Heat kernel expansion in the covariant perturbation theory

TL;DR

This paper develops a covariant perturbation theory approach to derive the heat kernel expansion for elliptic operators, providing explicit form factors and tensor invariants applicable to quantum field theories involving gravity and gauge fields.

Contribution

It introduces a new method for calculating the heat kernel expansion using covariant perturbation theory, including explicit form factors and tensor invariants up to second order.

Findings

Derived the coincidence limit of the heat kernel for a broad class of operators.

Obtained integral representations for form factors acting on tensor invariants.

Verified results through multiple independent methods including trace operations and Green function calculations.

Abstract

Working within the framework of the covariant perturbation theory, we obtain the coincidence limit of the heat kernel of an elliptic second order differential operator that is applicable to a large class of quantum field theories. The basis of tensor invariants of the curvatures of a gravity and gauge field background, to the second order, is derived, and the form factors acting on them are obtained in two integral representations. The results are verified by the functional trace operation, by the short proper time (Schwinger-DeWitt) expansions, as well as by the computation of the Green function for the two-dimensional scalar field model.