Many-body quantum mechanics as a symplectic dynamical system

An approach is formulated to the problem of obtaining approximate solutions to many-body quantum mechanics. The starting point is the representation of quantum mechanics as Hamiltonian mechanics on a symplectic manifold (phase space). It is shown that Dirac's variation of an action integral provides a natural mechanism for constraining the dynamics to symplectic submanifolds and gives rise to a hierarchy of approximate many-body theories of which Hartree-Fock, random-phase approximation, time-dependent Hartree-Fock, and the double commutator equations of motion formalism are special cases.