ATTEMPTS TO DEFINE QUASI-INTEGRABILITY

2012 ◽  
Vol 09 (06) ◽  
pp. 1261004 ◽  
Author(s):  
LUIZ A. FERREIRA ◽  
WOJTEK J. ZAKRZEWSKI
Keyword(s):  
Field Theory ◽  
Integrable Systems ◽  
Integrable Model ◽  
Conserved Quantities ◽  
Preliminary Results ◽  
Sine Gordon Model

In this paper we discuss some ideas on how to define the concept of quasi-integrability. Our ideas stem from the observation that many field theory models are "almost" integrable; i.e. they possess a large number of "almost" conserved quantities. Most of our discussion will involve a certain class of models which generalize the sine-Gordon model in (1 + 1) dimensions. As will be mentioned many field configurations of these models look like those of the integrable systems and so appear to be close to those in integrable model. We will then attempt to quantify these claims looking in particular, both analytically and numerically, at field configurations with scattering solitons. We will also discuss some preliminary results obtained in other models.

Lecture on SUSY Gauge Theories and Integrable Systems

1997 ◽  
Vol 11 (26n27) ◽  
pp. 3093-3124
Author(s):  
A. Marshakov
Keyword(s):  
Field Theory ◽  
Quantum Field Theory ◽  
Integrable Systems ◽  
Gauge Theories ◽  
Toda Chain ◽  
Quantum Field ◽  
Standard Quantum ◽  
Integrable Equations ◽  
Complex Curves ◽  
Periodic Toda Chain

I consider main features of the formulation of the finite-gap solutions to integrable equations in terms of complex curves and generating 1-differential. The example of periodic Toda chain solutions is considered in detail. Recently found exact nonperturbative solutions to [Formula: see text] SUSY gauge theories are formulated using the methods of the theory of integrable systems and where possible the parallels between standard quantum field theory results and solutions to the integrable systems are discussed.

Exact S-Matrices

2020 ◽  
pp. 676-743
Author(s):  
Giuseppe Mussardo
Keyword(s):  
Magnetic Field ◽  
Ising Model ◽  
Quantum Group ◽  
Scattering Theory ◽  
Integrable Model ◽  
Group Symmetry ◽  
Fusion Rules ◽  
Lee Model ◽  
Sine Gordon Model ◽  
Multiple Poles

The Ising model in a magnetic field is one of the most beautiful examples of an integrable model. This chapter presents its exact S-matrix and the exact spectrum of its excitations, which consist of eight particles of different masses. Similarly, it discusses the exact scattering theory behind the thermal deformation of the tricritical Ising model and the unusual features of the exact S-matrix of the non-unitary Yang–Lee model. Other examples are provided by O(n) invariant models, including the important Sine–Gordon model. It also discusses multiple poles, magnetic deformation, the E 8 Toda theory, bootstrap fusion rules, non-relativistic limits and quantum group symmetry of the Sine–Gordon model.

(1+1)-dimensional m-cKdV, g-cKdV integrable systems, and (2+1)-dimensional m-cKdV hierarchy

2008 ◽  
Vol 86 (12) ◽  
pp. 1367-1380 ◽  
Author(s):  
Y Zhang ◽  
H Tam
Keyword(s):  
Integrable Systems ◽  
Linear Functional ◽  
Hamiltonian Structure ◽  
Evolution Equations ◽  
Integrable Model ◽  
Lax Pair ◽  
The Self ◽  
Hamiltonian Structures ◽  
Yang Mills ◽  
Higher Dimensional

A few isospectral problems are introduced by referring to that of the cKdV equation hierarchy, for which two types of integrable systems called the (1 + 1)-dimensional m-cKdV hierarchy and the g-cKdV hierarchy are generated, respectively, whose Hamiltonian structures are also discussed by employing a linear functional and the quadratic-form identity. The corresponding expanding integrable models of the (1 + 1)-dimensional m-cKdV hierarchy and g-cKdV hierarchy are obtained. The Hamiltonian structure of the latter one is given by the variational identity, proposed by Ma Wen-Xiu as well. Finally, we use a Lax pair from the self-dual Yang–Mills equations to deduce a higher dimensional m-cKdV hierarchy of evolution equations and its Hamiltonian structure. Furthermore, its expanding integrable model is produced by the use of a enlarged Lie algebra.PACS Nos.: 02.30, 03.40.K

FROM INHOMOGENEOUS SIX-VERTEX LATTICE MODELS TO CONTINUUM QUANTUM INTEGRABLE MODELS

1992 ◽  
Vol 07 (25) ◽  
pp. 6385-6403
Author(s):  
Y.K. ZHOU
Keyword(s):  
Integrable Systems ◽  
Scaling Limit ◽  
Lattice Models ◽  
Integrable Models ◽  
Two Dimensional ◽  
Vertex Model ◽  
Quantum Integrable Systems ◽  
Vertex Models ◽  
Quantum Integrable Models ◽  
Sine Gordon Model

A method to find continuum quantum integrable systems from two-dimensional vertex models is presented. We explain the method with the example where the quantum sine-Gordon model is obtained from an inhomogeneous six-vertex model in its scaling limit. We also show that the method can be applied to other models.

scholarly journals BOUND STATE BOUNDARY S MATRIX OF THE SINE-GORDON MODEL

1994 ◽  
Vol 09 (27) ◽  
pp. 4801-4810 ◽  
Author(s):  
SUBIR GHOSHAL
Keyword(s):  
Field Theory ◽  
Bound States ◽  
Bound State ◽  
Two Dimensional ◽  
Integrable Field Theory ◽  
S Matrix ◽  
State Boundary ◽  
Sine Gordon Model ◽  
Integrable Field

We study the boundary S matrix for the reflection of bound states of the two-dimensional sine-Gordon integrable field theory in the presence of a boundary.

Perturbation Method for Boundary S-Matrix in 2D Quantum Field Theory

1997 ◽  
Vol 12 (38) ◽  
pp. 2951-2962 ◽  
Author(s):  
Nadia Topor
Keyword(s):  
Field Theory ◽  
Quantum Field Theory ◽  
Free Parameter ◽  
Exact Expression ◽  
Quantum Field ◽  
Boundary Interaction ◽  
S Matrix ◽  
Sine Gordon Model ◽  
The One ◽  
Perturbative Result

We develop a perturbation theory for evaluating the boundary S-matrix in 2D quantum field theory. We apply this approach to calculate the one-loop boundary S-matrix for the elementary particle of the sine–Gordon model with a boundary interaction. Our perturbative result agrees with the exact expression of the S-matrix conjectured by Goshal; it also allows us to derive the perturbative relation between the parameter ϑ in the S-matrix and the free parameter M in the boundary action, in the particular case in which its other free parameter φ0 is zero.

scholarly journals Conserved Quantities of Field Theory on Discrete Spacetime

1995 ◽  
Vol 93 (1) ◽  
pp. 173-184 ◽  
Author(s):  
H. Yamamoto ◽  
A. Hayashi ◽  
T. Hashimoto ◽  
M. Horibe
Keyword(s):  
Field Theory ◽  
Conserved Quantities ◽  
Discrete Spacetime

scholarly journals Chern-Simons field theory and completely integrable systems

1996 ◽  
Vol 378 (1-4) ◽  
pp. 175-180 ◽  
Author(s):  
L Martina ◽  
O.K Pashaev ◽  
G Soliani
Keyword(s):  
Field Theory ◽  
Integrable Systems ◽  
Completely Integrable Systems ◽  
Completely Integrable ◽  
Chern Simons
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