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Journal of Mathematical Physics

Star exponentials for any ordering of the elements of the inhomogeneous symplectic Lie algebra

Jean-Marie Maillard

Published Jan 22, 2004• 4 citations
Publisher

Abstract

We compute for any ordering the star exponentials of all polynomials of degree not greater than two on the 2ℓ-dimensional phase space of a quantum system with ℓ degrees of freedom, and we show in the particular case of the Moyal star product that the Weyl transform of the Moyal star exponential of the one-dimensional harmonic oscillator Hamiltonian is the evolution operator of this quantum system.

DOI
10.1063/1.1634352

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We compute for any ordering the star exponentials of all polynomials of degree not greater than two on the 2ℓ-dimensional phase space of a quantum system with ℓ degrees of freedom, and we show in the particular case of the Moyal star product that the Weyl transform of the Moyal star exponential of the one-dimensional harmonic oscillator Hamiltonian is the evolution operator of this quantum system.
The Bohr-Sommerfeld approximation to the eigenvalues of a one-dimensional quantum Hamiltonian is derived through order ℏ2\hbar^2ℏ2 (i.e., including the first correction term beyond the usual result) by means of the Moyal star product. The Hamiltonian need only have a Weyl transform (or symbol) that is a power series in ℏ\hbarℏ, starting with ℏ0\hbar^0ℏ0, with a generic fixed point in phase space. The Hamiltonian is not restricted to the kinetic-plus-potential form. The method involves transforming the Hamiltonian to a normal form, in which it becomes a function of the harmonic oscillator Hamiltonian. Diagrammatic and other techniques with potential applications to other normal form problems are presented for manipulating higher order terms in the Moyal series.
A normal form transformation is carried out on one-dimensional quantum Hamiltonians that transforms them into functions of the quantum harmonic oscillator. The method works with the Weyl transform (or 'symbol') of the Hamiltonian. The Moyal star product is used to carry out the normal form transformation at the level of symbols. Diagrammatic techniques are developed for handling the expressions that result from higher order terms in the Moyal series. Once the normal form is achieved, the Bohr–Sommerfeld formula for the eigenvalues, including higher order corrections, follows easily.
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.
Inspired by the formalism that relates the star-exponential with the quantum propagator for bosonic systems, in this work we introduce the analogous extension for the fermionic case. In particular, we analyse the problem of calculating the star-exponential (i.e., the symbol of the evolution operator) for Fermi systems within the deformation quantization program. Grassmann variables and coherent states are considered in order to obtain a closed-form expression for the fermionic star-exponential in terms of its associated propagator. As a primary application, a fermionic version of the Feynman-Kac formula is derived within this formalism, thus allowing a straightforward calculation of the ground state energy in phase space. Finally, the method is validated by successfully applying it to the simple harmonic and driven Fermi oscillators, for which the results developed here provide a powerful alternative computational tool for the study of fermionic systems.
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.