Modular polymer representations of the Weyl algebra

TL;DR

This paper introduces modular polymer representations of the Weyl algebra, combining features of loop quantum gravity and modular theory, which may offer new insights into quantum gravity and cosmology.

Contribution

It presents novel modular polymer representations of the Weyl algebra, unifying position and momentum aspects without well-defined operators, as candidates for new physics in quantum gravity.

Findings

Demonstrated the construction of modular polymer representations

Analyzed the harmonic oscillator dynamics within these representations

Explored potential applications to quantum cosmology

Abstract

One of the key conceptual challenges in quantum gravity is to understand how quantum theory should modify the very notion of spacetime. One way to investigate this question is to study the alternatives to Schr\"odinger quantum mechanics. The polymer representation, inspired by loop quantum gravity, can be understood as capturing features of discrete spatial geometry. The modular representation, on the other hand, has a built-in unification of position and momentum polarizations via a length scale. In this paper, we introduce the modular polymer representations of the Weyl algebra, in which neither position nor momentum exists as a well-defined operator. As inequivalent representations, they are candidates for describing new physics. We illustrate this by studying the dynamics of the harmonic oscillator as an example, with the prospect of eventually applying this representation to quantum cosmology.