# zbMATH — the first resource for mathematics

Quantum completely integrable models in field theory. (English) Zbl 0569.35064
This paper reviews the first state of the quantum version of the inverse scattering transform (QIST) method to investigate a special class of two- dimensional models in quantum field theory and statistical physics. In the first section the main aspects of the classical IST method is explained using examples of the sine-Gordon equation, nonlinear Schrödinger equation, Heisenberg ferromagnet model and Toda chain. In the second section the general program of the quantum approach is presented. The quantum models corresponding to the above equations are discussed. The general considerations are illustrated in section 3 by using the nonlinear Schrödinger model. Two other examples are discussed more briefly in section 4: these are the sine-Gordon model and XYZ-model of the quantum theory of magnetism.
A further development of the QIST method is reviewed in: ”Integrable models in $$I+I$$ dimensional quantum field theory”, CEN Saclay (France), 50 p. (1982). ”Recent development in gauge theory and integrable systems”, Kyoto Univ. Res. Inst. Math. Sci., 53-71 (1982).
Reviewer: P.Holod

##### MSC:
 35Q99 Partial differential equations of mathematical physics and other areas of application 81T08 Constructive quantum field theory 82B10 Quantum equilibrium statistical mechanics (general)