Cambridge Monographs on Mathematical Physics. Cambridge: Cambridge University Press (ISBN 978-0-521-86696-5/hbk; 978-0-511-53506-2/ebook). xx, 456 p. £ 80.00; $ 140.00/hbk (2006).

The monograph presents an approach allowing for the rigorous mathematical formulation of what is known in physics as Feynman path integral. The latter is formally defined as the exponential of the classical action along some path multiplied by the imaginary unit $i$ and “integrated” over all paths. Such a “path integral” provides an alternative formulation of quantum mechanics that is equivalent to the conventional operator formulation. More concretely, “integral over paths” is defined in physical literature by a limiting process of “skeletonization” of paths leading from piecewise linear paths to all continuous paths. Such a definition is not a special case of the standard mathematical definition of the concept of integral, or measure. The Wiener integral formally differs from the Feynman path integral by transition to “imaginary time”, $\tau = it$. This integral naturally appears in quantum statistics. Contrary to quantum-mechanical Feynman path integral, the Wiener path integral may be defined in terms of the Gaussian measure on the space of continuous functions (the so-called Kac integral). This definition has an advantage of being mathematically rigorous and being included in the general theory of measure and theory of integration as it is developed in mathematics. It is not easy to apply the definition of this type to Feynman path integrals. The book by Cartier and DeWitt-Morette deals with both these types of path integrals much in similar way. Most of the formulas are valid for various values of the parameter $s$ such that $s=1$ corresponds to quantum mechanics (Feynman path integral) and $s=i$ quantum statistics (Wiener integral). Many of the methods presented in the book are based on the works of its authors. The book includes six parts: [I] The physical and mathematical environment, [II] Quantum mechanics, [III] Methods from differential geometry, [IV] Non-Gaussian applications, [V] Problems in quantum field theory, [VI] Projects. In the first two parts the basics of path-integral techniques (including WKB approximation) are given and illustrated by simple examples of Brownian motion and dynamics of a forced harmonic oscillator. In the third part path integrals on manifolds with symmetry or non-trivial topology, and application of Grassmann analysis to path integrals are considered. Part four deals with Poisson processes and fixed-energy processes. Part five is devoted to the problem of renormalization arising in quantum field theory. Finally in the last part of the book those points of the theory are listed which are worth of further developing. The monograph of Cartier and DeWitt-Morette will be helpful for those mathematicians who are interested in physical applications of the general theory of measure (theory of integrals) and for the physicists who are interested in mathematically rigorous formulations of complicated problems in quantum physics.