Casimir Energy of 5D Electromagnetism and New Regularization Based on Minimal Area Principle
Abstract
We examine the Casimir energy of 5D electromagnetism in the recent standpoint. The bulk geometry is flat. Z2 symmetry and the periodic property, for the extra coordinate, are taken into account. After confirming the consistency with the past result, we do new things based on a new regularization. In the treatment of the divergences, we introduce IR and UV cut-offs and restrict the (4D momentum, extra coordinate)-integral region. The regularized configuration is the sphere lattice, in the 4D continuum space, which changes along the extra coordinate. The change (renormalization flow) is specified by the minimal area principle, hence this regularization configuration is string-like. We do the analysis not in the Kaluza-Klein expanded form but in a closed form. We do not use any perturbation. The formalism is based on the heat-kernel approach using the position/momentum propagator. Interesting relations between the heat-kernels and the P/M propagators are obtained, where we introduce the generalized P/M propagators. A useful expression of the Casimir energy, in terms of the P/M propagator, is obtained. The restricted-region approach is replaced by the weight-function approach in the latter-half description. Its meaning, in relation to the space-time quantization, is argued. Finite Casimir energy is numerically obtained. The compactification-size parameter (periodicity) suffers from the renormalization effect. Numerical evaluation is exploited. Especially the minimal surface lines in the 5D flat space are obtained both numerically using the Runge-Kutta method and analytically using the general solution.