Canonical bracket in quantum-classical hybrid systems

Abstract

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-$\frac{1}{2}$ is discussed and the spin-orbit dynamics is worked out.

Change in title, references added. To appear in Physical Review A

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Cited by in corpus (7)

• Entropy and canonical ensemble of hybrid quantum classical systems
• About the computation of finite temperatureensemble averages of hybrid quantum-classicalsystems with Molecular Dynamics
• Quantum-Classical Dynamical Brackets
• A New Approach to Generalised Uncertainty Relations
• Classical Dynamics from Self-Consistency Equations in Quantum Mechanics -- Extended Version
• Entanglement of Classical and Quantum Short-Range Dynamics in Mean-Field Systems
• An illustration of canonical quantum-classical dynamics