Infinity-free semiclassical evaluation of Casimir effects

Electromagnetic Casimir energies are a quantum effect proportional to $ħ.$ We show that in certain cases one can obtain an exact semiclassical expression for them that depends only on periodic orbits of the associated classical problem. A great merit of the approach is that infinities never appear if one considers only periodic orbits that make contact with the boundary surface. This notion is made more precise by classifying the closed orbits in a phase space with boundaries and identifying the classes that contribute to Casimir effects. A semiclassical evaluation of the path integral gives a systematic expansion of the Casimir energy in terms of the lengths of classical periodic orbits. For some simple geometries the semiclassical expansion can be summed and explicitly shown to reproduce known results. This is the case, for example, for the force per unit area between parallel plates at a separation small compared to their linear dimensions. A more interesting example for our purposes is the closely related problem of the force on a conducting sphere arbitrarily close to a conducting wall. We provide a rigorous proof of Derjaguin’s result for the leading contribution to the force. The semiclassical approach, which has never been truly exploited in Casimir studies, is relatively simple and transparent, and should have a wide range of applications. The methods presented, however, do not apply to cases where diffraction is important; diffraction can, in principle, also be described within this semiclassical approach, but its implementation presents some technical problems. In cases where diffraction is important, conventional methods of calculating the Casimir energy may often be simpler.