Schwinger--DeWitt expansion for the heat kernel of nonminimal operators in causal theories

TL;DR

This paper develops a systematic method to compute heat kernels for nonminimal operators in causal theories with null characteristic surfaces, using a pseudodifferential operator calculus and local curvature expansions.

Contribution

It introduces a new calculational scheme for nonminimal operators' heat kernels, extending minimal operator techniques to nonminimal cases in causal theories.

Findings

Provides a local expansion in powers of curvature and background fields.

Demonstrates the method on vector Proca and nondegenerate vector operators.

Discusses smoothness properties related to operator symbol nondegeneracy.

Abstract

We suggest a systematic calculational scheme for heat kernels of covariant nonminimal operators in causal theories whose characteristic surfaces are null with respect to a generic metric. The calculational formalism is based on a pseudodifferential operator calculus which allows one to build a linear operator map from the heat kernel of the minimal operator to the nonminimal one. This map is realized as a local expansion in powers of spacetime curvature, dimensional background fields, and their covariant derivatives with the coefficients -- the functions of the Synge world function and its derivatives. Finiteness of these functions, determined by multiple proper time integrals, is achieved by a special subtraction procedure which is an important part of the calculational scheme. We illustrate this technique on the examples of the vector Proca model and the vector field operator with a nondegenerate principal symbol. We also discuss smoothness properties of heat kernels of nonminimal operators in connection with the nondegenerate nature of their operator symbols.