We compute the Casimir energy for a massive scalar field constrained between two parallel planes (Dirichlet boundary conditions) in order to investigate its non-relativistic limit. Instead of employing the usual relativistic dispersion relation omega(p) = <img SRC="http:/img/fbpe/bjp/v31n1/08eq01.gif">, we use the non-relativistic one, omega(p) = p²/2m. It turns out that the Casimir energy is zero. We include the relativistic corrections perturbatively and show that at all orders the Casimir energy remains zero, since each term in the power series in 1/c² is proportional to the Riemann zeta function of a negative even integer. This puzzling result shows that, at least for the free massive scalar field, the Casimir effect is non-perturbative in the relativistic sense.

The influence of an external constant uniform magnetic field on the Casimir energy associated with a Dirac field under antiperiodic boundary condition is computed using Schwinger's method. The obtained result shows that the magnetic field enhances the fermionic Casimir energy, in oposition to the bosonic Casimir energy which is inhibited by the magnetic field.

We compute the Casimir pressure between an unusual pair of parallel plates, namely, a perfectly conducting plate (<FONT FACE="Symbol">e®¥)</font> and an infinitely permeable one (<FONT FACE="Symbol">m®¥</font>) with the generalized <FONT FACE="Symbol">z</font>-function method. The result for this problem, which has been rarely discussed in the literature, is a repulsive Casimir force. The <FONT FACE="Symbol">z</font>-function method provides a very compact and economic way of obtaining the final result.