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Path integrals in nonrelativistic stochastic quantum mechanics

R. Gagnon

Published Dec 1, 1985• 0 citations
OpenAlex

Abstract

It is shown that path integration can be formulated rigorously in stochastic phase spaces by means of Riemann-Stiltjes methods. The relationship of this formulation to the conventional Feynman path integration formulation in configuration representation is examined.

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Abstract Feynman developed the path integral approach to quantum mechanics as an alternative to wave or matrix mechanics that has found fruitful implementation in quantum mechanics, quantum electrodynamics, and statistical physics. Despite its many successes, the usual Feynman path integral formalism suffers from a lack of mathematical rigor. Questions of mathematical rigor still exist, but Mark Kac provided a rigorous path integral approach to quantum mechanics. The Kac formulation for quantum systems is given in terms of Wiener measure for a path integral and employs the Brownian motion distribution. In this paper, we review the Feynman–Kac approach and other stochastic processes related to Brownian motion. We also give an account of the successes and difficulties we faced during our journey through numerical path integral problems involved in physics.
When the space of paths is a certain Hilbert space H, we show how to extend the Feynman path integral ℱ of DeWitt and Albeverio and Hoegh-Krohn. Our extension enables us to integrate a wider class of functionals on H. We establish a new representation for the wavefunction in nonrelativistic quantum mechanics—the quasiclassical representation. Using our extension of ℱ and the quasiclassical representation, we discuss the problem of obtaining classical mechanics as the limiting case of quantum mechanics when h/→0.
We will extend Nonrelativistic Quantum Mechanics as a theory in $L\sp2$ to a theory in the space of distributions. We will provide 3 major theories of Nonrelativist Quantum Mechanics. First, we will extend the concept of an integral kernel for the evolution operator to a distribution kernel for the $L\sp2$ transition probability amplitude. Second, we will extend the $L\sp2$ Schrodinger's equation to a distributions Schrodinger's equation. Lastly, we will rigorously prove that; Feynman's original formulation of the real time, time-sliced path integral is well defined when formulated on the $L\sp2$ transition probability amplitude.