  Realizations of algebras in terms of canonical or bosonic variables can often
be used to simplify calculations and to exhibit underlying properties. There is
a long history of using such methods in order to study symmetry groups related
to collective motion, for instance in nuclear shell models. Here, related
questions are addressed for algebras obtained by turning the quantum commutator
into a Poisson bracket on moments of a quantum state, truncated to a given
order. In this application, canonical realizations allow one to express the
quantum back-reaction of moments on basic expectation values by means of
effective potentials. In order to match degrees of freedom, faithfulness of the
realization is important, which requires that, at least locally, the space of
moments as a Poisson manifold is realized by a complete set of Casimir-Darboux
coordinates in local charts. A systematic method to derive such variables is
presented and applied to certain sets of moments which are important for
physical applications. If only second-order moments are considered, their
Poisson-bracket relations are isomorphic to the Lie bracket of sp(2N,R),
providing an interesting link with realizations of nuclear shell models.
