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Journal of Physics A Mathematical and Theoretical

A generalized Wigner function for quantum systems with theSU(2) dynamical symmetry group

A. B. Klimov, José Luis Romero

Published Jan 23, 2008• 39 citations
Publisher

Abstract

We introduce a Wigner-like quasidistribution function to describe quantum systems with the SU (2) dynamic symmetry group. This function is defined in a three-dimensional group manifold and can be used to represent the states defined in several SU (2) invariant subspaces. The explicit differential Moyal-like form of the star product is found and analyzed in the semiclassical limit.

DOI
10.1088/1751-8113/41/5/055303

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We introduce a Wigner-like quasidistribution function to describe quantum systems with the SU (2) dynamic symmetry group. This function is defined in a three-dimensional group manifold and can be used to represent the states defined in several SU (2) invariant subspaces. The explicit differential Moyal-like form of the star product is found and analyzed in the semiclassical limit.
The star product is the basic tool used in the phase-space formulation of quantum mechanics. We find a differential form of the star product for a class of \mbox{sss-parametrized} SU (2) Stratonovich-Weyl symbols which appear in the phase-space representation of spin-like systems. The limit of large spin is considered and the asymptotic form of the differential operator defining the star product is obtained.
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.
We extend the Wigner-Weyl-Moyal phase-space formulation of quantum mechanics to general curved configuration spaces. The underlying phase space is based on the chosen coordinates of the manifold and their canonically conjugate momenta. The resulting Wigner function displays the axioms of a quasiprobability distribution, and any Weyl-ordered operator gets associated with the corresponding phase-space function, even in the absence of continuous symmetries. The corresponding quantum Liouville equation reduces to the classical curved space Liouville equation in the semiclassical limit. We demonstrate the formalism for a point particle moving on two-dimensional manifolds, such as a paraboloid or the surface of a sphere. The latter clarifies the treatment of compact coordinate spaces, as well as the relation of the presented phase-space representation to symmetry groups of the configuration space.
A new approach to perturbation theory in the quantum phase-space formalism is proposed, in order to devise some approximated description of the quantum phase-space dynamics, which is not directly related to the usual semi-classical approximation. A general class of equivalent quasi-distribution functions based on the Wigner-Moyal formalism is considered and a first-order invariant formulation of the dynamics is obtained. The relationship between the various phase-space representation is expressed in term of a pseudo-differential operator defined by the Moyal product. In particular, our theory is applied to the sub-class of representations obtained by a first order perturbation of the Wigner representation. Finally the connection of our approach with some well established gauge-invariant formulation of the Wigner dynamics in the presence of an external magnetic field is investigated.