Casimir energy of the massless conformal scalar field on S-2 by the point-splitting method

We calculate the Casimir energy of the massless conformal scalar held on the surface (S-2) of a 3 dimensional Riemann sphere by using the point-splitting, mode sum and the zeta-function renormalization methods. We also consider the half space case with both the Dirichlet and the Neumann boundary conditions. This problem is interesting since the Casimir energy could be calculated analytically by various methods, thus allowing us to compare different regularization schemes. (C) 1997 American Institute of Physics.