### Calculating Casimir Energies
in Renormalizable Quantum Field Theory

**Kimball A. Milton**

*Department of Physics and Astronomy*

*University of Oklahoma*

*Norman, OK 73019-0430
*

Quantum vacuum energy has been known
to have observable consequences since 1948 when Casimir calculated
the force of attraction between parallel uncharged plates, a phenomenon
confirmed experimentally with ever increasing precision. Casimir
himself suggested that a similar attractive

self-stress existed for a conducting spherical shell, but Boyer
obtained a repulsive stress. Other geometries and higher dimensions
have been considered over the years. Local effects, and divergences
associated with surfaces and edges have been considered by several
authors. Quite recently, Graham et al. have re-examined such calculations,
using conventional techniques of perturbative quantum field theory
to remove divergences, and have suggested that previous self-stress
results may be suspect. Here we show that the examples considered
in their work are misleading; in particular, it is well-known

that in two dimensions a circular boundary has a divergence in
the Casimir energy for massless fields, while for general dimension
$D$ not equal to an even integer the corresponding Casimir energy
arising from massless fields interior and exterior to a hyperspherical
shell is finite. It has also long been recognized that the Casimir
energy for massive fields is divergent for $D\ne1$. These conclusions
are reinforced by a calculation of the relevant leading Feynman
diagram in $D$ dimensions. There is therefore no doubt of the
validity of the conventional finite Casimir calculations.