Unified treatment of some Casimir energies and Lamb shifts: A dielectric between two ideal conductors

Lamb shifts and Casimir energies, often thought of as long-range and short-range effects, respectively, and studied separately, are each a manifestation of quantum electrodynamics and can be studied together. We do so for a dielectric medium between two parallel ideal conducting plates. We extract the usual Casimir energy and a bulk Lamb shift by studying the quantum fluctuations of the radiation field in the dielectric. We derive finite expressions for the Casimir energy and the bulk Lamb shift valid for any permittivity satisfying the Kramers-Kronig-relation; some of the Casimir shifts obtained are simpler in form than any in the literature. We separate the divergent and finite contributions to the bulk Lamb shift. For the dilute nonrelativistic gas we show that the divergent contribution to the bulk Lamb shift defines the bare electron mass in terms of the physically observed free-electron mass. Although we lack a physical interpretation for the necessary subtraction in the case of an arbitrary dielectric, it is natural to interpret the finite part of the bulk energy as a "Lamb shift" in this case too. We show that the derived finite bulk Lamb shift and Casimir energies in the limit of a dilute homogeneous gas are consistent with earlier results for a single atom between two ideal conductors, and for an atom near one of the walls. hs an application, the radiative contribution to the interaction energy of a single electron with uniform probability density between two ideal walls is obtained. [S1050-2947(98)01602-3].