On the role of the Barbero-Immirzi parameter in discrete quantum gravity

TL;DR

This paper investigates how the Barbero-Immirzi parameter influences discrete quantum gravity models, showing that its role depends on the chosen phase space, with implications for the dependence of physical predictions.

Contribution

It identifies two different phase space reductions from a discrete SO(4)-BF theory, revealing how the Barbero-Immirzi parameter's influence varies with the phase space choice.

Findings

Gamma-dependent symplectic structure in Twisted Geometries.

Gamma-independent symplectic structure in Regge geometries.

Physical predictions' dependence on phase space selection.

Abstract

The 1-parameter family of transformations identified by Barbero and Immirzi plays a significant role in non-perturbative approaches to quantum gravity, among them Loop Quantum Gravity and Spin Foams. It facilitates the loop quantization programme and subsequently the Barbero-Immirzi parameter (gamma) arises in both the spectra of geometrical operators and in the dynamics provided by Spin Foams. However, the debate continues as to whether quantum physics should be Barbero-Immirzi parameter dependent. Starting from a discrete SO(4)-BF theory phase space, we find two possible reductions with respect to a discrete form of the simplicity constraints. The first reduces to a phase space with gamma-dependent symplectic structure and more generally in agreement with the phase space underlying Loop Quantum Gravity restricted to a single graph - a.k.a. Twisted Geometries. The second, fuller reduction leads to a gamma-independent symplectic structure on the phase space of piecewise-flat-linear geometries - a.k.a. Regge geometries. Thus, the gamma-dependence of physical predictions is related to the choice of phase space underlying the quantization.