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The Journal of Chemical Physics

A quantum-classical bracket that satisfies the Jacobi identity

Oleg V. Prezhdo

Published May 22, 2006• 35 citations

Abstract

A quantum-classical bracket is proposed and is shown to satisfy the Jacobi identity, in contrast to previous definitions that obey this property only up to higher order terms in the Planck constant variant Planck's over 2pi. The Jacobi identity is required of a true Lie bracket and ensures that the Lie bracket of constants of motion is also a constant of motion. An explicit calculation of the Jacobi identity highlights the difference between the proposed and traditional definitions. A further example illustrates that the proposed bracket generates a more consistent quantum-classical dynamics than the traditional bracket. The traditional quantum-classical dynamics in the Henon-Heiles system diverges due to higher order variant Planck's over 2pi terms. The divergence is eliminated with the proposed bracket.

DOI: 10.1063/1.2200342

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Reply to “Comment on ‘A quantum-classical bracket that satisfies the Jacobi identity’ ” [J. Chem. Phys. 124, 201104 (2006)]

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The argument made in the cited Comment is based on a calculation that was not intended in the original paper. Starting from a fully quantum expression, application of the ℏ→0 limit after every bracket calculation as stated in the paper retains consistent orders of ℏ and does obey the Jacobi identity.

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It shown that the quantum-classical dynamical bracket recently proposed in J. Chem. Phys. 124, 201104 (2006) fails to satisfy the Jacobi identity.

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A quantum-classical bracket is proposed and is shown to satisfy the Jacobi identity, in contrast to previous definitions that obey this property only up to higher order terms in the Planck constant variant Planck's over 2pi. The Jacobi identity is required of a true Lie bracket and ensures that the Lie bracket of constants of motion is also a constant of motion. An explicit calculation of the Jacobi identity highlights the difference between the proposed and traditional definitions. A further example illustrates that the proposed bracket generates a more consistent quantum-classical dynamics than the traditional bracket. The traditional quantum-classical dynamics in the Henon-Heiles system diverges due to higher order variant Planck's over 2pi terms. The divergence is eliminated with the proposed bracket.

We study the problem of constructing a general hybrid quantum-classical bracket from a partial classical limit of a full quantum bracket. Introducing a hybrid composition product, we show that such a bracket is the commutator of that product. From this we see that the hybrid bracket will obey the Jacobi identity and the Leibniz rule provided the composition product is associative. This suggests that the set of hybrid variables belonging to an associative subalgebra with the composition product will have consistent quantum-classical dynamics. This restricts the class of allowed quantum-classical interaction Hamiltonians. Furthermore, we show that pure quantum or classical variables can interact in a consistent framework, unaffected by no-go theorems in the literature or the restrictions for hybrid variables. In the proposed scheme, quantum backreaction appears as quantum-dependent terms in the classical equations of motion.

We show how to write a set of brackets for the Langevin equation, describing the dissipative motion of a classical particle, subject to external random forces. The method does not rely on an action principle, and is based solely on the phenomenological description of the dissipative dynamics as given by the Langevin equation. The general expression for the brackets satisfied by the coordinates, as well as by the external random forces, at different times, is determined, and it turns out that they all satisfy the Jacobi identity. Upon quantization, these classical brackets are found to coincide with the commutation rules for the quantum Langevin equation, that have been obtained in the past, by appealing to microscopic conservative quantum models for the friction mechanism.

Quantum-classical equations of motion satisfying Jacobi identity were recently proposed by V. V. Kisil ( Europhys. Lett. , 72 (2005) 873), generalizing the Heisenberg group formulation of quantum mechanics. In the attempt to provide a physical interpretation for the resulting quantum-classical dynamics, we have reconstructed and analyzed the derivation, finding that there is a suitable representation from which the purely classical nature of the supposedly quantum-classical equations of motion clearly emerges.