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We examine the Casimir energy of 5D electromagnetism in the recentstandpoint. The bulk geometry is flat. Z$_2$ symmetry and the periodicproperty, for the extra coordinate, are taken into account. After confirmingthe consistency with the past result, we do new things based on a {\it newregularization}. In the treatment of the divergences, we introduce IR and UVcut-offs and {\it restrict} the (4D momentum, extra coordinate)-integralregion. The regularized configuration is the {\it sphere lattice}, in the 4Dcontinuum space, which changes along the extra coordinate. The change(renormalization flow) is specified by the {\it minimal area principle}, hencethis regularization configuration is string-like. We do the analysis not in theKaluza-Klein expanded form but in a {\it closed} form. We do {\it not} use anyperturbation. The formalism is based on the heat-kernel approach using the {\itposition/momentum propagator}. Interesting relations between the heat-kernelsand 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/Mpropagator, is obtained. The restricted-region approach is replaced by theweight-function approach in the latter-half description. Its meaning, inrelation to the {\it space-time quantization}, is argued. {\it Finite} Casimirenergy is numerically obtained. The compactification-size parameter(periodicity) suffers from the renormalization effect. Numerical evaluation isexploited. Especially the minimal surface lines in the 5D flat space areobtained both numerically using the Runge-Kutta method and analytically usingthe general solution.

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