Kaluza-Klein Pistons with non-Commutative Extra Dimensions

We calculate the scalar Casimir energy and Casimir force for a R×N Kaluza-Klein piston setup in which the extra dimensional space N contains a non-commutative 2sphere, SFZ . The cases to be studied are T d × SFZ and SFZ respectively as extra dimensional spaces, with T d the d dimensional commutative torus. The validity of the results and the regularization that the piston setup offers are examined in both cases. Finally we examine the 1-loop corrected Casimir energy for one piston chamber, due to the self interacting scalar field in the non-commutative geometry. The computation is done within some approximations. We compare this case for the same calculation done in Minkowski spacetime M. A discussion on the stabilization of the extra dimensional space within the piston setup follows at the end of the article. Introduction Casimir pistons have received much attention in recent years [2, 13, 12, 18, 28, 29, 20, 15]. This is due to the very attractive quantitative features that the piston setup has. Most of these are concentrated on the fact that although the Casimir energy of a scalar field between parallel plates can be singular, in the piston setup the Casimir force is regular. Thus we can have closed form formulas for the Casimir force. The Casimir piston can be realized by three parallel plates in which the one in the middle can be movable. Suppose between the plates a scalar field exists. The boundary conditions on the plates can be either Dirichlet or Neumann. In most cases it was found that the movable plate is attracted to the nearest end. This holds for rectangular cavities. However when closed cylinder with arbitrary cross section is considered with Dirichlet piston and ∗[email protected]