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Quantum probability. (English) Zbl 0652.60004
Probability and Mathematical Statistics. Boston, MA: Academic Press, Inc. xii, 316 p. $ 49.50 (1988).
It is an old and important tradition in quantum mechanics to construct rigorous mathematical models describing physical systems. So there exist today many expositions of the basic principles of quantum mechanics.
The first axiomatic framework to be developed was the traditional approach, which is mostly used by working physicists. One of the chapters of this book presents a brief survey of the traditional quantum mechanics. Among the other approaches, the so-called logical approach and the operational approach are presented and analysed in detail.
One chapter deals with amplitudes and transition amplitude functions and the corresponding quasimanuals. This method shows how Hilbert space comes into play in quantum mechanics as a derived concept, which is one of the oldest questions in quantum mechanics.
The last three chapters discuss applications of the model developed before. These applications include generalized measure theory, probability manifolds, and discrete quantum mechanics.
This volume would be of interest both to researchers following the development of new results and to people seeking an introduction into the field of abstract mathematical physics.

81-02 Research exposition (monographs, survey articles) pertaining to quantum theory
81P10 Logical foundations of quantum mechanics; quantum logic (quantum-theoretic aspects)
81Q35 Quantum mechanics on special spaces: manifolds, fractals, graphs, lattices
03G12 Quantum logic
60A05 Axioms; other general questions in probability