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.
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.
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@article{CRMATH_2007__344_10_663_0,
author = {Michel Fliess},
title = {Probabilit\'es et fluctuations quantiques},
journal = {Comptes Rendus. Math\'ematique},
pages = {663--668},
publisher = {Elsevier},
volume = {344},
number = {10},
year = {2007},
doi = {10.1016/j.crma.2007.04.001},
language = {fr},
}
Michel Fliess. Probabilités et fluctuations quantiques. Comptes Rendus. Mathématique, Volume 344 (2007) no. 10, pp. 663-668. doi : 10.1016/j.crma.2007.04.001. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2007.04.001/
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