Zero-point energy of a conducting spherical shell

The zero-point energy of a conducting spherical shell is evaluated by impos ing boundary conditions on the potential A(mu), and on the ghost fields. Th e scheme requires that temporal and tangential components of A(mu) perturba tions should vanish at the boundary, jointly with the gauge-averaging funct ional, first chosen to be of the Lorentz type. Gauge invariance of such bou ndary conditions is then obtained provided that the ghost fields vanish at the boundary. Normal and longitudinal modes of the potential obey an entang led system of eigenvalue equations, whose solution is a linear combination of Bessel functions under the above assumptions, and with the help of the F eynman choice for a dimensionless gauge parameter. Interestingly, ghost mod es cancel exactly the contribution to the Casimir energy resulting from tra nsverse and temporal modes of A(mu), jointly with the decoupled normal mode of A(mu). Moreover, normal and longitudinal components of A(mu) for the in terior and the exterior problem give a result in complete agreement with th e one first found by Boyer, who studied instead boundary conditions involvi ng TE and TM modes of the electromagnetic field. The coupled eigenvalue equ ations for perturbative modes of the potential are also analyzed in the axi al gauge, and for arbitrary values of the gauge parameter. The set of modes which contribute to the Casimir energy is then drastically changed, and co mparison with the case of a flat boundary sheds some light on the key featu res of the Casimir energy in noncovariant gauges.