Elementary approximate derivations of some retarded Casimir interactions involving one or two dielectric walls

The original derivation by Lifshitz [Sov. Phys. 2, 73 (1956)] of ${\mathit{P}}_{\mathit{D}\mathit{D}}$, the force per unit area between two plane parallel dielectric walls, is extremely complicated; the later derivations are simpler but still difficult. The standard derivation of the interaction ${\mathit{V}}_{\mathrm{At}}$ D of an atom and a dielectric wall uses the expression for ${\mathit{P}}_{\mathit{D}\mathit{D}}$ as its starting point. The results are valid for all values of the separation l. For l∼∞, where the interactions are retarded, we obtain resonably accurate approximate expressions for ${\mathit{P}}_{\mathit{DD}}$ and for ${\mathit{V}}_{\mathit{A}\mathrm{t}\mathrm{}\mathit{D}}$—and also for ${\mathit{V}}_{\mathit{E}\mathrm{l}\mathrm{}\mathit{D}}$, the retarded interaction of an electron and a dielectric wall—by the elementary procedure of assuming simple forms with one or two open parameters, adjusted to give the known results for retarded interactions which do not include dielectric walls. These include ${\mathit{P}}_{\mathit{M}\mathit{M}}$ (the force per unit area between two parallel plate metallic walls), ${\mathit{V}}_{\mathit{At}\mathrm{}\mathit{M}}$ (the atom-metal) interaction, ${\mathit{V}}_{\mathit{At}\mathrm{}\mathit{At}}$ (the atom-atom interaction), and ${\mathit{V}}_{\mathit{El}\mathrm{}\mathit{M}}$ (the electron-metal interaction). We also consider the possibility of obtaining an improved estimate of ${\mathit{P}}_{\mathit{D}\mathit{D}}$ by using known properties of ${\mathit{V}}_{\mathit{At}\mathrm{}\mathit{D}}$. The explicit results obtained by Lifshitz for the various interactions are also very complicated. The simple approximate forms of the interactions can be particularly useful for the wall-wall interaction, since ${\mathit{P}}_{\mathit{D}\mathit{D}}$ is a double integral with a complicated integrand which depends upon two parameters, the zero-frequency dielectric constants of each of the walls.