A new realization of quantum geometry

TL;DR

This paper introduces a novel realization of quantum geometry by quantizing the flux formulation of loop quantum gravity, leading to a new representation with bounded area operator and alternative Barbero-Immirzi parameter treatments.

Contribution

It presents a new representation of quantum geometry with a different vacuum state, impacting the spectra of geometric operators and expanding the framework of loop quantum gravity.

Findings

The area operator is bounded in this new realization.

The spectrum of geometric operators is modified by the new quantization.

Two methods for incorporating the Barbero-Immirzi parameter are identified.

Abstract

We construct in this article a new realization of quantum geometry, which is obtained by quantizing the recently-introduced flux formulation of loop quantum gravity. In this framework, the vacuum is peaked on flat connections, and states are built upon it by creating local curvature excitations. The inner product induces a discrete topology on the gauge group, which turns out to be an essential ingredient for the construction of a continuum limit Hilbert space. This leads to a representation of the full holonomy-flux algebra of loop quantum gravity which is unitarily-inequivalent to the one based on the Ashtekar-Isham-Lewandowski vacuum. It therefore provides a new notion of quantum geometry. We discuss how the spectra of geometric operators, including holonomy and area operators, are affected by this new quantization. In particular, we find that the area operator is bounded, and that there are two different ways in which the Barbero-Immirzi parameter can be taken into account. The methods introduced in this work open up new possibilities for investigating further realizations of quantum geometry based on different vacua.