Noncommutativity and θ-locality

In this paper, we introduce the condition of θ-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 commutators of observables behave at large spacelike separation like exp(−|x − y|2/θ), where θ 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 ⋆-product and prove that this space is a topological algebra under the star product. As an example, we consider the simplest normal ordered monomial : φ ⋆ φ : and show that it obeys the θ-locality condition.