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_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[ωε(ω)]∕dω and d[ωμ(ω)]∕dω. We find that, as W_disp refers thermodynamically to a nonclosed physical system, it is 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 nonclosed 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 space-time dimensions, D>4, and discuss briefly its dispersive generalization. This kind of application is essentially a generalization of the treatment of Alnes et al. J. Phys. A 40 F315 (2007) to the case of a medium-filled cavity between two hyperplanes.