Spectrum of the Casimir effect on a torus

Abstract

The finite (zeta-regularized) energy density of the Casimir effect is decomposed into finite contributions corresponding to a continuous spectrum of frequencies. This is done in a purely discrete way which respects at each step the quantized nature of the effect. The general equations that any such spectral decomposition must obey are derived, and several specific mathematical realizations of the same are given. As is explicitly shown with the physical Planck spectrum, this procedure opens the way to the obtaintion of some particular solutions which could be checked experimentally. In particular, expressions for the spectral decompositions are obtained for the cases of a circumferenceC ¹ and of a torusT ², and for the respectively related cases of one and two pairs of parallel plates, in 1+1 and 3+1 spacetime dimensions, for a massless scalar and for an electromagnetic field, respectively.