Particle-like solutions to classical noncommutative gauge theory

Abstract

We construct perturbative static solutions to the classical field equations of noncommutative U(1) gauge theory for the three cases: a) space-time noncommutativity, b) space-space noncommutativity and c) both a) and b). The solutions tend to the Coulumb solution at spatial infinity and are valid for intermediate values of the radial coordinate $r$. They yield a self-charge in a sphere of radius $r$ centered about the origin which increases with decreasing $r$ for case a), and decreases with decreasing $r$ for case b). For case a) this may mean that the exact solution screens an infinite charge at the origin, while for case b) it is plausible that the charge density is well behaved at the origin, as happens in Born-Infeld electrodynamics. For both cases a) and b) the self-energy in the intermediate region grows faster as $r$ tends to the origin than that of the Coulumb solution. It then appears that the divergence of the classical self-energy is more severe in the noncommutative theory than it is in the corresponding commutative theory. We compute the lowest order effects of these solutions on the hydrogen atom spectrum and use them to put experimental bounds on the space-time and space-space noncommutative scales. We find that cases a) and b) have different experimental signatures.