Asymptotic heat kernel expansion in the semi-classical limit

TL;DR

This paper derives an asymptotic expansion for the heat kernel of a semi-classical operator on a compact manifold, linking quantum and classical partition functions and providing bounds for the quantum case.

Contribution

It introduces a new asymptotic expansion for the heat kernel in the semi-classical limit and connects quantum and classical partition functions.

Findings

Asymptotic expansion of the heat kernel as h approaches 0.

Quantum partition function is asymptotic to the classical one.

Quantum partition function can be bounded by the classical partition function for positive h.

Abstract

Let $H_h = h^2 L +V$ where $L$ is a self-adjoint Laplace type operator acting on sections of a vector bundle over a compact Riemannian manifold and $V$ is a symmetric endomorphism field. We derive an asymptotic expansion for the heat kernel of $H_h$ as $h \to 0$. As a consequence we get an asymptotic expansion for the quantum partition function and we see that it is asymptotic to the classical partition function. Moreover, we show how to bound the quantum partition function for positive $h$ by the classical partition function.