Spectra of geometric operators in three-dimensional loop quantum gravity: From discrete to continuous

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 $\mathrm{SU}(2)$ Ashtekar-Barbero framework, the spectra are discrete and depend on the Barbero-Immirzi parameter $\ensuremath{\gamma}$ 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 $\ensuremath{\gamma}$ independent.