Microcausality of Dirac field on noncommutative spacetime

Abstract

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 ${\theta }^{0i}\ne 0$. 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.

21 pages, Latex, v3, some mistakes corrected