Separable Hilbert space for loop quantization

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.This work has been supported by the Spanish MICINN and MINECO research Grants No. FIS2009-11893, No. FIS2011-30145-C03-02, No. FIS2012-34379 and the Consolider-Ingenio 2010 Program CPAN (CSD2007-00042), Chilean FONDECYT regular Grant No. 1140335 as well as by the National Center for Science (NCN) of Poland research Grants No. 2012/05/E/ST2/03308 and No. 2011/02/A/ST2/00300. T. P. also acknowledges the financial support of UNAB via internal project No. DI-562-14/R