Non-Hamiltonian commutators in quantum mechanics

The symplectic structure of quantum commutators is first unveiled and then exploited to describe generalized non-Hamiltonian brackets in quantum mechanics. It is easily recognized that quantum-classical systems are described by a particular realization of such a bracket. In light of previous work, this paper explains a unified approach to classical and quantum-classical non-Hamiltonian dynamics. In order to illustrate the use of non-Hamiltonian commutators, it is shown how to define thermodynamic constraints in quantum-classical systems. In particular, quantum-classical Nose-Hoover equations of motion and the associated stationary density matrix are derived. The non-Hamiltonian commutators for both Nose-Hoover chains and Nose-Andersen (constant-pressure, constant-temperature) dynamics are also given. Perspectives of the formalism are discussed.