Algebraic Models: Coordinates, Scales, and Dynamical Symmetries

We discuss the variety of coordinates often used to characterize the coherent state classical limit of an algebraic model. We show selection of appropriate coordinates naturally motivates a procedure to generate a single particle Schr\"odinger hamiltonian which, for low energy states, gives equivalent results to a bosonic algebraic model to leading order in $N$. The process is used to study the associated geometries of the dynamical symmetries of U(3). By demanding that the inner product be preserved in the Schr\"odinger picture we conclude that different dynamical symmetries correspond to different scales.