Zeta function continuation and the Casimir energy on odd- and even-dimensional spheres

Abstract

The zeta function continuation method is applied to compute the Casimir energy on spheres SN. Both odd and even dimensional spheres are studied. For the appropriate conformally modified Laplacian A the Casimir energy $is shown to be finite for all dimensions even though, generically, it should diverge in odd dimensions due to the presence of a pole in the associated zeta function ζA(s). The residue of this pole is computed and shown to vanish in our case. An explicit analytic continuation of ζA(s) is constructed for all values of N. This enables us to calculate$ exactly and we find that the Casimir energy vanishes in all even dimensions. For odd dimensions δ is never zero but alternates in sign as N increases through odd values. Some results are also derived for the Casimir energy of other operators of Laplacian type