  In this paper we present a new quantum-trajectory based treatment of quantum
dynamics suitable for dissipative systems. Starting from a de Broglie/Bohm-like
representation of the quantum density matrix, we derive and define quantum
equations-of-motion for Liouville-space trajectories for a generalized system
coupled to a dissipative environment. Our theory includes a vector potential
which mixes forward and backwards propagating components and non-local quantum
potential which continuously produces coherences in the system. These
trajectories are then used to propagate an adaptive Lagrangian grid which
carries the density matrix, $\rho(x,y)$, and the action, $A(x,y)$, thereby
providing a complete hydrodynamic-like description of the dynamics.
