Noncommutative geometry of angular momentum space U(su(2))

Abstract

We study the standard angular momentum algebra $[x_i,x_j]=iλε_{ijk}x_k$ as a noncommutative manifold $R^3_λ$. We show that there is a natural 4D differential calculus and obtain its cohomology and Hodge * operator. We solve the spin 0 wave equation and some aspects of the Maxwell or electromagnetic theory including solutions for a uniform electric current density, and we find a natural Dirac operator. We embed $R^3_λ$ inside a 4D noncommutative spacetime which is the limit $q\to 1$ of q-Minkowski space and show that $R^3_λ$ has a natural quantum isometry group given by the quantum double $D(U(su(2)))$ as a singular limit of the $q$-Lorentz group. We view $\R^3_λ$ as a collection of all fuzzy spheres taken together. We also analyse the semiclassical limit via minimum uncertainty states $|j,θ,ϕ>$ approximating classical positions in polar coordinates.