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Form factors in completely integrable models of quantum field theory. (English) Zbl 0788.46077

Advanced Series in Mathematical Physics. 14. Singapore: World Scientific,. xi, 208 p. (1992).
Quantum field theory in two dimensions (one space one time dimension) is special in many ways. One important feature is the existence of completely integrable models which are nontrivial in the sense that their scattering operator (\(S\)-matrix) is not the identity. However, many- particle scattering is reduced to two-particle scattering owing to some factorizing property which in turn requires the validity of the Yang- Baxter triangle equation for the two-particle matrix elements. The latter condition puts the theory surprisingly close to the soluable lattice models of classical statistical mechanics in two dimensions. The present monograph concentrates on those aspects of the theory which deal with the so-called form factors using the Zamolodchikov-Faddeev approach. The term “form factor” is mainly used here to denote matrix elements of local operators between physical states of given particle momenta. The heart of the matter is seen in the formulation of four axioms that form factors should satisfy in order to guarantee locality (among other things) of the underlying field theory. More specifically, soliton form factors are studied for the sine-Gordon model and also form factors that relate to breather solutions. Then there is a chapter on the SU(2)-invariant Thirring model whose particles (so-called kinks) obey an unusual statistics: formally, their spins is 1/4. Still, S. S. Coleman proved that the sine-Gordon and the Thirring model are related by a transformation. Finally, form factors for the nonlinear O(3)-invariant \(\sigma\)-model are studied, and several formulas involving commutators in the sense of current algebra are derived.
The review seems competent and comprehensive. It will be of great help to those who look for a reliable source of the numerous detailed calculations that have been performed over the years by many experts.

MSC:

46N50 Applications of functional analysis in quantum physics
81T40 Two-dimensional field theories, conformal field theories, etc. in quantum mechanics
81T10 Model quantum field theories
46-02 Research exposition (monographs, survey articles) pertaining to functional analysis
81-02 Research exposition (monographs, survey articles) pertaining to quantum theory
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