A stochastic analysis of subcritical Euclidean fermionic field theories

TL;DR

This paper introduces a stochastic quantisation method for Grassmann measures using FBSDEs, enabling the construction of subcritical Euclidean fermionic field theories with proven exponential decay of correlations.

Contribution

It presents a novel stochastic approach to fermionic field theories that bypasses technical challenges of traditional flow equation analysis.

Findings

Constructed a family of weakly coupled subcritical Euclidean fermionic field theories

Proved exponential decay of correlations in these theories

Developed a stochastic quantisation framework for Grassmann measures

Abstract

Building on previous work on the stochastic analysis for Grassmann random variables, we introduce a forward-backward stochastic differential equation (FBSDE) which provides a stochastic quantisation of Grassmann measures. Our method is inspired by the so-called continuous renormalisation group, but avoids the technical difficulties encountered in the direct study of the flow equation for the effective potentials. As an application, we construct a family of weakly coupled subcritical Euclidean fermionic field theories and prove exponential decay of correlations.