Position-dependent noncommutative products: classical construction and field theory

TL;DR

This paper constructs and analyzes position-dependent noncommutative products in Euclidean space, exploring their mathematical properties and implications for quantum field theory, including UV divergence behavior in noncommutative $mbda\u03b4^4$ theory.

Contribution

It introduces a unique class of associative, position-dependent noncommutative products using Rieffel's deformation theory and applies them to field theory, extending previous constant-bla models.

Findings

Only one class of associative position-dependent noncommutative products exists for R^2 x R^2 topology.

The constructed products describe space-like or magnetic noncommutativity in Minkowski space.

In noncommutative mbda\u03b4^4 theory, two-point divergences are non-local, while four-point divergences are local.

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 $\th(x^1,x^2)$ 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 $\Theta$.