Alternative routes to equivalent classical models of a quantum system

Coarse-graining of some sort is the fundamental and unavoidable step in any attempt to derive the classical mechanical behavior from the quantum formalism. We utilize two-mode Bose-Hubbard model to illustrate how different coarse-grained systems can be naturally associated with a fixed quantum system if it is compatible with different dynamical algebras. Alternative coarse-grained systems generate different evolutions of the same physical quantities, and the difference becomes negligible only in the appropriate macro-limit.