Casimir energy of the universe and new regularization of higher dimensional quantum field theories

Abstract

Casimir energy is calculated for the 5D electromagnetism and 5D scalar theory in the warped geometry. It is compared with the flat case. A new regularization, called sphere lattice regularization, is taken. In the integration over the 5D space, we introduce two boundary curves (IR-surface and UV-surface) based on the minimal area principle. It is a direct realization of the geometrical approach to the renormalization group. The regularized configuration is closed-string like. We do not take the KK-expansion approach. Instead, the position/momentum propagator is exploited, combined with the heat-kernel method. All expressions are closed-form (not KK-expanded form). The generalized P/M propagators are introduced. We numerically evaluate Λ(4D UV-cutoff), ω(5D bulk curvature, warp parameter) and T(extra space IR parameter) dependence of the Casimir energy. We present two new ideas in order to define the 5D QFT: 1) the summation (integral) region over the 5D space is restricted by two minimal surfaces (IR-surface, UV-surface); or 2) we introduce a weight function and require the dominant contribution, in the summation, is given by the minimal surface. Based on these, 5D Casimir energy is finitely obtained after the proper renormalization procedure. The warp parameter ω suffers from the renormalization effect. The IR parameter T does not. We examine the meaning of the weight function and finally reach a new definition of the Casimir energy where the 4D momenta (or coordinates) are quantized with the extra coordinate as the Euclidean time (inverse temperature). We examine the cosmological constant problem and present an answer at the end. Dirac's large number naturally appears.