Particlelike solutions to classical noncommutative gauge theory

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 Coulomb solution at spatial infinity and are valid for intermediate values of the radial coordinate $r$. They yield a self-charge inside 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 Coulomb 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. For the former we get a significant improvement over previous bounds. We find that cases (a) and (b) have different experimental signatures.