Physics – High Energy Physics – High Energy Physics - Theory
Scientific paper
2005-04-04
Nucl.Phys.B727:513-536,2005
Physics
High Energy Physics
High Energy Physics - Theory
1+22 pages, no figures
Scientific paper
10.1016/j.nuclphysb.2005.08.016
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 $\th(x^1,x^2)$ 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 $\lambda\phi^4$ 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 $\Theta$.
Gayral Victor
Gracia-Bondia Jose M.
Ruiz Fernando Ruiz
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