The Shape of Compact Toroidal Dimensions $T^d_{\theta}$ and the Casimir Effect on $M^D\times T^d_{\theta}$ spacetime

Abstract

We study the influence of the shape of compact dimensions to the Casimir energy and Casimir force of a scalar field. We examine both the massive and the massless scalar field. The total spacetime topology is $M^D\times T^2_{\theta}$, where $M^D$ is the $D$ dimensional Minkowski spacetime and $T^2_{\theta}$ the twisted torus described by $R_1$, $R_2$ and $\theta$. For the case $R_1=R_2$ we found that the massive bulk scalar field Casimir energy is singular for $D$=even and this singularity is $R$-dependent and remains even when the force is calculated. Also the massless Casimir energy and force is regular only for D=4 (!). This is very interesting phenomenologically. We examine the energy and force as a function of $\theta$. Also we address the stabilization problem of the compact space. We also briefly discuss some phenomenological implications.