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Semiclassical methods of deformation quantisation in transport theory

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.