, in particular alrev ; crbook ; ttbook ; spinfoam-rev ). However, in my view there is still considerable ambiguity and none of the proposals is fully satisfactory. Nonetheless, over the last 2-3 years, considerable progress could be made by restricting oneself to subcases where detailed and explicit analysis is possible. These mini and midi superspaces are well adapted to analyze the deep conceptual tensions discussed above. For, they encompass the most interesting of classically singular space-times —Friedman-Lemaître-Robertson-Walker (FLRW) universes with the big bang singularity and black holes with the Schwarzschild-type singularity— and analyze them in detail using symmetry reduced versions of loop quantum gravity. In all cases studied so far, classical singularities are naturally resolved and the quantum space-time is vastly larger than what general relativity had us believe. As a result, there is a new paradigm to analyze the old questions. In my talk, I focused on cosmological singularities. The material I covered is discussed in greater detail in the original papers mb1 ; abl ; aps1 ; aps2 ; aps3 ; apsv ; aps4 ; acs ; cs and in more comprehensive reviews mb-rev ; aa-badhonef . Discussion of black hole singularities and the issue of information loss can be found either in the original papers ab1 ; ab2 ; bv ; atv or in a comprehensive, recent review addressed to non-experts aa-mink . Here, I will confine myself to a sketch of the singularity resolution in loop quantum cosmology (LQC). Section II will provide a conceptual setting, section III will summarize the main results and section IV will present the outlook.","role":"user"},{"content":"Abstract: A brief summary of loop quantum cosmology of homogeneous isotropic models is presented for non-experts with emphasis on the origin of and subtleties associated with the resolution of big bang and big crunch singularities. For further details, and a discussion of some frequently asked questions, the reader is directed to more comprehensive recent reviews.1","role":"assistant"}]

From Bell Inequalities to Tsirelson's Theorem

Correlations between observed measurements are a fundamental resource in quantum information. In his seminal 1964 paper, Bell [ 4 ] demonstrated an inequality that must be satisfied by correlations obtained classically, but may be violated by a quantum correlation experiment. Subsequently, the term Bell inequalities has come to describe the set of inequalities that characterize correlations that can be obtained between events that can be well described by classical physics. Since the 1980s, a series of experimental results have been published that apparently demonstrate that quantum correlations do in fact violate Bell inequalities [ 12 ] . This raises the question of whether a “good” mathematical characterization of quantum correlation vectors can be obtained. The first part of the paper is intended for the non-specialist. We begin by giving a survey of classical correlations, showing that they can be characterized using the well studied subject of L 1 superscript 𝐿 1 L^{1} -embeddings. Next we consider quantum correlations and give a statement of Tsirelson’s theorem [ 5 ] , which gives a partial characterization for the 2-party case. Interestingly, as we show, this characterization can be restated in terms of L 2 superscript 𝐿 2 L^{2} -embeddings. Tsirelson’s theorem has found many applications in quantum information, but a direct constructive proof of his theorem is not readily available. Although this theorem is powerful, even in the two-party case it does not solve the characterization problem completely. We discuss this, and give an additional necessary condition based on no-signalling. In many applications of Tsirelson’s theorem it is necessary to construct a quantum realization for a given correlation vector. An explicit construction is not given in his paper, but we give one here. We begin by giving a simple construction for the two dimensional case of Tsirelson’s theorem, which uses a two qubit state. This includes introducing many of the ideas needed for the general construction. In the second part of the paper, we give a new general construction, that is, states and operators which can represent all correlation functions. Our method is related to the theory of stabilizer states. We also demonstrate the optimality of this result, and give a scheme for a low dimensional approximation of an optimal representation. In the third part of the paper we discuss the problem of finding the maximum violation of a Bell inequality. We see that semidefinite programming (SDP) plays a crucial role in this optimization problem, and relate it to the results given earlier in the paper. Finally we review some very recent work involving an infinite hierarchy of SDPs that gives a non-polynomial time method of characterizing Bell inequalities and finding maximum violations.

The first part of this paper contains an introduction to Bell inequalities and Tsirelson’s theorem for the non-specialist. The next part gives an explicit optimum construction for the “hard” part of Tsirelson’s theorem. In the final part we describe how upper bounds on the maximal quantum violation of Bell inequalities can be obtained by an extension of Tsirelson’s theorem, and survey very recent results on how exact bounds may be obtained by solving an infinite series of semidefinite programs.

[{"content":"Help me to generate the abstract of a survey paper given the title:*From Bell Inequalities to Tsirelson's Theorem* and the introduction:Correlations between observed measurements are a fundamental resource in quantum information. In his seminal 1964 paper, Bell 4 demonstrated an inequality that must be satisfied by correlations obtained classically, but may be violated by a quantum correlation experiment. Subsequently, the term Bell inequalities has come to describe the set of inequalities that characterize correlations that can be obtained between events that can be well described by classical physics. Since the 1980s, a series of experimental results have been published that apparently demonstrate that quantum correlations do in fact violate Bell inequalities 12 . This raises the question of whether a good mathematical characterization of quantum correlation vectors can be obtained. The first part of the paper is intended for the non-specialist. We begin by giving a survey of classical correlations, showing that they can be characterized using the well studied subject of L 1 superscript 𝐿 1 L^{1} -embeddings. Next we consider quantum correlations and give a statement of Tsirelsons theorem 5 , which gives a partial characterization for the 2-party case. Interestingly, as we show, this characterization can be restated in terms of L 2 superscript 𝐿 2 L^{2} -embeddings. Tsirelsons theorem has found many applications in quantum information, but a direct constructive proof of his theorem is not readily available. Although this theorem is powerful, even in the two-party case it does not solve the characterization problem completely. We discuss this, and give an additional necessary condition based on no-signalling. In many applications of Tsirelsons theorem it is necessary to construct a quantum realization for a given correlation vector. An explicit construction is not given in his paper, but we give one here. We begin by giving a simple construction for the two dimensional case of Tsirelsons theorem, which uses a two qubit state. This includes introducing many of the ideas needed for the general construction. In the second part of the paper, we give a new general construction, that is, states and operators which can represent all correlation functions. Our method is related to the theory of stabilizer states. We also demonstrate the optimality of this result, and give a scheme for a low dimensional approximation of an optimal representation. In the third part of the paper we discuss the problem of finding the maximum violation of a Bell inequality. We see that semidefinite programming (SDP) plays a crucial role in this optimization problem, and relate it to the results given earlier in the paper. Finally we review some very recent work involving an infinite hierarchy of SDPs that gives a non-polynomial time method of characterizing Bell inequalities and finding maximum violations.","role":"user"},{"content":"Abstract: The first part of this paper contains an introduction to Bell inequalities and Tsirelsons theorem for the non-specialist. The next part gives an explicit optimum construction for the hard part of Tsirelsons theorem. In the final part we describe how upper bounds on the maximal quantum violation of Bell inequalities can be obtained by an extension of Tsirelsons theorem, and survey very recent results on how exact bounds may be obtained by solving an infinite series of semidefinite programs.","role":"assistant"}]

A SURVEY OF MEASURED GROUP THEORY

1. Introduction This survey concerns an area of mathematics which studies infinite countable groups using measure-theoretic tools, and studies Ergodic Theory of group actions, emphasizing the impact of group structure on the actions. Measured Group Theory is a particularly fitting title as it suggests an analogy with Geometric Group Theory . The origins of Measured Group Theory go back to the seminal paper of Robert Zimmer [ Zimmer:cocyclesuper:80 ] , which established a deep connection between questions on Orbit Equivalence in Ergodic Theory to the theory of lattices in semi-simple groups, specifically to Margulis’ celebrated superrigidity. The notion of amenable actions, introduced by Zimmer in an earlier work [ Zimmer:Amenable:78 ] , became an indispensable tool in the field. Zimmer continued to study orbit structures of actions of large groups in [ Zimmer:OE:1981 , Zimmer:1981:cohom , MR672181 , Zimmer:1982:Bull , Zimmer:1983:products , Zimmer:book:84 , Feldman+Sutherland+Zimmer:1989 , Cowling+Zimmer:89sp , Zimmer:1991:trans ] and [ Stuck+Zimmer:1994 ] . The monograph [ Zimmer:book:84 ] had a particularly big impact on both ergodic theorists and people studying big groups, as well as researchers in other areas, such as Operator Algebras and Descriptive Set Theory 1 1 1 Zimmer’s cocycle superrigidity proved in [ Zimmer:cocyclesuper:80 ] plays a central role in another area of research, vigorously pursued by Zimmer and other, concerning actions of large groups on manifolds. David Fisher surveys this direction in [ Fisher:survey ] in this volume. . In the recent years several new layers of results have been added to what we called Measured Group Theory, and this paper aims to give an overview of the current state of the subject. Such a goal is unattainable – any survey is doomed to be partial, biased, and outdated before it appears. Nevertheless, we shall try our best, hoping to encourage further interest in the subject. The reader is also referred to Gaboriau’s paper [ Gaboriau:2005exmps ] , which contains a very nice overview of some of the topics discussed here, and to Shalom’s survey [ Shalom:2005ECM ] which is even closer to the present paper (hence the similarity of the titles). The monographs by Kechris and Miller [ Kechris+Miller:2004book ] and the forthcoming one [ Kechris:new ] by Kechris include topics in Descriptive Set Theory related to Measured Group Theory. For topics related to von Neumann algebra we refer to Vaes’ [ Vaes:2007sb ] and Popa’s [ Popa:2007ICM ] and references therein. The scope of this paper is restricted to interaction of infinite Groups with Ergodic theory, leaving out the connections to the theory of von Neumann algebras and Descriptive Set Theory. When possible, we try to indicate proofs or ideas of proofs for the stated results. In particular, we chose to include a proof of one cocycle superrigidity theorem 5.20 , which enables a self-contained presentation of a number of important results: a very rigid equivalence relation (Theorem 4.19 ) with trivial fundamental group and outer automorphism group (Theorem 4.15 ), an equivalence relation not generated by an essentially free action of any group (§ 4.3.1 ). Disclaimer As usual, the quoted results are often presented not in full generality available, and the reader should consult the original papers for full details. The responsibility for inaccuracies, misquotes and other flaws lies with the author of these notes. Acknowledgements Organization of the paper The paper is organized as follows: the next section is devoted to a general introduction which emphasizes the relations between Measure Equivalence, Quasi-Isometry and Orbit Equivalence in Ergodic Theory. One may choose to skip most of this, but read Definition 2.1 and the following remarks. Section 3 concerns groups considered up to Measure Equivalence. Section 4 focuses on the notion of equivalence relations with orbit relations as a prime (but not only) example. In both of these sections we consider separately the invariants of the studied objects (groups and relations) and rigidity results, which pertain to classification. Section 5 describes the main techniques used in these theories (mostly for rigidity): a discussion of superrigidity phenomena and some of the ad hoc tools used in the subject; generalities on cocycles appear in the appendix A .

The title refers to the area of research which studies infinite groups using measure-theoretic tools, and studies the restrictions that group structure imposes on er- godic theory of their actions. The paper is a survey of recent developments focused on the notion of Measure Equivalence between groups, and Orbit Equivalence between group actions. We discuss known invariants and classification results (rigidity) in both areas.

[{"content":"Help me to generate the abstract of a survey paper given the title:*A SURVEY OF MEASURED GROUP THEORY* and the introduction:1. Introduction This survey concerns an area of mathematics which studies infinite countable groups using measure-theoretic tools, and studies Ergodic Theory of group actions, emphasizing the impact of group structure on the actions. Measured Group Theory is a particularly fitting title as it suggests an analogy with Geometric Group Theory . The origins of Measured Group Theory go back to the seminal paper of Robert Zimmer Zimmer:cocyclesuper:80 , which established a deep connection between questions on Orbit Equivalence in Ergodic Theory to the theory of lattices in semi-simple groups, specifically to Margulis celebrated superrigidity. The notion of amenable actions, introduced by Zimmer in an earlier work Zimmer:Amenable:78 , became an indispensable tool in the field. Zimmer continued to study orbit structures of actions of large groups in Zimmer:OE:1981 , Zimmer:1981:cohom , MR672181 , Zimmer:1982:Bull , Zimmer:1983:products , Zimmer:book:84 , Feldman+Sutherland+Zimmer:1989 , Cowling+Zimmer:89sp , Zimmer:1991:trans and Stuck+Zimmer:1994 . The monograph Zimmer:book:84 had a particularly big impact on both ergodic theorists and people studying big groups, as well as researchers in other areas, such as Operator Algebras and Descriptive Set Theory 1 1 1 Zimmers cocycle superrigidity proved in Zimmer:cocyclesuper:80 plays a central role in another area of research, vigorously pursued by Zimmer and other, concerning actions of large groups on manifolds. David Fisher surveys this direction in Fisher:survey in this volume. . In the recent years several new layers of results have been added to what we called Measured Group Theory, and this paper aims to give an overview of the current state of the subject. Such a goal is unattainable – any survey is doomed to be partial, biased, and outdated before it appears. Nevertheless, we shall try our best, hoping to encourage further interest in the subject. The reader is also referred to Gaboriaus paper Gaboriau:2005exmps , which contains a very nice overview of some of the topics discussed here, and to Shaloms survey Shalom:2005ECM which is even closer to the present paper (hence the similarity of the titles). The monographs by Kechris and Miller Kechris+Miller:2004book and the forthcoming one Kechris:new by Kechris include topics in Descriptive Set Theory related to Measured Group Theory. For topics related to von Neumann algebra we refer to Vaes Vaes:2007sb and Popas Popa:2007ICM and references therein. The scope of this paper is restricted to interaction of infinite Groups with Ergodic theory, leaving out the connections to the theory of von Neumann algebras and Descriptive Set Theory. When possible, we try to indicate proofs or ideas of proofs for the stated results. In particular, we chose to include a proof of one cocycle superrigidity theorem 5.20 , which enables a self-contained presentation of a number of important results: a very rigid equivalence relation (Theorem 4.19 ) with trivial fundamental group and outer automorphism group (Theorem 4.15 ), an equivalence relation not generated by an essentially free action of any group (§ 4.3.1 ). Disclaimer As usual, the quoted results are often presented not in full generality available, and the reader should consult the original papers for full details. The responsibility for inaccuracies, misquotes and other flaws lies with the author of these notes. Acknowledgements Organization of the paper The paper is organized as follows: the next section is devoted to a general introduction which emphasizes the relations between Measure Equivalence, Quasi-Isometry and Orbit Equivalence in Ergodic Theory. One may choose to skip most of this, but read Definition 2.1 and the following remarks. Section 3 concerns groups considered up to Measure Equivalence. Section 4 focuses on the notion of equivalence relations with orbit relations as a prime (but not only) example. In both of these sections we consider separately the invariants of the studied objects (groups and relations) and rigidity results, which pertain to classification. Section 5 describes the main techniques used in these theories (mostly for rigidity): a discussion of superrigidity phenomena and some of the ad hoc tools used in the subject; generalities on cocycles appear in the appendix A .","role":"user"},{"content":"Abstract: The title refers to the area of research which studies infinite groups using measure-theoretic tools, and studies the restrictions that group structure imposes on er- godic theory of their actions. The paper is a survey of recent developments focused on the notion of Measure Equivalence between groups, and Orbit Equivalence between group actions. We discuss known invariants and classification results (rigidity) in both areas.","role":"assistant"}]

Black holes and entropy in loop quantum gravity: An Overview

I Introduction Black holes (BH) have become rather prominent in fundamental physics ever since the fundamental results in the early 70’s showing that black holes satisfy some ‘thermodynamic-like laws’, summarized in the celebrated laws of black hole mechanics BCH , δ ​ M = κ 8 ​ π ​ G ​ δ ​ A , 𝛿 𝑀 𝜅 8 𝜋 𝐺 𝛿 𝐴 \delta M=\frac{\kappa}{8\pi G}\;\delta A\,, (1) From which one can formally relate, M ↔ E , κ ↔ T , A ↔ S , ↔ 𝑀 𝐸 𝜅 ↔ 𝑇 𝐴 ↔ 𝑆 M\leftrightarrow E,\qquad\kappa\leftrightarrow T\,,\qquad A\leftrightarrow S\,, where the relation between geometrical variables on Eq. ( 1 ) can be seen as the analogue of the first law of thermodynamics if the above association between geometric and thermodynamical objects is made. This analogy is further motivated by the fact that the surface gravity κ 𝜅 \kappa of a Killing horizon is constant and the area of an event horizon always grows. This observation, together with the proposal by Bekenstein and Hawking that BH possess a physical entropy and temperature, as confirmed by the computation of particle creation on black hole background, gave raise to a true identification between geometrical quantities and thermodynamical variables as follows BH : E = M T = κ ​ ℏ 2 ​ π and S = A 4 ​ G ​ ℏ . formulae-sequence 𝐸 𝑀 formulae-sequence 𝑇 𝜅 Planck-constant-over-2-pi 2 𝜋 and 𝑆 𝐴 4 𝐺 Planck-constant-over-2-pi E=M\qquad T=\frac{\kappa\;\hbar}{2\pi}\quad{\rm and}\quad S=\frac{A}{4\,G\hbar}\,. It is not unnatural to interpret that black holes must behave as thermodynamic systems, and in particular possess a non-zero temperature (that vanishes in the classical limit) and an entropy (that blows up). Quantum theory was needed in order to identify temperature and entropy with geometrical objects, by means of Planck’s constant ℏ Planck-constant-over-2-pi \hbar , suggesting that these identifications are quantum in nature. But, in order to have a full analogy, the question of what are the underlying degrees of freedom responsible for entropy became a pressing one. In other words, how can we account for the (huge) entropy associated to the black hole horizons? The standard wisdom is that only with a full marriage of Gravity and the Quantum will we be able to understand this issue. This is one of the main challenges that faces any candidate quantum theory of gravity. During the past 20 years there have been several attempts to identify those degrees of freedom. In particular one has to mention the success of string theory in explaining the entropy of extremal and near-extremal BH in several dimensions vafa . There have also been some proposals based on causal sets causal and on the use of entanglement entropy of matter fields entanglement . Within loop quantum gravity lqg ; lqg:easy , a leading candidate for a quantum theory of gravity, there has been some progress in describing black holes ‘in equilibrium’. In particular this implies that the objects to be studied are assumed to be isolated, in such a way that a study of its properties will guarantee that one can separate their description from that of the rest of the environment (as one normally does in thermodynamics). The resulting quantum picture is that the interaction between ‘bulk states’ as described by spin networks as they puncture the horizon, create horizon degrees of freedom that can (and do) fluctuate. These degrees of freedom are, on the one hand, independent of the bulk degrees of freedom, and on the other hand, fluctuate ‘in tandem’ with their bulk counterparts, as dictated by specific quantum conditions warranting the existence of the quantum horizon . The original program was developed in a series of papers ABCK ; ACK ; ABK and has been further studied AEV ; Dom:Lew ; majumdar ; meiss ; GM ; CDF ; CDF-2 ; ADF ; ABDFV and also widely reviewed in LR ; CDF-3 . The purpose of this contribution is to provide a bird’s eye view into the field, briefly summarizing the progress made in the past 12 years, including some recent results. This contribution can also be seen as a starting point and as a reading guide for those interested in more details. In what follows, we shall in particular try to answer the following questions: How do we characterize black holes in equilibrium? That is, what are the quantum horizon states? How do we know which states we should count? Can we learn how entropy behaves? Can we make contact, for large black holes, with the Bekenstein-Hawking entropy? Can we extend the formalism and consider small, Planck scale BH’s? How small is small? That is, where does the transition from the Planck scale to the ‘large area limit’ occurs? We shall not include topics such as the possibility of treating Hawking radiation carlo ; kirill or the criteria for dealing with black holes in thermal equilibrium majumdar2 .

Black holes in equilibrium and the counting of their entropy within Loop Quantum Gravity are reviewed. In particular, we focus on the conceptual setting of the formalism, briey summarizing the main results of the classical formalism and its quantization. We then focus on recent results for small, Planck scale, black holes, where new structures have been shown to arise, in particular an eective quantization of the entropy. We discuss recent results that employ in a very eective manner results from number theory, providing a complete solution to the counting of black hole entropy. We end with some comments on other approaches that are motivated by loop quantum gravity.

[{"content":"Help me to generate the abstract of a survey paper given the title:*Black holes and entropy in loop quantum gravity: An Overview* and the introduction:I Introduction Black holes (BH) have become rather prominent in fundamental physics ever since the fundamental results in the early 70s showing that black holes satisfy some thermodynamic-like laws, summarized in the celebrated laws of black hole mechanics BCH , δ ​ M = κ 8 ​ π ​ G ​ δ ​ A , 𝛿 𝑀 𝜅 8 𝜋 𝐺 𝛿 𝐴 \delta M=\frac{\kappa}{8\pi G}\;\delta A\,, (1) From which one can formally relate, M ↔ E , κ ↔ T , A ↔ S , ↔ 𝑀 𝐸 𝜅 ↔ 𝑇 𝐴 ↔ 𝑆 M\leftrightarrow E,\qquad\kappa\leftrightarrow T\,,\qquad A\leftrightarrow S\,, where the relation between geometrical variables on Eq. ( 1 ) can be seen as the analogue of the first law of thermodynamics if the above association between geometric and thermodynamical objects is made. This analogy is further motivated by the fact that the surface gravity κ 𝜅 \kappa of a Killing horizon is constant and the area of an event horizon always grows. This observation, together with the proposal by Bekenstein and Hawking that BH possess a physical entropy and temperature, as confirmed by the computation of particle creation on black hole background, gave raise to a true identification between geometrical quantities and thermodynamical variables as follows BH : E = M T = κ ​ ℏ 2 ​ π and S = A 4 ​ G ​ ℏ . formulae-sequence 𝐸 𝑀 formulae-sequence 𝑇 𝜅 Planck-constant-over-2-pi 2 𝜋 and 𝑆 𝐴 4 𝐺 Planck-constant-over-2-pi E=M\qquad T=\frac{\kappa\;\hbar}{2\pi}\quad{\rm and}\quad S=\frac{A}{4\,G\hbar}\,. It is not unnatural to interpret that black holes must behave as thermodynamic systems, and in particular possess a non-zero temperature (that vanishes in the classical limit) and an entropy (that blows up). Quantum theory was needed in order to identify temperature and entropy with geometrical objects, by means of Plancks constant ℏ Planck-constant-over-2-pi \hbar , suggesting that these identifications are quantum in nature. But, in order to have a full analogy, the question of what are the underlying degrees of freedom responsible for entropy became a pressing one. In other words, how can we account for the (huge) entropy associated to the black hole horizons? The standard wisdom is that only with a full marriage of Gravity and the Quantum will we be able to understand this issue. This is one of the main challenges that faces any candidate quantum theory of gravity. During the past 20 years there have been several attempts to identify those degrees of freedom. In particular one has to mention the success of string theory in explaining the entropy of extremal and near-extremal BH in several dimensions vafa . There have also been some proposals based on causal sets causal and on the use of entanglement entropy of matter fields entanglement . Within loop quantum gravity lqg ; lqg:easy , a leading candidate for a quantum theory of gravity, there has been some progress in describing black holes in equilibrium. In particular this implies that the objects to be studied are assumed to be isolated, in such a way that a study of its properties will guarantee that one can separate their description from that of the rest of the environment (as one normally does in thermodynamics). The resulting quantum picture is that the interaction between bulk states as described by spin networks as they puncture the horizon, create horizon degrees of freedom that can (and do) fluctuate. These degrees of freedom are, on the one hand, independent of the bulk degrees of freedom, and on the other hand, fluctuate in tandem with their bulk counterparts, as dictated by specific quantum conditions warranting the existence of the quantum horizon . The original program was developed in a series of papers ABCK ; ACK ; ABK and has been further studied AEV ; Dom:Lew ; majumdar ; meiss ; GM ; CDF ; CDF-2 ; ADF ; ABDFV and also widely reviewed in LR ; CDF-3 . The purpose of this contribution is to provide a birds eye view into the field, briefly summarizing the progress made in the past 12 years, including some recent results. This contribution can also be seen as a starting point and as a reading guide for those interested in more details. In what follows, we shall in particular try to answer the following questions: How do we characterize black holes in equilibrium? That is, what are the quantum horizon states? How do we know which states we should count? Can we learn how entropy behaves? Can we make contact, for large black holes, with the Bekenstein-Hawking entropy? Can we extend the formalism and consider small, Planck scale BHs? How small is small? That is, where does the transition from the Planck scale to the large area limit occurs? We shall not include topics such as the possibility of treating Hawking radiation carlo ; kirill or the criteria for dealing with black holes in thermal equilibrium majumdar2 .","role":"user"},{"content":"Abstract: Black holes in equilibrium and the counting of their entropy within Loop Quantum Gravity are reviewed. In particular, we focus on the conceptual setting of the formalism, briey summarizing the main results of the classical formalism and its quantization. We then focus on recent results for small, Planck scale, black holes, where new structures have been shown to arise, in particular an eective quantization of the entropy. We discuss recent results that employ in a very eective manner results from number theory, providing a complete solution to the counting of black hole entropy. We end with some comments on other approaches that are motivated by loop quantum gravity.","role":"assistant"}]

Review of Rotational Symmetry Breaking in Baby Skyrme Models

1 Breaking of Rotational Symmetry in Baby Skyrme Models The Skyrme model [ 1 , 2 ] is an SU(2)-valued nonlinear theory for pions in (3+1) dimensions with topological soliton solutions called skyrmions. Apart from a kinetic term, the Lagrangian of the model contains a ‘Skyrme’ term which is of the fourth order in derivatives, and is used to introduce scale to the model [ 3 ] . The existence of stable solutions in the Skyrme model is a consequence of the nontrivial topology of the mapping ℳ ℳ \mathcal{M} of the physical space into the field space at a given time, ℳ : S 3 → S ​ U ​ ( 2 ) ≅ S 3 , : ℳ → superscript 𝑆 3 𝑆 𝑈 2 superscript 𝑆 3 \quad\mathcal{M}:S^{3}\to SU(2)\cong S^{3},\quad where the physical space ℝ 3 superscript ℝ 3 \mathbb{R}^{3} is compactified to S 3 superscript 𝑆 3 S^{3} by requiring the spatial infinity to be equivalent in each direction. The topology which stems from this one-point compactification allows the classification of maps into equivalence classes, each of which has a unique conserved quantity called the topological charge. The Skyrme model has an analogue in (2+1) dimensions known as the baby Skyrme model, also admitting stable field configurations of a solitonic nature [ 4 ] . Due to its lower dimension, the baby Skyrme model serves as a simplification of the original model, but nonetheless it has a physical significance in its own right, having several applications in condensed-matter physics [ 5 ] , specifically in ferromagnetic quantum Hall systems [ 6 , 7 , 8 , 9 ] . There, baby skyrmions describe the excitations relative to ferromagnetic quantum Hall states, in terms of a gradient expansion in the spin density, a field with properties analogous to the pion field in the 3 3 3 D case [ 10 ] . The target manifold in the baby model is described by a three-dimensional vector ϕ = ( ϕ 1 , ϕ 2 , ϕ 3 ) bold-italic-ϕ subscript italic-ϕ 1 subscript italic-ϕ 2 subscript italic-ϕ 3 \bm{\phi}=(\phi_{1},\phi_{2},\phi_{3}) with the constraint ϕ ⋅ ϕ = 1 ⋅ bold-italic-ϕ bold-italic-ϕ 1 \bm{\phi}\cdot\bm{\phi}=1 . In analogy with the ( 3 + 1 ) 3 1 (3+1) D case, the domain of this model ℝ 2 superscript ℝ 2 \mathbb{R}^{2} is compactified to S 2 superscript 𝑆 2 S^{2} , yielding the topology required for the classification of its field configurations into classes with conserved topological charges. The Lagrangian density of the baby Skyrme model is given by: ℒ = 1 2 ​ ∂ μ ϕ ⋅ ∂ μ ϕ − κ 2 2 ​ [ ( ∂ μ ϕ ⋅ ∂ μ ϕ ) 2 − ( ∂ μ ϕ ⋅ ∂ ν ϕ ) ⋅ ( ∂ μ ϕ ⋅ ∂ ν ϕ ) ] − U ​ ( ϕ 3 ) , ℒ 1 2 subscript 𝜇 ⋅ bold-italic-ϕ superscript 𝜇 bold-italic-ϕ superscript 𝜅 2 2 delimited-[] superscript subscript 𝜇 ⋅ bold-italic-ϕ superscript 𝜇 bold-italic-ϕ 2 ⋅ subscript 𝜇 ⋅ bold-italic-ϕ subscript 𝜈 bold-italic-ϕ superscript 𝜇 ⋅ bold-italic-ϕ superscript 𝜈 bold-italic-ϕ 𝑈 subscript italic-ϕ 3 \displaystyle\mathcal{L}=\frac{1}{2}\partial_{\mu}\bm{\phi}\cdot\partial^{\mu}\bm{\phi}-\frac{\kappa^{2}}{2}\big{[}(\partial_{\mu}\bm{\phi}\cdot\partial^{\mu}\bm{\phi})^{2}-(\partial_{\mu}\bm{\phi}\cdot\partial_{\nu}\bm{\phi})\cdot(\partial^{\mu}\bm{\phi}\cdot\partial^{\nu}\bm{\phi})\big{]}-U(\phi_{3})\,, and consists of a kinetic term, a Skyrme term and a potential term. While in (3+1) dimensions the latter term is optional [ 11 ] , its presence in the (2+1)D model is necessary for the stability of the solutions. However, aside from the requirement that the potential vanishes at infinity for a given vacuum field value (normally taken to be ϕ ( 0 ) = ( 0 , 0 , 1 ) superscript bold-italic-ϕ 0 0 0 1 \bm{\phi}^{(0)}=(0,0,1) ), its exact form is arbitrary and gives rise to a rich family of possible baby-Skyrme models, several of which have been studied in detail in the literature. The simplest potential is the ‘holomorphic’ model with U ​ ( ϕ 3 ) = μ 2 ​ ( 1 − ϕ 3 ) 4 𝑈 subscript italic-ϕ 3 superscript 𝜇 2 superscript 1 subscript italic-ϕ 3 4 U(\phi_{3})=\mu^{2}(1-\phi_{3})^{4} [ 12 , 13 , 14 ] . It is known to have a stable solution only in the charge-one sector (the name refers to the fact that the stable solution has an analytic form in terms of holomorphic functions). The model with the potential U ​ ( ϕ 3 ) = μ 2 ​ ( 1 − ϕ 3 ) 𝑈 subscript italic-ϕ 3 superscript 𝜇 2 1 subscript italic-ϕ 3 U(\phi_{3})=\mu^{2}(1-\phi_{3}) (commonly referred to as the ‘old’ model) has also been extensively studied. This potential gives rise to very structured non-rotationally-symmetric multi-skyrmions [ 4 , 15 ] . Another model with U ​ ( ϕ 3 ) = μ 2 ​ ( 1 − ϕ 3 2 ) 𝑈 subscript italic-ϕ 3 superscript 𝜇 2 1 superscript subscript italic-ϕ 3 2 U(\phi_{3})=\mu^{2}(1-\phi_{3}^{2}) produces ring-like multi-skyrmions [ 16 ] . Other double-vacuum potentials which give rise to other types of solutions have also been studied [ 17 ] . Clearly, the form of the potential term has a decisive effect on the properties of the minimal energy configurations of the model. It is then worthwhile to see how the multisolitons of the baby Skyrme model look like for the one-parametric family of potentials U = μ 2 ​ ( 1 − ϕ 3 ) s 𝑈 superscript 𝜇 2 superscript 1 subscript italic-ϕ 3 𝑠 U=\mu^{2}(1-\phi_{3})^{s} which generalizes the ‘old‘ model ( s = 1 𝑠 1 s=1 ) and the holomorphic model ( s = 4 𝑠 4 s=4 ) [ 18 ] . As it turns out, the value of the parameter s 𝑠 s has dramatic effects on the static solutions of the model, both quantitatively and qualitatively, in the sense that it can be viewed as a ‘control’ parameter responsible for the repulsion or attraction between skyrmions, which in turn determines whether or not the minimal-energy configuration breaks rotational symmetry. The Lagrangian density is now: ℒ = 1 2 ​ ∂ μ ϕ ⋅ ∂ μ ϕ − κ 2 2 ​ ( ( ∂ μ ϕ ⋅ ∂ μ ϕ ) 2 − ( ∂ μ ϕ ⋅ ∂ ν ϕ ) ⋅ ( ∂ μ ϕ ⋅ ∂ ν ϕ ) ) − μ 2 ​ ( 1 − ϕ 3 ) s , ℒ 1 2 subscript 𝜇 ⋅ bold-italic-ϕ superscript 𝜇 bold-italic-ϕ superscript 𝜅 2 2 superscript subscript 𝜇 ⋅ bold-italic-ϕ superscript 𝜇 bold-italic-ϕ 2 ⋅ subscript 𝜇 ⋅ bold-italic-ϕ subscript 𝜈 bold-italic-ϕ superscript 𝜇 ⋅ bold-italic-ϕ superscript 𝜈 bold-italic-ϕ superscript 𝜇 2 superscript 1 subscript italic-ϕ 3 𝑠 \displaystyle\mathcal{L}=\frac{1}{2}\partial_{\mu}\bm{\phi}\cdot\partial^{\mu}\bm{\phi}-\frac{\kappa^{2}}{2}\left((\partial_{\mu}\bm{\phi}\cdot\partial^{\mu}\bm{\phi})^{2}-(\partial_{\mu}\bm{\phi}\cdot\partial_{\nu}\bm{\phi})\cdot(\partial^{\mu}\bm{\phi}\cdot\partial^{\nu}\bm{\phi})\right)-\mu^{2}(1-\phi_{3})^{s}\,, and contains three free parameters, namely κ , μ 𝜅 𝜇 \kappa,\mu and s 𝑠 s . Since either κ 𝜅 \kappa or μ 𝜇 \mu may be scaled away, the parameter space of this model is in fact only two dimensional. Our main goal here is to study the effects of these parameters on the static solutions of the model within each topological sector. The multi-skyrmions of our model are those field configurations which minimize the static energy functional within each topological sector. In polar coordinates the energy functional is given by E = ∫ r ​ d ​ r ​ d ​ θ ​ ( 1 2 ​ ( ∂ r ϕ ⋅ ∂ r ϕ + 1 r 2 ​ ∂ θ ϕ ⋅ ∂ θ ϕ ) + κ 2 2 ​ ( ∂ r ϕ × ∂ θ ϕ ) 2 r 2 + μ 2 ​ ( 1 − ϕ 3 ) s ) . 𝐸 𝑟 d 𝑟 d 𝜃 1 2 subscript 𝑟 ⋅ bold-italic-ϕ subscript 𝑟 bold-italic-ϕ 1 superscript 𝑟 2 subscript 𝜃 ⋅ bold-italic-ϕ subscript 𝜃 bold-italic-ϕ superscript 𝜅 2 2 superscript subscript 𝑟 bold-italic-ϕ subscript 𝜃 bold-italic-ϕ 2 superscript 𝑟 2 superscript 𝜇 2 superscript 1 subscript italic-ϕ 3 𝑠 \displaystyle E=\int r\,\text{d}r\,\text{d}\theta\left(\frac{1}{2}(\partial_{r}\bm{\phi}\cdot\partial_{r}\bm{\phi}+\frac{1}{r^{2}}\partial_{\theta}\bm{\phi}\cdot\partial_{\theta}\bm{\phi})+\frac{\kappa^{2}}{2}\frac{(\partial_{r}\bm{\phi}\times\partial_{\theta}\bm{\phi})^{2}}{r^{2}}+\mu^{2}(1-\phi_{3})^{s}\right)\;. The Euler-Lagrange equations derived from the energy functional ( 1 ) are nonlinear PDE ’s, so in most cases one must resort to numerical techniques in order to solve them. In our approach, the minimal energy configuration of a baby skyrmion of charge B and a given set of values μ , κ , s 𝜇 𝜅 𝑠 \mu,\kappa,s is found by a full-field relaxation method, which we describe in more detail in the Appendix. 1.1 Results Applying the minimization procedure, we obtain the static solutions of the model for 1 ≤ B ≤ 5 1 𝐵 5 1\leq B\leq 5 . Since the parameter space of the model is effectively two dimensional (as discussed earlier), without loss of generality we fix the potential strength at μ 2 = 0.1 superscript 𝜇 2 0.1 \mu^{2}=0.1 throughout, and the s 𝑠 s - κ 𝜅 \kappa parameter space is scanned in the region 0 < s ≤ 4 0 𝑠 4 0<s\leq 4 , 0.01 ≤ κ 2 ≤ 1 0.01 superscript 𝜅 2 1 0.01\leq\kappa^{2}\leq 1 . 1.1.1 Charge-one skyrmions In the charge-one sector, the solutions for every value of s 𝑠 s and κ 𝜅 \kappa are stable rotationally-symmetric configurations. Figure 1 a shows the obtained profile functions of the B = 1 𝐵 1 B=1 solution for different values of s 𝑠 s with κ 𝜅 \kappa fixed at κ 2 = 0.25 superscript 𝜅 2 0.25 \kappa^{2}=0.25 . Interestingly, the skyrmion energy as a function of s 𝑠 s is not monotonic, but acquires a minimum at s ≈ 2.2 𝑠 2.2 s\approx 2.2 , as is shown in Fig. 2 . Figure 1: Profile functions of the B = 1 𝐵 1 B=1 (left) and B = 2 𝐵 2 B=2 (right) skyrmions for s = 0.5 𝑠 0.5 s=0.5 (solid), s = 1 𝑠 1 s=1 (dotted) and s = 2 𝑠 2 s=2 (dot-dashed). Here κ 𝜅 \kappa is fixed at κ 2 = 0.25 superscript 𝜅 2 0.25 \kappa^{2}=0.25 . Figure 2: Total energies (divided by 4 ​ π ​ B 4 𝜋 𝐵 4\pi B ) of the charge-one ( ◆ ◆ \blacklozenge ) charge-two ( ■ ■ \blacksquare ) and charge-three ( ◆ ◆ \lozenge ) skyrmions as a function of the parameter s 𝑠 s for κ 2 = 0.05 superscript 𝜅 2 0.05 \kappa^{2}=0.05 . Each of the energy graphs attains a minimal value at some s 𝑠 s . At s ≈ 2 𝑠 2 s\approx 2 the energy-per-topological-charge of the charge-two and charge-three solutions reaches the charge-one energy (from below), and stable solutions are no longer observed. 1.1.2 Charge-two skyrmions Stable solutions also exist in the B = 2 𝐵 2 B=2 sector, but only for s < 2 𝑠 2 s<2 . They are rotationally-symmetric and ring-like, corresponding to two charge-one skyrmions on top of each other. Figure 1 b shows the profile functions of the stable solutions for different values of s 𝑠 s and κ 2 = 0.25 superscript 𝜅 2 0.25 \kappa^{2}=0.25 . As in the B = 1 𝐵 1 B=1 case, the energy of the charge-two skyrmion as a function of s 𝑠 s is non-monotonic and has a minimum around s = 1.3 𝑠 1.3 s=1.3 . As shown in Fig. 2 , at s ≈ 2 𝑠 2 s\approx 2 the energy of the ring-like configuration reaches the value of twice the energy of the charge-one skyrmion and stable configurations cease to exist. At this point, the skyrmion breaks apart into its constituent charge-one skyrmions, which in turn start drifting away from each other, thus breaking the rotational symmetry of the solution. Contour plots of the energy distribution of the B = 2 𝐵 2 B=2 skyrmion are shown in Fig. 3 for κ 2 = 1 superscript 𝜅 2 1 \kappa^{2}=1 and for two s 𝑠 s values. While for s = 1.5 𝑠 1.5 s=1.5 a ring-like stable configuration exists (Fig. 3 a), for s = 2.6 𝑠 2.6 s=2.6 the skyrmion breaks apart. The latter case is shown in Fig. 3 b which is a “snapshot” taken while the distance between the individual skyrmions kept growing. These results are in accord with corresponding results from previously known studies of both the ‘old’ ( s = 1 𝑠 1 s=1 ) model in which ring-like configurations have been observed [ 4 , 15 ] , and the holomorphic model for which no stable solutions have been found [ 12 , 13 ] . Figure 3: Contour plots of the energy distributions (ranging from violet – low density to red – high density) of the B = 2 𝐵 2 B=2 skyrmion for κ 2 = 1 superscript 𝜅 2 1 \kappa^{2}=1 . In the s < 2 𝑠 2 s<2 regime, ring-like rotationally-symmetric configurations exist, corresponding to two charge-one skyrmions on top of each other (left), whereas for s > 2 𝑠 2 s>2 , the charge-two skyrmion splits into two one-charge skyrmions drifting infinitely apart (right). Rotationally-symmetric charge-two locally stable solutions may also be observed in the large s 𝑠 s regime, including the ‘holomorphic’ s = 4 𝑠 4 s=4 case, in which case the global minimum in this regime corresponds to two infinitely separated charge-one skyrmions. The total energy of the rotationally symmetric solutions is larger than twice the energy of a charge-one skyrmion, indicating that the split skyrmion is an energetically more favorable configuration. We discuss this issue in more detail in the section 2.2 . 1.1.3 Charge-three and higher-charge skyrmions As with the B = 2 𝐵 2 B=2 skyrmion, the existence of stable charge-three skyrmions was also found to be s 𝑠 s dependent. For any tested value of κ 𝜅 \kappa in the range 0.01 ≤ κ 2 ≤ 1 0.01 superscript 𝜅 2 1 0.01\leq\kappa^{2}\leq 1 , we have found that above s ≈ 2 𝑠 2 s\approx 2 , no stable charge-three solutions exist; in this region the skyrmion breaks apart into individual skyrmions drifting further and further away from each other. In the s < 2 𝑠 2 s<2 region, where stable solutions exist, the energy distribution of the charge-three skyrmion turns out to be highly dependent on both s 𝑠 s and κ 𝜅 \kappa . Keeping κ 𝜅 \kappa fixed and varying s 𝑠 s , we find that in the small s 𝑠 s regime, ring-like rotationally-symmetric configurations exist. Increasing the value of s 𝑠 s results in stable minimal energy configurations with only ℤ ​ ( 2 ) ℤ 2 \mathbb{Z}(2) symmetry, corresponding to three partially-overlapping charge-one skyrmions in a row, as shown in Figs 4 b and 4 c. The energy of the charge-three skyrmion also has a minimum in s 𝑠 s , at around s = 1.5 𝑠 1.5 s=1.5 (as shown in Fig. 2 ). At s ≈ 2 𝑠 2 s\approx 2 the energy of the skyrmion becomes larger than three times the energy of a charge-one skyrmion and stable configurations are no longer obtainable. This is illustrated in Fig. 4 which shows contour plots of the energy distribution of the B = 3 𝐵 3 B=3 skyrmion for different values of s 𝑠 s and fixed κ 𝜅 \kappa . For s = 0.5 𝑠 0.5 s=0.5 (Fig. 4 a), the solution is rotationally symmetric and for s = 0.75 𝑠 0.75 s=0.75 and s = 1 𝑠 1 s=1 (Figs 4 b and 4 c respectively) the rotational symmetry of the solution is broken and only ℤ ​ ( 2 ) ℤ 2 \mathbb{Z}(2) symmetry remains. At s = 3 𝑠 3 s=3 , no stable solution exists. The latter case is shown in Fig. 4 d which is a “snapshot” taken while the distance between the individual skyrmions kept growing. The dependence of the skyrmion solutions on the value of κ 𝜅 \kappa with fixed s 𝑠 s show the following behavior: While for small κ 𝜅 \kappa the minimal energy configurations are rotationally-symmetric, increasing its value results in an increasingly larger rotational symmetry breaking. This is illustrated in Fig. 5 . Figure 4: Energy densities and corresponding contour plots (ranging from violet – low density to red – high density) of the B = 3 𝐵 3 B=3 skyrmion for fixed κ 𝜅 \kappa ( κ 2 = 0.01 superscript 𝜅 2 0.01 \kappa^{2}=0.01 ) and varying s 𝑠 s . In the s = 0.5 𝑠 0.5 s=0.5 case, the minimal energy configuration is rotationally symmetric, corresponding the three one-skyrmions on top of each other. For s = 0.75 𝑠 0.75 s=0.75 and s = 1 𝑠 1 s=1 the solutions exhibit only ℤ ​ ( 2 ) ℤ 2 \mathbb{Z}(2) symmetry, corresponding to partially-overlapping one-skyrmions. For s = 3 𝑠 3 s=3 no stable solution exists and the individual skyrmions are drifting apart. Figure 5: Energy densities and corresponding contour plots (ranging from violet – low density to red – high density) of the B = 3 𝐵 3 B=3 skyrmion for fixed s 𝑠 s ( s = 0.5 𝑠 0.5 s=0.5 ) and varying κ 𝜅 \kappa . At low κ 𝜅 \kappa , the minimal energy configuration is rotationally symmetric. As κ 𝜅 \kappa is increased, breaking of rotational symmetry appears, and only ℤ ​ ( 2 ) ℤ 2 \mathbb{Z}(2) symmetry remains. The B = 4 𝐵 4 B=4 and B = 5 𝐵 5 B=5 skyrmion solutions behave similarly to the B = 3 𝐵 3 B=3 solutions. This is illustrated in Fig. 6 , which shows the stable solutions that have been obtained in the s = 0.9 𝑠 0.9 s=0.9 case and the splitting of these skyrmions into their constituents in the s = 4 𝑠 4 s=4 case. Figure 6: Contour plots of the energy distributions (ranging from violet – low density to red – high density) of the B = 4 𝐵 4 B=4 and B = 5 𝐵 5 B=5 skyrmions for s = 0.9 𝑠 0.9 s=0.9 and s = 4 𝑠 4 s=4 ( κ 2 = 0.1 superscript 𝜅 2 0.1 \kappa^{2}=0.1 ). In the lower s 𝑠 s region stable solutions exist; the upper figures show a B = 4 𝐵 4 B=4 skyrmion in a bound state of two charge-two skyrmions (left), and a B = 5 𝐵 5 B=5 skyrmion in a two-one-two configuration. For values of s 𝑠 s higher than 2 2 2 , the multi-skyrmions split into individual one-skyrmions constantly drifting apart (lower figures).

We discuss one of the most interesting phenomena exhibited by baby skyrmions – breaking of rotational symmetry. The topics we will deal with here include the appearance of rotational symmetry breaking in the static solutions of baby Skyrme models, both in flat as well as in curved spaces, the zero-temperature crystalline structure of baby skyrmions, and finally, the appearance of spontaneous breaking of rotational symmetry in rotating baby skyrmions. 1.1. Breaking of Rotational Symmetry in Baby Skyrme Models The Skyrme model 1,2 is an SU(2)-valued nonlinear theory for pions in (3+1) dimensions with topological soliton solutions called skyrmions. Apart from a kinetic term, the Lagrangian of the model contains a ‘Skyrme’ term which is of the fourth order in derivatives, and is used to introduce scale to the model. 3 The existence of stable solutions in the Skyrme model is a consequence of the nontrivial topology of the mapping M of the physical space into the field space at a given time, M : S 3 → SU(2) ∼ S 3 , where the physical space R 3 is compactified to S 3 by requiring the spatial infinity to be equivalent in each direction. The topology which stems from this one-point compactification allows the classification of maps into equivalence classes, each of which has a unique conserved quantity called the topological charge. The Skyrme model has an analogue in (2+1) dimensions known as the baby Skyrme model, also admitting stable field configurations of a solitonic nature. 4 Due to its lower dimension, the baby Skyrme model serves as a simplification of the original model, but nonetheless it has a physical significance in its own right, having several applications in condensed-matter physics, 5 specifically in ferromagnetic quantum Hall systems. 6–9 There, baby skyrmions describe the excitations relative to ferromagnetic quantum Hall states, in terms of a gradient expansion in the spin density, a field with properties analogous to the pion field in the 3D case. 10 The target manifold in the baby model is described by a three-dimensional vector � = (�1,�2,�3) with the constraint �� � = 1. In analogy with the (3+1)D case, the domain of this model R 2 is compactified to S 2 , yielding the topology required for 1 to appear in: G. Brown and M. Rho, Eds., Multifaceted Skyrmions, (World Scientific, Singapore,

[{"content":"Help me to generate the abstract of a survey paper given the title:*Review of Rotational Symmetry Breaking in Baby Skyrme Models* and the introduction:1 Breaking of Rotational Symmetry in Baby Skyrme Models The Skyrme model 1 , 2 is an SU(2)-valued nonlinear theory for pions in (3+1) dimensions with topological soliton solutions called skyrmions. Apart from a kinetic term, the Lagrangian of the model contains a Skyrme term which is of the fourth order in derivatives, and is used to introduce scale to the model 3 . The existence of stable solutions in the Skyrme model is a consequence of the nontrivial topology of the mapping ℳ ℳ \mathcal{M} of the physical space into the field space at a given time, ℳ : S 3 → S ​ U ​ ( 2 ) ≅ S 3 , : ℳ → superscript 𝑆 3 𝑆 𝑈 2 superscript 𝑆 3 \quad\mathcal{M}:S^{3}\to SU(2)\cong S^{3},\quad where the physical space ℝ 3 superscript ℝ 3 \mathbb{R}^{3} is compactified to S 3 superscript 𝑆 3 S^{3} by requiring the spatial infinity to be equivalent in each direction. The topology which stems from this one-point compactification allows the classification of maps into equivalence classes, each of which has a unique conserved quantity called the topological charge. The Skyrme model has an analogue in (2+1) dimensions known as the baby Skyrme model, also admitting stable field configurations of a solitonic nature 4 . Due to its lower dimension, the baby Skyrme model serves as a simplification of the original model, but nonetheless it has a physical significance in its own right, having several applications in condensed-matter physics 5 , specifically in ferromagnetic quantum Hall systems 6 , 7 , 8 , 9 . There, baby skyrmions describe the excitations relative to ferromagnetic quantum Hall states, in terms of a gradient expansion in the spin density, a field with properties analogous to the pion field in the 3 3 3 D case 10 . The target manifold in the baby model is described by a three-dimensional vector ϕ = ( ϕ 1 , ϕ 2 , ϕ 3 ) bold-italic-ϕ subscript italic-ϕ 1 subscript italic-ϕ 2 subscript italic-ϕ 3 \bm{\phi}=(\phi_{1},\phi_{2},\phi_{3}) with the constraint ϕ ⋅ ϕ = 1 ⋅ bold-italic-ϕ bold-italic-ϕ 1 \bm{\phi}\cdot\bm{\phi}=1 . In analogy with the ( 3 + 1 ) 3 1 (3+1) D case, the domain of this model ℝ 2 superscript ℝ 2 \mathbb{R}^{2} is compactified to S 2 superscript 𝑆 2 S^{2} , yielding the topology required for the classification of its field configurations into classes with conserved topological charges. The Lagrangian density of the baby Skyrme model is given by: ℒ = 1 2 ​ ∂ μ ϕ ⋅ ∂ μ ϕ − κ 2 2 ​ ( ∂ μ ϕ ⋅ ∂ μ ϕ ) 2 − ( ∂ μ ϕ ⋅ ∂ ν ϕ ) ⋅ ( ∂ μ ϕ ⋅ ∂ ν ϕ ) − U ​ ( ϕ 3 ) , ℒ 1 2 subscript 𝜇 ⋅ bold-italic-ϕ superscript 𝜇 bold-italic-ϕ superscript 𝜅 2 2 delimited- superscript subscript 𝜇 ⋅ bold-italic-ϕ superscript 𝜇 bold-italic-ϕ 2 ⋅ subscript 𝜇 ⋅ bold-italic-ϕ subscript 𝜈 bold-italic-ϕ superscript 𝜇 ⋅ bold-italic-ϕ superscript 𝜈 bold-italic-ϕ 𝑈 subscript italic-ϕ 3 \displaystyle\mathcal{L}=\frac{1}{2}\partial_{\mu}\bm{\phi}\cdot\partial^{\mu}\bm{\phi}-\frac{\kappa^{2}}{2}\big{}(\partial_{\mu}\bm{\phi}\cdot\partial^{\mu}\bm{\phi})^{2}-(\partial_{\mu}\bm{\phi}\cdot\partial_{\nu}\bm{\phi})\cdot(\partial^{\mu}\bm{\phi}\cdot\partial^{\nu}\bm{\phi})\big{}-U(\phi_{3})\,, and consists of a kinetic term, a Skyrme term and a potential term. While in (3+1) dimensions the latter term is optional 11 , its presence in the (2+1)D model is necessary for the stability of the solutions. However, aside from the requirement that the potential vanishes at infinity for a given vacuum field value (normally taken to be ϕ ( 0 ) = ( 0 , 0 , 1 ) superscript bold-italic-ϕ 0 0 0 1 \bm{\phi}^{(0)}=(0,0,1) ), its exact form is arbitrary and gives rise to a rich family of possible baby-Skyrme models, several of which have been studied in detail in the literature. The simplest potential is the holomorphic model with U ​ ( ϕ 3 ) = μ 2 ​ ( 1 − ϕ 3 ) 4 𝑈 subscript italic-ϕ 3 superscript 𝜇 2 superscript 1 subscript italic-ϕ 3 4 U(\phi_{3})=\mu^{2}(1-\phi_{3})^{4} 12 , 13 , 14 . It is known to have a stable solution only in the charge-one sector (the name refers to the fact that the stable solution has an analytic form in terms of holomorphic functions). The model with the potential U ​ ( ϕ 3 ) = μ 2 ​ ( 1 − ϕ 3 ) 𝑈 subscript italic-ϕ 3 superscript 𝜇 2 1 subscript italic-ϕ 3 U(\phi_{3})=\mu^{2}(1-\phi_{3}) (commonly referred to as the old model) has also been extensively studied. This potential gives rise to very structured non-rotationally-symmetric multi-skyrmions 4 , 15 . Another model with U ​ ( ϕ 3 ) = μ 2 ​ ( 1 − ϕ 3 2 ) 𝑈 subscript italic-ϕ 3 superscript 𝜇 2 1 superscript subscript italic-ϕ 3 2 U(\phi_{3})=\mu^{2}(1-\phi_{3}^{2}) produces ring-like multi-skyrmions 16 . Other double-vacuum potentials which give rise to other types of solutions have also been studied 17 . Clearly, the form of the potential term has a decisive effect on the properties of the minimal energy configurations of the model. It is then worthwhile to see how the multisolitons of the baby Skyrme model look like for the one-parametric family of potentials U = μ 2 ​ ( 1 − ϕ 3 ) s 𝑈 superscript 𝜇 2 superscript 1 subscript italic-ϕ 3 𝑠 U=\mu^{2}(1-\phi_{3})^{s} which generalizes the old model ( s = 1 𝑠 1 s=1 ) and the holomorphic model ( s = 4 𝑠 4 s=4 ) 18 . As it turns out, the value of the parameter s 𝑠 s has dramatic effects on the static solutions of the model, both quantitatively and qualitatively, in the sense that it can be viewed as a control parameter responsible for the repulsion or attraction between skyrmions, which in turn determines whether or not the minimal-energy configuration breaks rotational symmetry. The Lagrangian density is now: ℒ = 1 2 ​ ∂ μ ϕ ⋅ ∂ μ ϕ − κ 2 2 ​ ( ( ∂ μ ϕ ⋅ ∂ μ ϕ ) 2 − ( ∂ μ ϕ ⋅ ∂ ν ϕ ) ⋅ ( ∂ μ ϕ ⋅ ∂ ν ϕ ) ) − μ 2 ​ ( 1 − ϕ 3 ) s , ℒ 1 2 subscript 𝜇 ⋅ bold-italic-ϕ superscript 𝜇 bold-italic-ϕ superscript 𝜅 2 2 superscript subscript 𝜇 ⋅ bold-italic-ϕ superscript 𝜇 bold-italic-ϕ 2 ⋅ subscript 𝜇 ⋅ bold-italic-ϕ subscript 𝜈 bold-italic-ϕ superscript 𝜇 ⋅ bold-italic-ϕ superscript 𝜈 bold-italic-ϕ superscript 𝜇 2 superscript 1 subscript italic-ϕ 3 𝑠 \displaystyle\mathcal{L}=\frac{1}{2}\partial_{\mu}\bm{\phi}\cdot\partial^{\mu}\bm{\phi}-\frac{\kappa^{2}}{2}\left((\partial_{\mu}\bm{\phi}\cdot\partial^{\mu}\bm{\phi})^{2}-(\partial_{\mu}\bm{\phi}\cdot\partial_{\nu}\bm{\phi})\cdot(\partial^{\mu}\bm{\phi}\cdot\partial^{\nu}\bm{\phi})\right)-\mu^{2}(1-\phi_{3})^{s}\,, and contains three free parameters, namely κ , μ 𝜅 𝜇 \kappa,\mu and s 𝑠 s . Since either κ 𝜅 \kappa or μ 𝜇 \mu may be scaled away, the parameter space of this model is in fact only two dimensional. Our main goal here is to study the effects of these parameters on the static solutions of the model within each topological sector. The multi-skyrmions of our model are those field configurations which minimize the static energy functional within each topological sector. In polar coordinates the energy functional is given by E = ∫ r ​ d ​ r ​ d ​ θ ​ ( 1 2 ​ ( ∂ r ϕ ⋅ ∂ r ϕ + 1 r 2 ​ ∂ θ ϕ ⋅ ∂ θ ϕ ) + κ 2 2 ​ ( ∂ r ϕ × ∂ θ ϕ ) 2 r 2 + μ 2 ​ ( 1 − ϕ 3 ) s ) . 𝐸 𝑟 d 𝑟 d 𝜃 1 2 subscript 𝑟 ⋅ bold-italic-ϕ subscript 𝑟 bold-italic-ϕ 1 superscript 𝑟 2 subscript 𝜃 ⋅ bold-italic-ϕ subscript 𝜃 bold-italic-ϕ superscript 𝜅 2 2 superscript subscript 𝑟 bold-italic-ϕ subscript 𝜃 bold-italic-ϕ 2 superscript 𝑟 2 superscript 𝜇 2 superscript 1 subscript italic-ϕ 3 𝑠 \displaystyle E=\int r\,\text{d}r\,\text{d}\theta\left(\frac{1}{2}(\partial_{r}\bm{\phi}\cdot\partial_{r}\bm{\phi}+\frac{1}{r^{2}}\partial_{\theta}\bm{\phi}\cdot\partial_{\theta}\bm{\phi})+\frac{\kappa^{2}}{2}\frac{(\partial_{r}\bm{\phi}\times\partial_{\theta}\bm{\phi})^{2}}{r^{2}}+\mu^{2}(1-\phi_{3})^{s}\right)\;. The Euler-Lagrange equations derived from the energy functional ( 1 ) are nonlinear PDE s, so in most cases one must resort to numerical techniques in order to solve them. In our approach, the minimal energy configuration of a baby skyrmion of charge B and a given set of values μ , κ , s 𝜇 𝜅 𝑠 \mu,\kappa,s is found by a full-field relaxation method, which we describe in more detail in the Appendix. 1.1 Results Applying the minimization procedure, we obtain the static solutions of the model for 1 ≤ B ≤ 5 1 𝐵 5 1\leq B\leq 5 . Since the parameter space of the model is effectively two dimensional (as discussed earlier), without loss of generality we fix the potential strength at μ 2 = 0.1 superscript 𝜇 2 0.1 \mu^{2}=0.1 throughout, and the s 𝑠 s - κ 𝜅 \kappa parameter space is scanned in the region 0 < s ≤ 4 0 𝑠 4 0<s\leq 4 , 0.01 ≤ κ 2 ≤ 1 0.01 superscript 𝜅 2 1 0.01\leq\kappa^{2}\leq 1 . 1.1.1 Charge-one skyrmions In the charge-one sector, the solutions for every value of s 𝑠 s and κ 𝜅 \kappa are stable rotationally-symmetric configurations. Figure 1 a shows the obtained profile functions of the B = 1 𝐵 1 B=1 solution for different values of s 𝑠 s with κ 𝜅 \kappa fixed at κ 2 = 0.25 superscript 𝜅 2 0.25 \kappa^{2}=0.25 . Interestingly, the skyrmion energy as a function of s 𝑠 s is not monotonic, but acquires a minimum at s ≈ 2.2 𝑠 2.2 s\approx 2.2 , as is shown in Fig. 2 . Figure 1: Profile functions of the B = 1 𝐵 1 B=1 (left) and B = 2 𝐵 2 B=2 (right) skyrmions for s = 0.5 𝑠 0.5 s=0.5 (solid), s = 1 𝑠 1 s=1 (dotted) and s = 2 𝑠 2 s=2 (dot-dashed). Here κ 𝜅 \kappa is fixed at κ 2 = 0.25 superscript 𝜅 2 0.25 \kappa^{2}=0.25 . Figure 2: Total energies (divided by 4 ​ π ​ B 4 𝜋 𝐵 4\pi B ) of the charge-one ( ◆ ◆ \blacklozenge ) charge-two ( ■ ■ \blacksquare ) and charge-three ( ◆ ◆ \lozenge ) skyrmions as a function of the parameter s 𝑠 s for κ 2 = 0.05 superscript 𝜅 2 0.05 \kappa^{2}=0.05 . Each of the energy graphs attains a minimal value at some s 𝑠 s . At s ≈ 2 𝑠 2 s\approx 2 the energy-per-topological-charge of the charge-two and charge-three solutions reaches the charge-one energy (from below), and stable solutions are no longer observed. 1.1.2 Charge-two skyrmions Stable solutions also exist in the B = 2 𝐵 2 B=2 sector, but only for s < 2 𝑠 2 s<2 . They are rotationally-symmetric and ring-like, corresponding to two charge-one skyrmions on top of each other. Figure 1 b shows the profile functions of the stable solutions for different values of s 𝑠 s and κ 2 = 0.25 superscript 𝜅 2 0.25 \kappa^{2}=0.25 . As in the B = 1 𝐵 1 B=1 case, the energy of the charge-two skyrmion as a function of s 𝑠 s is non-monotonic and has a minimum around s = 1.3 𝑠 1.3 s=1.3 . As shown in Fig. 2 , at s ≈ 2 𝑠 2 s\approx 2 the energy of the ring-like configuration reaches the value of twice the energy of the charge-one skyrmion and stable configurations cease to exist. At this point, the skyrmion breaks apart into its constituent charge-one skyrmions, which in turn start drifting away from each other, thus breaking the rotational symmetry of the solution. Contour plots of the energy distribution of the B = 2 𝐵 2 B=2 skyrmion are shown in Fig. 3 for κ 2 = 1 superscript 𝜅 2 1 \kappa^{2}=1 and for two s 𝑠 s values. While for s = 1.5 𝑠 1.5 s=1.5 a ring-like stable configuration exists (Fig. 3 a), for s = 2.6 𝑠 2.6 s=2.6 the skyrmion breaks apart. The latter case is shown in Fig. 3 b which is a snapshot taken while the distance between the individual skyrmions kept growing. These results are in accord with corresponding results from previously known studies of both the old ( s = 1 𝑠 1 s=1 ) model in which ring-like configurations have been observed 4 , 15 , and the holomorphic model for which no stable solutions have been found 12 , 13 . Figure 3: Contour plots of the energy distributions (ranging from violet – low density to red – high density) of the B = 2 𝐵 2 B=2 skyrmion for κ 2 = 1 superscript 𝜅 2 1 \kappa^{2}=1 . In the s < 2 𝑠 2 s<2 regime, ring-like rotationally-symmetric configurations exist, corresponding to two charge-one skyrmions on top of each other (left), whereas for s > 2 𝑠 2 s>2 , the charge-two skyrmion splits into two one-charge skyrmions drifting infinitely apart (right). Rotationally-symmetric charge-two locally stable solutions may also be observed in the large s 𝑠 s regime, including the holomorphic s = 4 𝑠 4 s=4 case, in which case the global minimum in this regime corresponds to two infinitely separated charge-one skyrmions. The total energy of the rotationally symmetric solutions is larger than twice the energy of a charge-one skyrmion, indicating that the split skyrmion is an energetically more favorable configuration. We discuss this issue in more detail in the section 2.2 . 1.1.3 Charge-three and higher-charge skyrmions As with the B = 2 𝐵 2 B=2 skyrmion, the existence of stable charge-three skyrmions was also found to be s 𝑠 s dependent. For any tested value of κ 𝜅 \kappa in the range 0.01 ≤ κ 2 ≤ 1 0.01 superscript 𝜅 2 1 0.01\leq\kappa^{2}\leq 1 , we have found that above s ≈ 2 𝑠 2 s\approx 2 , no stable charge-three solutions exist; in this region the skyrmion breaks apart into individual skyrmions drifting further and further away from each other. In the s < 2 𝑠 2 s<2 region, where stable solutions exist, the energy distribution of the charge-three skyrmion turns out to be highly dependent on both s 𝑠 s and κ 𝜅 \kappa . Keeping κ 𝜅 \kappa fixed and varying s 𝑠 s , we find that in the small s 𝑠 s regime, ring-like rotationally-symmetric configurations exist. Increasing the value of s 𝑠 s results in stable minimal energy configurations with only ℤ ​ ( 2 ) ℤ 2 \mathbb{Z}(2) symmetry, corresponding to three partially-overlapping charge-one skyrmions in a row, as shown in Figs 4 b and 4 c. The energy of the charge-three skyrmion also has a minimum in s 𝑠 s , at around s = 1.5 𝑠 1.5 s=1.5 (as shown in Fig. 2 ). At s ≈ 2 𝑠 2 s\approx 2 the energy of the skyrmion becomes larger than three times the energy of a charge-one skyrmion and stable configurations are no longer obtainable. This is illustrated in Fig. 4 which shows contour plots of the energy distribution of the B = 3 𝐵 3 B=3 skyrmion for different values of s 𝑠 s and fixed κ 𝜅 \kappa . For s = 0.5 𝑠 0.5 s=0.5 (Fig. 4 a), the solution is rotationally symmetric and for s = 0.75 𝑠 0.75 s=0.75 and s = 1 𝑠 1 s=1 (Figs 4 b and 4 c respectively) the rotational symmetry of the solution is broken and only ℤ ​ ( 2 ) ℤ 2 \mathbb{Z}(2) symmetry remains. At s = 3 𝑠 3 s=3 , no stable solution exists. The latter case is shown in Fig. 4 d which is a snapshot taken while the distance between the individual skyrmions kept growing. The dependence of the skyrmion solutions on the value of κ 𝜅 \kappa with fixed s 𝑠 s show the following behavior: While for small κ 𝜅 \kappa the minimal energy configurations are rotationally-symmetric, increasing its value results in an increasingly larger rotational symmetry breaking. This is illustrated in Fig. 5 . Figure 4: Energy densities and corresponding contour plots (ranging from violet – low density to red – high density) of the B = 3 𝐵 3 B=3 skyrmion for fixed κ 𝜅 \kappa ( κ 2 = 0.01 superscript 𝜅 2 0.01 \kappa^{2}=0.01 ) and varying s 𝑠 s . In the s = 0.5 𝑠 0.5 s=0.5 case, the minimal energy configuration is rotationally symmetric, corresponding the three one-skyrmions on top of each other. For s = 0.75 𝑠 0.75 s=0.75 and s = 1 𝑠 1 s=1 the solutions exhibit only ℤ ​ ( 2 ) ℤ 2 \mathbb{Z}(2) symmetry, corresponding to partially-overlapping one-skyrmions. For s = 3 𝑠 3 s=3 no stable solution exists and the individual skyrmions are drifting apart. Figure 5: Energy densities and corresponding contour plots (ranging from violet – low density to red – high density) of the B = 3 𝐵 3 B=3 skyrmion for fixed s 𝑠 s ( s = 0.5 𝑠 0.5 s=0.5 ) and varying κ 𝜅 \kappa . At low κ 𝜅 \kappa , the minimal energy configuration is rotationally symmetric. As κ 𝜅 \kappa is increased, breaking of rotational symmetry appears, and only ℤ ​ ( 2 ) ℤ 2 \mathbb{Z}(2) symmetry remains. The B = 4 𝐵 4 B=4 and B = 5 𝐵 5 B=5 skyrmion solutions behave similarly to the B = 3 𝐵 3 B=3 solutions. This is illustrated in Fig. 6 , which shows the stable solutions that have been obtained in the s = 0.9 𝑠 0.9 s=0.9 case and the splitting of these skyrmions into their constituents in the s = 4 𝑠 4 s=4 case. Figure 6: Contour plots of the energy distributions (ranging from violet – low density to red – high density) of the B = 4 𝐵 4 B=4 and B = 5 𝐵 5 B=5 skyrmions for s = 0.9 𝑠 0.9 s=0.9 and s = 4 𝑠 4 s=4 ( κ 2 = 0.1 superscript 𝜅 2 0.1 \kappa^{2}=0.1 ). In the lower s 𝑠 s region stable solutions exist; the upper figures show a B = 4 𝐵 4 B=4 skyrmion in a bound state of two charge-two skyrmions (left), and a B = 5 𝐵 5 B=5 skyrmion in a two-one-two configuration. For values of s 𝑠 s higher than 2 2 2 , the multi-skyrmions split into individual one-skyrmions constantly drifting apart (lower figures).","role":"user"},{"content":"Abstract: We discuss one of the most interesting phenomena exhibited by baby skyrmions – breaking of rotational symmetry. The topics we will deal with here include the appearance of rotational symmetry breaking in the static solutions of baby Skyrme models, both in flat as well as in curved spaces, the zero-temperature crystalline structure of baby skyrmions, and finally, the appearance of spontaneous breaking of rotational symmetry in rotating baby skyrmions. 1.1. Breaking of Rotational Symmetry in Baby Skyrme Models The Skyrme model 1,2 is an SU(2)-valued nonlinear theory for pions in (3+1) dimensions with topological soliton solutions called skyrmions. Apart from a kinetic term, the Lagrangian of the model contains a Skyrme term which is of the fourth order in derivatives, and is used to introduce scale to the model. 3 The existence of stable solutions in the Skyrme model is a consequence of the nontrivial topology of the mapping M of the physical space into the field space at a given time, M : S 3 → SU(2) ∼ S 3 , where the physical space R 3 is compactified to S 3 by requiring the spatial infinity to be equivalent in each direction. The topology which stems from this one-point compactification allows the classification of maps into equivalence classes, each of which has a unique conserved quantity called the topological charge. The Skyrme model has an analogue in (2+1) dimensions known as the baby Skyrme model, also admitting stable field configurations of a solitonic nature. 4 Due to its lower dimension, the baby Skyrme model serves as a simplification of the original model, but nonetheless it has a physical significance in its own right, having several applications in condensed-matter physics, 5 specifically in ferromagnetic quantum Hall systems. 6–9 There, baby skyrmions describe the excitations relative to ferromagnetic quantum Hall states, in terms of a gradient expansion in the spin density, a field with properties analogous to the pion field in the 3D case. 10 The target manifold in the baby model is described by a three-dimensional vector � = (�1,�2,�3) with the constraint �� � = 1. In analogy with the (3+1)D case, the domain of this model R 2 is compactified to S 2 , yielding the topology required for 1 to appear in: G. Brown and M. Rho, Eds., Multifaceted Skyrmions, (World Scientific, Singapore,","role":"assistant"}]

Cold nuclear effects on heavy flavours: a review

1 Foreword The understanding of normal nuclear effects on heavy flavours production recently became crucial to understand the J / ψ 𝐽 𝜓 J/\psi nuclear modification factors measured with the PHENIX experiment [ 1 ] . In particular, the observed rapidity dependence of J / ψ 𝐽 𝜓 J/\psi suppression is reversed with respect to what one would naively expect from density-induced suppression mechanisms, such as the colour screening originally proposed as a QGP signature [ 2 ] . J / ψ 𝐽 𝜓 J/\psi are more suppressed at higher rapidity, where the density is lower. The question is: Could cold nuclear matter effects explain this behaviour? Or do we need to invoke the coalescence of c 𝑐 c and c ¯ ¯ 𝑐 \bar{c} quarks coming from uncorrelated pairs instead? This question is so central that the two contributions on quarkonia suppression in heavy ion collisions we heard at this conference discussed it in details [ 3 , 4 ] . In this paper, I focus on cold nuclear matter only, on most of the available p+A like collisions (from SPS, FNAL, HERA-B and RHIC) and on all the heavy flavour observables ( J / ψ 𝐽 𝜓 J/\psi , ψ ′ superscript 𝜓 ′ \psi^{\prime} , Υ Υ \Upsilon and open charm…). In the following sections, I show the various dependencies of the heavy flavours productions, as they appeared chronologically. But before to look at data, one needs to define a somewhat arbitrary boundary between what we call ”hot” and ”cold”. Initial state effects (EMC effect, parton antishadowing, shadowing or saturation, energy loss and Cronin effect, intrinsic charm…) obviously belong to the cold sector. Among the final state effects, absorption by incoming nucleons is usually classified as a cold effect too. On the contrary, absorption by comovers (of often unknown partonic or hadronic nature) is an effect of the hot matter. Obviously, so are the long awaited colour screening and quark recombination or coalescence. In a short and pragmatic way, I define as cold what can be grasped in p+A like collisions, while hot is what happens in addition when one looks at A+A collisions. In this paper, we shall then mostly look at p+A like collisions.

Before wondering about the quark–gluon plasma (QGP), one has to take into account various cold (normal) nuclear matter effects, that can be probed through p+A like collisions. This paper aims at reviewing the current results (and understanding) of these effects on heavy quarks and quarkonia production.

[{"content":"Help me to generate the abstract of a survey paper given the title:*Cold nuclear effects on heavy flavours: a review* and the introduction:1 Foreword The understanding of normal nuclear effects on heavy flavours production recently became crucial to understand the J / ψ 𝐽 𝜓 J/\psi nuclear modification factors measured with the PHENIX experiment 1 . In particular, the observed rapidity dependence of J / ψ 𝐽 𝜓 J/\psi suppression is reversed with respect to what one would naively expect from density-induced suppression mechanisms, such as the colour screening originally proposed as a QGP signature 2 . J / ψ 𝐽 𝜓 J/\psi are more suppressed at higher rapidity, where the density is lower. The question is: Could cold nuclear matter effects explain this behaviour? Or do we need to invoke the coalescence of c 𝑐 c and c ¯ ¯ 𝑐 \bar{c} quarks coming from uncorrelated pairs instead? This question is so central that the two contributions on quarkonia suppression in heavy ion collisions we heard at this conference discussed it in details 3 , 4 . In this paper, I focus on cold nuclear matter only, on most of the available p+A like collisions (from SPS, FNAL, HERA-B and RHIC) and on all the heavy flavour observables ( J / ψ 𝐽 𝜓 J/\psi , ψ ′ superscript 𝜓 ′ \psi^{\prime} , Υ Υ \Upsilon and open charm…). In the following sections, I show the various dependencies of the heavy flavours productions, as they appeared chronologically. But before to look at data, one needs to define a somewhat arbitrary boundary between what we call hot and cold. Initial state effects (EMC effect, parton antishadowing, shadowing or saturation, energy loss and Cronin effect, intrinsic charm…) obviously belong to the cold sector. Among the final state effects, absorption by incoming nucleons is usually classified as a cold effect too. On the contrary, absorption by comovers (of often unknown partonic or hadronic nature) is an effect of the hot matter. Obviously, so are the long awaited colour screening and quark recombination or coalescence. In a short and pragmatic way, I define as cold what can be grasped in p+A like collisions, while hot is what happens in addition when one looks at A+A collisions. In this paper, we shall then mostly look at p+A like collisions.","role":"user"},{"content":"Abstract: Before wondering about the quark–gluon plasma (QGP), one has to take into account various cold (normal) nuclear matter effects, that can be probed through p+A like collisions. This paper aims at reviewing the current results (and understanding) of these effects on heavy quarks and quarkonia production.","role":"assistant"}]

REVIEWS AND PERSPECTIVES ON BLACK HOLE COMPLEMENTARITY

The black hole information loss problem is one of the most important issues in modern physics. If there is information loss, it may imply the fundamental limitation of the quantum theory of gravity. Thus, there is almost a consensus that accepts the conservation of information. The next natural question becomes, how can information come out of a black hole? Motivation and presence of black hole complementarity Let us assume two facts: first, the unitarity of quantum mechanics; second, the area of a black hole as its entropy. One may guess that, from the first assumption, a local observer should reconstruct all bits of information. If we accept the second assumption, from the information theoretical consideration, one notices that some information should come out of a black hole even if the black hole is sufficiently large. Even when a black hole is big, it should emit information by Hawking radiation after the information retention time. However, we know that free-falling information will touch the singularity, and information should be located at the center of the black hole. Then, the free-falling information is located at the center, and, at the same time, the information is located on the outside via the Hawking radiation. Of course, this is impossible according to the no cloning theorem. Now, black hole complementarity says that this can happen if there is no witness of the violation of natural laws. If the free-falling information and the Hawking radiation cannot communicate forever, there will be essentially no problem. This can resolve the information loss paradox in a fascinating way. As this author can safely assert, black hole complementarity is a quite general consensus between string theorists. Complementarity was a crucial step toward the speculation that produced the holographic principle. Moreover, during the 1990s, some important techniques were developed to realize the holographic principle: the D-brane picture and AdS/CFT. If AdS/CFT is true, a black hole in anti de Sitter space should be unitary. Also, from the D-brane picture, we understand that the black hole area is proportional to the statistical entropy. Then, because of some arguments from the information theory, black hole complementarity must be implemented in a black hole. Duplication experiment Black hole complementarity is a falsifiable hypothesis. In other words, it assumes that there is no witness of the duplication of information. Is this really true? For a Schwarzschild black hole, this could be checked. First, the outside observer cannot see the duplication, since the Hawking radiation is generated from the event horizon. Second, the inside observer cannot see the duplication, since the inside observer should collapse to the singularity quite quickly. 2 2 2 To see the duplication, the in-falling information should send a signal to the out-going direction during time Δ ​ t ∼ exp ⁡ ( − M 2 ) similar-to Δ 𝑡 superscript 𝑀 2 \Delta t\sim\exp(-M^{2}) , where M 𝑀 M is the black hole mass. Then, from the uncertainty principle, to send a signal with quantum information during Δ ​ t Δ 𝑡 \Delta t , it needs energy Δ ​ E ∼ exp ⁡ ( M 2 ) similar-to Δ 𝐸 superscript 𝑀 2 \Delta E\sim\exp(M^{2}) . However, it is greater than the black hole’s mass M 𝑀 M itself, and this thought experiment seems to be impossible. However, one can notice that this argument about the duplication experiment assumes some accidental facts: first, it assumes the singularity; second, it assumes that the Hawking radiation is generated at the event horizon. However, these assumptions are not true in general. In this paper, the author will discuss two models that violate one of these assumptions. First, we will drop the first assumption, and discuss a regular black hole. Second, we will drop the second assumption, and discuss a charged black hole.

If black hole complementarity is the correct idea to resolve the information loss problem, it should apply to general black holes. We suggest two models: Frolov, Markov, and Mukhanov's regular black hole and a charged black hole. These models can work as counterexamples to black hole complementarity. It has been mentioned that a large number of massless fields is an important condition to justify these models. The invalidity of this principle may imply that the holographic principle must be re-interpreted; the information loss problem, as well, should be re-considered.

[{"content":"Help me to generate the abstract of a survey paper given the title:*REVIEWS AND PERSPECTIVES ON BLACK HOLE COMPLEMENTARITY* and the introduction:The black hole information loss problem is one of the most important issues in modern physics. If there is information loss, it may imply the fundamental limitation of the quantum theory of gravity. Thus, there is almost a consensus that accepts the conservation of information. The next natural question becomes, how can information come out of a black hole? Motivation and presence of black hole complementarity Let us assume two facts: first, the unitarity of quantum mechanics; second, the area of a black hole as its entropy. One may guess that, from the first assumption, a local observer should reconstruct all bits of information. If we accept the second assumption, from the information theoretical consideration, one notices that some information should come out of a black hole even if the black hole is sufficiently large. Even when a black hole is big, it should emit information by Hawking radiation after the information retention time. However, we know that free-falling information will touch the singularity, and information should be located at the center of the black hole. Then, the free-falling information is located at the center, and, at the same time, the information is located on the outside via the Hawking radiation. Of course, this is impossible according to the no cloning theorem. Now, black hole complementarity says that this can happen if there is no witness of the violation of natural laws. If the free-falling information and the Hawking radiation cannot communicate forever, there will be essentially no problem. This can resolve the information loss paradox in a fascinating way. As this author can safely assert, black hole complementarity is a quite general consensus between string theorists. Complementarity was a crucial step toward the speculation that produced the holographic principle. Moreover, during the 1990s, some important techniques were developed to realize the holographic principle: the D-brane picture and AdS/CFT. If AdS/CFT is true, a black hole in anti de Sitter space should be unitary. Also, from the D-brane picture, we understand that the black hole area is proportional to the statistical entropy. Then, because of some arguments from the information theory, black hole complementarity must be implemented in a black hole. Duplication experiment Black hole complementarity is a falsifiable hypothesis. In other words, it assumes that there is no witness of the duplication of information. Is this really true? For a Schwarzschild black hole, this could be checked. First, the outside observer cannot see the duplication, since the Hawking radiation is generated from the event horizon. Second, the inside observer cannot see the duplication, since the inside observer should collapse to the singularity quite quickly. 2 2 2 To see the duplication, the in-falling information should send a signal to the out-going direction during time Δ ​ t ∼ exp ⁡ ( − M 2 ) similar-to Δ 𝑡 superscript 𝑀 2 \Delta t\sim\exp(-M^{2}) , where M 𝑀 M is the black hole mass. Then, from the uncertainty principle, to send a signal with quantum information during Δ ​ t Δ 𝑡 \Delta t , it needs energy Δ ​ E ∼ exp ⁡ ( M 2 ) similar-to Δ 𝐸 superscript 𝑀 2 \Delta E\sim\exp(M^{2}) . However, it is greater than the black holes mass M 𝑀 M itself, and this thought experiment seems to be impossible. However, one can notice that this argument about the duplication experiment assumes some accidental facts: first, it assumes the singularity; second, it assumes that the Hawking radiation is generated at the event horizon. However, these assumptions are not true in general. In this paper, the author will discuss two models that violate one of these assumptions. First, we will drop the first assumption, and discuss a regular black hole. Second, we will drop the second assumption, and discuss a charged black hole.","role":"user"},{"content":"Abstract: If black hole complementarity is the correct idea to resolve the information loss problem, it should apply to general black holes. We suggest two models: Frolov, Markov, and Mukhanov's regular black hole and a charged black hole. These models can work as counterexamples to black hole complementarity. It has been mentioned that a large number of massless fields is an important condition to justify these models. The invalidity of this principle may imply that the holographic principle must be re-interpreted; the information loss problem, as well, should be re-considered.","role":"assistant"}]

Self-Consistent-Field Method and τ-Functional Method on Group Manifold in Soliton Theory: a Review and New Results

1.1 Historical background on microscopic study of nuclear collective motions A standard description of fermion many-body systems starts with the most basic approximation that is based on an independent-particle (IP) picture, i.e., self-consistent-field (SCF) approximation for the fermions. The Hartree–Fock theory (HFT) is typical one of such an approximation for ground states of the systems. Excited states are treated with the random phase approximation (RPA). The time dependent Hartree–Fock (TDHF) equation and time dependent Hartree–Bogoliubov (TDHB) equation are nonlinear equations owing to their SCF characters and may have no unique solution. The HFT and HBT are given by variational method to optimize energy expectation value by a Slater determinant (S-det) and an HB wave function, respectively [ 3 ] . Particle-hole (p-h) operators of the fermions with N 𝑁 N single-particle states form a Lie algebra u ​ ( N ) 𝑢 𝑁 u(N) [ 4 ] and generate a Thouless transformation [ 5 ] which induces a representation of the corresponding Lie group U ​ ( N ) 𝑈 𝑁 U(N) . The U ​ ( N ) 𝑈 𝑁 U(N) canonical transformation transforms an S-det with M 𝑀 M particles to another S-det. Any S-det is obtained by such a transformation of a given reference S-det, i.e., Thouless theorem provides an exact wave function of fermion state vector which is the generalized coherent state representation (CS rep) of U ​ ( N ) 𝑈 𝑁 U(N) Lie group [ 6 ] . Following Yamamura and Kuriyama [ 7 ] , we give a brief history of methods extracting collective motions out of fully parametrized TDHF/TDHB manifolds in SCF. Arvieu and Veneroni, and Baranger and independently Marumori have proposed a theory for spherical even nuclei [ 8 ] called quasi-particle RPA (QRPA) and it has been a standard approximation for the excited states of the systems. In nuclei, there exist a short-range correlation and a long-range one [ 9 ] . The former is induced by a pairing interaction and generates a superconducting state. The excited state is classified by a seniority-scheme and described in terms of quasi-particles given by the BCS-Bogoliubov theory [ 10 ] . The latter is occured by p-h interactions and gives rise to collective motions related to a density fluctuation around equilibrium states. The p-h RPA (RPA) describes such collective motions like vibrational and rotational motions. It, however, stands on a harmonic approximation and should be extended to take some nonlinear effects into account. To solve such a problem, the boson expansion HB theory (BEHBT) has been developed by Belyaev and Zelevinsky [ 11 ] , and Marumori, Yamamura and Tokunaga [ 12 ] . The essence of the BEHBT is to express the fermion-pairs in terms of boson operators keeping a pure boson-character. The boson representation is constructed to reproduce the Lie algebra of the fermion-pairs. The state vector in the fermion Fock space corresponds to the one in the boson Fock space by one-to-one mapping. Such a boson representation makes any transition-matrix-valued quantity for the boson-state vectors coincide with that for the fermion-state ones. The algebra of fermion-pairs and the boson representation have been extensively investigated. The fermion-pairs form an algebra s ​ o ​ ( 2 ​ N ) 𝑠 𝑜 2 𝑁 so(2N) . As for the boson representation, e.g., da Providência and Weneser and Marshalek [ 13 ] have proposed boson operators basing on p-h pairs forming an algebra s ​ u ​ ( N ) 𝑠 𝑢 𝑁 su(N) . By Fukutome, Yamamura and Nishiyama [ 14 , 15 ] , the fermions were found to span the algebras s ​ o ​ ( 2 ​ N + 1 ) 𝑠 𝑜 2 𝑁 1 so(2N\!+\!1) and s ​ o ​ ( 2 ​ N + 2 ) 𝑠 𝑜 2 𝑁 2 so(2N\!+\!2) accompanying with u ​ ( N + 1 ) 𝑢 𝑁 1 u(N\!+\!1) . The BET expressed by Schwinger-type and Dyson-type bosons has been intensively studied by Fukutome and Nishiyama [ 16 , 17 , 18 , 19 ] . However, the above BET’s themselves do not contain any scheme under which collective degree of freedom can be selected from the whole degrees of freedom. On the contrary, we have a traditional approach to the microscopic theory of collective motion, the TDHF theory (TDHFT) and TDHB theory (TDHBT), e.g., [ 20 , 21 ] . The pioneering idea of the TDHFT was suggested by Marumori [ 8 ] for the case of small amplitude vibrational motions. Using this idea, one can determine the time dependence of any physical quantity, e.g., frequency of the small fluctuation around a static HF/HB field. The equation for the frequency has the same form as that given by RPA. A quantum energy given by this method means an excitation energy of the first excited state. Then, the RPA is a possible quantization of the TDHFT/TDHBT in the small amplitude limit. In fact, as was proved by Marshalek and Horzwarth [ 22 ] , the BEHBT is reduced to the TDHBT under the replacement of boson operators with classical canonical variables. Using a canonical transformation in a classical mechanics, it is expected to obtain a scheme for choosing the collective degree of freedom in the SCF. Historically, there was another stream, i.e., an adiabatic perturbation approach. This approach starts from an assumption that the speed of collective motion is much slower than that of any other non-collective motion. At an early stage of the study of this stream, the adiabatic treatment of the TDHFT (ATDHFT) was presented by Thouless and Valatin [ 23 ] . Such a theory has a feature common to the one of the theory for large-amplitude collective motion. Later the ATDHFT was developed mainly by Baranger and Veneroni, Brink, Villars, Goeke and Reinhard, and Mukherjee and Pal [ 24 ] . The most important point of the ATDHFT by Villars is in introducing a “collective path” into a phase space. A collective motion corresponds to a trajectory in the phase space which moves along the collective path. Standing on the same spirit, Holtzwarth and Yukawa, Rowe and Bassermann [ 25 ] , gave the TDHFT and Marumori, Maskawa, Sakata and Kuriyama so-called “maximally decoupled” method in a canonical form [ 26 ] . So, various techniques of classical mechanics are useful and then canonical quantization is expected. By solving the equation of collective path, one can obtain some corrections to the TDHF result. The TDHFT has a possibility to illustrate not only collective modes but also intrinsic modes. However, the following three points remain to be solved yet: (i) to determine a microscopic structure of collective motion, which may be a superposition of each particle motion, in relation to dynamics under consideration (ii) to determine IP motion which should be orthogonal to collective motion and (iii) to give a coupling between both the motions. The canonical-formed TDHFT enables us to select the collective motion in relation to the dynamics, though it makes no role to take IP motion into account, because the TD S-det contains only canonical variables to represent the collective motion. Along the same way as the TDHFT, Yamamura and Kuriyama have extended the TDHFT to that on a fermion CS constructed on the TD S-det. The CS rep contains not only the usual canonical variables but also the Grassmann variables. A classical image of fermions can be obtained by regarding the Grassmann variables as canonical ones [ 27 ] . The constraints governing the variables to remove the overcounted degrees of freedom were decided under the physical consideration. Owing to the Dirac’s canonical theory for a constrained system, the TDHFT was successfully developed for a unified description of collective and IP motions in the classical mechanics [ 28 ] . 1.2 Viewpoint of symmetry of evolution equations The TDHF/TDHB can be summarized to find optimal coordinate-systems on a group manifold basing on Lie algebras of the finite-dimensional fermion-pairs and to describe dynamics on the manifold. The boson operators in BET are generators occurring in the coordinate system of tangent space on the manifold in the fermion Fock space. But the BET’s themselves do not contain any scheme under which collective degrees of freedom can be selected from the whole degrees of freedom. Approaches to collective motions by the TDHFT suggest that the coordinate system on which collective motions is describable deeply relates not only to the global symmetry of the finite-dimensional group manifold itself but also to hidden local symmetries , besides the Hamiltonian. Various collective motions may be well understood by taking the local symmetries into account. The local symmetries may be closely connected with infinite-dimensional Lie algebras. However, there has been little attempts to manifestly understand collective motions in relation to the local symmetries. From the viewpoint of symmetry of evolution equations, we will study the algebro-geometric structures toward a unified understanding of both the collective and IP motions. The first issue is to investigate fundamental “curvature equations” to extract collective submanifolds out of the full TDHF/TDHB manifold. We show that the expression in a quasi-particle frame (QPF) of the zero-curvature equations described later becomes the nonlinear RPA which is the natural extension of the usual RPA. We abbreviate RPA and QRPA to only RPA. We had at first started from a question whether soliton equations exist in the TDHF/TDHB manifold or do not, in spite of the difference that the solitons are described in terms of infinite degrees of freedom and the RPA in terms of finite ones. We had met with the inverse-scattering-transform method by AKNS [ 29 ] and the differential geometrical approaches on group manifolds [ 30 ] . An integrable system is explained by the zero-curvature, i.e., integrability condition of connection on the corresponding Lie group. Approaches to collective motions had been little from the viewpoint of the curvature. If a collective submanifold is a collection of collective paths, an infinitesimal condition to transfer a path to another may be nothing but the integrability condition for the submanifold with respect to a parameter time t 𝑡 t describing a trajectory of an SCF Hamiltonian and to other parameters specifying any point on the submanifold. However the trajectory of the SCF Hamiltonian is unable to remain on the manifold. Then the curvature may be able to work as a criterion of effectiveness of the collective submanifold. From a wide viewpoint of symmetry the RPA is extended to any point on the manifold because an equilibrium state which we select as a starting point must be equipotent with any other point on the manifold. The well-known RPA had been introduced as a linear approximation to treat excited states around a ground state (the equilibrium state), which is essentially a harmonic approximation. When an amplitude of oscillation becomes larger and then an anharmonicity appears, then we have to treat the anharmonicity by taking nonlinear effects in the equation of motion into account. It is shown that equations defining the curvature of the collective submanifold becomes fundamental equations to treat the anharmonicity. We call them “the formal RPA equation”. It will be useful to understand algebro-geometric meanings of large-amplitude collective motions. The second issue is to go beyond the perturbative method with respect to the collective variables [ 26 ] . For this aim, we investigate an interrelation between the SCF method (SCFM) extracting collective motions and τ 𝜏 \tau -functional method ( τ 𝜏 \tau -FM) [ 31 ] constructing integrable equations in solitons. In a soliton theory on a group manifold, transformation groups governing solutions for soliton equations become infinite-dimensional Lie groups whose generators of the corresponding Lie algebras are expressed as infinite-order differential operators of affine Kac–Moody algebras. An infinite-dimensional fermion Fock space F ∞ subscript 𝐹 F_{\infty} is realized in terms of a space of complex polynomial algebra. The infinite-dimensional fermions are given in terms of the infinite-order differential operators and the soliton equation is nothing but the differential equation to determine the group orbit of the highest weight vector in the F ∞ subscript 𝐹 F_{\infty} [ 31 ] . The generalized CS rep gives a key to elucidate relationship of a HF wave function to a τ 𝜏 \tau -function in the soliton theory. This has been pointed out first by D’Ariano and Rasetti [ 32 ] for an infinite-dimensional harmonic electron gas. Standing on their observation, for the SCFM one can give a theoretical frame for an integrable sub-dynamics on an abstract F ∞ subscript 𝐹 F_{\infty} . The relation between SCFM in finite-dimensional fermions and τ 𝜏 \tau -FM in infinite ones, however, has not been investigated because dynamical descriptions of fermion systems by them have looked very different manners. In the papers [ 33 , 34 , 35 , 36 , 2 ] , we have first tried to clarify it using SCFM on U ​ ( N ) 𝑈 𝑁 U(N) group and τ 𝜏 \tau -FM on that group. To attain this object we will have to solve the following main problems: first, how we embed the finite-dimensional fermion system into a certain infinite one and how we rebuilt the TDHFT on it; second, how any algebraic mechanism works behind particle and collective motions and how any relation between collective variables and a spectral parameter in soliton theory is there; last, how the SCF Hamiltonian selects various subgroup-orbits and how a collective submanifold is made from them and further how the submanifold relates to the formal RPA. To understand microscopically cooperative phenomena, the concept of collective motion is introduced in relation to a TD variation of SC mean-field. IP motion is described in terms of particles referring to a stationary mean-field. The variation of a TD mean-field gives rise to couplings between collective and IP motions and couplings among quantum fluctuations of the TD mean-field itself [ 7 ] , while in τ 𝜏 \tau -FM a soliton equation is derived as follows: Consider an infinite-dimensional Lie algebra and its representation on a functional space. The group-orbit of the highest weight vector becomes an infinite-dimensional Grassmannian G ∞ subscript 𝐺 G_{\!\infty} . The bilinear equation (Plücker relation) is nothing else than the soliton equation. This means that a solution space of the soliton equation corresponds to a group-orbit of the vacuum state. The SCFM does not use the Plücker relation in the context of a bilinear differential equation defining finite-dimensional Grassmannian G M subscript 𝐺 𝑀 G_{M} but seems to use implicitly such a relation. In the SCFM a physical concept of quasi-particle and vacuum and a coset space is used instead. If we develop a perturbative theory for large-amplitude collective motion [ 26 ] , an infinite-dimensional Lie algebra might been necessarily used. The sub-group orbits consisting of several loop-group paths [ 37 ] classified by the Plücker relation exist innumerably in G M subscript 𝐺 𝑀 G_{M} so that the SCFM is related to the soliton theory in G ∞ subscript 𝐺 G_{\infty} . The Plücker relation in a coset space U ​ ( N ) U ​ ( M ) × U ​ ( N − M ) 𝑈 𝑁 𝑈 𝑀 𝑈 𝑁 𝑀 \frac{U(N)}{U(M)\times U(N-M)} [ 38 ] becomes analogous with the Hirota’s bilinear form [ 39 , 40 ] . Toward an ultimate goal we aim to reconstruct a theoretical frame for a υ 𝜐 \upsilon (external parameter)-dependent SCFM to describe more precisely the dynamics on the F ∞ subscript 𝐹 F_{\infty} . In the abstract fermion Fock space, we find common features in both SCFM and τ 𝜏 \tau -FM. (i) Each solution space is described as Grassmannian that is group orbit of the corresponding vacuum state. (ii) The former may implicitly explain the Plücker relation not in terms of bilinear differential equations defining G M subscript 𝐺 𝑀 G_{M} but in terms of the physical concept of quasi-particle and vacuum and mathematical language of coset space and coset variable. The various BETs are built on the Plücker relation to hold the Grassmannian. The latter asserts that the soliton equations are nothing but the bilinear differential equations giving a boson representation of the Plücker relation . The relation, however, has been unsatisfactorily investigated yet within the framework of the usual SCFM. We study it and show that both the methods stand on the common features, Plücker relation or bilinear differential equation defining the Grassmannian. On the contrary, we observe different points: (i) The former is built on a finite -dimensional Lie algebra but the latter on an infinite -dimensional one. (ii) The former has an SCF Hamiltonian consisting of a fermion one-body operator, which is derived from a functional derivative of an expectation value of a fermion Hamiltonian by a ground-state wave function. The latter introduces artificially a fermion Hamiltonian of one-body type operator as a boson mapping operator from states on fermion Fock space to corresponding ones on τ 𝜏 \tau -functional space ( τ 𝜏 \tau -FS). The last issue is, despite a difference due to the dimension of fermions, to aim at obtaining a close connection between concept of mean-field potential and gauge of fermions inherent in the SCFM and at making a role of a loop group [ 37 ] to be clear. Through the observation, we construct infinite-dimensional fermion operators from the finite-dimensional ones by Laurent expansion with respect to a circle S 1 superscript 𝑆 1 S^{1} . Then with the use of an affine Kac–Moody (KM) algebra according to the idea of Dirac’s positron theory [ 41 ] , we rebuilt a TDHFT in F ∞ subscript 𝐹 F_{\infty} . The TDHFT results in a gauge theory of fermions and the collective motion, fluctuation of the mean-field potential, appears as the motion of fermion gauges with a common factor. The physical concept of the quasi-particle and vacuum in the SCFM on the S 1 superscript 𝑆 1 S^{1} connects to the “Plücker relations” due to the Dirac theory, in other words, the algebraic mechanism extracting various sub-group orbits consisting of loop path out of the full TDHF manifold is just the “Hirota’s bilinear form” [ 40 ] which is an s ​ u ​ ( N ) ( ∈ s ​ l ​ ( N ) ) annotated 𝑠 𝑢 𝑁 absent 𝑠 𝑙 𝑁 su(N)(\in sl(N)) reduction of g ​ l ​ ( N ) 𝑔 𝑙 𝑁 gl(N) in the τ 𝜏 \tau -FM. As a result, it is shown that an infinite-dimensional fermion many-body system is also realizable in a finite-dimensional one and that roles of the soliton equation (Plücker relation) and the TDHF equation are made clear. We also understand an SCF dynamics through gauge of interacting infinite-dimensional fermions. A bilinear equation for the υ 𝜐 \upsilon -HFT has been transcribed onto the corresponding τ 𝜏 \tau -function using the regular representation for the group and the Schur polynomials. The υ 𝜐 \upsilon -HF SCFM on an infinite-dimensional Fock space F ∞ subscript 𝐹 F_{\infty} leads to a dynamics on an infinite-dimensional Grassmannian Gr ∞ subscript Gr {\rm Gr}_{\infty} and may describe more precisely such a dynamics on the group manifold. A finite-dimensional Grassmannian is identified with a Gr ∞ subscript Gr {\rm Gr}_{\infty} which is affiliated with the group manifold obtained by reducting g ​ l ​ ( ∞ ) 𝑔 𝑙 gl(\infty) to s ​ l ​ ( N ) 𝑠 𝑙 𝑁 sl(N) and s ​ u ​ ( N ) 𝑠 𝑢 𝑁 su(N) . We have given explicit expressions for Laurent coefficients of soliton solutions for s ​ l ^ ​ ( N ) ^ 𝑠 𝑙 𝑁 \widehat{sl}(N) and s ​ u ^ ​ ( N ) ^ 𝑠 𝑢 𝑁 \widehat{su}(N) on the Gr ∞ subscript Gr {\rm Gr}_{\infty} using Chevalley bases for s ​ l ​ ( N ) 𝑠 𝑙 𝑁 sl(N) and s ​ u ​ ( N ) 𝑠 𝑢 𝑁 su(N) [ 42 ] . As an illustration we will attempt to make a υ 𝜐 \upsilon -HFT approach to an infinite-dimensional matrix model extended from the finite-dimensional s ​ u ​ ( 2 ) 𝑠 𝑢 2 su(2) Lipkin–Meshkov–Glick (LMG) model [ 43 ] . For this aim, we give an affine KM algebra s ​ l ^ ​ ( 2 , C ) ^ 𝑠 𝑙 2 𝐶 \widehat{sl}(2,C) (complexification of s ​ u ^ ( 2 ) ) \widehat{su}(2)) to which the LMG generators subject, and their τ τ \uptau representations and the σ K subscript 𝜎 𝐾 \sigma_{K} mappings for them. We can represent an infinite-dimensional matrix of the LMG Hamiltonian and its HF Hamiltonian in terms of the Schur polynomials. Its infinite-dimensional HF operator is also given through the mapping σ M subscript 𝜎 𝑀 \sigma_{M} for ψ i ​ ψ j ∗ subscript 𝜓 𝑖 superscript subscript 𝜓 𝑗 \psi_{i}\psi_{j}^{*} of infinite-dimensional fermions ψ i subscript 𝜓 𝑖 \psi_{i} and ψ i ∗ superscript subscript 𝜓 𝑖 \psi_{i}^{*} , which is expressed by the Schur polynomials S k ​ ( x ) subscript 𝑆 𝑘 𝑥 S_{k}(x) and S k ​ ( ∂ x ) subscript 𝑆 𝑘 subscript 𝑥 S_{k}(\partial_{x}) . Further its τ 𝜏 \tau -function for a simple case is provided by the Plücker coordinates and Schur polynomials. In Section 2 , we propose curvature equations as fundamental equations to extract a collective submanifold out of the full TDHB manifold. Basing on these ideas, we construct the curvature equations and study the relation between the maximal decoupled method and the curvature equations. We further investigate the role of the non-zero curvature arising from the residual Hamiltonian. Making use of the expression of the zero-curvature equations in the QPF, we find the formal RPA equation. In Section 3 , we present a simply unified aspect for the SCFM and the τ 𝜏 \tau -FM and show a simple idea connecting both the methods. We study the algebraic relation between coset coordinate and Plücker coordinate. Basing on the above idea, we attempt to rebuilt the TDHFT in τ 𝜏 \tau -FS. We introduce υ 𝜐 \upsilon -dependent infinite-dimensional fermion operators and a F ∞ subscript 𝐹 F_{\infty} through Laurent expansion with respect to the degrees of freedom of the original fermions. The algebraic relation between both the methods is manifestly described. We embed a HF u ​ ( N ) 𝑢 𝑁 u(N) Lie algebra into a g ​ l ​ ( ∞ ) 𝑔 𝑙 gl(\infty) by means of infinite-dimensional fermions. The υ 𝜐 \upsilon -SCFM in τ 𝜏 \tau -FS is developed. The role of the shift operators in the τ 𝜏 \tau -FM is studied. As an illustration, explicit expressions for Laurent coefficients of soliton solutions for s ​ l ^ ​ ( N ) ^ 𝑠 𝑙 𝑁 \widehat{sl}(N) and s ​ u ^ ​ ( N ) ^ 𝑠 𝑢 𝑁 \widehat{su}(N) are presented. A problem related to a nonlinear Schrödinger equation is also discussed. In Section 4 , we construct a formal RPA equation on F ∞ subscript 𝐹 F_{\infty} and also argue about the relation between a l ​ o ​ o ​ p 𝑙 𝑜 𝑜 𝑝 loop collective path and a formal RPAEQ. Consequently, it can be proved that the usual perturbative method with respect to periodic collective variables in the TDHFT is involved in the present method which aims for constructing the TDHFT on the affine KM algebra. In Section 5 , we introduce infinite-dimensional “particle” and “hole” operators and operators K ^ 0 subscript ^ 𝐾 0 {\widehat{K}}_{0} and K ^ ± subscript ^ 𝐾 plus-or-minus {\widehat{K}}_{\pm} defined by infinite-dimensional “particle-hole” pair operators. Using these operators, we construct an infinite-dimensional Heisenberg subalgebra of the affine KM algebra s ​ l ^ ​ ( 2 , C ) ^ 𝑠 𝑙 2 𝐶 \widehat{sl}(2,C) . The LMG Hamiltonian and its HF Hamiltonian are expressed in terms of the Heisenberg basic-elements whose representations are isomorphic to those in the corresponding boson space. They are given in terms of infinite numbers of variables x k subscript 𝑥 𝑘 x_{k} and derivatives ∂ x k subscript subscript 𝑥 𝑘 \partial_{x_{k}} through the Schur polynomials S k ​ ( x ) subscript 𝑆 𝑘 𝑥 S_{k}(x) . We give also an infinite-dimensional representation of S ​ U ​ ( 2 ​ N ) ∞ 𝑆 𝑈 subscript 2 𝑁 SU(2N)_{\infty} transformation of the particle and hole operators. Finally, in Section 6 , we summarize and discuss the results and future problems. We give Appendices A – J . Especially in Appendix J , we show an explicit expression for Plücker coordinate for the LMG model and calculate a quantity, det ( 1 N + p † p ) ( p : coset variable ) \det(1_{N}+p^{\dagger}p)~{}(p:\mbox{coset variable}) for the LMG model, in terms of the Schur polynomials.

The maximally-decoupled method has been considered as a theory to apply an basic idea of an integrability condition to certain multiple parametrized symmetries. The method is regarded as a mathematical tool to describe a symmetry of a collective subma- nifold in which a canonicity condition makes the collective variables to be an orthogonal coordinate-system. For this aim we adopt a concept of curvature unfamiliar in the con- ventional time-dependent (TD) self-consistent field (SCF) theory. Our basic idea lies in the introduction of a sort of Lagrange manner familiar to fluid dynamics to describe a collective coordinate-system. This manner enables us to take a one-form which is linearly composed of a TD SCF Hamiltonian and infinitesimal generators induced by collective vari- able differentials of a canonical transformation on a group. The integrability condition of the system read the curvature C = 0. Our method is constructed manifesting itself the structure of the group under consideration. To go beyond the maximaly-decoupled method, we have aimed to construct an SCF theory, i.e., (external parameter)-dependent Hartree-Fock (HF) theory. Toward such an ultimate goal, the -HF theory has been recon- structed on an affine Kac-Moody algebra along the soliton theory, using infinite-dimensional fermion. An infinite-dimensional fermion operator is introduced through a Laurent ex- pansion of finite-dimensional fermion operators with respect to degrees of freedom of the fermions related to a -dependent potential with a -periodicity. A bilinear equation for the -HF theory has been transcribed onto the corresponding -function using the regu- lar representation for the group and the Schur-polynomials. The -HF SCF theory on an infinite-dimensional Fock space F1 leads to a dynamics on an infinite-dimensional Grass- mannian Gr1 and may describe more precisely such a dynamics on the group manifold. A finite-dimensional Grassmannian is identified with a Gr1 which is affiliated with the group manifold obtained by reducting gl(1) to sl(N) and su(N). As an illustration we will study an infinite-dimensional matrix model extended from the finite-dimensional su(2) Lipkin-Meshkov-Glick model which is a famous exactly-solvable model.

[{"content":"Help me to generate the abstract of a survey paper given the title:*Self-Consistent-Field Method and τ-Functional Method on Group Manifold in Soliton Theory: a Review and New Results* and the introduction:1.1 Historical background on microscopic study of nuclear collective motions A standard description of fermion many-body systems starts with the most basic approximation that is based on an independent-particle (IP) picture, i.e., self-consistent-field (SCF) approximation for the fermions. The Hartree–Fock theory (HFT) is typical one of such an approximation for ground states of the systems. Excited states are treated with the random phase approximation (RPA). The time dependent Hartree–Fock (TDHF) equation and time dependent Hartree–Bogoliubov (TDHB) equation are nonlinear equations owing to their SCF characters and may have no unique solution. The HFT and HBT are given by variational method to optimize energy expectation value by a Slater determinant (S-det) and an HB wave function, respectively 3 . Particle-hole (p-h) operators of the fermions with N 𝑁 N single-particle states form a Lie algebra u ​ ( N ) 𝑢 𝑁 u(N) 4 and generate a Thouless transformation 5 which induces a representation of the corresponding Lie group U ​ ( N ) 𝑈 𝑁 U(N) . The U ​ ( N ) 𝑈 𝑁 U(N) canonical transformation transforms an S-det with M 𝑀 M particles to another S-det. Any S-det is obtained by such a transformation of a given reference S-det, i.e., Thouless theorem provides an exact wave function of fermion state vector which is the generalized coherent state representation (CS rep) of U ​ ( N ) 𝑈 𝑁 U(N) Lie group 6 . Following Yamamura and Kuriyama 7 , we give a brief history of methods extracting collective motions out of fully parametrized TDHF/TDHB manifolds in SCF. Arvieu and Veneroni, and Baranger and independently Marumori have proposed a theory for spherical even nuclei 8 called quasi-particle RPA (QRPA) and it has been a standard approximation for the excited states of the systems. In nuclei, there exist a short-range correlation and a long-range one 9 . The former is induced by a pairing interaction and generates a superconducting state. The excited state is classified by a seniority-scheme and described in terms of quasi-particles given by the BCS-Bogoliubov theory 10 . The latter is occured by p-h interactions and gives rise to collective motions related to a density fluctuation around equilibrium states. The p-h RPA (RPA) describes such collective motions like vibrational and rotational motions. It, however, stands on a harmonic approximation and should be extended to take some nonlinear effects into account. To solve such a problem, the boson expansion HB theory (BEHBT) has been developed by Belyaev and Zelevinsky 11 , and Marumori, Yamamura and Tokunaga 12 . The essence of the BEHBT is to express the fermion-pairs in terms of boson operators keeping a pure boson-character. The boson representation is constructed to reproduce the Lie algebra of the fermion-pairs. The state vector in the fermion Fock space corresponds to the one in the boson Fock space by one-to-one mapping. Such a boson representation makes any transition-matrix-valued quantity for the boson-state vectors coincide with that for the fermion-state ones. The algebra of fermion-pairs and the boson representation have been extensively investigated. The fermion-pairs form an algebra s ​ o ​ ( 2 ​ N ) 𝑠 𝑜 2 𝑁 so(2N) . As for the boson representation, e.g., da Providência and Weneser and Marshalek 13 have proposed boson operators basing on p-h pairs forming an algebra s ​ u ​ ( N ) 𝑠 𝑢 𝑁 su(N) . By Fukutome, Yamamura and Nishiyama 14 , 15 , the fermions were found to span the algebras s ​ o ​ ( 2 ​ N + 1 ) 𝑠 𝑜 2 𝑁 1 so(2N\!+\!1) and s ​ o ​ ( 2 ​ N + 2 ) 𝑠 𝑜 2 𝑁 2 so(2N\!+\!2) accompanying with u ​ ( N + 1 ) 𝑢 𝑁 1 u(N\!+\!1) . The BET expressed by Schwinger-type and Dyson-type bosons has been intensively studied by Fukutome and Nishiyama 16 , 17 , 18 , 19 . However, the above BETs themselves do not contain any scheme under which collective degree of freedom can be selected from the whole degrees of freedom. On the contrary, we have a traditional approach to the microscopic theory of collective motion, the TDHF theory (TDHFT) and TDHB theory (TDHBT), e.g., 20 , 21 . The pioneering idea of the TDHFT was suggested by Marumori 8 for the case of small amplitude vibrational motions. Using this idea, one can determine the time dependence of any physical quantity, e.g., frequency of the small fluctuation around a static HF/HB field. The equation for the frequency has the same form as that given by RPA. A quantum energy given by this method means an excitation energy of the first excited state. Then, the RPA is a possible quantization of the TDHFT/TDHBT in the small amplitude limit. In fact, as was proved by Marshalek and Horzwarth 22 , the BEHBT is reduced to the TDHBT under the replacement of boson operators with classical canonical variables. Using a canonical transformation in a classical mechanics, it is expected to obtain a scheme for choosing the collective degree of freedom in the SCF. Historically, there was another stream, i.e., an adiabatic perturbation approach. This approach starts from an assumption that the speed of collective motion is much slower than that of any other non-collective motion. At an early stage of the study of this stream, the adiabatic treatment of the TDHFT (ATDHFT) was presented by Thouless and Valatin 23 . Such a theory has a feature common to the one of the theory for large-amplitude collective motion. Later the ATDHFT was developed mainly by Baranger and Veneroni, Brink, Villars, Goeke and Reinhard, and Mukherjee and Pal 24 . The most important point of the ATDHFT by Villars is in introducing a collective path into a phase space. A collective motion corresponds to a trajectory in the phase space which moves along the collective path. Standing on the same spirit, Holtzwarth and Yukawa, Rowe and Bassermann 25 , gave the TDHFT and Marumori, Maskawa, Sakata and Kuriyama so-called maximally decoupled method in a canonical form 26 . So, various techniques of classical mechanics are useful and then canonical quantization is expected. By solving the equation of collective path, one can obtain some corrections to the TDHF result. The TDHFT has a possibility to illustrate not only collective modes but also intrinsic modes. However, the following three points remain to be solved yet: (i) to determine a microscopic structure of collective motion, which may be a superposition of each particle motion, in relation to dynamics under consideration (ii) to determine IP motion which should be orthogonal to collective motion and (iii) to give a coupling between both the motions. The canonical-formed TDHFT enables us to select the collective motion in relation to the dynamics, though it makes no role to take IP motion into account, because the TD S-det contains only canonical variables to represent the collective motion. Along the same way as the TDHFT, Yamamura and Kuriyama have extended the TDHFT to that on a fermion CS constructed on the TD S-det. The CS rep contains not only the usual canonical variables but also the Grassmann variables. A classical image of fermions can be obtained by regarding the Grassmann variables as canonical ones 27 . The constraints governing the variables to remove the overcounted degrees of freedom were decided under the physical consideration. Owing to the Diracs canonical theory for a constrained system, the TDHFT was successfully developed for a unified description of collective and IP motions in the classical mechanics 28 . 1.2 Viewpoint of symmetry of evolution equations The TDHF/TDHB can be summarized to find optimal coordinate-systems on a group manifold basing on Lie algebras of the finite-dimensional fermion-pairs and to describe dynamics on the manifold. The boson operators in BET are generators occurring in the coordinate system of tangent space on the manifold in the fermion Fock space. But the BETs themselves do not contain any scheme under which collective degrees of freedom can be selected from the whole degrees of freedom. Approaches to collective motions by the TDHFT suggest that the coordinate system on which collective motions is describable deeply relates not only to the global symmetry of the finite-dimensional group manifold itself but also to hidden local symmetries , besides the Hamiltonian. Various collective motions may be well understood by taking the local symmetries into account. The local symmetries may be closely connected with infinite-dimensional Lie algebras. However, there has been little attempts to manifestly understand collective motions in relation to the local symmetries. From the viewpoint of symmetry of evolution equations, we will study the algebro-geometric structures toward a unified understanding of both the collective and IP motions. The first issue is to investigate fundamental curvature equations to extract collective submanifolds out of the full TDHF/TDHB manifold. We show that the expression in a quasi-particle frame (QPF) of the zero-curvature equations described later becomes the nonlinear RPA which is the natural extension of the usual RPA. We abbreviate RPA and QRPA to only RPA. We had at first started from a question whether soliton equations exist in the TDHF/TDHB manifold or do not, in spite of the difference that the solitons are described in terms of infinite degrees of freedom and the RPA in terms of finite ones. We had met with the inverse-scattering-transform method by AKNS 29 and the differential geometrical approaches on group manifolds 30 . An integrable system is explained by the zero-curvature, i.e., integrability condition of connection on the corresponding Lie group. Approaches to collective motions had been little from the viewpoint of the curvature. If a collective submanifold is a collection of collective paths, an infinitesimal condition to transfer a path to another may be nothing but the integrability condition for the submanifold with respect to a parameter time t 𝑡 t describing a trajectory of an SCF Hamiltonian and to other parameters specifying any point on the submanifold. However the trajectory of the SCF Hamiltonian is unable to remain on the manifold. Then the curvature may be able to work as a criterion of effectiveness of the collective submanifold. From a wide viewpoint of symmetry the RPA is extended to any point on the manifold because an equilibrium state which we select as a starting point must be equipotent with any other point on the manifold. The well-known RPA had been introduced as a linear approximation to treat excited states around a ground state (the equilibrium state), which is essentially a harmonic approximation. When an amplitude of oscillation becomes larger and then an anharmonicity appears, then we have to treat the anharmonicity by taking nonlinear effects in the equation of motion into account. It is shown that equations defining the curvature of the collective submanifold becomes fundamental equations to treat the anharmonicity. We call them the formal RPA equation. It will be useful to understand algebro-geometric meanings of large-amplitude collective motions. The second issue is to go beyond the perturbative method with respect to the collective variables 26 . For this aim, we investigate an interrelation between the SCF method (SCFM) extracting collective motions and τ 𝜏 \tau -functional method ( τ 𝜏 \tau -FM) 31 constructing integrable equations in solitons. In a soliton theory on a group manifold, transformation groups governing solutions for soliton equations become infinite-dimensional Lie groups whose generators of the corresponding Lie algebras are expressed as infinite-order differential operators of affine Kac–Moody algebras. An infinite-dimensional fermion Fock space F ∞ subscript 𝐹 F_{\infty} is realized in terms of a space of complex polynomial algebra. The infinite-dimensional fermions are given in terms of the infinite-order differential operators and the soliton equation is nothing but the differential equation to determine the group orbit of the highest weight vector in the F ∞ subscript 𝐹 F_{\infty} 31 . The generalized CS rep gives a key to elucidate relationship of a HF wave function to a τ 𝜏 \tau -function in the soliton theory. This has been pointed out first by DAriano and Rasetti 32 for an infinite-dimensional harmonic electron gas. Standing on their observation, for the SCFM one can give a theoretical frame for an integrable sub-dynamics on an abstract F ∞ subscript 𝐹 F_{\infty} . The relation between SCFM in finite-dimensional fermions and τ 𝜏 \tau -FM in infinite ones, however, has not been investigated because dynamical descriptions of fermion systems by them have looked very different manners. In the papers 33 , 34 , 35 , 36 , 2 , we have first tried to clarify it using SCFM on U ​ ( N ) 𝑈 𝑁 U(N) group and τ 𝜏 \tau -FM on that group. To attain this object we will have to solve the following main problems: first, how we embed the finite-dimensional fermion system into a certain infinite one and how we rebuilt the TDHFT on it; second, how any algebraic mechanism works behind particle and collective motions and how any relation between collective variables and a spectral parameter in soliton theory is there; last, how the SCF Hamiltonian selects various subgroup-orbits and how a collective submanifold is made from them and further how the submanifold relates to the formal RPA. To understand microscopically cooperative phenomena, the concept of collective motion is introduced in relation to a TD variation of SC mean-field. IP motion is described in terms of particles referring to a stationary mean-field. The variation of a TD mean-field gives rise to couplings between collective and IP motions and couplings among quantum fluctuations of the TD mean-field itself 7 , while in τ 𝜏 \tau -FM a soliton equation is derived as follows: Consider an infinite-dimensional Lie algebra and its representation on a functional space. The group-orbit of the highest weight vector becomes an infinite-dimensional Grassmannian G ∞ subscript 𝐺 G_{\!\infty} . The bilinear equation (Plücker relation) is nothing else than the soliton equation. This means that a solution space of the soliton equation corresponds to a group-orbit of the vacuum state. The SCFM does not use the Plücker relation in the context of a bilinear differential equation defining finite-dimensional Grassmannian G M subscript 𝐺 𝑀 G_{M} but seems to use implicitly such a relation. In the SCFM a physical concept of quasi-particle and vacuum and a coset space is used instead. If we develop a perturbative theory for large-amplitude collective motion 26 , an infinite-dimensional Lie algebra might been necessarily used. The sub-group orbits consisting of several loop-group paths 37 classified by the Plücker relation exist innumerably in G M subscript 𝐺 𝑀 G_{M} so that the SCFM is related to the soliton theory in G ∞ subscript 𝐺 G_{\infty} . The Plücker relation in a coset space U ​ ( N ) U ​ ( M ) × U ​ ( N − M ) 𝑈 𝑁 𝑈 𝑀 𝑈 𝑁 𝑀 \frac{U(N)}{U(M)\times U(N-M)} 38 becomes analogous with the Hirotas bilinear form 39 , 40 . Toward an ultimate goal we aim to reconstruct a theoretical frame for a υ 𝜐 \upsilon (external parameter)-dependent SCFM to describe more precisely the dynamics on the F ∞ subscript 𝐹 F_{\infty} . In the abstract fermion Fock space, we find common features in both SCFM and τ 𝜏 \tau -FM. (i) Each solution space is described as Grassmannian that is group orbit of the corresponding vacuum state. (ii) The former may implicitly explain the Plücker relation not in terms of bilinear differential equations defining G M subscript 𝐺 𝑀 G_{M} but in terms of the physical concept of quasi-particle and vacuum and mathematical language of coset space and coset variable. The various BETs are built on the Plücker relation to hold the Grassmannian. The latter asserts that the soliton equations are nothing but the bilinear differential equations giving a boson representation of the Plücker relation . The relation, however, has been unsatisfactorily investigated yet within the framework of the usual SCFM. We study it and show that both the methods stand on the common features, Plücker relation or bilinear differential equation defining the Grassmannian. On the contrary, we observe different points: (i) The former is built on a finite -dimensional Lie algebra but the latter on an infinite -dimensional one. (ii) The former has an SCF Hamiltonian consisting of a fermion one-body operator, which is derived from a functional derivative of an expectation value of a fermion Hamiltonian by a ground-state wave function. The latter introduces artificially a fermion Hamiltonian of one-body type operator as a boson mapping operator from states on fermion Fock space to corresponding ones on τ 𝜏 \tau -functional space ( τ 𝜏 \tau -FS). The last issue is, despite a difference due to the dimension of fermions, to aim at obtaining a close connection between concept of mean-field potential and gauge of fermions inherent in the SCFM and at making a role of a loop group 37 to be clear. Through the observation, we construct infinite-dimensional fermion operators from the finite-dimensional ones by Laurent expansion with respect to a circle S 1 superscript 𝑆 1 S^{1} . Then with the use of an affine Kac–Moody (KM) algebra according to the idea of Diracs positron theory 41 , we rebuilt a TDHFT in F ∞ subscript 𝐹 F_{\infty} . The TDHFT results in a gauge theory of fermions and the collective motion, fluctuation of the mean-field potential, appears as the motion of fermion gauges with a common factor. The physical concept of the quasi-particle and vacuum in the SCFM on the S 1 superscript 𝑆 1 S^{1} connects to the Plücker relations due to the Dirac theory, in other words, the algebraic mechanism extracting various sub-group orbits consisting of loop path out of the full TDHF manifold is just the Hirotas bilinear form 40 which is an s ​ u ​ ( N ) ( ∈ s ​ l ​ ( N ) ) annotated 𝑠 𝑢 𝑁 absent 𝑠 𝑙 𝑁 su(N)(\in sl(N)) reduction of g ​ l ​ ( N ) 𝑔 𝑙 𝑁 gl(N) in the τ 𝜏 \tau -FM. As a result, it is shown that an infinite-dimensional fermion many-body system is also realizable in a finite-dimensional one and that roles of the soliton equation (Plücker relation) and the TDHF equation are made clear. We also understand an SCF dynamics through gauge of interacting infinite-dimensional fermions. A bilinear equation for the υ 𝜐 \upsilon -HFT has been transcribed onto the corresponding τ 𝜏 \tau -function using the regular representation for the group and the Schur polynomials. The υ 𝜐 \upsilon -HF SCFM on an infinite-dimensional Fock space F ∞ subscript 𝐹 F_{\infty} leads to a dynamics on an infinite-dimensional Grassmannian Gr ∞ subscript Gr {\rm Gr}_{\infty} and may describe more precisely such a dynamics on the group manifold. A finite-dimensional Grassmannian is identified with a Gr ∞ subscript Gr {\rm Gr}_{\infty} which is affiliated with the group manifold obtained by reducting g ​ l ​ ( ∞ ) 𝑔 𝑙 gl(\infty) to s ​ l ​ ( N ) 𝑠 𝑙 𝑁 sl(N) and s ​ u ​ ( N ) 𝑠 𝑢 𝑁 su(N) . We have given explicit expressions for Laurent coefficients of soliton solutions for s ​ l ^ ​ ( N ) ^ 𝑠 𝑙 𝑁 \widehat{sl}(N) and s ​ u ^ ​ ( N ) ^ 𝑠 𝑢 𝑁 \widehat{su}(N) on the Gr ∞ subscript Gr {\rm Gr}_{\infty} using Chevalley bases for s ​ l ​ ( N ) 𝑠 𝑙 𝑁 sl(N) and s ​ u ​ ( N ) 𝑠 𝑢 𝑁 su(N) 42 . As an illustration we will attempt to make a υ 𝜐 \upsilon -HFT approach to an infinite-dimensional matrix model extended from the finite-dimensional s ​ u ​ ( 2 ) 𝑠 𝑢 2 su(2) Lipkin–Meshkov–Glick (LMG) model 43 . For this aim, we give an affine KM algebra s ​ l ^ ​ ( 2 , C ) ^ 𝑠 𝑙 2 𝐶 \widehat{sl}(2,C) (complexification of s ​ u ^ ( 2 ) ) \widehat{su}(2)) to which the LMG generators subject, and their τ τ \uptau representations and the σ K subscript 𝜎 𝐾 \sigma_{K} mappings for them. We can represent an infinite-dimensional matrix of the LMG Hamiltonian and its HF Hamiltonian in terms of the Schur polynomials. Its infinite-dimensional HF operator is also given through the mapping σ M subscript 𝜎 𝑀 \sigma_{M} for ψ i ​ ψ j ∗ subscript 𝜓 𝑖 superscript subscript 𝜓 𝑗 \psi_{i}\psi_{j}^{*} of infinite-dimensional fermions ψ i subscript 𝜓 𝑖 \psi_{i} and ψ i ∗ superscript subscript 𝜓 𝑖 \psi_{i}^{*} , which is expressed by the Schur polynomials S k ​ ( x ) subscript 𝑆 𝑘 𝑥 S_{k}(x) and S k ​ ( ∂ x ) subscript 𝑆 𝑘 subscript 𝑥 S_{k}(\partial_{x}) . Further its τ 𝜏 \tau -function for a simple case is provided by the Plücker coordinates and Schur polynomials. In Section 2 , we propose curvature equations as fundamental equations to extract a collective submanifold out of the full TDHB manifold. Basing on these ideas, we construct the curvature equations and study the relation between the maximal decoupled method and the curvature equations. We further investigate the role of the non-zero curvature arising from the residual Hamiltonian. Making use of the expression of the zero-curvature equations in the QPF, we find the formal RPA equation. In Section 3 , we present a simply unified aspect for the SCFM and the τ 𝜏 \tau -FM and show a simple idea connecting both the methods. We study the algebraic relation between coset coordinate and Plücker coordinate. Basing on the above idea, we attempt to rebuilt the TDHFT in τ 𝜏 \tau -FS. We introduce υ 𝜐 \upsilon -dependent infinite-dimensional fermion operators and a F ∞ subscript 𝐹 F_{\infty} through Laurent expansion with respect to the degrees of freedom of the original fermions. The algebraic relation between both the methods is manifestly described. We embed a HF u ​ ( N ) 𝑢 𝑁 u(N) Lie algebra into a g ​ l ​ ( ∞ ) 𝑔 𝑙 gl(\infty) by means of infinite-dimensional fermions. The υ 𝜐 \upsilon -SCFM in τ 𝜏 \tau -FS is developed. The role of the shift operators in the τ 𝜏 \tau -FM is studied. As an illustration, explicit expressions for Laurent coefficients of soliton solutions for s ​ l ^ ​ ( N ) ^ 𝑠 𝑙 𝑁 \widehat{sl}(N) and s ​ u ^ ​ ( N ) ^ 𝑠 𝑢 𝑁 \widehat{su}(N) are presented. A problem related to a nonlinear Schrödinger equation is also discussed. In Section 4 , we construct a formal RPA equation on F ∞ subscript 𝐹 F_{\infty} and also argue about the relation between a l ​ o ​ o ​ p 𝑙 𝑜 𝑜 𝑝 loop collective path and a formal RPAEQ. Consequently, it can be proved that the usual perturbative method with respect to periodic collective variables in the TDHFT is involved in the present method which aims for constructing the TDHFT on the affine KM algebra. In Section 5 , we introduce infinite-dimensional particle and hole operators and operators K ^ 0 subscript ^ 𝐾 0 {\widehat{K}}_{0} and K ^ ± subscript ^ 𝐾 plus-or-minus {\widehat{K}}_{\pm} defined by infinite-dimensional particle-hole pair operators. Using these operators, we construct an infinite-dimensional Heisenberg subalgebra of the affine KM algebra s ​ l ^ ​ ( 2 , C ) ^ 𝑠 𝑙 2 𝐶 \widehat{sl}(2,C) . The LMG Hamiltonian and its HF Hamiltonian are expressed in terms of the Heisenberg basic-elements whose representations are isomorphic to those in the corresponding boson space. They are given in terms of infinite numbers of variables x k subscript 𝑥 𝑘 x_{k} and derivatives ∂ x k subscript subscript 𝑥 𝑘 \partial_{x_{k}} through the Schur polynomials S k ​ ( x ) subscript 𝑆 𝑘 𝑥 S_{k}(x) . We give also an infinite-dimensional representation of S ​ U ​ ( 2 ​ N ) ∞ 𝑆 𝑈 subscript 2 𝑁 SU(2N)_{\infty} transformation of the particle and hole operators. Finally, in Section 6 , we summarize and discuss the results and future problems. We give Appendices A – J . Especially in Appendix J , we show an explicit expression for Plücker coordinate for the LMG model and calculate a quantity, det ( 1 N + p † p ) ( p : coset variable ) \det(1_{N}+p^{\dagger}p)~{}(p:\mbox{coset variable}) for the LMG model, in terms of the Schur polynomials.","role":"user"},{"content":"Abstract: The maximally-decoupled method has been considered as a theory to apply an basic idea of an integrability condition to certain multiple parametrized symmetries. The method is regarded as a mathematical tool to describe a symmetry of a collective subma- nifold in which a canonicity condition makes the collective variables to be an orthogonal coordinate-system. For this aim we adopt a concept of curvature unfamiliar in the con- ventional time-dependent (TD) self-consistent field (SCF) theory. Our basic idea lies in the introduction of a sort of Lagrange manner familiar to fluid dynamics to describe a collective coordinate-system. This manner enables us to take a one-form which is linearly composed of a TD SCF Hamiltonian and infinitesimal generators induced by collective vari- able differentials of a canonical transformation on a group. The integrability condition of the system read the curvature C = 0. Our method is constructed manifesting itself the structure of the group under consideration. To go beyond the maximaly-decoupled method, we have aimed to construct an SCF theory, i.e., (external parameter)-dependent Hartree-Fock (HF) theory. Toward such an ultimate goal, the -HF theory has been recon- structed on an affine Kac-Moody algebra along the soliton theory, using infinite-dimensional fermion. An infinite-dimensional fermion operator is introduced through a Laurent ex- pansion of finite-dimensional fermion operators with respect to degrees of freedom of the fermions related to a -dependent potential with a -periodicity. A bilinear equation for the -HF theory has been transcribed onto the corresponding -function using the regu- lar representation for the group and the Schur-polynomials. The -HF SCF theory on an infinite-dimensional Fock space F1 leads to a dynamics on an infinite-dimensional Grass- mannian Gr1 and may describe more precisely such a dynamics on the group manifold. A finite-dimensional Grassmannian is identified with a Gr1 which is affiliated with the group manifold obtained by reducting gl(1) to sl(N) and su(N). As an illustration we will study an infinite-dimensional matrix model extended from the finite-dimensional su(2) Lipkin-Meshkov-Glick model which is a famous exactly-solvable model.","role":"assistant"}]

Survey on aspherical manifolds

1. Homotopy theory of aspherical manifolds From the homotopy theory point of view an aspherical C ​ W 𝐶 𝑊 CW -complex is completely determined by its fundamental group. Namely Theorem 1.1 (Homotopy classification of aspherical spaces) . (i) Two aspherical C ​ W 𝐶 𝑊 CW -complexes are homotopy equivalent if and only if their fundamental groups are isomorphic; (ii) Let X 𝑋 X and Y 𝑌 Y be connected C ​ W 𝐶 𝑊 CW -complexes. Suppose that Y 𝑌 Y is aspherical. Then we obtain a bijection [ X , Y ] → ≅ [ Π ( X ) , Π ( Y ) ] , [ f ] ↦ [ Π ( f ) ) ] , [X,Y]\xrightarrow{\cong}[\Pi(X),\Pi(Y)],\quad[f]\mapsto[\Pi(f))], where [ X , Y ] 𝑋 𝑌 [X,Y] is the set of homotopy classes of maps from X 𝑋 X to Y 𝑌 Y , Π ​ ( X ) Π 𝑋 \Pi(X) , Π ​ ( Y ) Π 𝑌 \Pi(Y) are the fundamental groupoids, [ Π ​ ( X ) , Π ​ ( Y ) ] Π 𝑋 Π 𝑌 [\Pi(X),\Pi(Y)] is the set of natural equivalence classes of functors from Π ​ ( X ) Π 𝑋 \Pi(X) to Π ​ ( Y ) Π 𝑌 \Pi(Y) and Π ​ ( f ) : Π ​ ( X ) → Π ​ ( Y ) : Π 𝑓 → Π 𝑋 Π 𝑌 \Pi(f)\colon\Pi(X)\to\Pi(Y) is the functor induced by f : X → Y : 𝑓 → 𝑋 𝑌 f\colon X\to Y . Proof. (ii) One easily checks that the map is well-defined. For the proof of surjectivity and injectivity one constructs the desired preimage or the desired homotopy inductively over the skeletons of the source. (i) This follows directly from assertion (ii) . ∎ The description using fundamental groupoids is elegant and base point free, but a reader may prefer its more concrete interpretation in terms of fundamental groups, which we will give next: Choose base points x ∈ X 𝑥 𝑋 x\in X and y ∈ Y 𝑦 𝑌 y\in Y . Let hom ⁡ ( π 1 ​ ( X , x ) , π 1 ​ ( Y , y ) ) hom subscript 𝜋 1 𝑋 𝑥 subscript 𝜋 1 𝑌 𝑦 \hom(\pi_{1}(X,x),\pi_{1}(Y,y)) be the set of group homomorphisms from π 1 ​ ( X , x ) subscript 𝜋 1 𝑋 𝑥 \pi_{1}(X,x) to π 1 ​ ( Y , y ) subscript 𝜋 1 𝑌 𝑦 \pi_{1}(Y,y) . The group Inn ⁡ ( π 1 ​ ( Y , y ) ) Inn subscript 𝜋 1 𝑌 𝑦 \operatorname{Inn}\bigl{(}\pi_{1}(Y,y)\bigr{)} of inner automorphisms of π 1 ​ ( Y , y ) subscript 𝜋 1 𝑌 𝑦 \pi_{1}(Y,y) acts on hom ⁡ ( π 1 ​ ( X , x ) , π 1 ​ ( Y , y ) ) hom subscript 𝜋 1 𝑋 𝑥 subscript 𝜋 1 𝑌 𝑦 \hom\bigl{(}\pi_{1}(X,x),\pi_{1}(Y,y)\bigr{)} from the left by composition. We leave it to the reader to check that we obtain a bijection Inn ⁡ ( π 1 ​ ( Y , y ) ) \ hom ⁡ ( π 1 ​ ( X , x ) , π 1 ​ ( Y , y ) ) → ≅ [ Π ​ ( X ) , Π ​ ( Y ) ] , → \ Inn subscript 𝜋 1 𝑌 𝑦 hom subscript 𝜋 1 𝑋 𝑥 subscript 𝜋 1 𝑌 𝑦 Π 𝑋 Π 𝑌 \operatorname{Inn}\bigl{(}\pi_{1}(Y,y)\bigr{)}\backslash\hom\bigl{(}\pi_{1}(X,x),\pi_{1}(Y,y)\bigr{)}\xrightarrow{\cong}[\Pi(X),\Pi(Y)], under which the bijection appearing in Lemma 1.1 (ii) sends [ f ] delimited-[] 𝑓 [f] to the class of π 1 ​ ( f , x ) subscript 𝜋 1 𝑓 𝑥 \pi_{1}(f,x) for any choice of representative of f 𝑓 f with f ​ ( x ) = y 𝑓 𝑥 𝑦 f(x)=y . In the sequel we will often ignore base points especially when dealing with the fundamental group. Lemma 1.2 . A C ​ W 𝐶 𝑊 CW -complex X 𝑋 X is aspherical if and only if it is connected and its universal covering X ~ ~ 𝑋 \widetilde{X} is contractible. Proof. The projection p : X ~ → X : 𝑝 → ~ 𝑋 𝑋 p\colon\widetilde{X}\to X induces isomorphisms on the homotopy groups π n subscript 𝜋 𝑛 \pi_{n} for n ≥ 2 𝑛 2 n\geq 2 and a connected C ​ W 𝐶 𝑊 CW -complex is contractible if and only if all its homotopy groups are trivial (see [ 99 , Theorem IV.7.17 on page 182] . ∎ An aspherical C ​ W 𝐶 𝑊 CW -complex X 𝑋 X with fundamental group π 𝜋 \pi is the same as an Eilenberg Mac-Lane space K ​ ( π , 1 ) 𝐾 𝜋 1 K(\pi,1) of type ( π , 1 ) 𝜋 1 (\pi,1) and the same as the classifying space B ​ π 𝐵 𝜋 B\pi for the group π 𝜋 \pi .

This is a survey on known results and open problems about closed aspherical manifolds, i.e., connected closed manifolds whose universal coverings are contractible. Many examples come from certain kinds of non-positive curvature conditions. The property aspherical which is a purely homotopy theoretical condition implies many striking results about the geometry and analysis of the manifold or its universal covering, and the ring theoretic properties and the K- and L-theory of the group ring associated to its fundamental group. The Borel Conjecture predicts that closed aspherical manifolds are topologically rigid. The article contains new results about product decompositions of closed aspherical manifolds and an announcement of a result joint with Arthur Bartels and Shmuel Weinberger about hyperbolic groups with spheres of dimension greater or equal to six as boundary. At the end we describe (winking) our universe of closed manifolds.

[{"content":"Help me to generate the abstract of a survey paper given the title:*Survey on aspherical manifolds* and the introduction:1. Homotopy theory of aspherical manifolds From the homotopy theory point of view an aspherical C ​ W 𝐶 𝑊 CW -complex is completely determined by its fundamental group. Namely Theorem 1.1 (Homotopy classification of aspherical spaces) . (i) Two aspherical C ​ W 𝐶 𝑊 CW -complexes are homotopy equivalent if and only if their fundamental groups are isomorphic; (ii) Let X 𝑋 X and Y 𝑌 Y be connected C ​ W 𝐶 𝑊 CW -complexes. Suppose that Y 𝑌 Y is aspherical. Then we obtain a bijection X , Y → ≅ Π ( X ) , Π ( Y ) , f ↦ Π ( f ) ) , X,Y\xrightarrow{\cong}\Pi(X),\Pi(Y),\quadf\mapsto\Pi(f)), where X , Y 𝑋 𝑌 X,Y is the set of homotopy classes of maps from X 𝑋 X to Y 𝑌 Y , Π ​ ( X ) Π 𝑋 \Pi(X) , Π ​ ( Y ) Π 𝑌 \Pi(Y) are the fundamental groupoids, Π ​ ( X ) , Π ​ ( Y ) Π 𝑋 Π 𝑌 \Pi(X),\Pi(Y) is the set of natural equivalence classes of functors from Π ​ ( X ) Π 𝑋 \Pi(X) to Π ​ ( Y ) Π 𝑌 \Pi(Y) and Π ​ ( f ) : Π ​ ( X ) → Π ​ ( Y ) : Π 𝑓 → Π 𝑋 Π 𝑌 \Pi(f)\colon\Pi(X)\to\Pi(Y) is the functor induced by f : X → Y : 𝑓 → 𝑋 𝑌 f\colon X\to Y . Proof. (ii) One easily checks that the map is well-defined. For the proof of surjectivity and injectivity one constructs the desired preimage or the desired homotopy inductively over the skeletons of the source. (i) This follows directly from assertion (ii) . ∎ The description using fundamental groupoids is elegant and base point free, but a reader may prefer its more concrete interpretation in terms of fundamental groups, which we will give next: Choose base points x ∈ X 𝑥 𝑋 x\in X and y ∈ Y 𝑦 𝑌 y\in Y . Let hom ⁡ ( π 1 ​ ( X , x ) , π 1 ​ ( Y , y ) ) hom subscript 𝜋 1 𝑋 𝑥 subscript 𝜋 1 𝑌 𝑦 \hom(\pi_{1}(X,x),\pi_{1}(Y,y)) be the set of group homomorphisms from π 1 ​ ( X , x ) subscript 𝜋 1 𝑋 𝑥 \pi_{1}(X,x) to π 1 ​ ( Y , y ) subscript 𝜋 1 𝑌 𝑦 \pi_{1}(Y,y) . The group Inn ⁡ ( π 1 ​ ( Y , y ) ) Inn subscript 𝜋 1 𝑌 𝑦 \operatorname{Inn}\bigl{(}\pi_{1}(Y,y)\bigr{)} of inner automorphisms of π 1 ​ ( Y , y ) subscript 𝜋 1 𝑌 𝑦 \pi_{1}(Y,y) acts on hom ⁡ ( π 1 ​ ( X , x ) , π 1 ​ ( Y , y ) ) hom subscript 𝜋 1 𝑋 𝑥 subscript 𝜋 1 𝑌 𝑦 \hom\bigl{(}\pi_{1}(X,x),\pi_{1}(Y,y)\bigr{)} from the left by composition. We leave it to the reader to check that we obtain a bijection Inn ⁡ ( π 1 ​ ( Y , y ) ) \ hom ⁡ ( π 1 ​ ( X , x ) , π 1 ​ ( Y , y ) ) → ≅ Π ​ ( X ) , Π ​ ( Y ) , → \ Inn subscript 𝜋 1 𝑌 𝑦 hom subscript 𝜋 1 𝑋 𝑥 subscript 𝜋 1 𝑌 𝑦 Π 𝑋 Π 𝑌 \operatorname{Inn}\bigl{(}\pi_{1}(Y,y)\bigr{)}\backslash\hom\bigl{(}\pi_{1}(X,x),\pi_{1}(Y,y)\bigr{)}\xrightarrow{\cong}\Pi(X),\Pi(Y), under which the bijection appearing in Lemma 1.1 (ii) sends f delimited- 𝑓 f to the class of π 1 ​ ( f , x ) subscript 𝜋 1 𝑓 𝑥 \pi_{1}(f,x) for any choice of representative of f 𝑓 f with f ​ ( x ) = y 𝑓 𝑥 𝑦 f(x)=y . In the sequel we will often ignore base points especially when dealing with the fundamental group. Lemma 1.2 . A C ​ W 𝐶 𝑊 CW -complex X 𝑋 X is aspherical if and only if it is connected and its universal covering X ~ ~ 𝑋 \widetilde{X} is contractible. Proof. The projection p : X ~ → X : 𝑝 → ~ 𝑋 𝑋 p\colon\widetilde{X}\to X induces isomorphisms on the homotopy groups π n subscript 𝜋 𝑛 \pi_{n} for n ≥ 2 𝑛 2 n\geq 2 and a connected C ​ W 𝐶 𝑊 CW -complex is contractible if and only if all its homotopy groups are trivial (see 99 , Theorem IV.7.17 on page 182 . ∎ An aspherical C ​ W 𝐶 𝑊 CW -complex X 𝑋 X with fundamental group π 𝜋 \pi is the same as an Eilenberg Mac-Lane space K ​ ( π , 1 ) 𝐾 𝜋 1 K(\pi,1) of type ( π , 1 ) 𝜋 1 (\pi,1) and the same as the classifying space B ​ π 𝐵 𝜋 B\pi for the group π 𝜋 \pi .","role":"user"},{"content":"Abstract: This is a survey on known results and open problems about closed aspherical manifolds, i.e., connected closed manifolds whose universal coverings are contractible. Many examples come from certain kinds of non-positive curvature conditions. The property aspherical which is a purely homotopy theoretical condition implies many striking results about the geometry and analysis of the manifold or its universal covering, and the ring theoretic properties and the K- and L-theory of the group ring associated to its fundamental group. The Borel Conjecture predicts that closed aspherical manifolds are topologically rigid. The article contains new results about product decompositions of closed aspherical manifolds and an announcement of a result joint with Arthur Bartels and Shmuel Weinberger about hyperbolic groups with spheres of dimension greater or equal to six as boundary. At the end we describe (winking) our universe of closed manifolds.","role":"assistant"}]

Weight structures and motives; comotives, coniveau and Chow-weight spectral sequences, and mixed complexes of sheaves: a survey

This article is a survey of author’s results on Voevodsky’s motives and weight structures; yet it is supplied with detailed references. Weight structures are natural counterparts of t 𝑡 t -structures (for triangulated categories) introduced by the author in [ Bon07 ] (and also independently by D. Pauksztello in [ Pau08 ] ). They allow to to construct weight complexes , weight filtrations , and weight spectral sequences . Partial cases of the latter are: ’classical’ weight spectral sequences (for singular and étale cohomology), coniveau spectral sequences, and Atiyah-Hirzebruch spectral sequences (we mention all of these below). The details, proofs, and several more results could be found in [ Bon07 ] , [ Bon10a ] , and [ Bon09 ] (we also mention certain results of [ Bon10b ] , [ Heb10 ] , and [ Bon10p ] ). We describe more motivation for the theory of weight structures, and define weight structures in § 4 . Though our ’main’ weight structures will be defined on certain ’motivic’ categories, the author tried to make this survey accessible to readers that are rather interested in general triangulated categories (or possibly, the stable homotopy category in topology). Those readers may freely ignore all definitions and results that are related with algebraic geometry (and motives). On the other hand, the main motivic results (see § 3 ) could be understood without knowing anything about weight structures (after § 3 a ’motivic’ reader may proceed directly to § 9 to find some more motivation to study weight structures). Alternatively, it is quite possible for any reader to read section § 3 only after studying the general theory of weight structures (§§ 4 – 8 ). The author chose not to pay much attention to the differential graded approach to motives in this text; yet it is described in detail in [ Bon09 ] and in §6 of [ Bon07 ] (see also [ BeV08 ] and § 8.2 below). This text is based on the talks presented by the author at the conferences "Finiteness for motives and motivic cohomology" (Regensburg, 9–13th of February, 2009) and "Motivic homotopy theory" (Münster, 27-31st of July, 2009); yet some more recent topics are added. The author is deeply grateful to prof. Uwe Jannsen, prof. Eric Friedlander, and to other organizers of these conferences for their efforts.

This is a survey of author's results on weight structures and Voevodsky's motives. Weight structures are natural counterparts of t-structures (for triangulated categories) introduced by the author. They allow to construct weight complexes, weight filtrations, and weight spectral sequences for various cohomology theories. Partial cases of the latter are: 'classical' weight spectral sequences (for singular and etale cohomology), coniveau spectral sequences, and Atiyah-Hirzebruch spectral sequences. All of those are mentioned in the current paper. The details, proofs, and several more results could be found in other papers of the author (cited here). We also mention a certain triangulated category of comotives that contains reasonable (co)motives for all function fields (and also of other projective limits of smooth varieties).

[{"content":"Help me to generate the abstract of a survey paper given the title:*Weight structures and motives; comotives, coniveau and Chow-weight spectral sequences, and mixed complexes of sheaves: a survey* and the introduction:This article is a survey of authors results on Voevodskys motives and weight structures; yet it is supplied with detailed references. Weight structures are natural counterparts of t 𝑡 t -structures (for triangulated categories) introduced by the author in Bon07 (and also independently by D. Pauksztello in Pau08 ). They allow to to construct weight complexes , weight filtrations , and weight spectral sequences . Partial cases of the latter are: classical weight spectral sequences (for singular and étale cohomology), coniveau spectral sequences, and Atiyah-Hirzebruch spectral sequences (we mention all of these below). The details, proofs, and several more results could be found in Bon07 , Bon10a , and Bon09 (we also mention certain results of Bon10b , Heb10 , and Bon10p ). We describe more motivation for the theory of weight structures, and define weight structures in § 4 . Though our main weight structures will be defined on certain motivic categories, the author tried to make this survey accessible to readers that are rather interested in general triangulated categories (or possibly, the stable homotopy category in topology). Those readers may freely ignore all definitions and results that are related with algebraic geometry (and motives). On the other hand, the main motivic results (see § 3 ) could be understood without knowing anything about weight structures (after § 3 a motivic reader may proceed directly to § 9 to find some more motivation to study weight structures). Alternatively, it is quite possible for any reader to read section § 3 only after studying the general theory of weight structures (§§ 4 – 8 ). The author chose not to pay much attention to the differential graded approach to motives in this text; yet it is described in detail in Bon09 and in §6 of Bon07 (see also BeV08 and § 8.2 below). This text is based on the talks presented by the author at the conferences Finiteness for motives and motivic cohomology (Regensburg, 9–13th of February, 2009) and Motivic homotopy theory (Münster, 27-31st of July, 2009); yet some more recent topics are added. The author is deeply grateful to prof. Uwe Jannsen, prof. Eric Friedlander, and to other organizers of these conferences for their efforts.","role":"user"},{"content":"Abstract: This is a survey of author's results on weight structures and Voevodsky's motives. Weight structures are natural counterparts of t-structures (for triangulated categories) introduced by the author. They allow to construct weight complexes, weight filtrations, and weight spectral sequences for various cohomology theories. Partial cases of the latter are: 'classical' weight spectral sequences (for singular and etale cohomology), coniveau spectral sequences, and Atiyah-Hirzebruch spectral sequences. All of those are mentioned in the current paper. The details, proofs, and several more results could be found in other papers of the author (cited here). We also mention a certain triangulated category of comotives that contains reasonable (co)motives for all function fields (and also of other projective limits of smooth varieties).","role":"assistant"}]

CRITICAL REVIEWS ON HOLOGRAPHIC MEASURE OVER THE MULTIVERSE

If the paradigm of inflation is correct, then there would be no other option than to conclude that the global structure is the multiverse, a complex mixture of thermalized and inflating regions. In the context of the multiverse, we could explore the cosmic fine-tuning problem by asking for statistical distributions over the multiverse. An approach based on the statistical distribution is deeply related to anthropic reasoning. Whether our universe is typical among the possible pocket universes would be an important signature of the statistical inference and the reasoning of quantum cosmology. In general, the complexity of the global structure of the multiverse depends on the shape of inflation potentials. However, Guth pointed out that eternal inflation could be activated even with a simple potential, and there would be no proper way to quantify the statistical distribution based on the volume fractions due to the infinity and the slice-dependence [ 1 ] . Thus it implies that we may fail to handle the complexity and to specify the measure over the inflationary multiverse. This problem is called the ‘measure problem.’ Among many proposals to overcome the measure problem, there have been a remarkable category of attempts inspired by the similarities between physical properties (e.g. causal structures, thermodynamics, etc.) of black holes and those of de Sitter (dS) spaces. Susskind proposed that the complementarity principle should be involved in specifying the measure from his speculations for resolving the black hole information paradox [ 2 ] [ 3 ] . In the line of this thought, Bousso has developed a causal patch measure based on black hole complementarity [ 4 ] .

In this talk, the holographic measure based on black hole complementarity is critically reviewed. By noticing the similarities between the causal structure of an inflationary dS space and that of a black hole, we have considered the complementarity principle between the inside and the outside of the causal horizon as an attractive way to count the inflationary multiverse. Even though the holographic measure relieves the Boltzmann brain problem and stresses the physical reality based on observations, it could be challenged by the construction of counterexamples, both on regular black holes and charged black holes, to black hole complementarity.

[{"content":"Help me to generate the abstract of a survey paper given the title:*CRITICAL REVIEWS ON HOLOGRAPHIC MEASURE OVER THE MULTIVERSE* and the introduction:If the paradigm of inflation is correct, then there would be no other option than to conclude that the global structure is the multiverse, a complex mixture of thermalized and inflating regions. In the context of the multiverse, we could explore the cosmic fine-tuning problem by asking for statistical distributions over the multiverse. An approach based on the statistical distribution is deeply related to anthropic reasoning. Whether our universe is typical among the possible pocket universes would be an important signature of the statistical inference and the reasoning of quantum cosmology. In general, the complexity of the global structure of the multiverse depends on the shape of inflation potentials. However, Guth pointed out that eternal inflation could be activated even with a simple potential, and there would be no proper way to quantify the statistical distribution based on the volume fractions due to the infinity and the slice-dependence 1 . Thus it implies that we may fail to handle the complexity and to specify the measure over the inflationary multiverse. This problem is called the measure problem. Among many proposals to overcome the measure problem, there have been a remarkable category of attempts inspired by the similarities between physical properties (e.g. causal structures, thermodynamics, etc.) of black holes and those of de Sitter (dS) spaces. Susskind proposed that the complementarity principle should be involved in specifying the measure from his speculations for resolving the black hole information paradox 2 3 . In the line of this thought, Bousso has developed a causal patch measure based on black hole complementarity 4 .","role":"user"},{"content":"Abstract: In this talk, the holographic measure based on black hole complementarity is critically reviewed. By noticing the similarities between the causal structure of an inflationary dS space and that of a black hole, we have considered the complementarity principle between the inside and the outside of the causal horizon as an attractive way to count the inflationary multiverse. Even though the holographic measure relieves the Boltzmann brain problem and stresses the physical reality based on observations, it could be challenged by the construction of counterexamples, both on regular black holes and charged black holes, to black hole complementarity.","role":"assistant"}]

Perfect, Efficent, State Transfer and its Application as a Constructive Tool

The study of quantum mechanical systems is no longer restricted to the characterization of the properties of naturally occurring systems. Rather, with the advent of Quantum Information, the manipulation and engineering of these systems to suit our purposes has come to the fore. One of the simplest ways to understand these manipulations is to consider not universal Quantum Computation, but simpler sub-protocols. The sub-protocol of state transfer was originally proposed by Bose [ 1 ] , the motivation being that in a quantum computer based on, for instance, a solid state architecture, interactions are typically local, but we want to apply entangling gates between distant qubits. While a sequence of swap gates suffices to bring those distant qubits together, this is potentially prone to massive control errors, and it would be desirable to remove the need for such stringent control. One promising alternative is to integrate a ‘flying qubit’ such as a photon within the same system in order to transfer qubits quickly and easily. This, however, requires the successful implementation of two information processing realisations rather than just one, which is already a daunting task. Instead, the idea is to achieve state transfer by providing a pre-fabricated unit, made from the same solid state technology but with severely limited capabilities (thereby easing the fabrication) which takes as input a state in one location, and outputs it at another. In principle, since it is not necessary to interact with the device except at the input and output, it can, to some extent, be more isolated from the environment and less susceptible to decoherence. Although this was the original motivation, whether these schemes are ever practically realized is largely irrelevant; they have already provided a huge degree of insight into engineering more complex quantum protocols. In this paper, we provide a review of how quantum systems can be engineered precisely for the task of quantum state transfer with, in some sense, minimalistic properties; there should be no interaction with the system except at initialization and read-out, and the system Hamiltonian should remain fixed in time. We start, in Sec. 2 , by proving necessary and sufficient conditions for perfect state transfer in a one dimensional system, and showing that this is equivalent to the protocol of entanglement distribution. These conditions provide an infinite family of solutions, and we discuss how to design systems with specific properties, and demonstrate that one particular solution is optimal with respect to a variety of parameters. The understanding of this basic mechanism can then be applied to a variety of different systems, such as harmonic oscillators, and allow for the possibility of long-range couplings. In Sec. 3 , we consider what happens if more than a single excitation is present, enabling us to generate entanglement, or to encode such that initialisation of the state of the system is unnecessary. In Sec. 4 , we give a brief treatment of errors, although, if used as a constructive technique, this section is largely irrelevant. Sec. 5 describes the only known class of solutions for perfect state transfer in geometries beyond a one-dimensional chain such that the distances of transfer drops off no faster that a polynomial of the number of system qubits. Finally, in Sec. 6 , we show how perfect state transfer schemes can be be modified in order to generate different types of multipartite entanglement, such as GHZ or W-states, followed by the much stronger result of how to design a Hamiltonian to perfectly implement a quantum computation without any external control.

We review the subject of perfect state transfer — how one designs the (fixed) interactions of a chain of spins so that a quantum state, initially inserted on one end of the chain, is perfectly transferred to the opposite end in a fixed time. The perfect state transfer systems are then used as a constructive tool to design Hamiltonian implementations of other primitive protocols such as entanglement generation and signal amplification in measurements, before showing that, in fact, universal quantum computation can be implemented in this way.

[{"content":"Help me to generate the abstract of a survey paper given the title:*Perfect, Efficent, State Transfer and its Application as a Constructive Tool* and the introduction:The study of quantum mechanical systems is no longer restricted to the characterization of the properties of naturally occurring systems. Rather, with the advent of Quantum Information, the manipulation and engineering of these systems to suit our purposes has come to the fore. One of the simplest ways to understand these manipulations is to consider not universal Quantum Computation, but simpler sub-protocols. The sub-protocol of state transfer was originally proposed by Bose 1 , the motivation being that in a quantum computer based on, for instance, a solid state architecture, interactions are typically local, but we want to apply entangling gates between distant qubits. While a sequence of swap gates suffices to bring those distant qubits together, this is potentially prone to massive control errors, and it would be desirable to remove the need for such stringent control. One promising alternative is to integrate a flying qubit such as a photon within the same system in order to transfer qubits quickly and easily. This, however, requires the successful implementation of two information processing realisations rather than just one, which is already a daunting task. Instead, the idea is to achieve state transfer by providing a pre-fabricated unit, made from the same solid state technology but with severely limited capabilities (thereby easing the fabrication) which takes as input a state in one location, and outputs it at another. In principle, since it is not necessary to interact with the device except at the input and output, it can, to some extent, be more isolated from the environment and less susceptible to decoherence. Although this was the original motivation, whether these schemes are ever practically realized is largely irrelevant; they have already provided a huge degree of insight into engineering more complex quantum protocols. In this paper, we provide a review of how quantum systems can be engineered precisely for the task of quantum state transfer with, in some sense, minimalistic properties; there should be no interaction with the system except at initialization and read-out, and the system Hamiltonian should remain fixed in time. We start, in Sec. 2 , by proving necessary and sufficient conditions for perfect state transfer in a one dimensional system, and showing that this is equivalent to the protocol of entanglement distribution. These conditions provide an infinite family of solutions, and we discuss how to design systems with specific properties, and demonstrate that one particular solution is optimal with respect to a variety of parameters. The understanding of this basic mechanism can then be applied to a variety of different systems, such as harmonic oscillators, and allow for the possibility of long-range couplings. In Sec. 3 , we consider what happens if more than a single excitation is present, enabling us to generate entanglement, or to encode such that initialisation of the state of the system is unnecessary. In Sec. 4 , we give a brief treatment of errors, although, if used as a constructive technique, this section is largely irrelevant. Sec. 5 describes the only known class of solutions for perfect state transfer in geometries beyond a one-dimensional chain such that the distances of transfer drops off no faster that a polynomial of the number of system qubits. Finally, in Sec. 6 , we show how perfect state transfer schemes can be be modified in order to generate different types of multipartite entanglement, such as GHZ or W-states, followed by the much stronger result of how to design a Hamiltonian to perfectly implement a quantum computation without any external control.","role":"user"},{"content":"Abstract: We review the subject of perfect state transfer — how one designs the (fixed) interactions of a chain of spins so that a quantum state, initially inserted on one end of the chain, is perfectly transferred to the opposite end in a fixed time. The perfect state transfer systems are then used as a constructive tool to design Hamiltonian implementations of other primitive protocols such as entanglement generation and signal amplification in measurements, before showing that, in fact, universal quantum computation can be implemented in this way.","role":"assistant"}]

Algebras of unbounded operators and physical applications: a survey

I Introduction During the past 20 years a long series of papers concerning algebras of unbounded operators appeared in the literature, papers which, though being originally motivated by physical arguments, contain essentially no physics at all. On the contrary the mathematical aspects of these algebras have been analyzed in many details and this analysis produced, up to now, the monographs [ 40 ] and [ 2 ] . Some physics appeared first in [ 34 ] and [ 39 ] , in the attempt to describe systems with a very large ( 10 24 superscript 10 24 10^{24} ) number of degrees of freedom, following some general ideas originally proposed in the famous paper of Haag and Kastler, [ 33 ] . These authors consider, as widely discussed in the literature, [ 21 ] , systems with infinite degrees of freedom because, in this way, a simpler approach to, e.g., phase transitions and collective phenomena can be settled up. However, moving from a large but finite to an infinite number of degrees of freedom one has to build up a mathematical apparatus which is rather sophisticated and, as we will see, not yet completely fixed. More recently other physical applications of algebras of unbounded operators have been proposed by the present author and others, see [ 4 , 5 , 8 , 9 , 11 , 12 , 13 , 14 ] for instance. In our opinion it is time to review some of these results, trying to connect as much as possible these with the original results based on C*-algebras. The paper, which is meant to be very pedagogical, is organized as follows: in Section II we give an introduction to non relativistic ordinary quantum mechanics (i.e. quantum mechanics for systems with a finite number of degrees of freedom), useful to fix the notation and some preliminary ideas. Section III is devoted to a longer review to non relativistic quantum mechanics for systems with infinite degrees of freedom, with a particular interest for some physically relevant results and for open problems. In Section IV we introduce some mathematical definitions and results concerning algebras of unbounded operators, while their physical applications are given in Section V. Our conclusions and our future projects are finally contained in Section VI. To keep the paper self-contained we have also added two Appendices. In the first one we give the general construction of the algebraic settings which extends the Haag and Kastler’s construction, while in the second appendix we give a list of information of functional analysis which may be useful to some non particularly mathematically minded.

After an historical introduction on the standard algebraic approach to quantum mechanics of large systems we review the basic mathematical aspects of the algebras of unbounded operators. After that we discuss in some details their relevance in physical applications.

[{"content":"Help me to generate the abstract of a survey paper given the title:*Algebras of unbounded operators and physical applications: a survey* and the introduction:I Introduction During the past 20 years a long series of papers concerning algebras of unbounded operators appeared in the literature, papers which, though being originally motivated by physical arguments, contain essentially no physics at all. On the contrary the mathematical aspects of these algebras have been analyzed in many details and this analysis produced, up to now, the monographs 40 and 2 . Some physics appeared first in 34 and 39 , in the attempt to describe systems with a very large ( 10 24 superscript 10 24 10^{24} ) number of degrees of freedom, following some general ideas originally proposed in the famous paper of Haag and Kastler, 33 . These authors consider, as widely discussed in the literature, 21 , systems with infinite degrees of freedom because, in this way, a simpler approach to, e.g., phase transitions and collective phenomena can be settled up. However, moving from a large but finite to an infinite number of degrees of freedom one has to build up a mathematical apparatus which is rather sophisticated and, as we will see, not yet completely fixed. More recently other physical applications of algebras of unbounded operators have been proposed by the present author and others, see 4 , 5 , 8 , 9 , 11 , 12 , 13 , 14 for instance. In our opinion it is time to review some of these results, trying to connect as much as possible these with the original results based on C*-algebras. The paper, which is meant to be very pedagogical, is organized as follows: in Section II we give an introduction to non relativistic ordinary quantum mechanics (i.e. quantum mechanics for systems with a finite number of degrees of freedom), useful to fix the notation and some preliminary ideas. Section III is devoted to a longer review to non relativistic quantum mechanics for systems with infinite degrees of freedom, with a particular interest for some physically relevant results and for open problems. In Section IV we introduce some mathematical definitions and results concerning algebras of unbounded operators, while their physical applications are given in Section V. Our conclusions and our future projects are finally contained in Section VI. To keep the paper self-contained we have also added two Appendices. In the first one we give the general construction of the algebraic settings which extends the Haag and Kastlers construction, while in the second appendix we give a list of information of functional analysis which may be useful to some non particularly mathematically minded.","role":"user"},{"content":"Abstract: After an historical introduction on the standard algebraic approach to quantum mechanics of large systems we review the basic mathematical aspects of the algebras of unbounded operators. After that we discuss in some details their relevance in physical applications.","role":"assistant"}]

The little review on leptogenesis

An elegant framework to consistently address the observed Baryon Asymmetry in the Universe (BAU) [ 1 , 2 ] is leptogenesis [ 3 ] . According to the standard paradigm of leptogenesis [ 4 ] , minimal extensions of the Standard Model (SM) realize heavy Majorana neutrinos of masses related to the Grand Unified Theory (GUT) scale M GUT ∼ 10 16 similar-to subscript 𝑀 GUT superscript 10 16 M_{\rm GUT}\sim 10^{16} that decay out of equilibrium and create a net excess of lepton number ( L ) 𝐿 (L) . This excess in L 𝐿 L gets then reprocessed into the observed baryon number ( B ) 𝐵 (B) , through the ( B + L ) 𝐵 𝐿 (B+L) -violating sphaleron interactions [ 5 ] . The attractive feature of such a scenario is that the GUT-scale heavy Majorana neutrinos could also explain the observed smallness in mass of the SM light neutrinos by means of the so-called seesaw mechanism [ 6 ] . The original GUT-scale leptogenesis scenario, however, runs into certain difficulties, when one attempts to explain the flatness of the Universe and other cosmological data [ 2 ] within supergravity models of inflation. To avoid overproduction of gravitinos G ~ ~ 𝐺 \widetilde{G} whose late decays may ruin the successful predictions of Big Bang Nucleosynthesis (BBN), the reheat temperature T reh subscript 𝑇 reh T_{\rm reh} of the Universe should be lower than 10 9 superscript 10 9 10^{9} – 10 6 superscript 10 6 10^{6} GeV, for m G ~ = 8 subscript 𝑚 ~ 𝐺 8 m_{\widetilde{G}}=8 –0.2 TeV [ 7 ] . This implies that the heavy Majorana neutrinos should accordingly have masses as low as T reh ∼ < 10 9 superscript similar-to subscript 𝑇 reh superscript 10 9 T_{\rm reh}\stackrel{{\scriptstyle<}}{{{}_{\sim}}}10^{9} GeV, thereby rendering the relation of these particles with GUT-scale physics less natural. On the other hand, it proves very difficult to directly probe the heavy-neutrino sector of such a model at high-energy colliders, e.g. at the LHC or ILC, or in any other foreseeable experiment. A potentially interesting solution to the above problems may be obtained within the framework of resonant leptogenesis (RL) [ 8 ] . The key aspect of RL is that self-energy effects dominate the leptonic asymmetries [ 9 ] , when two heavy Majorana neutrinos happen to have a small mass difference with respect to their actual masses. If this mass difference becomes comparable to the heavy neutrino widths, a resonant enhancement of the leptonic asymmetries takes place that may reach values 𝒪 ​ ( 1 ) 𝒪 1 {\cal O}(1) [ 8 , 10 ] . An indispensable feature of RL models is that flavour effects due to the light-to-heavy neutrino Yukawa couplings [ 11 , 12 , 13 ] play a dramatic role and can modify the predictions for the BAU by many orders of magnitude [ 11 , 12 ] . Most importantly, these flavour effects enable the modelling [ 12 ] of minimal RL scenarios with electroweak-scale heavy Majorana neutrinos that could be tested at the LHC [ 14 , 15 , 16 ] and in other non-accelerator experiments, while maintaining agreement with the low-energy neutrino data. Many variants of RL have been proposed in the literature [ 17 , 18 ] , including soft leptogenesis [ 19 ] and radiative leptogenesis [ 20 ] . In this review we give a brief exposition of selected topics related to recent advancements of the field. In detail, in Section 2 , we remind ourselves about Sakharov’s basic conditions for baryogenesis and list a few classical scenarios that have been suggested in the literature, including leptogenesis. In Section 3 we discuss the importance of the various sources of flavour effects on leptogenesis. Section 4 presents the field theory of RL, including a discussion of its flavourdynamics. Section 5 discusses the potential particle-physics implications of RL. Finally, Section 6 summarizes the main points of this review.

This is a brief review on the scenario of baryogenesis through leptogenesis. Leptogenesis is an appealing scenario that may relate the observed baryon asymmetry in the Universe to the low-energy neutrino data. In this review talk, particular emphasis is put on recent developments on the field, such as the flavourdynamics of leptogenesis and resonant leptogenesis near the electroweak phase transition. It is illustrated how these recent developments enable the modelling of phenomenologically predictive scenarios that can directly be tested at the LHC and indirectly in low-energy experiments of lepton-number and lepton-flavour violation.

[{"content":"Help me to generate the abstract of a survey paper given the title:*The little review on leptogenesis* and the introduction:An elegant framework to consistently address the observed Baryon Asymmetry in the Universe (BAU) 1 , 2 is leptogenesis 3 . According to the standard paradigm of leptogenesis 4 , minimal extensions of the Standard Model (SM) realize heavy Majorana neutrinos of masses related to the Grand Unified Theory (GUT) scale M GUT ∼ 10 16 similar-to subscript 𝑀 GUT superscript 10 16 M_{\rm GUT}\sim 10^{16} that decay out of equilibrium and create a net excess of lepton number ( L ) 𝐿 (L) . This excess in L 𝐿 L gets then reprocessed into the observed baryon number ( B ) 𝐵 (B) , through the ( B + L ) 𝐵 𝐿 (B+L) -violating sphaleron interactions 5 . The attractive feature of such a scenario is that the GUT-scale heavy Majorana neutrinos could also explain the observed smallness in mass of the SM light neutrinos by means of the so-called seesaw mechanism 6 . The original GUT-scale leptogenesis scenario, however, runs into certain difficulties, when one attempts to explain the flatness of the Universe and other cosmological data 2 within supergravity models of inflation. To avoid overproduction of gravitinos G ~ ~ 𝐺 \widetilde{G} whose late decays may ruin the successful predictions of Big Bang Nucleosynthesis (BBN), the reheat temperature T reh subscript 𝑇 reh T_{\rm reh} of the Universe should be lower than 10 9 superscript 10 9 10^{9} – 10 6 superscript 10 6 10^{6} GeV, for m G ~ = 8 subscript 𝑚 ~ 𝐺 8 m_{\widetilde{G}}=8 –0.2 TeV 7 . This implies that the heavy Majorana neutrinos should accordingly have masses as low as T reh ∼ < 10 9 superscript similar-to subscript 𝑇 reh superscript 10 9 T_{\rm reh}\stackrel{{\scriptstyle<}}{{{}_{\sim}}}10^{9} GeV, thereby rendering the relation of these particles with GUT-scale physics less natural. On the other hand, it proves very difficult to directly probe the heavy-neutrino sector of such a model at high-energy colliders, e.g. at the LHC or ILC, or in any other foreseeable experiment. A potentially interesting solution to the above problems may be obtained within the framework of resonant leptogenesis (RL) 8 . The key aspect of RL is that self-energy effects dominate the leptonic asymmetries 9 , when two heavy Majorana neutrinos happen to have a small mass difference with respect to their actual masses. If this mass difference becomes comparable to the heavy neutrino widths, a resonant enhancement of the leptonic asymmetries takes place that may reach values 𝒪 ​ ( 1 ) 𝒪 1 {\cal O}(1) 8 , 10 . An indispensable feature of RL models is that flavour effects due to the light-to-heavy neutrino Yukawa couplings 11 , 12 , 13 play a dramatic role and can modify the predictions for the BAU by many orders of magnitude 11 , 12 . Most importantly, these flavour effects enable the modelling 12 of minimal RL scenarios with electroweak-scale heavy Majorana neutrinos that could be tested at the LHC 14 , 15 , 16 and in other non-accelerator experiments, while maintaining agreement with the low-energy neutrino data. Many variants of RL have been proposed in the literature 17 , 18 , including soft leptogenesis 19 and radiative leptogenesis 20 . In this review we give a brief exposition of selected topics related to recent advancements of the field. In detail, in Section 2 , we remind ourselves about Sakharovs basic conditions for baryogenesis and list a few classical scenarios that have been suggested in the literature, including leptogenesis. In Section 3 we discuss the importance of the various sources of flavour effects on leptogenesis. Section 4 presents the field theory of RL, including a discussion of its flavourdynamics. Section 5 discusses the potential particle-physics implications of RL. Finally, Section 6 summarizes the main points of this review.","role":"user"},{"content":"Abstract: This is a brief review on the scenario of baryogenesis through leptogenesis. Leptogenesis is an appealing scenario that may relate the observed baryon asymmetry in the Universe to the low-energy neutrino data. In this review talk, particular emphasis is put on recent developments on the field, such as the flavourdynamics of leptogenesis and resonant leptogenesis near the electroweak phase transition. It is illustrated how these recent developments enable the modelling of phenomenologically predictive scenarios that can directly be tested at the LHC and indirectly in low-energy experiments of lepton-number and lepton-flavour violation.","role":"assistant"}]

A review of volatility and option pricing

and Outline This paper provides a review of the most significant volatility models and their related option pricing models, where we survey the development from constant up to stochastic volatility. We define volatility, the volatility types and study the empirical characteristics e.g. leverage effect. We discuss the key attributes of each volatility modelling method, explaining how they capture theoretical and empirical characteristics of implied and realised volatility e.g. time scale variance. We also discuss less commonly known models. The study of volatility has become a significant area of research within financial mathematics. Firstly, volatility helps us understand price dynamics since it is one of the key variables in a stochastic differential equation governing an asset price. Secondly, volatility is the only variable in the Black-Scholes option pricing equation that is unobservable, hence the ability to model volatility is crucial to option pricing. Thirdly, volatility is a crucial factor in a wide range of research areas. For example, contagion effects involve the “transmission” of volatility from one country to another [ 9 ] . Volatility can explain extreme events as Blake [ 17 ] explains that the October 1987 crash could have resulted from volatility changes. Finally, volatility has a wide range of industrial applications from pricing exotic derivatives to asset pricing models [ 80 ] . Shiller [ 94 ] argues the market’s volatility dynamics can be applied to macroeconomic variables, particularly as the stock market is a well known leading indicator of the economy. Shiller [ 85 ] also claims volatility can be used as a measure of market efficiency. Option pricing in itself has become an important research area. Research interest in options pricing began with the Black-Scholes option pricing paper [ 24 ] ; since then the derivatives market has grown into a multi-trillion dollar industry [ 97 ] . Options have become important to industry, particularly as they can be used to hedge out risk. In fact in many situations it is more attractive to speculators and hedgers to trade an option rather than an underlying due to the limited loss. Additionally, option trading can normally be executed on a far higher level of leverage compared to trading stocks, therefore offering potentially higher returns for the same initial deposit. The outline of the paper is as follows. Firstly, we review basic financial mathematics theory, which is essential for the study of volatility modelling and option pricing. Next, we introduce the differing types of volatility and discuss their empirical behaviour e.g. leverage effect. We then discuss the key models of volatility and their associated option pricing methods. We finally end with a conclusion.

The literature on volatility modelling and option pricing is a large and diverse area due to its importance and applications. This paper provides a review of the most significant volatility models and option pricing methods, beginning with constant volatility models up to stochastic volatility. We also survey less commonly known models e.g., hybrid models. We explain various volatility types (e.g., realised and implied volatility) and discuss the empirical properties.

[{"content":"Help me to generate the abstract of a survey paper given the title:*A review of volatility and option pricing* and the introduction:and Outline This paper provides a review of the most significant volatility models and their related option pricing models, where we survey the development from constant up to stochastic volatility. We define volatility, the volatility types and study the empirical characteristics e.g. leverage effect. We discuss the key attributes of each volatility modelling method, explaining how they capture theoretical and empirical characteristics of implied and realised volatility e.g. time scale variance. We also discuss less commonly known models. The study of volatility has become a significant area of research within financial mathematics. Firstly, volatility helps us understand price dynamics since it is one of the key variables in a stochastic differential equation governing an asset price. Secondly, volatility is the only variable in the Black-Scholes option pricing equation that is unobservable, hence the ability to model volatility is crucial to option pricing. Thirdly, volatility is a crucial factor in a wide range of research areas. For example, contagion effects involve the transmission of volatility from one country to another 9 . Volatility can explain extreme events as Blake 17 explains that the October 1987 crash could have resulted from volatility changes. Finally, volatility has a wide range of industrial applications from pricing exotic derivatives to asset pricing models 80 . Shiller 94 argues the markets volatility dynamics can be applied to macroeconomic variables, particularly as the stock market is a well known leading indicator of the economy. Shiller 85 also claims volatility can be used as a measure of market efficiency. Option pricing in itself has become an important research area. Research interest in options pricing began with the Black-Scholes option pricing paper 24 ; since then the derivatives market has grown into a multi-trillion dollar industry 97 . Options have become important to industry, particularly as they can be used to hedge out risk. In fact in many situations it is more attractive to speculators and hedgers to trade an option rather than an underlying due to the limited loss. Additionally, option trading can normally be executed on a far higher level of leverage compared to trading stocks, therefore offering potentially higher returns for the same initial deposit. The outline of the paper is as follows. Firstly, we review basic financial mathematics theory, which is essential for the study of volatility modelling and option pricing. Next, we introduce the differing types of volatility and discuss their empirical behaviour e.g. leverage effect. We then discuss the key models of volatility and their associated option pricing methods. We finally end with a conclusion.","role":"user"},{"content":"Abstract: The literature on volatility modelling and option pricing is a large and diverse area due to its importance and applications. This paper provides a review of the most significant volatility models and option pricing methods, beginning with constant volatility models up to stochastic volatility. We also survey less commonly known models e.g., hybrid models. We explain various volatility types (e.g., realised and implied volatility) and discuss the empirical properties.","role":"assistant"}]

On the Numerical Evaluation of Distributions in Random Matrix Theory: A Review

1. Introduction Random Matrix Theory (RMT) has found many applications, most notably in physics, multivariate statistics, electrical engineering, and finance. As soon as there is the need for specific numbers, such as moments, quantiles, or correlations, the actual numerical evaluation of the underlying probability distributions becomes of interest. Without additional structure there would be, in general, only one method: Monte Carlo simulation. However, because of the universality of certain scaling limits (for a review see, e.g., ? ), a family of distinguished distribution functions enters which is derived from highly structured matrix models enjoying closed analytic solutions. These functions constitute a new class of special functions comparable in import to the classic distributions of probability theory. This paper addresses the accurate numerical evaluation 1 1 1 Limiting the means, as customary in numerical analysis for reasons of efficiency and strict adherence to numerical stability, to IEEE double precision hardware arithmetic (about 16 digits precision). of many of these functions on the one hand and shows, on the other hand, that such work facilitates numerical explorations that may lead, in the sense of Experimental Mathematics ( ? ), to new theoretical discoveries, see the results of Section 6 . 1.1. The Common Point of View The closed analytic solutions alluded to above are based (for deeper reasons or because of contingency) on two concurrent tools: Fredholm determinants of integral operators and Painlevé transcendents. Concerning the question which of them is better suited to be attacked numerically, there has been a prevailing point of view for the last 15 years or so, explicitly formulated by ? : “Without the Painlevé representation, the numerical evaluation of the Fredholm determinants is quite involved.” To understand the possible genesis of this point of view let us recall the results for the two most important scaling limits of the Gaussian Unitary Ensemble (GUE). 1.1.1. Level Spacing Function of GUE The large matrix limit of GUE, scaled for level spacing 1 1 1 in the bulk, yields the function E 2 ​ ( 0 ; s ) = ℙ ​ ( no levels lie in ( 0 , s ) ) . subscript 𝐸 2 0 𝑠 ℙ no levels lie in ( 0 , s ) E_{2}(0;s)={\mathbb{P}}(\text{no levels lie in $(0,s)$}). (1.1) ? showed that this function can be represented as a Fredholm determinant, namely, 2 2 2 We use the same symbol K 𝐾 K to denote both, the integral operator K ↾ X subscript ↾ 𝑋 𝐾 absent K\negthickspace\upharpoonright_{X} acting on the Hilbert space X 𝑋 X and its kernel function K ​ ( x , y ) 𝐾 𝑥 𝑦 K(x,y) . E 2 ( 0 ; s ) = det ( I − K sin ↾ L 2 ​ ( 0 , s ) ) , K sin ( x , y ) = sinc ( π ( x − y ) ) . E_{2}(0;s)=\det\left(I-K_{\sin}\negthickspace\upharpoonright_{L^{2}(0,s)}\right),\qquad K_{\sin}(x,y)={\operator@font sinc}(\pi(x-y)). (1.2) He proceeded by showing that the eigenfunctions of this selfadjoint integral operator are the radial prolate spheroidal wave functions with certain parameters. Using tables ( ? ) of these special functions he was finally able to evaluate E 2 ​ ( 0 ; s ) subscript 𝐸 2 0 𝑠 E_{2}(0;s) numerically. 3 3 3 Strictly speaking ? was concerned with evaluating the level spacing function E 1 ​ ( 0 ; s ) subscript 𝐸 1 0 𝑠 E_{1}(0;s) of GOE that he represented as the Fredholm determinant of the even sine kernel, see ( 5.8a ) below. However, the extension of Gaudin’s method to E 2 ​ ( 0 ; s ) subscript 𝐸 2 0 𝑠 E_{2}(0;s) is fairly straightforward, see ? for a determinantal formula that is equivalent to ( 1.2 ), namely ( 5.7 ) with k = 0 𝑘 0 k=0 , and ? for subsequent numerical work. As was pointed out by ? , who himself had calculated E 2 ​ ( 0 ; s ) subscript 𝐸 2 0 𝑠 E_{2}(0;s) by Gaudin’s method using Van Buren’s implementation of the prolate wave functions, Kahn’s tables, reproduced in the first 1967 edition of Mehta’s book, are rather inaccurate. In contrast, the tables in ? , reproduced in the second 1991 and third 2002 edition of Mehta’s book, are basically accurate with the proviso that the arguments have to be read not as the displayed four digit numbers but rather as s = 2 ​ t / π 𝑠 2 𝑡 𝜋 s=2t/\pi with t = 0.0 𝑡 0.0 t=0.0 , 0.1 0.1 0.1 , 0.2 0.2 0.2 , 0.3 0.3 0.3 , etc. A modern implementation of Gaudin’s method for E 2 ​ ( 0 ; s ) subscript 𝐸 2 0 𝑠 E_{2}(0;s) , using Mathematica’s fairly recent ability to evaluate the prolate wave functions, can be found in ? . On the other hand, in an admirably intricate analytic tour de force ? expressed the Fredholm determinant by E s ​ ( 0 ; s ) = exp ⁡ ( − ∫ 0 π ​ s σ ​ ( x ) x ​ 𝑑 x ) subscript 𝐸 𝑠 0 𝑠 superscript subscript 0 𝜋 𝑠 𝜎 𝑥 𝑥 differential-d 𝑥 E_{s}(0;s)=\exp\left(-\int_{0}^{\pi s}\frac{\sigma(x)}{x}\,dx\right) (1.3) in terms of the Jimbo–Miwa–Okamoto σ 𝜎 \sigma -form of Painlevé V, namely ( x ​ σ x ​ x ) 2 = 4 ​ ( σ − x ​ σ x ) ​ ( x ​ σ x − σ − σ x 2 ) , σ ​ ( x ) ≃ x π + x 2 π 2 ( x → 0 ) . formulae-sequence superscript 𝑥 subscript 𝜎 𝑥 𝑥 2 4 𝜎 𝑥 subscript 𝜎 𝑥 𝑥 subscript 𝜎 𝑥 𝜎 superscript subscript 𝜎 𝑥 2 similar-to-or-equals 𝜎 𝑥 𝑥 𝜋 superscript 𝑥 2 superscript 𝜋 2 → 𝑥 0 (x\sigma_{xx})^{2}=4(\sigma-x\sigma_{x})(x\sigma_{x}-\sigma-\sigma_{x}^{2}),\qquad\sigma(x)\simeq\frac{x}{\pi}+\frac{x^{2}}{\pi^{2}}\quad(x\to 0). (1.4) 1.1.2. The Tracy–Widom distribution The large matrix limit of GUE, scaled for the fluctuations at the soft edge (that is, the maximum eigenvalue), yields the function F 2 ​ ( s ) = ℙ ​ ( no levels lie in ( s , ∞ ) ) . subscript 𝐹 2 𝑠 ℙ no levels lie in ( s , ∞ ) F_{2}(s)={\mathbb{P}}(\text{no levels lie in $(s,\infty)$}). (1.5) Implicitly known for quite some time (see, e.g., ? , ? , ? ), the determinantal representation F 2 ( s ) = det ( I − K Ai ↾ L 2 ​ ( s , ∞ ) ) , K Ai ( x , y ) = Ai ​ ( x ) ​ Ai ′ ​ ( y ) − Ai ′ ​ ( x ) ​ Ai ​ ( y ) x − y , F_{2}(s)=\det\left(I-K_{{\operator@font Ai}}\negthickspace\upharpoonright_{L^{2}(s,\infty)}\right),\qquad K_{{\operator@font Ai}}(x,y)=\frac{{\operator@font Ai}(x){\operator@font Ai}^{\prime}(y)-{\operator@font Ai}^{\prime}(x){\operator@font Ai}(y)}{x-y}, (1.6) was spelt out by ? and by ? . The search for an analogue to Gaudin’s method remained unsuccessful since there is no solution of the corresponding eigenvalue problem known in terms of classic special functions ( ? , p. 453). It was therefore a major breakthrough when ? ( ? , ? ) derived their now famous representation F 2 ​ ( s ) = exp ⁡ ( − ∫ s ∞ ( x − s ) ​ u ​ ( x ) 2 ​ 𝑑 x ) subscript 𝐹 2 𝑠 superscript subscript 𝑠 𝑥 𝑠 𝑢 superscript 𝑥 2 differential-d 𝑥 F_{2}(s)=\exp\left(-\int_{s}^{\infty}(x-s)u(x)^{2}\,dx\right) (1.7) in terms of the Hastings–McLeod ( ? ) solution u ​ ( x ) 𝑢 𝑥 u(x) of Painlevé II, namely u x ​ x = 2 ​ u 3 + x ​ u , u ​ ( x ) ≃ Ai ​ ( x ) ( x → ∞ ) . formulae-sequence subscript 𝑢 𝑥 𝑥 2 superscript 𝑢 3 𝑥 𝑢 similar-to-or-equals 𝑢 𝑥 Ai 𝑥 → 𝑥 u_{xx}=2u^{3}+xu,\qquad u(x)\simeq{\operator@font Ai}(x)\quad(x\to\infty). (1.8) Subsequent numerical evaluations were then, until the recent work of ? , exclusively based on solving this asymptotic initial value problem. 1.2. Challenging the Common Point of View In this paper we challenge the common point of view that a Painlevé representation would be, at least numerically, preferable to a Fredholm determinant formula. We do so from the following angles: Simplicity, efficiency, accuracy, and extendibility. Let us briefly indicate the rationale for our point of view. (1) The numerical evaluation of Painlevé transcendents encountered in RMT is more involved as one would think at first sight. For reasons of numerical stability one needs additional, deep analytic knowledge, namely, asymptotic expansions of the corresponding connection formulae (see Section 3 ). (2) There is an extremely simple, fast, accurate, and general numerical method for evaluating Fredholm determinants (see Section 4 ). (3) Multivariate functions such as joint probability distributions often have a representation by a Fredholm determinant (see Section 8 ). On the other hand, if available at all, a representation in terms of a nonlinear partial differential equation is of very limited numerical use right now. 1.3. Outline of the Paper In Section 2 we collect some fundamental functions of RMT whose numerical solutions will play a role in the sequel. The intricate issues of a numerical solution of the Painlevé transcendents encountered in RMT are subject of Section 3 . An exposition of ? method for the numerical evaluation of Fredholm determinants is given in Section 4 . The numerical evaluation of the k 𝑘 k -level spacings in the bulk of GOE, GUE, and GSE, by using Fredholm determinants, is addressed in Section 5 . In Section 6 we get to new determinantal formulae for the distributions of the k 𝑘 k -th largest level in the soft edge scaling limit of GOE and GSE. These formulae rely on a determinantal identity that we found by extensive numerical experiments before proving it. By a powerful structural analogy, in Section 7 these formulae are easily extended to the k 𝑘 k -th smallest level in the hard edge scaling limit of LOE and LSE. In Section 8 , we discuss some examples of joint probabilities, like the one for the largest two eigenvalues of GUE at the soft edge or the one of the Airy process for two different times. Finally, in Section 9 , we give a short introduction into using the Matlab toolbox that comes with this paper.

In this paper we review and compare the numerical evaluation of those probability distributions in random matrix theory that are analytically represented in terms of Painleve transcendents or Fredholm determinants. Con- crete examples for the Gaussian and Laguerre (Wishart) β-ensembles and their various scaling limits are discussed. We argue that the numerical approximation of Fredholm determinants is the conceptually more simple and efficient of the two approaches, easily generalized to the computation of joint probabilities and correlations. Having the means for extensive numerical explorations at hand, we discovered new and surprising determinantal formulae for the kth largest (or smallest) level in the edge scaling limits of the Orthogonal and Symplec- tic Ensembles; formulae that in turn led to improved numerical evaluations. The paper comes with a toolbox of Matlab functions that facilitates further mathematical experiments by the reader.

[{"content":"Help me to generate the abstract of a survey paper given the title:*On the Numerical Evaluation of Distributions in Random Matrix Theory: A Review* and the introduction:1. Introduction Random Matrix Theory (RMT) has found many applications, most notably in physics, multivariate statistics, electrical engineering, and finance. As soon as there is the need for specific numbers, such as moments, quantiles, or correlations, the actual numerical evaluation of the underlying probability distributions becomes of interest. Without additional structure there would be, in general, only one method: Monte Carlo simulation. However, because of the universality of certain scaling limits (for a review see, e.g., ? ), a family of distinguished distribution functions enters which is derived from highly structured matrix models enjoying closed analytic solutions. These functions constitute a new class of special functions comparable in import to the classic distributions of probability theory. This paper addresses the accurate numerical evaluation 1 1 1 Limiting the means, as customary in numerical analysis for reasons of efficiency and strict adherence to numerical stability, to IEEE double precision hardware arithmetic (about 16 digits precision). of many of these functions on the one hand and shows, on the other hand, that such work facilitates numerical explorations that may lead, in the sense of Experimental Mathematics ( ? ), to new theoretical discoveries, see the results of Section 6 . 1.1. The Common Point of View The closed analytic solutions alluded to above are based (for deeper reasons or because of contingency) on two concurrent tools: Fredholm determinants of integral operators and Painlevé transcendents. Concerning the question which of them is better suited to be attacked numerically, there has been a prevailing point of view for the last 15 years or so, explicitly formulated by ? : Without the Painlevé representation, the numerical evaluation of the Fredholm determinants is quite involved. To understand the possible genesis of this point of view let us recall the results for the two most important scaling limits of the Gaussian Unitary Ensemble (GUE). 1.1.1. Level Spacing Function of GUE The large matrix limit of GUE, scaled for level spacing 1 1 1 in the bulk, yields the function E 2 ​ ( 0 ; s ) = ℙ ​ ( no levels lie in ( 0 , s ) ) . subscript 𝐸 2 0 𝑠 ℙ no levels lie in ( 0 , s ) E_{2}(0;s)={\mathbb{P}}(\text{no levels lie in $(0,s)$}). (1.1) ? showed that this function can be represented as a Fredholm determinant, namely, 2 2 2 We use the same symbol K 𝐾 K to denote both, the integral operator K ↾ X subscript ↾ 𝑋 𝐾 absent K\negthickspace\upharpoonright_{X} acting on the Hilbert space X 𝑋 X and its kernel function K ​ ( x , y ) 𝐾 𝑥 𝑦 K(x,y) . E 2 ( 0 ; s ) = det ( I − K sin ↾ L 2 ​ ( 0 , s ) ) , K sin ( x , y ) = sinc ( π ( x − y ) ) . E_{2}(0;s)=\det\left(I-K_{\sin}\negthickspace\upharpoonright_{L^{2}(0,s)}\right),\qquad K_{\sin}(x,y)={\operator@font sinc}(\pi(x-y)). (1.2) He proceeded by showing that the eigenfunctions of this selfadjoint integral operator are the radial prolate spheroidal wave functions with certain parameters. Using tables ( ? ) of these special functions he was finally able to evaluate E 2 ​ ( 0 ; s ) subscript 𝐸 2 0 𝑠 E_{2}(0;s) numerically. 3 3 3 Strictly speaking ? was concerned with evaluating the level spacing function E 1 ​ ( 0 ; s ) subscript 𝐸 1 0 𝑠 E_{1}(0;s) of GOE that he represented as the Fredholm determinant of the even sine kernel, see ( 5.8a ) below. However, the extension of Gaudins method to E 2 ​ ( 0 ; s ) subscript 𝐸 2 0 𝑠 E_{2}(0;s) is fairly straightforward, see ? for a determinantal formula that is equivalent to ( 1.2 ), namely ( 5.7 ) with k = 0 𝑘 0 k=0 , and ? for subsequent numerical work. As was pointed out by ? , who himself had calculated E 2 ​ ( 0 ; s ) subscript 𝐸 2 0 𝑠 E_{2}(0;s) by Gaudins method using Van Burens implementation of the prolate wave functions, Kahns tables, reproduced in the first 1967 edition of Mehtas book, are rather inaccurate. In contrast, the tables in ? , reproduced in the second 1991 and third 2002 edition of Mehtas book, are basically accurate with the proviso that the arguments have to be read not as the displayed four digit numbers but rather as s = 2 ​ t / π 𝑠 2 𝑡 𝜋 s=2t/\pi with t = 0.0 𝑡 0.0 t=0.0 , 0.1 0.1 0.1 , 0.2 0.2 0.2 , 0.3 0.3 0.3 , etc. A modern implementation of Gaudins method for E 2 ​ ( 0 ; s ) subscript 𝐸 2 0 𝑠 E_{2}(0;s) , using Mathematicas fairly recent ability to evaluate the prolate wave functions, can be found in ? . On the other hand, in an admirably intricate analytic tour de force ? expressed the Fredholm determinant by E s ​ ( 0 ; s ) = exp ⁡ ( − ∫ 0 π ​ s σ ​ ( x ) x ​ 𝑑 x ) subscript 𝐸 𝑠 0 𝑠 superscript subscript 0 𝜋 𝑠 𝜎 𝑥 𝑥 differential-d 𝑥 E_{s}(0;s)=\exp\left(-\int_{0}^{\pi s}\frac{\sigma(x)}{x}\,dx\right) (1.3) in terms of the Jimbo–Miwa–Okamoto σ 𝜎 \sigma -form of Painlevé V, namely ( x ​ σ x ​ x ) 2 = 4 ​ ( σ − x ​ σ x ) ​ ( x ​ σ x − σ − σ x 2 ) , σ ​ ( x ) ≃ x π + x 2 π 2 ( x → 0 ) . formulae-sequence superscript 𝑥 subscript 𝜎 𝑥 𝑥 2 4 𝜎 𝑥 subscript 𝜎 𝑥 𝑥 subscript 𝜎 𝑥 𝜎 superscript subscript 𝜎 𝑥 2 similar-to-or-equals 𝜎 𝑥 𝑥 𝜋 superscript 𝑥 2 superscript 𝜋 2 → 𝑥 0 (x\sigma_{xx})^{2}=4(\sigma-x\sigma_{x})(x\sigma_{x}-\sigma-\sigma_{x}^{2}),\qquad\sigma(x)\simeq\frac{x}{\pi}+\frac{x^{2}}{\pi^{2}}\quad(x\to 0). (1.4) 1.1.2. The Tracy–Widom distribution The large matrix limit of GUE, scaled for the fluctuations at the soft edge (that is, the maximum eigenvalue), yields the function F 2 ​ ( s ) = ℙ ​ ( no levels lie in ( s , ∞ ) ) . subscript 𝐹 2 𝑠 ℙ no levels lie in ( s , ∞ ) F_{2}(s)={\mathbb{P}}(\text{no levels lie in $(s,\infty)$}). (1.5) Implicitly known for quite some time (see, e.g., ? , ? , ? ), the determinantal representation F 2 ( s ) = det ( I − K Ai ↾ L 2 ​ ( s , ∞ ) ) , K Ai ( x , y ) = Ai ​ ( x ) ​ Ai ′ ​ ( y ) − Ai ′ ​ ( x ) ​ Ai ​ ( y ) x − y , F_{2}(s)=\det\left(I-K_{{\operator@font Ai}}\negthickspace\upharpoonright_{L^{2}(s,\infty)}\right),\qquad K_{{\operator@font Ai}}(x,y)=\frac{{\operator@font Ai}(x){\operator@font Ai}^{\prime}(y)-{\operator@font Ai}^{\prime}(x){\operator@font Ai}(y)}{x-y}, (1.6) was spelt out by ? and by ? . The search for an analogue to Gaudins method remained unsuccessful since there is no solution of the corresponding eigenvalue problem known in terms of classic special functions ( ? , p. 453). It was therefore a major breakthrough when ? ( ? , ? ) derived their now famous representation F 2 ​ ( s ) = exp ⁡ ( − ∫ s ∞ ( x − s ) ​ u ​ ( x ) 2 ​ 𝑑 x ) subscript 𝐹 2 𝑠 superscript subscript 𝑠 𝑥 𝑠 𝑢 superscript 𝑥 2 differential-d 𝑥 F_{2}(s)=\exp\left(-\int_{s}^{\infty}(x-s)u(x)^{2}\,dx\right) (1.7) in terms of the Hastings–McLeod ( ? ) solution u ​ ( x ) 𝑢 𝑥 u(x) of Painlevé II, namely u x ​ x = 2 ​ u 3 + x ​ u , u ​ ( x ) ≃ Ai ​ ( x ) ( x → ∞ ) . formulae-sequence subscript 𝑢 𝑥 𝑥 2 superscript 𝑢 3 𝑥 𝑢 similar-to-or-equals 𝑢 𝑥 Ai 𝑥 → 𝑥 u_{xx}=2u^{3}+xu,\qquad u(x)\simeq{\operator@font Ai}(x)\quad(x\to\infty). (1.8) Subsequent numerical evaluations were then, until the recent work of ? , exclusively based on solving this asymptotic initial value problem. 1.2. Challenging the Common Point of View In this paper we challenge the common point of view that a Painlevé representation would be, at least numerically, preferable to a Fredholm determinant formula. We do so from the following angles: Simplicity, efficiency, accuracy, and extendibility. Let us briefly indicate the rationale for our point of view. (1) The numerical evaluation of Painlevé transcendents encountered in RMT is more involved as one would think at first sight. For reasons of numerical stability one needs additional, deep analytic knowledge, namely, asymptotic expansions of the corresponding connection formulae (see Section 3 ). (2) There is an extremely simple, fast, accurate, and general numerical method for evaluating Fredholm determinants (see Section 4 ). (3) Multivariate functions such as joint probability distributions often have a representation by a Fredholm determinant (see Section 8 ). On the other hand, if available at all, a representation in terms of a nonlinear partial differential equation is of very limited numerical use right now. 1.3. Outline of the Paper In Section 2 we collect some fundamental functions of RMT whose numerical solutions will play a role in the sequel. The intricate issues of a numerical solution of the Painlevé transcendents encountered in RMT are subject of Section 3 . An exposition of ? method for the numerical evaluation of Fredholm determinants is given in Section 4 . The numerical evaluation of the k 𝑘 k -level spacings in the bulk of GOE, GUE, and GSE, by using Fredholm determinants, is addressed in Section 5 . In Section 6 we get to new determinantal formulae for the distributions of the k 𝑘 k -th largest level in the soft edge scaling limit of GOE and GSE. These formulae rely on a determinantal identity that we found by extensive numerical experiments before proving it. By a powerful structural analogy, in Section 7 these formulae are easily extended to the k 𝑘 k -th smallest level in the hard edge scaling limit of LOE and LSE. In Section 8 , we discuss some examples of joint probabilities, like the one for the largest two eigenvalues of GUE at the soft edge or the one of the Airy process for two different times. Finally, in Section 9 , we give a short introduction into using the Matlab toolbox that comes with this paper.","role":"user"},{"content":"Abstract: In this paper we review and compare the numerical evaluation of those probability distributions in random matrix theory that are analytically represented in terms of Painleve transcendents or Fredholm determinants. Con- crete examples for the Gaussian and Laguerre (Wishart) β-ensembles and their various scaling limits are discussed. We argue that the numerical approximation of Fredholm determinants is the conceptually more simple and efficient of the two approaches, easily generalized to the computation of joint probabilities and correlations. Having the means for extensive numerical explorations at hand, we discovered new and surprising determinantal formulae for the kth largest (or smallest) level in the edge scaling limits of the Orthogonal and Symplec- tic Ensembles; formulae that in turn led to improved numerical evaluations. The paper comes with a toolbox of Matlab functions that facilitates further mathematical experiments by the reader.","role":"assistant"}]

Algebraic Topology Foundations of Supersymmetry and Symmetry Breaking in Quantum Field Theory and Quantum Gravity: A Review

The theory of scattering by partially ordered, atomic or molecular, structures in terms of paracrystals and lattice convolutions was formulated by Hosemann and Bagchi in [ 122 ] using basic techniques of Fourier analysis and convolution products. A natural generalization of such molecular, partial symmetries and their corresponding analytical versions involves convolution algebras – a functional/distribution [ 198 , 199 ] based theory that we will discuss in the context of a more general and original concept of a convolution-algebroid of an extended symmetry groupoid of a paracrystal , of any molecular or nuclear system, or indeed, any quantum system in general, including quantum fields and local quantum net configurations that are endowed with either partially disordered or ‘completely’ ordered structures. Further specific applications of the paracrystal theory to X 𝑋 X -ray scattering, based on computer algorithms, programs and explicit numerical computations, were subsequently developed by the first author [ 12 ] for one-dimensional paracrystals, partially ordered membrane lattices [ 13 ] and other biological structures with partial structural disorder [ 19 ] . Such biological structures, ‘quasi-crystals’, and the paracrystals, in general, provide rather interesting physical examples of such extended symmetries (cf. [ 121 ] ). Further statistical analysis linked to structural symmetry and scattering theory considerations shows that a real paracrystal can be defined by a three dimensional convolution polynomial with a semi-empirically derived composition law, ∗ * , [ 123 ] . As was shown in [ 12 , 13 ] – supported with computed specific examples – several systems of convolution can be expressed analytically, thus allowing the numerical computation of X 𝑋 X -ray, or neutron, scattering by partially disordered layer lattices via complex Fourier transforms of one-dimensional structural models using fast digital computers. The range of paracrystal theory applications is however much wider than the one-dimensional lattices with disorder, thus spanning very diverse non-crystalline systems, from metallic glasses and spin glasses to superfluids, high-temperature superconductors, and extremely hot anisotropic plasmas such as those encountered in controlled nuclear fusion (for example, JET) experiments. Other applications – as previously suggested in [ 11 ] – may also include novel designs of ‘fuzzy’ quantum machines and quantum computers with extended symmetries of quantum state spaces. 1.1 Convolution product of groupoids and the convolution algebra of functions A salient, and well-fathomed concept from the mathematical perspective concerns that of a C ∗ superscript 𝐶 ∗ C^{\ast} -algebra of a (discrete) group (see, e.g., [ 74 ] ). The underlying vector space is that of complex valued functions with finite support, and the multiplication of the algebra is the fundamental convolution product which it is convenient for our purposes to write slightly differently from the common formula as ( f ∗ g ) ​ ( z ) = ∑ x ​ y = z f ​ ( x ) ​ g ​ ( y ) , ∗ 𝑓 𝑔 𝑧 subscript 𝑥 𝑦 𝑧 𝑓 𝑥 𝑔 𝑦 \displaystyle(f\ast g)(z)=\sum_{xy=z}f(x)g(y), and ∗ ∗ \ast -operation f ∗ ​ ( x ) = f ​ ( x − 1 ) ¯ . superscript 𝑓 ∗ 𝑥 ¯ 𝑓 superscript 𝑥 1 \displaystyle f^{\ast}(x)=\overline{f(x^{-1})}. The more usual expression of these formulas has a sum over the elements of the group. For topological groups, where the underlying vector space consists of continuous complex valued functions, this product requires the availability of some structure of measure and of measurable functions, with the sum replaced by an integral. Notice also that this algebra has an identity, the distribution function δ 1 subscript 𝛿 1 \delta_{1} , which has value 1 on the identity 1 of the group, and has zero value elsewhere. Given this convolution/distribution representation that combines crystalline (‘perfect’ or global-group, and/or group-like symmetries) with partial symmetries of paracrystals and glassy solids on the one hand, and also with non-commutative harmonic analysis [ 152 ] on the other hand, we propose that several extended quantum symmetries can be represented algebraically in terms of certain structured groupoids , their C ∗ superscript 𝐶 C^{*} -convolution quantum algebroids , paragroup/ quantized groups and/or other more general mathematical structures that will be introduced in this report. It is already known that such extensions to groupoid and algebroid/coalgebroid symmetries require also a generalization of non-commutative harmonic analysis which involves certain Haar measures, generalized Fourier–Stieltjes transforms and certain categorical duality relationships representing very general mathematical symmetries as well. Proceeding from the abstract structures endowed with extended symmetries to numerical applications in quantum physics always involves representations through specification of concrete elements, objects and transformations. Thus, groupoid and functorial representations that generalize group representations in several, meaningful ways are key to linking abstract, quantum operator algebras and symmetry properties with actual numerical computations of quantum eigenvalues and their eigenstates, as well as a wide variety of numerical factors involved in computing quantum dynamics. The well-known connection between groupoid convolution representations and matrices [ 213 ] is only one of the several numerical computations made possible via groupoid representations. A very promising approach to nonlinear (anharmonic) analysis of aperiodic quantum systems represented by rigged Hilbert space bundles may involve the computation of representation coefficients of Fourier–Stieltjes groupoid transforms that we will also discuss briefly in Section 7 . Currently, however, there are important aspects of quantum dynamics left out of the invariant, simplified picture provided by group symmetries and their corresponding representations of quantum operator algebras [ 103 ] . An alternative approach proposed in [ 115 ] employs differential forms to find symmetries. Often physicists deal with such problems in terms of either spontaneous symmetry breaking or approximate symmetries that require underlying explanations or ad-hoc dynamic restrictions that are semi-empirical. A well-studied example of this kind is that of the dynamic Jahn–Teller effect and the corresponding ‘theorem’ (Chapter 21 on pp. 807–831, as well as p. 735 of [ 2 ] ) which in its simplest form stipulates that a quantum state with electronic non-Kramers degeneracy may be unstable against small distortions of the surroundings, that would lower the symmetry of the crystal field and thus lift the degeneracy (i.e., cause observable splitting of the corresponding energy levels); this effect occurs in certain paramagnetic ion systems via dynamic distortions of the crystal field symmetries around paramagnetic or high-spin centers by moving ligands that are diamagnetic. The established physical explanation is that the Jahn–Teller coupling replaces a purely electronic degeneracy by a vibronic degeneracy (of exactly the same symmetry!). The dynamic, or spontaneous breaking of crystal field symmetry (for example, distortions of the octahedral or cubic symmetry) results in certain systems in the appearance of doublets of symmetry γ 3 subscript 𝛾 3 \gamma_{3} or singlets of symmetry γ 1 subscript 𝛾 1 \gamma_{1} or γ 2 subscript 𝛾 2 \gamma_{2} . Such dynamic systems could be locally expressed in terms of symmetry representations of a Lie algebroid, or globally in terms of a special Lie (or Lie–Weinstein) symmetry groupoid representations that can also take into account the spin exchange interactions between the Jahn–Teller centers exhibiting such quantum dynamic effects. Unlike the simple symmetries expressed by group representations, the latter can accommodate a much wider range of possible or approximate symmetries that are indeed characteristic of real, molecular systems with varying crystal field symmetry, as for example around certain transition ions dynamically bound to ligands in liquids where motional narrowing becomes very important. This well known example illustrates the importance of the interplay between symmetry and dynamics in quantum processes which is undoubtedly involved in many other instances including: quantum chromodynamics, superfluidity, spontaneous symmetry breaking, quantum gravity and Universe dynamics (i.e., the inflationary Universe). Therefore, the various interactions and interplay between the symmetries of quantum operator state space geometry and quantum dynamics at various levels leads to both algebraic and topological structures that are variable and complex, well beyond symmetry groups and well-studied group algebras (such as Lie algebras, see for example [ 103 ] ). A unified treatment of quantum phenomena/dynamics and structures may thus become possible with the help of algebraic topology, non-Abelian treatments; such powerful mathematical tools are capable of revealing novel, fundamental aspects related to extended symmetries and quantum dynamics through a detailed analysis of the variable geometry of (quantum) operator algebra state spaces. At the center stage of non-Abelian algebraic topology are groupoid and algebroid structures with their internal and external symmetries [ 213 ] that allow one to treat physical spacetime structures and dynamics within an unified categorical, higher dimensional algebra framework [ 45 ] . As already suggested in our previous report, the interplay between extended symmetries and dynamics generates higher dimensional structures of quantized spacetimes that exhibit novel properties not found in lower dimensional representations of groups, group algebras or Abelian groupoids. It is also our intention here to explore, uncover, and then develop, new links between several important but seemingly distinct mathematical approaches to extended quantum symmetries that were not considered in previous reports.

A novel algebraic topology approach to supersymmetry (SUSY) and symmet- ry breaking in quantum field and quantum gravity theories is presented with a view to developing a wide range of physical applications. These include: controlled nuclear fusion and other nuclear reaction studies in quantum chromodynamics, nonlinear physics at high energy densities, dynamic Jahn-Teller effects, superfluidity, high temperature superconduc- tors, multiple scattering by molecular systems, molecular or atomic paracrystal structures, nanomaterials, ferromagnetism in glassy materials, spin glasses, quantum phase transitions and supergravity. This approach requires a unified conceptual framework that utilizes ex- tended symmetries and quantum groupoid, algebroid and functorial representations of non- Abelian higher dimensional structures pertinent to quantized spacetime topology and state space geometry of quantum operator algebras. Fourier transforms, generalized Fourier- Stieltjes transforms, and duality relations link, respectively, the quantum groups and quan- tum groupoids with their dual algebraic structures; quantum double constructions are also discussed in this context in relation to quasi-triangular, quasi-Hopf algebras, bialgebroids, Grassmann-Hopf algebras and higher dimensional algebra. On the one hand, this quantum algebraic approach is known to provide solutions to the quantum Yang-Baxter equation. On the other hand, our novel approach to extended quantum symmetries and their associated representations is shown to be relevant to locally covariant general relativity theories that are consistent with either nonlocal quantum field theories or local bosonic (spin) models with the extended quantum symmetry of entangled, 'string-net condensed' (ground) states.

[{"content":"Help me to generate the abstract of a survey paper given the title:*Algebraic Topology Foundations of Supersymmetry and Symmetry Breaking in Quantum Field Theory and Quantum Gravity: A Review* and the introduction:The theory of scattering by partially ordered, atomic or molecular, structures in terms of paracrystals and lattice convolutions was formulated by Hosemann and Bagchi in 122 using basic techniques of Fourier analysis and convolution products. A natural generalization of such molecular, partial symmetries and their corresponding analytical versions involves convolution algebras – a functional/distribution 198 , 199 based theory that we will discuss in the context of a more general and original concept of a convolution-algebroid of an extended symmetry groupoid of a paracrystal , of any molecular or nuclear system, or indeed, any quantum system in general, including quantum fields and local quantum net configurations that are endowed with either partially disordered or completely ordered structures. Further specific applications of the paracrystal theory to X 𝑋 X -ray scattering, based on computer algorithms, programs and explicit numerical computations, were subsequently developed by the first author 12 for one-dimensional paracrystals, partially ordered membrane lattices 13 and other biological structures with partial structural disorder 19 . Such biological structures, quasi-crystals, and the paracrystals, in general, provide rather interesting physical examples of such extended symmetries (cf. 121 ). Further statistical analysis linked to structural symmetry and scattering theory considerations shows that a real paracrystal can be defined by a three dimensional convolution polynomial with a semi-empirically derived composition law, ∗ * , 123 . As was shown in 12 , 13 – supported with computed specific examples – several systems of convolution can be expressed analytically, thus allowing the numerical computation of X 𝑋 X -ray, or neutron, scattering by partially disordered layer lattices via complex Fourier transforms of one-dimensional structural models using fast digital computers. The range of paracrystal theory applications is however much wider than the one-dimensional lattices with disorder, thus spanning very diverse non-crystalline systems, from metallic glasses and spin glasses to superfluids, high-temperature superconductors, and extremely hot anisotropic plasmas such as those encountered in controlled nuclear fusion (for example, JET) experiments. Other applications – as previously suggested in 11 – may also include novel designs of fuzzy quantum machines and quantum computers with extended symmetries of quantum state spaces. 1.1 Convolution product of groupoids and the convolution algebra of functions A salient, and well-fathomed concept from the mathematical perspective concerns that of a C ∗ superscript 𝐶 ∗ C^{\ast} -algebra of a (discrete) group (see, e.g., 74 ). The underlying vector space is that of complex valued functions with finite support, and the multiplication of the algebra is the fundamental convolution product which it is convenient for our purposes to write slightly differently from the common formula as ( f ∗ g ) ​ ( z ) = ∑ x ​ y = z f ​ ( x ) ​ g ​ ( y ) , ∗ 𝑓 𝑔 𝑧 subscript 𝑥 𝑦 𝑧 𝑓 𝑥 𝑔 𝑦 \displaystyle(f\ast g)(z)=\sum_{xy=z}f(x)g(y), and ∗ ∗ \ast -operation f ∗ ​ ( x ) = f ​ ( x − 1 ) ¯ . superscript 𝑓 ∗ 𝑥 ¯ 𝑓 superscript 𝑥 1 \displaystyle f^{\ast}(x)=\overline{f(x^{-1})}. The more usual expression of these formulas has a sum over the elements of the group. For topological groups, where the underlying vector space consists of continuous complex valued functions, this product requires the availability of some structure of measure and of measurable functions, with the sum replaced by an integral. Notice also that this algebra has an identity, the distribution function δ 1 subscript 𝛿 1 \delta_{1} , which has value 1 on the identity 1 of the group, and has zero value elsewhere. Given this convolution/distribution representation that combines crystalline (perfect or global-group, and/or group-like symmetries) with partial symmetries of paracrystals and glassy solids on the one hand, and also with non-commutative harmonic analysis 152 on the other hand, we propose that several extended quantum symmetries can be represented algebraically in terms of certain structured groupoids , their C ∗ superscript 𝐶 C^{*} -convolution quantum algebroids , paragroup/ quantized groups and/or other more general mathematical structures that will be introduced in this report. It is already known that such extensions to groupoid and algebroid/coalgebroid symmetries require also a generalization of non-commutative harmonic analysis which involves certain Haar measures, generalized Fourier–Stieltjes transforms and certain categorical duality relationships representing very general mathematical symmetries as well. Proceeding from the abstract structures endowed with extended symmetries to numerical applications in quantum physics always involves representations through specification of concrete elements, objects and transformations. Thus, groupoid and functorial representations that generalize group representations in several, meaningful ways are key to linking abstract, quantum operator algebras and symmetry properties with actual numerical computations of quantum eigenvalues and their eigenstates, as well as a wide variety of numerical factors involved in computing quantum dynamics. The well-known connection between groupoid convolution representations and matrices 213 is only one of the several numerical computations made possible via groupoid representations. A very promising approach to nonlinear (anharmonic) analysis of aperiodic quantum systems represented by rigged Hilbert space bundles may involve the computation of representation coefficients of Fourier–Stieltjes groupoid transforms that we will also discuss briefly in Section 7 . Currently, however, there are important aspects of quantum dynamics left out of the invariant, simplified picture provided by group symmetries and their corresponding representations of quantum operator algebras 103 . An alternative approach proposed in 115 employs differential forms to find symmetries. Often physicists deal with such problems in terms of either spontaneous symmetry breaking or approximate symmetries that require underlying explanations or ad-hoc dynamic restrictions that are semi-empirical. A well-studied example of this kind is that of the dynamic Jahn–Teller effect and the corresponding theorem (Chapter 21 on pp. 807–831, as well as p. 735 of 2 ) which in its simplest form stipulates that a quantum state with electronic non-Kramers degeneracy may be unstable against small distortions of the surroundings, that would lower the symmetry of the crystal field and thus lift the degeneracy (i.e., cause observable splitting of the corresponding energy levels); this effect occurs in certain paramagnetic ion systems via dynamic distortions of the crystal field symmetries around paramagnetic or high-spin centers by moving ligands that are diamagnetic. The established physical explanation is that the Jahn–Teller coupling replaces a purely electronic degeneracy by a vibronic degeneracy (of exactly the same symmetry!). The dynamic, or spontaneous breaking of crystal field symmetry (for example, distortions of the octahedral or cubic symmetry) results in certain systems in the appearance of doublets of symmetry γ 3 subscript 𝛾 3 \gamma_{3} or singlets of symmetry γ 1 subscript 𝛾 1 \gamma_{1} or γ 2 subscript 𝛾 2 \gamma_{2} . Such dynamic systems could be locally expressed in terms of symmetry representations of a Lie algebroid, or globally in terms of a special Lie (or Lie–Weinstein) symmetry groupoid representations that can also take into account the spin exchange interactions between the Jahn–Teller centers exhibiting such quantum dynamic effects. Unlike the simple symmetries expressed by group representations, the latter can accommodate a much wider range of possible or approximate symmetries that are indeed characteristic of real, molecular systems with varying crystal field symmetry, as for example around certain transition ions dynamically bound to ligands in liquids where motional narrowing becomes very important. This well known example illustrates the importance of the interplay between symmetry and dynamics in quantum processes which is undoubtedly involved in many other instances including: quantum chromodynamics, superfluidity, spontaneous symmetry breaking, quantum gravity and Universe dynamics (i.e., the inflationary Universe). Therefore, the various interactions and interplay between the symmetries of quantum operator state space geometry and quantum dynamics at various levels leads to both algebraic and topological structures that are variable and complex, well beyond symmetry groups and well-studied group algebras (such as Lie algebras, see for example 103 ). A unified treatment of quantum phenomena/dynamics and structures may thus become possible with the help of algebraic topology, non-Abelian treatments; such powerful mathematical tools are capable of revealing novel, fundamental aspects related to extended symmetries and quantum dynamics through a detailed analysis of the variable geometry of (quantum) operator algebra state spaces. At the center stage of non-Abelian algebraic topology are groupoid and algebroid structures with their internal and external symmetries 213 that allow one to treat physical spacetime structures and dynamics within an unified categorical, higher dimensional algebra framework 45 . As already suggested in our previous report, the interplay between extended symmetries and dynamics generates higher dimensional structures of quantized spacetimes that exhibit novel properties not found in lower dimensional representations of groups, group algebras or Abelian groupoids. It is also our intention here to explore, uncover, and then develop, new links between several important but seemingly distinct mathematical approaches to extended quantum symmetries that were not considered in previous reports.","role":"user"},{"content":"Abstract: A novel algebraic topology approach to supersymmetry (SUSY) and symmet- ry breaking in quantum field and quantum gravity theories is presented with a view to developing a wide range of physical applications. These include: controlled nuclear fusion and other nuclear reaction studies in quantum chromodynamics, nonlinear physics at high energy densities, dynamic Jahn-Teller effects, superfluidity, high temperature superconduc- tors, multiple scattering by molecular systems, molecular or atomic paracrystal structures, nanomaterials, ferromagnetism in glassy materials, spin glasses, quantum phase transitions and supergravity. This approach requires a unified conceptual framework that utilizes ex- tended symmetries and quantum groupoid, algebroid and functorial representations of non- Abelian higher dimensional structures pertinent to quantized spacetime topology and state space geometry of quantum operator algebras. Fourier transforms, generalized Fourier- Stieltjes transforms, and duality relations link, respectively, the quantum groups and quan- tum groupoids with their dual algebraic structures; quantum double constructions are also discussed in this context in relation to quasi-triangular, quasi-Hopf algebras, bialgebroids, Grassmann-Hopf algebras and higher dimensional algebra. On the one hand, this quantum algebraic approach is known to provide solutions to the quantum Yang-Baxter equation. On the other hand, our novel approach to extended quantum symmetries and their associated representations is shown to be relevant to locally covariant general relativity theories that are consistent with either nonlocal quantum field theories or local bosonic (spin) models with the extended quantum symmetry of entangled, 'string-net condensed' (ground) states.","role":"assistant"}]

Internet Traffic Periodicities and Oscillations: A Brief Review

Internet traffic has exploded in the last fifteen years as an area of intense theoretical and experimental research. As the largest engineered infrastructure and information system in human history, the Internet’s staggering size and complexity are reinforced by its decentralized and self-organizing structure. Using packets of encapsulated data and a commonly agreed protocol suite, the Internet has far outgrown its origins as ARPANET whose traffic has demanded new models and ways of thinking to understand and predict. Amongst the earliest discoveries were the researches of Leland and Wilson [ 1 ] who identified the non-Poisson nature of Internet traffic. This was followed by the seminal paper of Leland, Taqqu, Willinger, and Wilson [ 2 ] which proved that Internet packet interarrival times are both self-similar and portray long-range dependence. Though self-similarity is present at all time scales, it is most well-defined when traffic is stationary, an assumption that can only last a few hours at the most. The lack of stationarity on long time scales is due to one of the most widely known periodicities (or oscillations) in Internet traffic, the diurnal cycle with 12 and 24 hour peaks. Internet periodicities are not new and have been well-studied since the earliest days of large-scale measurements of packet traffic, however, they rarely receive primary attention in discussions of traffic and are often mentioned only as an aside or a footnote. Gradually, however, they are gaining more attention. This new area of research has been dubbed network spectroscopy [ 3 ] or Internet spectroscopy . In this paper, they will take front and center as the most important periodicities, as well as the techniques to measure them, are described.

Internet traffic displays many persistent periodicities (oscillations) on a large range of time scales. This paper describes the measurement methodology to detect Internet traffic periodicities and also describes the main periodicities in Internet traffic.

[{"content":"Help me to generate the abstract of a survey paper given the title:*Internet Traffic Periodicities and Oscillations: A Brief Review* and the introduction:Internet traffic has exploded in the last fifteen years as an area of intense theoretical and experimental research. As the largest engineered infrastructure and information system in human history, the Internets staggering size and complexity are reinforced by its decentralized and self-organizing structure. Using packets of encapsulated data and a commonly agreed protocol suite, the Internet has far outgrown its origins as ARPANET whose traffic has demanded new models and ways of thinking to understand and predict. Amongst the earliest discoveries were the researches of Leland and Wilson 1 who identified the non-Poisson nature of Internet traffic. This was followed by the seminal paper of Leland, Taqqu, Willinger, and Wilson 2 which proved that Internet packet interarrival times are both self-similar and portray long-range dependence. Though self-similarity is present at all time scales, it is most well-defined when traffic is stationary, an assumption that can only last a few hours at the most. The lack of stationarity on long time scales is due to one of the most widely known periodicities (or oscillations) in Internet traffic, the diurnal cycle with 12 and 24 hour peaks. Internet periodicities are not new and have been well-studied since the earliest days of large-scale measurements of packet traffic, however, they rarely receive primary attention in discussions of traffic and are often mentioned only as an aside or a footnote. Gradually, however, they are gaining more attention. This new area of research has been dubbed network spectroscopy 3 or Internet spectroscopy . In this paper, they will take front and center as the most important periodicities, as well as the techniques to measure them, are described.","role":"user"},{"content":"Abstract: Internet traffic displays many persistent periodicities (oscillations) on a large range of time scales. This paper describes the measurement methodology to detect Internet traffic periodicities and also describes the main periodicities in Internet traffic.","role":"assistant"}]

Duality, correspondences and the Lefschetz map in equivariant KK-theory: a survey

1. Abstract duality and the Lefschetz map Throughout this paper, groupoid shall mean locally compact Hausdorff groupoid with Haar system. All topological spaces will be assumed paracompact, locally compact and Hausdorff. For the material in this section, see [ Emerson-Meyer:Dualities ] . For source material on equivariant KK KK \operatorname{KK} -theory for groupoids, see [ LeGall:KK_groupoid ] . One seems to be forced to consider groupoids, as opposed to groups, in equivariant Kasparov theory, even if one is ultimately only interested in groups. This will be explained later. Therefore we will work more or less uniformly with groupoids when discussing general theory. When we discuss topological equivariant Kasparov theory, we will further assume that all groupoids are proper . This restriction is needed for various geometric constructions. The additional assumption of properness involves no serious loss of generality for our purposes because the Baum-Connes isomorphism, when it applies, gives a method of replacing non-proper groupoids by proper ones. 1.1. Equivariant Kasparov theory for groupoids Let 𝒢 𝒢 \mathcal{G} be a groupoid. We let Z 𝑍 Z denote the base space. A 𝒢 𝒢 \mathcal{G} -C*-algebra is in particular a C*-algebra over Z 𝑍 Z . This means that there is given a non-degenerate equivariant *-homomorphism from 𝒞 0 ​ ( Z ) subscript 𝒞 0 𝑍 \mathcal{C}_{0}(Z) to central multipliers of A 𝐴 A . This identifies A 𝐴 A with the section algebra of a continuous bundle of C*-algebras over Z 𝑍 Z . For a groupoid action we require in addition an isomorphism r ∗ ​ ( A ) → s ∗ ​ ( A ) → superscript 𝑟 𝐴 superscript 𝑠 𝐴 r^{*}(A)\to s^{*}(A) which is compatible with the structure of r ∗ ​ ( A ) superscript 𝑟 𝐴 r^{*}(A) and s ∗ ​ ( A ) superscript 𝑠 𝐴 s^{*}(A) as C*-algebras over 𝒢 𝒢 \mathcal{G} . Here r : 𝒢 → Z : 𝑟 → 𝒢 𝑍 r\colon\mathcal{G}\to Z and s : 𝒢 → Z : 𝑠 → 𝒢 𝑍 s\colon\mathcal{G}\to Z are the range and source map of the groupoid, and r ∗ superscript 𝑟 r^{*} (and similarly s ∗ superscript 𝑠 s^{*} ) denotes the usual pullback operation of bundles. From the bundle point of view, all of this means that groupoid elements g 𝑔 g with s ​ ( g ) = x 𝑠 𝑔 𝑥 s(g)=x and r ​ ( g ) = y 𝑟 𝑔 𝑦 r(g)=y induce *-homomorphisms A x → A y → subscript 𝐴 𝑥 subscript 𝐴 𝑦 A_{x}\to A_{y} between the fibres of A 𝐴 A at x 𝑥 x and y 𝑦 y . In particular, if A 𝐴 A is commutative, then A 𝐴 A is the C*-algebra of continuous functions on a locally compact 𝒢 𝒢 \mathcal{G} -space X 𝑋 X , equipped with a map ϱ X : X → Z : subscript italic-ϱ 𝑋 → 𝑋 𝑍 \varrho_{X}\colon X\to Z called the anchor map for X 𝑋 X , and a homeomorphism 𝒢 × Z , s X → 𝒢 × Z , r X , ( g , x ) ↦ ( g , g ​ x ) formulae-sequence → subscript 𝑍 𝑠 𝒢 𝑋 subscript 𝑍 𝑟 𝒢 𝑋 maps-to 𝑔 𝑥 𝑔 𝑔 𝑥 \mathcal{G}\times_{Z,s}X\to\mathcal{G}\times_{Z,r}X,\hskip 28.45274pt(g,x)\mapsto(g,gx) where the domain and range of this homeomorphism (by abuse of notation) are respectively 𝒢 × Z , s X := { ( g , x ) ∈ 𝒢 × X ∣ s ​ ( g ) = ϱ X ​ ( x ) } , assign subscript 𝑍 𝑠 𝒢 𝑋 conditional-set 𝑔 𝑥 𝒢 𝑋 𝑠 𝑔 subscript italic-ϱ 𝑋 𝑥 \mathcal{G}\times_{Z,s}X\mathrel{\vcentcolon=}\{(g,x)\in\mathcal{G}\times X\mid s(g)=\varrho_{X}(x)\}, and similarly for 𝒢 × Z , r X subscript 𝑍 𝑟 𝒢 𝑋 \mathcal{G}\times_{Z,r}X using r 𝑟 r instead of s 𝑠 s . 1.2. Tensor products The category of 𝒢 𝒢 \mathcal{G} -C*-algebras has a symmetric monoidal structure given by tensor products. We describe this very briefly (see [ Emerson-Meyer:Dualities ] [Section 2] for details). Let A 𝐴 A and B 𝐵 B be two 𝒢 𝒢 \mathcal{G} -C*-algebras. Since they are each C*-algebras over Z 𝑍 Z , their external tensor product A ⊗ B tensor-product 𝐴 𝐵 A\otimes B is a C*-algebra over Z × Z 𝑍 𝑍 Z\times Z . We restrict this to a C*-algebra over the diagonal Z ⊂ Z × Z 𝑍 𝑍 𝑍 Z\subset Z\times Z . The result is called the tensor product of A 𝐴 A and B 𝐵 B over Z 𝑍 Z . The tensor product of A 𝐴 A and B 𝐵 B over Z 𝑍 Z carries a diagonal action of 𝒢 𝒢 \mathcal{G} . We leave it to the reader to check that we obtain a 𝒢 𝒢 \mathcal{G} -C*-algebra in this way. In order not to complicate notation, we write just A ⊗ B tensor-product 𝐴 𝐵 A\otimes B for the tensor product of A 𝐴 A and B 𝐵 B in the category of 𝒢 𝒢 \mathcal{G} -C*-algebras. We emphasize that the tensor product is over Z 𝑍 Z ; this is not the same as the tensor product in the category of C*-algebras. For commutative C*-algebras, i.e. for 𝒢 𝒢 \mathcal{G} -spaces, say X 𝑋 X and Y 𝑌 Y , with anchor maps as usual denoted ϱ X : X → Z : subscript italic-ϱ 𝑋 → 𝑋 𝑍 \varrho_{X}\colon X\to Z and ϱ Y : Y → Z : subscript italic-ϱ 𝑌 → 𝑌 𝑍 \varrho_{Y}\colon Y\to Z , the tensor product is Gelfand dual to the operation which forms from X 𝑋 X and Y 𝑌 Y the fibre product X × Z Y := { ( x , y ) ∈ X × Y ∣ ϱ X ​ ( x ) = ϱ Y ​ ( y ) } . assign subscript 𝑍 𝑋 𝑌 conditional-set 𝑥 𝑦 𝑋 𝑌 subscript italic-ϱ 𝑋 𝑥 subscript italic-ϱ 𝑌 𝑦 X\times_{Z}Y\mathrel{\vcentcolon=}\{(x,y)\in X\times Y\mid\varrho_{X}(x)=\varrho_{Y}(y)\}. The required anchor map ϱ X × Z Y : X × Z Y → Z : subscript italic-ϱ subscript 𝑍 𝑋 𝑌 → subscript 𝑍 𝑋 𝑌 𝑍 \varrho_{X\times_{Z}Y}\colon X\times_{Z}Y\to Z is of course the composition of the first coordinate projection and the anchor map for X 𝑋 X (or the analogue using Y 𝑌 Y ; they are equal). Of course groupoid elements act diagonally in the obvious way. Such coincidence spaces as the one just described will appear again and again in the theory of correspondences. Finally, for the record, we supply the following important definition. Definition 1.1 . Let 𝒢 𝒢 \mathcal{G} be a groupoid. A 𝒢 𝒢 \mathcal{G} -space X 𝑋 X is proper if the map 𝒢 × Z X → X , ( g , x ) ↦ ( g ​ x , x ) formulae-sequence → subscript 𝑍 𝒢 𝑋 𝑋 maps-to 𝑔 𝑥 𝑔 𝑥 𝑥 \mathcal{G}\times_{Z}X\to X,\;(g,x)\mapsto(gx,x) is a proper map, where 𝒢 × Z X := { ( g , x ) ∈ 𝒢 × X ∣ s ​ ( g ) = ϱ X ​ ( x ) } assign subscript 𝑍 𝒢 𝑋 conditional-set 𝑔 𝑥 𝒢 𝑋 𝑠 𝑔 subscript italic-ϱ 𝑋 𝑥 \mathcal{G}\times_{Z}X\mathrel{\vcentcolon=}\{(g,x)\in\mathcal{G}\times X\mid s(g)=\varrho_{X}(x)\} . A groupoid is itself called proper if it acts properly on its base space Z 𝑍 Z . Explicitly, the map 𝒢 → X × Z X , g ↦ ( r ​ ( g ) , s ​ ( g ) ) formulae-sequence → 𝒢 subscript 𝑍 𝑋 𝑋 maps-to 𝑔 𝑟 𝑔 𝑠 𝑔 \mathcal{G}\to X\times_{Z}X,\hskip 28.45274ptg\mapsto(r(g),s(g)) is required to be proper. 1.3. Equivariant Kasparov theory Le Gall has defined 𝒢 𝒢 \mathcal{G} -equivariant KK KK \operatorname{KK} -theory in [ LeGall:KK_groupoid ] . We briefly sketch the definitions. Let A 𝐴 A and B 𝐵 B be (possibly ℤ / 2 ℤ 2 \mathbb{Z}/2 -graded) 𝒢 𝒢 \mathcal{G} -C*-algebras. Then a cycle for KK 𝒢 ⁡ ( A , B ) superscript KK 𝒢 𝐴 𝐵 \operatorname{KK}^{\mathcal{G}}(A,B) is given by a ℤ / 2 ℤ 2 \mathbb{Z}/2 -graded 𝒢 𝒢 \mathcal{G} -equivariant Hilbert B 𝐵 B -module ℰ ℰ \mathcal{E} , together with a 𝒢 𝒢 \mathcal{G} -equivariant grading-preserving *-homomorphism from A 𝐴 A to the C*-algebra of bounded, adjointable operators on ℰ ℰ \mathcal{E} , and an essentially 𝒢 𝒢 \mathcal{G} -equivariant self-adjoint operator F 𝐹 F on ℰ ℰ \mathcal{E} which is graded odd and satisfies [ a , F ] 𝑎 𝐹 [a,F] and a ​ ( F 2 − 1 ) 𝑎 superscript 𝐹 2 1 a(F^{2}-1) are compact operators (essentially zero operators) for all a ∈ A 𝑎 𝐴 a\in A . Modulo an appropriate equivalence relation, the set of equivalence classes of cycles can be identified with the morphism set KK 𝒢 ⁡ ( A , B ) superscript KK 𝒢 𝐴 𝐵 \operatorname{KK}^{\mathcal{G}}(A,B) in an additive, symmetric monoidal category. Higher KK KK \operatorname{KK} -groups are defined using Clifford algebras, and since these are 2 2 2 -periodic, there are only two up to isomorphism. We denote by KK ∗ 𝒢 ⁡ ( A , B ) superscript subscript KK 𝒢 𝐴 𝐵 \operatorname{KK}_{*}^{\mathcal{G}}(A,B) the sum of these two groups. If A 𝐴 A and B 𝐵 B are 𝒢 𝒢 \mathcal{G} -C*-algebras, then the group RKK 𝒢 ⁡ ( X ; A , B ) superscript RKK 𝒢 𝑋 𝐴 𝐵 \operatorname{RKK}^{\mathcal{G}}(X;A,B) is by definition the groupoid-equivariant Kasparov group KK 𝒢 ⋉ X ⁡ ( 𝒞 0 ​ ( X ) ⊗ A , 𝒞 0 ​ ( X ) ⊗ B ) . superscript KK left-normal-factor-semidirect-product 𝒢 𝑋 tensor-product subscript 𝒞 0 𝑋 𝐴 tensor-product subscript 𝒞 0 𝑋 𝐵 \operatorname{KK}^{\mathcal{G}\ltimes X}(\mathcal{C}_{0}(X)\otimes A,\mathcal{C}_{0}(X)\otimes B). The tensor products are in the category of 𝒢 𝒢 \mathcal{G} -C*-algebras. This group differs from KK 𝒢 ⁡ ( A , 𝒞 0 ​ ( X ) ⊗ B ) superscript KK 𝒢 𝐴 tensor-product subscript 𝒞 0 𝑋 𝐵 \operatorname{KK}^{\mathcal{G}}(A,\mathcal{C}_{0}(X)\otimes B) only in the support condition on cycles. For example if 𝒢 𝒢 \mathcal{G} is trivial and A = B = ℂ 𝐴 𝐵 ℂ A=B=\mathbb{C} then KK 𝒢 ⁡ ( A , 𝒞 0 ​ ( X ) ⊗ B ) superscript KK 𝒢 𝐴 tensor-product subscript 𝒞 0 𝑋 𝐵 \operatorname{KK}^{\mathcal{G}}(A,\mathcal{C}_{0}(X)\otimes B) is the ordinary K K \operatorname{K} -theory of X 𝑋 X and RKK 𝒢 ⁡ ( X ; A , B ) superscript RKK 𝒢 𝑋 𝐴 𝐵 \operatorname{RKK}^{\mathcal{G}}(X;A,B) is the representable K K \operatorname{K} -theory of X 𝑋 X (a non-compactly supported theory.) We discuss these groups in more detail in the next section. Of course similar remarks hold for higher RKK 𝒢 superscript RKK 𝒢 \operatorname{RKK}^{\mathcal{G}} -groups. 1.4. Equivariant K K \operatorname{K} -theory In this section, we present an exceedingly brief overview of equivariant K K \operatorname{K} -theory, roughly sufficient for the theory of equivariant correspondences. For more details see [ Emerson-Meyer:Equivariant_K ] . Let X 𝑋 X be a proper 𝒢 𝒢 \mathcal{G} -space. Recall that a 𝒢 ⋉ X left-normal-factor-semidirect-product 𝒢 𝑋 \mathcal{G}\ltimes X -space consists of a 𝒢 𝒢 \mathcal{G} -space Y 𝑌 Y together with a 𝒢 𝒢 \mathcal{G} -equivariant map ϱ Y : Y → X : subscript italic-ϱ 𝑌 → 𝑌 𝑋 \varrho_{Y}:Y\to X serving as the anchor map for the 𝒢 ⋉ X left-normal-factor-semidirect-product 𝒢 𝑋 \mathcal{G}\ltimes X -action. Definition 1.2 . Let Y 𝑌 Y be a 𝒢 ⋉ X left-normal-factor-semidirect-product 𝒢 𝑋 \mathcal{G}\ltimes X -space. The 𝒢 𝒢 \mathcal{G} -equivariant representable K K \operatorname{K} -theory of Y 𝑌 Y with X 𝑋 X -compact supports is the group RK 𝒢 , X − ∗ ⁡ ( Y ) := KK ∗ 𝒢 ⋉ X ⁡ ( 𝒞 0 ​ ( X ) , 𝒞 0 ​ ( Y ) ) . assign subscript superscript RK absent 𝒢 𝑋 𝑌 subscript superscript KK left-normal-factor-semidirect-product 𝒢 𝑋 subscript 𝒞 0 𝑋 subscript 𝒞 0 𝑌 \operatorname{RK}^{-*}_{\mathcal{G},X}(Y)\mathrel{\vcentcolon=}\operatorname{KK}^{\mathcal{G}\ltimes X}_{*}\bigl{(}\mathcal{C}_{0}(X),\mathcal{C}_{0}(Y)\bigr{)}. The 𝒢 𝒢 \mathcal{G} -equivariant representable K K \operatorname{K} -theory of Y 𝑌 Y is RK 𝒢 ∗ ⁡ ( Y ) := RK 𝒢 , Y ∗ ⁡ ( Y ) . assign superscript subscript RK 𝒢 𝑌 subscript superscript RK 𝒢 𝑌 𝑌 \operatorname{RK}_{\mathcal{G}}^{*}(Y)\mathrel{\vcentcolon=}\operatorname{RK}^{*}_{\mathcal{G},Y}(Y). Cycles for KK 𝒢 ⋉ X ⁡ ( 𝒞 0 ​ ( X ) , 𝒞 0 ​ ( Y ) ) superscript KK left-normal-factor-semidirect-product 𝒢 𝑋 subscript 𝒞 0 𝑋 subscript 𝒞 0 𝑌 \operatorname{KK}^{\mathcal{G}\ltimes X}\bigl{(}\mathcal{C}_{0}(X),\mathcal{C}_{0}(Y)\bigr{)} consist of pairs ( ℋ , F ) ℋ 𝐹 (\mathcal{H},F) where ℰ ℰ \mathcal{E} is a countably generated ℤ / 2 ℤ 2 \mathbb{Z}/2 -graded 𝒢 ⋉ X left-normal-factor-semidirect-product 𝒢 𝑋 \mathcal{G}\ltimes X -equivariant right Hilbert 𝒞 0 ​ ( X ) subscript 𝒞 0 𝑋 \mathcal{C}_{0}(X) -module equipped with a 𝒢 ⋉ X left-normal-factor-semidirect-product 𝒢 𝑋 \mathcal{G}\ltimes X -equivariant non-degenerate *-homomorphism from 𝒞 0 ​ ( X ) subscript 𝒞 0 𝑋 \mathcal{C}_{0}(X) to the C*-algebra of bounded, adjointable operators on ℰ ℰ \mathcal{E} , and F 𝐹 F is a bounded, odd, self-adjoint, essentially 𝒢 𝒢 \mathcal{G} -equivariant adjointable operator on ℋ ℋ \mathcal{H} such that f ​ ( F 2 − 1 ) 𝑓 superscript 𝐹 2 1 f(F^{2}-1) is a compact operator, for all f ∈ 𝒞 0 ​ ( X ) 𝑓 subscript 𝒞 0 𝑋 f\in\mathcal{C}_{0}(X) . The properness of 𝒢 𝒢 \mathcal{G} implies that F 𝐹 F may be averaged to be actually 𝒢 𝒢 \mathcal{G} -equivariant, so we assume this in the following. The Hilbert 𝒞 0 ​ ( Y ) subscript 𝒞 0 𝑌 \mathcal{C}_{0}(Y) -module ℰ ℰ \mathcal{E} is the space of continuous sections of a continuous field of ℤ / 2 ℤ 2 \mathbb{Z}/2 -graded Hilbert spaces { ℋ y ∣ y ∈ Y } conditional-set subscript ℋ 𝑦 𝑦 𝑌 \{\mathcal{H}_{y}\mid y\in Y\} over Y 𝑌 Y . Since F 𝐹 F must be 𝒞 0 ​ ( Y ) subscript 𝒞 0 𝑌 \mathcal{C}_{0}(Y) -linear, it consists of a continuous family { F y ∣ y ∈ Y } conditional-set subscript 𝐹 𝑦 𝑦 𝑌 \{F_{y}\mid y\in Y\} of odd operators on these graded Hilbert spaces such that F y 2 − 1 subscript superscript 𝐹 2 𝑦 1 F^{2}_{y}-1 is a compact operator on ℋ y subscript ℋ 𝑦 \mathcal{H}_{y} for all y ∈ Y 𝑦 𝑌 y\in Y . By 𝒢 ⋉ X left-normal-factor-semidirect-product 𝒢 𝑋 \mathcal{G}\ltimes X -equivariance, the representation of 𝒞 0 ​ ( X ) subscript 𝒞 0 𝑋 \mathcal{C}_{0}(X) on ℰ ℰ \mathcal{E} must factor through the *-homomorphism 𝒞 0 ​ ( X ) → 𝒞 0 ​ ( Y ) → subscript 𝒞 0 𝑋 subscript 𝒞 0 𝑌 \mathcal{C}_{0}(X)\to\mathcal{C}_{0}(Y) Gelfand dual to the anchor map ϱ Y : Y → X : subscript italic-ϱ 𝑌 → 𝑌 𝑋 \varrho_{Y}\colon Y\to X . Therefore F 𝐹 F commutes with the action of 𝒞 0 ​ ( X ) subscript 𝒞 0 𝑋 \mathcal{C}_{0}(X) as well; in fact the induced representation of 𝒞 0 ​ ( X ) subscript 𝒞 0 𝑋 \mathcal{C}_{0}(X) on each Hilbert space ℋ y subscript ℋ 𝑦 \mathcal{H}_{y} sends a continuous function f ∈ 𝒞 0 ​ ( X ) 𝑓 subscript 𝒞 0 𝑋 f\in\mathcal{C}_{0}(X) to the operator of multiplication by the complex number f ​ ( ϱ Y ​ ( y ) ) 𝑓 subscript italic-ϱ 𝑌 𝑦 f\bigl{(}\varrho_{Y}(y)\bigr{)} . In particular, the only role of the representation of 𝒞 0 ​ ( X ) subscript 𝒞 0 𝑋 \mathcal{C}_{0}(X) is to relax the support condition on the compact-operator valued-function F 2 − 1 superscript 𝐹 2 1 F^{2}-1 from requiring it to vanish at ∞ \infty of Y 𝑌 Y to only requiring it to vanish at infinity along the fibres of ϱ Y : Y → X : subscript italic-ϱ 𝑌 → 𝑌 𝑋 \varrho_{Y}\colon Y\to X . If ϱ Y : Y → X : subscript italic-ϱ 𝑌 → 𝑌 𝑋 \varrho_{Y}\colon Y\to X is a proper map then RK 𝒢 , X ∗ ⁡ ( Y ) = RK 𝒢 ∗ ⁡ ( X ) := RK 𝒢 , X ∗ ⁡ ( X ) superscript subscript RK 𝒢 𝑋 𝑌 subscript superscript RK 𝒢 𝑋 assign subscript superscript RK 𝒢 𝑋 𝑋 \operatorname{RK}_{\mathcal{G},X}^{*}(Y)=\operatorname{RK}^{*}_{\mathcal{G}}(X)\mathrel{\vcentcolon=}\operatorname{RK}^{*}_{\mathcal{G},X}(X) ; these two groups have exactly the same cycles. Example 1.3 . Any 𝒢 𝒢 \mathcal{G} -equivariant complex vector bundle V 𝑉 V on Y 𝑌 Y yields a cycle for RK 𝒢 0 ⁡ ( Y ) subscript superscript RK 0 𝒢 𝑌 \operatorname{RK}^{0}_{\mathcal{G}}(Y) by choosing a 𝒢 𝒢 \mathcal{G} -invariant Hermitian metric on V 𝑉 V and forming the corresponding 𝒢 ⋉ Y left-normal-factor-semidirect-product 𝒢 𝑌 \mathcal{G}\ltimes Y -equivariant ℤ / 2 ℤ 2 \mathbb{Z}/2 -graded right Hilbert 𝒞 0 ​ ( Y ) subscript 𝒞 0 𝑌 \mathcal{C}_{0}(Y) module of sections, where the grading is the trivial one. We set the operator equal to zero. Example 1.4 . Let X 𝑋 X be a 𝒢 𝒢 \mathcal{G} -space and let V 𝑉 V be a 𝒢 𝒢 \mathcal{G} -equivariantly K K \operatorname{K} -oriented vector bundle over X 𝑋 X of (real) dimension n 𝑛 n . The 𝒢 𝒢 \mathcal{G} -equivariant vector bundle projection π V : V → X : subscript 𝜋 𝑉 → 𝑉 𝑋 \pi_{V}\colon V\to X gives V 𝑉 V the structure of a space over X 𝑋 X , so that V 𝑉 V becomes a 𝒢 ⋉ X left-normal-factor-semidirect-product 𝒢 𝑋 \mathcal{G}\ltimes X -space. Then the Thom isomorphism provides an invertible Thom class t V ∈ RK 𝒢 , X dim V ⁡ ( V ) := KK dim V 𝒢 ⋉ X ⁡ ( 𝒞 0 ​ ( X ) , 𝒞 0 ​ ( V ) ) . subscript 𝑡 𝑉 subscript superscript RK dimension 𝑉 𝒢 𝑋 𝑉 assign subscript superscript KK left-normal-factor-semidirect-product 𝒢 𝑋 dimension 𝑉 subscript 𝒞 0 𝑋 subscript 𝒞 0 𝑉 t_{V}\in\operatorname{RK}^{\dim V}_{\mathcal{G},X}(V)\mathrel{\vcentcolon=}\operatorname{KK}^{\mathcal{G}\ltimes X}_{\dim V}\bigl{(}\mathcal{C}_{0}(X),\mathcal{C}_{0}(V)\bigr{)}. In the case 𝒢 = Spin c ​ ( ℝ n ) 𝒢 superscript Spin 𝑐 superscript ℝ 𝑛 \mathcal{G}=\mathrm{Spin}^{c}(\mathbb{R}^{n}) and X = ⋆ 𝑋 ⋆ X=\star and V := ℝ n assign 𝑉 superscript ℝ 𝑛 V\mathrel{\vcentcolon=}\mathbb{R}^{n} with the representation Spin c ​ ( ℝ n ) → Spin ​ ( ℝ n ) → O ​ ( n , ℝ ) → superscript Spin 𝑐 superscript ℝ 𝑛 Spin superscript ℝ 𝑛 → O 𝑛 ℝ \mathrm{Spin}^{c}(\mathbb{R}^{n})\to\mathrm{Spin}(\mathbb{R}^{n})\to\mathrm{O}(n,\mathbb{R}) the class t ℝ n subscript 𝑡 superscript ℝ 𝑛 t_{\mathbb{R}^{n}} is the ‘Bott’ class figuring in equivariant Bott Periodicity. Certain further normalizations can be made in order to describe the groups RK 𝒢 , X ∗ ⁡ ( Y ) superscript subscript RK 𝒢 𝑋 𝑌 \operatorname{RK}_{\mathcal{G},X}^{*}(Y) . A standard one is to replace the ℤ / 2 ℤ 2 \mathbb{Z}/2 -grading on ℰ ℰ \mathcal{E} by the standard even grading, so that ℰ ℰ \mathcal{E} consists of the sum of two copies of the same Hilbert module. This means that F 𝐹 F can be taken to be of the form ( 0 F 1 ∗ F 1 0 ) matrix 0 superscript subscript 𝐹 1 subscript 𝐹 1 0 \begin{pmatrix}0&F_{1}^{*}\\ F_{1}&0\end{pmatrix} and the conditions involving F 𝐹 F are replaced by ones involving F 1 subscript 𝐹 1 F_{1} and F 1 ∗ superscript subscript 𝐹 1 F_{1}^{*} ; we may as well replace F 𝐹 F by F 1 subscript 𝐹 1 F_{1} . With this convention, the Fredholm conditions are that f ​ ( F ​ F ∗ − 1 ) 𝑓 𝐹 superscript 𝐹 1 f(FF^{*}-1) and f ( F ∗ F − 1 f(F^{*}F-1 are compact for all f ∈ 𝒞 0 ​ ( X ) 𝑓 subscript 𝒞 0 𝑋 f\in\mathcal{C}_{0}(X) . In other words, y ↦ F y maps-to 𝑦 subscript 𝐹 𝑦 y\mapsto F_{y} takes essentially unitary values in 𝔹 ​ ( ℋ y ) 𝔹 subscript ℋ 𝑦 \mathbb{B}(\mathcal{H}_{y}) for all y ∈ Y 𝑦 𝑌 y\in Y and the compact-operator-valued functions F ​ F ∗ − 1 𝐹 superscript 𝐹 1 FF^{*}-1 and F ∗ ​ F − 1 superscript 𝐹 𝐹 1 F^{*}F-1 vanish at infinity along the fibres of ϱ : Y → X : italic-ϱ → 𝑌 𝑋 \varrho\colon Y\to X . The equivariant stabilization theorem for Hilbert modules implies that we may take ℋ ℋ \mathcal{H} to have the special form L 2 ​ ( 𝒢 ) ∞ ⊗ 𝒞 0 ​ ( Z ) 𝒞 0 ​ ( Y ) subscript tensor-product subscript 𝒞 0 𝑍 superscript 𝐿 2 superscript 𝒢 subscript 𝒞 0 𝑌 L^{2}(\mathcal{G})^{\infty}\otimes_{\mathcal{C}_{0}(Z)}\mathcal{C}_{0}(Y) , where L 2 ​ ( 𝒢 ) superscript 𝐿 2 𝒢 L^{2}(\mathcal{G}) is the 𝒢 𝒢 \mathcal{G} -equivariant right Hilbert 𝒞 0 ​ ( Z ) subscript 𝒞 0 𝑍 \mathcal{C}_{0}(Z) - Hilbert module defined using the Haar system of 𝒢 𝒢 \mathcal{G} , and the superscript indicates the sum of countably many copies of L 2 ​ ( 𝒢 ) superscript 𝐿 2 𝒢 L^{2}(\mathcal{G}) . The corresponding field of Hilbert space has value L 2 ​ ( 𝒢 y ) ∞ superscript 𝐿 2 superscript superscript 𝒢 𝑦 L^{2}(\mathcal{G}^{y})^{\infty} at y ∈ Y 𝑦 𝑌 y\in Y where 𝒢 y superscript 𝒢 𝑦 \mathcal{G}^{y} denotes all G ∈ 𝒢 𝐺 𝒢 G\in\mathcal{G} ending in y 𝑦 y , on which we have a given measure specified by the Haar system of 𝒢 𝒢 \mathcal{G} . This leads to a description of RK 𝒢 , Y 0 ⁡ ( X ) subscript superscript RK 0 𝒢 𝑌 𝑋 \operatorname{RK}^{0}_{\mathcal{G},Y}(X) as the group of homotopy-classes of 𝒢 𝒢 \mathcal{G} -equivariant continuous maps from Y 𝑌 Y to the space ℱ 𝒢 subscript ℱ 𝒢 \mathcal{F}_{\mathcal{G}} of Fredholm operators on the Hilbert spaces L 2 ​ ( 𝒢 y ) ∞ superscript 𝐿 2 superscript superscript 𝒢 𝑦 L^{2}(\mathcal{G}^{y})^{\infty} , but topologizing the space ℱ 𝒢 subscript ℱ 𝒢 \mathcal{F}_{\mathcal{G}} is somewhat delicate. Similarly, the relative groups RK 𝒢 , X ∗ ⁡ ( Y ) subscript superscript RK 𝒢 𝑋 𝑌 \operatorname{RK}^{*}_{\mathcal{G},X}(Y) are maps to Fredholm operators with compact vertical support with respect to the map Y → X → 𝑌 𝑋 Y\to X , where the support of a map to Fredholm operators is by definition the complement of the set where the map takes invertible values. Remark 1.5 . If 𝒢 𝒢 \mathcal{G} acts properly and co-compactly on X 𝑋 X , A 𝐴 A is a trivial 𝒢 𝒢 \mathcal{G} -C*-algebra and B 𝐵 B is a 𝒢 ⋉ X left-normal-factor-semidirect-product 𝒢 𝑋 \mathcal{G}\ltimes X -C*-algebra, then there is a canonical isomorphism KK 𝒢 ⋉ X ⁡ ( 𝒞 0 ​ ( X ) ⊗ A , B ) ≅ KK ⁡ ( A , 𝒢 ⋉ B ) . superscript KK left-normal-factor-semidirect-product 𝒢 𝑋 tensor-product subscript 𝒞 0 𝑋 𝐴 𝐵 KK 𝐴 left-normal-factor-semidirect-product 𝒢 𝐵 \operatorname{KK}^{\mathcal{G}\ltimes X}(\mathcal{C}_{0}(X)\otimes A,B)\cong\operatorname{KK}(A,\mathcal{G}\ltimes B). In particular, the 𝒢 𝒢 \mathcal{G} -equivariant representable K K \operatorname{K} -theory of X 𝑋 X agrees with the K K \operatorname{K} -theory of the corresponding cross-product. Under this identification, classes in RK 𝒢 0 ⁡ ( X ) subscript superscript RK 0 𝒢 𝑋 \operatorname{RK}^{0}_{\mathcal{G}}(X) which are represented by equivariant vector bundles on X 𝑋 X correspond to classes in K 0 ⁡ ( 𝒢 ⋉ 𝒞 0 ​ ( X ) ) subscript K 0 left-normal-factor-semidirect-product 𝒢 subscript 𝒞 0 𝑋 \operatorname{K}_{0}(\mathcal{G}\ltimes\mathcal{C}_{0}(X)) which are represented by projections in the stabilisation of 𝒢 ⋉ 𝒞 0 ​ ( X ) left-normal-factor-semidirect-product 𝒢 subscript 𝒞 0 𝑋 \mathcal{G}\ltimes\mathcal{C}_{0}(X) . See [ Emerson-Meyer:Equivariant_K ] for more information. Thus, even if the reader is only interested in groups, or the trivial group, it is convenient to introduce groupoids to some extent in order to describe cohomology theories with different support conditions. 1.5. Tensor and forgetful functors The following simple functor will play an important role. If P 𝑃 P is a 𝒢 ⋉ X left-normal-factor-semidirect-product 𝒢 𝑋 \mathcal{G}\ltimes X -algebra, we denote by T P subscript 𝑇 𝑃 T_{P} the map RKK 𝒢 ⁡ ( X ; A , B ) := KK 𝒢 ⋉ X ⁡ ( 𝒞 0 ​ ( X ) ⊗ A , 𝟙 X ⊗ B ) → KK 𝒢 ⁡ ( P ⊗ A , P ⊗ B ) assign superscript RKK 𝒢 𝑋 𝐴 𝐵 superscript KK left-normal-factor-semidirect-product 𝒢 𝑋 tensor-product subscript 𝒞 0 𝑋 𝐴 tensor-product subscript 1 𝑋 𝐵 → superscript KK 𝒢 tensor-product 𝑃 𝐴 tensor-product 𝑃 𝐵 \operatorname{RKK}^{\mathcal{G}}(X;A,B)\mathrel{\vcentcolon=}\operatorname{KK}^{\mathcal{G}\ltimes X}(\mathcal{C}_{0}(X)\otimes A,\mathds{1}_{X}\otimes B)\to\operatorname{KK}^{\mathcal{G}}(P\otimes A,P\otimes B) which sends a 𝒢 ⋉ X left-normal-factor-semidirect-product 𝒢 𝑋 \mathcal{G}\ltimes X -equivariant right Hilbert 𝒞 0 ​ ( X ) ⊗ B tensor-product subscript 𝒞 0 𝑋 𝐵 \mathcal{C}_{0}(X)\otimes B -Hilbert module ℰ ℰ \mathcal{E} to ℰ ⊗ X P subscript tensor-product 𝑋 ℰ 𝑃 \mathcal{E}\otimes_{X}P , the tensor product being in the category of 𝒢 ⋉ X left-normal-factor-semidirect-product 𝒢 𝑋 \mathcal{G}\ltimes X -algebras (we accordingly use a subscript for emphasis) and sends F ∈ 𝔹 ​ ( ℰ ) 𝐹 𝔹 ℰ F\in\mathbb{B}(\mathcal{E}) to the operator F ⊗ X id P subscript tensor-product 𝑋 𝐹 subscript id 𝑃 F\otimes_{X}\textup{id}_{P} . This definition makes sense since F 𝐹 F commutes with the 𝒞 0 ​ ( X ) subscript 𝒞 0 𝑋 \mathcal{C}_{0}(X) -structure on ℰ ℰ \mathcal{E} . The functor T P subscript 𝑇 𝑃 T_{P} is the composition of external product ␣ ⊗ X 1 P : KK 𝒢 ⋉ X ⁡ ( 𝒞 0 ​ ( X ) ⊗ A , 𝒞 0 ​ ( X ) ⊗ B ) → KK 𝒢 ⋉ X ⁡ ( A ⊗ X P , B ⊗ X P ) : subscript tensor-product 𝑋 ␣ subscript 1 𝑃 → superscript KK left-normal-factor-semidirect-product 𝒢 𝑋 tensor-product subscript 𝒞 0 𝑋 𝐴 tensor-product subscript 𝒞 0 𝑋 𝐵 superscript KK left-normal-factor-semidirect-product 𝒢 𝑋 subscript tensor-product 𝑋 𝐴 𝑃 subscript tensor-product 𝑋 𝐵 𝑃 \textup{\textvisiblespace}\otimes_{X}1_{P}\colon\operatorname{KK}^{\mathcal{G}\ltimes X}(\mathcal{C}_{0}(X)\otimes A,\mathcal{C}_{0}(X)\otimes B)\to\operatorname{KK}^{\mathcal{G}\ltimes X}(A\otimes_{X}P,B\otimes_{X}P) (where the X 𝑋 X -structure on A ⊗ P tensor-product 𝐴 𝑃 A\otimes P etc . is on the P 𝑃 P factor), and the forgetful map KK 𝒢 ⋉ X ⁡ ( A ⊗ X P , B X ⊗ P ) → KK 𝒢 ⁡ ( A ⊗ X P , B ⊗ X P ) → superscript KK left-normal-factor-semidirect-product 𝒢 𝑋 subscript tensor-product 𝑋 𝐴 𝑃 tensor-product subscript 𝐵 𝑋 𝑃 superscript KK 𝒢 subscript tensor-product 𝑋 𝐴 𝑃 subscript tensor-product 𝑋 𝐵 𝑃 \operatorname{KK}^{\mathcal{G}\ltimes X}(A\otimes_{X}P,B_{X}\otimes P)\to\operatorname{KK}^{\mathcal{G}}(A\otimes_{X}P,B\otimes_{X}P) which maps a 𝒢 ⋉ X left-normal-factor-semidirect-product 𝒢 𝑋 \mathcal{G}\ltimes X -algebra or Hilbert module to the underlying 𝒢 𝒢 \mathcal{G} -algebra, or Hilbert module, thus forgetting the X 𝑋 X -structure. 1.6. Kasparov duals We begin our discussion of duality by by formalizing some duality calculations of Kasparov, c.f. [ Kasparov:Novikov ] *Theorem 4.9. Explicit examples will be discussed later. For convenience of notation we will often write 𝟙 := 𝒞 0 ​ ( Z ) assign 1 subscript 𝒞 0 𝑍 \mathds{1}\mathrel{\vcentcolon=}\mathcal{C}_{0}(Z) . This notation expresses the fact that 𝒞 0 ​ ( Z ) subscript 𝒞 0 𝑍 \mathcal{C}_{0}(Z) is the tensor unit in the tensor category of 𝒢 𝒢 \mathcal{G} -C*-algebras. Similarly, if 𝒢 𝒢 \mathcal{G} acts on a space X 𝑋 X then we sometimes denote by 𝟙 X subscript 1 𝑋 \mathds{1}_{X} the 𝒢 𝒢 \mathcal{G} -C*-algebra 𝒞 0 ​ ( X ) subscript 𝒞 0 𝑋 \mathcal{C}_{0}(X) ; thus 𝟙 X subscript 1 𝑋 \mathds{1}_{X} is the tensor unit in the category of 𝒢 ⋉ X left-normal-factor-semidirect-product 𝒢 𝑋 \mathcal{G}\ltimes X C*-algebras, X 𝑋 X being the base of 𝒢 ⋉ X left-normal-factor-semidirect-product 𝒢 𝑋 \mathcal{G}\ltimes X . This notation is consistent with the source of this material (see [ Emerson-Meyer:Dualities ] .) Definition 1.6 . Let n ∈ ℤ 𝑛 ℤ n\in\mathbb{Z} . An n 𝑛 n -dimensional 𝒢 𝒢 \mathcal{G} -equivariant Kasparov dual for the 𝒢 𝒢 \mathcal{G} -space X 𝑋 X is a triple ( P , D , Θ ) 𝑃 𝐷 Θ (P,D,\Theta) , where • P 𝑃 P is a (possibly ℤ / 2 ℤ 2 \mathbb{Z}/2 -graded) 𝒢 ⋉ X left-normal-factor-semidirect-product 𝒢 𝑋 \mathcal{G}\ltimes X - C ∗ superscript C \textup{C}^{*} -algebra, • D ∈ KK − n 𝒢 ⁡ ( P , 𝟙 ) 𝐷 subscript superscript KK 𝒢 𝑛 𝑃 1 D\in\operatorname{KK}^{\mathcal{G}}_{-n}(P,\mathds{1}) , and • Θ ∈ RKK n 𝒢 ⁡ ( X ; 𝟙 , P ) Θ subscript superscript RKK 𝒢 𝑛 𝑋 1 𝑃 \Theta\in\operatorname{RKK}^{\mathcal{G}}_{n}(X;\mathds{1},P) , subject to the following conditions: (1) Θ ⊗ P D = id 𝟙 subscript tensor-product 𝑃 Θ 𝐷 subscript id 1 \Theta\otimes_{P}D=\textup{id}_{\mathds{1}} in RKK 0 𝒢 ⁡ ( X ; 𝟙 , 𝟙 ) subscript superscript RKK 𝒢 0 𝑋 1 1 \operatorname{RKK}^{\mathcal{G}}_{0}(X;\mathds{1},\mathds{1}) ; (2) Θ ⊗ f = Θ ⊗ P T P ​ ( f ) tensor-product Θ 𝑓 subscript tensor-product 𝑃 Θ subscript 𝑇 𝑃 𝑓 \Theta\otimes f=\Theta\otimes_{P}T_{P}(f) in RKK ∗ + n 𝒢 ⁡ ( X ; A , B ⊗ P ) subscript superscript RKK 𝒢 absent 𝑛 𝑋 𝐴 tensor-product 𝐵 𝑃 \operatorname{RKK}^{\mathcal{G}}_{*+n}(X;A,B\otimes P) for all 𝒢 𝒢 \mathcal{G} - C ∗ superscript C \textup{C}^{*} -algebras A 𝐴 A and B 𝐵 B and all f ∈ RKK ∗ 𝒢 ⁡ ( X ; A , B ) 𝑓 subscript superscript RKK 𝒢 𝑋 𝐴 𝐵 f\in\operatorname{RKK}^{\mathcal{G}}_{*}(X;A,B) ; (3) T P ​ ( Θ ) ⊗ P ⊗ P Φ P = ( − 1 ) n ​ T P ​ ( Θ ) subscript tensor-product tensor-product 𝑃 𝑃 subscript 𝑇 𝑃 Θ subscript Φ 𝑃 superscript 1 𝑛 subscript 𝑇 𝑃 Θ T_{P}(\Theta)\otimes_{P\otimes P}\Phi_{P}=(-1)^{n}T_{P}(\Theta) in KK n 𝒢 ⁡ ( P , P ⊗ P ) subscript superscript KK 𝒢 𝑛 𝑃 tensor-product 𝑃 𝑃 \operatorname{KK}^{\mathcal{G}}_{n}(P,P\otimes P) , where Φ P subscript Φ 𝑃 \Phi_{P} is the flip automorphism on P ⊗ P tensor-product 𝑃 𝑃 P\otimes P . The following theorem is proved in [ Emerson-Meyer:Dualities ] . Theorem 1.7 . Let n ∈ ℤ 𝑛 ℤ n\in\mathbb{Z} , let P 𝑃 P be a 𝒢 ⋉ X left-normal-factor-semidirect-product 𝒢 𝑋 \mathcal{G}\ltimes X - C ∗ superscript C \textup{C}^{*} -algebra, D ∈ KK − n 𝒢 ⁡ ( P , 𝟙 ) 𝐷 subscript superscript KK 𝒢 𝑛 𝑃 1 D\in\operatorname{KK}^{\mathcal{G}}_{-n}(P,\mathds{1}) , and Θ ∈ RKK n 𝒢 ⁡ ( X ; 𝟙 , P ) Θ subscript superscript RKK 𝒢 𝑛 𝑋 1 𝑃 \Theta\in\operatorname{RKK}^{\mathcal{G}}_{n}(X;\mathds{1},P) . Define two natural transformations PD : KK i − n 𝒢 ⁡ ( P ⊗ A , B ) → RKK i 𝒢 ⁡ ( X ; A , B ) , : absent → subscript superscript KK 𝒢 𝑖 𝑛 tensor-product 𝑃 𝐴 𝐵 subscript superscript RKK 𝒢 𝑖 𝑋 𝐴 𝐵 \displaystyle\colon\operatorname{KK}^{\mathcal{G}}_{i-n}(P\otimes A,B)\to\operatorname{RKK}^{\mathcal{G}}_{i}(X;A,B), f 𝑓 \displaystyle\qquad f ↦ Θ ⊗ P f , maps-to absent subscript tensor-product 𝑃 Θ 𝑓 \displaystyle\mapsto\Theta\otimes_{P}f, PD ∗ superscript PD \displaystyle\textup{PD}^{*} : RKK i 𝒢 ⁡ ( X ; A , B ) → KK i − n 𝒢 ⁡ ( P ⊗ A , B ) , : absent → subscript superscript RKK 𝒢 𝑖 𝑋 𝐴 𝐵 subscript superscript KK 𝒢 𝑖 𝑛 tensor-product 𝑃 𝐴 𝐵 \displaystyle\colon\operatorname{RKK}^{\mathcal{G}}_{i}(X;A,B)\to\operatorname{KK}^{\mathcal{G}}_{i-n}(P\otimes A,B), g 𝑔 \displaystyle\qquad g ↦ ( − 1 ) i ​ n ​ T P ​ ( g ) ⊗ P D , maps-to absent subscript tensor-product 𝑃 superscript 1 𝑖 𝑛 subscript 𝑇 𝑃 𝑔 𝐷 \displaystyle\mapsto(-1)^{in}T_{P}(g)\otimes_{P}D, These two are inverse to each other if and only if ( P , D , Θ ) 𝑃 𝐷 Θ (P,D,\Theta) is an n 𝑛 n -dimensional 𝒢 𝒢 \mathcal{G} -equivariant Kasparov dual for X 𝑋 X . 1.7. Abstract duals The reader may have noticed that the only place the 𝒞 0 ​ ( X ) subscript 𝒞 0 𝑋 \mathcal{C}_{0}(X) -structure on P 𝑃 P comes into play in the conditions listed in Definition 1.6 , and in the statement of Theorem 1.7 , is via the functor T P subscript 𝑇 𝑃 T_{P} . In particular, if one has a Kasparov dual ( P , D , Θ ) 𝑃 𝐷 Θ (P,D,\Theta) and if one changes the 𝒞 0 ​ ( X ) subscript 𝒞 0 𝑋 \mathcal{C}_{0}(X) -structure on P 𝑃 P , for example by composing it with a 𝒢 𝒢 \mathcal{G} -equivariant homeomorphism of X 𝑋 X , then the map PD of Theorem 1.7 does not change; since by the theorem PD ∗ superscript PD \textup{PD}^{*} is its inverse map, it would not change either, strangely, since its definition uses T P subscript 𝑇 𝑃 T_{P} . In fact it turns out that the functor T P subscript 𝑇 𝑃 T_{P} can be reconstructed from PD if one knows that PD is an isomorphism. This is an important idea in connection with the Lefschetz map and suggests the following useful definition. Definition 1.8 . An n 𝑛 n -dimensional abstract dual for X 𝑋 X is a pair ( P , Θ ) 𝑃 Θ (P,\Theta) , where P 𝑃 P is a 𝒢 𝒢 \mathcal{G} - C ∗ superscript C \textup{C}^{*} -algebra and Θ ∈ RKK n 𝒢 ⁡ ( X ; 𝟙 , P ) Θ subscript superscript RKK 𝒢 𝑛 𝑋 1 𝑃 \Theta\in\operatorname{RKK}^{\mathcal{G}}_{n}(X;\mathds{1},P) , such that the map PD defined as in Theorem 1.7 is an isomorphism for all 𝒢 𝒢 \mathcal{G} - C ∗ superscript C \textup{C}^{*} -algebras A 𝐴 A and B 𝐵 B . This definition is shorter, and, as mentioned, is useful for theoretical reasons, but it seems like it should be difficult to check in practise. In any case, it is clear from Theorem 1.7 that a pair ( P , Θ ) 𝑃 Θ (P,\Theta) is an abstract dual if it is part of a Kasparov dual ( P , D , Θ ) 𝑃 𝐷 Θ (P,D,\Theta) . Proposition 1.9 . An abstract dual for a space X 𝑋 X is unique up to a canonical KK 𝒢 superscript KK 𝒢 \operatorname{KK}^{\mathcal{G}} -equivalence if it exists, and even covariantly functorial in the following sense. Let X 𝑋 X and Y 𝑌 Y be two 𝒢 𝒢 \mathcal{G} -spaces and let f : X → Y : 𝑓 → 𝑋 𝑌 f\colon X\to Y be a 𝒢 𝒢 \mathcal{G} -equivariant continuous map. Let ( P X , Θ X ) subscript 𝑃 𝑋 subscript Θ 𝑋 (P_{X},\Theta_{X}) and ( P Y , Θ Y ) subscript 𝑃 𝑌 subscript Θ 𝑌 (P_{Y},\Theta_{Y}) be abstract duals for X 𝑋 X and Y 𝑌 Y of dimensions n X subscript 𝑛 𝑋 n_{X} and n Y subscript 𝑛 𝑌 n_{Y} , respectively. Then there is a unique P f ∈ KK n Y − n X 𝒢 ⁡ ( P X , P Y ) subscript 𝑃 𝑓 subscript superscript KK 𝒢 subscript 𝑛 𝑌 subscript 𝑛 𝑋 subscript 𝑃 𝑋 subscript 𝑃 𝑌 P_{f}\in\operatorname{KK}^{\mathcal{G}}_{n_{Y}-n_{X}}(P_{X},P_{Y}) with Θ X ⊗ P X P f = f ∗ ​ ( Θ Y ) subscript tensor-product subscript 𝑃 𝑋 subscript Θ 𝑋 subscript 𝑃 𝑓 superscript 𝑓 subscript Θ 𝑌 \Theta_{X}\otimes_{P_{X}}P_{f}=f^{*}(\Theta_{Y}) . Given two composable maps between three spaces with duals, we have P f ∘ g = P f ∘ P g subscript 𝑃 𝑓 𝑔 subscript 𝑃 𝑓 subscript 𝑃 𝑔 P_{f\circ g}=P_{f}\circ P_{g} . If X = Y 𝑋 𝑌 X=Y , f = id X 𝑓 subscript id 𝑋 f=\textup{id}_{X} , and ( P X , Θ X ) = ( P Y , Θ Y ) subscript 𝑃 𝑋 subscript Θ 𝑋 subscript 𝑃 𝑌 subscript Θ 𝑌 (P_{X},\Theta_{X})=(P_{Y},\Theta_{Y}) , then P f = id P X subscript 𝑃 𝑓 subscript id subscript 𝑃 𝑋 P_{f}=\textup{id}_{P_{X}} . If only X = Y 𝑋 𝑌 X=Y , f = id X 𝑓 subscript id 𝑋 f=\textup{id}_{X} , then P f subscript 𝑃 𝑓 P_{f} is a KK 𝒢 superscript KK 𝒢 \operatorname{KK}^{\mathcal{G}} -equivalence between the two duals of X 𝑋 X . Although the map f : X → Y : 𝑓 → 𝑋 𝑌 f\colon X\to Y appearing in Proposition 1.9 does not have to be proper, it nonetheless yields a morphism P f subscript 𝑃 𝑓 P_{f} in KK 𝒢 superscript KK 𝒢 \operatorname{KK}^{\mathcal{G}} . 1.8. Duality co-algebra Let ( P , Θ ) 𝑃 Θ (P,\Theta) be an n 𝑛 n -dimensional abstract dual for a 𝒢 𝒢 \mathcal{G} -space X 𝑋 X . By the Yoneda Lemma, another abstract dual ( P ′ , Θ ′ ) superscript 𝑃 ′ superscript Θ ′ (P^{\prime},\Theta^{\prime}) also for X 𝑋 X and say of dimension n ′ superscript 𝑛 ′ n^{\prime} is related to ( P , Θ ) 𝑃 Θ (P,\Theta) by an invertible element (1.10) ψ ∈ KK n ′ − n 𝒢 ⁡ ( P , P ′ ) , such that ​ Θ ⊗ P ψ = Θ ′ . formulae-sequence 𝜓 subscript superscript KK 𝒢 superscript 𝑛 ′ 𝑛 𝑃 superscript 𝑃 ′ subscript tensor-product 𝑃 such that Θ 𝜓 superscript Θ ′ \psi\in\operatorname{KK}^{\mathcal{G}}_{n^{\prime}-n}(P,P^{\prime}),\;\textup{such that}\;\Theta\otimes_{P}\psi=\Theta^{\prime}. We repeat for emphasis that since ( P , Θ ) 𝑃 Θ (P,\Theta) is only an abstract dual, we are not assuming that there is a 𝒢 ⋉ X left-normal-factor-semidirect-product 𝒢 𝑋 \mathcal{G}\ltimes X -structure on P 𝑃 P . However, we are going to attempt to reconstruct what we might consider to be a 𝒢 ⋉ X left-normal-factor-semidirect-product 𝒢 𝑋 \mathcal{G}\ltimes X -structure on P 𝑃 P at the level of KK KK \operatorname{KK} -theory . Along the way we will keep track of how the change in dual from ( P , Θ ) 𝑃 Θ (P,\Theta) to ( P ′ , Θ ′ ) superscript 𝑃 ′ superscript Θ ′ (P^{\prime},\Theta^{\prime}) affects our constructions. Define D ∈ KK − n 𝒢 ⁡ ( P , 𝟙 ) 𝐷 subscript superscript KK 𝒢 𝑛 𝑃 1 D\in\operatorname{KK}^{\mathcal{G}}_{-n}(P,\mathds{1}) by the requirement (1.11) PD ​ ( D ) := Θ ⊗ P D = 1 𝟙 in RKK 0 𝒢 ⁡ ( X ; 𝟙 , 𝟙 ) . formulae-sequence assign PD 𝐷 subscript tensor-product 𝑃 Θ 𝐷 subscript 1 1 in RKK 0 𝒢 ⁡ ( X ; 𝟙 , 𝟙 ) . \textup{PD}(D)\mathrel{\vcentcolon=}\Theta\otimes_{P}D=1_{\mathds{1}}\qquad\text{in $\operatorname{RKK}^{\mathcal{G}}_{0}(X;\mathds{1},\mathds{1})$.} as in the first condition in Definition 1.6 ). The class D 𝐷 D should thus play the role of the class named D 𝐷 D in a Kasparov dual. It is routine to check that when we change the dual, as above, D 𝐷 D is replaced by ψ − 1 ⊗ P D subscript tensor-product 𝑃 superscript 𝜓 1 𝐷 \psi^{-1}\otimes_{P}D . We call D 𝐷 D counit of the duality because it plays the algebraic role of a counit in the theory of adjoint functors (see [ Emerson-Meyer:Dualities ] and also Remark 3.3 below). Define ∇ ∈ KK n 𝒢 ⁡ ( P , P ⊗ P ) ∇ subscript superscript KK 𝒢 𝑛 𝑃 tensor-product 𝑃 𝑃 \nabla\in\operatorname{KK}^{\mathcal{G}}_{n}(P,P\otimes P) by the requirement that PD ​ ( ∇ ) := Θ ⊗ P ∇ = Θ ⊗ X Θ in RKK 2 ​ n 𝒢 ⁡ ( X ; 𝟙 , P ⊗ P ) . formulae-sequence assign PD ∇ subscript tensor-product 𝑃 Θ ∇ subscript tensor-product 𝑋 Θ Θ in RKK 2 n 𝒢 ⁡ ( X ; 𝟙 , P ⊗ P ) . \textup{PD}(\nabla)\mathrel{\vcentcolon=}\Theta\otimes_{P}\nabla=\Theta\otimes_{X}\Theta\qquad\text{in $\operatorname{RKK}^{\mathcal{G}}_{2n}(X;\mathds{1},P\otimes P)$.} We call ∇ ∇ \nabla the comultiplication of the duality . When we change the dual, ∇ ∇ \nabla is replaced by ( − 1 ) n ​ ( n ′ − n ) ​ ψ − 1 ⊗ P ∇ ⊗ P ⊗ P ( ψ ⊗ ψ ) ∈ KK n ′ 𝒢 ⁡ ( P ′ , P ′ ⊗ P ′ ) . subscript tensor-product tensor-product 𝑃 𝑃 subscript tensor-product 𝑃 superscript 1 𝑛 superscript 𝑛 ′ 𝑛 superscript 𝜓 1 ∇ tensor-product 𝜓 𝜓 subscript superscript KK 𝒢 superscript 𝑛 ′ superscript 𝑃 ′ tensor-product superscript 𝑃 ′ superscript 𝑃 ′ (-1)^{n(n^{\prime}-n)}\psi^{-1}\otimes_{P}\nabla\otimes_{P\otimes P}(\psi\otimes\psi)\in\operatorname{KK}^{\mathcal{G}}_{n^{\prime}}(P^{\prime},P^{\prime}\otimes P^{\prime}). Remark 1.12 . If n = 0 𝑛 0 n=0 then the object P 𝑃 P of KK 𝒢 superscript KK 𝒢 \operatorname{KK}^{\mathcal{G}} with counit D 𝐷 D and comultiplication ∇ ∇ \nabla is a cocommutative, counital coalgebra object in the tensor category KK 𝒢 superscript KK 𝒢 \operatorname{KK}^{\mathcal{G}} : (1.13) ∇ ⊗ P ⊗ P ( ∇ ⊗ 1 P ) = ∇ ⊗ P ⊗ P ( 1 P ⊗ ∇ ) , subscript tensor-product tensor-product 𝑃 𝑃 ∇ tensor-product ∇ subscript 1 𝑃 subscript tensor-product tensor-product 𝑃 𝑃 ∇ tensor-product subscript 1 𝑃 ∇ \displaystyle\nabla\otimes_{P\otimes P}(\nabla\otimes 1_{P})=\nabla\otimes_{P\otimes P}(1_{P}\otimes\nabla), (1.14) ∇ ⊗ P ⊗ P Φ P = ∇ , subscript tensor-product tensor-product 𝑃 𝑃 ∇ subscript Φ 𝑃 ∇ \displaystyle\nabla\otimes_{P\otimes P}\Phi_{P}=\nabla, (1.15) ∇ ⊗ P ⊗ P ( D ⊗ 1 P ) = 1 P = ∇ ⊗ P ⊗ P ( 1 P ⊗ D ) . subscript tensor-product tensor-product 𝑃 𝑃 ∇ tensor-product 𝐷 subscript 1 𝑃 subscript 1 𝑃 subscript tensor-product tensor-product 𝑃 𝑃 ∇ tensor-product subscript 1 𝑃 𝐷 \displaystyle\nabla\otimes_{P\otimes P}(D\otimes 1_{P})=1_{P}=\nabla\otimes_{P\otimes P}(1_{P}\otimes D). Equation ( 1.13 ) holds in KK 2 ​ n 𝒢 ⁡ ( P , P ⊗ 3 ) subscript superscript KK 𝒢 2 𝑛 𝑃 superscript 𝑃 tensor-product absent 3 \operatorname{KK}^{\mathcal{G}}_{2n}(P,P^{\otimes 3}) , equation ( 1.14 ) holds in KK n 𝒢 ⁡ ( P , P ⊗ P ) subscript superscript KK 𝒢 𝑛 𝑃 tensor-product 𝑃 𝑃 \operatorname{KK}^{\mathcal{G}}_{n}(P,P\otimes P) , and ( 1.15 ) holds in KK 0 𝒢 ⁡ ( P , P ) subscript superscript KK 𝒢 0 𝑃 𝑃 \operatorname{KK}^{\mathcal{G}}_{0}(P,P) . Now, for 𝒢 𝒢 \mathcal{G} - C ∗ superscript C \textup{C}^{*} -algebras A 𝐴 A and B 𝐵 B , we define T P ′ : RKK ∗ 𝒢 ⁡ ( X ; A , B ) → KK ∗ 𝒢 ⁡ ( P ⊗ A , P ⊗ B ) , f ↦ ∇ ⊗ P PD − 1 ​ ( f ) , : subscript superscript 𝑇 ′ 𝑃 formulae-sequence → subscript superscript RKK 𝒢 𝑋 𝐴 𝐵 superscript subscript KK 𝒢 tensor-product 𝑃 𝐴 tensor-product 𝑃 𝐵 maps-to 𝑓 subscript tensor-product 𝑃 ∇ superscript PD 1 𝑓 T^{\prime}_{P}\colon\operatorname{RKK}^{\mathcal{G}}_{*}(X;A,B)\to\operatorname{KK}_{*}^{\mathcal{G}}(P\otimes A,P\otimes B),\qquad f\mapsto\nabla\otimes_{P}\textup{PD}^{-1}(f), where PD is the duality isomorphism, ∇ ∇ \nabla is the comultiplication of the duality, and ⊗ P subscript tensor-product 𝑃 \otimes_{P} operates on the second copy of P 𝑃 P in the target P ⊗ P tensor-product 𝑃 𝑃 P\otimes P of ∇ ∇ \nabla . A computation yields that (1.16) PD ​ ( T P ′ ​ ( f ) ) = Θ ⊗ X f in RKK i + n 𝒢 ⁡ ( X ; A , P ⊗ B ) PD subscript superscript 𝑇 ′ 𝑃 𝑓 subscript tensor-product 𝑋 Θ 𝑓 in RKK i + n 𝒢 ⁡ ( X ; A , P ⊗ B ) \textup{PD}\bigl{(}T^{\prime}_{P}(f)\bigr{)}=\Theta\otimes_{X}f\qquad\text{in $\operatorname{RKK}^{\mathcal{G}}_{i+n}(X;A,P\otimes B)$} for all f ∈ RKK i 𝒢 ⁡ ( X ; A , B ) 𝑓 subscript superscript RKK 𝒢 𝑖 𝑋 𝐴 𝐵 f\in\operatorname{RKK}^{\mathcal{G}}_{i}(X;A,B) . It follows that T P ′ ​ ( f ) = T P ​ ( f ) subscript superscript 𝑇 ′ 𝑃 𝑓 subscript 𝑇 𝑃 𝑓 T^{\prime}_{P}(f)=T_{P}(f) if ( P , Θ ) 𝑃 Θ (P,\Theta) is part of a Kasparov dual, and thus T P subscript 𝑇 𝑃 T_{P} is in fact independent of the 𝒢 ⋉ X left-normal-factor-semidirect-product 𝒢 𝑋 \mathcal{G}\ltimes X -structure on P 𝑃 P , verifying our guess above. When we change the dual, we replace T P ′ subscript superscript 𝑇 ′ 𝑃 T^{\prime}_{P} by the map (1.17) RKK i 𝒢 ⁡ ( X ; A , B ) ∋ f ↦ ( − 1 ) i ​ ( n − n ′ ) ​ ψ − 1 ⊗ P T P ​ ( f ) ⊗ P ψ ∈ KK i 𝒢 ⁡ ( P ′ ⊗ A , P ′ ⊗ B ) . contains subscript superscript RKK 𝒢 𝑖 𝑋 𝐴 𝐵 𝑓 maps-to subscript tensor-product 𝑃 subscript tensor-product 𝑃 superscript 1 𝑖 𝑛 superscript 𝑛 ′ superscript 𝜓 1 subscript 𝑇 𝑃 𝑓 𝜓 subscript superscript KK 𝒢 𝑖 tensor-product superscript 𝑃 ′ 𝐴 tensor-product superscript 𝑃 ′ 𝐵 \operatorname{RKK}^{\mathcal{G}}_{i}(X;A,B)\ni f\mapsto(-1)^{i(n-n^{\prime})}\psi^{-1}\otimes_{P}T_{P}(f)\otimes_{P}\psi\in\operatorname{KK}^{\mathcal{G}}_{i}(P^{\prime}\otimes A,P^{\prime}\otimes B). In fact, one can check that the maps T P ′ subscript superscript 𝑇 ′ 𝑃 T^{\prime}_{P} above define a functor T P ′ : RKK 𝒢 ⁡ ( X ) → KK 𝒢 . : subscript superscript 𝑇 ′ 𝑃 → superscript RKK 𝒢 𝑋 superscript KK 𝒢 T^{\prime}_{P}\colon\operatorname{RKK}^{\mathcal{G}}(X)\to\operatorname{KK}^{\mathcal{G}}. This is a KK 𝒢 superscript KK 𝒢 \operatorname{KK}^{\mathcal{G}} -functor in the sense that it is compatible with the tensor products ⊗ tensor-product \otimes , and it is left adjoint to the functor p X ∗ : KK 𝒢 → RKK 𝒢 : superscript subscript 𝑝 𝑋 → superscript KK 𝒢 superscript RKK 𝒢 p_{X}^{*}\colon\operatorname{KK}^{\mathcal{G}}\to\operatorname{RKK}^{\mathcal{G}} induced from the groupoid homomorphism 𝒢 ⋉ X → 𝒢 → left-normal-factor-semidirect-product 𝒢 𝑋 𝒢 \mathcal{G}\ltimes X\to\mathcal{G} . It follows that we can write the inverse duality map involved in an abstract dual ( P , Θ ) 𝑃 Θ (P,\Theta) as: (1.18) PD − 1 ​ ( f ) = ( − 1 ) i ​ n ​ T P ′ ​ ( f ) ⊗ P D in KK i − n 𝒢 ⁡ ( P ⊗ A , B ) superscript PD 1 𝑓 subscript tensor-product 𝑃 superscript 1 𝑖 𝑛 subscript superscript 𝑇 ′ 𝑃 𝑓 𝐷 in KK i − n 𝒢 ⁡ ( P ⊗ A , B ) \textup{PD}^{-1}(f)=(-1)^{in}T^{\prime}_{P}(f)\otimes_{P}D\qquad\text{in $\operatorname{KK}^{\mathcal{G}}_{i-n}(P\otimes A,B)$} for f ∈ RKK i 𝒢 ⁡ ( X ; A , B ) 𝑓 subscript superscript RKK 𝒢 𝑖 𝑋 𝐴 𝐵 f\in\operatorname{RKK}^{\mathcal{G}}_{i}(X;A,B) . By the above discussion this formula agrees with the map PD ∗ superscript PD \textup{PD}^{*} when we have a Kasparov dual. 1.9. The Lefschetz map The formal computations summarized in the previous section allows us to single out an interesting invariant of a 𝒢 𝒢 \mathcal{G} -space X 𝑋 X , at least under the hypothesis that X 𝑋 X has some abstract dual. For any 𝒢 𝒢 \mathcal{G} -space X 𝑋 X the diagonal embedding X → X × Z X → 𝑋 subscript 𝑍 𝑋 𝑋 X\to X\times_{Z}X is a proper map and hence induces a ∗ -homomorphism 𝟙 X ⊗ 𝟙 X ≅ 𝒞 0 ​ ( X × Z X ) → 𝒞 0 ​ ( X ) = 𝟙 X . tensor-product subscript 1 𝑋 subscript 1 𝑋 subscript 𝒞 0 subscript 𝑍 𝑋 𝑋 → subscript 𝒞 0 𝑋 subscript 1 𝑋 \mathds{1}_{X}\otimes\mathds{1}_{X}\cong\mathcal{C}_{0}(X\times_{Z}X)\to\mathcal{C}_{0}(X)=\mathds{1}_{X}. This map is 𝒢 ⋉ X left-normal-factor-semidirect-product 𝒢 𝑋 \mathcal{G}\ltimes X -equivariant and hence yields Δ X ∈ RKK 𝒢 ⁡ ( X ; 𝟙 X , 𝟙 ) ≅ KK 𝒢 ⋉ X ⁡ ( 𝒞 0 ​ ( X × Z X ) , 𝒞 0 ​ ( X ) ) . subscript Δ 𝑋 superscript RKK 𝒢 𝑋 subscript 1 𝑋 1 superscript KK left-normal-factor-semidirect-product 𝒢 𝑋 subscript 𝒞 0 subscript 𝑍 𝑋 𝑋 subscript 𝒞 0 𝑋 \Delta_{X}\in\operatorname{RKK}^{\mathcal{G}}(X;\mathds{1}_{X},\mathds{1})\cong\operatorname{KK}^{\mathcal{G}\ltimes X}\bigl{(}\mathcal{C}_{0}(X\times_{Z}X),\mathcal{C}_{0}(X)\bigr{)}. We call this the diagonal restriction class . It yields a canonical map (1.19) ␣ ⊗ 𝟙 X Δ X : KK 𝒢 ⁡ ( 𝟙 X ⊗ A , 𝟙 X ⊗ B ) → RKK 𝒢 ⁡ ( X ; 𝟙 X ⊗ A , B ) . : subscript tensor-product subscript 1 𝑋 ␣ subscript Δ 𝑋 → superscript KK 𝒢 tensor-product subscript 1 𝑋 𝐴 tensor-product subscript 1 𝑋 𝐵 superscript RKK 𝒢 𝑋 tensor-product subscript 1 𝑋 𝐴 𝐵 \textup{\textvisiblespace}\otimes_{\mathds{1}_{X}}\Delta_{X}\colon\operatorname{KK}^{\mathcal{G}}(\mathds{1}_{X}\otimes A,\mathds{1}_{X}\otimes B)\to\operatorname{RKK}^{\mathcal{G}}(X;\mathds{1}_{X}\otimes A,B). In particular, this contains a map KK 𝒢 ⁡ ( 𝟙 X , 𝟙 X ) → RKK 𝒢 ⁡ ( X ; 𝟙 X , 𝟙 ) → superscript KK 𝒢 subscript 1 𝑋 subscript 1 𝑋 superscript RKK 𝒢 𝑋 subscript 1 𝑋 1 \operatorname{KK}^{\mathcal{G}}(\mathds{1}_{X},\mathds{1}_{X})\to\operatorname{RKK}^{\mathcal{G}}(X;\mathds{1}_{X},\mathds{1}) . Example 1.20 . If f : X → X : 𝑓 → 𝑋 𝑋 f\colon X\to X is a proper, continuous, 𝒢 𝒢 \mathcal{G} -equivariant map, then [ f ] ⊗ 𝟙 X Δ X ∈ RKK 𝒢 ⁡ ( X ; 𝟙 X , 𝟙 ) subscript tensor-product subscript 1 𝑋 delimited-[] 𝑓 subscript Δ 𝑋 superscript RKK 𝒢 𝑋 subscript 1 𝑋 1 [f]\otimes_{\mathds{1}_{X}}\Delta_{X}\in\operatorname{RKK}^{\mathcal{G}}(X;\mathds{1}_{X},\mathds{1}) is the class of the ∗ -homomorphism induced by ( id X , f ) : X → X × Z X : subscript id 𝑋 𝑓 → 𝑋 subscript 𝑍 𝑋 𝑋 (\textup{id}_{X},f)\colon X\to X\times_{Z}X . Now drop the assumption that f 𝑓 f be proper. Then ( id X , f ) subscript id 𝑋 𝑓 (\textup{id}_{X},f) is still a proper, continuous, 𝒢 𝒢 \mathcal{G} -equivariant map. The class of the ∗ -homomorphism it induces is equal to f ∗ ​ ( Δ X ) superscript 𝑓 subscript Δ 𝑋 f^{*}(\Delta_{X}) , where we use the maps f ∗ : RKK ∗ 𝒢 ⁡ ( X ; A , B ) → RKK ∗ 𝒢 ⁡ ( X ; A , B ) : superscript 𝑓 → subscript superscript RKK 𝒢 𝑋 𝐴 𝐵 subscript superscript RKK 𝒢 𝑋 𝐴 𝐵 f^{*}\colon\operatorname{RKK}^{\mathcal{G}}_{*}(X;A,B)\to\operatorname{RKK}^{\mathcal{G}}_{*}(X;A,B) for A = 𝟙 X 𝐴 subscript 1 𝑋 A=\mathds{1}_{X} , B = 𝟙 𝐵 1 B=\mathds{1} induced by f : X → X : 𝑓 → 𝑋 𝑋 f\colon X\to X (the functor X ↦ RKK 𝒢 ⁡ ( X ; A , B ) maps-to 𝑋 superscript RKK 𝒢 𝑋 𝐴 𝐵 X\mapsto\operatorname{RKK}^{\mathcal{G}}(X;A,B) is functorial with respect to arbitrary 𝒢 𝒢 \mathcal{G} -maps, not just proper ones.) This suggests that we can think of RKK 𝒢 ⁡ ( X ; 𝟙 X , 𝟙 ) superscript RKK 𝒢 𝑋 subscript 1 𝑋 1 \operatorname{RKK}^{\mathcal{G}}(X;\mathds{1}_{X},\mathds{1}) as generalized, possibly non-proper self-maps of X 𝑋 X . In fact if the anchor map X → Z → 𝑋 𝑍 X\to Z is a proper map, so that X 𝑋 X is a bundle of compact spaces over Z 𝑍 Z , then ␣ ⊗ 𝟙 X Δ X subscript tensor-product subscript 1 𝑋 ␣ subscript Δ 𝑋 \textup{\textvisiblespace}\otimes_{\mathds{1}_{X}}\Delta_{X} is an isomophism (an easy exercise in the definitions.) Now let T P ′ subscript superscript 𝑇 ′ 𝑃 T^{\prime}_{P} be the tensor functor and Δ X subscript Δ 𝑋 \Delta_{X} the diagonal restriction class of an abstract dual. We define the multiplication class of P 𝑃 P by (1.21) [ m ] := T P ′ ​ ( Δ X ) ∈ KK 0 𝒢 ⁡ ( P ⊗ 𝟙 X , P ) . assign delimited-[] 𝑚 subscript superscript 𝑇 ′ 𝑃 subscript Δ 𝑋 subscript superscript KK 𝒢 0 tensor-product 𝑃 subscript 1 𝑋 𝑃 [m]\mathrel{\vcentcolon=}T^{\prime}_{P}(\Delta_{X})\in\operatorname{KK}^{\mathcal{G}}_{0}(P\otimes\mathds{1}_{X},P). A change of dual as in ( 1.10 ) replaces [ m ] delimited-[] 𝑚 [m] by ψ − 1 ⊗ P [ m ] ⊗ P ψ subscript tensor-product 𝑃 subscript tensor-product 𝑃 superscript 𝜓 1 delimited-[] 𝑚 𝜓 \psi^{-1}\otimes_{P}[m]\otimes_{P}\psi . Lemma 1.22 . Let ( P , D , Θ ) 𝑃 𝐷 Θ (P,D,\Theta) be a Kasparov dual. Then [ m ] delimited-[] 𝑚 [m] is the class in KK 𝒢 superscript KK 𝒢 \operatorname{KK}^{\mathcal{G}} of the multiplication homomorphism 𝒞 0 ​ ( X ) ⊗ Z P → P → subscript tensor-product 𝑍 subscript 𝒞 0 𝑋 𝑃 𝑃 \mathcal{C}_{0}(X)\otimes_{Z}P\to P that describes the X 𝑋 X -structure on P 𝑃 P ( up to commuting the tensor factors ) . We now have enough theoretical development to define the Lefschetz map and sketch the proof of its homotopy invariance. Let X 𝑋 X be a 𝒢 𝒢 \mathcal{G} -space and ( P , Θ ) 𝑃 Θ (P,\Theta) an n 𝑛 n -dimensional abstract dual for X 𝑋 X , PD and PD − 1 superscript PD 1 \textup{PD}^{-1} the duality isomorphisms. As before, we write 𝟙 := 𝒞 0 ​ ( Z ) assign 1 subscript 𝒞 0 𝑍 \mathds{1}\mathrel{\vcentcolon=}\mathcal{C}_{0}(Z) , 𝟙 X := 𝒞 0 ​ ( X ) assign subscript 1 𝑋 subscript 𝒞 0 𝑋 \mathds{1}_{X}\mathrel{\vcentcolon=}\mathcal{C}_{0}(X) and Δ X ∈ RKK 𝒢 ⁡ ( X ; 𝟙 X , 𝟙 ) = KK 𝒢 ⋉ X ⁡ ( 𝟙 X ⊗ 𝟙 X , 𝟙 X ) subscript Δ 𝑋 superscript RKK 𝒢 𝑋 subscript 1 𝑋 1 superscript KK left-normal-factor-semidirect-product 𝒢 𝑋 tensor-product subscript 1 𝑋 subscript 1 𝑋 subscript 1 𝑋 \Delta_{X}\in\operatorname{RKK}^{\mathcal{G}}(X;\mathds{1}_{X},\mathds{1})=\operatorname{KK}^{\mathcal{G}\ltimes X}(\mathds{1}_{X}\otimes\mathds{1}_{X},\mathds{1}_{X}) the diagonal restriction class and Θ ¯ := forget X ​ ( Θ ) ∈ KK n 𝒢 ⁡ ( 𝟙 X , P ⊗ 𝟙 X ) . assign ¯ Θ subscript forget 𝑋 Θ subscript superscript KK 𝒢 𝑛 subscript 1 𝑋 tensor-product 𝑃 subscript 1 𝑋 \bar{\Theta}\mathrel{\vcentcolon=}\textup{forget}_{X}(\Theta)\in\operatorname{KK}^{\mathcal{G}}_{n}(\mathds{1}_{X},P\otimes\mathds{1}_{X}). Definition 1.23 . The equivariant Lefschetz map Lef : RKK ∗ 𝒢 ⁡ ( X ; 𝟙 X , 𝟙 ) → KK ∗ 𝒢 ⁡ ( 𝟙 X , 𝟙 ) : Lef → subscript superscript RKK 𝒢 𝑋 subscript 1 𝑋 1 superscript subscript KK 𝒢 subscript 1 𝑋 1 \operatorname{Lef}\colon\operatorname{RKK}^{\mathcal{G}}_{*}(X;\mathds{1}_{X},\mathds{1})\to\operatorname{KK}_{*}^{\mathcal{G}}(\mathds{1}_{X},\mathds{1}) for a 𝒢 𝒢 \mathcal{G} -space X 𝑋 X is defined as the composite map RKK i 𝒢 ⁡ ( X ; 𝟙 X , 𝟙 ) → PD − 1 KK i − n 𝒢 ⁡ ( P ⊗ 𝟙 X , 𝟙 ) → Θ ¯ ⊗ P ⊗ 𝟙 X ␣ KK i 𝒢 ⁡ ( 𝟙 X , 𝟙 ) . superscript PD 1 → subscript superscript RKK 𝒢 𝑖 𝑋 subscript 1 𝑋 1 subscript superscript KK 𝒢 𝑖 𝑛 tensor-product 𝑃 subscript 1 𝑋 1 subscript tensor-product tensor-product 𝑃 subscript 1 𝑋 ¯ Θ ␣ → subscript superscript KK 𝒢 𝑖 subscript 1 𝑋 1 \operatorname{RKK}^{\mathcal{G}}_{i}(X;\mathds{1}_{X},\mathds{1})\xrightarrow{\textup{PD}^{-1}}\operatorname{KK}^{\mathcal{G}}_{i-n}(P\otimes\mathds{1}_{X},\mathds{1})\xrightarrow{\bar{\Theta}\otimes_{P\otimes\mathds{1}_{X}}\textup{\textvisiblespace}}\operatorname{KK}^{\mathcal{G}}_{i}(\mathds{1}_{X},\mathds{1}). The equivariant Euler characteristic of X 𝑋 X is Eul X := Lef ⁡ ( Δ X ) ∈ KK 0 𝒢 ⁡ ( 𝟙 X , 𝟙 ) = KK 0 𝒢 ⁡ ( 𝒞 0 ​ ( X ) , 𝒞 0 ​ ( Z ) ) . assign subscript Eul 𝑋 Lef subscript Δ 𝑋 subscript superscript KK 𝒢 0 subscript 1 𝑋 1 subscript superscript KK 𝒢 0 subscript 𝒞 0 𝑋 subscript 𝒞 0 𝑍 \operatorname{Eul}_{X}\mathrel{\vcentcolon=}\operatorname{Lef}(\Delta_{X})\in\operatorname{KK}^{\mathcal{G}}_{0}(\mathds{1}_{X},\mathds{1})=\operatorname{KK}^{\mathcal{G}}_{0}\bigl{(}\mathcal{C}_{0}(X),\mathcal{C}_{0}(Z)\bigr{)}. Let f ∈ RKK i 𝒢 ⁡ ( X ; 𝟙 X , 𝟙 ) 𝑓 subscript superscript RKK 𝒢 𝑖 𝑋 subscript 1 𝑋 1 f\in\operatorname{RKK}^{\mathcal{G}}_{i}(X;\mathds{1}_{X},\mathds{1}) . Equations ( 1.18 ) and ( 1.21 ) yield (1.24) Lef ⁡ ( f ) Lef 𝑓 \displaystyle\operatorname{Lef}(f) = ( − 1 ) i ​ n ​ Θ ¯ ⊗ P ⊗ 𝟙 X T P ′ ​ ( f ) ⊗ P D , absent subscript tensor-product 𝑃 subscript tensor-product tensor-product 𝑃 subscript 1 𝑋 superscript 1 𝑖 𝑛 ¯ Θ subscript superscript 𝑇 ′ 𝑃 𝑓 𝐷 \displaystyle=(-1)^{in}\bar{\Theta}\otimes_{P\otimes\mathds{1}_{X}}T^{\prime}_{P}(f)\otimes_{P}D, (1.25) Eul X subscript Eul 𝑋 \displaystyle\operatorname{Eul}_{X} = ( − 1 ) i ​ n ​ Θ ¯ ⊗ P ⊗ 𝟙 X [ m ] ⊗ P D . absent subscript tensor-product 𝑃 subscript tensor-product tensor-product 𝑃 subscript 1 𝑋 superscript 1 𝑖 𝑛 ¯ Θ delimited-[] 𝑚 𝐷 \displaystyle=(-1)^{in}\bar{\Theta}\otimes_{P\otimes\mathds{1}_{X}}[m]\otimes_{P}D. We have already established that if ( P , Θ ) 𝑃 Θ (P,\Theta) is part of a Kasparov dual, then T P ′ = T P subscript superscript 𝑇 ′ 𝑃 subscript 𝑇 𝑃 T^{\prime}_{P}=T_{P} and [ m ] delimited-[] 𝑚 [m] is the KK KK \operatorname{KK} -class of the multiplication ∗ -homomorphism 𝒞 0 ​ ( X , P ) → P → subscript 𝒞 0 𝑋 𝑃 𝑃 \mathcal{C}_{0}(X,P)\to P , so that ( 1.24 ) yields explicit formulas for Lef ⁡ ( f ) Lef 𝑓 \operatorname{Lef}(f) and Eul X subscript Eul 𝑋 \operatorname{Eul}_{X} . This is extremely important because otherwise it would not be possible to compute these invariants. Let X 𝑋 X and X ′ superscript 𝑋 ′ X^{\prime} be 𝒢 𝒢 \mathcal{G} -spaces, and let f : X → X ′ : 𝑓 → 𝑋 superscript 𝑋 ′ f\colon X\to X^{\prime} be a 𝒢 𝒢 \mathcal{G} -homotopy equivalence. Then f 𝑓 f induces an equivalence of categories RKK 𝒢 ⁡ ( X ′ ) ≅ RKK 𝒢 ⁡ ( X ) superscript RKK 𝒢 superscript 𝑋 ′ superscript RKK 𝒢 𝑋 \operatorname{RKK}^{\mathcal{G}}(X^{\prime})\cong\operatorname{RKK}^{\mathcal{G}}(X) , that is, we get invertible maps f ∗ : RKK ∗ 𝒢 ⁡ ( X ′ ; A , B ) → RKK ∗ 𝒢 ⁡ ( X ; A , B ) : superscript 𝑓 → subscript superscript RKK 𝒢 superscript 𝑋 ′ 𝐴 𝐵 subscript superscript RKK 𝒢 𝑋 𝐴 𝐵 f^{*}\colon\operatorname{RKK}^{\mathcal{G}}_{*}(X^{\prime};A,B)\to\operatorname{RKK}^{\mathcal{G}}_{*}(X;A,B) for all 𝒢 𝒢 \mathcal{G} - C ∗ superscript C \textup{C}^{*} -algebras A 𝐴 A and B 𝐵 B . Now assume, in addition, that f 𝑓 f is proper; we do not need the inverse map or the homotopies to be proper. Then f 𝑓 f induces a ∗ -homomorphism f ! : 𝒞 0 ​ ( X ′ ) → 𝒞 0 ​ ( X ) : superscript 𝑓 → subscript 𝒞 0 superscript 𝑋 ′ subscript 𝒞 0 𝑋 f^{!}\colon\mathcal{C}_{0}(X^{\prime})\to\mathcal{C}_{0}(X) , which yields [ f ! ] ∈ KK 𝒢 ⁡ ( 𝒞 0 ​ ( X ′ ) , 𝒞 0 ​ ( X ) ) delimited-[] superscript 𝑓 superscript KK 𝒢 subscript 𝒞 0 superscript 𝑋 ′ subscript 𝒞 0 𝑋 [f^{!}]\in\operatorname{KK}^{\mathcal{G}}\bigl{(}\mathcal{C}_{0}(X^{\prime}),\mathcal{C}_{0}(X)\bigr{)} . We write [ f ! ] delimited-[] superscript 𝑓 [f^{!}] instead of [ f ∗ ] delimited-[] superscript 𝑓 [f^{*}] to better distinguish this from the map f ∗ superscript 𝑓 f^{*} above. Unless f 𝑓 f is a proper 𝒢 𝒢 \mathcal{G} -homotopy equivalence, [ f ! ] delimited-[] superscript 𝑓 [f^{!}] need not be invertible. Theorem 1.26 . Let X 𝑋 X and X ′ superscript 𝑋 ′ X^{\prime} be 𝒢 𝒢 \mathcal{G} -spaces with abstract duals, and let f : X → X ′ : 𝑓 → 𝑋 superscript 𝑋 ′ f\colon X\to X^{\prime} be both a proper map and a 𝒢 𝒢 \mathcal{G} -homotopy equivalence. Then [ f ! ] ⊗ 𝒞 0 ​ ( X ) Eul X = Eul X ′ in KK 0 𝒢 ⁡ ( 𝒞 0 ​ ( X ′ ) , 𝟙 ) subscript tensor-product subscript 𝒞 0 𝑋 delimited-[] superscript 𝑓 subscript Eul 𝑋 subscript Eul superscript 𝑋 ′ in KK 0 𝒢 ⁡ ( 𝒞 0 ( X ′ ) , 𝟙 ) [f^{!}]\otimes_{\mathcal{C}_{0}(X)}\operatorname{Eul}_{X}=\operatorname{Eul}_{X^{\prime}}\qquad\text{in $\operatorname{KK}^{\mathcal{G}}_{0}(\mathcal{C}_{0}(X^{\prime}),\mathds{1})$} and the Lefschetz maps for X 𝑋 X and X ′ superscript 𝑋 ′ X^{\prime} are related by a commuting diagram RKK ∗ 𝒢 ⁡ ( X ; 𝒞 0 ​ ( X ) , 𝟙 ) subscript superscript RKK 𝒢 𝑋 subscript 𝒞 0 𝑋 1 \textstyle{\operatorname{RKK}^{\mathcal{G}}_{*}(X;\mathcal{C}_{0}(X),\mathds{1})\ignorespaces\ignorespaces\ignorespaces\ignorespaces} Lef X subscript Lef 𝑋 \scriptstyle{\operatorname{Lef}_{X}} RKK ∗ 𝒢 ⁡ ( X ′ ; 𝒞 0 ​ ( X ) , 𝟙 ) subscript superscript RKK 𝒢 superscript 𝑋 ′ subscript 𝒞 0 𝑋 1 \textstyle{\ignorespaces\ignorespaces\ignorespaces\ignorespaces\operatorname{RKK}^{\mathcal{G}}_{*}(X^{\prime};\mathcal{C}_{0}(X),\mathds{1})\ignorespaces\ignorespaces\ignorespaces\ignorespaces} f ∗ superscript 𝑓 \scriptstyle{f^{*}} ≅ \scriptstyle{\cong} [ f ! ] ∗ superscript delimited-[] superscript 𝑓 \scriptstyle{[f^{!}]^{*}} RKK ∗ 𝒢 ⁡ ( X ′ ; 𝒞 0 ​ ( X ′ ) , 𝟙 ) subscript superscript RKK 𝒢 superscript 𝑋 ′ subscript 𝒞 0 superscript 𝑋 ′ 1 \textstyle{\operatorname{RKK}^{\mathcal{G}}_{*}(X^{\prime};\mathcal{C}_{0}(X^{\prime}),\mathds{1})\ignorespaces\ignorespaces\ignorespaces\ignorespaces} Lef X ′ subscript Lef superscript 𝑋 ′ \scriptstyle{\operatorname{Lef}_{X^{\prime}}} KK ∗ 𝒢 ⁡ ( 𝒞 0 ​ ( X ) , 𝟙 ) superscript subscript KK 𝒢 subscript 𝒞 0 𝑋 1 \textstyle{\operatorname{KK}_{*}^{\mathcal{G}}(\mathcal{C}_{0}(X),\mathds{1})\ignorespaces\ignorespaces\ignorespaces\ignorespaces} [ f ! ] ∗ superscript delimited-[] superscript 𝑓 \scriptstyle{[f^{!}]^{*}} KK ∗ 𝒢 ⁡ ( 𝒞 0 ​ ( X ′ ) , 𝟙 ) , superscript subscript KK 𝒢 subscript 𝒞 0 superscript 𝑋 ′ 1 \textstyle{\operatorname{KK}_{*}^{\mathcal{G}}(\mathcal{C}_{0}(X^{\prime}),\mathds{1}),} where [ f ! ] ∗ superscript delimited-[] superscript 𝑓 [f^{!}]^{*} denotes composition with [ f ! ] delimited-[] superscript 𝑓 [f^{!}] . In particular, Eul X subscript Eul 𝑋 \operatorname{Eul}_{X} and the map Lef X subscript Lef 𝑋 \operatorname{Lef}_{X} do not depend on the chosen dual. The proof relies on the discussion preceding the theorem. Theorem 1.26 implies that the Lefschetz maps for properly 𝒢 𝒢 \mathcal{G} -homotopy equivalent spaces are equivalent because then [ f ! ] delimited-[] superscript 𝑓 [f^{!}] is invertible, so that all horizontal maps in the diagram in Theorem 1.26 are invertible. In this sense, the Lefschetz map and the Euler class are invariants of the proper 𝒢 𝒢 \mathcal{G} -homotopy type of X 𝑋 X . The construction in Example 1.20 associates a class [ Δ f ] ∈ RKK 0 𝒢 ⁡ ( X ; 𝒞 0 ​ ( X ) , 𝟙 ) delimited-[] subscript Δ 𝑓 subscript superscript RKK 𝒢 0 𝑋 subscript 𝒞 0 𝑋 1 [\Delta_{f}]\in\operatorname{RKK}^{\mathcal{G}}_{0}(X;\mathcal{C}_{0}(X),\mathds{1}) to any continuous, 𝒢 𝒢 \mathcal{G} -equivariant map f : X → X : 𝑓 → 𝑋 𝑋 f\colon X\to X ; it does not matter whether f 𝑓 f is proper. We abbreviate Lef ⁡ ( f ) := Lef ⁡ ( [ Δ f ] ) assign Lef 𝑓 Lef delimited-[] subscript Δ 𝑓 \operatorname{Lef}(f)\mathrel{\vcentcolon=}\operatorname{Lef}([\Delta_{f}]) and call this the Lefschetz invariant of f 𝑓 f . Of course, equivariantly homotopic self-maps induce the same class in RKK 𝒢 ⁡ ( X ; 𝒞 0 ​ ( X ) , 𝟙 ) superscript RKK 𝒢 𝑋 subscript 𝒞 0 𝑋 1 \operatorname{RKK}^{\mathcal{G}}(X;\mathcal{C}_{0}(X),\mathds{1}) and therefore have the same Lefschetz invariant. We have Lef ⁡ ( id X ) = Eul X Lef subscript id 𝑋 subscript Eul 𝑋 \operatorname{Lef}(\textup{id}_{X})=\operatorname{Eul}_{X} . More generally, specializing ( 1.19 ) gives a map ␣ ⊗ 𝟙 X Δ X : KK ∗ 𝒢 ⁡ ( 𝒞 0 ​ ( X ) , 𝒞 0 ​ ( X ) ) → RKK ∗ 𝒢 ⁡ ( X ; 𝒞 0 ​ ( X ) , 𝟙 ) , : subscript tensor-product subscript 1 𝑋 ␣ subscript Δ 𝑋 → superscript subscript KK 𝒢 subscript 𝒞 0 𝑋 subscript 𝒞 0 𝑋 subscript superscript RKK 𝒢 𝑋 subscript 𝒞 0 𝑋 1 \textup{\textvisiblespace}\otimes_{\mathds{1}_{X}}\Delta_{X}\colon\operatorname{KK}_{*}^{\mathcal{G}}\bigl{(}\mathcal{C}_{0}(X),\mathcal{C}_{0}(X)\bigr{)}\to\operatorname{RKK}^{\mathcal{G}}_{*}(X;\mathcal{C}_{0}(X),\mathds{1}), which we compose with the Lefschetz map; abusing notation, we still denote this composition by Lef : KK ∗ 𝒢 ⁡ ( 𝒞 0 ​ ( X ) , 𝒞 0 ​ ( X ) ) → KK ∗ 𝒢 ⁡ ( 𝒞 0 ​ ( X ) , 𝟙 ) : Lef → superscript subscript KK 𝒢 subscript 𝒞 0 𝑋 subscript 𝒞 0 𝑋 superscript subscript KK 𝒢 subscript 𝒞 0 𝑋 1 \operatorname{Lef}\colon\operatorname{KK}_{*}^{\mathcal{G}}\bigl{(}\mathcal{C}_{0}(X),\mathcal{C}_{0}(X)\bigr{)}\to\operatorname{KK}_{*}^{\mathcal{G}}(\mathcal{C}_{0}(X),\mathds{1}) Finally, we record that Lefschetz invariants for elements of RKK ∗ 𝒢 ⁡ ( X ; 𝒞 0 ​ ( X ) , 𝟙 ) subscript superscript RKK 𝒢 𝑋 subscript 𝒞 0 𝑋 1 \operatorname{RKK}^{\mathcal{G}}_{*}(X;\mathcal{C}_{0}(X),\mathds{1}) can be arbitrarily complicated: the Lefschetz map is rather easily seen to be split surjective. The splitting is given by specializing the inflation map (1.27) p X ∗ : KK ∗ 𝒢 ⁡ ( A , B ) → KK 𝒢 ⋉ X ⁡ ( 𝟙 X ⊗ A , 𝟙 X ⊗ B ) : superscript subscript 𝑝 𝑋 → superscript subscript KK 𝒢 𝐴 𝐵 superscript KK left-normal-factor-semidirect-product 𝒢 𝑋 tensor-product subscript 1 𝑋 𝐴 tensor-product subscript 1 𝑋 𝐵 p_{X}^{*}\colon\operatorname{KK}_{*}^{\mathcal{G}}(A,B)\to\operatorname{KK}^{\mathcal{G}\ltimes X}(\mathds{1}_{X}\otimes A,\mathds{1}_{X}\otimes B) to A := 𝟙 X assign 𝐴 subscript 1 𝑋 A\mathrel{\vcentcolon=}\mathds{1}_{X} and B := 𝟙 assign 𝐵 1 B\mathrel{\vcentcolon=}\mathds{1} . The fundamental example of a Kasparov dual is provided by the vertical tangent space to a bundle of smooth manifolds over the base Z 𝑍 Z of a groupoid, in which morphisms act smoothly. We come back to this in § 4 .

We survey work by the author and Ralf Meyer on equivariant KK-theory. Duality plays a key role in our approach. We organize the survey around the objective of computing a certain homotopy-invariant of a space equipped with a proper action of a group or groupoid called the Lefschetz map. The Lefschetz map associates an equivariant K-homology class to an equivariant Kasparov self-morphism of a space X admitting a dual. We want to describe it explicitly in the setting of bundles of smooth manifolds over the base space of a proper groupoid, in which groupoid elements act by diffeomorphisms between fibres. To get the required description we describe a topological model of equivariant KK-theory by way of a theory of correspondences, building on ideas of Paul Baum, Alain Connes and Georges Skandalis that appeared in the 1980's. This model agrees with the analytic model for bundles of smooth manifolds under some technical conditions related to the existence of equivariant vector bundles. Subject to these conditions we obtain a computation of the Lefschetz map in purely topological terms.

[{"content":"Help me to generate the abstract of a survey paper given the title:*Duality, correspondences and the Lefschetz map in equivariant KK-theory: a survey* and the introduction:1. Abstract duality and the Lefschetz map Throughout this paper, groupoid shall mean locally compact Hausdorff groupoid with Haar system. All topological spaces will be assumed paracompact, locally compact and Hausdorff. For the material in this section, see Emerson-Meyer:Dualities . For source material on equivariant KK KK \operatorname{KK} -theory for groupoids, see LeGall:KK_groupoid . One seems to be forced to consider groupoids, as opposed to groups, in equivariant Kasparov theory, even if one is ultimately only interested in groups. This will be explained later. Therefore we will work more or less uniformly with groupoids when discussing general theory. When we discuss topological equivariant Kasparov theory, we will further assume that all groupoids are proper . This restriction is needed for various geometric constructions. The additional assumption of properness involves no serious loss of generality for our purposes because the Baum-Connes isomorphism, when it applies, gives a method of replacing non-proper groupoids by proper ones. 1.1. Equivariant Kasparov theory for groupoids Let 𝒢 𝒢 \mathcal{G} be a groupoid. We let Z 𝑍 Z denote the base space. A 𝒢 𝒢 \mathcal{G} -C*-algebra is in particular a C*-algebra over Z 𝑍 Z . This means that there is given a non-degenerate equivariant *-homomorphism from 𝒞 0 ​ ( Z ) subscript 𝒞 0 𝑍 \mathcal{C}_{0}(Z) to central multipliers of A 𝐴 A . This identifies A 𝐴 A with the section algebra of a continuous bundle of C*-algebras over Z 𝑍 Z . For a groupoid action we require in addition an isomorphism r ∗ ​ ( A ) → s ∗ ​ ( A ) → superscript 𝑟 𝐴 superscript 𝑠 𝐴 r^{*}(A)\to s^{*}(A) which is compatible with the structure of r ∗ ​ ( A ) superscript 𝑟 𝐴 r^{*}(A) and s ∗ ​ ( A ) superscript 𝑠 𝐴 s^{*}(A) as C*-algebras over 𝒢 𝒢 \mathcal{G} . Here r : 𝒢 → Z : 𝑟 → 𝒢 𝑍 r\colon\mathcal{G}\to Z and s : 𝒢 → Z : 𝑠 → 𝒢 𝑍 s\colon\mathcal{G}\to Z are the range and source map of the groupoid, and r ∗ superscript 𝑟 r^{*} (and similarly s ∗ superscript 𝑠 s^{*} ) denotes the usual pullback operation of bundles. From the bundle point of view, all of this means that groupoid elements g 𝑔 g with s ​ ( g ) = x 𝑠 𝑔 𝑥 s(g)=x and r ​ ( g ) = y 𝑟 𝑔 𝑦 r(g)=y induce *-homomorphisms A x → A y → subscript 𝐴 𝑥 subscript 𝐴 𝑦 A_{x}\to A_{y} between the fibres of A 𝐴 A at x 𝑥 x and y 𝑦 y . In particular, if A 𝐴 A is commutative, then A 𝐴 A is the C*-algebra of continuous functions on a locally compact 𝒢 𝒢 \mathcal{G} -space X 𝑋 X , equipped with a map ϱ X : X → Z : subscript italic-ϱ 𝑋 → 𝑋 𝑍 \varrho_{X}\colon X\to Z called the anchor map for X 𝑋 X , and a homeomorphism 𝒢 × Z , s X → 𝒢 × Z , r X , ( g , x ) ↦ ( g , g ​ x ) formulae-sequence → subscript 𝑍 𝑠 𝒢 𝑋 subscript 𝑍 𝑟 𝒢 𝑋 maps-to 𝑔 𝑥 𝑔 𝑔 𝑥 \mathcal{G}\times_{Z,s}X\to\mathcal{G}\times_{Z,r}X,\hskip 28.45274pt(g,x)\mapsto(g,gx) where the domain and range of this homeomorphism (by abuse of notation) are respectively 𝒢 × Z , s X := { ( g , x ) ∈ 𝒢 × X ∣ s ​ ( g ) = ϱ X ​ ( x ) } , assign subscript 𝑍 𝑠 𝒢 𝑋 conditional-set 𝑔 𝑥 𝒢 𝑋 𝑠 𝑔 subscript italic-ϱ 𝑋 𝑥 \mathcal{G}\times_{Z,s}X\mathrel{\vcentcolon=}\{(g,x)\in\mathcal{G}\times X\mid s(g)=\varrho_{X}(x)\}, and similarly for 𝒢 × Z , r X subscript 𝑍 𝑟 𝒢 𝑋 \mathcal{G}\times_{Z,r}X using r 𝑟 r instead of s 𝑠 s . 1.2. Tensor products The category of 𝒢 𝒢 \mathcal{G} -C*-algebras has a symmetric monoidal structure given by tensor products. We describe this very briefly (see Emerson-Meyer:Dualities Section 2 for details). Let A 𝐴 A and B 𝐵 B be two 𝒢 𝒢 \mathcal{G} -C*-algebras. Since they are each C*-algebras over Z 𝑍 Z , their external tensor product A ⊗ B tensor-product 𝐴 𝐵 A\otimes B is a C*-algebra over Z × Z 𝑍 𝑍 Z\times Z . We restrict this to a C*-algebra over the diagonal Z ⊂ Z × Z 𝑍 𝑍 𝑍 Z\subset Z\times Z . The result is called the tensor product of A 𝐴 A and B 𝐵 B over Z 𝑍 Z . The tensor product of A 𝐴 A and B 𝐵 B over Z 𝑍 Z carries a diagonal action of 𝒢 𝒢 \mathcal{G} . We leave it to the reader to check that we obtain a 𝒢 𝒢 \mathcal{G} -C*-algebra in this way. In order not to complicate notation, we write just A ⊗ B tensor-product 𝐴 𝐵 A\otimes B for the tensor product of A 𝐴 A and B 𝐵 B in the category of 𝒢 𝒢 \mathcal{G} -C*-algebras. We emphasize that the tensor product is over Z 𝑍 Z ; this is not the same as the tensor product in the category of C*-algebras. For commutative C*-algebras, i.e. for 𝒢 𝒢 \mathcal{G} -spaces, say X 𝑋 X and Y 𝑌 Y , with anchor maps as usual denoted ϱ X : X → Z : subscript italic-ϱ 𝑋 → 𝑋 𝑍 \varrho_{X}\colon X\to Z and ϱ Y : Y → Z : subscript italic-ϱ 𝑌 → 𝑌 𝑍 \varrho_{Y}\colon Y\to Z , the tensor product is Gelfand dual to the operation which forms from X 𝑋 X and Y 𝑌 Y the fibre product X × Z Y := { ( x , y ) ∈ X × Y ∣ ϱ X ​ ( x ) = ϱ Y ​ ( y ) } . assign subscript 𝑍 𝑋 𝑌 conditional-set 𝑥 𝑦 𝑋 𝑌 subscript italic-ϱ 𝑋 𝑥 subscript italic-ϱ 𝑌 𝑦 X\times_{Z}Y\mathrel{\vcentcolon=}\{(x,y)\in X\times Y\mid\varrho_{X}(x)=\varrho_{Y}(y)\}. The required anchor map ϱ X × Z Y : X × Z Y → Z : subscript italic-ϱ subscript 𝑍 𝑋 𝑌 → subscript 𝑍 𝑋 𝑌 𝑍 \varrho_{X\times_{Z}Y}\colon X\times_{Z}Y\to Z is of course the composition of the first coordinate projection and the anchor map for X 𝑋 X (or the analogue using Y 𝑌 Y ; they are equal). Of course groupoid elements act diagonally in the obvious way. Such coincidence spaces as the one just described will appear again and again in the theory of correspondences. Finally, for the record, we supply the following important definition. Definition 1.1 . Let 𝒢 𝒢 \mathcal{G} be a groupoid. A 𝒢 𝒢 \mathcal{G} -space X 𝑋 X is proper if the map 𝒢 × Z X → X , ( g , x ) ↦ ( g ​ x , x ) formulae-sequence → subscript 𝑍 𝒢 𝑋 𝑋 maps-to 𝑔 𝑥 𝑔 𝑥 𝑥 \mathcal{G}\times_{Z}X\to X,\;(g,x)\mapsto(gx,x) is a proper map, where 𝒢 × Z X := { ( g , x ) ∈ 𝒢 × X ∣ s ​ ( g ) = ϱ X ​ ( x ) } assign subscript 𝑍 𝒢 𝑋 conditional-set 𝑔 𝑥 𝒢 𝑋 𝑠 𝑔 subscript italic-ϱ 𝑋 𝑥 \mathcal{G}\times_{Z}X\mathrel{\vcentcolon=}\{(g,x)\in\mathcal{G}\times X\mid s(g)=\varrho_{X}(x)\} . A groupoid is itself called proper if it acts properly on its base space Z 𝑍 Z . Explicitly, the map 𝒢 → X × Z X , g ↦ ( r ​ ( g ) , s ​ ( g ) ) formulae-sequence → 𝒢 subscript 𝑍 𝑋 𝑋 maps-to 𝑔 𝑟 𝑔 𝑠 𝑔 \mathcal{G}\to X\times_{Z}X,\hskip 28.45274ptg\mapsto(r(g),s(g)) is required to be proper. 1.3. Equivariant Kasparov theory Le Gall has defined 𝒢 𝒢 \mathcal{G} -equivariant KK KK \operatorname{KK} -theory in LeGall:KK_groupoid . We briefly sketch the definitions. Let A 𝐴 A and B 𝐵 B be (possibly ℤ / 2 ℤ 2 \mathbb{Z}/2 -graded) 𝒢 𝒢 \mathcal{G} -C*-algebras. Then a cycle for KK 𝒢 ⁡ ( A , B ) superscript KK 𝒢 𝐴 𝐵 \operatorname{KK}^{\mathcal{G}}(A,B) is given by a ℤ / 2 ℤ 2 \mathbb{Z}/2 -graded 𝒢 𝒢 \mathcal{G} -equivariant Hilbert B 𝐵 B -module ℰ ℰ \mathcal{E} , together with a 𝒢 𝒢 \mathcal{G} -equivariant grading-preserving *-homomorphism from A 𝐴 A to the C*-algebra of bounded, adjointable operators on ℰ ℰ \mathcal{E} , and an essentially 𝒢 𝒢 \mathcal{G} -equivariant self-adjoint operator F 𝐹 F on ℰ ℰ \mathcal{E} which is graded odd and satisfies a , F 𝑎 𝐹 a,F and a ​ ( F 2 − 1 ) 𝑎 superscript 𝐹 2 1 a(F^{2}-1) are compact operators (essentially zero operators) for all a ∈ A 𝑎 𝐴 a\in A . Modulo an appropriate equivalence relation, the set of equivalence classes of cycles can be identified with the morphism set KK 𝒢 ⁡ ( A , B ) superscript KK 𝒢 𝐴 𝐵 \operatorname{KK}^{\mathcal{G}}(A,B) in an additive, symmetric monoidal category. Higher KK KK \operatorname{KK} -groups are defined using Clifford algebras, and since these are 2 2 2 -periodic, there are only two up to isomorphism. We denote by KK ∗ 𝒢 ⁡ ( A , B ) superscript subscript KK 𝒢 𝐴 𝐵 \operatorname{KK}_{*}^{\mathcal{G}}(A,B) the sum of these two groups. If A 𝐴 A and B 𝐵 B are 𝒢 𝒢 \mathcal{G} -C*-algebras, then the group RKK 𝒢 ⁡ ( X ; A , B ) superscript RKK 𝒢 𝑋 𝐴 𝐵 \operatorname{RKK}^{\mathcal{G}}(X;A,B) is by definition the groupoid-equivariant Kasparov group KK 𝒢 ⋉ X ⁡ ( 𝒞 0 ​ ( X ) ⊗ A , 𝒞 0 ​ ( X ) ⊗ B ) . superscript KK left-normal-factor-semidirect-product 𝒢 𝑋 tensor-product subscript 𝒞 0 𝑋 𝐴 tensor-product subscript 𝒞 0 𝑋 𝐵 \operatorname{KK}^{\mathcal{G}\ltimes X}(\mathcal{C}_{0}(X)\otimes A,\mathcal{C}_{0}(X)\otimes B). The tensor products are in the category of 𝒢 𝒢 \mathcal{G} -C*-algebras. This group differs from KK 𝒢 ⁡ ( A , 𝒞 0 ​ ( X ) ⊗ B ) superscript KK 𝒢 𝐴 tensor-product subscript 𝒞 0 𝑋 𝐵 \operatorname{KK}^{\mathcal{G}}(A,\mathcal{C}_{0}(X)\otimes B) only in the support condition on cycles. For example if 𝒢 𝒢 \mathcal{G} is trivial and A = B = ℂ 𝐴 𝐵 ℂ A=B=\mathbb{C} then KK 𝒢 ⁡ ( A , 𝒞 0 ​ ( X ) ⊗ B ) superscript KK 𝒢 𝐴 tensor-product subscript 𝒞 0 𝑋 𝐵 \operatorname{KK}^{\mathcal{G}}(A,\mathcal{C}_{0}(X)\otimes B) is the ordinary K K \operatorname{K} -theory of X 𝑋 X and RKK 𝒢 ⁡ ( X ; A , B ) superscript RKK 𝒢 𝑋 𝐴 𝐵 \operatorname{RKK}^{\mathcal{G}}(X;A,B) is the representable K K \operatorname{K} -theory of X 𝑋 X (a non-compactly supported theory.) We discuss these groups in more detail in the next section. Of course similar remarks hold for higher RKK 𝒢 superscript RKK 𝒢 \operatorname{RKK}^{\mathcal{G}} -groups. 1.4. Equivariant K K \operatorname{K} -theory In this section, we present an exceedingly brief overview of equivariant K K \operatorname{K} -theory, roughly sufficient for the theory of equivariant correspondences. For more details see Emerson-Meyer:Equivariant_K . Let X 𝑋 X be a proper 𝒢 𝒢 \mathcal{G} -space. Recall that a 𝒢 ⋉ X left-normal-factor-semidirect-product 𝒢 𝑋 \mathcal{G}\ltimes X -space consists of a 𝒢 𝒢 \mathcal{G} -space Y 𝑌 Y together with a 𝒢 𝒢 \mathcal{G} -equivariant map ϱ Y : Y → X : subscript italic-ϱ 𝑌 → 𝑌 𝑋 \varrho_{Y}:Y\to X serving as the anchor map for the 𝒢 ⋉ X left-normal-factor-semidirect-product 𝒢 𝑋 \mathcal{G}\ltimes X -action. Definition 1.2 . Let Y 𝑌 Y be a 𝒢 ⋉ X left-normal-factor-semidirect-product 𝒢 𝑋 \mathcal{G}\ltimes X -space. The 𝒢 𝒢 \mathcal{G} -equivariant representable K K \operatorname{K} -theory of Y 𝑌 Y with X 𝑋 X -compact supports is the group RK 𝒢 , X − ∗ ⁡ ( Y ) := KK ∗ 𝒢 ⋉ X ⁡ ( 𝒞 0 ​ ( X ) , 𝒞 0 ​ ( Y ) ) . assign subscript superscript RK absent 𝒢 𝑋 𝑌 subscript superscript KK left-normal-factor-semidirect-product 𝒢 𝑋 subscript 𝒞 0 𝑋 subscript 𝒞 0 𝑌 \operatorname{RK}^{-*}_{\mathcal{G},X}(Y)\mathrel{\vcentcolon=}\operatorname{KK}^{\mathcal{G}\ltimes X}_{*}\bigl{(}\mathcal{C}_{0}(X),\mathcal{C}_{0}(Y)\bigr{)}. The 𝒢 𝒢 \mathcal{G} -equivariant representable K K \operatorname{K} -theory of Y 𝑌 Y is RK 𝒢 ∗ ⁡ ( Y ) := RK 𝒢 , Y ∗ ⁡ ( Y ) . assign superscript subscript RK 𝒢 𝑌 subscript superscript RK 𝒢 𝑌 𝑌 \operatorname{RK}_{\mathcal{G}}^{*}(Y)\mathrel{\vcentcolon=}\operatorname{RK}^{*}_{\mathcal{G},Y}(Y). Cycles for KK 𝒢 ⋉ X ⁡ ( 𝒞 0 ​ ( X ) , 𝒞 0 ​ ( Y ) ) superscript KK left-normal-factor-semidirect-product 𝒢 𝑋 subscript 𝒞 0 𝑋 subscript 𝒞 0 𝑌 \operatorname{KK}^{\mathcal{G}\ltimes X}\bigl{(}\mathcal{C}_{0}(X),\mathcal{C}_{0}(Y)\bigr{)} consist of pairs ( ℋ , F ) ℋ 𝐹 (\mathcal{H},F) where ℰ ℰ \mathcal{E} is a countably generated ℤ / 2 ℤ 2 \mathbb{Z}/2 -graded 𝒢 ⋉ X left-normal-factor-semidirect-product 𝒢 𝑋 \mathcal{G}\ltimes X -equivariant right Hilbert 𝒞 0 ​ ( X ) subscript 𝒞 0 𝑋 \mathcal{C}_{0}(X) -module equipped with a 𝒢 ⋉ X left-normal-factor-semidirect-product 𝒢 𝑋 \mathcal{G}\ltimes X -equivariant non-degenerate *-homomorphism from 𝒞 0 ​ ( X ) subscript 𝒞 0 𝑋 \mathcal{C}_{0}(X) to the C*-algebra of bounded, adjointable operators on ℰ ℰ \mathcal{E} , and F 𝐹 F is a bounded, odd, self-adjoint, essentially 𝒢 𝒢 \mathcal{G} -equivariant adjointable operator on ℋ ℋ \mathcal{H} such that f ​ ( F 2 − 1 ) 𝑓 superscript 𝐹 2 1 f(F^{2}-1) is a compact operator, for all f ∈ 𝒞 0 ​ ( X ) 𝑓 subscript 𝒞 0 𝑋 f\in\mathcal{C}_{0}(X) . The properness of 𝒢 𝒢 \mathcal{G} implies that F 𝐹 F may be averaged to be actually 𝒢 𝒢 \mathcal{G} -equivariant, so we assume this in the following. The Hilbert 𝒞 0 ​ ( Y ) subscript 𝒞 0 𝑌 \mathcal{C}_{0}(Y) -module ℰ ℰ \mathcal{E} is the space of continuous sections of a continuous field of ℤ / 2 ℤ 2 \mathbb{Z}/2 -graded Hilbert spaces { ℋ y ∣ y ∈ Y } conditional-set subscript ℋ 𝑦 𝑦 𝑌 \{\mathcal{H}_{y}\mid y\in Y\} over Y 𝑌 Y . Since F 𝐹 F must be 𝒞 0 ​ ( Y ) subscript 𝒞 0 𝑌 \mathcal{C}_{0}(Y) -linear, it consists of a continuous family { F y ∣ y ∈ Y } conditional-set subscript 𝐹 𝑦 𝑦 𝑌 \{F_{y}\mid y\in Y\} of odd operators on these graded Hilbert spaces such that F y 2 − 1 subscript superscript 𝐹 2 𝑦 1 F^{2}_{y}-1 is a compact operator on ℋ y subscript ℋ 𝑦 \mathcal{H}_{y} for all y ∈ Y 𝑦 𝑌 y\in Y . By 𝒢 ⋉ X left-normal-factor-semidirect-product 𝒢 𝑋 \mathcal{G}\ltimes X -equivariance, the representation of 𝒞 0 ​ ( X ) subscript 𝒞 0 𝑋 \mathcal{C}_{0}(X) on ℰ ℰ \mathcal{E} must factor through the *-homomorphism 𝒞 0 ​ ( X ) → 𝒞 0 ​ ( Y ) → subscript 𝒞 0 𝑋 subscript 𝒞 0 𝑌 \mathcal{C}_{0}(X)\to\mathcal{C}_{0}(Y) Gelfand dual to the anchor map ϱ Y : Y → X : subscript italic-ϱ 𝑌 → 𝑌 𝑋 \varrho_{Y}\colon Y\to X . Therefore F 𝐹 F commutes with the action of 𝒞 0 ​ ( X ) subscript 𝒞 0 𝑋 \mathcal{C}_{0}(X) as well; in fact the induced representation of 𝒞 0 ​ ( X ) subscript 𝒞 0 𝑋 \mathcal{C}_{0}(X) on each Hilbert space ℋ y subscript ℋ 𝑦 \mathcal{H}_{y} sends a continuous function f ∈ 𝒞 0 ​ ( X ) 𝑓 subscript 𝒞 0 𝑋 f\in\mathcal{C}_{0}(X) to the operator of multiplication by the complex number f ​ ( ϱ Y ​ ( y ) ) 𝑓 subscript italic-ϱ 𝑌 𝑦 f\bigl{(}\varrho_{Y}(y)\bigr{)} . In particular, the only role of the representation of 𝒞 0 ​ ( X ) subscript 𝒞 0 𝑋 \mathcal{C}_{0}(X) is to relax the support condition on the compact-operator valued-function F 2 − 1 superscript 𝐹 2 1 F^{2}-1 from requiring it to vanish at ∞ \infty of Y 𝑌 Y to only requiring it to vanish at infinity along the fibres of ϱ Y : Y → X : subscript italic-ϱ 𝑌 → 𝑌 𝑋 \varrho_{Y}\colon Y\to X . If ϱ Y : Y → X : subscript italic-ϱ 𝑌 → 𝑌 𝑋 \varrho_{Y}\colon Y\to X is a proper map then RK 𝒢 , X ∗ ⁡ ( Y ) = RK 𝒢 ∗ ⁡ ( X ) := RK 𝒢 , X ∗ ⁡ ( X ) superscript subscript RK 𝒢 𝑋 𝑌 subscript superscript RK 𝒢 𝑋 assign subscript superscript RK 𝒢 𝑋 𝑋 \operatorname{RK}_{\mathcal{G},X}^{*}(Y)=\operatorname{RK}^{*}_{\mathcal{G}}(X)\mathrel{\vcentcolon=}\operatorname{RK}^{*}_{\mathcal{G},X}(X) ; these two groups have exactly the same cycles. Example 1.3 . Any 𝒢 𝒢 \mathcal{G} -equivariant complex vector bundle V 𝑉 V on Y 𝑌 Y yields a cycle for RK 𝒢 0 ⁡ ( Y ) subscript superscript RK 0 𝒢 𝑌 \operatorname{RK}^{0}_{\mathcal{G}}(Y) by choosing a 𝒢 𝒢 \mathcal{G} -invariant Hermitian metric on V 𝑉 V and forming the corresponding 𝒢 ⋉ Y left-normal-factor-semidirect-product 𝒢 𝑌 \mathcal{G}\ltimes Y -equivariant ℤ / 2 ℤ 2 \mathbb{Z}/2 -graded right Hilbert 𝒞 0 ​ ( Y ) subscript 𝒞 0 𝑌 \mathcal{C}_{0}(Y) module of sections, where the grading is the trivial one. We set the operator equal to zero. Example 1.4 . Let X 𝑋 X be a 𝒢 𝒢 \mathcal{G} -space and let V 𝑉 V be a 𝒢 𝒢 \mathcal{G} -equivariantly K K \operatorname{K} -oriented vector bundle over X 𝑋 X of (real) dimension n 𝑛 n . The 𝒢 𝒢 \mathcal{G} -equivariant vector bundle projection π V : V → X : subscript 𝜋 𝑉 → 𝑉 𝑋 \pi_{V}\colon V\to X gives V 𝑉 V the structure of a space over X 𝑋 X , so that V 𝑉 V becomes a 𝒢 ⋉ X left-normal-factor-semidirect-product 𝒢 𝑋 \mathcal{G}\ltimes X -space. Then the Thom isomorphism provides an invertible Thom class t V ∈ RK 𝒢 , X dim V ⁡ ( V ) := KK dim V 𝒢 ⋉ X ⁡ ( 𝒞 0 ​ ( X ) , 𝒞 0 ​ ( V ) ) . subscript 𝑡 𝑉 subscript superscript RK dimension 𝑉 𝒢 𝑋 𝑉 assign subscript superscript KK left-normal-factor-semidirect-product 𝒢 𝑋 dimension 𝑉 subscript 𝒞 0 𝑋 subscript 𝒞 0 𝑉 t_{V}\in\operatorname{RK}^{\dim V}_{\mathcal{G},X}(V)\mathrel{\vcentcolon=}\operatorname{KK}^{\mathcal{G}\ltimes X}_{\dim V}\bigl{(}\mathcal{C}_{0}(X),\mathcal{C}_{0}(V)\bigr{)}. In the case 𝒢 = Spin c ​ ( ℝ n ) 𝒢 superscript Spin 𝑐 superscript ℝ 𝑛 \mathcal{G}=\mathrm{Spin}^{c}(\mathbb{R}^{n}) and X = ⋆ 𝑋 ⋆ X=\star and V := ℝ n assign 𝑉 superscript ℝ 𝑛 V\mathrel{\vcentcolon=}\mathbb{R}^{n} with the representation Spin c ​ ( ℝ n ) → Spin ​ ( ℝ n ) → O ​ ( n , ℝ ) → superscript Spin 𝑐 superscript ℝ 𝑛 Spin superscript ℝ 𝑛 → O 𝑛 ℝ \mathrm{Spin}^{c}(\mathbb{R}^{n})\to\mathrm{Spin}(\mathbb{R}^{n})\to\mathrm{O}(n,\mathbb{R}) the class t ℝ n subscript 𝑡 superscript ℝ 𝑛 t_{\mathbb{R}^{n}} is the Bott class figuring in equivariant Bott Periodicity. Certain further normalizations can be made in order to describe the groups RK 𝒢 , X ∗ ⁡ ( Y ) superscript subscript RK 𝒢 𝑋 𝑌 \operatorname{RK}_{\mathcal{G},X}^{*}(Y) . A standard one is to replace the ℤ / 2 ℤ 2 \mathbb{Z}/2 -grading on ℰ ℰ \mathcal{E} by the standard even grading, so that ℰ ℰ \mathcal{E} consists of the sum of two copies of the same Hilbert module. This means that F 𝐹 F can be taken to be of the form ( 0 F 1 ∗ F 1 0 ) matrix 0 superscript subscript 𝐹 1 subscript 𝐹 1 0 \begin{pmatrix}0&F_{1}^{*}\\ F_{1}&0\end{pmatrix} and the conditions involving F 𝐹 F are replaced by ones involving F 1 subscript 𝐹 1 F_{1} and F 1 ∗ superscript subscript 𝐹 1 F_{1}^{*} ; we may as well replace F 𝐹 F by F 1 subscript 𝐹 1 F_{1} . With this convention, the Fredholm conditions are that f ​ ( F ​ F ∗ − 1 ) 𝑓 𝐹 superscript 𝐹 1 f(FF^{*}-1) and f ( F ∗ F − 1 f(F^{*}F-1 are compact for all f ∈ 𝒞 0 ​ ( X ) 𝑓 subscript 𝒞 0 𝑋 f\in\mathcal{C}_{0}(X) . In other words, y ↦ F y maps-to 𝑦 subscript 𝐹 𝑦 y\mapsto F_{y} takes essentially unitary values in 𝔹 ​ ( ℋ y ) 𝔹 subscript ℋ 𝑦 \mathbb{B}(\mathcal{H}_{y}) for all y ∈ Y 𝑦 𝑌 y\in Y and the compact-operator-valued functions F ​ F ∗ − 1 𝐹 superscript 𝐹 1 FF^{*}-1 and F ∗ ​ F − 1 superscript 𝐹 𝐹 1 F^{*}F-1 vanish at infinity along the fibres of ϱ : Y → X : italic-ϱ → 𝑌 𝑋 \varrho\colon Y\to X . The equivariant stabilization theorem for Hilbert modules implies that we may take ℋ ℋ \mathcal{H} to have the special form L 2 ​ ( 𝒢 ) ∞ ⊗ 𝒞 0 ​ ( Z ) 𝒞 0 ​ ( Y ) subscript tensor-product subscript 𝒞 0 𝑍 superscript 𝐿 2 superscript 𝒢 subscript 𝒞 0 𝑌 L^{2}(\mathcal{G})^{\infty}\otimes_{\mathcal{C}_{0}(Z)}\mathcal{C}_{0}(Y) , where L 2 ​ ( 𝒢 ) superscript 𝐿 2 𝒢 L^{2}(\mathcal{G}) is the 𝒢 𝒢 \mathcal{G} -equivariant right Hilbert 𝒞 0 ​ ( Z ) subscript 𝒞 0 𝑍 \mathcal{C}_{0}(Z) - Hilbert module defined using the Haar system of 𝒢 𝒢 \mathcal{G} , and the superscript indicates the sum of countably many copies of L 2 ​ ( 𝒢 ) superscript 𝐿 2 𝒢 L^{2}(\mathcal{G}) . The corresponding field of Hilbert space has value L 2 ​ ( 𝒢 y ) ∞ superscript 𝐿 2 superscript superscript 𝒢 𝑦 L^{2}(\mathcal{G}^{y})^{\infty} at y ∈ Y 𝑦 𝑌 y\in Y where 𝒢 y superscript 𝒢 𝑦 \mathcal{G}^{y} denotes all G ∈ 𝒢 𝐺 𝒢 G\in\mathcal{G} ending in y 𝑦 y , on which we have a given measure specified by the Haar system of 𝒢 𝒢 \mathcal{G} . This leads to a description of RK 𝒢 , Y 0 ⁡ ( X ) subscript superscript RK 0 𝒢 𝑌 𝑋 \operatorname{RK}^{0}_{\mathcal{G},Y}(X) as the group of homotopy-classes of 𝒢 𝒢 \mathcal{G} -equivariant continuous maps from Y 𝑌 Y to the space ℱ 𝒢 subscript ℱ 𝒢 \mathcal{F}_{\mathcal{G}} of Fredholm operators on the Hilbert spaces L 2 ​ ( 𝒢 y ) ∞ superscript 𝐿 2 superscript superscript 𝒢 𝑦 L^{2}(\mathcal{G}^{y})^{\infty} , but topologizing the space ℱ 𝒢 subscript ℱ 𝒢 \mathcal{F}_{\mathcal{G}} is somewhat delicate. Similarly, the relative groups RK 𝒢 , X ∗ ⁡ ( Y ) subscript superscript RK 𝒢 𝑋 𝑌 \operatorname{RK}^{*}_{\mathcal{G},X}(Y) are maps to Fredholm operators with compact vertical support with respect to the map Y → X → 𝑌 𝑋 Y\to X , where the support of a map to Fredholm operators is by definition the complement of the set where the map takes invertible values. Remark 1.5 . If 𝒢 𝒢 \mathcal{G} acts properly and co-compactly on X 𝑋 X , A 𝐴 A is a trivial 𝒢 𝒢 \mathcal{G} -C*-algebra and B 𝐵 B is a 𝒢 ⋉ X left-normal-factor-semidirect-product 𝒢 𝑋 \mathcal{G}\ltimes X -C*-algebra, then there is a canonical isomorphism KK 𝒢 ⋉ X ⁡ ( 𝒞 0 ​ ( X ) ⊗ A , B ) ≅ KK ⁡ ( A , 𝒢 ⋉ B ) . superscript KK left-normal-factor-semidirect-product 𝒢 𝑋 tensor-product subscript 𝒞 0 𝑋 𝐴 𝐵 KK 𝐴 left-normal-factor-semidirect-product 𝒢 𝐵 \operatorname{KK}^{\mathcal{G}\ltimes X}(\mathcal{C}_{0}(X)\otimes A,B)\cong\operatorname{KK}(A,\mathcal{G}\ltimes B). In particular, the 𝒢 𝒢 \mathcal{G} -equivariant representable K K \operatorname{K} -theory of X 𝑋 X agrees with the K K \operatorname{K} -theory of the corresponding cross-product. Under this identification, classes in RK 𝒢 0 ⁡ ( X ) subscript superscript RK 0 𝒢 𝑋 \operatorname{RK}^{0}_{\mathcal{G}}(X) which are represented by equivariant vector bundles on X 𝑋 X correspond to classes in K 0 ⁡ ( 𝒢 ⋉ 𝒞 0 ​ ( X ) ) subscript K 0 left-normal-factor-semidirect-product 𝒢 subscript 𝒞 0 𝑋 \operatorname{K}_{0}(\mathcal{G}\ltimes\mathcal{C}_{0}(X)) which are represented by projections in the stabilisation of 𝒢 ⋉ 𝒞 0 ​ ( X ) left-normal-factor-semidirect-product 𝒢 subscript 𝒞 0 𝑋 \mathcal{G}\ltimes\mathcal{C}_{0}(X) . See Emerson-Meyer:Equivariant_K for more information. Thus, even if the reader is only interested in groups, or the trivial group, it is convenient to introduce groupoids to some extent in order to describe cohomology theories with different support conditions. 1.5. Tensor and forgetful functors The following simple functor will play an important role. If P 𝑃 P is a 𝒢 ⋉ X left-normal-factor-semidirect-product 𝒢 𝑋 \mathcal{G}\ltimes X -algebra, we denote by T P subscript 𝑇 𝑃 T_{P} the map RKK 𝒢 ⁡ ( X ; A , B ) := KK 𝒢 ⋉ X ⁡ ( 𝒞 0 ​ ( X ) ⊗ A , 𝟙 X ⊗ B ) → KK 𝒢 ⁡ ( P ⊗ A , P ⊗ B ) assign superscript RKK 𝒢 𝑋 𝐴 𝐵 superscript KK left-normal-factor-semidirect-product 𝒢 𝑋 tensor-product subscript 𝒞 0 𝑋 𝐴 tensor-product subscript 1 𝑋 𝐵 → superscript KK 𝒢 tensor-product 𝑃 𝐴 tensor-product 𝑃 𝐵 \operatorname{RKK}^{\mathcal{G}}(X;A,B)\mathrel{\vcentcolon=}\operatorname{KK}^{\mathcal{G}\ltimes X}(\mathcal{C}_{0}(X)\otimes A,\mathds{1}_{X}\otimes B)\to\operatorname{KK}^{\mathcal{G}}(P\otimes A,P\otimes B) which sends a 𝒢 ⋉ X left-normal-factor-semidirect-product 𝒢 𝑋 \mathcal{G}\ltimes X -equivariant right Hilbert 𝒞 0 ​ ( X ) ⊗ B tensor-product subscript 𝒞 0 𝑋 𝐵 \mathcal{C}_{0}(X)\otimes B -Hilbert module ℰ ℰ \mathcal{E} to ℰ ⊗ X P subscript tensor-product 𝑋 ℰ 𝑃 \mathcal{E}\otimes_{X}P , the tensor product being in the category of 𝒢 ⋉ X left-normal-factor-semidirect-product 𝒢 𝑋 \mathcal{G}\ltimes X -algebras (we accordingly use a subscript for emphasis) and sends F ∈ 𝔹 ​ ( ℰ ) 𝐹 𝔹 ℰ F\in\mathbb{B}(\mathcal{E}) to the operator F ⊗ X id P subscript tensor-product 𝑋 𝐹 subscript id 𝑃 F\otimes_{X}\textup{id}_{P} . This definition makes sense since F 𝐹 F commutes with the 𝒞 0 ​ ( X ) subscript 𝒞 0 𝑋 \mathcal{C}_{0}(X) -structure on ℰ ℰ \mathcal{E} . The functor T P subscript 𝑇 𝑃 T_{P} is the composition of external product ␣ ⊗ X 1 P : KK 𝒢 ⋉ X ⁡ ( 𝒞 0 ​ ( X ) ⊗ A , 𝒞 0 ​ ( X ) ⊗ B ) → KK 𝒢 ⋉ X ⁡ ( A ⊗ X P , B ⊗ X P ) : subscript tensor-product 𝑋 ␣ subscript 1 𝑃 → superscript KK left-normal-factor-semidirect-product 𝒢 𝑋 tensor-product subscript 𝒞 0 𝑋 𝐴 tensor-product subscript 𝒞 0 𝑋 𝐵 superscript KK left-normal-factor-semidirect-product 𝒢 𝑋 subscript tensor-product 𝑋 𝐴 𝑃 subscript tensor-product 𝑋 𝐵 𝑃 \textup{\textvisiblespace}\otimes_{X}1_{P}\colon\operatorname{KK}^{\mathcal{G}\ltimes X}(\mathcal{C}_{0}(X)\otimes A,\mathcal{C}_{0}(X)\otimes B)\to\operatorname{KK}^{\mathcal{G}\ltimes X}(A\otimes_{X}P,B\otimes_{X}P) (where the X 𝑋 X -structure on A ⊗ P tensor-product 𝐴 𝑃 A\otimes P etc . is on the P 𝑃 P factor), and the forgetful map KK 𝒢 ⋉ X ⁡ ( A ⊗ X P , B X ⊗ P ) → KK 𝒢 ⁡ ( A ⊗ X P , B ⊗ X P ) → superscript KK left-normal-factor-semidirect-product 𝒢 𝑋 subscript tensor-product 𝑋 𝐴 𝑃 tensor-product subscript 𝐵 𝑋 𝑃 superscript KK 𝒢 subscript tensor-product 𝑋 𝐴 𝑃 subscript tensor-product 𝑋 𝐵 𝑃 \operatorname{KK}^{\mathcal{G}\ltimes X}(A\otimes_{X}P,B_{X}\otimes P)\to\operatorname{KK}^{\mathcal{G}}(A\otimes_{X}P,B\otimes_{X}P) which maps a 𝒢 ⋉ X left-normal-factor-semidirect-product 𝒢 𝑋 \mathcal{G}\ltimes X -algebra or Hilbert module to the underlying 𝒢 𝒢 \mathcal{G} -algebra, or Hilbert module, thus forgetting the X 𝑋 X -structure. 1.6. Kasparov duals We begin our discussion of duality by by formalizing some duality calculations of Kasparov, c.f. Kasparov:Novikov *Theorem 4.9. Explicit examples will be discussed later. For convenience of notation we will often write 𝟙 := 𝒞 0 ​ ( Z ) assign 1 subscript 𝒞 0 𝑍 \mathds{1}\mathrel{\vcentcolon=}\mathcal{C}_{0}(Z) . This notation expresses the fact that 𝒞 0 ​ ( Z ) subscript 𝒞 0 𝑍 \mathcal{C}_{0}(Z) is the tensor unit in the tensor category of 𝒢 𝒢 \mathcal{G} -C*-algebras. Similarly, if 𝒢 𝒢 \mathcal{G} acts on a space X 𝑋 X then we sometimes denote by 𝟙 X subscript 1 𝑋 \mathds{1}_{X} the 𝒢 𝒢 \mathcal{G} -C*-algebra 𝒞 0 ​ ( X ) subscript 𝒞 0 𝑋 \mathcal{C}_{0}(X) ; thus 𝟙 X subscript 1 𝑋 \mathds{1}_{X} is the tensor unit in the category of 𝒢 ⋉ X left-normal-factor-semidirect-product 𝒢 𝑋 \mathcal{G}\ltimes X C*-algebras, X 𝑋 X being the base of 𝒢 ⋉ X left-normal-factor-semidirect-product 𝒢 𝑋 \mathcal{G}\ltimes X . This notation is consistent with the source of this material (see Emerson-Meyer:Dualities .) Definition 1.6 . Let n ∈ ℤ 𝑛 ℤ n\in\mathbb{Z} . An n 𝑛 n -dimensional 𝒢 𝒢 \mathcal{G} -equivariant Kasparov dual for the 𝒢 𝒢 \mathcal{G} -space X 𝑋 X is a triple ( P , D , Θ ) 𝑃 𝐷 Θ (P,D,\Theta) , where • P 𝑃 P is a (possibly ℤ / 2 ℤ 2 \mathbb{Z}/2 -graded) 𝒢 ⋉ X left-normal-factor-semidirect-product 𝒢 𝑋 \mathcal{G}\ltimes X - C ∗ superscript C \textup{C}^{*} -algebra, • D ∈ KK − n 𝒢 ⁡ ( P , 𝟙 ) 𝐷 subscript superscript KK 𝒢 𝑛 𝑃 1 D\in\operatorname{KK}^{\mathcal{G}}_{-n}(P,\mathds{1}) , and • Θ ∈ RKK n 𝒢 ⁡ ( X ; 𝟙 , P ) Θ subscript superscript RKK 𝒢 𝑛 𝑋 1 𝑃 \Theta\in\operatorname{RKK}^{\mathcal{G}}_{n}(X;\mathds{1},P) , subject to the following conditions: (1) Θ ⊗ P D = id 𝟙 subscript tensor-product 𝑃 Θ 𝐷 subscript id 1 \Theta\otimes_{P}D=\textup{id}_{\mathds{1}} in RKK 0 𝒢 ⁡ ( X ; 𝟙 , 𝟙 ) subscript superscript RKK 𝒢 0 𝑋 1 1 \operatorname{RKK}^{\mathcal{G}}_{0}(X;\mathds{1},\mathds{1}) ; (2) Θ ⊗ f = Θ ⊗ P T P ​ ( f ) tensor-product Θ 𝑓 subscript tensor-product 𝑃 Θ subscript 𝑇 𝑃 𝑓 \Theta\otimes f=\Theta\otimes_{P}T_{P}(f) in RKK ∗ + n 𝒢 ⁡ ( X ; A , B ⊗ P ) subscript superscript RKK 𝒢 absent 𝑛 𝑋 𝐴 tensor-product 𝐵 𝑃 \operatorname{RKK}^{\mathcal{G}}_{*+n}(X;A,B\otimes P) for all 𝒢 𝒢 \mathcal{G} - C ∗ superscript C \textup{C}^{*} -algebras A 𝐴 A and B 𝐵 B and all f ∈ RKK ∗ 𝒢 ⁡ ( X ; A , B ) 𝑓 subscript superscript RKK 𝒢 𝑋 𝐴 𝐵 f\in\operatorname{RKK}^{\mathcal{G}}_{*}(X;A,B) ; (3) T P ​ ( Θ ) ⊗ P ⊗ P Φ P = ( − 1 ) n ​ T P ​ ( Θ ) subscript tensor-product tensor-product 𝑃 𝑃 subscript 𝑇 𝑃 Θ subscript Φ 𝑃 superscript 1 𝑛 subscript 𝑇 𝑃 Θ T_{P}(\Theta)\otimes_{P\otimes P}\Phi_{P}=(-1)^{n}T_{P}(\Theta) in KK n 𝒢 ⁡ ( P , P ⊗ P ) subscript superscript KK 𝒢 𝑛 𝑃 tensor-product 𝑃 𝑃 \operatorname{KK}^{\mathcal{G}}_{n}(P,P\otimes P) , where Φ P subscript Φ 𝑃 \Phi_{P} is the flip automorphism on P ⊗ P tensor-product 𝑃 𝑃 P\otimes P . The following theorem is proved in Emerson-Meyer:Dualities . Theorem 1.7 . Let n ∈ ℤ 𝑛 ℤ n\in\mathbb{Z} , let P 𝑃 P be a 𝒢 ⋉ X left-normal-factor-semidirect-product 𝒢 𝑋 \mathcal{G}\ltimes X - C ∗ superscript C \textup{C}^{*} -algebra, D ∈ KK − n 𝒢 ⁡ ( P , 𝟙 ) 𝐷 subscript superscript KK 𝒢 𝑛 𝑃 1 D\in\operatorname{KK}^{\mathcal{G}}_{-n}(P,\mathds{1}) , and Θ ∈ RKK n 𝒢 ⁡ ( X ; 𝟙 , P ) Θ subscript superscript RKK 𝒢 𝑛 𝑋 1 𝑃 \Theta\in\operatorname{RKK}^{\mathcal{G}}_{n}(X;\mathds{1},P) . Define two natural transformations PD : KK i − n 𝒢 ⁡ ( P ⊗ A , B ) → RKK i 𝒢 ⁡ ( X ; A , B ) , : absent → subscript superscript KK 𝒢 𝑖 𝑛 tensor-product 𝑃 𝐴 𝐵 subscript superscript RKK 𝒢 𝑖 𝑋 𝐴 𝐵 \displaystyle\colon\operatorname{KK}^{\mathcal{G}}_{i-n}(P\otimes A,B)\to\operatorname{RKK}^{\mathcal{G}}_{i}(X;A,B), f 𝑓 \displaystyle\qquad f ↦ Θ ⊗ P f , maps-to absent subscript tensor-product 𝑃 Θ 𝑓 \displaystyle\mapsto\Theta\otimes_{P}f, PD ∗ superscript PD \displaystyle\textup{PD}^{*} : RKK i 𝒢 ⁡ ( X ; A , B ) → KK i − n 𝒢 ⁡ ( P ⊗ A , B ) , : absent → subscript superscript RKK 𝒢 𝑖 𝑋 𝐴 𝐵 subscript superscript KK 𝒢 𝑖 𝑛 tensor-product 𝑃 𝐴 𝐵 \displaystyle\colon\operatorname{RKK}^{\mathcal{G}}_{i}(X;A,B)\to\operatorname{KK}^{\mathcal{G}}_{i-n}(P\otimes A,B), g 𝑔 \displaystyle\qquad g ↦ ( − 1 ) i ​ n ​ T P ​ ( g ) ⊗ P D , maps-to absent subscript tensor-product 𝑃 superscript 1 𝑖 𝑛 subscript 𝑇 𝑃 𝑔 𝐷 \displaystyle\mapsto(-1)^{in}T_{P}(g)\otimes_{P}D, These two are inverse to each other if and only if ( P , D , Θ ) 𝑃 𝐷 Θ (P,D,\Theta) is an n 𝑛 n -dimensional 𝒢 𝒢 \mathcal{G} -equivariant Kasparov dual for X 𝑋 X . 1.7. Abstract duals The reader may have noticed that the only place the 𝒞 0 ​ ( X ) subscript 𝒞 0 𝑋 \mathcal{C}_{0}(X) -structure on P 𝑃 P comes into play in the conditions listed in Definition 1.6 , and in the statement of Theorem 1.7 , is via the functor T P subscript 𝑇 𝑃 T_{P} . In particular, if one has a Kasparov dual ( P , D , Θ ) 𝑃 𝐷 Θ (P,D,\Theta) and if one changes the 𝒞 0 ​ ( X ) subscript 𝒞 0 𝑋 \mathcal{C}_{0}(X) -structure on P 𝑃 P , for example by composing it with a 𝒢 𝒢 \mathcal{G} -equivariant homeomorphism of X 𝑋 X , then the map PD of Theorem 1.7 does not change; since by the theorem PD ∗ superscript PD \textup{PD}^{*} is its inverse map, it would not change either, strangely, since its definition uses T P subscript 𝑇 𝑃 T_{P} . In fact it turns out that the functor T P subscript 𝑇 𝑃 T_{P} can be reconstructed from PD if one knows that PD is an isomorphism. This is an important idea in connection with the Lefschetz map and suggests the following useful definition. Definition 1.8 . An n 𝑛 n -dimensional abstract dual for X 𝑋 X is a pair ( P , Θ ) 𝑃 Θ (P,\Theta) , where P 𝑃 P is a 𝒢 𝒢 \mathcal{G} - C ∗ superscript C \textup{C}^{*} -algebra and Θ ∈ RKK n 𝒢 ⁡ ( X ; 𝟙 , P ) Θ subscript superscript RKK 𝒢 𝑛 𝑋 1 𝑃 \Theta\in\operatorname{RKK}^{\mathcal{G}}_{n}(X;\mathds{1},P) , such that the map PD defined as in Theorem 1.7 is an isomorphism for all 𝒢 𝒢 \mathcal{G} - C ∗ superscript C \textup{C}^{*} -algebras A 𝐴 A and B 𝐵 B . This definition is shorter, and, as mentioned, is useful for theoretical reasons, but it seems like it should be difficult to check in practise. In any case, it is clear from Theorem 1.7 that a pair ( P , Θ ) 𝑃 Θ (P,\Theta) is an abstract dual if it is part of a Kasparov dual ( P , D , Θ ) 𝑃 𝐷 Θ (P,D,\Theta) . Proposition 1.9 . An abstract dual for a space X 𝑋 X is unique up to a canonical KK 𝒢 superscript KK 𝒢 \operatorname{KK}^{\mathcal{G}} -equivalence if it exists, and even covariantly functorial in the following sense. Let X 𝑋 X and Y 𝑌 Y be two 𝒢 𝒢 \mathcal{G} -spaces and let f : X → Y : 𝑓 → 𝑋 𝑌 f\colon X\to Y be a 𝒢 𝒢 \mathcal{G} -equivariant continuous map. Let ( P X , Θ X ) subscript 𝑃 𝑋 subscript Θ 𝑋 (P_{X},\Theta_{X}) and ( P Y , Θ Y ) subscript 𝑃 𝑌 subscript Θ 𝑌 (P_{Y},\Theta_{Y}) be abstract duals for X 𝑋 X and Y 𝑌 Y of dimensions n X subscript 𝑛 𝑋 n_{X} and n Y subscript 𝑛 𝑌 n_{Y} , respectively. Then there is a unique P f ∈ KK n Y − n X 𝒢 ⁡ ( P X , P Y ) subscript 𝑃 𝑓 subscript superscript KK 𝒢 subscript 𝑛 𝑌 subscript 𝑛 𝑋 subscript 𝑃 𝑋 subscript 𝑃 𝑌 P_{f}\in\operatorname{KK}^{\mathcal{G}}_{n_{Y}-n_{X}}(P_{X},P_{Y}) with Θ X ⊗ P X P f = f ∗ ​ ( Θ Y ) subscript tensor-product subscript 𝑃 𝑋 subscript Θ 𝑋 subscript 𝑃 𝑓 superscript 𝑓 subscript Θ 𝑌 \Theta_{X}\otimes_{P_{X}}P_{f}=f^{*}(\Theta_{Y}) . Given two composable maps between three spaces with duals, we have P f ∘ g = P f ∘ P g subscript 𝑃 𝑓 𝑔 subscript 𝑃 𝑓 subscript 𝑃 𝑔 P_{f\circ g}=P_{f}\circ P_{g} . If X = Y 𝑋 𝑌 X=Y , f = id X 𝑓 subscript id 𝑋 f=\textup{id}_{X} , and ( P X , Θ X ) = ( P Y , Θ Y ) subscript 𝑃 𝑋 subscript Θ 𝑋 subscript 𝑃 𝑌 subscript Θ 𝑌 (P_{X},\Theta_{X})=(P_{Y},\Theta_{Y}) , then P f = id P X subscript 𝑃 𝑓 subscript id subscript 𝑃 𝑋 P_{f}=\textup{id}_{P_{X}} . If only X = Y 𝑋 𝑌 X=Y , f = id X 𝑓 subscript id 𝑋 f=\textup{id}_{X} , then P f subscript 𝑃 𝑓 P_{f} is a KK 𝒢 superscript KK 𝒢 \operatorname{KK}^{\mathcal{G}} -equivalence between the two duals of X 𝑋 X . Although the map f : X → Y : 𝑓 → 𝑋 𝑌 f\colon X\to Y appearing in Proposition 1.9 does not have to be proper, it nonetheless yields a morphism P f subscript 𝑃 𝑓 P_{f} in KK 𝒢 superscript KK 𝒢 \operatorname{KK}^{\mathcal{G}} . 1.8. Duality co-algebra Let ( P , Θ ) 𝑃 Θ (P,\Theta) be an n 𝑛 n -dimensional abstract dual for a 𝒢 𝒢 \mathcal{G} -space X 𝑋 X . By the Yoneda Lemma, another abstract dual ( P ′ , Θ ′ ) superscript 𝑃 ′ superscript Θ ′ (P^{\prime},\Theta^{\prime}) also for X 𝑋 X and say of dimension n ′ superscript 𝑛 ′ n^{\prime} is related to ( P , Θ ) 𝑃 Θ (P,\Theta) by an invertible element (1.10) ψ ∈ KK n ′ − n 𝒢 ⁡ ( P , P ′ ) , such that ​ Θ ⊗ P ψ = Θ ′ . formulae-sequence 𝜓 subscript superscript KK 𝒢 superscript 𝑛 ′ 𝑛 𝑃 superscript 𝑃 ′ subscript tensor-product 𝑃 such that Θ 𝜓 superscript Θ ′ \psi\in\operatorname{KK}^{\mathcal{G}}_{n^{\prime}-n}(P,P^{\prime}),\;\textup{such that}\;\Theta\otimes_{P}\psi=\Theta^{\prime}. We repeat for emphasis that since ( P , Θ ) 𝑃 Θ (P,\Theta) is only an abstract dual, we are not assuming that there is a 𝒢 ⋉ X left-normal-factor-semidirect-product 𝒢 𝑋 \mathcal{G}\ltimes X -structure on P 𝑃 P . However, we are going to attempt to reconstruct what we might consider to be a 𝒢 ⋉ X left-normal-factor-semidirect-product 𝒢 𝑋 \mathcal{G}\ltimes X -structure on P 𝑃 P at the level of KK KK \operatorname{KK} -theory . Along the way we will keep track of how the change in dual from ( P , Θ ) 𝑃 Θ (P,\Theta) to ( P ′ , Θ ′ ) superscript 𝑃 ′ superscript Θ ′ (P^{\prime},\Theta^{\prime}) affects our constructions. Define D ∈ KK − n 𝒢 ⁡ ( P , 𝟙 ) 𝐷 subscript superscript KK 𝒢 𝑛 𝑃 1 D\in\operatorname{KK}^{\mathcal{G}}_{-n}(P,\mathds{1}) by the requirement (1.11) PD ​ ( D ) := Θ ⊗ P D = 1 𝟙 in RKK 0 𝒢 ⁡ ( X ; 𝟙 , 𝟙 ) . formulae-sequence assign PD 𝐷 subscript tensor-product 𝑃 Θ 𝐷 subscript 1 1 in RKK 0 𝒢 ⁡ ( X ; 𝟙 , 𝟙 ) . \textup{PD}(D)\mathrel{\vcentcolon=}\Theta\otimes_{P}D=1_{\mathds{1}}\qquad\text{in $\operatorname{RKK}^{\mathcal{G}}_{0}(X;\mathds{1},\mathds{1})$.} as in the first condition in Definition 1.6 ). The class D 𝐷 D should thus play the role of the class named D 𝐷 D in a Kasparov dual. It is routine to check that when we change the dual, as above, D 𝐷 D is replaced by ψ − 1 ⊗ P D subscript tensor-product 𝑃 superscript 𝜓 1 𝐷 \psi^{-1}\otimes_{P}D . We call D 𝐷 D counit of the duality because it plays the algebraic role of a counit in the theory of adjoint functors (see Emerson-Meyer:Dualities and also Remark 3.3 below). Define ∇ ∈ KK n 𝒢 ⁡ ( P , P ⊗ P ) ∇ subscript superscript KK 𝒢 𝑛 𝑃 tensor-product 𝑃 𝑃 \nabla\in\operatorname{KK}^{\mathcal{G}}_{n}(P,P\otimes P) by the requirement that PD ​ ( ∇ ) := Θ ⊗ P ∇ = Θ ⊗ X Θ in RKK 2 ​ n 𝒢 ⁡ ( X ; 𝟙 , P ⊗ P ) . formulae-sequence assign PD ∇ subscript tensor-product 𝑃 Θ ∇ subscript tensor-product 𝑋 Θ Θ in RKK 2 n 𝒢 ⁡ ( X ; 𝟙 , P ⊗ P ) . \textup{PD}(\nabla)\mathrel{\vcentcolon=}\Theta\otimes_{P}\nabla=\Theta\otimes_{X}\Theta\qquad\text{in $\operatorname{RKK}^{\mathcal{G}}_{2n}(X;\mathds{1},P\otimes P)$.} We call ∇ ∇ \nabla the comultiplication of the duality . When we change the dual, ∇ ∇ \nabla is replaced by ( − 1 ) n ​ ( n ′ − n ) ​ ψ − 1 ⊗ P ∇ ⊗ P ⊗ P ( ψ ⊗ ψ ) ∈ KK n ′ 𝒢 ⁡ ( P ′ , P ′ ⊗ P ′ ) . subscript tensor-product tensor-product 𝑃 𝑃 subscript tensor-product 𝑃 superscript 1 𝑛 superscript 𝑛 ′ 𝑛 superscript 𝜓 1 ∇ tensor-product 𝜓 𝜓 subscript superscript KK 𝒢 superscript 𝑛 ′ superscript 𝑃 ′ tensor-product superscript 𝑃 ′ superscript 𝑃 ′ (-1)^{n(n^{\prime}-n)}\psi^{-1}\otimes_{P}\nabla\otimes_{P\otimes P}(\psi\otimes\psi)\in\operatorname{KK}^{\mathcal{G}}_{n^{\prime}}(P^{\prime},P^{\prime}\otimes P^{\prime}). Remark 1.12 . If n = 0 𝑛 0 n=0 then the object P 𝑃 P of KK 𝒢 superscript KK 𝒢 \operatorname{KK}^{\mathcal{G}} with counit D 𝐷 D and comultiplication ∇ ∇ \nabla is a cocommutative, counital coalgebra object in the tensor category KK 𝒢 superscript KK 𝒢 \operatorname{KK}^{\mathcal{G}} : (1.13) ∇ ⊗ P ⊗ P ( ∇ ⊗ 1 P ) = ∇ ⊗ P ⊗ P ( 1 P ⊗ ∇ ) , subscript tensor-product tensor-product 𝑃 𝑃 ∇ tensor-product ∇ subscript 1 𝑃 subscript tensor-product tensor-product 𝑃 𝑃 ∇ tensor-product subscript 1 𝑃 ∇ \displaystyle\nabla\otimes_{P\otimes P}(\nabla\otimes 1_{P})=\nabla\otimes_{P\otimes P}(1_{P}\otimes\nabla), (1.14) ∇ ⊗ P ⊗ P Φ P = ∇ , subscript tensor-product tensor-product 𝑃 𝑃 ∇ subscript Φ 𝑃 ∇ \displaystyle\nabla\otimes_{P\otimes P}\Phi_{P}=\nabla, (1.15) ∇ ⊗ P ⊗ P ( D ⊗ 1 P ) = 1 P = ∇ ⊗ P ⊗ P ( 1 P ⊗ D ) . subscript tensor-product tensor-product 𝑃 𝑃 ∇ tensor-product 𝐷 subscript 1 𝑃 subscript 1 𝑃 subscript tensor-product tensor-product 𝑃 𝑃 ∇ tensor-product subscript 1 𝑃 𝐷 \displaystyle\nabla\otimes_{P\otimes P}(D\otimes 1_{P})=1_{P}=\nabla\otimes_{P\otimes P}(1_{P}\otimes D). Equation ( 1.13 ) holds in KK 2 ​ n 𝒢 ⁡ ( P , P ⊗ 3 ) subscript superscript KK 𝒢 2 𝑛 𝑃 superscript 𝑃 tensor-product absent 3 \operatorname{KK}^{\mathcal{G}}_{2n}(P,P^{\otimes 3}) , equation ( 1.14 ) holds in KK n 𝒢 ⁡ ( P , P ⊗ P ) subscript superscript KK 𝒢 𝑛 𝑃 tensor-product 𝑃 𝑃 \operatorname{KK}^{\mathcal{G}}_{n}(P,P\otimes P) , and ( 1.15 ) holds in KK 0 𝒢 ⁡ ( P , P ) subscript superscript KK 𝒢 0 𝑃 𝑃 \operatorname{KK}^{\mathcal{G}}_{0}(P,P) . Now, for 𝒢 𝒢 \mathcal{G} - C ∗ superscript C \textup{C}^{*} -algebras A 𝐴 A and B 𝐵 B , we define T P ′ : RKK ∗ 𝒢 ⁡ ( X ; A , B ) → KK ∗ 𝒢 ⁡ ( P ⊗ A , P ⊗ B ) , f ↦ ∇ ⊗ P PD − 1 ​ ( f ) , : subscript superscript 𝑇 ′ 𝑃 formulae-sequence → subscript superscript RKK 𝒢 𝑋 𝐴 𝐵 superscript subscript KK 𝒢 tensor-product 𝑃 𝐴 tensor-product 𝑃 𝐵 maps-to 𝑓 subscript tensor-product 𝑃 ∇ superscript PD 1 𝑓 T^{\prime}_{P}\colon\operatorname{RKK}^{\mathcal{G}}_{*}(X;A,B)\to\operatorname{KK}_{*}^{\mathcal{G}}(P\otimes A,P\otimes B),\qquad f\mapsto\nabla\otimes_{P}\textup{PD}^{-1}(f), where PD is the duality isomorphism, ∇ ∇ \nabla is the comultiplication of the duality, and ⊗ P subscript tensor-product 𝑃 \otimes_{P} operates on the second copy of P 𝑃 P in the target P ⊗ P tensor-product 𝑃 𝑃 P\otimes P of ∇ ∇ \nabla . A computation yields that (1.16) PD ​ ( T P ′ ​ ( f ) ) = Θ ⊗ X f in RKK i + n 𝒢 ⁡ ( X ; A , P ⊗ B ) PD subscript superscript 𝑇 ′ 𝑃 𝑓 subscript tensor-product 𝑋 Θ 𝑓 in RKK i + n 𝒢 ⁡ ( X ; A , P ⊗ B ) \textup{PD}\bigl{(}T^{\prime}_{P}(f)\bigr{)}=\Theta\otimes_{X}f\qquad\text{in $\operatorname{RKK}^{\mathcal{G}}_{i+n}(X;A,P\otimes B)$} for all f ∈ RKK i 𝒢 ⁡ ( X ; A , B ) 𝑓 subscript superscript RKK 𝒢 𝑖 𝑋 𝐴 𝐵 f\in\operatorname{RKK}^{\mathcal{G}}_{i}(X;A,B) . It follows that T P ′ ​ ( f ) = T P ​ ( f ) subscript superscript 𝑇 ′ 𝑃 𝑓 subscript 𝑇 𝑃 𝑓 T^{\prime}_{P}(f)=T_{P}(f) if ( P , Θ ) 𝑃 Θ (P,\Theta) is part of a Kasparov dual, and thus T P subscript 𝑇 𝑃 T_{P} is in fact independent of the 𝒢 ⋉ X left-normal-factor-semidirect-product 𝒢 𝑋 \mathcal{G}\ltimes X -structure on P 𝑃 P , verifying our guess above. When we change the dual, we replace T P ′ subscript superscript 𝑇 ′ 𝑃 T^{\prime}_{P} by the map (1.17) RKK i 𝒢 ⁡ ( X ; A , B ) ∋ f ↦ ( − 1 ) i ​ ( n − n ′ ) ​ ψ − 1 ⊗ P T P ​ ( f ) ⊗ P ψ ∈ KK i 𝒢 ⁡ ( P ′ ⊗ A , P ′ ⊗ B ) . contains subscript superscript RKK 𝒢 𝑖 𝑋 𝐴 𝐵 𝑓 maps-to subscript tensor-product 𝑃 subscript tensor-product 𝑃 superscript 1 𝑖 𝑛 superscript 𝑛 ′ superscript 𝜓 1 subscript 𝑇 𝑃 𝑓 𝜓 subscript superscript KK 𝒢 𝑖 tensor-product superscript 𝑃 ′ 𝐴 tensor-product superscript 𝑃 ′ 𝐵 \operatorname{RKK}^{\mathcal{G}}_{i}(X;A,B)\ni f\mapsto(-1)^{i(n-n^{\prime})}\psi^{-1}\otimes_{P}T_{P}(f)\otimes_{P}\psi\in\operatorname{KK}^{\mathcal{G}}_{i}(P^{\prime}\otimes A,P^{\prime}\otimes B). In fact, one can check that the maps T P ′ subscript superscript 𝑇 ′ 𝑃 T^{\prime}_{P} above define a functor T P ′ : RKK 𝒢 ⁡ ( X ) → KK 𝒢 . : subscript superscript 𝑇 ′ 𝑃 → superscript RKK 𝒢 𝑋 superscript KK 𝒢 T^{\prime}_{P}\colon\operatorname{RKK}^{\mathcal{G}}(X)\to\operatorname{KK}^{\mathcal{G}}. This is a KK 𝒢 superscript KK 𝒢 \operatorname{KK}^{\mathcal{G}} -functor in the sense that it is compatible with the tensor products ⊗ tensor-product \otimes , and it is left adjoint to the functor p X ∗ : KK 𝒢 → RKK 𝒢 : superscript subscript 𝑝 𝑋 → superscript KK 𝒢 superscript RKK 𝒢 p_{X}^{*}\colon\operatorname{KK}^{\mathcal{G}}\to\operatorname{RKK}^{\mathcal{G}} induced from the groupoid homomorphism 𝒢 ⋉ X → 𝒢 → left-normal-factor-semidirect-product 𝒢 𝑋 𝒢 \mathcal{G}\ltimes X\to\mathcal{G} . It follows that we can write the inverse duality map involved in an abstract dual ( P , Θ ) 𝑃 Θ (P,\Theta) as: (1.18) PD − 1 ​ ( f ) = ( − 1 ) i ​ n ​ T P ′ ​ ( f ) ⊗ P D in KK i − n 𝒢 ⁡ ( P ⊗ A , B ) superscript PD 1 𝑓 subscript tensor-product 𝑃 superscript 1 𝑖 𝑛 subscript superscript 𝑇 ′ 𝑃 𝑓 𝐷 in KK i − n 𝒢 ⁡ ( P ⊗ A , B ) \textup{PD}^{-1}(f)=(-1)^{in}T^{\prime}_{P}(f)\otimes_{P}D\qquad\text{in $\operatorname{KK}^{\mathcal{G}}_{i-n}(P\otimes A,B)$} for f ∈ RKK i 𝒢 ⁡ ( X ; A , B ) 𝑓 subscript superscript RKK 𝒢 𝑖 𝑋 𝐴 𝐵 f\in\operatorname{RKK}^{\mathcal{G}}_{i}(X;A,B) . By the above discussion this formula agrees with the map PD ∗ superscript PD \textup{PD}^{*} when we have a Kasparov dual. 1.9. The Lefschetz map The formal computations summarized in the previous section allows us to single out an interesting invariant of a 𝒢 𝒢 \mathcal{G} -space X 𝑋 X , at least under the hypothesis that X 𝑋 X has some abstract dual. For any 𝒢 𝒢 \mathcal{G} -space X 𝑋 X the diagonal embedding X → X × Z X → 𝑋 subscript 𝑍 𝑋 𝑋 X\to X\times_{Z}X is a proper map and hence induces a ∗ -homomorphism 𝟙 X ⊗ 𝟙 X ≅ 𝒞 0 ​ ( X × Z X ) → 𝒞 0 ​ ( X ) = 𝟙 X . tensor-product subscript 1 𝑋 subscript 1 𝑋 subscript 𝒞 0 subscript 𝑍 𝑋 𝑋 → subscript 𝒞 0 𝑋 subscript 1 𝑋 \mathds{1}_{X}\otimes\mathds{1}_{X}\cong\mathcal{C}_{0}(X\times_{Z}X)\to\mathcal{C}_{0}(X)=\mathds{1}_{X}. This map is 𝒢 ⋉ X left-normal-factor-semidirect-product 𝒢 𝑋 \mathcal{G}\ltimes X -equivariant and hence yields Δ X ∈ RKK 𝒢 ⁡ ( X ; 𝟙 X , 𝟙 ) ≅ KK 𝒢 ⋉ X ⁡ ( 𝒞 0 ​ ( X × Z X ) , 𝒞 0 ​ ( X ) ) . subscript Δ 𝑋 superscript RKK 𝒢 𝑋 subscript 1 𝑋 1 superscript KK left-normal-factor-semidirect-product 𝒢 𝑋 subscript 𝒞 0 subscript 𝑍 𝑋 𝑋 subscript 𝒞 0 𝑋 \Delta_{X}\in\operatorname{RKK}^{\mathcal{G}}(X;\mathds{1}_{X},\mathds{1})\cong\operatorname{KK}^{\mathcal{G}\ltimes X}\bigl{(}\mathcal{C}_{0}(X\times_{Z}X),\mathcal{C}_{0}(X)\bigr{)}. We call this the diagonal restriction class . It yields a canonical map (1.19) ␣ ⊗ 𝟙 X Δ X : KK 𝒢 ⁡ ( 𝟙 X ⊗ A , 𝟙 X ⊗ B ) → RKK 𝒢 ⁡ ( X ; 𝟙 X ⊗ A , B ) . : subscript tensor-product subscript 1 𝑋 ␣ subscript Δ 𝑋 → superscript KK 𝒢 tensor-product subscript 1 𝑋 𝐴 tensor-product subscript 1 𝑋 𝐵 superscript RKK 𝒢 𝑋 tensor-product subscript 1 𝑋 𝐴 𝐵 \textup{\textvisiblespace}\otimes_{\mathds{1}_{X}}\Delta_{X}\colon\operatorname{KK}^{\mathcal{G}}(\mathds{1}_{X}\otimes A,\mathds{1}_{X}\otimes B)\to\operatorname{RKK}^{\mathcal{G}}(X;\mathds{1}_{X}\otimes A,B). In particular, this contains a map KK 𝒢 ⁡ ( 𝟙 X , 𝟙 X ) → RKK 𝒢 ⁡ ( X ; 𝟙 X , 𝟙 ) → superscript KK 𝒢 subscript 1 𝑋 subscript 1 𝑋 superscript RKK 𝒢 𝑋 subscript 1 𝑋 1 \operatorname{KK}^{\mathcal{G}}(\mathds{1}_{X},\mathds{1}_{X})\to\operatorname{RKK}^{\mathcal{G}}(X;\mathds{1}_{X},\mathds{1}) . Example 1.20 . If f : X → X : 𝑓 → 𝑋 𝑋 f\colon X\to X is a proper, continuous, 𝒢 𝒢 \mathcal{G} -equivariant map, then f ⊗ 𝟙 X Δ X ∈ RKK 𝒢 ⁡ ( X ; 𝟙 X , 𝟙 ) subscript tensor-product subscript 1 𝑋 delimited- 𝑓 subscript Δ 𝑋 superscript RKK 𝒢 𝑋 subscript 1 𝑋 1 f\otimes_{\mathds{1}_{X}}\Delta_{X}\in\operatorname{RKK}^{\mathcal{G}}(X;\mathds{1}_{X},\mathds{1}) is the class of the ∗ -homomorphism induced by ( id X , f ) : X → X × Z X : subscript id 𝑋 𝑓 → 𝑋 subscript 𝑍 𝑋 𝑋 (\textup{id}_{X},f)\colon X\to X\times_{Z}X . Now drop the assumption that f 𝑓 f be proper. Then ( id X , f ) subscript id 𝑋 𝑓 (\textup{id}_{X},f) is still a proper, continuous, 𝒢 𝒢 \mathcal{G} -equivariant map. The class of the ∗ -homomorphism it induces is equal to f ∗ ​ ( Δ X ) superscript 𝑓 subscript Δ 𝑋 f^{*}(\Delta_{X}) , where we use the maps f ∗ : RKK ∗ 𝒢 ⁡ ( X ; A , B ) → RKK ∗ 𝒢 ⁡ ( X ; A , B ) : superscript 𝑓 → subscript superscript RKK 𝒢 𝑋 𝐴 𝐵 subscript superscript RKK 𝒢 𝑋 𝐴 𝐵 f^{*}\colon\operatorname{RKK}^{\mathcal{G}}_{*}(X;A,B)\to\operatorname{RKK}^{\mathcal{G}}_{*}(X;A,B) for A = 𝟙 X 𝐴 subscript 1 𝑋 A=\mathds{1}_{X} , B = 𝟙 𝐵 1 B=\mathds{1} induced by f : X → X : 𝑓 → 𝑋 𝑋 f\colon X\to X (the functor X ↦ RKK 𝒢 ⁡ ( X ; A , B ) maps-to 𝑋 superscript RKK 𝒢 𝑋 𝐴 𝐵 X\mapsto\operatorname{RKK}^{\mathcal{G}}(X;A,B) is functorial with respect to arbitrary 𝒢 𝒢 \mathcal{G} -maps, not just proper ones.) This suggests that we can think of RKK 𝒢 ⁡ ( X ; 𝟙 X , 𝟙 ) superscript RKK 𝒢 𝑋 subscript 1 𝑋 1 \operatorname{RKK}^{\mathcal{G}}(X;\mathds{1}_{X},\mathds{1}) as generalized, possibly non-proper self-maps of X 𝑋 X . In fact if the anchor map X → Z → 𝑋 𝑍 X\to Z is a proper map, so that X 𝑋 X is a bundle of compact spaces over Z 𝑍 Z , then ␣ ⊗ 𝟙 X Δ X subscript tensor-product subscript 1 𝑋 ␣ subscript Δ 𝑋 \textup{\textvisiblespace}\otimes_{\mathds{1}_{X}}\Delta_{X} is an isomophism (an easy exercise in the definitions.) Now let T P ′ subscript superscript 𝑇 ′ 𝑃 T^{\prime}_{P} be the tensor functor and Δ X subscript Δ 𝑋 \Delta_{X} the diagonal restriction class of an abstract dual. We define the multiplication class of P 𝑃 P by (1.21) m := T P ′ ​ ( Δ X ) ∈ KK 0 𝒢 ⁡ ( P ⊗ 𝟙 X , P ) . assign delimited- 𝑚 subscript superscript 𝑇 ′ 𝑃 subscript Δ 𝑋 subscript superscript KK 𝒢 0 tensor-product 𝑃 subscript 1 𝑋 𝑃 m\mathrel{\vcentcolon=}T^{\prime}_{P}(\Delta_{X})\in\operatorname{KK}^{\mathcal{G}}_{0}(P\otimes\mathds{1}_{X},P). A change of dual as in ( 1.10 ) replaces m delimited- 𝑚 m by ψ − 1 ⊗ P m ⊗ P ψ subscript tensor-product 𝑃 subscript tensor-product 𝑃 superscript 𝜓 1 delimited- 𝑚 𝜓 \psi^{-1}\otimes_{P}m\otimes_{P}\psi . Lemma 1.22 . Let ( P , D , Θ ) 𝑃 𝐷 Θ (P,D,\Theta) be a Kasparov dual. Then m delimited- 𝑚 m is the class in KK 𝒢 superscript KK 𝒢 \operatorname{KK}^{\mathcal{G}} of the multiplication homomorphism 𝒞 0 ​ ( X ) ⊗ Z P → P → subscript tensor-product 𝑍 subscript 𝒞 0 𝑋 𝑃 𝑃 \mathcal{C}_{0}(X)\otimes_{Z}P\to P that describes the X 𝑋 X -structure on P 𝑃 P ( up to commuting the tensor factors ) . We now have enough theoretical development to define the Lefschetz map and sketch the proof of its homotopy invariance. Let X 𝑋 X be a 𝒢 𝒢 \mathcal{G} -space and ( P , Θ ) 𝑃 Θ (P,\Theta) an n 𝑛 n -dimensional abstract dual for X 𝑋 X , PD and PD − 1 superscript PD 1 \textup{PD}^{-1} the duality isomorphisms. As before, we write 𝟙 := 𝒞 0 ​ ( Z ) assign 1 subscript 𝒞 0 𝑍 \mathds{1}\mathrel{\vcentcolon=}\mathcal{C}_{0}(Z) , 𝟙 X := 𝒞 0 ​ ( X ) assign subscript 1 𝑋 subscript 𝒞 0 𝑋 \mathds{1}_{X}\mathrel{\vcentcolon=}\mathcal{C}_{0}(X) and Δ X ∈ RKK 𝒢 ⁡ ( X ; 𝟙 X , 𝟙 ) = KK 𝒢 ⋉ X ⁡ ( 𝟙 X ⊗ 𝟙 X , 𝟙 X ) subscript Δ 𝑋 superscript RKK 𝒢 𝑋 subscript 1 𝑋 1 superscript KK left-normal-factor-semidirect-product 𝒢 𝑋 tensor-product subscript 1 𝑋 subscript 1 𝑋 subscript 1 𝑋 \Delta_{X}\in\operatorname{RKK}^{\mathcal{G}}(X;\mathds{1}_{X},\mathds{1})=\operatorname{KK}^{\mathcal{G}\ltimes X}(\mathds{1}_{X}\otimes\mathds{1}_{X},\mathds{1}_{X}) the diagonal restriction class and Θ ¯ := forget X ​ ( Θ ) ∈ KK n 𝒢 ⁡ ( 𝟙 X , P ⊗ 𝟙 X ) . assign ¯ Θ subscript forget 𝑋 Θ subscript superscript KK 𝒢 𝑛 subscript 1 𝑋 tensor-product 𝑃 subscript 1 𝑋 \bar{\Theta}\mathrel{\vcentcolon=}\textup{forget}_{X}(\Theta)\in\operatorname{KK}^{\mathcal{G}}_{n}(\mathds{1}_{X},P\otimes\mathds{1}_{X}). Definition 1.23 . The equivariant Lefschetz map Lef : RKK ∗ 𝒢 ⁡ ( X ; 𝟙 X , 𝟙 ) → KK ∗ 𝒢 ⁡ ( 𝟙 X , 𝟙 ) : Lef → subscript superscript RKK 𝒢 𝑋 subscript 1 𝑋 1 superscript subscript KK 𝒢 subscript 1 𝑋 1 \operatorname{Lef}\colon\operatorname{RKK}^{\mathcal{G}}_{*}(X;\mathds{1}_{X},\mathds{1})\to\operatorname{KK}_{*}^{\mathcal{G}}(\mathds{1}_{X},\mathds{1}) for a 𝒢 𝒢 \mathcal{G} -space X 𝑋 X is defined as the composite map RKK i 𝒢 ⁡ ( X ; 𝟙 X , 𝟙 ) → PD − 1 KK i − n 𝒢 ⁡ ( P ⊗ 𝟙 X , 𝟙 ) → Θ ¯ ⊗ P ⊗ 𝟙 X ␣ KK i 𝒢 ⁡ ( 𝟙 X , 𝟙 ) . superscript PD 1 → subscript superscript RKK 𝒢 𝑖 𝑋 subscript 1 𝑋 1 subscript superscript KK 𝒢 𝑖 𝑛 tensor-product 𝑃 subscript 1 𝑋 1 subscript tensor-product tensor-product 𝑃 subscript 1 𝑋 ¯ Θ ␣ → subscript superscript KK 𝒢 𝑖 subscript 1 𝑋 1 \operatorname{RKK}^{\mathcal{G}}_{i}(X;\mathds{1}_{X},\mathds{1})\xrightarrow{\textup{PD}^{-1}}\operatorname{KK}^{\mathcal{G}}_{i-n}(P\otimes\mathds{1}_{X},\mathds{1})\xrightarrow{\bar{\Theta}\otimes_{P\otimes\mathds{1}_{X}}\textup{\textvisiblespace}}\operatorname{KK}^{\mathcal{G}}_{i}(\mathds{1}_{X},\mathds{1}). The equivariant Euler characteristic of X 𝑋 X is Eul X := Lef ⁡ ( Δ X ) ∈ KK 0 𝒢 ⁡ ( 𝟙 X , 𝟙 ) = KK 0 𝒢 ⁡ ( 𝒞 0 ​ ( X ) , 𝒞 0 ​ ( Z ) ) . assign subscript Eul 𝑋 Lef subscript Δ 𝑋 subscript superscript KK 𝒢 0 subscript 1 𝑋 1 subscript superscript KK 𝒢 0 subscript 𝒞 0 𝑋 subscript 𝒞 0 𝑍 \operatorname{Eul}_{X}\mathrel{\vcentcolon=}\operatorname{Lef}(\Delta_{X})\in\operatorname{KK}^{\mathcal{G}}_{0}(\mathds{1}_{X},\mathds{1})=\operatorname{KK}^{\mathcal{G}}_{0}\bigl{(}\mathcal{C}_{0}(X),\mathcal{C}_{0}(Z)\bigr{)}. Let f ∈ RKK i 𝒢 ⁡ ( X ; 𝟙 X , 𝟙 ) 𝑓 subscript superscript RKK 𝒢 𝑖 𝑋 subscript 1 𝑋 1 f\in\operatorname{RKK}^{\mathcal{G}}_{i}(X;\mathds{1}_{X},\mathds{1}) . Equations ( 1.18 ) and ( 1.21 ) yield (1.24) Lef ⁡ ( f ) Lef 𝑓 \displaystyle\operatorname{Lef}(f) = ( − 1 ) i ​ n ​ Θ ¯ ⊗ P ⊗ 𝟙 X T P ′ ​ ( f ) ⊗ P D , absent subscript tensor-product 𝑃 subscript tensor-product tensor-product 𝑃 subscript 1 𝑋 superscript 1 𝑖 𝑛 ¯ Θ subscript superscript 𝑇 ′ 𝑃 𝑓 𝐷 \displaystyle=(-1)^{in}\bar{\Theta}\otimes_{P\otimes\mathds{1}_{X}}T^{\prime}_{P}(f)\otimes_{P}D, (1.25) Eul X subscript Eul 𝑋 \displaystyle\operatorname{Eul}_{X} = ( − 1 ) i ​ n ​ Θ ¯ ⊗ P ⊗ 𝟙 X m ⊗ P D . absent subscript tensor-product 𝑃 subscript tensor-product tensor-product 𝑃 subscript 1 𝑋 superscript 1 𝑖 𝑛 ¯ Θ delimited- 𝑚 𝐷 \displaystyle=(-1)^{in}\bar{\Theta}\otimes_{P\otimes\mathds{1}_{X}}m\otimes_{P}D. We have already established that if ( P , Θ ) 𝑃 Θ (P,\Theta) is part of a Kasparov dual, then T P ′ = T P subscript superscript 𝑇 ′ 𝑃 subscript 𝑇 𝑃 T^{\prime}_{P}=T_{P} and m delimited- 𝑚 m is the KK KK \operatorname{KK} -class of the multiplication ∗ -homomorphism 𝒞 0 ​ ( X , P ) → P → subscript 𝒞 0 𝑋 𝑃 𝑃 \mathcal{C}_{0}(X,P)\to P , so that ( 1.24 ) yields explicit formulas for Lef ⁡ ( f ) Lef 𝑓 \operatorname{Lef}(f) and Eul X subscript Eul 𝑋 \operatorname{Eul}_{X} . This is extremely important because otherwise it would not be possible to compute these invariants. Let X 𝑋 X and X ′ superscript 𝑋 ′ X^{\prime} be 𝒢 𝒢 \mathcal{G} -spaces, and let f : X → X ′ : 𝑓 → 𝑋 superscript 𝑋 ′ f\colon X\to X^{\prime} be a 𝒢 𝒢 \mathcal{G} -homotopy equivalence. Then f 𝑓 f induces an equivalence of categories RKK 𝒢 ⁡ ( X ′ ) ≅ RKK 𝒢 ⁡ ( X ) superscript RKK 𝒢 superscript 𝑋 ′ superscript RKK 𝒢 𝑋 \operatorname{RKK}^{\mathcal{G}}(X^{\prime})\cong\operatorname{RKK}^{\mathcal{G}}(X) , that is, we get invertible maps f ∗ : RKK ∗ 𝒢 ⁡ ( X ′ ; A , B ) → RKK ∗ 𝒢 ⁡ ( X ; A , B ) : superscript 𝑓 → subscript superscript RKK 𝒢 superscript 𝑋 ′ 𝐴 𝐵 subscript superscript RKK 𝒢 𝑋 𝐴 𝐵 f^{*}\colon\operatorname{RKK}^{\mathcal{G}}_{*}(X^{\prime};A,B)\to\operatorname{RKK}^{\mathcal{G}}_{*}(X;A,B) for all 𝒢 𝒢 \mathcal{G} - C ∗ superscript C \textup{C}^{*} -algebras A 𝐴 A and B 𝐵 B . Now assume, in addition, that f 𝑓 f is proper; we do not need the inverse map or the homotopies to be proper. Then f 𝑓 f induces a ∗ -homomorphism f ! : 𝒞 0 ​ ( X ′ ) → 𝒞 0 ​ ( X ) : superscript 𝑓 → subscript 𝒞 0 superscript 𝑋 ′ subscript 𝒞 0 𝑋 f^{!}\colon\mathcal{C}_{0}(X^{\prime})\to\mathcal{C}_{0}(X) , which yields f ! ∈ KK 𝒢 ⁡ ( 𝒞 0 ​ ( X ′ ) , 𝒞 0 ​ ( X ) ) delimited- superscript 𝑓 superscript KK 𝒢 subscript 𝒞 0 superscript 𝑋 ′ subscript 𝒞 0 𝑋 f^{!}\in\operatorname{KK}^{\mathcal{G}}\bigl{(}\mathcal{C}_{0}(X^{\prime}),\mathcal{C}_{0}(X)\bigr{)} . We write f ! delimited- superscript 𝑓 f^{!} instead of f ∗ delimited- superscript 𝑓 f^{*} to better distinguish this from the map f ∗ superscript 𝑓 f^{*} above. Unless f 𝑓 f is a proper 𝒢 𝒢 \mathcal{G} -homotopy equivalence, f ! delimited- superscript 𝑓 f^{!} need not be invertible. Theorem 1.26 . Let X 𝑋 X and X ′ superscript 𝑋 ′ X^{\prime} be 𝒢 𝒢 \mathcal{G} -spaces with abstract duals, and let f : X → X ′ : 𝑓 → 𝑋 superscript 𝑋 ′ f\colon X\to X^{\prime} be both a proper map and a 𝒢 𝒢 \mathcal{G} -homotopy equivalence. Then f ! ⊗ 𝒞 0 ​ ( X ) Eul X = Eul X ′ in KK 0 𝒢 ⁡ ( 𝒞 0 ​ ( X ′ ) , 𝟙 ) subscript tensor-product subscript 𝒞 0 𝑋 delimited- superscript 𝑓 subscript Eul 𝑋 subscript Eul superscript 𝑋 ′ in KK 0 𝒢 ⁡ ( 𝒞 0 ( X ′ ) , 𝟙 ) f^{!}\otimes_{\mathcal{C}_{0}(X)}\operatorname{Eul}_{X}=\operatorname{Eul}_{X^{\prime}}\qquad\text{in $\operatorname{KK}^{\mathcal{G}}_{0}(\mathcal{C}_{0}(X^{\prime}),\mathds{1})$} and the Lefschetz maps for X 𝑋 X and X ′ superscript 𝑋 ′ X^{\prime} are related by a commuting diagram RKK ∗ 𝒢 ⁡ ( X ; 𝒞 0 ​ ( X ) , 𝟙 ) subscript superscript RKK 𝒢 𝑋 subscript 𝒞 0 𝑋 1 \textstyle{\operatorname{RKK}^{\mathcal{G}}_{*}(X;\mathcal{C}_{0}(X),\mathds{1})\ignorespaces\ignorespaces\ignorespaces\ignorespaces} Lef X subscript Lef 𝑋 \scriptstyle{\operatorname{Lef}_{X}} RKK ∗ 𝒢 ⁡ ( X ′ ; 𝒞 0 ​ ( X ) , 𝟙 ) subscript superscript RKK 𝒢 superscript 𝑋 ′ subscript 𝒞 0 𝑋 1 \textstyle{\ignorespaces\ignorespaces\ignorespaces\ignorespaces\operatorname{RKK}^{\mathcal{G}}_{*}(X^{\prime};\mathcal{C}_{0}(X),\mathds{1})\ignorespaces\ignorespaces\ignorespaces\ignorespaces} f ∗ superscript 𝑓 \scriptstyle{f^{*}} ≅ \scriptstyle{\cong} f ! ∗ superscript delimited- superscript 𝑓 \scriptstyle{f^{!}^{*}} RKK ∗ 𝒢 ⁡ ( X ′ ; 𝒞 0 ​ ( X ′ ) , 𝟙 ) subscript superscript RKK 𝒢 superscript 𝑋 ′ subscript 𝒞 0 superscript 𝑋 ′ 1 \textstyle{\operatorname{RKK}^{\mathcal{G}}_{*}(X^{\prime};\mathcal{C}_{0}(X^{\prime}),\mathds{1})\ignorespaces\ignorespaces\ignorespaces\ignorespaces} Lef X ′ subscript Lef superscript 𝑋 ′ \scriptstyle{\operatorname{Lef}_{X^{\prime}}} KK ∗ 𝒢 ⁡ ( 𝒞 0 ​ ( X ) , 𝟙 ) superscript subscript KK 𝒢 subscript 𝒞 0 𝑋 1 \textstyle{\operatorname{KK}_{*}^{\mathcal{G}}(\mathcal{C}_{0}(X),\mathds{1})\ignorespaces\ignorespaces\ignorespaces\ignorespaces} f ! ∗ superscript delimited- superscript 𝑓 \scriptstyle{f^{!}^{*}} KK ∗ 𝒢 ⁡ ( 𝒞 0 ​ ( X ′ ) , 𝟙 ) , superscript subscript KK 𝒢 subscript 𝒞 0 superscript 𝑋 ′ 1 \textstyle{\operatorname{KK}_{*}^{\mathcal{G}}(\mathcal{C}_{0}(X^{\prime}),\mathds{1}),} where f ! ∗ superscript delimited- superscript 𝑓 f^{!}^{*} denotes composition with f ! delimited- superscript 𝑓 f^{!} . In particular, Eul X subscript Eul 𝑋 \operatorname{Eul}_{X} and the map Lef X subscript Lef 𝑋 \operatorname{Lef}_{X} do not depend on the chosen dual. The proof relies on the discussion preceding the theorem. Theorem 1.26 implies that the Lefschetz maps for properly 𝒢 𝒢 \mathcal{G} -homotopy equivalent spaces are equivalent because then f ! delimited- superscript 𝑓 f^{!} is invertible, so that all horizontal maps in the diagram in Theorem 1.26 are invertible. In this sense, the Lefschetz map and the Euler class are invariants of the proper 𝒢 𝒢 \mathcal{G} -homotopy type of X 𝑋 X . The construction in Example 1.20 associates a class Δ f ∈ RKK 0 𝒢 ⁡ ( X ; 𝒞 0 ​ ( X ) , 𝟙 ) delimited- subscript Δ 𝑓 subscript superscript RKK 𝒢 0 𝑋 subscript 𝒞 0 𝑋 1 \Delta_{f}\in\operatorname{RKK}^{\mathcal{G}}_{0}(X;\mathcal{C}_{0}(X),\mathds{1}) to any continuous, 𝒢 𝒢 \mathcal{G} -equivariant map f : X → X : 𝑓 → 𝑋 𝑋 f\colon X\to X ; it does not matter whether f 𝑓 f is proper. We abbreviate Lef ⁡ ( f ) := Lef ⁡ ( Δ f ) assign Lef 𝑓 Lef delimited- subscript Δ 𝑓 \operatorname{Lef}(f)\mathrel{\vcentcolon=}\operatorname{Lef}(\Delta_{f}) and call this the Lefschetz invariant of f 𝑓 f . Of course, equivariantly homotopic self-maps induce the same class in RKK 𝒢 ⁡ ( X ; 𝒞 0 ​ ( X ) , 𝟙 ) superscript RKK 𝒢 𝑋 subscript 𝒞 0 𝑋 1 \operatorname{RKK}^{\mathcal{G}}(X;\mathcal{C}_{0}(X),\mathds{1}) and therefore have the same Lefschetz invariant. We have Lef ⁡ ( id X ) = Eul X Lef subscript id 𝑋 subscript Eul 𝑋 \operatorname{Lef}(\textup{id}_{X})=\operatorname{Eul}_{X} . More generally, specializing ( 1.19 ) gives a map ␣ ⊗ 𝟙 X Δ X : KK ∗ 𝒢 ⁡ ( 𝒞 0 ​ ( X ) , 𝒞 0 ​ ( X ) ) → RKK ∗ 𝒢 ⁡ ( X ; 𝒞 0 ​ ( X ) , 𝟙 ) , : subscript tensor-product subscript 1 𝑋 ␣ subscript Δ 𝑋 → superscript subscript KK 𝒢 subscript 𝒞 0 𝑋 subscript 𝒞 0 𝑋 subscript superscript RKK 𝒢 𝑋 subscript 𝒞 0 𝑋 1 \textup{\textvisiblespace}\otimes_{\mathds{1}_{X}}\Delta_{X}\colon\operatorname{KK}_{*}^{\mathcal{G}}\bigl{(}\mathcal{C}_{0}(X),\mathcal{C}_{0}(X)\bigr{)}\to\operatorname{RKK}^{\mathcal{G}}_{*}(X;\mathcal{C}_{0}(X),\mathds{1}), which we compose with the Lefschetz map; abusing notation, we still denote this composition by Lef : KK ∗ 𝒢 ⁡ ( 𝒞 0 ​ ( X ) , 𝒞 0 ​ ( X ) ) → KK ∗ 𝒢 ⁡ ( 𝒞 0 ​ ( X ) , 𝟙 ) : Lef → superscript subscript KK 𝒢 subscript 𝒞 0 𝑋 subscript 𝒞 0 𝑋 superscript subscript KK 𝒢 subscript 𝒞 0 𝑋 1 \operatorname{Lef}\colon\operatorname{KK}_{*}^{\mathcal{G}}\bigl{(}\mathcal{C}_{0}(X),\mathcal{C}_{0}(X)\bigr{)}\to\operatorname{KK}_{*}^{\mathcal{G}}(\mathcal{C}_{0}(X),\mathds{1}) Finally, we record that Lefschetz invariants for elements of RKK ∗ 𝒢 ⁡ ( X ; 𝒞 0 ​ ( X ) , 𝟙 ) subscript superscript RKK 𝒢 𝑋 subscript 𝒞 0 𝑋 1 \operatorname{RKK}^{\mathcal{G}}_{*}(X;\mathcal{C}_{0}(X),\mathds{1}) can be arbitrarily complicated: the Lefschetz map is rather easily seen to be split surjective. The splitting is given by specializing the inflation map (1.27) p X ∗ : KK ∗ 𝒢 ⁡ ( A , B ) → KK 𝒢 ⋉ X ⁡ ( 𝟙 X ⊗ A , 𝟙 X ⊗ B ) : superscript subscript 𝑝 𝑋 → superscript subscript KK 𝒢 𝐴 𝐵 superscript KK left-normal-factor-semidirect-product 𝒢 𝑋 tensor-product subscript 1 𝑋 𝐴 tensor-product subscript 1 𝑋 𝐵 p_{X}^{*}\colon\operatorname{KK}_{*}^{\mathcal{G}}(A,B)\to\operatorname{KK}^{\mathcal{G}\ltimes X}(\mathds{1}_{X}\otimes A,\mathds{1}_{X}\otimes B) to A := 𝟙 X assign 𝐴 subscript 1 𝑋 A\mathrel{\vcentcolon=}\mathds{1}_{X} and B := 𝟙 assign 𝐵 1 B\mathrel{\vcentcolon=}\mathds{1} . The fundamental example of a Kasparov dual is provided by the vertical tangent space to a bundle of smooth manifolds over the base Z 𝑍 Z of a groupoid, in which morphisms act smoothly. We come back to this in § 4 .","role":"user"},{"content":"Abstract: We survey work by the author and Ralf Meyer on equivariant KK-theory. Duality plays a key role in our approach. We organize the survey around the objective of computing a certain homotopy-invariant of a space equipped with a proper action of a group or groupoid called the Lefschetz map. The Lefschetz map associates an equivariant K-homology class to an equivariant Kasparov self-morphism of a space X admitting a dual. We want to describe it explicitly in the setting of bundles of smooth manifolds over the base space of a proper groupoid, in which groupoid elements act by diffeomorphisms between fibres. To get the required description we describe a topological model of equivariant KK-theory by way of a theory of correspondences, building on ideas of Paul Baum, Alain Connes and Georges Skandalis that appeared in the 1980's. This model agrees with the analytic model for bundles of smooth manifolds under some technical conditions related to the existence of equivariant vector bundles. Subject to these conditions we obtain a computation of the Lefschetz map in purely topological terms.","role":"assistant"}]

Holographic entanglement entropy: an overview

In recent developments of string theory, the idea of holography has obviously played crucial roles. Holography claims that the degrees of freedom in ( d + 2 ) 𝑑 2 (d+2) -dimensional quantum gravity are much more reduced than we naively think, and will be comparable to those of quantum many body systems in d + 1 𝑑 1 d+1 dimensions [ 2 , 3 ] . This was essentially found by remembering that the entropy of a black hole is not proportional to its volume, but to its area of the event horizon Σ Σ \Sigma (the Bekenstein-Hawking formula [ 4 ] ): S B ​ H = Area ​ ( Σ ) 4 ​ G N , subscript 𝑆 𝐵 𝐻 Area Σ 4 subscript 𝐺 𝑁 S_{BH}={\mbox{Area}(\Sigma)\over 4G_{N}}\ , (1.1) where G N subscript 𝐺 𝑁 G_{N} is the Newton constant. Owing to the discovery of the AdS/CFT correspondence [ 5 ] , we know explicit examples where the holography is manifestly realized. The AdS/CFT argues that the quantum gravity on ( d + 2 ) 𝑑 2 (d+2) -dimensional anti-de Sitter spacetime (AdS d+2 ) is equivalent to a certain conformal field theory in d + 1 𝑑 1 d+1 dimensions (CFT d+1 ) [ 5 , 6 , 7 , 8 ] . Even after quite active researches of AdS/CFT for these ten years, fundamental mechanism of the AdS/CFT correspondence still remains a mystery, in spite of so many of evidences in various examples. In particular, we cannot answer which region of AdS is responsible to particular information in the dual CFT. To make modest progresses for this long standing problem, we believe that it is important to understand and formulate the holography in terms of a universal observable, rather than quantities which depend on the details of theories such as specific operators or Wilson loops etc. We only expect that a quantum gravity in some spacetime is dual to (i.e. equivalent to) a certain theory which is governed by the law of quantum mechanics. We would like to propose that an appropriate quantity which can be useful in this universal viewpoint is the entanglement entropy. Indeed, we can always define the entanglement entropy in any quantum mechanical system. The entanglement entropy S A subscript 𝑆 𝐴 S_{A} in quantum field theories or quantum many body systems is a non-local quantity as opposed to correlation functions. It is defined as the von Neumann entropy S A subscript 𝑆 𝐴 S_{A} of the reduced density matrix when we ‘trace out’ (or smear out) degrees of freedom inside a d 𝑑 d -dimensional space-like submanifold B 𝐵 B in a given ( d + 1 ) 𝑑 1 (d+1) -dimensional QFT, which is a complement of A 𝐴 A . S A subscript 𝑆 𝐴 S_{A} measures how the subsystems A 𝐴 A and B 𝐵 B are correlated with each other. Intuitively we can also say that this is the entropy for an observer in A 𝐴 A who is not accessible to B 𝐵 B as the information is lost by the smearing out in region B 𝐵 B . This origin of entropy looks analogous to the black hole entropy. Indeed, this was the historical motivation of considering the entanglement entropy in quantum field theories [ 9 , 10 , 11 ] . Interestingly, the leading divergence of S A subscript 𝑆 𝐴 S_{A} is proportional to the area of the subsystem A 𝐴 A , called the area law [ 10 , 11 ] (refer also to the review articles [ 12 , 13 , 14 , 15 , 16 , 17 , 18 , 19 ] ). Since S A subscript 𝑆 𝐴 S_{A} is defined as a von Neumann entropy, we expect that the entanglement entropy is directly related to the degrees of freedom. Indeed, in two-dimensional conformal field theory, the entanglement entropy is proportional to the central charge in two-dimensional conformal field theories (2D CFTs) as shown in [ 20 , 21 ] , where a general prescription of computing the entropy in 2D CFTs is given. Also in the mass perturbed CFTs (massive QFTs) the same conclusion holds [ 22 , 23 , 21 ] . Furthermore, our holographic result shows that the similar statement is also true in four or higher even-dimensional CFTs. As opposed to the thermal entropy, the entanglement entropy is non-vanishing at zero temperature. Therefore we can employ it to probe the quantum properties of the ground state for a given quantum system. It is also a useful order parameter of quantum phase transition at zero temperature as will be explained in Sec. 4 . Now we come back to our original question where in AdS given information in CFT is saved. Since the information included in a subsystem B 𝐵 B is evaluated by the entanglement entropy S A subscript 𝑆 𝐴 S_{A} , we can formulate this question more concretely as follows: “Which part of AdS space is responsible for the calculation of S A subscript 𝑆 𝐴 S_{A} in the dual gravity side ?” Two of the authors of this article proposed a holographic formula of the entanglement entropy in [ 24 , 25 ] : S A = Area ​ ( γ A ) 4 ​ G N ( d + 2 ) , subscript 𝑆 𝐴 Area subscript 𝛾 𝐴 4 subscript superscript 𝐺 𝑑 2 𝑁 S_{A}={\mbox{Area}(\gamma_{A})\over 4G^{(d+2)}_{N}}\ , (1.2) where γ A subscript 𝛾 𝐴 \gamma_{A} is the d 𝑑 d -dimensional minimal surface γ A subscript 𝛾 𝐴 \gamma_{A} whose boundary is given by the ( d − 1 ) 𝑑 1 (d-1) -dimensional manifold ∂ γ A = ∂ A subscript 𝛾 𝐴 𝐴 \partial\gamma_{A}=\partial A (see Fig. 3 ); the constant G N ( d + 2 ) subscript superscript 𝐺 𝑑 2 𝑁 G^{(d+2)}_{N} is the Newton constant of the general gravity in AdS d+2 . This formula can be applied equally well to asymptotically AdS static spacetimes. Originally, this formula ( 1.2 ) is speculated from the Bekenstein-Hawking formula ( 1.1 ). Indeed, since the minimal surface tends to wrap the horizon in the presence of event horizon, our formula ( 1.2 ) can be regarded as a generalization of the well-known formula ( 1.1 ). Also the area law of S A subscript 𝑆 𝐴 S_{A} [ 10 , 11 ] can be automatically derived from our holographic description. The purpose of this article is to explain this holographic description and then to review its current status with recent progresses and applications [ 26 ] - [ 83 ] . In AdS / 3 {}_{3}/ CFT 2 , we can confirm that the formula ( 1.2 ) is precisely true by comparing the holographic result with the known 2D CFT results [ 24 , 25 ] . In higher-dimensional cases, however, the proposed formula ( 1.2 ) has not been derived rigorously from the bulk to boundary relation in AdS/CFT [ 6 , 7 ] at present. Also the direct calculations of the entanglement entropy in the CFT side is very complicated in higher dimensions. Nevertheless, a heuristic derivation has been presented in [ 28 ] and many evidences [ 25 , 29 , 30 , 33 , 37 , 58 , 73 ] have been found. Our holographic formula has also been successfully applied to the explanation of black hole and de-Sitter entropy [ 26 , 27 , 29 , 45 ] (see also [ 84 , 85 ] for earlier pioneering ideas on the entanglement entropy in AdS/CFT with event horizon; see also [ 86 ] ), and to an order parameter of a confinement/deconfinement phase transition [ 33 , 44 , 46 , 59 , 65 , 66 , 69 , 74 ] . In condensed matter physics, the entanglement entropy is expected to be a key quantity to understand several aspects of quantum many-body physics. A central question in quantum many-body physics is how we can characterize different phases and phase transitions. While microscopic Hamiltonians in condensed matter systems (electronic systems, in particular) are quantum mechanical, a wide range of quantum phases turn out to have a classical analogue, and if so, they can be understood in terms of symmetry breaking of some kind, and in terms of classical order parameters. On the other hand, this paradigm, known from Landau and Ginzburg, does not always apply when phases of our interest are inherently quantum. Indeed, one of main foci in modern condensed matter physics is to understand quantum phases of matter and phase transitions between them, which are beyond the Landau-Ginzburg paradigm. To name a few, relatively well-understood examples, the fractional quantum Hall effect, and quantum magnets on some geometrically frustrated lattices have attracted a lot of interest. Many-body wavefunctions of quantum ground states in these phases look featureless when one looks at correlation functions of local operators; They cannot be characterized by classical order parameters of some kind. Indeed, they should be distinguished by their pattern of entanglement rather than their pattern of symmetry breaking [ 88 ] . Thus, the entanglement entropy is potentially useful to characterize these exotic phases 4 4 4 Recently, there have been a number of progresses on holographic descriptions of various phase transitions analogous to the ones in condensed matter physics, e.g. refer to the review [ 87 ] . One can ask these questions from a slightly more practical, but ultimately fundamental, point of view; how can we simulate quantum states of matter efficiently by classical computers? The total dimension of the Hilbert space increases exponentially as we increase the system size, and hence bruteforce approaches (e.g., exact diagonalization) to quantum many-body systems are destined to fail. It turns out having a good understanding on how local regions of the whole quantum system are entangled to each other would help to find good algorithms for quantum many-body problems, such as the density matrix renormalization group (DMRG) [ 89 ] . To be more precise, the scaling of the entanglement entropy as a function of the size of a given subregion of the system of interest gives us a criterion for efficient approximability. In other word, the entanglement entropy tells us amount of information and degrees of freedom necessary to represent a quantum ground state efficiently. Reversing the logic, one can distinguish different phases of quantum matter according to their computational complexity and hence from the scaling of the entanglement entropy. After all, what makes simulation of quantum systems by classical computers difficult is nothing but entanglement. Indeed, this idea has been pushed extensively in recent couple of years for several 1D quantum systems. It has been revealed that several quantum phases in 1D spin chains can be distinguished by different scaling of the entanglement entropy. See, for example, [ 22 , 23 , 90 , 91 , 92 ] and references in [ 21 ] . For higher-dimensional condensed matter systems, there have been many recent attempts in this direction. In particular, the entanglement entropy was applied for so-called topological phases in 2+1 dimensions [ 93 , 94 ] . Typically, these phases have a finite gap and are accompanied by many exotic features such as fractionalization of quantum numbers, non-Abelian statistics of quasi-particles, topological degeneracy, etc. They can be also useful for fault tolerant quantum computations. On the other hand, unconventional quantum liquid phases with gapless excitations, such as gapless spin liquid phases, seem to be, at least at present, more difficult to characterize in higher dimensions. Our results from the AdS/CFT correspondence can be useful to study these gapless (spin liquid) states (some of these phases have been suspected to be described by a relativistic gauge field theory of some sort [ 88 ] ). The organization of this paper is as follows: In Sec. 2 , we go through some basic properties of the entanglement entropy. In particular, we discuss how the entanglement entropy scales as a function of the size of the subsystem in quantum field theories and many-body systems. Sec. 3 presents our basic formula of the holographic entanglement entropy via AdS/CFT. Many results mentioned in Sec. 2 are reproduced from the holographic point of view. In Sec. 4 , we apply the entanglement entropy as a non-local order parameter to confinement/deconfinement phase transitions. Sec. 5 reviews two connections between the entanglement entropy and the black hole entropy obtained from the holographic calculation of the entanglement entropy. In Sec. 6 , we will explain a covariant formulation of the holographic entanglement entropy. We conclude in Sec. 7 with a summary and with possible future directions.

In this paper, we review recent progress on the holographic understanding of the entanglement entropy in the anti-de Sitter space/conformal field theory (AdS/CFT) correspondence. In general, the AdS/CFT relates physical observables in strongly coupled quantum many-body systems to certain classical quantities in gravity plus matter theories. In the case of our holographic entanglement entropy, its gravity dual turns out to be purely geometric, i.e. the area of minimal area surfaces in AdS spaces. One interesting application is to study various phase transitions by regarding the entanglement entropy as order parameters. Indeed we will see that our holographic calculations nicely reproduce the confinement/deconfinement transition. Our results can also be applied to understanding the microscopic origins of black hole entropy.

[{"content":"Help me to generate the abstract of a survey paper given the title:*Holographic entanglement entropy: an overview* and the introduction:In recent developments of string theory, the idea of holography has obviously played crucial roles. Holography claims that the degrees of freedom in ( d + 2 ) 𝑑 2 (d+2) -dimensional quantum gravity are much more reduced than we naively think, and will be comparable to those of quantum many body systems in d + 1 𝑑 1 d+1 dimensions 2 , 3 . This was essentially found by remembering that the entropy of a black hole is not proportional to its volume, but to its area of the event horizon Σ Σ \Sigma (the Bekenstein-Hawking formula 4 ): S B ​ H = Area ​ ( Σ ) 4 ​ G N , subscript 𝑆 𝐵 𝐻 Area Σ 4 subscript 𝐺 𝑁 S_{BH}={\mbox{Area}(\Sigma)\over 4G_{N}}\ , (1.1) where G N subscript 𝐺 𝑁 G_{N} is the Newton constant. Owing to the discovery of the AdS/CFT correspondence 5 , we know explicit examples where the holography is manifestly realized. The AdS/CFT argues that the quantum gravity on ( d + 2 ) 𝑑 2 (d+2) -dimensional anti-de Sitter spacetime (AdS d+2 ) is equivalent to a certain conformal field theory in d + 1 𝑑 1 d+1 dimensions (CFT d+1 ) 5 , 6 , 7 , 8 . Even after quite active researches of AdS/CFT for these ten years, fundamental mechanism of the AdS/CFT correspondence still remains a mystery, in spite of so many of evidences in various examples. In particular, we cannot answer which region of AdS is responsible to particular information in the dual CFT. To make modest progresses for this long standing problem, we believe that it is important to understand and formulate the holography in terms of a universal observable, rather than quantities which depend on the details of theories such as specific operators or Wilson loops etc. We only expect that a quantum gravity in some spacetime is dual to (i.e. equivalent to) a certain theory which is governed by the law of quantum mechanics. We would like to propose that an appropriate quantity which can be useful in this universal viewpoint is the entanglement entropy. Indeed, we can always define the entanglement entropy in any quantum mechanical system. The entanglement entropy S A subscript 𝑆 𝐴 S_{A} in quantum field theories or quantum many body systems is a non-local quantity as opposed to correlation functions. It is defined as the von Neumann entropy S A subscript 𝑆 𝐴 S_{A} of the reduced density matrix when we trace out (or smear out) degrees of freedom inside a d 𝑑 d -dimensional space-like submanifold B 𝐵 B in a given ( d + 1 ) 𝑑 1 (d+1) -dimensional QFT, which is a complement of A 𝐴 A . S A subscript 𝑆 𝐴 S_{A} measures how the subsystems A 𝐴 A and B 𝐵 B are correlated with each other. Intuitively we can also say that this is the entropy for an observer in A 𝐴 A who is not accessible to B 𝐵 B as the information is lost by the smearing out in region B 𝐵 B . This origin of entropy looks analogous to the black hole entropy. Indeed, this was the historical motivation of considering the entanglement entropy in quantum field theories 9 , 10 , 11 . Interestingly, the leading divergence of S A subscript 𝑆 𝐴 S_{A} is proportional to the area of the subsystem A 𝐴 A , called the area law 10 , 11 (refer also to the review articles 12 , 13 , 14 , 15 , 16 , 17 , 18 , 19 ). Since S A subscript 𝑆 𝐴 S_{A} is defined as a von Neumann entropy, we expect that the entanglement entropy is directly related to the degrees of freedom. Indeed, in two-dimensional conformal field theory, the entanglement entropy is proportional to the central charge in two-dimensional conformal field theories (2D CFTs) as shown in 20 , 21 , where a general prescription of computing the entropy in 2D CFTs is given. Also in the mass perturbed CFTs (massive QFTs) the same conclusion holds 22 , 23 , 21 . Furthermore, our holographic result shows that the similar statement is also true in four or higher even-dimensional CFTs. As opposed to the thermal entropy, the entanglement entropy is non-vanishing at zero temperature. Therefore we can employ it to probe the quantum properties of the ground state for a given quantum system. It is also a useful order parameter of quantum phase transition at zero temperature as will be explained in Sec. 4 . Now we come back to our original question where in AdS given information in CFT is saved. Since the information included in a subsystem B 𝐵 B is evaluated by the entanglement entropy S A subscript 𝑆 𝐴 S_{A} , we can formulate this question more concretely as follows: Which part of AdS space is responsible for the calculation of S A subscript 𝑆 𝐴 S_{A} in the dual gravity side ? Two of the authors of this article proposed a holographic formula of the entanglement entropy in 24 , 25 : S A = Area ​ ( γ A ) 4 ​ G N ( d + 2 ) , subscript 𝑆 𝐴 Area subscript 𝛾 𝐴 4 subscript superscript 𝐺 𝑑 2 𝑁 S_{A}={\mbox{Area}(\gamma_{A})\over 4G^{(d+2)}_{N}}\ , (1.2) where γ A subscript 𝛾 𝐴 \gamma_{A} is the d 𝑑 d -dimensional minimal surface γ A subscript 𝛾 𝐴 \gamma_{A} whose boundary is given by the ( d − 1 ) 𝑑 1 (d-1) -dimensional manifold ∂ γ A = ∂ A subscript 𝛾 𝐴 𝐴 \partial\gamma_{A}=\partial A (see Fig. 3 ); the constant G N ( d + 2 ) subscript superscript 𝐺 𝑑 2 𝑁 G^{(d+2)}_{N} is the Newton constant of the general gravity in AdS d+2 . This formula can be applied equally well to asymptotically AdS static spacetimes. Originally, this formula ( 1.2 ) is speculated from the Bekenstein-Hawking formula ( 1.1 ). Indeed, since the minimal surface tends to wrap the horizon in the presence of event horizon, our formula ( 1.2 ) can be regarded as a generalization of the well-known formula ( 1.1 ). Also the area law of S A subscript 𝑆 𝐴 S_{A} 10 , 11 can be automatically derived from our holographic description. The purpose of this article is to explain this holographic description and then to review its current status with recent progresses and applications 26 - 83 . In AdS / 3 {}_{3}/ CFT 2 , we can confirm that the formula ( 1.2 ) is precisely true by comparing the holographic result with the known 2D CFT results 24 , 25 . In higher-dimensional cases, however, the proposed formula ( 1.2 ) has not been derived rigorously from the bulk to boundary relation in AdS/CFT 6 , 7 at present. Also the direct calculations of the entanglement entropy in the CFT side is very complicated in higher dimensions. Nevertheless, a heuristic derivation has been presented in 28 and many evidences 25 , 29 , 30 , 33 , 37 , 58 , 73 have been found. Our holographic formula has also been successfully applied to the explanation of black hole and de-Sitter entropy 26 , 27 , 29 , 45 (see also 84 , 85 for earlier pioneering ideas on the entanglement entropy in AdS/CFT with event horizon; see also 86 ), and to an order parameter of a confinement/deconfinement phase transition 33 , 44 , 46 , 59 , 65 , 66 , 69 , 74 . In condensed matter physics, the entanglement entropy is expected to be a key quantity to understand several aspects of quantum many-body physics. A central question in quantum many-body physics is how we can characterize different phases and phase transitions. While microscopic Hamiltonians in condensed matter systems (electronic systems, in particular) are quantum mechanical, a wide range of quantum phases turn out to have a classical analogue, and if so, they can be understood in terms of symmetry breaking of some kind, and in terms of classical order parameters. On the other hand, this paradigm, known from Landau and Ginzburg, does not always apply when phases of our interest are inherently quantum. Indeed, one of main foci in modern condensed matter physics is to understand quantum phases of matter and phase transitions between them, which are beyond the Landau-Ginzburg paradigm. To name a few, relatively well-understood examples, the fractional quantum Hall effect, and quantum magnets on some geometrically frustrated lattices have attracted a lot of interest. Many-body wavefunctions of quantum ground states in these phases look featureless when one looks at correlation functions of local operators; They cannot be characterized by classical order parameters of some kind. Indeed, they should be distinguished by their pattern of entanglement rather than their pattern of symmetry breaking 88 . Thus, the entanglement entropy is potentially useful to characterize these exotic phases 4 4 4 Recently, there have been a number of progresses on holographic descriptions of various phase transitions analogous to the ones in condensed matter physics, e.g. refer to the review 87 . One can ask these questions from a slightly more practical, but ultimately fundamental, point of view; how can we simulate quantum states of matter efficiently by classical computers? The total dimension of the Hilbert space increases exponentially as we increase the system size, and hence bruteforce approaches (e.g., exact diagonalization) to quantum many-body systems are destined to fail. It turns out having a good understanding on how local regions of the whole quantum system are entangled to each other would help to find good algorithms for quantum many-body problems, such as the density matrix renormalization group (DMRG) 89 . To be more precise, the scaling of the entanglement entropy as a function of the size of a given subregion of the system of interest gives us a criterion for efficient approximability. In other word, the entanglement entropy tells us amount of information and degrees of freedom necessary to represent a quantum ground state efficiently. Reversing the logic, one can distinguish different phases of quantum matter according to their computational complexity and hence from the scaling of the entanglement entropy. After all, what makes simulation of quantum systems by classical computers difficult is nothing but entanglement. Indeed, this idea has been pushed extensively in recent couple of years for several 1D quantum systems. It has been revealed that several quantum phases in 1D spin chains can be distinguished by different scaling of the entanglement entropy. See, for example, 22 , 23 , 90 , 91 , 92 and references in 21 . For higher-dimensional condensed matter systems, there have been many recent attempts in this direction. In particular, the entanglement entropy was applied for so-called topological phases in 2+1 dimensions 93 , 94 . Typically, these phases have a finite gap and are accompanied by many exotic features such as fractionalization of quantum numbers, non-Abelian statistics of quasi-particles, topological degeneracy, etc. They can be also useful for fault tolerant quantum computations. On the other hand, unconventional quantum liquid phases with gapless excitations, such as gapless spin liquid phases, seem to be, at least at present, more difficult to characterize in higher dimensions. Our results from the AdS/CFT correspondence can be useful to study these gapless (spin liquid) states (some of these phases have been suspected to be described by a relativistic gauge field theory of some sort 88 ). The organization of this paper is as follows: In Sec. 2 , we go through some basic properties of the entanglement entropy. In particular, we discuss how the entanglement entropy scales as a function of the size of the subsystem in quantum field theories and many-body systems. Sec. 3 presents our basic formula of the holographic entanglement entropy via AdS/CFT. Many results mentioned in Sec. 2 are reproduced from the holographic point of view. In Sec. 4 , we apply the entanglement entropy as a non-local order parameter to confinement/deconfinement phase transitions. Sec. 5 reviews two connections between the entanglement entropy and the black hole entropy obtained from the holographic calculation of the entanglement entropy. In Sec. 6 , we will explain a covariant formulation of the holographic entanglement entropy. We conclude in Sec. 7 with a summary and with possible future directions.","role":"user"},{"content":"Abstract: In this paper, we review recent progress on the holographic understanding of the entanglement entropy in the anti-de Sitter space/conformal field theory (AdS/CFT) correspondence. In general, the AdS/CFT relates physical observables in strongly coupled quantum many-body systems to certain classical quantities in gravity plus matter theories. In the case of our holographic entanglement entropy, its gravity dual turns out to be purely geometric, i.e. the area of minimal area surfaces in AdS spaces. One interesting application is to study various phase transitions by regarding the entanglement entropy as order parameters. Indeed we will see that our holographic calculations nicely reproduce the confinement/deconfinement transition. Our results can also be applied to understanding the microscopic origins of black hole entropy.","role":"assistant"}]

The Levi problem on Stein spaces with singularities. A survey

1. A brief history of the smooth case In 1910 E. E. Levi [ Lev ] noticed that a domain of holomorphy Ω Ω \Omega in ℂ n superscript ℂ 𝑛 {\mathbb{C}}^{n} , with smooth C 2 superscript 𝐶 2 C^{2} boundary, should satisfy some pseudocovexity condition on the boundary points. More precisely he showed that if ρ 𝜌 \rho is a C 2 superscript 𝐶 2 C^{2} defining function for the boundary ∂ Ω Ω \partial\Omega of Ω Ω \Omega then the associated quadratic form L ​ ρ 𝐿 𝜌 L\rho ( we shall call it, as usual, the Levi form of ρ 𝜌 \rho ) is necessarily positive semi-definite on the holomorphic tangent space T z ​ ( ∂ Ω ) := { w ∈ ℂ n ∣ Σ i n ​ w i ​ ∂ ρ / ∂ z i ​ ( z ) = 0 } assign subscript 𝑇 𝑧 Ω conditional-set 𝑤 superscript ℂ 𝑛 superscript subscript Σ 𝑖 𝑛 subscript 𝑤 𝑖 𝜌 subscript 𝑧 𝑖 𝑧 0 T_{z}(\partial\Omega):=\{w\in{\mathbb{C}}^{n}\mid\Sigma_{i}^{n}w_{i}{\partial\rho}/{\partial z_{i}(z)}=0\} for any point z ∈ ∂ Ω 𝑧 Ω z\in\partial\Omega . O. Blumenthal [ Blu ] raised the important and difficult question on the validity of the converse of this statement , i.e. if a domain Ω ⊂ ℂ n Ω superscript ℂ 𝑛 \Omega\subset{\mathbb{C}}^{n} with smooth pseudoconvex boundary is necessarily a Stein domain. This problem, called also the Levi problem, was open for a long time, untill 1953, when K. Oka [ O ] solved it completely in the affirmative (an indepedent proof of this result was also obtained by F. Norguet [ No ] and by H.J. Bremermann [ Brem ] ). More generally, K. Oka considered unbranched Riemann domaines π : Ω → ℂ n : 𝜋 → Ω superscript ℂ 𝑛 \pi:\Omega\to{\mathbb{C}}^{n} (i.e. π 𝜋 \pi is locally biholomorphic) and proved that Ω Ω \Omega is Stein iff − log ⁡ d 𝑑 -\log d is a plurisubharmonic function on Ω Ω \Omega where d 𝑑 d denotes the boundary distance on Ω Ω \Omega . Note that Riemann unbranched domains over ℂ n superscript ℂ 𝑛 {\mathbb{C}}^{n} appear naturally as domains of existence of families of holomorphc functions defined on open subsets in ℂ n superscript ℂ 𝑛 {\mathbb{C}}^{n} . Oka’s result shows in particular that the Steiness of Ω Ω \Omega is a local property of its boundary. To be more precise we shall call a holomorphic map p : Ω → X : 𝑝 → Ω 𝑋 p:\Omega\to X of complex manifolds (or, more generally, of complex spaces) a Stein morphism if every point x ∈ X 𝑥 𝑋 x\in X has a neighborhood V = V ​ ( x ) 𝑉 𝑉 𝑥 V=V(x) such that p − 1 ​ ( V ) superscript 𝑝 1 𝑉 p^{-1}(V) is Stein. For example, if we consider the inclusion map i : Ω → X : 𝑖 → Ω 𝑋 i:\Omega\to X of an open subset Ω Ω \Omega of X 𝑋 X , then Ω Ω \Omega is called locally Stein (in X 𝑋 X ) iff the map i 𝑖 i is a Stein morphism, or equivalently each point x ∈ ∂ Ω 𝑥 Ω x\in\partial\Omega has a neighborhood V = V ​ ( x ) 𝑉 𝑉 𝑥 V=V(x) such that V ∩ Ω 𝑉 Ω V\cap\Omega is Stein. Oka’s theorem can therefore be stated as follows: an open subset Ω ⊂ ℂ n Ω superscript ℂ 𝑛 \Omega\subset{\mathbb{C}}^{n} is Stein if and only if i 𝑖 i is a Stein morphism, or, more generally, a Riemann unbranched morphism π : Ω → ℂ n : 𝜋 → Ω superscript ℂ 𝑛 \pi:\Omega\to{\mathbb{C}}^{n} is Stein iff π 𝜋 \pi is a Stein morphism. If π : Ω → ℂ n : 𝜋 → Ω superscript ℂ 𝑛 \pi:\Omega\to{\mathbb{C}}^{n} is a Stein morphism which has discrete fibers (even finite) but π 𝜋 \pi is not assumed to be locally biholomorphic (i.e. it is a branched Riemann domain) then Ω Ω \Omega could be non Stein (see [ F 2 ] , [ C-D4 ] ). Having in mind Oka’s result, H. Cartan at the important Colloque sur les fonctions de plusieurs variables held in Brussels in 1953 [ Car ] raised the following problem: let X 𝑋 X be a Stein manifold and Ω ⊂ X Ω 𝑋 \Omega\subset X a locally Stein open subset. Does it folow that Ω Ω \Omega is itself Stein? (the Local Steiness Problem in the smooth case, the manifold case). A positive answer to this question has been given by H. Grauert and F. Doquier [ D-G ] . Additionally, they solved the more general case of Riemann unbranched domains over Stein manifolds by proving : If π : Ω → X : 𝜋 → Ω 𝑋 \pi:\Omega\to X is a Riemann unbranched domain over a Stein manifold X 𝑋 X and if π 𝜋 \pi is a Stein morphism, then Ω Ω \Omega is itself Stein. Grauert’s method [ D-G ] is essentilally based on the following result: if X ⊂ ℂ n 𝑋 superscript ℂ 𝑛 X\subset{\mathbb{C}}^{n} is a complex closed submanifold then there exists an open neighborhood V 𝑉 V of X 𝑋 X ( which can be chosen to be Stein) and a holomorphic retract ρ : V → X : 𝜌 → 𝑉 𝑋 \rho:V\to X . This result can be used to reduce the general case of Riemann unbranched domains over Stein manifolds to the case of Riemann unbranched domains spread over ℂ n superscript ℂ 𝑛 {\mathbb{C}}^{n} . Such a holomorphic retract does not exist if X 𝑋 X has singularities. In fact Rossi [ Ro ] showed that the existence of a holomorphic retract onto X 𝑋 X as above, even of a neighborhood of X 𝑋 X minus a point, onto X 𝑋 X minus a point, implies the smoothness of X 𝑋 X . Therefore, in order to study the Levi problem in the case of Stein spaces with singularities, one needs other methods and new ideas.

We discuss the well-known open problems : the Local Steiness Problem and the Union Problem.

[{"content":"Help me to generate the abstract of a survey paper given the title:*The Levi problem on Stein spaces with singularities. A survey* and the introduction:1. A brief history of the smooth case In 1910 E. E. Levi Lev noticed that a domain of holomorphy Ω Ω \Omega in ℂ n superscript ℂ 𝑛 {\mathbb{C}}^{n} , with smooth C 2 superscript 𝐶 2 C^{2} boundary, should satisfy some pseudocovexity condition on the boundary points. More precisely he showed that if ρ 𝜌 \rho is a C 2 superscript 𝐶 2 C^{2} defining function for the boundary ∂ Ω Ω \partial\Omega of Ω Ω \Omega then the associated quadratic form L ​ ρ 𝐿 𝜌 L\rho ( we shall call it, as usual, the Levi form of ρ 𝜌 \rho ) is necessarily positive semi-definite on the holomorphic tangent space T z ​ ( ∂ Ω ) := { w ∈ ℂ n ∣ Σ i n ​ w i ​ ∂ ρ / ∂ z i ​ ( z ) = 0 } assign subscript 𝑇 𝑧 Ω conditional-set 𝑤 superscript ℂ 𝑛 superscript subscript Σ 𝑖 𝑛 subscript 𝑤 𝑖 𝜌 subscript 𝑧 𝑖 𝑧 0 T_{z}(\partial\Omega):=\{w\in{\mathbb{C}}^{n}\mid\Sigma_{i}^{n}w_{i}{\partial\rho}/{\partial z_{i}(z)}=0\} for any point z ∈ ∂ Ω 𝑧 Ω z\in\partial\Omega . O. Blumenthal Blu raised the important and difficult question on the validity of the converse of this statement , i.e. if a domain Ω ⊂ ℂ n Ω superscript ℂ 𝑛 \Omega\subset{\mathbb{C}}^{n} with smooth pseudoconvex boundary is necessarily a Stein domain. This problem, called also the Levi problem, was open for a long time, untill 1953, when K. Oka O solved it completely in the affirmative (an indepedent proof of this result was also obtained by F. Norguet No and by H.J. Bremermann Brem ). More generally, K. Oka considered unbranched Riemann domaines π : Ω → ℂ n : 𝜋 → Ω superscript ℂ 𝑛 \pi:\Omega\to{\mathbb{C}}^{n} (i.e. π 𝜋 \pi is locally biholomorphic) and proved that Ω Ω \Omega is Stein iff − log ⁡ d 𝑑 -\log d is a plurisubharmonic function on Ω Ω \Omega where d 𝑑 d denotes the boundary distance on Ω Ω \Omega . Note that Riemann unbranched domains over ℂ n superscript ℂ 𝑛 {\mathbb{C}}^{n} appear naturally as domains of existence of families of holomorphc functions defined on open subsets in ℂ n superscript ℂ 𝑛 {\mathbb{C}}^{n} . Okas result shows in particular that the Steiness of Ω Ω \Omega is a local property of its boundary. To be more precise we shall call a holomorphic map p : Ω → X : 𝑝 → Ω 𝑋 p:\Omega\to X of complex manifolds (or, more generally, of complex spaces) a Stein morphism if every point x ∈ X 𝑥 𝑋 x\in X has a neighborhood V = V ​ ( x ) 𝑉 𝑉 𝑥 V=V(x) such that p − 1 ​ ( V ) superscript 𝑝 1 𝑉 p^{-1}(V) is Stein. For example, if we consider the inclusion map i : Ω → X : 𝑖 → Ω 𝑋 i:\Omega\to X of an open subset Ω Ω \Omega of X 𝑋 X , then Ω Ω \Omega is called locally Stein (in X 𝑋 X ) iff the map i 𝑖 i is a Stein morphism, or equivalently each point x ∈ ∂ Ω 𝑥 Ω x\in\partial\Omega has a neighborhood V = V ​ ( x ) 𝑉 𝑉 𝑥 V=V(x) such that V ∩ Ω 𝑉 Ω V\cap\Omega is Stein. Okas theorem can therefore be stated as follows: an open subset Ω ⊂ ℂ n Ω superscript ℂ 𝑛 \Omega\subset{\mathbb{C}}^{n} is Stein if and only if i 𝑖 i is a Stein morphism, or, more generally, a Riemann unbranched morphism π : Ω → ℂ n : 𝜋 → Ω superscript ℂ 𝑛 \pi:\Omega\to{\mathbb{C}}^{n} is Stein iff π 𝜋 \pi is a Stein morphism. If π : Ω → ℂ n : 𝜋 → Ω superscript ℂ 𝑛 \pi:\Omega\to{\mathbb{C}}^{n} is a Stein morphism which has discrete fibers (even finite) but π 𝜋 \pi is not assumed to be locally biholomorphic (i.e. it is a branched Riemann domain) then Ω Ω \Omega could be non Stein (see F 2 , C-D4 ). Having in mind Okas result, H. Cartan at the important Colloque sur les fonctions de plusieurs variables held in Brussels in 1953 Car raised the following problem: let X 𝑋 X be a Stein manifold and Ω ⊂ X Ω 𝑋 \Omega\subset X a locally Stein open subset. Does it folow that Ω Ω \Omega is itself Stein? (the Local Steiness Problem in the smooth case, the manifold case). A positive answer to this question has been given by H. Grauert and F. Doquier D-G . Additionally, they solved the more general case of Riemann unbranched domains over Stein manifolds by proving : If π : Ω → X : 𝜋 → Ω 𝑋 \pi:\Omega\to X is a Riemann unbranched domain over a Stein manifold X 𝑋 X and if π 𝜋 \pi is a Stein morphism, then Ω Ω \Omega is itself Stein. Grauerts method D-G is essentilally based on the following result: if X ⊂ ℂ n 𝑋 superscript ℂ 𝑛 X\subset{\mathbb{C}}^{n} is a complex closed submanifold then there exists an open neighborhood V 𝑉 V of X 𝑋 X ( which can be chosen to be Stein) and a holomorphic retract ρ : V → X : 𝜌 → 𝑉 𝑋 \rho:V\to X . This result can be used to reduce the general case of Riemann unbranched domains over Stein manifolds to the case of Riemann unbranched domains spread over ℂ n superscript ℂ 𝑛 {\mathbb{C}}^{n} . Such a holomorphic retract does not exist if X 𝑋 X has singularities. In fact Rossi Ro showed that the existence of a holomorphic retract onto X 𝑋 X as above, even of a neighborhood of X 𝑋 X minus a point, onto X 𝑋 X minus a point, implies the smoothness of X 𝑋 X . Therefore, in order to study the Levi problem in the case of Stein spaces with singularities, one needs other methods and new ideas.","role":"user"},{"content":"Abstract: We discuss the well-known open problems : the Local Steiness Problem and the Union Problem.","role":"assistant"}]

Review of theta(13) and prospects for Double Chooz

I Physics Motivations The neutrino oscillation phenomenon has been clearly established by the study of solar, atmospheric, reactor and beam neutrinos. The PMNS mixing matrix relates the three neutrino mass eigenstates to the three neutrino flavor eigenstates. This can be parametrized by three mixing angles ( θ 12 subscript 𝜃 12 \theta_{12} , θ 13 subscript 𝜃 13 \theta_{13} , θ 23 subscript 𝜃 23 \theta_{23} ) and one CP violating phase δ C ​ P subscript 𝛿 𝐶 𝑃 \delta_{CP} (if neutrinos are Dirac particles). During the last years, tremendous progress has been achieved in the experimental field trying to measure the values of θ i ​ j subscript 𝜃 𝑖 𝑗 \theta_{ij} and the two squared mass differences Δ ​ m i ​ j 2 = m i 2 − m j 2 Δ subscript superscript 𝑚 2 𝑖 𝑗 subscript superscript 𝑚 2 𝑖 subscript superscript 𝑚 2 𝑗 \Delta m^{2}_{ij}=m^{2}_{i}-m^{2}_{j} which govern the oscillation probabilities. The two mass differences and the θ 12 subscript 𝜃 12 \theta_{12} and θ 23 subscript 𝜃 23 \theta_{23} mixing angles have been measured with good precision GonzalezGarcia:2007ib . However, the θ 13 subscript 𝜃 13 \theta_{13} angle, the sign of Δ ​ m 31 2 Δ subscript superscript 𝑚 2 31 \Delta m^{2}_{31} (mass hierarchy) and the δ C ​ P subscript 𝛿 𝐶 𝑃 \delta_{CP} phase are still unknown. In particular, only an upper limit on the value of θ 13 subscript 𝜃 13 \theta_{13} has been established indicating that the angle is very small compared to the other mixing angles. A three-flavor global analysis of the existing data provides a constraint on θ 13 subscript 𝜃 13 \theta_{13} being s ​ i ​ n 2 ​ θ 13 𝑠 𝑖 superscript 𝑛 2 subscript 𝜃 13 sin^{2}\theta_{13} < < 0.028 at 90% C.L Schwetz:2006dh . This limit is essentially dominated by the result obtained by the CHOOZ reactor experiment Apollonio:2002gd in France. This experiment measured the fraction of ν e ¯ ¯ subscript 𝜈 𝑒 \bar{\nu_{e}} ’s surviving at a distance of 1.05 km from the reactor cores to be R = 1.01 ± plus-or-minus \pm 2.8% (stat) ± plus-or-minus \pm 2.7% (syst). This result was mainly limited by the systematic uncertainties induced by the imperfect knowledge of the neutrino production and interaction. The measurement of this angle is of fundamental interest not only for the final understanding of neutrino oscillations but because it determines the possibilities to observe CP violation in the leptonic sector with the forthcoming neutrino experiments.

The Double Chooz reactor neutrino experiment will be the next detector to search for a non vanishing theta(13) mixing angle with unprecedented sensitivity, which might open the way to unveiling CP violation in the leptonic sector. The measurement of this angle will be based on a precise comparison of the antineutrino spectrum at two identical detectors located at different distances from the Chooz nuclear reactor cores in France. Double Chooz is particularly attractive because of its capability to measure sin^2(2theta(13)) to 3 sigma significance if sin^2(2theta(13)) > 0.05 or to exclude sin^2(2theta(13)) down to 0.03 at 90% C.L. for Dm^2 = 2.5 x 10^(-3) eV^2 in three years of data taking with both detectors. The installation of the far detector started in May 2008 and the first neutrino results are expected in 2009. The advantages of reactor neutrino experiments to measure the theta(13) mixing angle are described in this article and in particular, the design, current status and expected performance of the Double Chooz detector.

[{"content":"Help me to generate the abstract of a survey paper given the title:*Review of theta(13) and prospects for Double Chooz* and the introduction:I Physics Motivations The neutrino oscillation phenomenon has been clearly established by the study of solar, atmospheric, reactor and beam neutrinos. The PMNS mixing matrix relates the three neutrino mass eigenstates to the three neutrino flavor eigenstates. This can be parametrized by three mixing angles ( θ 12 subscript 𝜃 12 \theta_{12} , θ 13 subscript 𝜃 13 \theta_{13} , θ 23 subscript 𝜃 23 \theta_{23} ) and one CP violating phase δ C ​ P subscript 𝛿 𝐶 𝑃 \delta_{CP} (if neutrinos are Dirac particles). During the last years, tremendous progress has been achieved in the experimental field trying to measure the values of θ i ​ j subscript 𝜃 𝑖 𝑗 \theta_{ij} and the two squared mass differences Δ ​ m i ​ j 2 = m i 2 − m j 2 Δ subscript superscript 𝑚 2 𝑖 𝑗 subscript superscript 𝑚 2 𝑖 subscript superscript 𝑚 2 𝑗 \Delta m^{2}_{ij}=m^{2}_{i}-m^{2}_{j} which govern the oscillation probabilities. The two mass differences and the θ 12 subscript 𝜃 12 \theta_{12} and θ 23 subscript 𝜃 23 \theta_{23} mixing angles have been measured with good precision GonzalezGarcia:2007ib . However, the θ 13 subscript 𝜃 13 \theta_{13} angle, the sign of Δ ​ m 31 2 Δ subscript superscript 𝑚 2 31 \Delta m^{2}_{31} (mass hierarchy) and the δ C ​ P subscript 𝛿 𝐶 𝑃 \delta_{CP} phase are still unknown. In particular, only an upper limit on the value of θ 13 subscript 𝜃 13 \theta_{13} has been established indicating that the angle is very small compared to the other mixing angles. A three-flavor global analysis of the existing data provides a constraint on θ 13 subscript 𝜃 13 \theta_{13} being s ​ i ​ n 2 ​ θ 13 𝑠 𝑖 superscript 𝑛 2 subscript 𝜃 13 sin^{2}\theta_{13} < < 0.028 at 90% C.L Schwetz:2006dh . This limit is essentially dominated by the result obtained by the CHOOZ reactor experiment Apollonio:2002gd in France. This experiment measured the fraction of ν e ¯ ¯ subscript 𝜈 𝑒 \bar{\nu_{e}} s surviving at a distance of 1.05 km from the reactor cores to be R = 1.01 ± plus-or-minus \pm 2.8% (stat) ± plus-or-minus \pm 2.7% (syst). This result was mainly limited by the systematic uncertainties induced by the imperfect knowledge of the neutrino production and interaction. The measurement of this angle is of fundamental interest not only for the final understanding of neutrino oscillations but because it determines the possibilities to observe CP violation in the leptonic sector with the forthcoming neutrino experiments.","role":"user"},{"content":"Abstract: The Double Chooz reactor neutrino experiment will be the next detector to search for a non vanishing theta(13) mixing angle with unprecedented sensitivity, which might open the way to unveiling CP violation in the leptonic sector. The measurement of this angle will be based on a precise comparison of the antineutrino spectrum at two identical detectors located at different distances from the Chooz nuclear reactor cores in France. Double Chooz is particularly attractive because of its capability to measure sin^2(2theta(13)) to 3 sigma significance if sin^2(2theta(13)) > 0.05 or to exclude sin^2(2theta(13)) down to 0.03 at 90% C.L. for Dm^2 = 2.5 x 10^(-3) eV^2 in three years of data taking with both detectors. The installation of the far detector started in May 2008 and the first neutrino results are expected in 2009. The advantages of reactor neutrino experiments to measure the theta(13) mixing angle are described in this article and in particular, the design, current status and expected performance of the Double Chooz detector.","role":"assistant"}]

A Survey on Cross-Site Scripting Attacks

The use of the web paradigm is becoming an emerging strategy for application software companies [ 9 ] . It allows the design of pervasive applications which can be potentially used by thousands of customers from simple web clients. Moreover, the existence of new technologies for the improvement of web features (e.g., Ajax [ 10 ] ) allows software engineers the conception of new tools which are not longer restricted to specific operating systems (such as web based document processors [ 12 ] , social network services [ 13 ] , collaborative encyclopedias [ 40 ] and weblogs [ 41 ] ). However, the inclusion of effective security mechanisms on those web applications is an increasing concern [ 39 ] . Besides the expected value that the applications are offering to their potential users, reliable mechanisms for the protection of those data and resources associated to the web application should also be offered. Existing approaches to secure traditional applications are not always sufficient when addressing the web paradigm and often leave end users responsible for the protection of key aspects of a service. This situation must be avoided since, if not well managed, it could allow inappropriate uses of a web application and lead to a violation of its security requirements. We focus in this paper on the specific case of Cross-Site Scripting attacks (XSS attacks for short) against the security of web applications. This attack relays on the injection of a malicious code into a web application, in order to compromise the trust relationship between a user and the web application’s site. If the vulnerability is successfully exploited, the malicious user who injected the code may then bypass, for instance, those controls that guarantee the privacy of its users, or even the integrity of the application itself. There exist in the literature different types of XSS attacks and possible exploitable scenarios. We survey in this paper the two most representative XSS attacks that can actually affect current web applications, and we discuss existing approaches for its prevention, such as filtering of web content, analysis of scripts and runtime enforcement of web browsers. Some alternative categorizations, both of the types of XSS attacks and of the prevention mechanisms, may be found in [ 14 ] . We discuss these approaches and their limitations, as well as their deployment and applicability. The rest of this paper is organized as follows. In Section 2 we further present our motivation problem and show some representative examples. We then survey in Section 3 related solutions and overview their main drawbacks. Finally, Section 4 closes the paper with a list of conclusions.

Web applications are becoming truly pervasive in all kinds of business models and organizations. Today, most critical systems such as those related to health care, banking, or even emergency response, are relying on these applications. They must therefore include, in addition to the expected value offered to their users, reliable mechanisms to ensure their security. In this paper, we focus on the specific problem of cross-site scripting attacks against web applications. We present a study of this kind of attacks, and survey current approaches for their prevention. Applicability and limitations of each proposal are also discussed.

[{"content":"Help me to generate the abstract of a survey paper given the title:*A Survey on Cross-Site Scripting Attacks* and the introduction:The use of the web paradigm is becoming an emerging strategy for application software companies 9 . It allows the design of pervasive applications which can be potentially used by thousands of customers from simple web clients. Moreover, the existence of new technologies for the improvement of web features (e.g., Ajax 10 ) allows software engineers the conception of new tools which are not longer restricted to specific operating systems (such as web based document processors 12 , social network services 13 , collaborative encyclopedias 40 and weblogs 41 ). However, the inclusion of effective security mechanisms on those web applications is an increasing concern 39 . Besides the expected value that the applications are offering to their potential users, reliable mechanisms for the protection of those data and resources associated to the web application should also be offered. Existing approaches to secure traditional applications are not always sufficient when addressing the web paradigm and often leave end users responsible for the protection of key aspects of a service. This situation must be avoided since, if not well managed, it could allow inappropriate uses of a web application and lead to a violation of its security requirements. We focus in this paper on the specific case of Cross-Site Scripting attacks (XSS attacks for short) against the security of web applications. This attack relays on the injection of a malicious code into a web application, in order to compromise the trust relationship between a user and the web applications site. If the vulnerability is successfully exploited, the malicious user who injected the code may then bypass, for instance, those controls that guarantee the privacy of its users, or even the integrity of the application itself. There exist in the literature different types of XSS attacks and possible exploitable scenarios. We survey in this paper the two most representative XSS attacks that can actually affect current web applications, and we discuss existing approaches for its prevention, such as filtering of web content, analysis of scripts and runtime enforcement of web browsers. Some alternative categorizations, both of the types of XSS attacks and of the prevention mechanisms, may be found in 14 . We discuss these approaches and their limitations, as well as their deployment and applicability. The rest of this paper is organized as follows. In Section 2 we further present our motivation problem and show some representative examples. We then survey in Section 3 related solutions and overview their main drawbacks. Finally, Section 4 closes the paper with a list of conclusions.","role":"user"},{"content":"Abstract: Web applications are becoming truly pervasive in all kinds of business models and organizations. Today, most critical systems such as those related to health care, banking, or even emergency response, are relying on these applications. They must therefore include, in addition to the expected value offered to their users, reliable mechanisms to ensure their security. In this paper, we focus on the specific problem of cross-site scripting attacks against web applications. We present a study of this kind of attacks, and survey current approaches for their prevention. Applicability and limitations of each proposal are also discussed.","role":"assistant"}]

A short review on entanglement in quantum spin systems

I Introduction Quantum systems are ultimately characterized by the observable correlations they exhibit. For instance, an observable such as the correlation function between two spins in a typical spin chain may decay exponentially as a function of the distance separating them or, in the case the system undergoes a phase transition, algebraically. The correct assessment of these quantum correlations is tantamount to understanding how entanglement is distributed in the state of the system. This is easily understood as follows. Let us consider a connected correlation ⟨ Ψ | bra Ψ \displaystyle\langle\Psi| O i ​ O j ​ | Ψ ⟩ c ≡ subscript 𝑂 𝑖 subscript 𝑂 𝑗 subscript ket Ψ 𝑐 absent \displaystyle O_{i}O_{j}|\Psi\rangle_{c}\equiv (1) ⟨ Ψ | O i ​ O j | Ψ ⟩ − ⟨ Ψ | O i | Ψ ⟩ ​ ⟨ Ψ | O j | Ψ ⟩ , quantum-operator-product Ψ subscript 𝑂 𝑖 subscript 𝑂 𝑗 Ψ quantum-operator-product Ψ subscript 𝑂 𝑖 Ψ quantum-operator-product Ψ subscript 𝑂 𝑗 Ψ \displaystyle\langle\Psi|O_{i}O_{j}|\Psi\rangle-\langle\Psi|O_{i}|\Psi\rangle\langle\Psi|O_{j}|\Psi\rangle\ , where O i subscript 𝑂 𝑖 O_{i} and O j subscript 𝑂 𝑗 O_{j} are operators at sites i 𝑖 i and j 𝑗 j respectively. This connected correlator would vanish identically for any product state | Ψ ⟩ = ⊗ i | ψ i ⟩ |\Psi\rangle=\otimes_{i}|\psi_{i}\rangle . That is, O i ⊗ O j tensor-product subscript 𝑂 𝑖 subscript 𝑂 𝑗 O_{i}\otimes O_{j} is a product operator and, consequently, its correlations can only come from the amount of entanglement in the state | Ψ ⟩ ket Ψ |\Psi\rangle . It follows that the ground state of any interesting system will be highly correlated and, as a particular case, even the vacuum displays a non-trivial entanglement structure in quantum field theories. Notice that, at this point, our emphasis has moved from Hamiltonians to states. It is perfectly sensible to analyse entanglement properties of specific states per se, which may be artificially created using a post-selection mechanism or may effectively be obtained in different ways using various interactions. We are, thus, concerned with the entanglement properties that characterize a quantum state. Yet, we shall focus on states that are physically relevant. In particular, we shall study the entanglement properties of ground states of Hamiltonians that describe the interaction present in spin chains. It is clear that the property of entanglement can be made apparent by studying correlations functions on a given state. We could consider two-, three- or n-point connected correlation functions. Any of them would manifest how the original interactions in the Hamiltonian have operated in the system to achieve the observed degree of entanglement. For instance, free particles (Gaussian Hamiltonians) produce n-point correlators that reduce to products of two-point correlators via Wick’s theorem. Nonetheless, the study of specific correlation functions is model dependent. How can we compare the correlations of a Heisenberg Model with those of Quantum Chromodynamics? Each theory brings its own set of local and non-local operators that close an Operator Product Expansion. Different theories will carry different sets of operators, so that a naive comparison is hopeless. A wonderful possibility to quantify degrees of entanglement for unrelated theories emerges from the use of Renormalization Group ideas and the study of universal properties. For instance, a system may display exponential decays in its correlations functions which is globally controlled by a common correlation length. A model with a larger correlation length is expected to present stronger long-distance quantum correlations. We may as well try to find a universal unique figure of merit that would allow for a fair comparison of the entanglement present in e. g. the ground state of all possible theories. Such a figure of merit cannot be attached to the correlations properties of model-dependent operators since it would not allow for comparison among different theories. The way to overcome this problem is to look for an operator which is defined in every theory. It turns out that there is only one such operator: the stress tensor. To be more precise, we can use the language of conformal field theory which establishes that there always is a highest weight operator that we call the Identity. The Identity will bring a tower of descendants, the stress tensor being its first representative. Indeed, the stress tensor is always defined in any theory since it corresponds to the operator that measures the energy content of the system and it is the operator that couples the system to gravity. Correlators of stress tensor operators are naturally related to entanglement. In particular, the coefficient of the two-point stress tensor correlator in a conformal field theory in two dimensions corresponds to the central charge of the theory. There is a second option to measure entanglement in a given state with a single measure of entanglement which is closer in spirit to the ideas of Quantum Information. The basic idea consists of using the von Neumann (entanglement) entropy of the reduced density matrix of a sub-part of the system which is analysed. Indeed, the entanglement entropy quantifies the amount of surprise that a sub-part of a system finds when discovering it is correlated to the rest of the system. Therefore, entanglement entropy is a bona fide measure of the correlations in the system. The advantage of the von Neumann entropy of entanglement is that it can be defined for any system. We expect its general properties, as the way it scales with the size of the sub-part of the system we are considering, should characterized the quantum state in a quite refined way. It is tantalizing to exhaustively explore the behaviour of the entropy of entanglement in relevant physical systems. For instance, will the entropy of entanglement scale differently at a critical point as compare to a non-critical one? Will scaling properties depend on the dimensionality of the system. Is disorder relevant for long-distance correlations? Are there non-local systems where entropy obeys some singular behaviour? How does entanglement renormalize? How does entanglement evolve dynamically? We can even go further away from standard dynamical models and question whether entanglement is somehow related to computational complexity problems, both NP-complete and QMA-complete. We shall now briefly review some of these questions.

We review some of the recent progress on the study of entropy of entanglement in many-body quantum systems. Emphasis is placed on the scaling properties of entropy for one-dimensional multi-partite models at quantum phase transitions and, more generally, on the concept of area law. We also briefly describe the relation between entanglement and the presence of impurities, the idea of particle entanglement, the evolution of entanglement along renormalization group trajectories, the dynamical evolution of entanglement and the fate of entanglement along a quantum computation.

[{"content":"Help me to generate the abstract of a survey paper given the title:*A short review on entanglement in quantum spin systems* and the introduction:I Introduction Quantum systems are ultimately characterized by the observable correlations they exhibit. For instance, an observable such as the correlation function between two spins in a typical spin chain may decay exponentially as a function of the distance separating them or, in the case the system undergoes a phase transition, algebraically. The correct assessment of these quantum correlations is tantamount to understanding how entanglement is distributed in the state of the system. This is easily understood as follows. Let us consider a connected correlation ⟨ Ψ | bra Ψ \displaystyle\langle\Psi| O i ​ O j ​ | Ψ ⟩ c ≡ subscript 𝑂 𝑖 subscript 𝑂 𝑗 subscript ket Ψ 𝑐 absent \displaystyle O_{i}O_{j}|\Psi\rangle_{c}\equiv (1) ⟨ Ψ | O i ​ O j | Ψ ⟩ − ⟨ Ψ | O i | Ψ ⟩ ​ ⟨ Ψ | O j | Ψ ⟩ , quantum-operator-product Ψ subscript 𝑂 𝑖 subscript 𝑂 𝑗 Ψ quantum-operator-product Ψ subscript 𝑂 𝑖 Ψ quantum-operator-product Ψ subscript 𝑂 𝑗 Ψ \displaystyle\langle\Psi|O_{i}O_{j}|\Psi\rangle-\langle\Psi|O_{i}|\Psi\rangle\langle\Psi|O_{j}|\Psi\rangle\ , where O i subscript 𝑂 𝑖 O_{i} and O j subscript 𝑂 𝑗 O_{j} are operators at sites i 𝑖 i and j 𝑗 j respectively. This connected correlator would vanish identically for any product state | Ψ ⟩ = ⊗ i | ψ i ⟩ |\Psi\rangle=\otimes_{i}|\psi_{i}\rangle . That is, O i ⊗ O j tensor-product subscript 𝑂 𝑖 subscript 𝑂 𝑗 O_{i}\otimes O_{j} is a product operator and, consequently, its correlations can only come from the amount of entanglement in the state | Ψ ⟩ ket Ψ |\Psi\rangle . It follows that the ground state of any interesting system will be highly correlated and, as a particular case, even the vacuum displays a non-trivial entanglement structure in quantum field theories. Notice that, at this point, our emphasis has moved from Hamiltonians to states. It is perfectly sensible to analyse entanglement properties of specific states per se, which may be artificially created using a post-selection mechanism or may effectively be obtained in different ways using various interactions. We are, thus, concerned with the entanglement properties that characterize a quantum state. Yet, we shall focus on states that are physically relevant. In particular, we shall study the entanglement properties of ground states of Hamiltonians that describe the interaction present in spin chains. It is clear that the property of entanglement can be made apparent by studying correlations functions on a given state. We could consider two-, three- or n-point connected correlation functions. Any of them would manifest how the original interactions in the Hamiltonian have operated in the system to achieve the observed degree of entanglement. For instance, free particles (Gaussian Hamiltonians) produce n-point correlators that reduce to products of two-point correlators via Wicks theorem. Nonetheless, the study of specific correlation functions is model dependent. How can we compare the correlations of a Heisenberg Model with those of Quantum Chromodynamics? Each theory brings its own set of local and non-local operators that close an Operator Product Expansion. Different theories will carry different sets of operators, so that a naive comparison is hopeless. A wonderful possibility to quantify degrees of entanglement for unrelated theories emerges from the use of Renormalization Group ideas and the study of universal properties. For instance, a system may display exponential decays in its correlations functions which is globally controlled by a common correlation length. A model with a larger correlation length is expected to present stronger long-distance quantum correlations. We may as well try to find a universal unique figure of merit that would allow for a fair comparison of the entanglement present in e. g. the ground state of all possible theories. Such a figure of merit cannot be attached to the correlations properties of model-dependent operators since it would not allow for comparison among different theories. The way to overcome this problem is to look for an operator which is defined in every theory. It turns out that there is only one such operator: the stress tensor. To be more precise, we can use the language of conformal field theory which establishes that there always is a highest weight operator that we call the Identity. The Identity will bring a tower of descendants, the stress tensor being its first representative. Indeed, the stress tensor is always defined in any theory since it corresponds to the operator that measures the energy content of the system and it is the operator that couples the system to gravity. Correlators of stress tensor operators are naturally related to entanglement. In particular, the coefficient of the two-point stress tensor correlator in a conformal field theory in two dimensions corresponds to the central charge of the theory. There is a second option to measure entanglement in a given state with a single measure of entanglement which is closer in spirit to the ideas of Quantum Information. The basic idea consists of using the von Neumann (entanglement) entropy of the reduced density matrix of a sub-part of the system which is analysed. Indeed, the entanglement entropy quantifies the amount of surprise that a sub-part of a system finds when discovering it is correlated to the rest of the system. Therefore, entanglement entropy is a bona fide measure of the correlations in the system. The advantage of the von Neumann entropy of entanglement is that it can be defined for any system. We expect its general properties, as the way it scales with the size of the sub-part of the system we are considering, should characterized the quantum state in a quite refined way. It is tantalizing to exhaustively explore the behaviour of the entropy of entanglement in relevant physical systems. For instance, will the entropy of entanglement scale differently at a critical point as compare to a non-critical one? Will scaling properties depend on the dimensionality of the system. Is disorder relevant for long-distance correlations? Are there non-local systems where entropy obeys some singular behaviour? How does entanglement renormalize? How does entanglement evolve dynamically? We can even go further away from standard dynamical models and question whether entanglement is somehow related to computational complexity problems, both NP-complete and QMA-complete. We shall now briefly review some of these questions.","role":"user"},{"content":"Abstract: We review some of the recent progress on the study of entropy of entanglement in many-body quantum systems. Emphasis is placed on the scaling properties of entropy for one-dimensional multi-partite models at quantum phase transitions and, more generally, on the concept of area law. We also briefly describe the relation between entanglement and the presence of impurities, the idea of particle entanglement, the evolution of entanglement along renormalization group trajectories, the dynamical evolution of entanglement and the fate of entanglement along a quantum computation.","role":"assistant"}]

The Busy Beaver Competition: a historical survey

1.1 Noncomputable functions In 1936, Turing succeeded in making formal the intuitive notion of a function computable by a finite, mechanical, procedure. He defined what is now called a Turing machine and stated that a function on integers is intuitively computable if and only if it is computable by a Turing machine. Other authors, such as Church, Kleene, Post, and, later, Markov, defined other models of computation that turn out to compute the same functions as Turing machines do. See Soare (1996, 2007, 2009) for more details about the history of the Church-Turing Thesis , as is now named the capture of the intuitive notion of computability by the formal notion of Turing machine. Given a model of computation, a noncomputable function can easily be defined by diagonalization . The list of all computable functions is written, and then a function is defined such that it is distinct from each function in the list. Then this function is noncomputable. Such a definition by diagonalization leaves too much room in the choice of the list and in the choice of the values of the final function. What is needed is a function whose definition is simple, natural and without ambiguousness. In 1962, Rado succeeded in providing a natural definition for noncomputable functions on the integers. He defined a Busy Beaver game, leading to two functions Σ Σ \mathnormal{\Sigma} and S 𝑆 S which are still the best examples of noncomputable functions that one can give nowadays. The values Σ ​ ( n ) Σ 𝑛 \mathnormal{\Sigma}(n) and S ​ ( n ) 𝑆 𝑛 S(n) are defined by considering the finite set of carefully defined Turing machines with two symbols and n 𝑛 n states, and picking among these machines those with some maximal behavior. It makes sense to compute the values Σ ​ ( n ) Σ 𝑛 \mathnormal{\Sigma}(n) , S ​ ( n ) 𝑆 𝑛 S(n) of these functions on small integers n = 1 , 2 , … 𝑛 1 2 … n=1,2,\ldots . We have Σ ​ ( 1 ) = S ​ ( 1 ) = 1 Σ 1 𝑆 1 1 \mathnormal{\Sigma}(1)=S(1)=1 , trivially. Lin and Rado (1965) gave proofs for the values Σ ​ ( 2 ) Σ 2 \mathnormal{\Sigma}(2) , S ​ ( 2 ) 𝑆 2 S(2) , Σ ​ ( 3 ) Σ 3 \mathnormal{\Sigma}(3) and S ​ ( 3 ) 𝑆 3 S(3) , and Brady (1983) did for Σ ​ ( 4 ) Σ 4 \mathnormal{\Sigma}(4) and S ​ ( 4 ) 𝑆 4 S(4) . Only lower bounds had been provided for Σ ​ ( 5 ) Σ 5 \mathnormal{\Sigma}(5) and S ​ ( 5 ) 𝑆 5 S(5) , by the works of Green, Lynn, Schult, Uhing and eventually Marxen and Buntrock. The lower bounds for Σ ​ ( 6 ) Σ 6 \mathnormal{\Sigma}(6) and S ​ ( 6 ) 𝑆 6 S(6) are still an ongoing quest. The initial Busy Beaver game, as defined by Rado, used Turing machines with two symbols. Brady (1988) generalized the problem to Turing machines with k 𝑘 k symbols, k ≥ 3 𝑘 3 k\geq 3 . He defined a function S ​ ( n , k ) 𝑆 𝑛 𝑘 S(n,k) such that S ​ ( n , 2 ) 𝑆 𝑛 2 S(n,2) is Rado’s S ​ ( n ) 𝑆 𝑛 S(n) , and gave some lower bounds. Michel (2004) resumed the computation of lower bounds for S ​ ( n , k ) 𝑆 𝑛 𝑘 S(n,k) and another function Σ ​ ( n , k ) Σ 𝑛 𝑘 \mathnormal{\Sigma}(n,k) , and the search is going on, with the works of Brady, Souris, Lafitte and Papazian, T. and S. Ligocki. Since 2004, results are sent by email to Marxen and to Michel, who record them on their websites. This paper aims to give a published version of these records. 1.2 Big numbers Consider Rado’s functions S 𝑆 S and Σ Σ \mathnormal{\Sigma} . Not only they are not computable, but they grow faster than any computable function. That is, for any computable function f 𝑓 f , there exists an integer N 𝑁 N such that, for all n > N 𝑛 𝑁 n>N , we have S ​ ( n ) > f ​ ( n ) 𝑆 𝑛 𝑓 𝑛 S(n)>f(n) . This property can be used to write big numbers. For example, if S k ​ ( n ) superscript 𝑆 𝑘 𝑛 S^{k}(n) denotes S ​ ( S ​ ( … ​ S ​ ( n ) ​ … ) ) 𝑆 𝑆 … 𝑆 𝑛 … S(S(\ldots S(n)\ldots)) , iterated k 𝑘 k times, then S 9 9 ​ ( 9 ) superscript 𝑆 superscript 9 9 9 S^{9^{9}}(9) is a very big number, bigger than any number that was written with six symbols before the definition of the S 𝑆 S function. Bigger numbers can be obtained by defining functions growing much faster than Rado’s busy beaver functions. A natural idea to get such functions is to define Turing machines of order k 𝑘 k as follows. Turing machines of order 1 are usual Turing machines without oracle, and, for k ≥ 2 𝑘 2 k\geq 2 , Turing machines of order k 𝑘 k are Turing machines with oracle, where the oracle is the halting problem for Turing machines of order k − 1 𝑘 1 k-1 . Then the k 𝑘 k -th busy beaver function B k ​ ( n ) subscript 𝐵 𝑘 𝑛 B_{k}(n) is the maximum number of steps taken by a Turing machine of order k 𝑘 k with n 𝑛 n states and two symbols that stops when it is launched on a blank tape. So B 1 ​ ( n ) = S ​ ( n ) subscript 𝐵 1 𝑛 𝑆 𝑛 B_{1}(n)=S(n) , and B k ​ ( n ) subscript 𝐵 𝑘 𝑛 B_{k}(n) grows faster than any function computable by a Turing machine of order k 𝑘 k . Unfortunately, there is no canonical way to define a Turing machine with oracle, so Scott Aaronson, in his paper Who can name the bigger number? (see the website), asked for naturally defined functions growing as fast as the k 𝑘 k -th busy beaver functions for k ≥ 2 𝑘 2 k\geq 2 . Such functions were found by Nabutovsky and Weinberger (2007). By using homology of groups, they defined a function growing as fast as the third busy beaver function, and another one growing as fast as the fifth busy beaver function. Michel (2010) went on studying these functions. Scott Aaronson (2020) defined the beeping busy beaver function , which grows as fast as the second busy beaver function. See the definition and values in Section 7.5. 1.3 Contents The paper is structured as follows. 1. Introduction. 2. Preliminaries. 3. Historical overview. 4. Historical survey (lower bounds for S ​ ( n , k ) 𝑆 𝑛 𝑘 S(n,k) and Σ ​ ( n , k ) Σ 𝑛 𝑘 \mathnormal{\Sigma}(n,k) , and tables of the Turing machines that achieve these lower bounds). 5. Behaviors of busy beavers. We also display the relations between these behaviors and open problems in mathematics called Collatz-like problems and we resume some machines with non-Collatz-like behaviors. We also present pairs of machines that have the same behaviors, but not the same numbers of states and symbols. 6. Properties of the busy beaver functions and relations between S ​ ( n ) 𝑆 𝑛 S(n) and Σ ​ ( n ) Σ 𝑛 \mathnormal{\Sigma}(n) . 7. Variants of busy beavers: - Busy beavers defined by 4-tuples. - Busy beavers whose head can stand still. - Busy beavers on a one-way infinite tape. - Two-dimensional busy beavers. - Beeping busy beavers 8. The methods. 9. Busy beavers and unprovability.

Tibor Rado defined the Busy Beaver Competition in 1962. He used Turing machines to give explicit definitions for some functions that are not computable and grow faster than any computable function. He put forward the problem of computing the values of these functions on numbers 1, 2, 3, ... More and more powerful computers have made possible the computation of lower bounds for these values. In 1988, Brady extended the definitions to functions on two variables. We give a historical survey of these works. The successive record holders in the Busy Beaver Competition are displayed, with their discoverers, the date they were found, and, for some of them, an analysis of their behavior.

[{"content":"Help me to generate the abstract of a survey paper given the title:*The Busy Beaver Competition: a historical survey* and the introduction:1.1 Noncomputable functions In 1936, Turing succeeded in making formal the intuitive notion of a function computable by a finite, mechanical, procedure. He defined what is now called a Turing machine and stated that a function on integers is intuitively computable if and only if it is computable by a Turing machine. Other authors, such as Church, Kleene, Post, and, later, Markov, defined other models of computation that turn out to compute the same functions as Turing machines do. See Soare (1996, 2007, 2009) for more details about the history of the Church-Turing Thesis , as is now named the capture of the intuitive notion of computability by the formal notion of Turing machine. Given a model of computation, a noncomputable function can easily be defined by diagonalization . The list of all computable functions is written, and then a function is defined such that it is distinct from each function in the list. Then this function is noncomputable. Such a definition by diagonalization leaves too much room in the choice of the list and in the choice of the values of the final function. What is needed is a function whose definition is simple, natural and without ambiguousness. In 1962, Rado succeeded in providing a natural definition for noncomputable functions on the integers. He defined a Busy Beaver game, leading to two functions Σ Σ \mathnormal{\Sigma} and S 𝑆 S which are still the best examples of noncomputable functions that one can give nowadays. The values Σ ​ ( n ) Σ 𝑛 \mathnormal{\Sigma}(n) and S ​ ( n ) 𝑆 𝑛 S(n) are defined by considering the finite set of carefully defined Turing machines with two symbols and n 𝑛 n states, and picking among these machines those with some maximal behavior. It makes sense to compute the values Σ ​ ( n ) Σ 𝑛 \mathnormal{\Sigma}(n) , S ​ ( n ) 𝑆 𝑛 S(n) of these functions on small integers n = 1 , 2 , … 𝑛 1 2 … n=1,2,\ldots . We have Σ ​ ( 1 ) = S ​ ( 1 ) = 1 Σ 1 𝑆 1 1 \mathnormal{\Sigma}(1)=S(1)=1 , trivially. Lin and Rado (1965) gave proofs for the values Σ ​ ( 2 ) Σ 2 \mathnormal{\Sigma}(2) , S ​ ( 2 ) 𝑆 2 S(2) , Σ ​ ( 3 ) Σ 3 \mathnormal{\Sigma}(3) and S ​ ( 3 ) 𝑆 3 S(3) , and Brady (1983) did for Σ ​ ( 4 ) Σ 4 \mathnormal{\Sigma}(4) and S ​ ( 4 ) 𝑆 4 S(4) . Only lower bounds had been provided for Σ ​ ( 5 ) Σ 5 \mathnormal{\Sigma}(5) and S ​ ( 5 ) 𝑆 5 S(5) , by the works of Green, Lynn, Schult, Uhing and eventually Marxen and Buntrock. The lower bounds for Σ ​ ( 6 ) Σ 6 \mathnormal{\Sigma}(6) and S ​ ( 6 ) 𝑆 6 S(6) are still an ongoing quest. The initial Busy Beaver game, as defined by Rado, used Turing machines with two symbols. Brady (1988) generalized the problem to Turing machines with k 𝑘 k symbols, k ≥ 3 𝑘 3 k\geq 3 . He defined a function S ​ ( n , k ) 𝑆 𝑛 𝑘 S(n,k) such that S ​ ( n , 2 ) 𝑆 𝑛 2 S(n,2) is Rados S ​ ( n ) 𝑆 𝑛 S(n) , and gave some lower bounds. Michel (2004) resumed the computation of lower bounds for S ​ ( n , k ) 𝑆 𝑛 𝑘 S(n,k) and another function Σ ​ ( n , k ) Σ 𝑛 𝑘 \mathnormal{\Sigma}(n,k) , and the search is going on, with the works of Brady, Souris, Lafitte and Papazian, T. and S. Ligocki. Since 2004, results are sent by email to Marxen and to Michel, who record them on their websites. This paper aims to give a published version of these records. 1.2 Big numbers Consider Rados functions S 𝑆 S and Σ Σ \mathnormal{\Sigma} . Not only they are not computable, but they grow faster than any computable function. That is, for any computable function f 𝑓 f , there exists an integer N 𝑁 N such that, for all n > N 𝑛 𝑁 n>N , we have S ​ ( n ) > f ​ ( n ) 𝑆 𝑛 𝑓 𝑛 S(n)>f(n) . This property can be used to write big numbers. For example, if S k ​ ( n ) superscript 𝑆 𝑘 𝑛 S^{k}(n) denotes S ​ ( S ​ ( … ​ S ​ ( n ) ​ … ) ) 𝑆 𝑆 … 𝑆 𝑛 … S(S(\ldots S(n)\ldots)) , iterated k 𝑘 k times, then S 9 9 ​ ( 9 ) superscript 𝑆 superscript 9 9 9 S^{9^{9}}(9) is a very big number, bigger than any number that was written with six symbols before the definition of the S 𝑆 S function. Bigger numbers can be obtained by defining functions growing much faster than Rados busy beaver functions. A natural idea to get such functions is to define Turing machines of order k 𝑘 k as follows. Turing machines of order 1 are usual Turing machines without oracle, and, for k ≥ 2 𝑘 2 k\geq 2 , Turing machines of order k 𝑘 k are Turing machines with oracle, where the oracle is the halting problem for Turing machines of order k − 1 𝑘 1 k-1 . Then the k 𝑘 k -th busy beaver function B k ​ ( n ) subscript 𝐵 𝑘 𝑛 B_{k}(n) is the maximum number of steps taken by a Turing machine of order k 𝑘 k with n 𝑛 n states and two symbols that stops when it is launched on a blank tape. So B 1 ​ ( n ) = S ​ ( n ) subscript 𝐵 1 𝑛 𝑆 𝑛 B_{1}(n)=S(n) , and B k ​ ( n ) subscript 𝐵 𝑘 𝑛 B_{k}(n) grows faster than any function computable by a Turing machine of order k 𝑘 k . Unfortunately, there is no canonical way to define a Turing machine with oracle, so Scott Aaronson, in his paper Who can name the bigger number? (see the website), asked for naturally defined functions growing as fast as the k 𝑘 k -th busy beaver functions for k ≥ 2 𝑘 2 k\geq 2 . Such functions were found by Nabutovsky and Weinberger (2007). By using homology of groups, they defined a function growing as fast as the third busy beaver function, and another one growing as fast as the fifth busy beaver function. Michel (2010) went on studying these functions. Scott Aaronson (2020) defined the beeping busy beaver function , which grows as fast as the second busy beaver function. See the definition and values in Section 7.5. 1.3 Contents The paper is structured as follows. 1. Introduction. 2. Preliminaries. 3. Historical overview. 4. Historical survey (lower bounds for S ​ ( n , k ) 𝑆 𝑛 𝑘 S(n,k) and Σ ​ ( n , k ) Σ 𝑛 𝑘 \mathnormal{\Sigma}(n,k) , and tables of the Turing machines that achieve these lower bounds). 5. Behaviors of busy beavers. We also display the relations between these behaviors and open problems in mathematics called Collatz-like problems and we resume some machines with non-Collatz-like behaviors. We also present pairs of machines that have the same behaviors, but not the same numbers of states and symbols. 6. Properties of the busy beaver functions and relations between S ​ ( n ) 𝑆 𝑛 S(n) and Σ ​ ( n ) Σ 𝑛 \mathnormal{\Sigma}(n) . 7. Variants of busy beavers: - Busy beavers defined by 4-tuples. - Busy beavers whose head can stand still. - Busy beavers on a one-way infinite tape. - Two-dimensional busy beavers. - Beeping busy beavers 8. The methods. 9. Busy beavers and unprovability.","role":"user"},{"content":"Abstract: Tibor Rado defined the Busy Beaver Competition in 1962. He used Turing machines to give explicit definitions for some functions that are not computable and grow faster than any computable function. He put forward the problem of computing the values of these functions on numbers 1, 2, 3, ... More and more powerful computers have made possible the computation of lower bounds for these values. In 1988, Brady extended the definitions to functions on two variables. We give a historical survey of these works. The successive record holders in the Busy Beaver Competition are displayed, with their discoverers, the date they were found, and, for some of them, an analysis of their behavior.","role":"assistant"}]