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Journal of Physics A Mathematical and General

Moyal-like form of the star product for generalizedSU(2) Stratonovich-Weyl symbols

A. B. Klimov, P.B. Espinoza

Published Sep 24, 2002• 39 citations
Publisher

Abstract

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{$s$-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.

DOI
10.1088/0305-4470/35/40/305

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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.
In this paper, we formulate the phase space description of qubit systems using coadjoint orbits of SU(2)SU(2)SU(2) and the Stratonovich-Weyl correspondence, yielding a deformation quantization on the sphere. The resulting star product reproduces the operator algebra of complexified quaternions and its antisymmetric part induces the Lie-Poisson structure associated with the Kirillov-Kostant-Souriau symplectic form. We show that quantum dynamics can be expressed entirely in phase space through star exponentials of Hamiltonian symbols, leading to an explicit representation of the propagator. Further, we establish the equivalence between the coherent-state path integral formulation on S2S^2S2 and the algebraic description in terms of star exponentials. Some examples illustrating the construction of the star-exponential functions and the resulting Poisson structure are included.
In Moyal's formulation of quantum mechanics, a quantum spin s is described in terms of continuous symbols, i.e., by smooth functions on a two-dimensional sphere. Such prescriptions to associate operators with Wigner functions, P or Q symbols, are conveniently expressed in terms of operator kernels satisfying the Stratonovich-Weyl postulates. In analogy to this approach, a discrete Moyal formalism is defined on the basis of a modified set of postulates. It is shown that appropriately modified postulates single out a well-defined set of kernels that give rise to discrete symbols. Now operators are represented by functions taking values on (2s+1)2{(2s+1)}^{2}(2s+1)2 points of the sphere. The discrete symbols contain no redundant information, contrary to the continuous ones. The properties of the resulting discrete Moyal formalism for a quantum spin are worked out in detail and compared to the continuous formalism.
Abstract The Weyl‐Wigner‐Moyal formalism for quantum particle with discrete internal degrees of freedom is developed. A one to one correspondence between operators in the Hilbert space and functions on the phase space is found. The expressions for the Stratonovich‐Weyl quantiser, star product and Wigner functions of such systems for arbitrary values of spin are obtained in detail. As examples the Landau levels and the corresponding Wigner functions for a spin nonrelativistic particle as well as the magnetic resonance for a spin nonrelativistic uncharged particle are analysed.
The Moyal description of quantum mechanics, based on the Wigner-Weyl isomorphism between operators and symbols, provides a comprehensive phase space representation of dynamics. The Weyl symbol image of the Heisenberg picture evolution operator is regular in ħ and so presents a preferred foundation for semiclassical analysis. Its semiclassical expansion "coefficients," acting on symbols that represent observables, are simple, globally defined (phase space) differential operators constructed in terms of the classical flow. The first of two presented methods introduces a cluster-graph expansion for the symbol of an exponentiated operator, which extends Groenewold′s formula for the Weyl product of two symbols and has ħ as its natural small parameter. This Poisson bracket based cluster expansion determines the Jacobi equations for the semiclassical expansion of "quantum trajectories." Their Green function solutions construct the regular ħ ⇓ 0 asymptotic series for the Heisenberg-Weyl evolution map. The second method directly substitutes such a series into the Moyal equation of motion and determines the ħ coefficients recursively. In contrast to the WKB approximation for propagators, the Heisenberg-Weyl description of evolution involves no essential singularity in ħ, no Hamilton-Jacobi equation to solve for the action, and no multiple trajectories, caustics, or Maslov indices.
The Weyl-Wigner-Moyal formalism is developed for spin by means of a correspondence between spherical harmonics and spherical-harmonic tensor operators. The exact asymptotic relation among the P , Q , and Weyl symbols is found, and the analogue of the Moyal expansion is developed for the Weyl symbol of the product of two operators in terms of the symbols for the individual operators. It is shown that in the classical limit, the Weyl symbol for a commutator equals i times the Poisson bracket of the corresponding Weyl symbols.