Covariant approximation schemes for calculation of the heat kernel in quantum field theory

TL;DR

This paper presents a new covariant algebraic approach to calculating the heat kernel in quantum field theory and quantum gravity, especially effective in the low-energy approximation with slowly varying background fields.

Contribution

It introduces a covariant algebraic method for heat kernel calculation that simplifies the process and provides explicit formulas in the low-energy regime.

Findings

Developed a covariant algebraic framework for heat kernel calculation.

Derived explicit covariant formulas for the heat kernel diagonal.

Applied the method to low-energy approximations with covariantly constant backgrounds.

Abstract

This paper is an overview on our recent results in the calculation of the heat kernel in quantum field theory and quantum gravity. We introduce a deformation of the background fields (including the metric of a curved spacetime manifold) and study various asymptotic expansions of the heat kernel diagonal associated with this deformation. Especial attention is payed to the low-energy approximation corresponding to the strong slowly varying background fields. We develop a new covariant purely algebraic approach for calculating the heat kernel diagonal in low-energy approximation by taking into account a finite number of low-order covariant derivatives of the background fields, and neglecting all covariant derivatives of higher orders. Then there exist a set of covariant differential operators that together with the background fields and their low-order derivatives generate a finite dimensional Lie algebra. In the zeroth order of the low-energy perturbation theory, determined by covariantly constant background, we use this algebraic structure to present the heat operator in the form of an average over the corresponding Lie group. This simplifies considerably the calculations and allows to obtain closed explicitly covariant formulas for the heat kernel diagonal. These formulas serve as the generating functions for the whole sequence of the Hadamard-Minakshisundaram- De Witt-Seeley coefficients in the low-energy approximation.