Local and Global Casimir Energies in a Green's Function Approach

Abstract

The effects of quantum fluctuations in fields confined by background configurations may be simply and transparently computed using the Green's function approach pioneered by Schwinger. Not only can total energies and surface forces be computed in this way, but local energy densities, and in general, all components of the vacuum expectation value of the energy-momentum tensor may be calculated. For simple geometries this approach may be carried out exactly, which yields insight into what happens in less tractable situations. In this talk I will concentrate on the example of a scalar field in a circular cylindrical delta-function background. This situation is quite similar to that of a spherical delta-function background. The local energy density in these cases diverges as the surface of the background is approached, but these divergences are integrable. The total energy is finite in strong coupling, but in weak coupling a divergence occurs in third order. This universal feature is shown to reflect a divergence in the energy associated with the surface, the integrated local energy density within the shell itself, which surface energy should be removable by a process of renormalization.

Vacuum energy of scalar fields on spherical shells with general matching conditions

“…Once the analytically continued expression for ( 11) is obtained, one can use it to analyze, in particular, the vacuum energy of the system [2,6]. Due to the spherical symmetry of the system, the spectral zeta function (11) can be written as…”

“…In this paper, we focus on the analysis of the Casimir pressure on a single sphere, paying particular attention to those cases where this pressure can be defined unambiguously, thus continuing the work of [10], which focused on the an analysis of the interaction energy for a massless scalar field in the presence of two concentric δ-δ ′ spheres. It is important to point out that there are only a few configurations for which the Casimir self-energy is well-defined without the need for renormalization [11]. Some important examples of such cases include the dilute limit for spheres and cylinders [12,13], a magnetodielectric object where the speed of light is the same inside and outside [14,15], a perfectly conducting spherical or cylindrical shell [16,17], and the δ-potential weak limit for massless scalar fields [18].…”