THE VACUUM ENERGY DENSITY FOR SPHERICAL AND CYLINDRICAL UNIVERSES

The vacuum energy density (Casimir energy) corresponding to a massless scalar quantum field living in different universes (mainly no-boundar y ones), in several dimensions, is calculated. Hawking's zeta function regularization procedure supplemented with binomial expansion is show n to be a rigorous and well suited method for performing the analysis. It is compared with other more involved techniques. The principal-par t prescription is used to deal with the poles that eventually appear. Results of the analysis are the absence of poles at four dimensions (4 D) (for a 4D Riemann sphere and for a 4D cylinder of 3D Riemann spheri cal section), the total coincidence of the results corresponding to a 3D and a 4D cylinder (the first after pole subtraction), and the fact that the vacuum energy density for cylinders is (in absolute value) ov er an order of magnitude smaller than for spheres of the same dimensio n.