Canonical bracket in quantum-classical hybrid systems

TL;DR

This paper develops a canonical Lie bracket for hybrid quantum-classical systems, ensuring consistent dynamics, and explores its properties and limitations, including a specific example involving spin-$rac{1}{2}$ particles.

Contribution

It constructs a unique hybrid bracket satisfying strong postulates for finite-dimensional quantum systems, advancing the theoretical framework of quantum-classical hybrid dynamics.

Findings

Constructed a unique hybrid bracket under strong postulates.

Showed that positivity preservation of the density matrix is not guaranteed.

Analyzed spin-orbit dynamics for a particle with classical position/momentum and quantum spin.

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.