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Geometry of quantum theory. 2nd ed. (English) Zbl 0581.46061
New York etc.: Springer-Verlag. XVIII, 412 p. DM 178.00 (1985).
An account of numerous successful encounters of mathematics and physics can hardly become obsolate. Hence the present re-publication of ”Geometry of quantum theory” is a most welcome event. Out of the two original volumes [(1968; Zbl 0155.568) and (1970; Zbl 0194.288)] now emerges one, the contents of which give the reader a better chance of embracing the comprehensive material. Thus, wisely, the original chapters on projective geometry appear radically compressed. The book takes the reader on a travel through five principal encounters between mathematics and physics - within geometry, logic, algebra and analysis. Since by ”geometry”, the author understands an irreducible complemented modular lattice, it becomes clear that the notion of geometry plays the leading role. Chapters from one to four present an interplay of geometry and logic. Chapter five discusses group action on topological spaces, and chapter six gives the reader a fair account of the Mackey systems of imprimitivity. Projective representations are discussed in chapter seven. Chapter eight discusses kinematics and dynamics in the light of the abstract Schrödinger equation, and the last chapter is devoted to the relativistic free particle.
Each chapter is accompanied by notes making the content more up-to-date and supplying old concepts - as for instance that of Fock representation which turned out to be of principal importance but was not mentioned in the original edition.
Reviewer: W.Słowikowski

46N99 Miscellaneous applications of functional analysis
81P10 Logical foundations of quantum mechanics; quantum logic (quantum-theoretic aspects)
06C20 Complemented modular lattices, continuous geometries
46-02 Research exposition (monographs, survey articles) pertaining to functional analysis