Quantum-classical dynamical brackets

We study the problem of constructing a general hybrid quantum-classical bracket from partial classical limit full quantum bracket. Introducing composition product, we show that such is commutator product. From this see will obey Jacobi identity and Leibniz rule provided product associative. This suggests set variables belonging to an associative subalgebra with have consistent dynamics. restricts class allowed interaction Hamiltonians. Furthermore, pure or can interact in framework, unaffected by no-go theorems literature restrictions for variables. In proposed scheme, backreaction appears as quantum-dependent terms equations motion.