  We provide an introduction to deformation quantisation and discuss the
application of the formalism in solving the evolution problem for many-body
systems in terms of semiclassical expansion. In any fixed order of expansion
over the Planck's constant, the evolution problem can be reduced to a
statistical-mechanics problem of calculating an ensemble of quantum
characteristics in the phase space and their Jacobi fields. In comparison with
the corresponding rules of classical statistical mechanics, the rules for
computing the probabilities and time-dependent averages of observables are
modified. The evolution equations represent a finite system of first-order
ordinary differential equations for quantum trajectories in the phase space and
the associated Jacobi fields. Quantum characteristics allow for the consistent
inclusion of specific quantum effects, such as non-localities and coherence, in
the description of the propagation of particles in the transport models. Since
the late 1980s - early 1990s, when the first transport models were created for
modeling heavy-ion collisions, computing power has increased by about five
orders of magnitude. This dramatic rise in computing power makes it possible to
include Jacobi fields in the collision dynamics, to extend beyond the classical
treatment of phase-space trajectories, currently adopted in all of the
transport models.
