On the non-local heat kernel expansion

TL;DR

This paper introduces a new diagrammatic method to derive the non-local heat kernel expansion for Laplace-type operators, providing explicit form factors up to second order in curvature and enabling broader generalizations.

Contribution

A novel diagrammatic derivation of the non-local heat kernel expansion, simplifying previous approaches and extending applicability to various differential operators.

Findings

Explicit form factors obtained up to second order in curvature

Method can be generalized to other differential operators

Simplifies derivation of non-local heat kernel expansion

Abstract

We propose a novel derivation of the non-local heat kernel expansion, first studied by Barvinsky, Vilkovisky and Avramidi, based on simple diagrammatic equations satisfied by the heat kernel. For Laplace-type differential operators we obtain the explicit form of the non-local heat kernel form factors to second order in the curvature. Our method can be generalized easily to the derivation of the non-local heat kernel expansion of a wide class of differential operators.