Fluctuations as stochastic deformation

TL;DR

This paper introduces a novel concept of stochastic deformation, developing an algebraic procedure analogous to deformation quantization but with an imaginary parameter, and applies it to various physical models to derive stochastic equations.

Contribution

The paper develops the theory of stochastic deformation and demonstrates its application to relativistic and nonrelativistic models, deriving stochastic equations like Fokker-Planck and Klein-Kramers.

Findings

Derivation of stochastic models from classical systems using stochastic deformation.

Connection of stochastic deformation to known equations like Fokker-Planck and Klein-Kramers.

Application to electromagnetic and scalar fields, revealing new stochastic descriptions.

Abstract

A notion of stochastic deformation is introduced and the corresponding algebraic deformation procedure is developed. This procedure is analogous to the deformation of an algebra of observables like deformation quantization, but for an imaginary deformation parameter (the Planck constant). This method is demonstrated on diverse relativistic and nonrelativistic models with finite and infinite degrees of freedom. It is shown that under stochastic deformation the model of a nonrelativistic particle interacting with the electromagnetic field on a curved background passes into the stochastic model described by the Fokker-Planck equation with the diffusion tensor being the inverse metric tensor. The first stochastic correction to the Newton equations for this system is found. The Klein-Kramers equation is also derived as the stochastic deformation of a certain classical model. Relativistic generalizations of the Fokker-Planck and Klein-Kramers equations are obtained by applying the procedure of stochastic deformation to appropriate relativistic classical models. The analog of the Fokker-Planck equation associated with the stochastic Lorentz-Dirac equation is derived too. The stochastic deformation of the models of a free scalar field and an electromagnetic field is investigated. It turns out that in the latter case the obtained stochastic model describes a fluctuating electromagnetic field in a transparent medium.