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Journal of Mathematical Physics

An introduction to the semiclassical limit of Euclidean quantum mechanics

Torbjörn Kolsrud, Jean-Claude Zambrini

Published Apr 1, 1992• 12 citations
Publisher

Abstract

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.

DOI
10.1063/1.529707

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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 general description of Bernstein processes, a class of diffusion processes, relevant to the probabilistic counterpart of quantum theory known as Euclidean Quantum Mechanics, is given. It is compatible with finite or infinite dimensional state spaces and singular interactions. Although the relations with statistical physics concepts (Gibbs measure, entropy,…) is stressed here, recent developments requiring Feynman’s quantum mechanical tools (action functional, path integrals, Noether’s Theorem,…) are also mentioned and suggest new research directions, especially in the geometrical structure of our approach.
We give a Hilbert space picture of a novel probabilistic interpretation of the classical heat equation, realizing an idea of Schrodinger. By this we also obtain an Euclidean version of non relativistic Quantum Mechanics, distinct from the one obtained using the Feynman-Kac formula On donne une version hilbertienne d'une nouvelle interpretation probabiliste de l'equation classique de la chaleur, ce qui realise une idee de Schrodinger. On obtient une version euclidienne de la mecanique quantique non relativiste distincte de celle obtenue par la formule de Feynman-Kac
Bernstein diffusions belong to a new class of time symmetric (but not time homogeneous) stochastic processes associated with the quantum dynamics of nonrelativistic particles in potentials. It is shown in which sense Feynman’s formal path integral method can be interpreted in terms of those processes, specially for the subset of Gaussian Bernstein diffusions. The familiar Ornstein-Uhlenbeck process becomes, in this framework, a particular Gaussian process in a large class of Bernstein diffusions, all associated with the same dynamics. The method is also illustrated for some Hilbert space valued Gaussian Diffusions These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
In Quantum Physics, the term “Euclidean” means that one, compared to the conventional formulation, works in “imaginary time.” It is a general fact that objects associated with Euclidean as opposed to Minkowski space are easier to deal with; this is seen for instance when comparing the usual Laplacian, an elliptic operator, with its Minkowski space counterpart, the hyperbolic d’Alembertian. In Quantum Mechanics, the term “Euclidean” is usually associated with approaches based on the so-called Feynman-Kac formula. The latter originates in attempts to give Feynman’s ideas [10, 11] on path integrals a solid mathematical founding.
The interpretation provided by Schrödinger of quantum mechanics through classical stochastic processes is the root of Euclidean quantum mechanics founded on Bernstein processes. Open quantum systems have been extensively investigated during the last three decades and have motivated new probabilistic interpretations for quantum dynamics. This article, mainly addressed to classical probabilists, gives a short introduction to quantum dynamical semigroups and quantum Markov flows as a natural extension of Schrödinger ideas in noncommutative stochastic analysis. These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.