On a special picture of dynamical evolution of nonlinear quantum systems in the phase-space representation

A special picture of quantum evolution in the phase-space representation is derived. In this picture, both the symbols of operators as well as the distribution density, carry part of the time dependence. The symbols of operators evolve classically, whereas the distribution density carries the remaining time dependence, which is needed to recover the full time dependence of quantum averages. Such representation is free of semiclassical expansions or a mixed classical-quantum description. The spin-boson Hamiltonian system is explicitly studied; its phase-space structure is established by obtaining analytical solutions for the three stable regions. Quantum averages are calculated.