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

E. Batista; S. Majid

doi:10.1063/1.1517395

paper

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

arXiv:hep-th/0205128 · doi:10.1063/1.1517395

Abstract

We study the standard angular momentum algebra $[x_{i}, x_{j}] = iλ ε_{ij k} x_{k}$ as a noncommutative manifold $R_{λ}^{3}$ . 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 $R_{λ}^{3}$ inside a 4D noncommutative spacetime which is the limit $q \to 1$ of q-Minkowski space and show that $R_{λ}^{3}$ has a natural quantum isometry group given by the quantum double $D (U (s u (2)))$ as a singular limit of the $q$ -Lorentz group. We view $R_{λ}^{3}$ 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.

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

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

Abstract

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