Quantum field theory on noncommutative space-time: The relation between ultraviolet behaviour, topology and symmetry

We study properties of a scalar quantum field theory on two-dimensional non commutative spacetimes. We show that ultraviolet behaviour of a field theor y on noncommutative spaces is sensitive to the topology of the space time, namely to its compactness. The compactness of space-like coordinates implie s discreteness of the time variable which leads to appearance of unphysical modes and violation of unitarity even in the absence of a star-product in the interaction terms. Thus, this conclusion holds also for other quantum f ield theories with discrete time.