Casimir Energies: Temperature Dependence, Dispersion, and Anomalies

TL;DR

This paper analyzes the physical meaning of Casimir energy in dispersive media, emphasizing that the dispersive energy $W_{disp}$ should not be equated with thermodynamic internal energy, and explores anomalies in higher-dimensional Casimir calculations.

Contribution

It clarifies the thermodynamic interpretation of dispersive Casimir energies and extends the analysis of surface divergences to higher dimensions with dispersive media.

Findings

Dispersive Casimir energy $W_{disp}$ is not equivalent to thermodynamic internal energy.

Surface divergences in Casimir energies exhibit anomalies in higher dimensions.

Dispersive effects modify the interpretation of electromagnetic energy in Casimir systems.

Abstract

Assuming the conventional Casimir setting with two thick parallel perfectly conducting plates of large extent with a homogeneous and isotropic medium between them, we discuss the physical meaning of the electromagnetic field energy $W_{\rm disp}$ when the intervening medium is weakly dispersive but nondissipative. The presence of dispersion means that the energy density contains terms of the form $d[\omega\epsilon(\omega)] /d\omega$ and $d[\omega\mu(\omega)] /d\omega$. We find that, as $W_{\rm disp}$ refers thermodynamically to a non-closed physical system, it is {\it not} to be identified with the internal thermodynamic energy $U$ following from the free energy $F$, or the electromagnetic energy $W$, when the last-mentioned quantities are calculated without such dispersive derivatives. To arrive at this conclusion, we adopt a model in which the system is a capacitor, linked to an external self-inductance $L$ such that stationary oscillations become possible. Therewith the model system becomes a non-closed one. As an introductory step, we review the meaning of the nondispersive energies, $F, U,$ and $W$. As a final topic, we consider an anomaly connected with local surface divergences encountered in Casimir energy calculations for higher spacetime dimensions, $D>4$, and discuss briefly its dispersive generalization. This kind of application is essentially a generalization of the treatment of Alnes {\it et al.} [J. Phys. A: Math. Theor. {\bf 40}, F315 (2007)] to the case of a medium-filled cavity between two hyperplanes.