Partition function for a singular background

TL;DR

This paper introduces a new approximation scheme called the local Born approximation for evaluating the partition function of a scalar field in a delta-function potential background, especially effective at high temperatures.

Contribution

It develops a novel local Born approximation method to compute the partition function in inhomogeneous backgrounds, extending analysis beyond traditional derivative expansion limitations.

Findings

Derived high temperature expansion for the partition function.

Obtained an analytic expression for the zero point energy.

Validated the local Born approximation as effective at high temperatures.

Abstract

We present a method for evaluating the partition function in a varying external field. Specifically, we look at the case of a non-interacting, charged, massive scalar field at finite temperature with an associated chemical potential in the background of a delta-function potential. Whilst we present a general method, valid at all temperatures, we only give the result for the leading order term in the high temperature limit. Although the derivative expansion breaks down for inhomogeneous backgrounds we are able to obtain the high temperature expansion, as well as an analytic expression for the zero point energy, by way of a different approximation scheme, which we call the {\it local Born approximation} (LBA).