Casimir Energies: Temperature Dependence, Dispersion, and Anomalies

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.