We consider a model which effectively restricts the functional integral of Yang-Mills theories to the fundamental modular region. Using algebraic arguments, we prove that this theory has the same divergences as ordinary Yang Mills theory in the Landau gauge and that it is unitary. The restriction of the functional integral is interpreted as a kind of spontaneous breakdown of the BRS symmetry.

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.

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We derive boundary conditions for electromagnetic fields on a -function plate. The optical properties of such a plate are shown to necessarily be anisotropic in that they only depend on the transverse properties of the plate. We unambiguously obtain the boundary conditions for a perfectly conducting -function plate in the limit of infinite dielectric response. We show that a material does not ''optically vanish'' in the thin-plate limit. The thin-plate limit of a plasma slab of thickness d with plasma frequency ! 2 p ¼ p =d reduces to a -function plate for frequencies (( 1. We show that the Casimir interaction energy between two parallel perfectly conducting -function plates is the same as that for parallel perfectly conducting slabs. Similarly, we show that the interaction energy between an atom and a perfect electrically conducting -function plate is the usual Casimir-Polder energy, which is verified by considering the thin-plate limit of dielectric slabs. The ''thick'' and ''thin'' boundary conditions considered by Bordag are found to be identical in the sense that they lead to the same electromagnetic fields.