Quantum field theory on non-commutative space-times and the persistence of ultraviolet divergences

We study properties of a scalar quantum field theory on two-dimensional non-commutative space-times. Contrary to the common belief that non-commutativity of space-time would be a key to remove the ultraviolet divergences, we show that field theories on a non-commutative plane with the most natural Heisenberg-like commutation relations among coordinates or even on a non-commutative quantum plane with Eq(2) symmetry have ultraviolet divergences, while the theory on a non-commutative cylinder is ultraviolet finite. Thus, ultraviolet behavior of a field theory on non-commutative 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 symmetry transformations of physical states on non-commutative space-times.