Space-time noncommutativity, discreteness of time and unitarity

Violation of unitarity for noncommutative field theory on compact space-tim es is considered. Although such theories are free of ultraviolet divergence s they still violate unitarity, while in a usual field theory such a violat ion occurs when the theory is nonrenormalizable. The compactness of spaceli ke coordinates implies discreteness of the time variable which leads to the appearance of unphysical modes and violation of unitarity even in the abse nce of a star-product in the interaction terms. Thus, this conclusion holds also for other quantum field theories with discrete time. Violation of cau sality., among others, occurs also in the case of the nonvanishing of the c ommutation relations between observables at space-like distances with a typ ical scale of noncommutativity. While this feature allows for a possible vi olation of the spin-statistics theorem, such a violation does not rescue th e situation but makes causality violation scale as the inverse of the mass appearing in the considered model, i.e., it becomes even more severe. We al so stress the role of smearing over the noncommutative coordinates entering the field operator symbols.