Attractive or repulsive nature of Casimir force in D-dimensional Minkowski spacetime

The dependence of Casimir energy (associated with a massless scalar field) on spacetime dimensionality (D) is shown to be strongly entangled with the type of geometric bounds and the kind of macroscopic boundary conditions imposed on the field. In the case of a massless scalar field satisfying Dirichlet boundary conditions in the presence of a hyperparallelepipedal cavity with p sides of finite length L and D-p-1 sides with length much greater than L, a new compact integral formula, more suitable to analyze the nature of the Casimir force, is obtained. The force is attractive if p is odd or for very large even values of p, irrespective of D. For each small even p there exists a critical spacetime dimension Dc(p) such that the force is repulsive if D<Dc and attractive otherwise. As a consequence, the instability of the semiclassical Abraham-Lorentz-Casimir model of the electron is proved to depend on the spacetime dimensionality