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.
