A note on the mechanics emerged from systems with a stochastic process of the time variable

Stochastic mechanics is an attempt to build a well-defined probabilistic formulation of quantum theory. This formulation proposes a deeper origin for the (non-relativistic) Schrödinger equation using a hypothesis that a massive particle with mass m is influenced by a Brownian motion having a diffusi...

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Bibliographic Details
Published in:Physica A 2023-01, Vol.609, p.128334, Article 128334
Main Author: Shushi, Tomer
Format: Article
Language:English
Subjects:
Brownian motion
Ito process
Itô’s lemma
Quantum mechanics
Stochastic mechanics
Time
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Summary:Stochastic mechanics is an attempt to build a well-defined probabilistic formulation of quantum theory. This formulation proposes a deeper origin for the (non-relativistic) Schrödinger equation using a hypothesis that a massive particle with mass m is influenced by a Brownian motion having a diffusion coefficient of ħ/2m and no friction. However, the hypothetical ether (as suggested by Nelson) that causes such a Brownian motion remains uncleared. While most of the literature on stochastic mechanics follows the original interpretation given by Nelson, we present an alternative interpretation of stochastic mechanics, where the stochasticity of time, and not space, is responsible for the emergence of the quantum particle. In particular, we show that stochastic mechanics (in one dimension) can be deduced by assuming that time is a stochastic process of a universal absolute time. This replaces the proposed ether with a new notion of time for particles in the quantum regime.
ISSN:0378-4371
1873-2119
DOI:10.1016/j.physa.2022.128334