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

TL;DR

This paper develops a multidimensional approach to analyze the Casimir piston, demonstrating the cancellation of hypersurface divergences and providing explicit formulas for the Casimir force in various dimensions, confirmed by numerical results.

Contribution

It introduces a method to handle hypersurface divergences in multidimensional Casimir calculations and derives explicit force expressions applicable in any dimension.

Findings

Hypersurface divergences cancel in the Casimir piston scenario across all dimensions.

Two formulas for the Casimir force are derived, suitable for different plate separations.

Numerical agreement with previous optical path results confirms the validity of the approach.

Abstract

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. We apply the d-dimensional formulas (both expressions) to the two and three-dimensional Casimir piston with Neumann boundary conditions. The 3D Neumann results are in numerical agreement with those recently derived in arXiv:0705.0139 using an optical path technique providing an independent confirmation of our multidimensional approach. We limit our study to massless scalar fields.