  Zero-frequency Casimir theory is analyzed from different viewpoints, focusing
on the Drude-plasma issue that turns up when one considers thermal corrections
to the Casimir force. The problem is that the plasma model, although leaving
out dissipation in the material, apparently gives the best agreement with
recent experiments. We consider a dielectric plate separated from a dielectric
half-space by a vacuum gap, both media being similar. We consider the following
categories: (1) Making use of the statistical mechanical method developed by
H{\o}ye and Brevik (1998), implying that the quantized electromagnetic field is
replaced by interaction between dipole moments oscillating in harmonic
potentials, we first verify that the Casimir force is in agreement with the
Drude prediction. No use of Fresnel's reflection coefficients is made at this
stage. (2) Then turning to the field theoretical description implying use of
the reflection coefficients, we derive results in agreement with the forgoing
when first setting the frequency equal to zero, before letting the permittivity
becoming large. With the plasma relation the reflection coefficient for TE zero
frequency modes depend on the component of the wave vector parallel to the
surfaces and lies between 0 and 1. This contradicts basic electrostatic theory.
(3) Turning to high permeability magnetic materials the TE zero frequency mode
describes the static magnetic field in the same way as the TM zero frequency
modes describe the static electric fields in electrostatics. With the plasma
model magnetic fields, except for a small part, can not pass through metals.
i.e.~metals effectively become superconductors. However, recent experimental
results clearly favor the plasma model. We shortly discuss a possible
explanation for this apparent conflict with electrostatics.
