Scattering in a three-dimensional fuzzy space

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.

Classical dynamics on three-dimensional fuzzy space

“…The noncommutative quantum mechanics on three-dimensional fuzzy space has been studied extensively in Refs. [8,[14][15][16]. In these studies it was shown that this formulation reduces to commutative quantum mechanics in the commutative limit and that it is a realistic description of the physics at low energies.…”

Classical dynamics on three dimensional fuzzy space: Connecting the short and long length scales

“…In this section we study the modified classical dynamics on three dimensional fuzzy space. The non-commutative quantum mechanics on three dimensional fuzzy space has been studied extensively in [8,[14][15][16]. In these studies it was shown that this formulation reduces to commutative quantum mechanics in the commutative limit and that it is a realistic description of the physics at low energies.…”

Commutative–non-commutative dualities

“…for the space-time commutation relations is normally adopted, where θ µν are constants. Generalizations to fuzzy-and Snyder-space have been explored in [9][10][11][12] and [13] respectively. The latter offers the possibility of avoiding the breaking of rotational and Lorentz symmetries.…”