Scattering in three-dimensional fuzzy space

TL;DR

This paper develops a scattering theory in three-dimensional non-commutative space, revealing how spatial non-commutativity affects scattering processes, phase shifts, and bound states, with notable deviations from classical results at low energies.

Contribution

It introduces a framework for scattering in non-commutative space using a positive operator valued measure, providing explicit formulas and analyzing the effects on spectra and cross-sections.

Findings

Low-energy kinematics matches commutative scattering

Phase shifts deviate strongly due to non-commutativity

Potential becomes effectively repulsive at large depths

Abstract

We develop scattering theory in a non-commutative space defined by a $su(2)$ 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 spacial non-commutativity 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 non-commutativity 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 wave-length inside the well becomes comparable to the non-commutative length-scale.