Casimir and Vacuum Energy of 5D Warped System and Sphere Lattice Regularization

We examine the Casimir energy of 5D electro-magnetism in the recent standpoint. Z$_2$ symmetry is taken into account. After confirming the consistency with the past result, we do new things based on a new regularization. The regularization is based on the minimal area principle and the regularized configuration is the {\it sphere lattice}. We do it not in the Kaluza-Klein expanded form but in the closed form. The formalism is based on the heat-kernel approach using the position/momentum propagator. A useful expression of the Casimir energy, in terms of the P/M propagator, is obtained. Renormalization flow is realized as the change along the extra-axis.