  We discuss, within the simplified context provided by the polymeric harmonic
oscillator, a construction leading to a separable Hilbert space that preserves
some of the most important features of the spectrum of the Hamiltonian
operator. This construction may be applied to other polymer quantum mechanical
systems, including those of loop quantum cosmology, and is likely generalizable
to certain formulations of full loop quantum gravity. It is helpful to sidestep
some of the physically relevant issues that appear in that context, in
particular those related to superselection and the definition of suitable
ensembles for the statistical mechanics of these types of systems.
