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 $W_{\mathrm{disp}}$ when the intervening medium is weakly dispersive but nondissipative. The presence of dispersion means that the energy density contains terms of the form $d\left[\omega \epsilon \left(\omega \right)\right]∕d\omega$ and $d\left[\omega \mu \left(\omega \right)\right]∕d\omega$. We find that, as $W_{\mathrm{disp}}$ 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.