Casimir Energy of 5D Electromagnetism and New Regularization Based on Minimal Area Principle

We examine the Casimir energy of 5D electromagnetism in the recent standpoint. The bulk geometry is flat. Z$_2$ 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 {\it new regularization}. In the treatment of the divergences, we introduce IR and UV cut-offs and {\it restrict} the (4D momentum, extra coordinate)-integral region. The regularized configuration is the {\it sphere lattice}, in the 4D continuum space, which changes along the extra coordinate. The change (renormalization flow) is specified by the {\it minimal area principle}, hence this regularization configuration is string-like. We do the analysis not in the Kaluza-Klein expanded form but in a {\it closed} form. We do {\it not} use any perturbation. The formalism is based on the heat-kernel approach using the {\it position/momentum propagator}. Interesting relations between the heat-kernels and the P/M propagators are obtained, where we introduce the {\it 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 {\it space-time quantization}, is argued. {\it 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.