  Violation of unitarity for noncommutative field theory on compact space-times
is considered. Although such theories are free of ultraviolet divergences, they
still violate unitarity while in a usual field theory such a violation occurs
when the theory is nonrenormalizable. 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 the absence of a
star-product in the interaction terms. Thus, this conclusion holds also for
other quantum field theories with discrete time. Violation of causality, among
others, occurs also as the nonvanishing of the commutation relations between
observables at space-like distances with a typical scale of noncommutativity.
While this feature allows for a possible violation of the spin-statistics
theorem, such a violation does not rescue the situation but makes the scale of
causality violation as the inverse of the mass appearing in the considered
model, i.e., even more severe. We also stress the role of smearing over the
noncommutative coordinates entering the field operator symbols.

High Energy Physics - Theory