Generalized Casimir Energies for Systems with Arbitrary (ǫ(z)) Planar Geometry

The Casimir force between parallel plates of arbitrary kind is shown to be simply related to the plates transmission and reflection coefficient. A trivial application of this general relation leads to known Lifshitz force between dielectrics as well as its generalizations. The simple two ideally conducting parallel plates Casimir force F = − π 240 h̄c a4 [1] recently been measured experimentally[9, 10]. It has various generalizations to different kinds of fields, boundary conditions and geometries whose calculation is in general quite complicated[6, 7, 8][2]. A new variant in which each of the plates conducts in specific in general different direction has been exactly evaluated recently[3] and can hopefully be seen experimentally[16]. Another generalization is to replace the ideal conducting plates by dielectrics. Thus we may for example consider the vacuum energy corresponding to electromagnetic action in matter 1 8π ∫ dx(εE − 1 μ B) as some very complicated functional of ε(x), μ(x). In general this energy will contain divergent terms. When we consider the energy difference between two configurations (given by ε(x), μ(x)) the divergent terms should cancel provided that the configurations may be related by a physical process which does not involve a change in internal matter properties (otherwise such a change would bring about a dependence on the physical UV cutoff). In particular relative displacement of the different dielectrics in space should result in a (finite) force. The special case of the force between two infinite parallel dielectrics (corresponding to ε = 