  An approach to the quantum-classical mechanics of phase space dependent
operators, which has been proposed recently, is remodeled as a formalism for
wave fields. Such wave fields obey a system of coupled non-linear equations
that can be written by means of a suitable non-Hamiltonian bracket. As an
example, the theory is applied to the relaxation dynamics of the spin-boson
model. In the adiabatic limit, a good agreement with calculations performed by
the operator approach is obtained. Moreover, the theory proposed in this paper
can take nonadiabatic effects into account without resorting to surface-hopping
approximations. Hence, the results obtained follow qualitatively those of
previous surface-hopping calculations and increase by a factor of (at least)
two the time length over which nonadiabatic dynamics can be propagated with
small statistical errors. Moreover, it is worth to note that the dynamics of
quantum-classical wave fields here proposed is a straightforward
non-Hamiltonian generalization of the formalism for non-linear quantum
mechanics that Weinberg introduced recently.
