General principles of quantum field theory. Transl. from the Russian by G. G. Gould. (English) Zbl 0732.46040

Bogolyubov, N. N.; Logunov, A. A.; Oksak, A. I.; Todorov, I. T.

Mathematical Physics and Applied Mathematics, 10. Dordrecht etc.: Kluwer Academic Publishers. xi, 694 p. Dfl. 420.00; £140.00; $ 198.00 (1990).

This book provides a rather comprehensive account of general quantum field theory, with an emphasis of model-independent methods. It is a considerably expanded revision of “Introduction to axiomatic quantum field theory” (Bogolubov, Logunov and Todorov, Benjamin/Cummings, 1975).
Compared with this earlier book, some of the new features are some emphasis on algebraic view point, due attention to the gauge theories with indefinite metric, inclusion of some soluble examples in 2-space- time dimension, discussion of euclidean Green’s functions (Schwinger functions), detailed discussion of analytic properties and high energy behaviors of scattering amplitudes.
The original Russian book was published in 1987. The book consists of 5 parts and 17 chapters.
Part I, occupying about 230 pages, is an account of various mathematical methods such as loclly convex spaces, Hilbert spaces and linear operators, and \(C^*\)-algebras in Chapter 1, generalized functions in Chapter 2, Lorentz covariant generalized functions in Chapter 3 with an appendix on Lie groups and their representations, the Jost-Lehmann-Dyson representation in Chapter 4 and analytic functions of several complex variables in Chapter 5.
Part II is an account of algebraic formulation of quantum theory in Chapter 6 and of relativistic invariance in Chapter 7 including description of Fock spaces.
Part III is an account of local quantum field theory with the Wightman formalism in Chapter 8, various discussions using analytic properties of Weightman functions in the coordinate space in Chapter 9 including TCP- theorem, Borchers classes, connection between spin and statistics, equal- time commutation relations, Haag’s theorem, Euclidean Green’s functions, parastatistics and infinite components fields, somewhat short discussion of fields in an indefinite metric in Chapter 10 and explicitly soluble two-dimensional models in Chapter 11.
Part IV is an account of collision theory including Haag-Ruelle scattering theory in Chapter 12, LSZ formalism in Chapter 13 and the S- matrix method in Chapter 14.
Part V is an account of the analytic properties of the scattering amplitudes as a consequence of causality and the spectral property, including discussion of Lehmann small ellipse and dispersion relations in Chapter 15, analytic propertiesof the 4-point Green’s function in Chapter 16 and consequences for high-energy behaviors of cross sections in Chapter 17.
Commentary on the bibliography and references along with a rather exhaustive list of references takes over 50 pages and should be very useful.

Reviewer: H.Araki (Kyoto)

MSC:

46N50	Applications of functional analysis in quantum physics
47N50	Applications of operator theory in the physical sciences
81T05	Axiomatic quantum field theory; operator algebras
46-02	Research exposition (monographs, survey articles) pertaining to functional analysis
81U20	\(S\)-matrix theory, etc. in quantum theory
81-02	Research exposition (monographs, survey articles) pertaining to quantum theory
46L60	Applications of selfadjoint operator algebras to physics

Keywords:

general quantum field theory; gauge theories with indefinite metric; euclidean Green’s functions; Schwinger functions; loclly convex spaces; Hilbert spaces; \(C^ *\)-algebras; Lorentz covariant generalized functions; Lie groups and their representations; Jost-Lehmann-Dyson representation; relativistic invariance; Fock spaces; local quantum field theory with the Wightman formalism; Weightman functions; TCP-theorem; Borchers classes; spin and statistics; equal-time commutation relations; Haag’s theorem; Euclidean Green’s functions; parastatistics; infinite components fields; collision theory; Haag-Ruelle scattering theory; LSZ formalism; S-matrix method; scattering amplitudes; causality; spectral property; Lehmann small ellipse; dispersion relations; 4-point Green’s function; high-energy behaviors of cross sections