Weyl's symbols of Heisenberg operators of canonical coordinates and momenta as quantum characteristics

The knowledge of quantum phase flow induced under the Weyl's association rule by the evolution of Heisenberg operators of canonical coordinates and momenta allows to find the evolution of symbols of generic Heisenberg operators. The quantum phase flow trajectories obey the quantum Hamilton's equations and play the role of quantum characteristics. At any fixed level of accuracy of semiclassical expansion, quantum characteristics can be constructed by solving a coupled system of first-order ordinary differential equations for quantum trajectories and generalized Jacobi fields. Quantum phase flow satisfies the star-composition law and preserves hamiltonian and constraint star-functions. We present semiclassical expansion of quantum characteristics for many-body scattering problem and provide tools for calculation of average values of time-dependent physical observables and cross sections. The method of quantum characteristics admits the consistent incorporation of specific quantum effects, such as non-locality and coherence in propagation of particles, into semiclassical transport models.