Position-dependent noncommutative products: Classical construction and field theory

We look in Euclidean R4 for associative star products realizing the commutation relation [xμ,xν]=iΘμν(x), where the noncommutativity parameters Θμν depend on the position coordinates x. We do this by adopting Rieffel's deformation theory (originally formulated for constant Θ and which includes the Moyal product as a particular case) and find that, for a topology R2×R2, there is only one class of such products which are associative. It corresponds to a noncommutativity matrix whose canonical form has components Θ12=−Θ21=0 and Θ34=−Θ43=θ(x1,x2), with θ(x1,x2) 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 n⩾3 arbitrary dimensions and use it to find traveling noncommutative lumps generalizing noncommutative solitons discussed in the literature. Next we consider Euclidean λϕ4 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.