Local zeta regularization and the Casimir effect

The local zeta regularization allows to treat local divergences appearing in quantum field theory; these are renormalized by pure analytic continuation (in the parameter of the regulator), with no need to remove or subtract divergent terms. This approach can be applied to the stress-energy tensor of the Casimir effect, and works as well on curved space-times. It is not useless to illustrate the power and elegance of this method in a simple case. In the present paper, our attention is devoted to the case of a neutral, massless scalar field in flat space-time, on a space domain with suitable (e.g., Dirichlet) boundary conditions. After a general outline of the local zeta method for the Casimir effect, we exemplify it in the typical case of a (Dirichlet) field between two parallel plates, or outside them. The results agree with the ones obtained by more popular methods, such as point splitting regularization. Connections with the existing literature on this subject are indicated.