On choice of connection in loop quantum gravity

TL;DR

This work addresses the ambiguity in selecting a Lorentz connection for Wilson lines in covariant loop quantum gravity by showing a two-parameter family of Lorentz connections that diagonalize the quantum area operator. Imposing 4D diffeomorphism invariance selects a unique spacetime connection (the AV result) whose area spectrum is independent of the Immirzi parameter $\beta$, removing the parameter from physical predictions. The standard SU(2) loop gravity results appear as a Lorentz-covariant extension of Ashtekar-Barbero, but those connections are not spacetime connections and break diffeomorphism invariance, reintroducing the Immirzi ambiguity. The findings advocate using the spacetime Lorentz connection for a fully symmetric quantization, while providing a Lorentz-covariant extension of the area spectrum and guiding future work on Hilbert space structure and entropy calculations in this framework.

Abstract

We investigate the quantum area operator in the loop approach based on the Lorentz covariant hamiltonian formulation of general relativity. We show that there exists a two-parameter family of Lorentz connections giving rise to Wilson lines which are eigenstates of the area operator. For each connection the area spectrum is evaluated. In particular, the results of the su(2) approach turn out to be included in the formalism. However, only one connection from the family is a spacetime connection ensuring that the 4d diffeomorphism invariance is preserved under quantization. It leads to the area spectrum independent of the Immirzi parameter. As a consequence, we conclude that the su(2) approach must be modified accordingly to the results obtained since it breaks one of the classical symmetries.