A note on the heat kernel method applied to fermions

TL;DR

This paper proves that the spectrum of fermionic operators in quantum field theory is independent of the sign of the fermion mass, providing a rigorous foundation for a commonly used assumption and illustrating it with explicit calculations.

Contribution

It offers a rigorous proof that the fermionic spectrum does not depend on the fermion mass sign, using an alternative basis for gamma matrices.

Findings

Spectrum independence from fermion mass sign established

Calculated the coincidence limit of coefficient a_2(x,x') on general backgrounds

Validated a widely used but previously unproven assumption

Abstract

The spectrum of the fermionic operators depending on external fields is an important object in Quantum Field Theory. In this paper we prove, using transition to the alternative basis for the $\gamma$-matrices, that this spectrum does not depend on the sign of the fermion mass, up to a constant factor. This assumption has been extensively used, but usually without proof. As an illustration, we calculated the coincidence limit of the coefficient $a_2(x,x^\prime)$ on the general metric background, vector and axial vector fields.