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

TL;DR

This paper explores the noncommutative geometry of angular momentum space, developing differential calculus, solving wave equations, and analyzing symmetries, revealing new structures and quantum symmetries in a noncommutative setting.

Contribution

It introduces a natural 4D differential calculus on the noncommutative angular momentum space and identifies its quantum isometry group as the quantum double D(U(su(2))).

Findings

Developed a 4D differential calculus and computed its cohomology and Hodge * operator.

Solved the spin 0 wave equation and analyzed electromagnetic solutions in the noncommutative setting.

Identified the quantum isometry group as the quantum double D(U(su(2))) and connected the space to fuzzy spheres.

Abstract

We study the standard angular momentum algebra $[x_i,x_j]=i\lambda \epsilon_{ijk}x_k$ as a noncommutative manifold $R^3_\lambda$. 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 $R^3_\lambda$ inside a 4D noncommutative spacetime which is the limit $q\to 1$ of q-Minkowski space and show that $R^3_\lambda$ 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 $\R^3_\lambda$ as a collection of all fuzzy spheres taken together. We also analyse the semiclassical limit via minimum uncertainty states $|j,\theta,\phi>$ approximating classical positions in polar coordinates.