Position-dependent noncommutative products: classical construction and field theory

Abstract

We look in Euclidean $R^4$ for associative star products realizing the commutation relation $[x^{\mu },x^{\nu }]=i{\Theta }^{\mu \nu }(x)$, where the noncommutativity parameters ${\Theta }^{\mu \nu }$ depend on the position coordinates $x$. We do this by adopting Rieffel's deformation theory (originally formulated for constant $\Theta$ 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 ${\Theta }^{12}=-{\Theta }^{21}=0$ and ${\Theta }^{34}=-{\Theta }^{43}=\theta (x^1,x^2)$, 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 $n\ge 3$ arbitrary dimensions and use it to find traveling noncommutative lumps generalizing noncommutative solitons discussed in the literature. Next we consider Euclidean $\lambda {\varphi }^4$ 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 $\Theta$.

1+22 pages, no figures

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