Exact integrability in quantum field theory and statistical systems
The properties of exactly integrable two-dimensional quantum systems are reviewed and discussed. The nature of exact integrability as a physical phenomenon and various aspects of the mathematical formalism are explored by discussing several examples, including detailed treatments of the nonlinear Schroedinger (delta-function gas) model, the massive Thirring model, and the six-vertex (ice) model. The diagonalization of a Hamiltonian by Bethe's Ansatz is illustrated for the nonlinear Schroedinger model, and the integral equation method of Lieb for obtaining the spectrum of the many-body system from periodic boundary conditions is reviewed. Similar methods are applied to the massive Thirring model, where the fermion-antifermion and bound-state spectrum are obtained explicitly by the integral equation method. After a brief review of the classical inverse scattering method, the quantum inverse method for the nonlinear Schroedinger model is introduced and shown to be an algebraization of the Bethe Ansatz technique. In the quantum inverse method, an auxiliary linear problem is used to define nonlocal operators which are functionals of the original local field on a fixed-time string of arbitrary length. The particular operators for which the string is infinitely long (free boundary conditions) or forms a closed loop around a cylinder (periodic boundary conditions) correspond to the quantized scattering data and have a special significance. One of them creates the Bethe eigenstates, while the other is the generating function for an infinite number of conservation laws. The analogous operators on a lattice are constructed for the symmetric six-vertex model, where the object which corresponds to a solution of the auxiliary linear problem is a string of vertices contracted over horizontal links (arrows). The relationship between the quantum inverse method and the transfer matrix formalism is exhibited.
- Research Organization:
- Fermi National Accelerator Laboratory, P.O. Box 500, Batavia, Illinois 60510
- OSTI ID:
- 6615609
- Journal Information:
- Rev. Mod. Phys.; (United States), Vol. 53:2
- Country of Publication:
- United States
- Language:
- English
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71 CLASSICAL AND QUANTUM MECHANICS
GENERAL PHYSICS
QUANTUM FIELD THEORY
CONSERVATION LAWS
TWO-DIMENSIONAL CALCULATIONS
THIRRING MODEL
BOUND STATE
BOUNDARY CONDITIONS
EIGENSTATES
HAMILTONIANS
INTEGRAL TRANSFORMATIONS
INTEGRALS
INVERSE SCATTERING PROBLEM
NONLINEAR PROBLEMS
REVIEWS
SCHROEDINGER EQUATION
WAVE FUNCTIONS
DIFFERENTIAL EQUATIONS
DOCUMENT TYPES
EQUATIONS
FIELD THEORIES
FUNCTIONS
MATHEMATICAL OPERATORS
PARTIAL DIFFERENTIAL EQUATIONS
QUANTUM OPERATORS
TRANSFORMATIONS
WAVE EQUATIONS
645400* - High Energy Physics- Field Theory
657002 - Theoretical & Mathematical Physics- Classical & Quantum Mechanics