  Time-resolved measurement techniques are opening a window on nonequilibrium
quantum phenomena that is radically different from the traditional picture in
the frequency domain. The simulation and interpretation of nonequilibrium
dynamics is a conspicuous challenge for theory. This paper presents a novel
approach to quantum many-body dynamics that is based on a Hamiltonian
formulation of the Bogoliubov-Born-Green-Kirkwood-Yvon (BBGKY) hierarchy of
equations of motion for reduced density matrices. These equations have an
underlying symplectic structure, and we write them in the form of the classical
Hamilton equations for canonically conjugate variables. Applying canonical
perturbation theory or the Krylov-Bogoliubov averaging method to the resulting
equations yields a systematic approximation scheme. The possibility of using
memory-dependent functional approximations to close the Hamilton equations at a
particular level of the hierarchy is discussed. The geometric structure of the
equations gives rise to reduced geometric phases that are observable even for
noncyclic evolutions of the many-body state. The formalism is applied to a
finite Hubbard chain which undergoes a quench in on-site interaction energy U.
Canonical perturbation theory, carried out to second order, fully captures the
nontrivial real-time dynamics of the model, including resonance phenomena and
the coupling of fast and slow variables.
