Kaluza–Klein pistons with noncommutative extra dimensions

We calculate the scalar Casimir energy and Casimir force for an $\mathbb R^3{\times}N$ Kaluza–Klein piston setup in which the extra-dimensional space $N$ contains a noncommutative two-dimensional sphere $S_\text{fz}$. We study the cases with $T^d\times S_\text{fz}$ and $S_\text{fz}$ as the extra-dimensional spaces, where $T^d$ is the $d$-dimensional commutative torus, and examine the validity of the results and the regularization obtained in the piston setup in each case. We examine the Casimir energy with one-loop corrections for one piston chamber due to the self-interacting scalar field in the noncommutative geometry. We compute with some approximations. We compare the obtained results with the results of analogous computations for the Minkowski space–time $M^D$. In conclusion, we discuss the stabilization of the extra-dimensional space in the piston setup.