Failure of microcausality in quantum field theory on noncommutative spacetime

The commutator of $:\varphi (x)\star \varphi (x):$ with ${\partial }_{\mu }^y:\varphi (y)\star \varphi (y):$ fails to vanish at equal times and thus also fails to obey microcausality at spacelike separation even for the case in which ${\theta }^{0i}=0$. The failure to obey microcausality for these sample observables implies that this form of noncommutative field theory fails to obey microcausality in general. This result holds generally when there are time derivatives in the observables. We discuss possible responses to this problem.