Failure of microcausality in noncommutative field theories

We revisit the question of microcausality violations in quantum field theory on noncommutative spacetime, taking $\mathcal{O}(x)=\mathrm{\text{:}}\varphi \star \varphi \mathrm{\text{:}}(x)$ as a sample observable. Using methods of the theory of distributions, we precisely describe the support properties of the commutator $[\mathcal{O}(x),\mathcal{O}(y)]$ and prove that, in the case of space-space noncommutativity, it does not vanish at spacelike separation in the noncommuting directions. However, the matrix elements of this commutator exhibit a rapid falloff along an arbitrary spacelike direction irrespective of the type of noncommutativity. We also consider the star commutator for this observable and show that it fails to vanish even at spacelike separation in the commuting directions and completely violates causality. We conclude with a brief discussion about the modified Wightman functions which are vacuum expectation values of the star products of fields at different spacetime points.