Scalar Casimir energies inM4≥Nfor even N

Abstract

We construct a Green's-function formalism for computing vacuum-fluctuation energies of scalar fields in 4+N dimensions, where the extra N dimensions are compactified into a hypersphere ${S}^{N}$ of radius a. In all cases a leading cosmological energy term ${u}_{\mathrm{cosmo}\mathrm{\ensuremath{\propto}}{a}^{N}/{b}^{4+N}}$ results. Here b is an ultraviolet cutoff at the Planck scale. In all cases an unambiguous Casimir energy is computed. For odd N these energies agree with those calculated by Candelas and Weinberg. For even N, the Casimir energy is logarithmically divergent: ${u}_{\mathrm{Casimir}\mathrm{\ensuremath{\sim}}({\ensuremath{\alpha}}_{N}/{a}^{4})\mathrm{l}\mathrm{n}(\mathit{a}/\mathit{b})}$. The coefficients ${\ensuremath{\alpha}}_{N}$ are computed in terms of Bernoulli numbers.