Local and Global Casimir Energies for a Semitransparent Cylindrical Shell

TL;DR

This paper investigates the local and global Casimir energies for a scalar field around a semitransparent cylindrical shell, revealing that the total energy vanishes at weak coupling and divergences are associated with surface energies.

Contribution

It provides a detailed analysis of Casimir energies in cylindrical geometry, including the first proof of a global divergence at third order related to surface energy.

Findings

Global energy vanishes at weak coupling to second order.

Divergences occur at third order, linked to surface energy.

Local energy density diverges near the shell surface.

Abstract

The local Casimir energy density and the global Casimir energy for a massless scalar field associated with a $\lambda\delta$-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 $\mathcal{O}(\lambda^2)$, 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 delta-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 $\mathcal{O}(\lambda^3)$, as anticipated, but the proof is supplied here for the first time; this divergence is entirely associated with the surface energy, and does {\em not} reflect divergences in the local energy density as the surface is approached.