Attractive or repulsive nature of the Casimir force for rectangular cavity

Abstract

The Casimir effect giving rise to an attractive or repulsive force between the configuration boundaries that confine the massless scalar field is reexamined for a $(D\ensuremath{-}1)$-dimensional rectangular cavity with unequal finite $p$ edges and different spacetime dimensions $D$ in this paper. With periodic or Neumann boundary conditions, the energy is always negative. The case of Dirichlet boundary conditions is more complicated. The sign of the Casimir energy satisfying Dirichlet conditions on the surface of a hypercube (a cavity with equal finite $p$ edges) depends on whether $p$ is even or odd. In the general case (a cavity with unequal $p$ edges), however, we show that the sign of the Casimir energy does not only depend on whether $p$ is odd or even. Furthermore, we find that the Casimir force is always attractive if the edges are chosen appropriately. It is interesting that the Casimir force may be repulsive for odd $p$ cavity with unequal edges, in contrast with the same problem in a hypercube case.