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

Abstract

The summation of zero-point energies defined by \(\sum_{m,n}(\pi/2)\sqrt{(n+\beta)^{2}/a^{2}+m^{2}/b^{2}}\) 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.