POISSON STRUCTURE AND INTEGRABILITY OF THE NEUMANN-MOSER-UHLENBECK MODEL

1990 ◽  
Vol 05 (23) ◽  
pp. 4477-4488 ◽  
Author(s):  
J. AVAN ◽  
M. TALON
Keyword(s):  
Poisson Structure ◽  
Point Of View ◽  
Conserved Quantities ◽  
Poisson Brackets ◽  
Quantum Level ◽  
Quantum Integrability ◽  
Commuting Operators ◽  
Lax Operator

Neumann’s model, describing the motion of a particle on an N-sphere under harmonic forces, is studied from the point of view of classical and quantum integrability. Classical integrability is derived from a generalized structure, “R-S couple” or “D-matrix” for the Poisson brackets of the Lax operator. The already-known set of conserved quantities for this model turns out to follow straightforwardly from this structure. It gives rise to a set of commuting operators at the quantum level, and the algebra of Lax operators directly follows from the classical one.

scholarly journals LIE-POISSON STRUCTURE OF CONFORMAL NON-ABELIAN THIRRING MODELS

1994 ◽  
Vol 09 (06) ◽  
pp. 483-489 ◽  
Author(s):  
OLEG A. SOLOVIEV
Keyword(s):  
Hamiltonian Systems ◽  
Poisson Structure ◽  
Algebraic Structure ◽  
Coupling Constants ◽  
Poisson Brackets ◽  
Quantum Level ◽  
Hamiltonian Method ◽  
Special Values ◽  
Lie Algebraic Structure ◽  
Hamiltonian Quantization

It is shown that non-Abelian Thirring models can be formulated as the Hamiltonian systems with Poisson brackets of the Lie algebraic structure. This fact allows Thirring models to be quantized by the Hamiltonian method. We show that the classical Lie-Poisson structure can be promoted to the quantum level in two different ways corresponding to different phases of non-Abelian Thirring models. There are special values of coupling constants at which the Hamiltonian quantization of Thirring models can be carried out consistently with the conformal invariance. These fixed couplings appear to be the solutions of the Virasoro master equation.

ALGEBRAIC STUDY ON THE POISSON STRUCTURE OF LAX OPERATOR OF B AND C TYPE AND SUPER LAX OPERATOR.

1992 ◽  
Vol 07 (supp01a) ◽  
pp. 405-418
Author(s):  
KAORU IKEDA
Keyword(s):  
Poisson Structure ◽  
Algebraic Structure ◽  
Lax Equation ◽  
Lax Operator

The Poisson structure of Lax operator of B and C type and super Lax operator which has odd parity are studied. The algebraic structure of Poisson structure as a background of Lax equation is thrown light on.

CANONICAL BÄCKLUND TRANSFORMATION FOR A DISCRETE INTEGRABLE CHAIN AND ITS ASSOCIATED PROPERTIES

2003 ◽  
Vol 18 (16) ◽  
pp. 1127-1139
Author(s):  
A. GHOSE CHOUDHURY ◽  
BARUN KHANRA ◽  
A. ROY CHOWDHURY
Keyword(s):  
Poisson Structure ◽  
Algebraic Structure ◽  
Bäcklund Transformation ◽  
Inverse Transformation ◽  
Strict Sense ◽  
Backlund Transformation ◽  
Lax Equation ◽  
R Matrix ◽  
Lax Operator ◽  
Time Part

The concept of a canonical Bäcklund transformation as laid down by Sklyanin is extended to a discrete integral chain, with a Poisson structure which is not canonical in the strict sense. The transformation is induced by an auxiliary Lax operator with a classical r-matrix which is similar in its algebraic structure to that of the original Lax operator governing the dynamics of the chain. Moreover, the transformation can be obtained from a suitable generating function. It is also shown how successive transformations can be composed to construct a new transformation. Finally an inverse transformation is also constructed. The compatibility of the transformation with the "time" part of the Lax equation is explicitly demonstrated. It is also shown that the Bianchi theorem of permutability holds good.

A DIRECT HEAT KERNEL METHOD FOR GENERATING THE CONSERVED QUANTITIES OF THE ZS-AKNS SYSTEM

1991 ◽  
Vol 06 (01) ◽  
pp. 163-170
Author(s):  
HANS J. WOSPAKRIK
Keyword(s):  
Heat Kernel ◽  
Kernel Method ◽  
Direct Method ◽  
Recursive Formula ◽  
Dimensional System ◽  
Conserved Quantities ◽  
Diagonal Part ◽  
Lax Operator ◽  
Akns System ◽  
Two Dimensional System

A heat kernel formulation of an isospectral Lax operator for generating the infinite conserved quantities of the ZS-AKNS two-dimensional system is presented. Using the Nepomechie direct method for computing the asymptotic expansion of the diagonal part of the heat kernel, one obtains a non-recursive formula for the conserved quantities.

scholarly journals A perspective on quantum integrability in many-body-localized and Yang–Baxter systems

2017 ◽  
Vol 375 (2108) ◽  
pp. 20160429 ◽  
Author(s):  
Joel E. Moore
Keyword(s):  
Conservation Laws ◽  
Local Equilibrium ◽  
Delta Function ◽  
Conserved Quantities ◽  
Many Body ◽  
Quantum Integrability ◽  
Hydrodynamic Evolution ◽  
Interacting Systems ◽  
Global Equilibrium ◽  
Quantum Integrable Models

Two of the most active areas in quantum many-particle dynamics involve systems with an unusually large number of conservation laws. Many-body-localized systems generalize ideas of Anderson localization by disorder to interacting systems. While localization still exists with interactions and inhibits thermalization, the interactions between conserved quantities lead to some dramatic differences from the Anderson case. Quantum integrable models such as the XXZ spin chain or Bose gas with delta-function interactions also have infinite sets of conservation laws, again leading to modifications of conventional thermalization. A practical way to treat the hydrodynamic evolution from local equilibrium to global equilibrium in such models is discussed. This paper expands upon a presentation at a discussion meeting of the Royal Society on 7 February 2017. The work described was carried out with a number of collaborators, including Jens Bardarson, Vir Bulchandani, Roni Ilan, Christoph Karrasch, Siddharth Parameswaran, Frank Pollmann and Romain Vasseur. This article is part of the themed issue ‘Breakdown of ergodicity in quantum systems: from solids to synthetic matter’.

scholarly journals Non-Integrable Dynamics and Physics Beyond the Standard Model

2021 ◽  
Author(s):  
Ervin Goldfain
Keyword(s):  
Standard Model ◽  
Chaotic Behavior ◽  
Beyond The Standard Model ◽  
Conserved Quantities ◽  
Poisson Brackets ◽  
Deterministic Models ◽  
Classical Systems ◽  
Mathematical Background ◽  
The Standard Model

The evolution of integrable classical systems leads to conserved quantities and vanishing Poisson brackets. In contrast, such invariants do not exist in the dynamics of non-integrable systems, which include (but are not limited to) deterministic models with long-term chaotic behavior. The object of this review is to briefly survey the mathematical background of nonintegrability and its role in the physics unfolding well-above the Standard Model (SM) scale.

scholarly journals Non-Integrable Dynamics and Physics Beyond the Standard Model

2021 ◽  
Author(s):  
Ervin Goldfain
Keyword(s):  
Standard Model ◽  
Chaotic Behavior ◽  
Beyond The Standard Model ◽  
Conserved Quantities ◽  
Poisson Brackets ◽  
Deterministic Models ◽  
Classical Systems ◽  
Mathematical Background ◽  
The Standard Model

The evolution of integrable classical systems leads to conserved quantities and vanishing Poisson brackets. In contrast, such invariants do not exist in the dynamics of non-integrable systems, which include (but are not limited to) deterministic models with long-term chaotic behavior. The object of this review is to briefly survey the mathematical background of nonintegrability and its role in the physics unfolding well-above the Standard Model (SM) scale.

scholarly journals EXACT BETHE ANSATZ SOLUTION OF NON-ULTRALOCAL QUANTUM mKdV MODEL

1995 ◽  
Vol 10 (38) ◽  
pp. 2955-2966 ◽  
Author(s):  
ANJAN KUNDU
Keyword(s):  
Eigenvalue Problem ◽  
Bethe Ansatz ◽  
Mkdv Equation ◽  
Generalized Equation ◽  
Baxter Equation ◽  
Quantum Level ◽  
Exact Integrability ◽  
Xxz Chain ◽  
Lax Operator ◽  
Bethe Ansatz Solution

A lattice regularized Lax operator for the non-ultralocal modified Korteweg de Vries (mKdV) equation is proposed at the quantum level with the basic operators satisfying a q-deformed braided algebra. From the associated quantum R and Z-matrices the exact integrability of the model is proved through the braided quantum Yang-Baxter equation which is a suitably generalized equation for the non-ultralocal models. Using the algebraic Bethe ansatz the eigenvalue problem of the quantum mKdV model is exactly solved and its connection with the spin-1/2 XXZ chain is established, facilitating the investigation of the corresponding conformal properties.

The Geosciences Applied to Lunar Exploration

1962 ◽  
Vol 14 ◽  
pp. 169-257 ◽  
Author(s):  
J. Green
Keyword(s):  
Lunar Surface ◽  
Probable Mechanism ◽  
Point Of View ◽  
Lunar Exploration ◽  
Geological Model ◽  
Apply Geophysics ◽  
Surface Features ◽  
The Impact

The term geo-sciences has been used here to include the disciplines geology, geophysics and geochemistry. However, in order to apply geophysics and geochemistry effectively one must begin with a geological model. Therefore, the science of geology should be used as the basis for lunar exploration. From an astronomical point of view, a lunar terrain heavily impacted with meteors appears the more reasonable; although from a geological standpoint, volcanism seems the more probable mechanism. A surface liberally marked with volcanic features has been advocated by such geologists as Bülow, Dana, Suess, von Wolff, Shaler, Spurr, and Kuno. In this paper, both the impact and volcanic hypotheses are considered in the application of the geo-sciences to manned lunar exploration. However, more emphasis is placed on the volcanic, or more correctly the defluidization, hypothesis to account for lunar surface features.

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