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arXiv:quant-ph/0003005

Symmetric Classical Mechanics

Nuno Costa Dias, João Nuno Prata

Published Mar 1, 2000• 0 citations
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Abstract

We investigate the possibility that the semiclassical limit of quantum mechanics might be correctly described by a classical dynamical theory, other than standard classical mechanics. Using a set of classicality criteria proposed in a related paper, we show that the time evolution of a set of quantum initial data satisfying these criteria is fully consistent with the predictions of a new theory of classical dynamics. The dynamical structure of the new theory is given by the Moyal bracket. This is a Lie bracket that was first derived as the dynamical structure of the Moyal-Weyl-Wigner formulation of quantum mechanics. We present a new derivation of the Moyal bracket, this time in the context of the semiclassical limit of quantum mechanics and thus prove that both classical and quantum dynamics can be formulated in terms of the same canonical structure.

DOI
10.48550/arxiv.quant-ph/0003005

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We investigate the possibility that the semiclassical limit of quantum mechanics might be correctly described by a classical dynamical theory, other than standard classical mechanics. Using a set of classicality criteria proposed in a related paper, we show that the time evolution of a set of quantum initial data satisfying these criteria is fully consistent with the predictions of a new theory of classical dynamics. The dynamical structure of the new theory is given by the Moyal bracket. This is a Lie bracket that was first derived as the dynamical structure of the Moyal-Weyl-Wigner formulation of quantum mechanics. We present a new derivation of the Moyal bracket, this time in the context of the semiclassical limit of quantum mechanics and thus prove that both classical and quantum dynamics can be formulated in terms of the same canonical structure.
A canonical formulation of coupled classical-quantum dynamics is presented. The theory is named symmetric hybrid dynamics. It is proved that under some general conditions its predictions are consistent with the full quantum ones. Moreover symmetric hybrid dynamics displays a fully consistent canonical structure. Namely, it is formulated over a Lie algebra of observables, time evolution is unitary and the solution of a hybrid type Schrödinger equation. A quantization prescription from classical mechanics to hybrid dynamics is presented. The quantization map is a Lie algebra isomorphism. Finally some possible applications of the theory are succinctly suggested.
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.
We systematically investigate master equations involving double-bracket dissipators in the classical limit. Dissipators defined by double commutators arise naturally in dephasing dynamics, while those defined by double anticommutators are typically associated with noisy Hamiltonians. The classical limit of such master equations is derived by reformulating the dissipative open dynamics in phase space using the Wigner-Weyl transform and Moyal bracket formalism, and by identifying the leading-order terms in a systematic ℏ\hbarℏ-expansion. We first analyze a double-commutator master equation associated with energy diffusion, and then turn to master equations containing a double anticommutator with the system Hamiltonian, recently derived in the context of noisy non-Hermitian systems. For both classes of double-bracket equations, we establish a gradient-flow representation of the dynamics within both the quantum and semiclassical formalisms. We further study the semiclassical evolution generated by double-bracket master equations for the harmonic (integrable) and driven anharmonic (chaotic) oscillators, considering both classical and quantum initial states. The dynamics are characterized through several observables, including mean position, momentum, and energy. The quantumness of the time-evolved state is assessed via the Wigner logarithmic negativity, and the interplay between double-bracket dissipation and chaotic dynamics is examined in detail. Finally, we extend our analysis to generalized master equations involving higher-order nested brackets, which arise naturally as a time-continuous formulation of spectral filtering techniques widely used in the numerical simulation of quantum systems.
By making use of the Weyl-Wigner-Groenewold-Moyal association rules, a commutative product and a new quantum bracket are constructed in the ring of operators \cal{F}(H). In this way, an isomorphism between Lie algebra of classical observables (with Poisson bracket) and the Lie algebra of quantum observables with this new bracket is established. By these observations, a formulation of the classical mechanics in \cal{F}(H} is obtained and is shown to be \hbar\to 0 limit of the Heisenberg picture formulation of the quantum mechanics.
Classical mechanics is formulated in complex Hilbert space with the introduction of a commutative product of operators, an antisymmetric bracket and a quasidensity operator that is not positive definite. These are analogues of the star product, the Moyal bracket, and the Wigner function in the phase space formulation of quantum mechanics. Quantum mechanics is then viewed as a limiting form of classical mechanics, as Planck's constant approaches zero, rather than the other way around. The forms of semiquantum approximations to classical mechanics, analogous to semiclassical approximations to quantum mechanics, are indicated.
We present two possible criteria quantifying the degree of classicality of an arbitrary (finite dimensional) dynamical system. The inputs for these criteria are the classical dynamical structure of the system together with the quantum and the classical data providing the two alternative descriptions of its initial time configuration. It is proved that a general quantum system satisfying the criteria up to some extend displays a time evolution consistent with the classical predictions up to some degree and thus it is argued that the criteria provide a suitable measure of classicality. The features of the formalism are illustrated through two simple examples.