  The effect of edges and apertures on the Casimir energy of an arrangement of
plates and boundaries can be calculated in terms of an effective nonlocal
lower-dimensional field theory that lives on the boundary. This formalism has
been developed in a number of previous papers and applied to specific examples
with Dirichlet boundary conditions. Here we generalize the formalism to
arbitrary boundary conditions. As a specific example, the geometry of a flat
plate and a half-plate placed parallel to it is considered for a number of
different boundary conditions and the area-dependent and edge dependent
contributions to the Casimir energy are evaluated. While our results agree with
known results for those special cases (such as the Dirichlet and Neumann
limits) for which other methods of calculation have been used, our formalism is
suitable for general boundary conditions, especially for the diffractive
effects.
