  We derive boundary conditions for electromagnetic fields on a
$\delta$-function plate. The optical properties of such a plate are shown to
necessarily be anisotropic in that they only depend on the transverse
properties of the plate. We unambiguously obtain the boundary conditions for a
perfectly conducting $\delta$-function plate in the limit of infinite
dielectric response. We show that a material does not "optically vanish" in the
thin-plate limit. The thin-plate limit of a plasma slab of thickness $d$ with
plasma frequency $\omega_p^2=\zeta_p/d$ reduces to a $\delta$-function plate
for frequencies ($\omega=i\zeta$) satisfying $\zeta d \ll \sqrt{\zeta_p d} \ll
1$. We show that the Casimir interaction energy between two parallel perfectly
conducting $\delta$-function plates is the same as that for parallel perfectly
conducting slabs. Similarly, we show that the interaction energy between an
atom and a perfect electrically conducting $\delta$-function plate is the usual
Casimir-Polder energy, which is verified by considering the thin-plate limit of
dielectric slabs. The "thick" and "thin" boundary conditions considered by
Bordag are found to be identical in the sense that they lead to the same
electromagnetic fields.
