Casimir energy density of the scalar field on two-dimensional spherical surface

In this thesis, the Casimir energy of a massless conformal scalar field and a massless non-conformal scalar field on the surface of two-dimensional full and half spheres has been recalculated. Firstly, we have re-evaluated the Casimir energy of a massless conformal scalar field on the full and half surface S2 with both the Dirichlet and the Neumann boundry condition of a 3-dimensional Riemann sphere by using the mode sum methods. Then, the Casimir energy density of a massless conformal scalar field on the surface of the full sphere has been recalculated by using the point-splitting method. Finally, the Casimir energy of a massles non-conformal scalar field on the full and half sphere S2 with both the Dirichlet and Neumann boundry conditions has been obtained by using zeta function regularization.