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(upsilon)] i Theta(mu upsilon)(x), where the noncommutativity parameters Theta(mu upsilon) 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 x 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 theta(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 >= 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 nonlocal, the four-point UV divergences are local, in accordance with recent results for constant Ѳ.