Vacuum Casimir energy densities and field divergences at boundaries

Abstract

We consider and review the emergence of singular field fluctuations or energy densities at sharp boundaries or point-like field sources in the vacuum. The presence of singular energy densities of a field may be relevant from a conceptual point of view, because they contribute to the self-energy of the system. They could also generate significant gravitational effects. We first consider the case of the interface between a metallic boundary and the vacuum, and obtain the structure of the singular electric and magnetic energy densities at the interface through an appropriate limit from a dielectric to an ideal conductor. Then, we consider the case of a nondispersive and nondissipative point-like source of the electromagnetic field, described by its polarizability, and show that also in this case the electric and magnetic energy densities show a singular structure at the source position. We discuss how, in both cases, these singularities give an essential contribution to the electromagnetic self-energy of the system; moreover, they solve an apparent inconsistency between the space integral of the field energy density and the average value of the field Hamiltonian. The singular behavior we find is softened, or even eliminated, for boundaries fluctuating in space and for extended field sources. We discuss in detail the case in which a reflecting boundary is not fixed in space but is allowed to move around an equilibrium position, under the effect of quantum fluctuations of its position. Specifically, we consider the simple case of a one-dimensional massless scalar field in a cavity with one fixed and one mobile wall described quantum-mechanically. We investigate how the possible motion of the wall changes the vacuum fluctuations and the energy density of the field, compared with the fixed-wall case. Also, we explicitly show how the fluctuating motion of the wall smears out the singular behaviour of the field energy density at the boundary.