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Journal of Physics A Mathematical and Theoretical

Generalized SU (2) covariant Wigner functions and some of their applications

A. B. Klimov, José Luis Romero, Hubert de Guise

Published Jul 13, 2017• 60 citations
Publisher

Abstract

Abstract We survey some applications of SU (2) covariant maps to the phase space quantum mechanics of systems with fixed or variable spin. A generalization to SU (3) symmetry is also briefly discussed in framework of the axiomatic Stratonovich–Weyl formulation.
DOI
10.1088/1751-8121/50/32/323001

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The semiclassical evolution of the generalized SU (2) Wigner function for quantum systems with a variable number of excitations is discussed in detail. It is shown that in the framework of the Liouvillian approach quantum dynamics can be approximately described in terms of classical trajectories on an appropriate four-dimensional symplectic manifold. The analysis of this manifold allows the introduction of a phase-like observable that can be interpreted as a relative phase ‘between’ invariant subspaces.
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The Weyl–Wigner map yields the entire structure of Moyal quantum mechanics directly from the standard operator formulation. The covariant generalization of Moyal theory, also known as Vey quantum mechanics, was presented in the literature many years ago. However, a derivation of the formalism directly from standard operator quantum mechanics, clarifying the relation between the two formulations, is still missing. In this article we present a covariant generalization of the Weyl order prescription and of the Weyl–Wigner map and use them to derive Vey quantum mechanics directly from the standard operator formulation. The procedure displays some interesting features: it yields all the key ingredients and provides a more straightforward interpretation of the Vey theory including a direct implementation of unitary operator transformations as phase space coordinate transformations in the Vey idiom. These features are illustrated through a simple example.
As is well known, there are several formulations of quantum mechanics. One of them, closely connected with the Weyl representation of operators [ 1 ] and Wigner functions [ 2 ], use functions on the phase space to represent both observables and quantum states. For working in this representation it is necessary to replace operators by their Weyl symbols and density matrices by Wigner functions [ 3 , 4 ]. After fundamental works of Weyl, Wigner and Kirkwood [ 1 , 2 , 5 ] properties of the representation were studied and used to calculate the properties of various physical systems (see [ 6–23 ]), including systems with magnetic field [ 15–23 ].