Radiative Correction to the Casimir Force on a Sphere

TL;DR

This paper calculates the first radiative correction to the Casimir energy of a conducting spherical shell using covariant perturbation theory, revealing a correction term with a logarithmic dependence on the sphere's radius.

Contribution

It extends the calculation of Casimir energy corrections to spherical geometries using a covariant perturbation approach with boundary constraints.

Findings

Radiative correction is of order α/(R^2 m_e)

Correction has a logarithmic dependence on m_e R

Sign of correction is opposite to that for parallel plates

Abstract

The first radiative correction to the Casimir energy of a perfectly conducting spherical shell is calculated. The calculation is performed in the framework of covariant perturbation theory with the boundary conditions implemented as constraints. The formalism is briefly reviewed and its use is explained by deriving the known results for two parallel planes. The ultraviolet divergencies are shown to have the same structure as those for a massive field in zeroth order of $\alpha$. In the zeta-functional regularization employed by us no divergencies appear. If the radius of the sphere is large compared to the Compton wavelength of the electron the radiative correction is of order $\alpha/(R^2m_e)$ and contains a logarithmic dependence on $m_eR$. It has the opposite sign but the same order of magnitude as in the case of two parallel planes.