A Functional Approach to the Heat Kernel in Curved Space

TL;DR

This paper introduces a new classical-variable approach to analyze the heat kernel in curved space, providing an unambiguous expression and demonstrating its application by computing the heat kernel at lowest order in curvature.

Contribution

It adapts Onofri's classical-variable method to curved space, clarifying the mathematical definition of the heat kernel and enabling explicit calculations.

Findings

Derived an unambiguous expression for the heat kernel in curved space.

Applied the method to compute the heat kernel at lowest order in curvature.

Showed the approach's effectiveness through explicit calculation.

Abstract

The heat kernel $M_{xy} = <x\mid exp [ 1/\sqrt{g} \partial_\mu g^{\mu\nu} \sqrt{g} \partial_\nu ]t \mid y>$ is of central importance when studying the propagation of a scalar particle in curved space. It is quite convenient to analyze this quantity in terms of classical variables by use of the quantum mechanical path integral; regrettably it is not entirely clear how this path integral can be mathematically well defined in curved space. An alternate approach to studying the heat kernel in terms of classical variables was introduced by Onofri. This technique is shown to be applicable to problems in curved space; an unambiguous expression for $M_{xy}$ is obtained which involves functional derivatives of a classical quantity. We illustrate how this can be used by computing $M_{xx}$ to lowest order in the curvature scalar R.