Casimir effect as a sum over modes in dissipative systems

The aim of this paper is to show that within the open-system framework the sum-over-modes approach of Casimir leads to the Lifshitz formula for the Casimir free energy. A general result applicable to arbitrary geometries is obtained through the use of Ford, Lewis, and O'Connell's remarkable formula. Additionally, we address the possibility of obtaining the Casimir energy as a sum over complex "modes." We show in this case that the standard sum-over-modes formula must be suitably generalized to avert unphysical complex energies. Finally, we apply our results to several standard examples.