Microcausality of Dirac field on noncommutative spacetime

We study the microcausality of free Dirac field on noncommutative spacetime. We calculate the vacuum and non-vacuum state expectation values for the Moyal commutator $[\bar{\psi}_{\alpha}(x)\star\psi_{\beta}(x),\bar{\psi}_ {\sigma}(x^{\prime})\star\psi_{\tau}(x^{\prime})]_{\star}$ of Dirac field on noncommutative spacetime. We find that they do not vanish for some cases of the indexes for an arbitrary spacelike interval, no matter whether $\theta^{0i}=0$ or $\theta^{0i}\neq0$. However for the physical observable quantities of Dirac field such as the Lorentz scalar $:\bar{\psi}(x)\star\psi(x):$ and the current $j^{\mu}(x)=:\bar{\psi}(x)\gamma^{\mu}\star\psi(x):$ etc., we find that they still satisfy the microcausality. Therefore microcausality is satisfied for Dirac field on noncommutative spacetime.