The Feynman Integral and Feynman’s Operational Calculus Gerald W. Johnson and Michel L. Lapidus Oxford U. Press, New York, 2000. $160.00 (771 pp.). ISBN 0-19-853574-0
I have been interested in mathematical approaches to the Feynman path integral ever since I attempted in the 1950s to define functional Fourier transforms in terms of Wiener–Hermite expansions. Therefore, I was intrigued by the appearance of this monumental volume. I confess first thinking of the famous quotation by Goethe, who said, “Mathematicians are like Frenchmen: You tell them something, they translate it into their own language, and then it becomes something completely different.” (This is particularly the case since both authors are mathematicians and one of them is French). I must say at the outset, however, that large parts of Gerald W. Johnson and Michel L. Lapidus’s The Feynman Integral and Feynman’s Operational Calculus are accessible to less mathematically inclined theoretical physicists, who can certainly skip some of the mathematical detail).
The idea behind the Feynman path integral goes back to a paper by P. A. M. Dirac published in 1933 in Physikalische Zeitschrift der Sowjetunion. It formed the core of Richard Feynman’s space–time approach to quantum mechanics and quantum electrodynamics. Although the path integral was not mathematically well defined, it was widely used in quantum field theory, statistical mechanics, and string theory. Recently, path integrals have been the heuristic guide to spectacular developments in pure mathematics.
It was clear to Feynman that his “path integral” was no integral in the ordinary sense of the word, and that what he called its “summation over histories” did not involve a measure in the usual sense. Furthermore, the Lagrangian of a classical particle involves its velocity, whereas the paths over which the “integral” is extended are just continuous, not necessarily differentiable. The imaginary exponential is a highly oscillatory function, and thus most contributions cancel (except near the stationary values of the action—the classical trajectories of the particle).
In the 1920s, Norbert Wiener, extending the early work on Brownian motion by Albert Einstein and Marian Smoluchowski, introduced the notion of integration in function spaces that now goes under the name of Wiener measure, or Wiener integral. It plays a fundamental role in the theory of stochastic processes and, as distinguished from Feynman’s path integral, has a rigorous mathematical underpinning. In 1947, Mark Kac realized that if one replaces the time parameter in the imaginary exponent in the Dirac–Feynman expression for the action with a purely imaginary time, the Feynman path integral becomes a Wiener integral; the Schrödinger equation (with its imaginary time-derivative) turns into the parabolic diffusion or heat equation, for which the Wiener integral provides a solution. This solution for the Schrödinger equation has become known as the Feynman–Kac formula. It has played a fundamental role in Euclidean constructive field theory and in statistical mechanics.
The book by Johnson and Lapidus deals with various approaches to making the Feynman path integral into a mathematically meaningful object. The first few chapters present a considerable amount of background material on measure theory, functional analysis, and the traditional formulation of quantum mechanics, as well as two chapters on Wiener measure and stochastic processes. This material can be particularly useful for a theoretical physicist whose mathematics may be a bit rusty.
Chapter 7 contains a detailed heuristic introduction to Feynman path integrals. It turns out that there are several independent approaches to a mathematically satisfactory definition: Edward Nelson’s approach via the Lie–Trotter product formula, Kac’s original analytic-continuation-in-time approach, as well as those developed by the authors, based on analytic continuation in mass and imaginary resolvents (which form the subject of later chapters). The detailed mathematical treatment is often interspersed with interesting remarks and heuristic material that eases the flow. Throughout the book one finds examples of application to problems in nonrelativistic quantum mechanics.
The second topic of this book is the Feynman operational calculus. It was invented by Feynman in 1951 in an attempt to “disentangle” exponentials of noncommuting operators such as often occur in time-ordered perturbation theory (commonly known as the Dyson time-ordered exponential). Feynman realized that his highly heuristic approach poses serious mathematical problems, and this book appears to be a first systematic, mathematically rigorous study of this subject.
The authors discuss several methods of making sense of the Feynman heuristics: via the path integral, via a “generalized Dyson series,” and via a more general noncommutative calculus. These chapters may well be of interest to physicists involved in “noncommutative geometry.”
The last chapter deals with other work related to the book’s topics, ranging from alternative approaches to the path integral (so-called Fresnel integrals) to a very readable survey of the influence of Feynman integrals on contemporary mathematics and physics. In particular, the authors discuss low-dimensional topology and Edward Witten’s approach to knot invariants, and they end with a discussion of Maxim Kontsevich’s work on deformation quantization.
The list of references is quite extensive (though the acronyms, such as “GelKLLRT” are sometimes distracting, forcing the curious reader to flip back to the bibliography to find out to whom the authors refer).
I would recommend this book to serious students of the subject, if it were not for the prohibitive price; let’s hope that the publishers will release a more reasonably priced paperback, accessible to graduate students and emerit(ae)i.
