Quantum field theory on noncommutative space-time and its implication on spin-statistics theorem

We study properties of a scalar quantum field theory on noncommutative space-times. We show that field theories on a noncommutative plane have ultraviolet divergences, while the theory on a noncommutative cylinder is ultraviolet finite. Thus, ultraviolet behavior of a field theory on noncommutative spaces is sensitive to the topology of the space-time, namely to its compactness, We present general arguments for the case of higher space-time dimensions and as well discuss the implication of the noncommutativity on the spin-statistics theorem.