A Natural Basis of States for the Noncommutative Sphere and its Moyal bracket

Details A Natural Basis of States for the Noncommutative Sphere and its Moyal bracket A Natural Basis of States for the Noncommutative Sphere and its Moyal bracket

1997-03-21

arxiv.org/abs/q-alg/9703038v1

Mathematics

Quantum Algebra

18 pages Latex, No figures, Submitted to Journal of Mathematical Physics, March 1997

Scientific paper

10.1063/1.532003

An infinite dimensional algebra which is a non-decomposable reducible representation of $su(2)$ is given. This algebra is defined with respect to two real parameters. If one of these parameters is zero the algebra is the commutative algebra of functions on the sphere, otherwise it is a noncommutative analogue. This is an extension of the algebra normally refered to as the (Berezin) quantum sphere or ``fuzzy'' sphere. A natural indefinite ``inner'' product and a basis of the algebra orthogonal with respect to it are given. The basis elements are homogenious polynomials, eigenvectors of a Laplacian, and related to the Hahn polynomials. It is shown that these elements tend to the spherical harmonics for the sphere. A Moyal bracket is constructed and shown to be the standard Moyal bracket for the sphere.

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