Position-dependent noncommutative products: Classical construction and field theory Original Research Article

We look in Euclidean image for associative star products realizing the commutation relation image, where the noncommutativity parameters image depend on the position coordinates x. We do this by adopting Rieffelʹs deformation theory (originally formulated for constant Θ and which includes the Moyal product as a particular case) and find that, for a topology image, there is only one class of such products which are associative. It corresponds to a noncommutativity matrix whose canonical form has components image and image, with image an arbitrary positive smooth bounded function. In Minkowski space–time, this describes a position-dependent space-like or magnetic noncommutativity. We show how to generalize our construction to image arbitrary dimensions and use it to find traveling noncommutative lumps generalizing noncommutative solitons discussed in the literature. Next we consider Euclidean image field theory on such a noncommutative background. Using a zeta-like regulator, the covariant perturbation method and working in configuration space, we explicitly compute the UV singularities. We find that, while the two-point UV divergences are nonlocal, the four-point UV divergences are local, in accordance with recent results for constant Θ.