The Fuzzy Sphere: From The Uncertainty Relation To The Stereographic Projection

TL;DR

This paper introduces a new class of states on the fuzzy sphere that satisfy a weaker uncertainty relation, providing a natural deformation of stereographic projection and connecting to coherent states in the large limit.

Contribution

It proposes a novel family of states on the fuzzy sphere that encode position information and generalize stereographic projection, bridging to coherent states on the non-commutative plane.

Findings

States satisfy a weaker uncertainty relation

States reproduce properties of coherent states in large limit

Deformation of stereographic projection achieved

Abstract

On the fuzzy sphere, no state saturates simultaneously all the Heisenberg uncertainties. We propose a weaker uncertainty for which this holds. The family of states so obtained is physically motivated because it encodes information about positions in this fuzzy context. In particular, these states realize in a natural way a deformation of the stereographic projection. Surprisingly, in the large $j$ limit, they reproduce some properties of the ordinary coherent states on the non commutative plane.