Casimir energies in M⁴≥^N for even N. Green’s-function and zeta-function techniques

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_Casimir=(1/a⁴)[α_Nlna/b)+β_N], where N (even) is the dimension of the internal sphere, a is its radius, and b^-1 is an ultraviolet cutoff (presumably at the Planck scale). The coefficient of the divergent logarithm, α_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 ζ-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 ζ-function calculation of the Casimir energy for even N is pointed out.