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Book review

Author(s) Egorov, A.D.
Sobolevsky, P.I.
Yanovich, L.A.
Title Functional integrals: Approximate evaluation and applications
Publisher Kluwer
Year of publication 1993
Reviewed by Günter Vojta

Functional integrals represent a relatively new area of mathematics which governs large parts of modern mathematical and theoretical physics. The historical development started with the rigorous theory of Brownian motion by N. Wiener (Wiener integrals, 1923) and the nonrigorous quantum mechanics and quantum field theory by R.P. Feynman (Feynman path integrals, 1945). However, the foundation of a general rigorous theory was hindered by a lot of serious problems due to the fact that the analysis and particularly the measure theory on infinite-dimensional spaces is, by far, not yet finished. Therefore, it has been highly important that approximative methods for the theory and the practical evaluation of functional integrals have now been developed to a large extent.

The research monograph under review is devoted to this field. It contains extensive material partly founded on the research of the authors themselves - about the numerical calculation of functional integrals including the topics of stochastic processes and of quantum theory. Essentially, it is a work about measures on function spaces and their applications.

The book is divided into 15 chapters the first chapter being a summary of basic facts of analysis on linear topological spaces. The next chapters deal with integrals over Gaussian measures and some quasimeasures, integrals on special topological spaces and interpolation-type formulae. Integrals based on characteristic functionals, correlation functionals, conditional Wiener processes and other stochastic processes including processes of the Ornstein-Uhlenbeck type are then considered. Then follow important chapters on approximations coupled with diagram expansions, i.e. graph representations of perturbation series in quantum field theory, on interpolation of measures, on integrals over manifolds and on martingale problems. Chapters on Monte-Carlo methods, on Gaussian multiple integrals and on some special problems conclude the work.

The volume is well organized and produced. The text is supplemented by a bibliography of 192 references with titles, largely of Russian authors, and a meagre index. This truly impressive book is an important contribution to the growing literature on functional integrals and their applications in such fields as theory of stochastic processes, quantum theory and statistical physics. It deserves a broad audience.