A Generalized Mode Summation Formula of the Zero-Point Energy in a Cavity

Abstract

The summation of zero-point energies defined by ∑_m,n(π⁄2)\\sqrt(n+β)²⁄a²+m²⁄b² is studied. The parameter β is a positive real number and the zero-point energy of a massless scalar field is given by setting β=0 as a special case. The Casimir energy is obtained by the generalized Abel–Plana formula as a function of β and it is found that the Casimir energy density in the limit b→∞ changes from negative to positive as increasing β. The Casimir effect in the three dimensional case is also considered.