Casimir Energies for Spheres in n-dimensional Minkowski Space and Generalized Bernoulli Polynomials

We consider a mathematical version of conformal quantum field theory in n-dimensions which give a unified approach to quantum field theory on Minkowski space and the Einstein universe. This unifying treatment enables us to relate massless fields in Minkowski space with massless fields in the Einstein universe into which such fields uniquely and conformally extend. Apart from certain global differences, fields in the Einstein universe approximate fields in Minkowski space as the radius of the Einstein universe tends to infinity and we utilize this fact to describe what seems to be a precise method for determining Casimir energies on spheres of arbitrary radii in n-dimensional Minkowski space. Specifically, our exact results for Casimir energies of massless scalar fields in the n dimensional Einstein universe involving generalized Bernoulli polynomials together with results on conformal covariance properties of massless fields and a scaling property of the Casimir energy for n even should enable us to obtain exact results for Casimir energies for massless scalar fields with Dirichlet boundary conditions on spheres in n dimensional Minkowski space, at least for n even. We illustrate our calculational method in detail for the n = 2 case, which is completely representative of the arbitrary n case (n even).