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The special theory of relativity: a mathematical exposition. (English) Zbl 0816.53056
Universitext. New York, NY: Springer-Verlag. xii, 214 p. (1993).
The book presents a university course devoted to special relativity, relativistic quantum mechanics and classical field theories given in a modern rigorous mathematical exposition. It contains the axioms of the pseudo-Euclidean four-dimensional vector space $$V_ 4$$, and definitions of Minkowski tensors. The difference between Lorentz transformations of coordinate charts and Lorentz mapping of tangent (Minkowski) vector space is displayed. The Lorentz group of space-symmetry transformations and its spinor, bispinor, tensor and spin-tensor representations as well as the Pauli and Dirac matrix algebras are described. On this basis the systematical investigation of the relativistic quantum mechanics and the basic introduction into classical linear and nonlinear field theory of particles with spins are presented in both Lagrangian and Hamiltonian formulation of these theories. The main statements are considered related to the general variational formalism (minimal action principle, Nöther theorem) and to the free classical fields with spins 0, 1/2, 1, as well as to the interaction of Dirac bispinor field with Abelian and general non-abelian gauge fields.
The exposition presented in the book may be treated as a good foundation for classical (without making use of the second quantization procedure) field theoretical description of interactions of elementary particles in the framework of the causal Green function approach (Feynman, 1949; Bjorken, Drell, 1964; Bogush, Moroz, 1968; Bogush, 1987). The last chapter 7 deals with special research topics concerned with the Born (1938, 1949) principle of reciprocity in the classical field theory of relativistic particles and pursued by some scientists (Yukawa, 1949, 1950; Das, 1966, 1980, 1988, 1989; Caianello et al. 1979, 1990; Tomilchik, Tarakanov, 1981 and others). The theory of spin-0 and spin-1/2 particles is studied in detail in the framework of the extended relativistic eight-dimensional (coordinate-momentum) phase space.
Reviewer: A.A.Bogush (Minsk)

MSC:
 53Z05 Applications of differential geometry to physics 83-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to relativity and gravitational theory 81V25 Other elementary particle theory in quantum theory 00A79 Physics (Use more specific entries from Sections 70-XX through 86-XX when possible)