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Journal of Mathematical Physics

Euclidean quantum mechanics in the momentum representation

Nicolas Privault, Jean-Claude Zambrini

Published Feb 8, 2005• 6 citations
Publisher

Abstract

A time reversible probabilistic representation of solutions of the (Euclidean) Schrödinger equation in momentum representation is constructed using Lévy processes and bridges. Each diffusion in the position representation is associated with a jump diffusion in the momentum space. Our method can be looked upon as a rigorous version of Feynman’s path integral approach. Several examples are studied.

DOI
10.1063/1.1850178

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A time reversible probabilistic representation of solutions of the (Euclidean) Schrödinger equation in momentum representation is constructed using Lévy processes and bridges. Each diffusion in the position representation is associated with a jump diffusion in the momentum space. Our method can be looked upon as a rigorous version of Feynman’s path integral approach. Several examples are studied.
We prove a version of the Feynman-Kac formula for Levy processes and integro-differential operators, with application to the momentum representation of suitable quantum (Euclidean) systems whose Hamiltonians involve Lévy-type potentials. Large deviation techniques are used to obtain the limiting behavior of the systems as the Planck constant approaches zero. It turns out that the limiting behavior coincides with fresh aspects of the semiclassical limit of (Euclidean) quantum mechanics. Non-trivial examples of Levy processes are considered as illustrations and precise asymptotics are given for the terms in both configuration and momentum representations.
A new class of Gaussian (‘‘Bernstein’’) diffusion processes associated with semiclassical quantum dynamics is introduced and studied according to a new Euclidean strategy for the quantization of systems of particles in a potential (‘‘Euclidean quantum mechanics’’). All the measures known by other means, up to now, are included as particular cases of the present framework. It is shown that the theory may be regarded as a well-defined probabilistic version of Feynman’s space-time approach distinct, in particular, from the one due to Kac, and with much closer relations with classical Lagrangian dynamics. Many explicit examples are considered.
A nonstandard path space is constructed that gives the mathematically rigorous formulation for the path integral representation of the fundamental solution to the Cauchy problem for the Dirac equation in (1+1)-dimensional space-time. Nonstandard analysis makes the mathematical concepts elementary, consequently, the procedures to prove theorems are considerably simplified. A difference scheme with infinitesimal spacing is available in determining the probability distribution over the path-space and this method is also available for the heat equation and for the free Schrödinger equation.