A note on the heat kernel method applied to fermions

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 $γ$-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.