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AIP conference proceedings

A General Approach to Describe Nonlinear Semiquantum Motion

Claudia M. Sarris, Miryam Patricia Sassano, Araceli N. Proto

Published Jan 1, 2007• 0 citations
Publisher

Abstract

We describe a general method to study the nonlinear semiquantum systems: as the dynamics of the system develops in a semiquantum space V spanned by the mean values of a Complete Set of Noncommuting observables (CSNCO) and the conjugate classical variables, X, P, it is possible to define a positive definite metric for the quantum manifold (QM). Through the components of the second rank covariant metric tensor, a close connection between this metric and the uncertainty principle can be established, meanwhile, the classical manifold (CM) of the system, is preserved as a symplectic one. We present an example of an entangled nonlinear semiquantum system where different dynamics take place.

DOI
10.1063/1.2746721

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We describe a general method to study the nonlinear semiquantum systems: as the dynamics of the system develops in a semiquantum space V spanned by the mean values of a Complete Set of Noncommuting observables (CSNCO) and the conjugate classical variables, X, P, it is possible to define a positive definite metric for the quantum manifold (QM). Through the components of the second rank covariant metric tensor, a close connection between this metric and the uncertainty principle can be established, meanwhile, the classical manifold (CM) of the system, is preserved as a symplectic one. We present an example of an entangled nonlinear semiquantum system where different dynamics take place.
We describe how, departing from the Shannon entropy, it is possible to deal with semiquantum time-independent nonlinear Hamiltonians. The interplay between the quantal and classical degrees of freedom can be easily seen, and the set of differential equations that govern the temporal evolution of the quantal mean values and the classical variables is obtained. We find invariants of motion and, particularly, we describe under which conditions, the uncertainty principle remains as an invariant of motion too. Through the analysis of these invariants, it is possible to follow the transition of the system from quantum to classical regime and to conclude that the uncertainty principle behaves as an indicator telling us whether the system is in regular or irregular regime. A simple example is shown.
In this paper, a nonlinear semiquantum Hamiltonian associated to the special unitary group SU(2) Lie algebra is studied so as to analyze its dynamics. The treatment here applied allows for a reduction in: 1) the system’s dimension, as well as 2) the number of system’s parameters (to only three). We can now discern clear patterns in: 1) the complete characterization of the system’s fixed points and 2) their stability. It is shown that the parameter associated to the uncertainty principle, which constitutes a very strong constraint, is the key one in determining the presence of fixed points and bifurcation curves in the parameter’s space.
We develop a mathematically well-defined path integral formalism for general symplectic manifolds. We argue that in order to make a path integral quantization covariant under general coordinate transformations on the phase space and involve a genuine functional measure that is both finite and countably additive, the phase space manifold should be equipped with a Riemannian structure (metric). A suitable method to calculate the metric is also proposed. 1. Most quantization formulations lead to correct results only when the system in question is expressed in one of a family of special (Cartesian) coordinate systems [1]. However, for Hamiltonian systems on a generic symplectic manifold, the canonical variables do not admit such special coordinates. Moreover classical Hamiltonian dynamics exhibits covariance under general coordinate transformations on the symplectic manifold, and, in this sense, is coordinate-free. This covariance is lost upon a formal phase-space path integral quantization. The reason is that the Hamiltonian action involves terms linear in the time derivatives and, therefore, its exponential (even in Euclidean time) does not induce a
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.
Quantum mechanics has been formulated in phase space, with the Wigner function as the representative of the quantum density operator, and classical mechanics has been formulated in Hilbert space, with the Groenewold operator as the representative of the classical Liouville density function. Semiclassical approximations to the quantum evolution of the Wigner function have been defined, enabling the quantum evolution to be approached from a classical starting point. Now analogous semiquantum approximations to the classical evolution of the Groenewold operator are defined, enabling the classical evolution to be approached from a quantum starting point. Simple nonlinear systems with one degree of freedom are considered, whose Hamiltonians are polynomials in the Hamiltonian of the simple harmonic oscillator. The behavior of expectation values of simple observables and of eigenvalues of the Groenewold operator are calculated numerically and compared for the various semiclassical and semiquantum approximations.