Cancellation of nonrenormalizable hypersurface divergences and the d-dimensional Casimir piston

Understand

Using a multidimensional cut-off technique, we obtain expressions for the cut-off dependent part of the vacuum energy for parallelepiped geometries in any spatial dimension d.

• The cut-off part yields nonrenormalizable hypersurface divergences and we show explicitly that they cancel in the Casimir piston scenario in all dimensions.
• We obtain two different expressions for the d-dimensional Casimir force on the piston where one expression is more convenient to use when the plate separation a is large and the other when a is small (a useful $a \to 1/a$ duality).
• The Casimir force on the piston is found to be attractive (negative) for any dimension d.