Finite temperature Casimir effect on spherical shells in (D+1)-dimensional spacetime and its high temperature limit

TL;DR

This paper analyzes the finite temperature Casimir effect on spherical shells in (D+1)-dimensional spacetime, deriving explicit formulas for the free energy and its high temperature behavior for various boundary conditions.

Contribution

It provides a comprehensive calculation of the renormalized Casimir free energy in higher dimensions with different boundary conditions, including explicit high temperature limits.

Findings

Explicit formulas for Casimir free energy and $ta'(0)$ derived.

High temperature limit of free energy shown to be proportional to $-c_D T \, \ln T$.

Graphical dependence of free energy on temperature presented.

Abstract

We consider the finite temperature Casimir free energy acting on a spherical shell in (D+1)-dimensional Minkowski spacetime due to the vacuum fluctuations of scalar and electromagnetic fields. Dirichlet, Neumann, perfectly conducting and infinitely permeable boundary conditions are considered. The Casimir free energy is regularized using zeta functional regularization technique. To renormalize the Casimir free energy, we compute the heat kernel coefficients $c_n$, $0\leq n\leq D+1$, from the zeta function $\zeta(s)$. After renormalization, the high temperature leading term of the Casimir free energy is $-c_DT\ln T-T \zeta'(0)/2$. Explicit expressions for the renormalized Casimir free energy and $\zeta'(0)$ are derived. The dependence of the renormalized Casimir free energy on temperature is shown graphically.