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Journal of Mathematical Physics

Comment on momentum in stochastic mechanics

Simon Golin

Published Jun 1, 1986• 5 citations
Publisher

Abstract

Stochastic mechanics is a probabilistic description of quantum systems in terms of stochastic differential equations. Davidson [M. Davidson, Lett. Math. Phys. 5, 523 (1981)] and de Falco, De Martino, and De Siena [D. de Falco, S. De Martino, and S. De Siena, Lett. Nuovo Cimento 36, 457 (1983)] have introduced momentum variables into this scheme. In this paper a discussion of this attempt is presented and some difficulties concerning the physical interpretation are pointed out.

DOI
10.1063/1.527065

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Stochastic mechanics is a probabilistic description of quantum systems in terms of stochastic differential equations. Davidson [M. Davidson, Lett. Math. Phys. 5, 523 (1981)] and de Falco, De Martino, and De Siena [D. de Falco, S. De Martino, and S. De Siena, Lett. Nuovo Cimento 36, 457 (1983)] have introduced momentum variables into this scheme. In this paper a discussion of this attempt is presented and some difficulties concerning the physical interpretation are pointed out.
Stochastic mechanics---the study of classical stochastic systems governed by things like master equations and Fokker-Planck equations---exhibits striking mathematical parallels to quantum mechanics. In this article, we make those parallels more transparent by presenting a quantum mechanics-like formalism for deriving a path integral description of systems described by stochastic differential equations. Our formalism expediently recovers the usual path integrals (the Martin-Siggia-Rose-Janssen-De Dominicis and Onsager-Machlup forms) and is flexible enough to account for different variable domains (e.g. real line versus compact interval), stochastic interpretations, arbitrary numbers of variables, explicit time-dependence, dimensionful control parameters, and more. We discuss the implications of our formalism for stochastic biology.
Stochastic mechanics provides a probabilistic scheme for the description of quantum systems. Grabert, Haaumlnggi, and Talkner, and Nelson have pointed out that its multitime correlations seem to be in disagreement with quantum-mechanical predictions. We show that these difficulties are removed upon a careful analysis of repeated measurements in stochastic mechanics. The wave-packet reduction is naturally described in the stochastic framework, and the predictions for repeated measurements consequently agree in both theories.
Introduction. There exists a very comprehensive theory concerning the time development of probabilistic systems. This is the Theory of Stochastic processes (1). Quantum mechanics as described by the Schrödinger equation rests very heavily on probabilistic interpretation, and yet, after forty years of remarkably rapid advances and successes, it is not clear how it is related to stochastic theory. The formal similarity of certain aspects of, and equations used in these two theories are well known. The diffusion equation corresponds to the Schrödinger equation, a correspondence only marred by a square root of minus unity appearing with the time derivative in the latter equation. The Kolmogoroff–Chapman relation for probabilities in Brownian motion has an analogue (2) in quantum mechanics, not for probabilities though, but for the complex probability amplitudes. The uncertainty relation also has a stochastic analogue (3). Some of these analogous characteristics are actually symptomatic of the gulf which separates these two theories. There are solutions to the diffusion equation which have a very clear physical probabilistic meaning. This is not the case with the corresponding and formally similar solutions to the Schrödinger equation and is partly due to the √ − 1 mentioned earlier. The probabilistic interpretation of the solutions to the diffusion equation is direct and immediate. On the other hand, the probabilistic interpretation of the solutions, ψ of the Schrödinger equation involve the introduction of the probability density, p = ψ*ψ. One great difference between these two theories lies in their each having distinct principles of linear superposition. In Stochastic Theory probabilities are superposed and in quantum mechanics amplitudes or wave functions are superposed.
Previously [2004, AIP Conf. Proc., 750, 361], we considered a stochastic mechanics in the form of a stochastic differential equation mdxdt = μ(t) where μ(t) is a stochastic process defined by the set of Bohmian momentum time histories from an ensemble of particles. In this paper, we consider another stochastic differential equation d2xdt2 = 1m[−dV(x)dx + ζ(t)] where the stochastic force ζ(t) is a stochastic process defined by the set of quantum force time histories from an ensemble of particles. We show that, if the stochastic process is a purely random process characterized by an n‐th order joint probability density in the form of products of delta functions, then the stochastic mechanics with the stochastic force ζ(t) is equivalent to quantum mechanics in the sense that the former yields the same position probability density as the latter. However, for a particular non‐purely random process, we show that the stochastic mechanics with the stochastic force ζ(t) is not equivalent to quantum mechanics. Therefore, this stochastic mechanics is also generally not equivalent to quantum mechanics.