Opposites Attract: A Theorem about the Casimir Force

Abstract

We consider the Casimir interaction between (nonmagnetic) dielectric bodies or conductors. Our main result is a proof that the Casimir force between two bodies related by reflection is always attractive, independent of the exact form of the bodies or dielectric properties. Apart from being a fundamental property of fields, the theorem and its corollaries also rule out a class of suggestions to obtain repulsive forces, such as the two hemisphere repulsion suggestion and its relatives. DOI: 10.1103/PhysRevLett.97.160401 PACS numbers: 12.20.ÿm, 03.65.ÿw, 03.70.+k The Casimir effect has been a fundamental issue in quantum physics since its prediction [1]. The effect has become increasingly approachable in recent years with the achievement of precise experimental measurements of the effect [2 -5], probing the detailed dependence of the force on the properties of the materials, and measuring new variants such as corrugation effects. The theory and experiment have good agreement for simple geometries.In spite of the vast body of work on the subject (for a review, see [6] ), some properties of the force are yet under controversy. Because of the computational complexity of the problem, the main body of work on the effect is a collection of explicit calculations for simple geometries. In this Letter we resolve one of these controversies and supply general statements about Casimir forces, applicable to a broad class of geometries.The interest in repulsive Casimir and van der Waals forces has grown substantially recently due to possible practical importance in nanoscience, where such forces may play a role as a solution to stiction problems. It is known that repulsive forces are possible between molecules immersed in a medium whose properties are intermediate between the properties of two polarizable molecules [7]. Conditions for repulsion between paramagnetic materials and dielectrics without recourse for an intermediate medium were given in [8]. However, the prospect of realizing materials with nontrivial permeability on a large enough frequency range is unclear [9].It is common knowledge, based on the Casimir-Polder interaction, that small dielectric bodies interacting at large distance attract [10]. Based on summation of two-body forces one may speculate that any two dielectrics would attract at all distances. In this Letter we show that at least for the case of a symmetric configuration of two dielectrics or conductors this prediction holds independently of their distance and shape for models which can be described by a local dielectric function. Of course, in any real material as distances become small enough, i.e., compared with interatomic distances, Casimir treatment of the problem is not adequate anymore.We first emphasize that the two-body picture is not enough to prove this. Calculations of the interaction between macroscopic bodies by summation of pair interactions are only justified within second order perturbation theory. Indeed, in [8] it was demonstrated how summing two-body forces may give wrong predic...

Casimir energy of smooth compact surfaces

“…Routes to the calculation of the Casimir interaction between objects have been given previously [10][11][12][13][14][15]. We believe these all develop along the same lines, differing mainly in notation and in how the authors propose to turn their formal results into numerical values.…”

Material dependence of the wire-particle Casimir interaction

“…For identical objects or mirror symmetric configurations, this type of interaction is attractive [2] and can cause stiction in micromotors and other similar structures [3]. More generally, if the permittivities of the objects are higher or lower than those of the surrounding medium, any equilibrium position of the objects is unstable due to the Casimir interactions [4].…”

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