Noncommutative geometry of angular momentum space U(su(2))
Abstract
We study the standard angular momentum algebra [x_{i},x_{j}]=ıλɛ_{ijk}x_{k} as a noncommutative manifold Rλ3. 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 Rλ3 inside a 4D noncommutative spacetime which is the limit q→1 of qMinkowski space and show that Rλ3 has a natural quantum isometry group given by the quantum double C_{(}^{SU(2))⋊U(su(2)) which is a singular limit of the qLorentz group. We view Rλ3} as a collection of all fuzzy spheres taken together. We also analyze the semiclassical limit via minimum uncertainty states j,θ,φ> approximating classical positions in polar coordinates.
 Publication:

Journal of Mathematical Physics
 Pub Date:
 January 2003
 DOI:
 10.1063/1.1517395
 arXiv:
 arXiv:hepth/0205128
 Bibcode:
 2003JMP....44..107B
 Keywords:

 11.30.Hv;
 02.20.Qs;
 02.40.Gh;
 11.10.z;
 Flavor symmetries;
 General properties structure and representation of Lie groups;
 Noncommutative geometry;
 Field theory;
 High Energy Physics  Theory;
 Mathematics  Quantum Algebra
 EPrint:
 Minor revision to add reference [11]. 37 pages latex