A Functional Approach to the Heat Kernel in Curved Space

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.