Microcausality violation of scalar field on noncommutative spacetime with the time-space noncommutativity

TL;DR

This paper investigates how noncommutative spacetime affects the microcausality of free scalar fields, finding violations when time-space noncommutativity is present, unlike the space-space case where microcausality holds.

Contribution

It provides a detailed analysis of microcausality violations in scalar fields on noncommutative spacetime with time-space noncommutativity, extending previous understanding.

Findings

Microcausality holds for space-space noncommutativity.

Microcausality is violated for time-space noncommutativity.

Explicit calculations of Moyal commutators support these conclusions.

Abstract

Quantum field theories on noncommutative spacetime have many different properties from those on commutative spacetime. In this paper, we study the microcausality of free scalar field on noncommutative spacetime. We expand the scalar field in the form of usual Lorentz invariant measure in noncommutative spacetime. Then we calculate the expectation values of the Moyal commutators for the quadratic operators, such as $\phi(x)\star\phi(x)$, $\pi({\bf x},t)\star\pi({\bf x},t)$, $\partial_{i}\phi({\bf x},t)\star\partial_{i}\phi({\bf x},t)$, and $\partial_{i}\phi({\bf x},t)\star\pi({\bf x},t)$. We obtain that for space-space noncommutativity, i.e., $\theta^{0i}=0$ in $\theta^{\mu\nu}$, microcausality of free scalar field on noncommutative spacetime is satisfied. For time-space noncommutativity, i.e., $\theta^{0i}\neq0$ in $\theta^{\mu\nu}$, microcausality of free scalar field on noncommutative spacetime is violated.