Perturbative Casimir energies of dispersive spheres, cubes and cylinders

The quantum-electrodynamic binding energies B are determined perturbatively to order (n alpha)(2) for single macroscopic bodies (quasi-continua mimicking atomic solids) having the dispersive dielectric function epsilon(omega) similar or equal to {1 + 4 pin alpha Omega (2)/(Omega (2) - (omega (2) - i0)(2))}, as if each atom were an oscillator of frequency Omega, and n the number density of atoms (pairwise separations p). The familiar divergences all persist although they are modified by dispersion (finite rather than infinite Omega); they must be controlled instead by imposing the condition rho > lambda similar to (minimum lattice spacing) much less thanc/Omega. QED gives identically the same B = -(n alpha)(2)(1/2) integral (infinity)(lambda) d rho rho (2) f(rho )g(rho) as one obtains from the properly retarded attraction -alpha (2) f(rho) between atoms, with g(rho) a correlation function defined purely by the geometry of the body. The first three terms of the Taylor series for g are determined, respectively, by volume V, surface area S and any sharp edges. To order (na)2, but not beyond, the results for solid bodies lead directly to those for cavities of the same shape and size in otherwise unbounded material. Unlike the attraction between disjoint bodies, B for any single finite body (typical linear dimensions a much greater than c/ Omega) is dominated by components proportional, respectively, to (n alpha)(2)h Omega x {-V/lambda (3), +S/lambda (2), -a/lambda (if there are edges) and +/- log(c/2 Omega lambda)}. These always tend to induce collapse rather than expansion. The pure Casimir components are of order (n alpha)(2)hc/a, and (like the logarithmic terms) sometimes positive, which makes them impossible to understand if the dominant terms are disregarded. The B are found in closed form for spheres, spherical shells and cubes, up to corrections vanishing with lambda. For unit length of an infinitely long right circular cylinder of radius a, the standard V- and S-proportional terms are corrected only by -(n alpha)(2)(pi (2)h Omega /128a) log(c/2 Omega lambda); the pure Casimir component, which would be proportional to (n alpha)(2)hc/a(2), vanishes through apparently accidental cancellations peculiar to order (n alpha)(2).