Heisenberg dynamics of mixed quantum-classical systems

We consider the dynamics of interacting quantum and classical systems in the Heisenberg representation. Unlike the usual construction in standard quantum mechanics, mixed quantum-classical systems involve the interplay of unitary operators acting on the quantum observables and the Lagrangian trajectories sweeping the classical degrees of freedom. This interplay reflects an intricate structure which is made particularly challenging by the backreaction excerpted on the classical trajectories by the quantum degrees of freedom. While the backreaction is underestimated in the common Ehrenfest model, more recent methodologies succeed in capturing this important effect by resorting to Koopman wavefunctions in classical mechanics. Luckily, both Ehrenfest and Koopman models enjoy a variational framework which is exploited here to unfold the geometric structure underlying quantum-classical coupling. A special role is played by the action of the diffeomorphic Lagrangian paths on a non-Abelian pure-gauge potential which comprises statistical correlations. After presenting the treatment in the simple case of Ehrenfest dynamics, we move on to the Koopman model and present the role of the backreaction terms therein. Finally, we compare both models in the context of pure-dephasing systems.