In this study, the Casimir energy for massive scalar field with periodic boundary condition was calculated on spherical surfaces with S1, S2, and S3 topologies. To obtain the Casimir energy on a spherical surface, the contribution of the vacuum energy of Minkowski space is usually subtracted from that of the original system. In the large mass limit for surface S2, however, some divergences would eventually remain in the obtained result. To remove these remaining divergences, a secondary renormalization program was manually performed. In the present work, a direct approach for calculation of the Casimir energy has been introduced. In this approach, two similar configurations were considered and then the vacuum energies of these configurations were subtracted from each other. This method provides more physical meaning than the other common methods. Additionally, in the large mass limit for surface S2, it provides a situation in which the second renormalization program is automatically conducted in the calculation procedure, and there was no longer a need to do so manually. Finally, by plotting the obtained values for the Casimir energy of the topologies and investigating their appropriate limits, the logic agreement between the results of our scheme and those of previous studies was discussed.
Large mass limit of the continuum theories in Kaplan’s formulation
