Quantum Integrable Systems: 1st Edition (Paperback) book cover

Quantum Integrable Systems

1st Edition

By Asesh Roy Chowdhury, Aninlya Ghose Choudhury

Chapman and Hall/CRC

424 pages | 36 B/W Illus.

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pub: 2004-01-28
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Description

The study of integrable systems has opened new horizons in classical physics over the past few decades, particularly in the subatomic world. Yet despite the field now having reached a level of maturity, very few books provide an introduction to the field accessible to specialists and nonspecialists alike, and none offer a systematic survey of the more recent advances.

This book presents and clarifies the developments of the last ten years in quantum integrable systems. After a preliminary discussion of the fundamentals of classical nonlinear integrable systems, the authors explore the quantum domain. Their approach emphasizes physical systems and the use of concrete examples, and they take care to establish the relationship between new and older methods. The presentation includes the first comprehensive discussion of the quantum Bäcklund transformation Q-operator and various techniques related to algebraic Bethe Ansatz that are not available elsewhere in book form.

In Quantum Integrable Systems, researchers active in the field have an up-to-date source for recent advances and new techniques, and nonspecialists finally have an accessible introduction to the concepts and basic tools they need to explore and exploit the wide-ranging applicability of the subject.

Reviews

"The authors present the recent studies of the quantum inverse scattering method and problems related to it."

-Zentralblatt MATH

Table of Contents

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

BÄCKLUND TRANSFORMATIONS

Introduction

Permutability Theorem

Bäcklund Transformations and Classical Inverse Scattering

Bäcklund Transformations from Riccati Equation

Darboux Bäcklund Transformations

The Exponential Lattice

Canonical Transformations

Group Property of Bäcklund Transformations

Recent Developments in Bäcklund Transformation Theory

Sklyanin's Formalism for Canonical Bäcklund

Transformations

Extended Phase Space Method

Quantization of Bäcklund 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

About the Series

Chapman & Hall/CRC Research Notes in Mathematics Series

Learn more…

Subject Categories

BISAC Subject Codes/Headings:
SCI040000
SCIENCE / Mathematical Physics
SCI057000
SCIENCE / Quantum Theory