2005/04/04 by Victor Gayral, V. Gayral, J. M. Gracia-Bondia +2 · 4 citations
Mathematics · Physics and Astronomy · #Algebraic structures and combinatorial models #Black Holes and Theoretical Physics #Noncommutative and Quantum Gravity Theories #hep-th
paper · pdf · doi:10.1016/j.nuclphysb.2005.08.016
published as Nucl.Phys.B727:513-536,2005 · 1+22 pages, no figures
arxiv created 2005/04/04 · openalex publication_date 2005/09/01 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/04
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 þ(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 non-local, the four-point UV divergences are local, in accordance with recent results for constant Θ.