Retarded Dispersion Energy between Macroscopic Bodies

We calculate the change in self-energy of the electromagnetic radiation field in the presence of two dielectric bodies $A$ and $B$. Starting from Maxwell's equations, the perturbed radiation field is expanded in terms of plane waves. The perturbed frequencies are obtained by applying quantum-mechanical perturbation theory. The sum over the perturbed minus the unperturbed frequencies, giving the retarded dispersion energy between the two bodies, is evaluated explicitly for the case of two spheres $A$ and $B$. It is shown to assume a finite value at zero separation $d$ of the spheres, so that no assumptions regarding an appropriate minimum separation are necessary. The total dispersion energy between bodies $A$ and $B$, which includes also the energy gain of the material modes, is found via a complex integral transform. The total dispersion energy at small and medium separations is in agreement with previous results, i.e., we obtain a $d^{-1\mathrm{}}$ relationship at separations smaller than the radii of the spheres, and a $d^{-6\mathrm{}}$ relationship at separations larger than the radii of the spheres. In the retarded case, i.e., at separations $d$ large compared with the characteristic wavelengths in the absorption spectra of the dielectrics, the dispersion energy is found to obey a $d^{-2\mathrm{}}$ law at separations $d$ smaller than the radii of the spheres and the Casimir-Polder $d^{-7\mathrm{}}$ law at separations $d$ larger than the radii.