Modular invariance in finite temperature Casimir effect

TL;DR

This paper extends the temperature inversion symmetry of the Casimir effect's partition function to full modular invariance using a purely imaginary chemical potential, revealing deep mathematical structures and equivalences.

Contribution

It introduces a full modular invariance framework for the Casimir effect's partition function by incorporating an imaginary chemical potential and expressing it via Eisenstein series.

Findings

Partition function expressed as a real analytic Eisenstein series.

Equivalence shown between Maxwell's theory with conducting plates and a scalar field with periodic boundary conditions.

Extended symmetry provides new insights into the mathematical structure of the Casimir effect.

Abstract

The temperature inversion symmetry of the partition function of the electromagnetic field in the set-up of the Casimir effect is extended to full modular transformations by turning on a purely imaginary chemical potential for adapted spin angular momentum. The extended partition function is expressed in terms of a real analytic Eisenstein series. These results become transparent after explicitly showing equivalence of the partition functions for Maxwell's theory between perfectly conducting parallel plates and for a massless scalar with periodic boundary conditions.