Vacuum Energy of the Laplacian on the Spheres

Abstract

Let Δg be the Laplacian on smooth functions on a compact Riemannian manifold (M, g)  and ζg the associated spectral zeta function. Some special values of the spectral zeta function and their generalisations such as the spectral height and spectral determinant usually defined in terms of the spectral zeta function to be ζ'g (0) and exp(ζ'g(0)) respectively, have been computed explicitly, see e.g [1,2] and [3]. Another special value of the spectral zeta function which has been a fundamental issue in quantum field theory is  the Vacuum (Casimir) energy. Casimir energy is defined, mathematically, via the spectral zeta function as a function on the set of metrics on the manifold by ζg (-1/2) [4,5] and [6]. In this paper, a general technique for computing the Casimir energy of the Laplacian on the unit n -dimensional sphere, Sn by factoring the spectral zeta function through the Riemann zeta function ζR is presented.