Noncommutative deformations of quantum field theories, locality, and causality

We investigate noncommutative deformations of quantum field theories for different star products, particularly emphasizing the locality properties and the regularity of the deformed fields. Using functional analysis methods, we describe the basic structural features of the vacuum expectation values of star-modified products of fields and field commutators. As an alternative to microcausality, we introduce a notion of $\theta$-locality, where $\theta$ is the noncommutativity parameter. We also analyze the conditions for the convergence and continuity of star products and define the function algebra that is most suitable for the Moyal and Wick–Voros products. This algebra corresponds to the concept of strict deformation quantization and is a useful tool for constructing quantum field theories on a noncommutative space–time.