Systems of Distinct Representatives for Random Sets

Abstract

We compute the one loop Casimir energy of an interacting scalar field in a compact non-commutative space of R 1,d × T 2 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-Klein theory, where some of the extra dimensions are non commutative.1 This infinity can of course be removed by proper-dimensional regularization.