A stochastic version of the Noether Theorem

TL;DR

This paper extends Noether's Theorem to stochastic systems with random forces, demonstrating conservation of mean drift momentum and implications for causal correlations in separated systems, with potential links to experimental results.

Contribution

It introduces a stochastic version of Noether's Theorem using moment generating functionals and applies it to different types of random covariant forces, revealing conservation laws in stochastic dynamics.

Findings

Conservation of mean drift momentum in stochastic forces

Random systems can produce causal correlations over distance

Application to electrodynamic and proper-time dependent forces

Abstract

A stochastic version of the Noether Theorem is derived for systems under the action of external random forces. The concept of moment generating functional is employed to describe the symmetry of the stochastic forces. The theorem is applied to two kinds of random covariant forces. One of them generated in an electrodynamic way and the other defined in the rest frame of the particle as a function of the proper time. For both of them, it is shown the conservation of the mean value of a random drift momentum. The validity of the theorem makes clear that random systems can produce causal stochastic correlations between two faraway separated systems, that had interacted in the past. In addition possible connections of the discussion with the Ives Couder's experimental results are remarked.