Spectra of geometric operators in three-dimensional LQG: From discrete to continuous

TL;DR

This paper compares the spectra of geometric operators in different formulations of 3D Lorentzian loop quantum gravity, revealing discrete spectra with Barbero-Immirzi dependence in one formulation and continuous, gamma-independent spectra in another.

Contribution

It demonstrates how the choice of variables and reality conditions in 3D LQG affects the spectra of geometric operators, bridging discrete and continuous spectra.

Findings

Discrete spectra depend on the Barbero-Immirzi parameter in the SU(2) framework.

Self-dual variables lead to continuous, gamma-independent spectra.

Comparison clarifies the impact of variable choice on quantum geometry.

Abstract

We study and compare the spectra of geometric operators (length and area) in the quantum kinematics of two formulations of three-dimensional Lorentzian loop quantum gravity. In the SU(2) Ashtekar-Barbero framework, the spectra are discrete and depend on the Barbero-Immirzi parameter $\gamma$ exactly like in the four-dimensional case. However, we show that when working with the self-dual variables and imposing the reality conditions the spectra become continuous and $\gamma$-independent.