Diffusion Processes in Phase Spaces and Quantum Mechanics

TL;DR

This paper models a diffusion process in phase space that leads to quantum-like behavior, deriving the Schrödinger equation from a classical diffusion framework with complex inner states.

Contribution

It introduces a novel diffusion process in phase space with complex inner states, connecting classical diffusion to quantum mechanics and deriving the Schrödinger equation.

Findings

Fast motion reduces wave functions to a subspace of complex functions.

Slow motion in this subspace obeys the Schrödinger equation.

Estimated fast motion duration is approximately 10^{-11} seconds.

Abstract

A diffusion process for charge distributions in a phase space is examined. The corresponding charge moves in a force field and under an action of a random field. There are the diffusion motions for coordinates and for momenta. In our model, an inner state of the charge is defined by a complex vector. The vector rotates with a great constant angular velocity with respect to the proper time of the charge. A state of the diffusion process is a (complex-valued) wave function on the phase space. As in quantum mechanics, we assume that, for the wave functions, the superposition principle holds. The diffusion process averages out vectors of inner states from different points of the phase space. A differential equation for this diffusion process is founded and examined. We demonstrate that the motion (described by this process) decomposes into a fast motion and a slow motion. The fast motion reduces an arbitrary wave function to a function from a subspace whose elements are parameterized by complex-valued functions of coordinates. The slow motion occurs in this subspace and it is described by the Schr\''odinger equation. The parameters of the suggested model are estimated. The duration of the fast motion is of order $10^{-11}$ s.