Probabilités et fluctuations quantiques

Abstract

Reçu le 30 juin 2006 ; accepté après révision le 4 avril 2007 Disponible sur Internet le 11 mai 2007 Présenté par Yves Meyer RésuméCette 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. Pour citer cet article : M. Fliess, C. R. Acad. Sci. Paris, Ser. I 344 (2007). AbstractProbabilities and quantum fluctuations. 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. To cite this article: M. Fliess, C. R. Acad. Sci. Paris, Ser. I 344 (2007). Abridged English versionWe are sketching a 'simple and natural' explanation of the probabilistic nature of quantum mechanics. This is achieved by utilizing the mathematical formalism of nonstandard analysis and Feynman's interpretation of the Heisenberg uncertainty principle, i.e., of the quantum fluctuations. Nontechnical presentation of the main ideasThis summary is intended for readers who are not familiar with nonstandard analysis. As often in nonstandard analysis (see, e.g., [1,26]), we replace a continuous time interval by an infinite 'discrete' set of infinitely closed time instants. Substituting m x t for p in the well known expression of the uncertainty principle, where x is the position, m the mass, p the momentum, h = 2πh the Planck constant, yields Eq. (1). We rewrite it by postulating that the Adresse quantity (2), where δt > 0 is a given infinitesimal, is limited and appreciable, i.e., it is neither infinitely large nor infinitely small. Those computations are stemming from Feynman's interpretation [13,14] of the uncertainty principle (see, also, [21,27]): The 'quantum trajectories' are fractal curves, of Hausdorff dimension 2. 'Weak' mathematical assumptions permit to derive the infinitesimal difference equation (3). The lack of any further physical assumption yields the equiprobability of +1 and −1. If x is Markov, i.e., if b and σ are functions of t and x(t), and not of {x(τ ) | 0 τ < t}, the corresponding infinitesimal random walk is 'equivalent' to a stochastic differential equation in the usual sense (see, e.g., [1,5]). Remark 1.More or less analogous random walks have already been introduced in the literature (see, e.g., [3,19,28,29]), but in another context. Nonstandard analysisReplace the interval [0, ...

Spin-Orbit Interaction in particles' motion as a model of quantum computation

“…In stochastic mechanics each particle follows a continuous path which is random but has a well-defined probability distribution [9], [10]. Stochastic mechanics has been recognized as a self-consistent formulation of quantum mechanics in the framework of stochastic processes and is established as part of stochastic control theory [11], [12], [13], [14]. The particles' kinematic model is associated to the model of a quantum oscillator, taking the particle to be a 3-DOF variable (position in a cartesian coordinates frame) [15], [16].…”

Questioning the signal to noise ratioin digital communications

“…Remarque 3 La possibilité de liens entre théorie de l'information et mécanique quantique a été examinée par divers auteurs (voir, par exemple, [4,5,24]). Rappelons à ce propos que l'approche du bruit en [11] a déjà conduit à une tentative nouvelle de formalisation du quantique [12], qui sera complétée grâce à un résultat remarquable et tout récent, dû à Charreton [6].…”