Position-dependent noncommutative products: classical construction and field theory

Abstract

We look in Euclidean $R^4$ for associative star products realizing the commutation relation $[x^μ,x^ν]=iΘ^{μν}(x)$, where the noncommutativity parameters $Θ^{μν}$ 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 $R^2 \times R^2$, there is only one class of such products which are associative. It corresponds to a noncommutativity matrix whose canonical form has components $Θ^{12}=-Θ^{21}=0$ and $Θ^{34}=-Θ^{43}= θ(x^1,x^2)$, with $þ(x^1,x^2)$ 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 $n\geq 3$ arbitrary dimensions and use it to find traveling noncommutative lumps generalizing noncommutative solitons discussed in the literature. Next we consider Euclidean $λϕ^4$ 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 non-local, the four-point UV divergences are local, in accordance with recent results for constant $Θ$.