  We discuss a phase space representation of quantum dynamics of systems with
many degrees of freedom. This representation is based on a perturbative
expansion in quantum fluctuations around one of the classical limits. We
explicitly analyze expansions around three such limits: (i) corpuscular or
Newtonian limit in the coordinate-momentum representation, (ii) wave or
Gross-Pitaevskii limit for interacting bosons in the coherent state
representation, and (iii) Bloch limit for the spin systems. We discuss both the
semiclassical (truncated Wigner) approximation and further quantum corrections
appearing in the form of either stochastic quantum jumps along the classical
trajectories or the nonlinear response to such jumps. We also discuss how
quantum jumps naturally emerge in the analysis of non-equal time correlation
functions. This representation of quantum dynamics is closely related to the
phase space methods based on the Wigner-Weyl quantization and to the Keldysh
technique. We show how such concepts as the Wigner function, Weyl symbol, Moyal
product, Bopp operators, and others automatically emerge from the Feynmann's
path integral representation of the evolution in the Heisenberg representation.
We illustrate the applicability of this expansion with various examples mostly
in the context of cold atom systems including sine-Gordon model, one- and
two-dimensional Bose Hubbard model, Dicke model and others.
