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arXiv:quant-ph/0612033

Approche intrinsèque des fluctuations quantiques en mécanique stochastique (An intrinsic approach of the quantum fluctuations in stochastic mechanics)

Michel Fliess

Published Dec 5, 2006• 0 citations
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Abstract

This note is answering an old questioning about the Fényes-Nelson stochastic mechanics. The Brownian nature of the quantum fluctuations, which are associated to this mechanics, is deduced from Feynman's interpretation of the Heisenberg uncertainty principle via infinitesimal random walks stemming from nonstandard analysis. It is therefore no more necessary to combine those fluctuations with a background field, which has never been well understood. Most of the technical details are contained in an extended english abstract.

DOI
10.48550/arxiv.quant-ph/0612033

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This note is answering an old questioning about the Fényes-Nelson stochastic mechanics. The Brownian nature of the quantum fluctuations, which are associated to this mechanics, is deduced from Feynman's interpretation of the Heisenberg uncertainty principle via infinitesimal random walks stemming from nonstandard analysis. It is therefore no more necessary to combine those fluctuations with a background field, which has never been well understood. Most of the technical details are contained in an extended english abstract.
This note is sketching a simple and natural mathematical construction for explaining the probabilistic nature of quantum mechanics. It employs nonstandard analysis and is based on Feynman's interpretation of the Heisenberg uncertainty principle, i.e., of the quantum fluctuations, which was brought to the forefront in some fractal approaches. It results, as in Nelson's stochastic mechanics, in stochastic differential equations which are deduced from infinitesimal random walks. An extended english abstract gives most of the details.
Cette Note esquisse une construction mathématique simple et naturelle du caractère probabiliste de la mécanique quantique. Elle utilise l'analyse non standard et repose sur l'interprétation due à Feynman, mise en avant dans certaines approches fractales, du principe d'incertitude de Heisenberg, c'est-à-dire des fluctuations quantiques. On aboutit ainsi à des équations différentielles stochastiques, comme dans la mécanique stochastique de Nelson, découlant de marches aléatoires infinitésimales.
Introduction The apparent generality of the title to this chapter is appropriately reduced by sufficiently precisely defining the terms used. We have in mind the natural extension to quantum mechanical settings of the ideas usually covered in a discussion of stochastic processes in classical mechanical settings. In classical mechanical settings, the fundamental process is Brownian motion, which describes the influence of external fluctuating forces on a small particle. The mathematical apparatus developed for the study of Brownian motion has been generalized and applied to irreversible thermodynamics (Onsager and Machlup, 1953), fluctuating hydrodynamics (Fox and Uhlenbeck, 1970), light scattering from fluids and mixtures (Berne and Pecora, 1976) and chemical reactions (McQuarrie and Keizer, 1981). In each case, the stochastic process involves fluctuations generated by thermal motions. We are not considering the stochastic trajectories (chaos) of classically deterministic motion in non-linear systems (Lichtenberg and Lieberman, 1983), which is an entirely separate context for stochastic processes. Our goal in this chapter is to study the effects of thermal fluctuations in quantum mechanical settings. Therefore, we exclude a discussion of interpreting quantum mechanics as a classical stochastic motion (Nelson, 1966), an idea which should have been put to rest by Grabert, Hänggi and Talkner (1979). We also exclude a discussion of quantum chaos, which is currently a very active area of research (Zaslavsky, 1981). Just as Brownian motion provides the paradigm for stochastic processes in classical mechanical settings, the Anderson–Kubo oscillator (Anderson, 1954; Kubo, 1954; Kubo, Toda and Nashitsume, 1985) is the paradigm for stochastic processes in quantum mechanical settings.