Noncommutativity and theta-locality

TL;DR

This paper introduces theta-locality as a substitute for microcausality in noncommutative quantum field theory, demonstrating its implications and establishing a test function space that forms a topological algebra under the star product.

Contribution

It defines theta-locality, relates it to asymptotic commutativity, and proves the test function space forms a topological algebra under the Moyal star product.

Findings

Theta-locality behaves like exponential decay at large spacelike separations.

The test function space is a topological algebra under the star product.

Normal ordered monomials satisfy theta-locality.

Abstract

In this paper, we introduce the condition of theta-locality which can be used as a substitute for microcausality in quantum field theory on noncommutative spacetime. This condition is closely related to the asymptotic commutativity which was previously used in nonlocal QFT. Heuristically, it means that the commutator of observables behaves at large spacelike separation like $\exp(-|x-y|^2/\theta)$, where $\theta$ is the noncommutativity parameter. The rigorous formulation given in the paper implies averaging fields with suitable test functions. We define a test function space which most closely corresponds to the Moyal star product and prove that this space is a topological algebra under the star product. As an example, we consider the simplest normal ordered monomial $:\phi\star\phi:$ and show that it obeys the theta-locality condition.