  A new approximate solution to the quantum-classical Liouville equation is
derived starting from the formal solution of this equation in forward-backward
form. The time evolution of a mixed quantum-classical system described by this
equation is obtained in a coherent state basis using the mapping
representation, which expresses $N$ quantum degrees of freedom in a
2N-dimensional phase space. The solution yields a simple non-Hamiltonian
dynamics in which a set of $N$ coherent state coordinates evolve in forward and
backward trajectories while the bath coordinates evolve under the influence of
the mean potential that depends on these forward and backward trajectories. It
is shown that the solution satisfies the differential form of the
quantum-classical Liouville equation exactly. Relations to other mixed
quantum-classical and semi-classical schemes are discussed.
