We give an exact series expansion of the Casimir force between plane and spherical metallic surfaces in the non trivial situation where the sphere radius R, the plane-sphere distance L and the plasma wavelength λP have arbitrary relative values. We then present numerical evaluation of this expansion for not too small values of L/R. For metallic nanospheres where R, L and λP have comparable values, we interpret our results in terms of a correlation between the effects of geometry beyond the proximity force approximation (PFA) and of finite reflectivity due to material properties. We also discuss the interest of our results for the current Casimir experiments performed with spheres of large radius R ≫ L.The Casimir force is a striking macroscopic effect of quantum vacuum fluctuations which has been seen in a number of dedicated experiments in the last decade (see for example [1,2] and references therein). One aim of the Casimir force experiments is to investigate the presence of hypothetical weak forces predicted by unification models through a careful comparison of the measurements with quantum electrodynamics predictions. This aim can only be reached if theoretical computations are able to take into account a realistic and reliable modeling of the experimental conditions. Among the effects to be taken into account are the material properties and the surface geometry, these effects being also able to produce phenomena of interest in nanosystems [3,4].A number of Casimir measurements have been performed with gold-covered plane and spherical surfaces separated by distances L of the order of the plasma wavelength (λ P ≃ 136nm for gold), making material properties important in their analysis [5]. As those measurements use spheres with a radius R ≫ L, they are commonly analyzed through the Proximity Force Approximation (PFA) [6], which amounts to a trivial integration over the sphere-plate distances. An exception is the Purdue experiment dedicated to the investigation of the accuracy of PFA in the sphere-plate geometry [7], the result of which will be given as a precise statement below.In the present letter, we give for the first time an exact series expansion of the Casimir force between a plane and a sphere in electromagnetic vacuum, taking into account the material properties via the plasma model (see Fig. 1). We present numerical evaluation of this expansion which are limited to not too small values of L/R, because of the multipolar nature of the series. We show below that these new results lead to a striking correlation between the effects of geometry and imperfect reflection when evaluated for nanospheres, with R, L and λ P having comparable values. In the end of this letter, we also discuss the interest of these results for the Casimir experiments performed with large spheres R ≫ L [7].Our starting point is a general scattering formula for the Casimir energy [8]. Using suitable plane-wave and multipole bases, we deduce the Casimir energy E PS be- tween a plane and a spherical metallic surface in electromagnetic...

In this paper we calculate the Casimir energy for a dielectric-diamagnetic cylinder with the speed of light differing on the inside and outside. Although the result is in general divergent, special cases are meaningful. The well-known results for a uniform speed of light are reproduced. The self-stress on a purely dielectric cylinder is shown to vanish through second order in the deviation of the permittivity from its vacuum value, in agreement with the result calculated from the sum of van der Waals forces. These results are unambiguously separated from divergent terms.

The local Casimir energy density and the global Casimir energy for a massless scalar field associated with a λδ-function potential in a 3+1 dimensional circular cylindrical geometry are considered. The global energy is examined for both weak and strong coupling, the latter being the well-studied Dirichlet cylinder case. For weak-coupling, through O(λ 2 ), the total energy is shown to vanish by both analytic and numerical arguments, based both on Green's-function and zeta-function techniques. Divergences occurring in the calculation are shown to be absorbable by renormalization of physical parameters of the model. The global energy may be obtained by integrating the local energy density only when the latter is supplemented by an energy term residing precisely on the surface of the cylinder. The latter is identified as the integrated local energy density of the cylindrical shell when the latter is physically expanded to have finite thickness. Inside and outside the δ-function shell, the local energy density diverges as the surface of the shell is approached; the divergence is weakest when the conformal stress tensor is used to define the energy density. A real global divergence first occurs in O(λ 3 ), as anticipated, but the proof is supplied here for the first time; this divergence is entirely associated with the surface energy, and does not reflect divergences in the local energy density as the surface is approached.

The local Casimir energy density for a massless scalar field associated with step-function potentials in a 3 + 1 dimensional spherical geometry is considered. The potential is chosen to be zero except in a shell of thickness δ, where it has height h, with the constraint hδ = 1. In the limit of zero thickness, an ideal δ-function shell is recovered. In this limit, the behavior of the energy density as the surface of the shell is approached is studied in both the strong and weak coupling regimes. The former case corresponds to the well-known Dirichlet shell limit. New results, which shed light on the nature of surface divergences and on the energy contained within the shell, are obtained in the weak coupling limit, and for a shell of finite thickness. In the case of zero thickness, the energy has a contribution not only from the local energy density, but from an energy term residing entirely on the surface. It is shown that the latter coincides with the integrated local energy density within the shell. We also study the dependence of local and global quantities on the conformal parameter. In particular new insight is provided on the reason for the divergence in the global Casimir energy in third order in the coupling.

We calculate the lateral Casimir force between corrugated parallel plates, described by -function potentials, interacting through a scalar field, using the multiple scattering formalism. The contributions to the Casimir energy due to uncorrugated parallel plates is treated as a background from the outset. We derive the leading-and next-to-leading-order contribution to the lateral Casimir force for the case when the corrugation amplitudes are small in comparison to corrugation wavelengths. We present explicit results in terms of finite integrals for the case of the Dirichlet limit, and exact results for the weak-coupling limit, for the leading-and next-to-leading-orders. The correction due to the next-to-leading contribution is significant. In the weak coupling limit we calculate the lateral Casimir force exactly in terms of a single integral which we evaluate numerically. Exact results for the case of the weak limit allows us to estimate the error in the perturbative results. We show that the error in the lateral Casimir force, in the weak coupling limit, when the next-to-leading order contribution is included is remarkably low when the corrugation amplitudes are small in comparison to corrugation wavelengths. We expect similar conclusions to hold for the Dirichlet case. The analogous calculation for the electromagnetic case should reduce the theoretical error sufficiently for comparison with the experiments.

The Casimir interaction between two concentric corrugated cylinders provides the mechanism for non-contact gears. To this end, we calculate the Casimir torque between two such cylinders, described by δ-potentials, which interact through a scalar field. We derive analytic expressions for the Casimir torque for the case when the corrugation amplitudes are small in comparison to the corrugation wavelengths. We derive explicit results for the Dirichlet case, and exact results for the weak coupling limit, in the leading order. The results for the corrugated cylinders approach the corresponding expressions for the case of corrugated parallel plates in the limit of large radii of cylinders (relative to the difference in their radii) while keeping the corrugation wavelength fixed. * Electronic address: Ines.Cavero-Pelaez@spectro.jussieu.fr † Electronic address: milton@nhn.ou.edu; URL: http://www.nhn.ou.edu/%7Emilton ‡ Electronic address: prachi@nhn.ou.edu § Electronic address: shajesh@nhn.ou.edu; URL: http://www.nhn.ou.edu/%7Eshajesh

Continuing a program of examining the behavior of the vacuum expectation value of the stress tensor in a background which varies only in a single direction, we here study the electromagnetic stress tensor in a medium with permittivity depending on a single spatial coordinate, specifically, a planar dielectric halfspace facing a vacuum region. There are divergences occurring that are regulated by temporal and spatial point splitting, which have a universal character for both transverse electric and transverse magnetic modes. The nature of the divergences depends on the model of dispersion adopted. And there are singularities occurring at the edge between the dielectric and vacuum regions, which also have a universal character, depending on the structure of the discontinuities in the material properties there. Remarks are offered concerning renormalization of such models, and the significance of the stress tensor. The ambiguity in separating "bulk" and "scattering" parts of the stress tensor is discussed.

Abstract. Although Casimir, or quantum vacuum, forces between distinct bodies, or self-stresses of individual bodies, have been calculated by a variety of different methods since 1948, they have always been plagued by divergences. Some of these divergences are associated with the volume, and so may be more or less unambiguously removed, while other divergences are associated with the surface. The interpretation of these has been quite controversial. Particularly mysterious is the contradiction between finite total self-energies and surface divergences in the local energy density. In this paper we clarify the role of surface divergences.