Infinity-free semiclassical evaluation of Casimir effects

Abstract

Electromagnetic Casimir energies are a quantum effect proportional to ប. 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.

New developments in the Casimir effect

“…The contribution from periodic trajectories decreases with their length (the contribution from each trajectory is inversely proportional to the third power of its length). As shown in [164] the semiclassical approximation reproduces the value of the Casimir energy for a large class of configurations. It was applied to configurations of two spheres of radii R 1 , R 2 , a distance a ≪ R 1 , R 2 apart, and also of a sphere (lens) above a disk under the condition a ≪ R. In both cases the results are the same as were obtained earlier by the application of the Proximity Force Theorem or the additive summation with normalization (for a sphere above a disk this result is given by Eq.…”

“…What this means is that trajectories which are going around a sphere make non-negligible contributions to the result. The inclusion of diffraction into the semiclassical theory of the Casimir effect is an outstanding question to be solved in the future [164,165].…”

“…In the papers [164,165] the semiclassical approach was proposed for calculation of the Casimir energies and forces between curved conducting surfaces. In the framework of this approach the Casimir energy is explained in terms of the classical periodic trajectories along which the virtual photons are traveling between the walls.…”

Focusing vacuum fluctuations

“…In this case, one would not expect geometric optics to be a good approximation. Nonetheless, Schaden and Spruch [5] have argued that one can often obtain reasonable results from a semiclassical approximation involing a sum over periodic classical orbits. Our use and justification of a geometric optics approximation is perhaps more akin to that of Hawking [6] in his derivation of black hole evaporation.…”

Effect of the heterogeneity of metamaterials on the Casimir-Lifshitz interaction

“…The effect of geometry has been studied using a variety of techniques, which include perturbative expansion around ideal geometries [4][5][6] and in dielectric contrast [7][8][9][10][11], semiclassical [12] and classical ray-optics [13] approximations, multiple scattering [14,15] and multipole expansions [16][17][18], world-line method [19], exact numerical diagonalization methods [20], and the method of numerical calculation of the Green function [21]. These studies have significantly advanced our understanding of the subtle effect of geometry on Casimir-Lifshitz interactions, and have led to proposals for using the knowledge in designing useful nano-scale mechanical devices [22].…”