Quantum-classical path integral: A rigorous approach to condensed phase dynamics

A rigorous methodology that combines a fully quantum mechanical treatment of a small system with a classical trajectory description of a large number of degrees of freedom is discussed in connection with the mechanism of decoherence. The need for mean field approximations in traditional quantum-classical calculations is removed by abandoning the delocalized wavefunction description of the quantum particle in favor of Feynman's formulation, which is based on local paths. The effects of the environment enter the quantum-classical path integral (QCPI) expression in terms of phase factors along classical trajectories, whose number grows exponentially with the number of time steps. It is argued that memory loss allows termination of trajectory branching and that the main contribution to decoherence arises from a single classical trajectory (from each sampled initial condition). Exploiting these ideas allows a dramatic reduction in the required number of trajectories, making QCPI calculations feasible.