Scattering in a three-dimensional fuzzy space

We develop scattering theory in a noncommutative space defined by an su(2) coordinate algebra. By introducing a positive operator valued measure as a replacement for strong position measurements, we are able to derive explicit expressions for the probability current, differential and total cross sections. We show that at low incident energies the kinematics of these expressions is identical to that of commutative scattering theory. The consequences of spatial noncommutativity are found to be more pronounced at the dynamical level where, even at low incident energies, the phase shifts of the partial waves can deviate strongly from commutative results. This is demonstrated for scattering from a spherical well. The impact of noncommutativity on the well’s spectrum and on the properties of its bound and scattering states are considered in detail. It is found that for sufficiently large well depths the potential effectively becomes repulsive and that the cross section tends towards that of hard sphere scattering. This can occur even at low incident energies when the particle’s wavelength inside the well becomes comparable to the noncommutative length scale.