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

TL;DR

This paper derives boundary conditions for electromagnetic fields on a delta-function plate, analyzes its optical properties, and calculates Casimir interaction energies, showing equivalence with traditional slab models and extending understanding of thin-plate limits.

Contribution

It provides a rigorous derivation of boundary conditions for delta-function plates and demonstrates their electromagnetic and Casimir interaction properties, unifying thin-plate limits with existing models.

Findings

Boundary conditions depend only on transverse properties.

Casimir energy between delta-plates matches that of slabs.

Thin-plate limit reproduces Casimir-Polder energy.

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.