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

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

2002-05-14

arxiv.org/abs/hep-th/0205128v2

J.Math.Phys. 44 (2003) 107-137

Physics

High Energy Physics

High Energy Physics - Theory

Minor revision to add reference [11]. 37 pages latex

Scientific paper

10.1063/1.1517395

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.

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