Noncommutative quantum field theory: unitarity and discrete time

The unitarity condition for noncommutative field theories on compact space–times is considered. Although such theories are free of ultraviolet divergences, the compactness of space-like coordinates implies discreteness of the time variable which leads to appearance of unphysical modes and violation of unitarity even in absence of the star-product in the interaction terms. Thus, this conclusion holds also for other quantum field theories with discrete time. We also discuss violation of causality in terms of commutators for observables and possibility for a violation of the spin–statistics theorem.