  A novel technique based on Schwinger's proper time method is applied to the
Casimir problem of the M.I.T. bag model. Calculations of the regularized vacuum
energies of massless scalar and Dirac spinor fields confined to a static and
spherical cavity are presented in a consistent manner. While our results agree
partly with previous calculations based on asymptotic methods, the main
advantage of our technique is that the numerical errors are under control.
Interpreting the bag constant as a vacuum expectation value, we investigate
potential cancellations of boundary divergences between the canonical energy
and its bag constant counterpart in the fermionic case. It is found that such
cancellations do not occur.
