Classical probability waves

TL;DR

This paper explores how classical probability waves, when modified with a minimum spatial distance, exhibit energy dispersion akin to quantum objects, suggesting a transition from complex to real probability waves in quantum Brownian systems.

Contribution

It introduces a novel approach linking classical probability waves with quantum energy dispersion through a minimum distance in coordinate space.

Findings

Probability waves in classical equations can acquire quantum-like energy dispersion.

At finite temperature, probability density waves propagate with sound velocity.

A potential transition from complex to real probability waves in quantum Brownian particles is proposed.

Abstract

Probability waves in the configuration space are associated with coherent solutions of the classical Liouville or Fokker-Planck equations. Distributions localized in the momentum space provide action waves, specified by the probability density and the generating function of the Hamilton-Jacobi theory. It is shown that by introducing a minimum distance in the coordinate space, the action distributions aquire the energy dispersion specific to the quantum objects. At finite temperature, probability density waves propagating with the sound velocity are obtained as nonstationary solutions of the classical Fokker-Planck equation. The results suggest that in a system of quantum Brownian particles, a transition from complex to real probability waves could be observed.