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The Zeta Function Approach to Casimir Energy Calculations in Higher Dimensions

The vacuum fluctuations give rise to a number of phenomena; however, the the Casimir Effect is arguably the most salient manifestation of the quantum vacuum. In its most basic form it is realized through the interaction of a pair of neutral parallel conducting plates. The presence of the plates modifies the quantum vacuum, and this modifcation causes the plates to be pulled toward each other. The Casimir Effect has also been explored in the context of higher dimensional theories. The non-trivial boundary conditions imposed by compactified periodic higher dimensions is know to alter the vacuum in a quantifiable way, and is a possible solution to the issue of modulus stabilization, namely - the stabilization of higher dimensions. Typical in Casimir energy calculations are renormalization techniques which are used to tame the infinite sums and integrals that arise. These calculations are usally fairly involved, and explicit pedagogical material is sparse. The purpose of this paper is to introduce Dimensional Regularization techniques specific to Casimir energy calculations with an additional compactified spatial dimension.