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PCT, spin and statistics, and all that. Classic ed. (English) Zbl 0704.53058
Amsterdam: Addison-Wesley. xvi, 207 p. $ 32.50 (1989).
The editors of the reprint series “Advanced Book Classics” have committed themselves to sustaining the lifespan of books with historic significance, timeless quality, and an appeal to a worldwide audience. There is no doubt that the book under review meets these standards. It is the second edition (1978) of this now classical book on the mathematical foundations of quantum field theory which is reprinted here without alterations. It differs from the first edition (1964; Zbl 0135.443) by a couple of minor corrections and modifications, and by an appendix addressing some developments brought forward between 1964 and 1978 in so far as they have changed the authors’ outlook on the foundations of quantum field theory as described in the text. The character of the book is chiefly determined by the plan (i) to present the material in the form of mathematical theorems (and, consequently, to omit everything of a “heuristic” nature that cannot be put in such a form); and (ii) to presuppose nothing beyond some basic knowledge about Hilbert spaces and ordinary quantum mechanics.
The first chapter on “Relativistic transformation laws” is of a preliminary character, as well as the second one which presents “Some mathematical tools” from functional analysis and from complex analysis. The third chapter “Fields and vacuum expectation values” introduces the basic notions of relativistic quantum field theory in a manner which has come to be known as “Wightman axiomatics”. The fourth chapter “Some general theorems and relativistic quantum field theory” presents the main results which can be proven within this framework, viz. the PCT theorem, the connection between spin and statistics, and Haag’s theorem. The book ends with the above mentioned appendix on constructive quantum field theory, the theory of local algebras, and the theory of superselection rules.
Many readers, both physicists and mathematicians, will take pleasure in this new hardcover edition of a very special book, available at a comparatively moderate price.
Reviewer: V.Perlick

53B50 Applications of local differential geometry to the sciences
81-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to quantum theory
47N50 Applications of operator theory in the physical sciences
46N50 Applications of functional analysis in quantum physics