Showing papers in "Symmetry Integrability and Geometry-methods and Applications in 2007"
TL;DR: In this paper, an explicit representation of the double affine Hecke algebra (DAHA) related to sym-metric and non-symmetric Askey-Wilson polynomials is presented.
Abstract: Zhedanov's algebra AW(3) is considered with explicit structure constants such that, in the basic representation, the first generator becomes the second order q-difference operator for the Askey-Wilson polynomials. It is proved that this representation is faith- ful for a certain quotient of AW(3) such that the Casimir operator is equal to a special constant. Some explicit aspects of the double affine Hecke algebra (DAHA) related to sym- metric and non-symmetric Askey-Wilson polynomials are presented and proved without requiring knowledge of general DAHA theory. Finally a central extension of this quotient of AW(3) is introduced which can be embedded in the DAHA by means of the faithful basic representations of both algebras.
80 citations
TL;DR: Antisymmetric orbit functions on the Euclidean space are antisymmetrized exponential functions as discussed by the authors, which are closely related to irreducible characters of a compact semisimple Lie group.
Abstract: In the paper, properties of antisymmetric orbit functions are reviewed and further developed. Antisymmetric orbit functions on the Euclidean space $E_n$ are antisymmetrized exponential functions. Antisymmetrization is fulfilled by a Weyl group, corresponding to a Coxeter-Dynkin diagram. Properties of such functions are described. These functions are closely related to irreducible characters of a compact semisimple Lie group $G$ of rank $n$. Up to a sign, values of antisymmetric orbit functions are repeated on copies of the fundamental domain $F$ of the affine Weyl group (determined by the initial Weyl group) in the entire Euclidean space $E_n$. Antisymmetric orbit functions are solutions of the corresponding Laplace equation in $E_n$, vanishing on the boundary of the fundamental domain $F$. Antisymmetric orbit functions determine a so-called antisymmetrized Fourier transform which is closely related to expansions of central functions in characters of irreducible representations of the group $G$. They also determine a transform on a finite set of points of $F$ (the discrete antisymmetric orbit function transform). Symmetric and antisymmetric multivariate exponential, sine and cosine discrete transforms are given.
79 citations
TL;DR: In this article, the concept of finite-temperature form factor was introduced in the context of the Majorana theory and applied to the calculation of correlation functions in the quantum Ising model.
Abstract: We review the concept of finite-temperature form factor that was introduced re- cently by the author in the context of the Majorana theory. Finite-temperature form factors can be used to obtain spectral decompositions of finite-temperature correlation functions in a way that mimics the form-factor expansion of the zero temperature case. We develop the concept in the general factorised scattering set-up of integrable quantum field theory, list certain expected properties and present the full construction in the case of the massive Majorana theory, including how it can be applied to the calculation of correlation functions in the quantum Ising model. In particular, we include the "twisted construction", which was not developed before and which is essential for the application to the quantum Ising model.
48 citations
TL;DR: In this paper, the authors present basics of conceptually new-type way for explaining of the origin, evolution and current physical properties of our Universe from the graviton-matter gas view-point.
Abstract: We present basics of conceptually new-type way for explaining of the origin, evolution and current physical properties of our Universe from the graviton-matter gas view- point. Quantization method for the Friedmann-Lemaˆitre Universe based on the canonical Hamilton equations of motion is proposed and quantum information theory way to physics of the Universe is showed. The current contribution from the graviton-matter gas temperature in quintessence approximation is discussed.
46 citations
TL;DR: In this article, the second-order nonlinear equations of Rabelo which describe pseudospherical surfaces were studied and their general solutions were obtained by transforming these equations to the constant-characteristic form.
Abstract: We study four distinct second-order nonlinear equations of Rabelo which describe pseudospherical surfaces. By transforming these equations to the constant-characteristic form we relate them to some well-studied integrable equations. Two of the Rabelo equations are found to be related to the sine-Gordon equation. The other two are transformed into a linear equation and the Liouville equation, and in this way their general solutions are obtained.
45 citations
TL;DR: In this article, the authors review some recent results on the theory of Lagrangian systems on Lie algebroids and consider the symplectic and variational formalism and show how to reduce Pontryagin maximum principle.
Abstract: We review some recent results on the theory of Lagrangian systems on Lie algebroids. In particular we consider the symplectic and variational formalism and we study reduction. Finally we also consider optimal control systems on Lie algebroids and we show how to reduce Pontryagin maximum principle.
37 citations
TL;DR: In this paper, it was shown that specific subsets of the generalized Pauli operators of two qubits can also be associated with the points and lines of the four-dimensional projective space over the Galois field with two elements -the Veldkamp space of W(2).
Abstract: Given a remarkable representation of the generalized Pauli operators of two- qubits in terms of the points of the generalized quadrangle of order two, W(2), it is shown that specific subsets of these operators can also be associated with the points and lines of the four-dimensional projective space over the Galois field with two elements - the so-called Veldkamp space of W(2). An intriguing novelty is the recognition of (uni- and tri-centric) triads and specific pentads of the Pauli operators in addition to the "classical" subsets answering to geometric hyperplanes of W(2).
37 citations
TL;DR: In this paper, an exact solvable position-dependent mass Schrodinger equation in two di-mensions, depicting a particle moving in a semi-infinite layer, is re-examined in the light of recent theories describing superintegrable two-dimensional systems with integrals of motion that are quadratic functions of the momenta.
Abstract: An exactly solvable position-dependent mass Schrodinger equation in two di- mensions, depicting a particle moving in a semi-infinite layer, is re-examined in the light of recent theories describing superintegrable two-dimensional systems with integrals of motion that are quadratic functions of the momenta. To get the energy spectrum a quadratic algebra approach is used together with a realization in terms of deformed parafermionic oscillator operators. In this process, the importance of supplementing algebraic considerations with a proper treatment of boundary conditions for selecting physical wavefunctions is stressed. Some new results for matrix elements are derived. This example emphasizes the interest of a quadratic algebra approach to position-dependent mass Schrodinger equations.
35 citations
TL;DR: In this article, the integrals of completely integrable quantum systems associated with classical root systems were studied and a conjecture was made that the quantum systems with enough integrals to achieve complete integrability coincide with the systems that have integrals with constant principal symbols corresponding to the homogeneous generators of the Bn-invariants.
Abstract: We study integrals of completely integrable quantum systems associated with classical root systems. We review integrals of the systems invariant under the corresponding Weyl group and as their limits we construct enough integrals of the non-invariant systems, which include systems whose complete integrability will be first established in this paper. We also present a conjecture claiming that the quantum systems with enough integrals given in this note coincide with the systems that have the integrals with constant principal symbols corresponding to the homogeneous generators of the Bn-invariants. We review conditions supporting the conjecture and give a new condition assuring it.
35 citations
TL;DR: The relationship between super-integrability and exact solvability is discussed in this article, where the authors consider some examples of quantum superintegrable systems and the as- sociated nonlinear extensions of Lie algebras.
Abstract: We consider some examples of quantum super-integrable systems and the as- sociated nonlinear extensions of Lie algebras. The intimate relationship between super- integrability and exact solvability is illustrated. Eigenfunctions are constructed through the action of the commuting operators. Finite dimensional representations of the quadratic al- gebras are thus constructed in a way analogous to that of the highest weight representations of Lie algebras.
34 citations
TL;DR: In this paper, a conformal description of Poincare-Einstein manifolds is developed: these structures are seen to be a special case of a natural weakening of the Einstein condition termed an almost Einstein structure.
Abstract: A conformal description of Poincare-Einstein manifolds is developed: these structures are seen to be a special case of a natural weakening of the Einstein condition termed an almost Einstein structure. This is used for two purposes: to shed light on the relationship between the scattering construction of Graham-Zworski and the higher order conformal Dirichlet-Neumann maps of Branson and the author; to sketch a new construction of non-local (Dirichlet-to-Neumann type) conformal operators between tensor bundles.
TL;DR: In this article, a brief review of the Somos sequences is provided, with particular focus being made on the integrable structure of Somos-4 recurrences, and on the Laurent property.
Abstract: This article is dedicated to the memory of Vadim Kuznetsov, and begins with some of the author's recollections of him. Thereafter, a brief review of Somos sequences is provided, with particular focus being made on the integrable structure of Somos-4 recurrences, and on the Laurent property. Subsequently a family of fourth-order recurrences that share the Laurent property are considered, which are equivalent to Poisson maps in four dimensions. Two of these maps turn out to be superintegrable, and their iteration furnishes infinitely many solutions of some associated quartic Diophantine equations.
TL;DR: In this paper, a new derivation is given of Branson's factorization formula for the confor- mally invariant operator on the sphere whose principal part is the k-th power of the scalar Laplacian.
Abstract: A new derivation is given of Branson's factorization formula for the confor- mally invariant operator on the sphere whose principal part is the k-th power of the scalar Laplacian. The derivation deduces Branson's formula from knowledge of the correspon- ding conformally invariant operator on Euclidean space (the k-th power of the Euclidean Laplacian) via conjugation by the stereographic projection mapping.
TL;DR: In this article, it was shown that the universal difference equation Df = 0 for an M-valued function f has a basis of solutions consisting of quasi-exponentials.
Abstract: Let M be the tensor product of finite-dimensional polynomial evaluation Y (gl N )- modules. Consider the universal difference operator D = N P k=0 ( 1) k Tk(u)e k@u whose coef- ficients Tk(u) : M ! M are the XXX transfer matrices associated with M. We show that the difference equation Df = 0 for an M-valued function f has a basis of solutions consisting of quasi-exponentials. We prove the same for the universal differential operator D = N
TL;DR: This work considers the problem of varying conformally the metric of a four dimensional manifold in order to obtain constant Q-curvature, and shows how the problem leads naturally to consider the set of formal barycenters of the manifold.
Abstract: We consider the problem of varying conformally the metric of a four dimensional manifold in order to obtain constant Q-curvature. The problem is variational, and solutions are in general found as critical points of saddle type. We show how the problem leads naturally to consider the set of formal barycenters of the manifold.
TL;DR: In this paper, it was shown that the Casimir operator acting on sections of a homogeneous vector bundle over a generalized flag manifold naturally extends to an invariant differential operator on arbitrary parabolic geometries.
Abstract: We prove that the Casimir operator acting on sections of a homogeneous vector bundle over a generalized flag manifold naturally extends to an invariant differential operator on arbitrary parabolic geometries. We study some properties of the resulting invariant operators and compute their action on various special types of natural bundles. As a first application, we give a very general construction of splitting operators for parabolic geometries. Then we discuss the curved Casimir operators on differential forms with values in a tractor bundle, which nicely relates to the machinery of BGG sequences. This also gives a nice interpretation of the resolution of a finite dimensional representation by (spaces of smooth vectors in) principal series representations provided by a BGG sequence.
TL;DR: For both the conformal and projective groups, all the differential invariants of a generic surface in 3D space can be written as combinations of the invariant derivatives of a single differential invariant as discussed by the authors.
Abstract: We show that, for both the conformal and projective groups, all the differential invariants of a generic surface in three-dimensional space can be written as combinations of the invariant derivatives of a single differential invariant. The proof is based on the equivariant method of moving frames.
TL;DR: In this article, the graph description of Teichmuller theory of surfaces with marked points on boundary components (bordered surfaces) is presented in terms of hyperbolic geometry.
Abstract: We propose the graph description of Teichmuller theory of surfaces with marked points on boundary components (bordered surfaces). Introducing new parameters, we formu- late this theory in terms of hyperbolic geometry. We can then describe both classical and quantum theories having the proper number of Thurston variables (foliation-shear coordi- nates), mapping-class group invariance (both classical and quantum), Poisson and quantum algebra of geodesic functions, and classical and quantum braid-group relations. These new algebras can be defined on the double of the corresponding graph related (in a novel way) to a double of the Riemann surface (which is a Riemann surface with holes, not a smooth Riemann surface). We enlarge the mapping class group allowing transformations relating different Teichmuller spaces of bordered surfaces of the same genus, same number of boun- dary components, and same total number of marked points but with arbitrary distributions of marked points among the boundary components. We describe the classical and quantum algebras and braid group relations for particular sets of geodesic functions corresponding to An and Dn algebras and discuss briefly the relation to the Thurston theory.
TL;DR: In this paper, two generalizations of Kempf's quadratic canonical commutation relation in one dimension are considered and the corresponding nonzero minimal uncertainties in position and momentum are determined and the effect on the energy spectrum and eigenfunctions of the harmonic oscillator in an electric field is studied.
Abstract: Two generalizations of Kempf's quadratic canonical commutation relation in one dimension are considered. The first one is the most general quadratic commutation relation. The corresponding nonzero minimal uncertainties in position and momentum are determined and the effect on the energy spectrum and eigenfunctions of the harmonic oscillator in an electric field is studied. The second extension is a function-dependent generalization of the simplest quadratic commutation relation with only a nonzero minimal uncertainty in po- sition. Such an uncertainty now becomes dependent on the average position. With each function-dependent commutation relation we associate a family of potentials whose spectrum can be exactly determined through supersymmetric quantum mechanical and shape invari- ance techniques. Some representations of the generalized Heisenberg algebras are proposed in terms of conventional position and momentum operatorsx, p. The resulting Hamiltonians contain a contribution proportional to p 4 and their p-dependent terms may also be functions
TL;DR: In this paper, the quasi-commutative approximation to a noncommutativity geo-metry defined as a generalization of the moving frame formalism is considered. And the relation which exists between noncomutativity and geometry is used to study the properties of the high frequency waves on the flat background.
Abstract: We consider the quasi-commutative approximation to a noncommutative geo- metry defined as a generalization of the moving frame formalism. The relation which exists between noncommutativity and geometry is used to study the properties of the high- frequency waves on the flat background.
TL;DR: In this article, the rank 2 deformation of the Fourier-Jacobi Lie algebra of sp(2,R) has been generalized to arbitrary tensor and spinor bundles using supersymmetric quantum mechanics.
Abstract: Lichnerowicz's algebra of differential geometric operators acting on symmetric tensors can be obtained from generalized geodesic motion of an observer carrying a complex tangent vector. This relation is based upon quantizing the classical evolution equations, and identifying wavefunctions with sections of the symmetric tensor bundle and Noether charges with geometric operators. In general curved spaces these operators obey a deformation of the Fourier-Jacobi Lie algebra of sp(2,R). These results have already been generalized by the authors to arbitrary tensor and spinor bundles using supersymmetric quantum mechanical models and have also been applied to the theory of higher spin particles. These Proceedings review these results in their simplest, symmetric tensor setting. New results on a novel and extremely useful reformulation of the rank 2 deformation of the Fourier-Jacobi Lie algebra in terms of an associative algebra are also presented. This new algebra was originally motivated by studies of operator orderings in enveloping algebras. It provides a new method that is superior in many respects to common techniques such as Weyl or normal ordering.
TL;DR: In this paper, a mini-review of the heat kernel expansion for generalized Laplacians on various noncommutative spaces is presented, including applications to the spectral action principle, renormaliza- tion of non-convex theories and anomalies.
Abstract: This is a mini-review of the heat kernel expansion for generalized Laplacians on various noncommutative spaces. Applications to the spectral action principle, renormaliza- tion of noncommutative theories and anomalies are also considered.
TL;DR: In this paper, the Hamiltonian treatment of classical and quantum properties of Liouville field theory on a timelike strip in 2D Minkowski space is devoted to the Hamiltonians treatment of Hamiltonian properties.
Abstract: The paper is devoted to the Hamiltonian treatment of classical and quantum properties of Liouville field theory on a timelike strip in 2d Minkowski space. We give a complete description of classical solutions regular in the interior of the strip and obeying constant conformally invariant conditions on both boundaries. Depending on the values of the two boundary parameters these solutions may have different monodromy properties and are related to bound or scattering states. By Bohr-Sommerfeld quantization we find the quasiclassical discrete energy spectrum for the bound states in agreement with the corresponding limit of spectral data obtained previously by conformal bootstrap methods in Euclidean space. The full quantum version of the special vertex operator e ' in terms of free field exponentials is constructed in the hyperbolic sector.
TL;DR: In this paper, the authors describe the reduction procedure for a symplectic Lie algebroid by a Lie sub-group and a symmetry Lie group and obtain the corresponding reduced Hamiltonian dynamics.
Abstract: We describe the reduction procedure for a symplectic Lie algebroid by a Lie sub- algebroid and a symmetry Lie group. Moreover, given an invariant Hamiltonian function we obtain the corresponding reduced Hamiltonian dynamics. Several examples illustrate the generality of the theory.
TL;DR: In this paper, a bispectral family of polynomials in two variables was introduced as the eigenfunctions of the transition matrix for a nontrivial Markov chain due to M. Hoare and M. Rahman.
Abstract: In a very recent paper, M. Rahman introduced a remarkable family of polynomials in two variables as the eigenfunctions of the transition matrix for a nontrivial Markov chain due to M. Hoare and M. Rahman. I indicate here that these polynomials are bispectral. This should be just one of the many remarkable properties enjoyed by these polynomials. For several challenges, including finding a general proof of some of the facts displayed here the reader should look at the last section of this paper.
TL;DR: This article showed that the trigonometric solitons of the KP hierarchy enjoy a differential difference bispectral property, which becomes transparent when translated on two suitable spaces of pairs of matrices satisfying certain rank one conditions.
Abstract: We show that the trigonometric solitons of the KP hierarchy enjoy a differential- difference bispectral property, which becomes transparent when translated on two suitable spaces of pairs of matrices satisfying certain rank one conditions. The result can be seen as a non-self-dual illustration of Wilson's fundamental idea (Invent. Math. 133 (1998), 1-41) for understanding the (self-dual) bispectral property of the rational solutions of the KP hierarchy. It also gives a bispectral interpretation of a (dynamical) duality between the hyperbolic Calogero-Moser system and the rational Ruijsenaars-Schneider system, which was first observed by Ruijsenaars (Comm. Math. Phys. 115 (1988), 127-165).
TL;DR: The (2k)-th Gauss-Bonnet curvature is a generalization to higher dimensions of the 2k-dimensional GaussBonnet integrand, it coincides with the usual scalar curvature for k = 1.
Abstract: The (2k)-th Gauss-Bonnet curvature is a generalization to higher dimensions of the (2k)-dimensional Gauss-Bonnet integrand, it coincides with the usual scalar curvature for k = 1. The Gauss-Bonnet curvatures are used in theoretical physics to describe gravity in higher dimensional space times where they are known as the Lagrangian of Lovelock gravity, Gauss-Bonnet Gravity and Lanczos gravity. In this paper we present various aspects of these curvature invariants and review their variational properties. In particular, we discuss natural generalizations of the Yamabe problem, Einstein metrics and minimal submanifolds.
TL;DR: The linearizability of differential equations was first considered by Lie as discussed by the authors for scalar second order semi-linear ordinary differential equations and the connection between isometries and symmetries of the system of geodesic equations criteria were established for second order quadratically and cubically semilinear equations and for systems of equations.
Abstract: The linearizability of differential equations was first considered by Lie for scalar second order semi-linear ordinary differential equations. Since then there has been consi- derable work done on the algebraic classification of linearizable equations and even on sys- tems of equations. However, little has been done in the way of providing explicit criteria to determine their linearizability. Using the connection between isometries and symmetries of the system of geodesic equations criteria were established for second order quadratically and cubically semi-linear equations and for systems of equations. The connection was proved for maximally symmetric spaces and a conjecture was put forward for other cases. Here the criteria are briefly reviewed and the conjecture is proved.
TL;DR: In this paper, the authors provide background and motivation for the treatment of higher-dimensional systems (self-adjoint second-order partial differential operators) by semiclassical approximations and other methods.
Abstract: Quantum vacuum energy (Casimir energy) is reviewed for a mathematical audience as a topic in spectral theory. Then some one-dimensional systems are solved exactly, in terms of closed classical paths and periodic orbits. The relations among local spectral densities, energy densities, global eigenvalue densities, and total energies are demonstrated. This material provides background and motivation for the treatment of higher-dimensional systems (self-adjoint second-order partial differential operators) by semiclassical approxi- mation and other methods.
TL;DR: In this paper, a discretization of the n-dimensional generalisation of the Chaplygin sphere problem is presented, which preserves the same first integrals as the continuous model, except the energy.
Abstract: The celebrated problem of a non-homogeneous sphere rolling over a horizontal plane was proved to be integrable and was reduced to quadratures by Chaplygin. Applying the formalism of variational integrators (discrete Lagrangian systems) with nonholonomic constraints and introducing suitable discrete constraints, we construct a discretization of the n-dimensional generalization of the Chaplygin sphere problem, which preserves the same first integrals as the continuous model, except the energy. We then study the discretization of the classical 3-dimensional problem for a class of special initial conditions, when an analog of the energy integral does exist and the corresponding map is given by an addition law on elliptic curves. The existence of the invariant measure in this case is also discussed.