  In their simplest formulations, classical dynamics is the study of
Hamiltonian flows and quantum mechanics that of propagators. Both are linked,
and emerge from the datum of a single classical concept, the Hamiltonian
function. We study and emphasize the analogies between Hamiltonian flows and
quantum propagators; this allows us to verify G. Mackey's observation that
quantum mechanics (in its Weyl formulation) is a refinement of Hamiltonian
mechanics. We discuss in detail the metaplectic representation, which very
explicitly shows the close relationship between classical mechanics and quantum
mechanics, the latter emerging from the first by lifting Hamiltonian flows to
the double covering of the symplectic group. We also give explicit formulas for
the factorization of Hamiltonian flows into simpler flows, and prove a quantum
counterpart of these results.
