Scalar and spinor Casimir energies in even-dimensional Kaluza-Klein spaces of the form M4 x S/sup N/1 x S/sup N/2 x xxx

We use two methods of computing the unique logarithmically divergent part of the Casimir energy for massive scalar and spinor fields defined on even-dimensional Kaluza-Klein spaces of the form M4 x S/sup N/1 x S/sup N/2 x xxx. Both methods (heat kernel and direct) give identical results. The first evaluates the required internal zeta function by identifying it in the asymptotic expansion of the trace of the heat kernel, and the second evaluates the zeta function directly using the Euler-Maclaurin sum formula. In Appendix C we tabulate these energies for all spaces of total internal dimension ≤6. These methods are easily applied to vector and tensor fields needed in computing one-loop vacuum gravitational energies on these spaces. Stable solutions are given for internal structure S2 x S2