Probabilit\'es et fluctuations quantiques (Probabilities and quantum fluctuations)

TL;DR

This paper proposes a mathematical framework using nonstandard analysis and stochastic differential equations to explain quantum probabilities and fluctuations, inspired by Feynman's interpretation of the uncertainty principle.

Contribution

It introduces a novel approach combining nonstandard analysis with stochastic mechanics to model quantum phenomena.

Findings

Derivation of stochastic differential equations from infinitesimal random walks.

Connection of quantum fluctuations with Feynman's interpretation of the uncertainty principle.

A new mathematical construction for quantum probabilistic behavior.

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.