  We present a Hamiltonian method of constructing BBGKY-like hierarchies for
quantum field theories. With suitable choices, our method creates a
hierarchical system of evolution equations for the k-th order reduced density
matrices. These equations can be closed at finite order using methods developed
for the classical BBGKY hierarchy to give non-perturbative approximations for
the full quantum equations of motion. Classical observables can then be
numerically computed from these approximate equations, providing an
analytically tractable method of modeling high-energy environments where
quantum effects play a pronounced role.
