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

We study properties of a scalar quantum field theory on two-dimensional non -commutative space-times. Contrary to the common belief that non-commutativ ity 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 H eisenberg-like commutation relations among coordinates or even on a non-com mutative quantum plane with E-q(2) symmetry have ultraviolet divergences, w hile the theory on a non-commutative cylinder is ultraviolet finite. Thus, ultraviolet behavior of a field theory on non-commutative spaces is sensiti ve 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. (C) 2000 Elsevier Science B.V. Ail rights reserved.