Casimir force in compact non-commutative extra dimensions and radius stabilization

We compute the one loop Casimir energy of an interacting scalar field in a compact non-commutative space of ℝ^1,d x double-struck T sign²_θ, where we have ordinary flat (1 + d)-dimensional Minkowski space and two-dimensional non-commuative torus. We find that next order correction due to the noncommutativity still contributes an attractive force and thus will have a quantum instability. However, vector field with periodic boundary condition gives repulsive force for d > 5 and we get a stabilized radius. This suggests a stabilization mechanism for a senario in Kaluza-Klem theory, where some of the extra dimensions are non commutative.