Wigner–Weyl isomorphism for quantum mechanics on Lie groups

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Abstract
The Wigner–Weyl isomorphism for quantum mechanics on a compact simple Lie group G is developed in detail. Several features are shown to arise which have no counterparts in the familiar Cartesian case. Notable among these is the notion of a semiquantized phase space, a structure on which the Weyl symbols of operators turn out to be naturally defined and, figuratively speaking, located midway between the classical phase space T*G and the Hilbert space of square integrable functions on G. General expressions for the star product for Weyl symbols are presented and explicitly worked out for the angle-angular momentum case.