Supersymmetry in Stochastic Quantization Method and Field-Dependent Kernel

TL;DR

This paper explores the formulation of a discretized Langevin equation using Stratonovich calculus, examines the conditions for a generating functional with a field-dependent kernel, and investigates the existence of supersymmetry in the stochastic action.

Contribution

It introduces a mid-point prescription for the generating functional with a field-dependent kernel and analyzes supersymmetry properties in this context.

Findings

Mid-point prescription is valid only in Stratonovich calculus.

Field-dependent kernels can be incorporated in the generating functional.

Supersymmetry of the stochastic action may not always exist with field-dependent kernels.

Abstract

We define a discretized Langevin equation in Stratonovich-{\it type} calculus. We show that a generating functional with a field-dependent kernel can be written in mid-point prescription only when we calculate in the calculus. Moreover we investigate whether supersymmetry of the stochastic action with field-dependent kernel exists or not.