Towards polymer quantum mechanics for fermionic systems

Polymer quantum mechanics is based on models that mimic the loop quantization of gravity. It coincides with the results of the standard quantum mechanical treatment for such models when a certain length scale parameter is considered to be small. In this work we present some steps in the construction of the polymer representation of a Fermi oscillator, the fermonic counterpart of the harmonic oscillator. It is suggested that the non regular character of the bosonic polymer representation has as a counterpart the non superanalytic character of the fermonic polymer case. We propose a candidate Hamiltonian operator and investigate and contrast its energy spectrum with the standard one.