Casimir Energy of the Laplacian on a Riemannian Manifold

Abstract

Special values of spectral zeta function on Riemannian manifolds have been computed using various numerical approximation schemes. The roles of some of those values are of fundamental importance in quantum field theory. A particular value of interest in this chapter is the Casimir energy defined, mathematically, via the spectral zeta function as a function on the set of metrics on the manifold by \(\zeta_g (-\frac{1}{2})\) [1,2] and [3]. In this chapter, a general method 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 \(\zeta_R\) is  addressed. The spectral zeta function of the Laplacian can be computed using this method on a variety of different Riemannian manifolds.