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Functional integration and quantum physics. 2nd ed. (English) Zbl 1061.28010
Providence, RI: AMS Chelsea Publishing (ISBN 0-8218-3582-3/hbk). xiv, 306 p. (2005).
This new edition is a reprint of the highly successful book on path integral techniques by Barry Simon first published in 1979 (cf. Zbl 0434.28013), essentially unaltered but supplemented by a brief bibliographic note on recent developments in the field. Throughout the book Simon limits himself to the well-defined Wiener-Kac formulation of the path integral rather than the formal (often ill-defined) Feynman version of it. In the development of the method, one discovers a striking interplay between stochastic processes, statistical physics, and quantum mechanics. At present the path integral approach acquired a firm position within the fabric of theoretical physics, especially in modern field theory.
The volume has seven chapters. The first chapter lists typical problems in conventional quantum mechanics which can be handled using functional integration and provides fundamental tools of probability theory, while the second chapter passes on to stochastic processes relevant to Kac’s formula for \(\exp(-tH)\) where \(H\) denotes the Schrödinger operator. The third chapter starts from the Birman-Schwinger kernel to get bounds on the number of bound states, derives Lieb’s formula, discusses the Dyson-Lenard answer to the problem of stability of matter (improved by Lieb-Thirring in 1975), and recovers an early result of H. Weyl from 1911. Correlation inequalities are the subject of the next chapter, followed by Itô’s integral to incorporate magnetic fields. The implications of the Donsker-Varadhan theory regarding the asymptotic behavior of Markov expectations for large time are demonstrated in one chapter while perturbation theory and the existence of the wave operator, relevant for the description of scattering, are discussed in the final chapter.
Unfortunately, printing errors of the previous edition have not been corrected.

28C20 Set functions and measures and integrals in infinite-dimensional spaces (Wiener measure, Gaussian measure, etc.)
46N99 Miscellaneous applications of functional analysis
81P20 Stochastic mechanics (including stochastic electrodynamics)
81S40 Path integrals in quantum mechanics