Dynamics of mixed quantum-classical wavefunctions

Despite the widespread use of mixed quantum-classical (QC) methods in nonadiabatic dynamics, the formulation of a consistent QC coupling model continues to pose several challenges. The most accredited model, the QC Liouville equation, fails to recover Heisenberg's uncertainty principle and other proposals suffer from similar drawbacks.

In this work, we exploit an old idea by George Sudarshan: as shown by Koopman, classical mechanics can be envisioned as unitary dynamics of phase-space wavefunctions (WFs) and thus one can construct mixed QCWFs to capture QC correlations. However, identifying a consistent dynamics of QCWFs is far from easy and Sudarshan's original proposal led to several issues.

Here, we revisit this approach by modifying the standard Koopman method to overcome some of its ambiguities. Upon resorting to van Hove's theory of unitary representations in classical mechanics, we formulate a hybrid wave equation for the dynamics of QCWFs. The resulting model is Hamiltonian, leads to a positive quantum density matrix, and reduces to mean-field models in absence of correlations. In addition, the classical density is made positive-definite by enforcing gauge-invariance with respect to local phase factors on phase-space. A proposal for a numerical algorithm is also presented.