Development of a Stochastic Interpretation of Quantum Mechanics by E. Nelson. Derivation of the Schrodinger-Euler-Poisson Equations

TL;DR

This paper develops a stochastic interpretation of quantum mechanics by combining principles of least action and maximum entropy, deriving stochastic Schrödinger-Euler-Poisson equations that describe quantum states across scales.

Contribution

It introduces a novel approach by integrating two fundamental principles into a unified extremum principle, leading to stochastic equations that generalize quantum mechanics.

Findings

Derived stochastic Schrödinger-Euler-Poisson equations

Connected the equations to the standard Schrödinger equations with coefficients

Expressed the Planck constant to mass ratio through averaged stochastic characteristics

Abstract

The aim of the article is to develop the stochastic interpretation of quantum mechanics by E. Nelson on the basis of balancing the intra-systemic contradiction (i.e., antisymmetry) between "order" and "chaos". For the set task, it is proposed to combine two mutually opposite system-forming principles: "the principle of least action" and "the principle of maximum entropy" into one the "principle of averaged efficiency extremum". In a detailed consideration of the averaged states of a chaotically wandering particle, the time-independent (stationary) and time-dependent stochastic Schrodinger-Euler-Poisson equations are obtained as conditions for finding the extremals of the functional of the globally averaged efficiency functional of the stochastic system under study. The resulting stochastic equations coincides with the corresponding Schrodinger equations up to coefficients. In this case, the ratio of the reduced Planck constant to the particle mass is expressed through the averaged characteristics of a three-dimensional random process in which the considered wandering particle participates. The obtained stochastic equations are suitable for describing the quantum states of stochastic systems of any scale.