A microlocal investigation of stochastic partial differential equations for spinors with an application to the Thirring model

TL;DR

This paper develops a microlocal analysis framework for stochastic PDEs involving spinor fields on Riemannian manifolds, extending previous scalar methods, and applies it to analyze a stochastic Thirring model, showing subcritical behavior in low dimensions.

Contribution

It introduces a microlocal analysis approach for stochastic spinor PDEs, extending scalar techniques to spinors, and applies this to the stochastic Thirring model, demonstrating subcriticality in certain dimensions.

Findings

Framework for stochastic spinor PDEs using microlocal analysis.

Perturbative computation of expectations and correlations.

Stochastic Thirring model is subcritical for dimensions ≤ 2.

Abstract

On a $d$-dimensional Riemannian, spin manifold $(M,g)$ we consider non-linear, stochastic partial differential equations for spinor fields, driven by a Dirac operator and coupled to an additive Gaussian, vector-valued white noise. We extend to the case in hand a procedure, introduced in [DDRZ20] for the scalar counterpart, which allows to compute at a perturbative level the expectation value of the solutions as well as the associated correlation functions accounting intrinsically for the underlying renormalization freedoms. This framework relies strongly on tools proper of microlocal analysis and it is inspired by the algebraic approach to quantum field theory. As a concrete example we apply it to a stochastic version of the Thirring model proving in particular that it lies in the subcritical regime if $d\leq 2$.