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Results obtained previously by the authors on the inclusion of Feynman path integrals in the momentum representation into general integration theory are generalized. Feynman path integrals that are not chronological products are considered. A (generalized) Feynman measure is introduced in the coordinate representation.

Виктор Павлович Маслов, Alexander Mikhailovich Chebotarev

Published Jan 1, 1976• 0 citations
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Efforts to give an improved mathematical meaning to Feynman's path integral formulation of quantum mechanics started soon after its introduction and continue to this day. In the present paper, one common thread of development is followed over many years, with contributions made by various authors. The present version of this line of development involves a continuous-time regularization for a general phase space path integral and provides, in the author's opinion at least, perhaps the optimal formulation of the path integral.
"Feynman path integrals are ubiquitous in quantum physics, even if a large part of the scientific community still considers them as a heuristic tool that lacks a sound mathematical definition. Our book aims to refute this prejudice, providing an extensive and self-contained description of the mathematical theory of Feynman path integration, from the earlier attempts to the latest developments, as well as its applications to quantum mechanics. This second edition presents a detailed discussion of the general theory of complex integration on infinite dimensional spaces, providing on one hand a unified view of the various existing approaches to the mathematical construction of Feynman path integrals and on the other hand a connection with the classical theory of stochastic processes. Moreover, new chapters containing recent applications to several dynamical systems have been added. This book bridges between the realms of stochastic analysis and the theory of Feynman path integration. It is accessible to both mathematicians and physicists"--
The definitions of the Feynman path integral for the Pauli equation and more general equations in configuration space and in phase space are proposed, probably for the first time. Then it is proved rigorously that the Feynman path integrals are well-defined and are the solutions to the corresponding equations. These Feynman path integrals are defined by the time-slicing method through broken line paths, which is familiar in physics. Our definitions of these Feynman path integrals and our results give the extension of ones for the Schrödinger equation.
Based on the equivalence between momentum representation and coordinate representation,an alternative method is presented,where the momentum representation and the Feynman's path integral are employed to calculate the propagator function of a particle in a constant force field,then the result is transformed into the coordinate representation. We present a way to solve the Feynman's path integral,which exhibits the high value of momentum representation if the solution of one- dimensional motion of a particle in a constant force field.