Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

Physica Scripta

Poisson structure and classical (r-s) matrix for SO(3) symmetric sine-Gordon model

A. Ghose Choudhury, Asesh Roy Chowdhury

Published Jun 1, 1993• 0 citations
Publisher

Abstract

Poisson Structure associated with the SO(3) symmetric sine-Gordon system is analysed. The system though similar to WZW model, yet is seen to give rise to Poisson-brackets of monodromy matrices which are cubic in the matrix itself. This may be due to the different non-ultralocal character of the bracket { L (λ) ⊗ , L (μ)}. Lastly those of the scattering data is seen to have the usual ( r, s ) structure.

DOI
10.1088/0031-8949/47/6/003

Related papers

Classical Poisson Bracket Algebra and Quantum Affine Gaudin Model

Asesh Roy Chowdhury, A. Ghose Chowdhury

We have analysed an affine Gaudin Model with the help of a L -operator with entries belonging to affine Gaudin algebra. It is observed that the usual classical r -matrix does not exist, but an extended version due to Maillet and Vega (called the ( r - s ) algebra) can be defined. The subsequent quadratic algebra of the monodromy matrices are then used to set up the Algebraic Bethe Ansatz for the model. Relation to other similar model are discussed.

Journal of the Physical Society of Japan•Jan 15, 1994•13 citations

ALGEBRAIC BETHE ANSATZ SCHEME FOR RELATIVISTIC INTEGRABLE FIELD THEORIES IN CONTINUUM-: SINE-GORDON MODEL

Gautam Bhattacharya, Sasanka Ghosh

The linear problem associated with the Lax operator of the classical sine-Gordon theory can be recast into the monodromy matrix form that can be extended to quantum theory as well. Product of the quantum monodromy matrices will have contributions from the singularities arising out of the operator product expansions of sine-Gordon field. This enables one to find the star-triangle relations. This is a generalization of the method used by Thacker for the non-relativistic nonlinear Schrödinger field theory. In the infinite volume limit, it leads to an unambiguous description of the algebra involving the scattering data operators. Starting from a vacuum the module of physical states are constructed by the application of chains of the scattering operators and they turn out to have definite eigenvalues of energy and momentum. The mass-spectra and two-particle S matrix elements can be calculated straightforwardly and they coincide with the well-known results of Zamolodchikov and Zamolodchikov. It is also shown that the particle states can be created by a generalized Jordon-Wigner transformation of the free fermionic creation operators — thus establishing a link between the S matrix element and certain co-cycles.

International Journal of Modern Physics A•Feb 1, 1989•6 citations

Classical r -matrix structure for elliptic Ruijsenaars chain and 1 + 1 field analogue of Ruijsenaars–Schneider model

Dmitrii Murinov, Andrei Vladimirovich Zotov

Abstract The classical dynamical r -matrix structure for the periodic elliptic Ruijsenaars chain is described. The Poisson brackets for the monodromy matrix are calculated as well, thus providing Liouville integrability of the model. Next, we study its continuous non-relativistic limit and reproduce the Maillet type non-ultralocal r -matrix structure for the field analogue of the elliptic Calogero–Moser model.

arXiv:2508.12656•Journal of Physics A Mathematical and Theoretical•Dec 8, 2025•2 citations

On the Poisson-Bracket algebra for the scattering data of a non-ultralocal field theory with soliton solutions

Asesh Roy Chowdhury, Shibani Sen

A preliminary investigation is done about the Poisson-Bracket algebra obeyed by the scattering data associated with a non-ultralocal field theory sustaining soliton solutions. The result is quite distinct from the usual Poisson-bracket relations usually observed in ultralocal field theory. This important difference is reflected in the changed Zamoldachikov algebra of the R ( K )-reflection operator, crucial in the quantization progress of the nonlinear system in two dimensions. Our results though not conclusive, yet is useful in the future progress of quantization of such field theories.

Physica Scripta•Jul 1, 1987•2 citations

Algebra of quantum mechanics via classical phonons. II: Klein-Gordon dynamics, the Heisenberg formalism, the Dirac canonical commutation rule and the Poincare algebra through the continuous Poisson bracket formalism

Emmanuel Giner

In the first part of this series we have shown how the Schrodinger equation for a single particle and the corresponding non relativistic quantum observables can be obtained from a purely classical phonon model through the Newtonian equations of motion. In this work we focus instead on how the classical Hamiltonian formalism applied to the same phonon system allows to recover the feature of relativistic quantum mechanics for a single spinless particle. Using the classical nature of the phonon model, we naturally define continuous Poisson brackets between classical observables, which allows to recover the dynamics of such observables, i.e. the Ehrenfest relations associated to real-valued Klein-Gordon fields. The Poisson brackets also permits to obtain the generic form of constants of motions, thus generalizing the concept of inner products and momentum on Klein-Gordon fields. We then connect the formalism of real-valued classical functionals with that of hermitian operators and complex-valued wave functions. This is done through the introduction of a non-local complex-valued change of variables which allows to rewrite the real-valued Klein-Gordon equation in a form akin to the Schrodinger equation, and the classical observables as quantum expectation values. Then, we show how this change of variables allows to rewrite the classical Poisson brackets as commutators of hermitian operators. This points out the strict equivalence between the Heisenberg formalism and the formalism of classical Poisson bracket. Eventually, we illustrate how the Poisson brackets allows to recover the transformations of Poincare group in 1+1 dimension together with its algebra. The latter makes the link between the Lorentz invariant inner product of Mostafazadeh and the Casimir invariant associated to the mass of particle.

quant-pharXiv:2607.03909•Jul 7, 2026•0 citations

Classical r-matrix, new integrable system and finite boundary condition

A Gouse Choudhury, Asesh Roy Chowdhury

We have analysed the classical r-matrix structure of a new integrable model in two-dimensional coupled Liouville-Thirring model. Due to the non-ultralocal character of the system, a new form of (r, -s) structure is obtained. It is also proved that the integrability of the model is not destroyed if non-trivial finite boundary conditions are imposed. An equation determining the form of the matrices K + and K - is deduced which is a simple generalization of that of Sklyanin for the ultralocal case.

Journal of Physics A Mathematical and General•Jan 21, 1995•1 citation
Poisson Structure associated with the SO(3) symmetric sine-Gordon system is analysed. The system though similar to WZW model, yet is seen to give rise to Poisson-brackets of monodromy matrices which are cubic in the matrix itself. This may be due to the different non-ultralocal character of the bracket { L (λ) ⊗ , L (μ)}. Lastly those of the scattering data is seen to have the usual ( r, s ) structure.
The linear problem associated with the Lax operator of the classical sine-Gordon theory can be recast into the monodromy matrix form that can be extended to quantum theory as well. Product of the quantum monodromy matrices will have contributions from the singularities arising out of the operator product expansions of sine-Gordon field. This enables one to find the star-triangle relations. This is a generalization of the method used by Thacker for the non-relativistic nonlinear Schrödinger field theory. In the infinite volume limit, it leads to an unambiguous description of the algebra involving the scattering data operators. Starting from a vacuum the module of physical states are constructed by the application of chains of the scattering operators and they turn out to have definite eigenvalues of energy and momentum. The mass-spectra and two-particle S matrix elements can be calculated straightforwardly and they coincide with the well-known results of Zamolodchikov and Zamolodchikov. It is also shown that the particle states can be created by a generalized Jordon-Wigner transformation of the free fermionic creation operators — thus establishing a link between the S matrix element and certain co-cycles.
A preliminary investigation is done about the Poisson-Bracket algebra obeyed by the scattering data associated with a non-ultralocal field theory sustaining soliton solutions. The result is quite distinct from the usual Poisson-bracket relations usually observed in ultralocal field theory. This important difference is reflected in the changed Zamoldachikov algebra of the R ( K )-reflection operator, crucial in the quantization progress of the nonlinear system in two dimensions. Our results though not conclusive, yet is useful in the future progress of quantization of such field theories.
In the first part of this series we have shown how the Schrodinger equation for a single particle and the corresponding non relativistic quantum observables can be obtained from a purely classical phonon model through the Newtonian equations of motion. In this work we focus instead on how the classical Hamiltonian formalism applied to the same phonon system allows to recover the feature of relativistic quantum mechanics for a single spinless particle. Using the classical nature of the phonon model, we naturally define continuous Poisson brackets between classical observables, which allows to recover the dynamics of such observables, i.e. the Ehrenfest relations associated to real-valued Klein-Gordon fields. The Poisson brackets also permits to obtain the generic form of constants of motions, thus generalizing the concept of inner products and momentum on Klein-Gordon fields. We then connect the formalism of real-valued classical functionals with that of hermitian operators and complex-valued wave functions. This is done through the introduction of a non-local complex-valued change of variables which allows to rewrite the real-valued Klein-Gordon equation in a form akin to the Schrodinger equation, and the classical observables as quantum expectation values. Then, we show how this change of variables allows to rewrite the classical Poisson brackets as commutators of hermitian operators. This points out the strict equivalence between the Heisenberg formalism and the formalism of classical Poisson bracket. Eventually, we illustrate how the Poisson brackets allows to recover the transformations of Poincare group in 1+1 dimension together with its algebra. The latter makes the link between the Lorentz invariant inner product of Mostafazadeh and the Casimir invariant associated to the mass of particle.