Stochastic Quantization of $(\lambda\phi^{4})_d$ Scalar Theory: Generalized Langevin Equation with Memory Kernel

TL;DR

This paper reviews stochastic quantization of scalar field theories, extending it to include Langevin equations with memory kernels, and demonstrates convergence to the non-regularized theory under certain conditions.

Contribution

It introduces a formalism for stochastic quantization with memory kernels and shows convergence to the non-regularized theory using colored noise.

Findings

Stochastic perturbation theory up to one-loop level for scalar fields.

Implementation of stochastic regularization with colored noise.

Convergence to the non-regularized theory with memory kernels.

Abstract

We review the method of stochastic quantization for a scalar field theory. We first give a brief survey for the case of self-interacting scalar fields, implementing the stochastic perturbation theory up to the one-loop level. The divergences therein are taken care of by employing the usual prescription of the stochastic regularization, introducing a colored random noise in the Einstein relations. We then extend this formalism to the case where we assume a Langevin equation with a memory kernel. We have shown that, if we also maintain the Einstein's relations with a colored noise, there is convergence to a non-regularized theory.