Electromagnetic semitransparent $\delta$-function plate: Casimir interaction energy between parallel infinitesimally thin plates

Abstract

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.

21 pages, 7 figures, references added

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