Quantum Groups and Polymer Quantum Mechanics

TL;DR

This paper demonstrates that q-oscillator structures and q-calculus naturally fit within Polymer Quantum Mechanics, revealing a topological basis for sector inequivalence and enabling the definition of deformed creation and annihilation operators.

Contribution

It introduces a q-oscillator framework for Polymer Quantum Mechanics, connecting q-calculus with the theory and analyzing the topological nature of sector inequivalence.

Findings

q-oscillator structure can be defined on H_Poly

q-calculus is natural for Polymer Quantum Mechanics

Superselected sectors' inequivalence is topological

Abstract

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 ( H_Poly ). It is henceforth deemed impossible to define standard creation and annihilation operators. In this letter we show that a q-oscillator structure, and hence q-deformed creation/annihilation operators, can be naturally defined on H_Poly , 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 H Poly is of topological nature.