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arXiv:quant-ph/9812046 · Physical Review A

Impediments to mixing classical and quantum dynamics

José Caro, L. L. Salcedo

Published Aug 1, 1999• 113 citations
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Abstract

The dynamics of systems composed of a classical sector plus a quantum sector is studied. We show that, even in the simplest cases, (i) the existence of a consistent canonical description for such mixed systems is incompatible with very basic requirements related to the time evolution of the two sectors when they are decoupled, (ii) the classical sector cannot inherit quantum fluctuations from the quantum sector, and (iii) a coupling among the two sectors is incompatible with the requirement of physical positivity of the theory, i.e., there would be positive observables with a nonpositive expectation value.

DOI
10.1103/physreva.60.842

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The dynamics of systems composed of a classical sector plus a quantum sector is studied. We show that, even in the simplest cases, (i) the existence of a consistent canonical description for such mixed systems is incompatible with very basic requirements related to the time evolution of the two sectors when they are decoupled, (ii) the classical sector cannot inherit quantum fluctuations from the quantum sector, and (iii) a coupling among the two sectors is incompatible with the requirement of physical positivity of the theory, i.e., there would be positive observables with a nonpositive expectation value.
The dynamics of systems composed of a classical sector plus a quantum sector is studied. We show that, even in the simplest cases, (i) the existence of a consistent canonical description for such mixed systems is incompatible with very basic requirements related to the time evolution of the two sectors when they are decoupled. (ii) The classical sector cannot inherit quantum fluctuations from the quantum sector. And, (iii) a coupling among the two sectors is incompatible with the requirement of physical positivity of the theory, i.e., there would be positive observables with a non positive expectation value. PACS numbers: 03.65.Bz, 03.65.Sq, 03.65.Fd Typeset using REVTEXI. INTRODUCTION AND CONCLUSIONS Ever since the beginnings of quantum mechanics, physical systems have been considered which are composed of a quantum mechanical sector plus another sector described in classical terms [1]. For instance, this issue is central in the quantum theory of the measurement when the apparatus is treated classically. The same situation appears also at a less fundamental
A recent proposal for mixed dynamics of classical and quantum ensembles is shown, in contrast to other proposals, to satisfy the minimal algebraic requirements proposed by Salcedo for any consistent formulation of such dynamics. Generalized Ehrenfest relations for the expectation values of classical and quantum observables are also obtained. It is further shown that additional desirable requirements, related to separability, may be satisfied under the assumption that only the configuration of the classical component is directly accessible to measurement, e.g., via a classical pointer. Although the mixed dynamics is formulated in terms of ensembles on configuration space, thermodynamic mixtures of such ensembles may be defined which are equivalent to canonical phase-space ensembles on the classical sector. Hence, the formulation appears to be both consistent and physically complete.
We study compound systems with a classical sector and a quantum sector. Among other consistency conditions we require a canonical structure, that is, a Lie bracket for the dynamical evolution of hybrid observables in the Heisenberg picture, interpolating between the Poisson bracket and the commutator. Weak and strong postulates are proposed. We explicitly construct one such hybrid bracket when the Hilbert space of the quantum sector is finite dimensional and show that it is unique if the strong postulates are enforced. The adjoint bracket for the Schrodinger picture version of the dynamics is also obtained. Unfortunately, preservation of the positivity of the density matrix under the evolution is not guaranteed. The case of a particle with classical position and momentum and quantum spin-12\frac{1}{2}21​ is discussed and the spin-orbit dynamics is worked out.
We present a consistent formalism to describe the dynamics of hybrid systems with mixed classical and quantum degrees of freedom. The probability function of the system, which, in general, will be a combination of the classical distribution function and the quantum density matrix, is described in terms of its corresponding moments. We then define a hybrid Poisson bracket, such that the dynamics of the moments is ruled by an effective Hamiltonian. In particular, a closed formula for the Poisson brackets between any two moments for an arbitrary number of degrees of freedom is presented, which corrects previous expressions derived in the literature for the purely quantum case. This formula is of special relevance for practical applications of the formalism. Finally, we study the dynamics of a particular hybrid system given by two coupled oscillators, one being quantum and the other classical. Due to the coupling, specific quantum and classical properties are transferred between different sectors. In particular, the quantum sector is allowed to violate the uncertainty relation, though we explicitly show that there exists a minimum positive bound of the total uncertainty of the hybrid system.
A general dynamical system composed of two coupled sectors is considered. The initial-time configuration of one of these sectors is described by a set of classical data while the other is described by standard quantum data. These dynamical systems will be named half quantum and the aim of this paper is to derive their dynamical evolution. The standard approach would be to use quantum mechanics to make predictions for the time evolution of the half quantum initial data. The main problem is how quantum mechanics can be applied to a dynamical system whose initial time configuration is not described by a set of fully quantum data. A solution to this problem is presented and used as a guideline to obtain a general formulation of coupled classical-quantum dynamics. Finally, a quantization prescription mapping a given classical theory to the corresponding half quantum one is presented.