Quantum groups and polymer quantum mechanics

In Polymer Quantum Mechanics, a quantization scheme that naturally emerges from Loop Quantum Gravity, position and momentum operators cannot be both well defined on the Hilbert space (Script capital HPoly). It is henceforth deemed impossible to define standard creation and annihilation operators. In this paper, we show that a q-oscillator structure, and hence q-deformed creation/annihilation operators, can be naturally defined on Script capital HPoly, which is then mapped into the sum of many copies of the q-oscillator Hilbert space. This shows that the q-calculus is a natural calculus for Polymer Quantum Mechanics. Moreover, we show that the inequivalence of different superselected sectors of Script capital HPoly is of topological nature.