Casimir energies in M4 >= N for even N. Green's-function and zeta-function techniques.

Abstract

The Green's-function technique developed in the first paper in this series is generalized to apply to massive scalar, vector, second-order tensor, and Dirac spinor fields, as a preliminary to a full graviton calculation. The Casimir energies are of the form ${u}_{\mathrm{Casimir}}$=(1/${a}^{4}$)[${\ensuremath{\alpha}}_{N}$lna/b)+${\ensuremath{\beta}}_{N}$], where N (even) is the dimension of the internal sphere, a is its radius, and ${b}^{\mathrm{\ensuremath{-}}1}$ is an ultraviolet cutoff (presumably at the Planck scale). The coefficient of the divergent logarithm, ${\ensuremath{\alpha}}_{N}$, is unambiguously obtained for each field considered. The Green's-function technique gives rise to no difficulties in the evaluation of imaginary-mass-mode contributions to the Casimir energy. In addition, a new, simplified \ensuremath{\zeta}-function technique is presented which is very easily implemented by symbolic programs, and which, of course, gives the same results. An error in a previous \ensuremath{\zeta}-function calculation of the Casimir energy for even N is pointed out.