Wigner-Weyl isomorphism for quantum mechanics on Lie groups

Abstract

The Wigner-Weyl isomorphism for quantum mechanics on a compact simple Lie group $G$ is developed in detail. Several New features are shown to arise which have no counterparts in the familiar Cartesian case. Notable among these is the notion of a `semiquantised 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 $T^{\ast }G$ 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.

32 pages, Latex2e

References in corpus (8)

• Phase-space formulation of quantum mechanics and quantum state reconstruction for physical systems with Lie-group symmetries
• Deformation quantization in the teaching of quantum mechanics
• A General Theory of Phase-Space Quasiprobability Distributions
• Wigner functions for curved spaces I: On hyperboloids
• Quantum Mechanics Another Way
• Wigner distributions and quantum mechanics on Lie groups: the case of the regular representation
• Wigner distributions for non Abelian finite groups of odd order
• Group Theory and Quasiprobability Integrals of Wigner Functions

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