The Stochastic Quantization Method for Systems with Dissipation

TL;DR

This paper explores a stochastic quantization approach for dissipative systems, revealing that restricting Fourier transforms to positive domains breaks time-reversal symmetry and relates to the Caldeira-Leggett method.

Contribution

It introduces a stochastic quantization framework for dissipative systems that incorporates Fourier domain restrictions and analyzes its connection to existing path-integral methods.

Findings

Fourier transform restriction is necessary for stochastic quantization of dissipative systems.

Time-reversal invariance is broken by the Fourier domain restriction.

The stochastic approach relates to the Caldeira-Leggett path-integral method.

Abstract

The stochastic quantization of dissipative systems is discussed. It is shown that in order to stochastically quantize a system with dissipation, one has to restrict the Fourier transform of the space-time variable to the positive half domain in the complex plane. This breaks the time-reversal invariance, which manifests in the formulation through the resulting noninvariant forms for the propagators. The relation of the stochastic approach with the Caldeira and Leggett path-integral method is also analyzed.