The Stochastic Quantization Method in Phase Space and a New Gauge Fixing Procedure

TL;DR

This paper explores stochastic quantization in phase space for systems with first class constraints, showing gauge fixing emerges naturally from stochastic consistency, aligning equilibrium solutions with path integral distributions.

Contribution

It introduces a novel approach where gauge fixing is automatically determined through stochastic consistency, eliminating the need for explicit gauge fixing procedures.

Findings

Gauge fixing conditions are naturally selected by stochastic consistency.

Equilibrium solutions match the path integral distribution.

The method simplifies quantization of constrained systems.

Abstract

We study the stochastic quantization of the system with first class constraints in phase space. Though the Langevin equations of the canonical variables are defined without ordinary gauge fixing procedure, gauge fixing conditions are automatically selected and introduced by imposing stochastic consistency conditions upon the first class constraints. Then the equilibrium solution of the Fokker-Planck equation is identical with corresponding path integral distribution.