Faithful realizations of semiclassical truncations

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.