Local and global Casimir energies for a semitransparent cylindrical shell

Abstract

The local Casimir energy density and the global Casimir energy for a massless scalar field associated with a λδ-function potential in a 3+1 dimensional circular cylindrical geometry are considered. The global energy is examined for both weak and strong coupling, the latter being the well-studied Dirichlet cylinder case. For weak-coupling, through O(λ 2 ), the total energy is shown to vanish by both analytic and numerical arguments, based both on Green's-function and zeta-function techniques. Divergences occurring in the calculation are shown to be absorbable by renormalization of physical parameters of the model. The global energy may be obtained by integrating the local energy density only when the latter is supplemented by an energy term residing precisely on the surface of the cylinder. The latter is identified as the integrated local energy density of the cylindrical shell when the latter is physically expanded to have finite thickness. Inside and outside the δ-function shell, the local energy density diverges as the surface of the shell is approached; the divergence is weakest when the conformal stress tensor is used to define the energy density. A real global divergence first occurs in O(λ 3 ), as anticipated, but the proof is supplied here for the first time; this divergence is entirely associated with the surface energy, and does not reflect divergences in the local energy density as the surface is approached.

Recent developments in the Casimir effect

“…which we recognize as just the semitransparent cylinder problem with m → ζ = −iω and κ → k [48]. The reduced Green's function for single plate thus is In the limit λ → ∞ we recover the well-known Green's function subject to Dirichlet boundary conditions.…”

“…These objections have recently been most forcefully presented by Graham, Jaffe, et al [45], but they date back to Deutsch and Candelas [46]. In fact, it now appears that these surface divergences can be dealt with successfully in a process of renormalization, and that finite self-energies in the sense of Boyer may be extracted [47,48].…”

Vacuum stress and closed paths in rectangles, pistons and pistols

“…This theme has been repeatedly visited since then [22,23,15,24,25,26] and will play a major role in the present work.…”

Casimir energies of cylinders: Universal function

“…Of course, there are well-known analytic results for circular cylinders [10,11,27,44]. In these cases, however, both interior and exterior contributions to the cylindrical shell are included, which is necessary, because otherwise a finite energy cannot be calculated.…”