Derivative expansion of the heat kernel at finite temperature

TL;DR

This paper extends the covariant symbol method to finite temperature space-times, providing explicit heat kernel formulas up to fourth order and emphasizing the Polyakov loop's role, with applications to effective actions.

Contribution

It introduces a covariant derivative expansion of the heat kernel at finite temperature, including explicit formulas and extensions of Chan's effective action formula.

Findings

Explicit heat kernel formulas up to fourth order at finite temperature.

Extension of covariant symbol method to spaces with topology $ ^n imes S^1$.

Formal applicability to a broader class of $h$-spaces.

Abstract

The method of covariant symbols of Pletnev and Banin is extended to space-times with topology $\R^n\times S^1\times ... \times S^1$. By means of this tool, we obtain explicit formulas for the diagonal matrix elements and the trace of the heat kernel at finite temperature to fourth order in a strict covariant derivative expansion. The role of the Polyakov loop is emphasized. Chan's formula for the effective action to one loop is similarly extended. The expressions obtained formally apply to a larger class of spaces, $h$-spaces, with an arbitrary weight function $h(p)$ in the integration over the momentum of the loop.