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American Mathematical Society eBooks

Integrable systems : from classical to quantum : proceedings of the 38th session of the séminaire de mathématiques supérieures, July 26 - August 6, 1999 Montréal, Québec, Canada

J. Harnad, Gert Sabidussi, Pavel Winternitz A

Published Jan 1, 2000• 0 citations
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Abstract

On the chiral WZNW phase space, exchange r-matrices and Poisson-Lie groupoids by J. Balog, L. Feher, and L. Palla Loop groups, $R$-matrices and separation of variables by J. Harnad The geometry of generalised Hitchin systems by J. C. Hurtubise Determinant representation for form factors by V. E. Korepin Isomonodromic deformations in genus zero and one: Algebro-geometric solutions and Schlesinger transformations by D. A. Korotkin Quantum inverse scattering problem and correlation functions of integrable models by J.-M. Maillet Multiseparability and superintegrability for classical and quantum systems by W. Miller, Jr. Integrability and symmetry of the XXZ model by T. Miwa Characteristic systems on Poisson Lie groups and their quantization by N. Reshetikhin Special functions associated with Calogero-Moser type quantum systems by S. N. M. Ruijsenaars Backlund transformations and Baster's $Q$-operator by E. K. Sklyanin Universality of the distribution functions of random matrix theory by C. A. Tracy and H. Widom.

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Perspectives on quantization : proceedings of a 1996 AMS-IMS-SIAM Joint Summer Research Conference, July 7-11, 1996, Mt. Holyoke College

L. A. Coburn, Marc A. Rieffel

Discrete series representations and integration over boundary orbits of symmetric domains by J. Arazy and H. Upmeier Microlocal techniques for semiclassical problems in geometric quantization by D. Borthwick A non-Gaussian fixed point for the renormalization group by J. Dimock Quantum mechanics in phase space by B. C. Hall Nonstandard quantum groups associated to certain Belavin-Drinfeld triples by T. J. Hodges Ergodic theorems for quantum Kronecker flows by S. Klimek and A. Lesniewski Quantum maps by S. Klimek and A. Lesniewski The relationship between classical mechanics and quantum mechanics by G. W. Mackey Deformation quantization and $K$-theory by G. Nagy A $q$-analogue of the first homology group of a 3-manifold by J. H. Przytycki Constructive non-linear quantum field theory in four space-time dimensions by I. Segal Groupoids and quantization by A. J. L. Sheu Quantization, symmetries, and relativity by A. Unterberger Quantizing Poisson manifolds by A. A. Voronov.

American Mathematical Society eBooks•Jan 1, 1998•0 citations

Quantum integrable systems

Steve Zelditch

NONLINEAR SYSTEMS AND CLASSICAL IST Introduction Definition of Integrability Lax Pair Technique Inverse Scattering Transform Hamiltonian Structure COORDINATE BETHE ANSATZ Introduction Nonlinear Systems and the CBA Fermionic System Boundary Condition in Bethe Ansatz Heisenberg Spin Chain Spin of the Bethe Ansatz State Other Integrable Models YANG-BAXTER EQUATION Introduction General Description Factorized Scattering Baxter's Star Triangle Relation Vertex Models Reflection Equation Algebra CONTINUOUS INTEGRABLE SYSTEMS Introduction Quantum Continuous Integrable Systems Conserved Quantities Nonultralocal systems and the YBE Operator Product Expansion and YBE Finite Boundary Conditions Modified Classical Yang-Baxter Equation ALGEBRAIC BETHE ANSATZ Introduction Discrete Self Trapping Model Asymmetric XXZ Model in a Magnetic Field Analytical Bethe Ansatz Off-Shell Bethe Ansatz Nested Bethe Ansatz Fusion Procedure Fusion Procedure for Open chains Fusion Procedure for Transfer Matrices Application of Fusion Procedure INTEGRABLE LONG-RANGE MODELS Introduction Long-Range Models from the ABA Symmetry Transformation Calogero-Moser Models SEPARATION OF VARIABLES Introduction Hamilton-Jacobi Equation Sklyanin's Method for SoV Goryachev-Chaplygin Top Quantum Case and the Role of Lie Algebra Bi-Hamiltonian Structure and SoV SoV for GCM Model SoV and Boundary Conditions BACKLUND TRANSFORMATIONS Introduction Permutability Theorem Backlund Transformations and Classical Inverse Scattering Backlund Transformations from Riccati Equation Darboux Backlund Transformations The Exponential Lattice Canonical Transformations Group Property of Backlund Transformations Recent Developments in Backlund Transformation Theory Sklyanin's Formalism for Canonical Backlund Transformations Extended Phase Space Method Quantization of Backlund Transformations Method of Projection Operators QUANTUM GLM EQUATION Introduction Quantum GLM Equation Quantum Floquet Function Exact Quantization Quantum GLM Equation in a Continuous System Bound States and an Alternative Approach APPENDICES Direct Product Calculus Grassman Algebra Bethe Ansatz Equation AKNS Problem BIBLIOGRAPHY INDEX

Regional conference series in mathematics•Dec 12, 2017•18 citations

Integrable quantum field theories and their applications : proceedings of the APCTP Winter School : Cheju Island, Korea, 28 February-4 March 2000

Changrim Ahn, Chaiho P. Rim, 隆 佐々木

Applications of Reflection Amplitudes in Toda-Type Theories (C Ahn et al.) Lax Pairs and Involutive Hamiltonians for CN and BCN Ruijsenaars-Schneider Models (K Chen et al.) Fateev's Models and Their Applications (D Controzzi & A M Tsvelik) The ODE/IM Correspondence (P Dorey et al.) Integrable Sigma Models (P Fendley) Lorentz Lattice Gases and Spin Chains (M J Martins) Quantum Calogero-Moser Models for Any Root System (R Sasaki) Quasi-Particles in Conformal Field Theories for Fractional Quantum Hall Systems (K Schoutens & R A J van Elburg) Towards Form Factors in Finite Volume (F A Smirnov) Static and Dynamic Properties of Trapped Bost-Einstein Condensates (T Tsurumi et al.) Integrability of the Calogero Model: Conserved Quantitles, the Classical General Solution and the Quantum Orthogonal Basis (H Ujino et al.) Conformal Boundary Conditions (J B Zuber).

WORLD SCIENTIFIC eBooks•Jan 1, 2001•0 citations

Quantization and coherent states methods : proceedings of the XI Workshop on Geometric Methods in Physics : Białowieza, Poland, 7-14 July 1992

S. Twareque Ali, Ivaïlo M. Mladenov, Anatol Odzijewicz

Half-forms and Berry's phase, N.M.J. Woodhouse quantum particle on a Torus in a magnetic field, H.D. Doebner and J. Tolar loop spaces and a geometrical approach to path integral quantization, R.F. Picken on the connection between orbits and representations, N.V. Pedersen geometric quantization of Riemann rotors, G. Rosensteel and E. Ihrig large on the decomposition of the oscillator representation, A. Strasburger symplectic and Kahler coherent state representations of unimodular Lie groups, W. Lisiecki local and non-local probabilities in Einstein-Podolsky-Rosen correlations, K. Wodkiewicz the Lorentz force equation in twistor terms - a symplectic framework, A. Bette Green's function for crossed time-dependent-electric and magnetic fields phase-space quantum mechanics approach, L.M. Nieto quantization on coadjoint orbits of diffeomorphism groups - some research directions, G.A. Goldin semiclassical behaviour of coherent states on the circle, S. de Bievre and J.A. Gonzalez.

•Jan 1, 1993•0 citations

Baecklund transformations and Baxter's Q-operator

Evgeny Konstantinovich Sklyanin

The course of 5 lectures given at the seminar "Integrable Systems: from Classical to Quantum" (Universite de Montreal, Jul 26 -- Aug 6, 1999) contains a detailed comment on the recently discovered (Gaudin-Pasquier, 1992) connection between Baecklund transformations in the theory of classical integrable systems on one hand, and Baxter's Q-operator for quantum integrable systems, on the other hand. We restrict our attention to the systems with finite number of degrees of freedom. Our main illustrative example is the periodic Toda lattice. We present a general construction of Q-operator for models governed by the SL(2)-invariant R-matrix and apply it to our example. We discuss also applications of BT and Q-operators to the separation of variables and theory of special functions.

arXiv:nlin/0009009•Sep 4, 2000•10 citations

Classical functional Bethe ansatz for SL(N): Separation of variables for the magnetic chain

D. R. D. Scott

It is desirable for one to be able to construct separation variables for systems with SL(N) R matrices or Lax pairs. This article relates and extends the two known methods of construction. The Sklyanin’s functional Bethe ansatz (FBA) [Commun. Math. Phys. 150, 181–191 (1992)] was previously only known for SL(2) and SL(3) (and associated) R matrices. The algebraic geometric method of Adams, Harnad, and Hurtubise [Commun. Math. Phys. 155, 385–413 (1993)] has been shown to work for SL(N) but up until now only for linear Poisson brackets. In this article Sklyanin’s program is advanced by giving the FBA for certain systems with SL(N) R matrices. This is achieved by constructing rational functions 𝒜(u) and ℬ(u) of the matrix elements of T(u), so that, in the generic case, the zeros xi of ℬ(u) are the separation coordinates and the Pi=𝒜(xi) provide their conjugate momenta. It is shown that the separation variables thus defined are the same as those given (as the simultaneous solutions of several equations in two variables) by the other method. The crucial calculation of Adams et al. of the commutation relations is also adapted to the case of quadratic Poisson brackets. The method is illustrated with the magnetic chain and the Gaudin model, and its wider applicability is discussed. No knowledge of algebraic geometry is required.

arXiv:hep-th/9403030•Journal of Mathematical Physics•Nov 1, 1994•38 citations
On the chiral WZNW phase space, exchange r-matrices and Poisson-Lie groupoids by J. Balog, L. Feher, and L. Palla Loop groups, RRR-matrices and separation of variables by J. Harnad The geometry of generalised Hitchin systems by J. C. Hurtubise Determinant representation for form factors by V. E. Korepin Isomonodromic deformations in genus zero and one: Algebro-geometric solutions and Schlesinger transformations by D. A. Korotkin Quantum inverse scattering problem and correlation functions of integrable models by J.-M. Maillet Multiseparability and superintegrability for classical and quantum systems by W. Miller, Jr. Integrability and symmetry of the XXZ model by T. Miwa Characteristic systems on Poisson Lie groups and their quantization by N. Reshetikhin Special functions associated with Calogero-Moser type quantum systems by S. N. M. Ruijsenaars Backlund transformations and Baster's QQQ-operator by E. K. Sklyanin Universality of the distribution functions of random matrix theory by C. A. Tracy and H. Widom.
Discrete series representations and integration over boundary orbits of symmetric domains by J. Arazy and H. Upmeier Microlocal techniques for semiclassical problems in geometric quantization by D. Borthwick A non-Gaussian fixed point for the renormalization group by J. Dimock Quantum mechanics in phase space by B. C. Hall Nonstandard quantum groups associated to certain Belavin-Drinfeld triples by T. J. Hodges Ergodic theorems for quantum Kronecker flows by S. Klimek and A. Lesniewski Quantum maps by S. Klimek and A. Lesniewski The relationship between classical mechanics and quantum mechanics by G. W. Mackey Deformation quantization and KKK-theory by G. Nagy A qqq-analogue of the first homology group of a 3-manifold by J. H. Przytycki Constructive non-linear quantum field theory in four space-time dimensions by I. Segal Groupoids and quantization by A. J. L. Sheu Quantization, symmetries, and relativity by A. Unterberger Quantizing Poisson manifolds by A. A. Voronov.
NONLINEAR SYSTEMS AND CLASSICAL IST Introduction Definition of Integrability Lax Pair Technique Inverse Scattering Transform Hamiltonian Structure COORDINATE BETHE ANSATZ Introduction Nonlinear Systems and the CBA Fermionic System Boundary Condition in Bethe Ansatz Heisenberg Spin Chain Spin of the Bethe Ansatz State Other Integrable Models YANG-BAXTER EQUATION Introduction General Description Factorized Scattering Baxter's Star Triangle Relation Vertex Models Reflection Equation Algebra CONTINUOUS INTEGRABLE SYSTEMS Introduction Quantum Continuous Integrable Systems Conserved Quantities Nonultralocal systems and the YBE Operator Product Expansion and YBE Finite Boundary Conditions Modified Classical Yang-Baxter Equation ALGEBRAIC BETHE ANSATZ Introduction Discrete Self Trapping Model Asymmetric XXZ Model in a Magnetic Field Analytical Bethe Ansatz Off-Shell Bethe Ansatz Nested Bethe Ansatz Fusion Procedure Fusion Procedure for Open chains Fusion Procedure for Transfer Matrices Application of Fusion Procedure INTEGRABLE LONG-RANGE MODELS Introduction Long-Range Models from the ABA Symmetry Transformation Calogero-Moser Models SEPARATION OF VARIABLES Introduction Hamilton-Jacobi Equation Sklyanin's Method for SoV Goryachev-Chaplygin Top Quantum Case and the Role of Lie Algebra Bi-Hamiltonian Structure and SoV SoV for GCM Model SoV and Boundary Conditions BACKLUND TRANSFORMATIONS Introduction Permutability Theorem Backlund Transformations and Classical Inverse Scattering Backlund Transformations from Riccati Equation Darboux Backlund Transformations The Exponential Lattice Canonical Transformations Group Property of Backlund Transformations Recent Developments in Backlund Transformation Theory Sklyanin's Formalism for Canonical Backlund Transformations Extended Phase Space Method Quantization of Backlund Transformations Method of Projection Operators QUANTUM GLM EQUATION Introduction Quantum GLM Equation Quantum Floquet Function Exact Quantization Quantum GLM Equation in a Continuous System Bound States and an Alternative Approach APPENDICES Direct Product Calculus Grassman Algebra Bethe Ansatz Equation AKNS Problem BIBLIOGRAPHY INDEX
Applications of Reflection Amplitudes in Toda-Type Theories (C Ahn et al.) Lax Pairs and Involutive Hamiltonians for CN and BCN Ruijsenaars-Schneider Models (K Chen et al.) Fateev's Models and Their Applications (D Controzzi & A M Tsvelik) The ODE/IM Correspondence (P Dorey et al.) Integrable Sigma Models (P Fendley) Lorentz Lattice Gases and Spin Chains (M J Martins) Quantum Calogero-Moser Models for Any Root System (R Sasaki) Quasi-Particles in Conformal Field Theories for Fractional Quantum Hall Systems (K Schoutens & R A J van Elburg) Towards Form Factors in Finite Volume (F A Smirnov) Static and Dynamic Properties of Trapped Bost-Einstein Condensates (T Tsurumi et al.) Integrability of the Calogero Model: Conserved Quantitles, the Classical General Solution and the Quantum Orthogonal Basis (H Ujino et al.) Conformal Boundary Conditions (J B Zuber).
Half-forms and Berry's phase, N.M.J. Woodhouse quantum particle on a Torus in a magnetic field, H.D. Doebner and J. Tolar loop spaces and a geometrical approach to path integral quantization, R.F. Picken on the connection between orbits and representations, N.V. Pedersen geometric quantization of Riemann rotors, G. Rosensteel and E. Ihrig large on the decomposition of the oscillator representation, A. Strasburger symplectic and Kahler coherent state representations of unimodular Lie groups, W. Lisiecki local and non-local probabilities in Einstein-Podolsky-Rosen correlations, K. Wodkiewicz the Lorentz force equation in twistor terms - a symplectic framework, A. Bette Green's function for crossed time-dependent-electric and magnetic fields phase-space quantum mechanics approach, L.M. Nieto quantization on coadjoint orbits of diffeomorphism groups - some research directions, G.A. Goldin semiclassical behaviour of coherent states on the circle, S. de Bievre and J.A. Gonzalez.
The course of 5 lectures given at the seminar "Integrable Systems: from Classical to Quantum" (Universite de Montreal, Jul 26 -- Aug 6, 1999) contains a detailed comment on the recently discovered (Gaudin-Pasquier, 1992) connection between Baecklund transformations in the theory of classical integrable systems on one hand, and Baxter's Q-operator for quantum integrable systems, on the other hand. We restrict our attention to the systems with finite number of degrees of freedom. Our main illustrative example is the periodic Toda lattice. We present a general construction of Q-operator for models governed by the SL(2)-invariant R-matrix and apply it to our example. We discuss also applications of BT and Q-operators to the separation of variables and theory of special functions.
It is desirable for one to be able to construct separation variables for systems with SL(N) R matrices or Lax pairs. This article relates and extends the two known methods of construction. The Sklyanin’s functional Bethe ansatz (FBA) [Commun. Math. Phys. 150, 181–191 (1992)] was previously only known for SL(2) and SL(3) (and associated) R matrices. The algebraic geometric method of Adams, Harnad, and Hurtubise [Commun. Math. Phys. 155, 385–413 (1993)] has been shown to work for SL(N) but up until now only for linear Poisson brackets. In this article Sklyanin’s program is advanced by giving the FBA for certain systems with SL(N) R matrices. This is achieved by constructing rational functions 𝒜(u) and ℬ(u) of the matrix elements of T(u), so that, in the generic case, the zeros xi of ℬ(u) are the separation coordinates and the Pi=𝒜(xi) provide their conjugate momenta. It is shown that the separation variables thus defined are the same as those given (as the simultaneous solutions of several equations in two variables) by the other method. The crucial calculation of Adams et al. of the commutation relations is also adapted to the case of quadratic Poisson brackets. The method is illustrated with the magnetic chain and the Gaudin model, and its wider applicability is discussed. No knowledge of algebraic geometry is required.