Microcausality of Dirac field on noncommutative spacetime

TL;DR

This paper investigates the microcausality of free Dirac fields on noncommutative spacetime, finding that while some field operators do not vanish at spacelike intervals, physical observables still satisfy microcausality.

Contribution

It demonstrates that Dirac field observables maintain microcausality on noncommutative spacetime despite non-vanishing commutators of field operators.

Findings

Vacuum and non-vacuum expectation values of Moyal commutator do not vanish at spacelike intervals.

Physical observables like scalar and current operators satisfy microcausality.

Microcausality holds for Dirac fields 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 $\theta^{0i}\neq0$. 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.