Treatment of Constraints in Stochastic Quantization Method and Covariantized Langevin Equation

TL;DR

This paper enhances stochastic quantization for constrained systems by refining the Langevin equation using Ito calculus, ensuring correct path integral quantization and addressing divergences in the $O(N)$ non-linear sigma model.

Contribution

It introduces an improved Langevin equation that correctly handles constraints and divergences, aligning with covariant formalisms and improving stochastic quantization methods.

Findings

Cancellation of $ ext{delta}^n(0)$ divergences at one loop order

Equivalence of the improved Langevin equation to covariant formalisms

Successful application to the $O(N)$ non-linear sigma model

Abstract

We study the treatment of the constraints in stochastic quantization method. We improve the treatment of the stochastic consistency condition proposed by Namiki et al. by suitably taking account of the Ito calculus. Then we obtain an improved Langevin equation and the Fokker-Planck equation which naturally leads to the correct path integral quantization of the constrained system as the stochastic equilibrium state. This treatment is applied to $O(N)$ non-linear $\sigma$ model and it is shown that singular terms appearing in the improved Langevin equation cancel out the $\delta^n(0)$ divergences in one loop order. We also ascertain that the above Langevin equation, rewritten in terms of independent variablesis, actually equivalent to the one in the general-coordinate-transformation-covariant and vielbein-rotation-invariant formalism.