Electromagnetic Casimir effect on the boundary of a D-dimensional cavity and the high temperature asymptotics

TL;DR

This paper analyzes the electromagnetic Casimir effect in D-dimensional cavities at finite temperature, revealing divergence issues and high-temperature behavior depending on the dimension, with specific results for three-dimensional cases.

Contribution

It establishes the mathematical equivalence of boundary conditions and derives the high temperature asymptotics of the Casimir free energy based on heat kernel coefficients.

Findings

Casimir stress is divergence-free only when D=3

High temperature behavior depends on heat kernel coefficients

Renormalization needed for D>3 at high temperatures

Abstract

We consider the finite temperature Casimir stress acting on the boundary of a D>=3 dimensional cavity due to the vacuum fluctuations of electromagnetic fields. Both perfectly conducting and infinitely permeable boundary conditions are considered, and it is proved that they correspond mathematically to the relative and absolute boundary conditions. The divergence terms of the Casimir free energy are related to the heat kernel coefficients of the Laplace operator. It is shown that the Casimir stress is free of divergence if and only if D is exactly three. The high temperature asymptotics of the regularized Casimir free energy are also found to depend on the heat kernel coefficients. When D>3, renormalization is required to remove terms of order higher than or equal to T^2.