Relativistic diffusion equation from stochastic quantization

TL;DR

This paper introduces a novel stochastic quantization scheme that generalizes the Fokker-Planck equation to relativistic particles, resulting in a relativistic diffusion equation with applications to particles in potential fields.

Contribution

The authors propose a new stochastic quantization method equivalent to deformation quantization with an imaginary Planck constant, extending it to relativistic particles and deriving a relativistic diffusion equation.

Findings

Relativistic generalization of the Fokker-Planck equation derived.

Stationary distributions for particles in barriers and harmonic potentials obtained.

The method links stochastic quantization to deformation quantization with an imaginary parameter.

Abstract

The new scheme of stochastic quantization is proposed. This quantization procedure is equivalent to the deformation of an algebra of observables in the manner of deformation quantization with an imaginary deformation parameter (the Planck constant). We apply this method to the models of nonrelativistic and relativistic particles interacting with an electromagnetic field. In the first case we establish the equivalence of such a quantization to the Fokker-Planck equation with a special force. The application of the proposed quantization procedure to the model of a relativistic particle results in a relativistic generalization of the Fokker-Planck equation in the coordinate space, which in the absence of the electromagnetic field reduces to the relativistic diffusion (heat) equation. The stationary probability distribution functions for a stochastically quantized particle diffusing under a barrier and a particle in the potential of a harmonic oscillator are derived.