On choice of connection in loop quantum gravity

TL;DR

This paper explores the quantum area operator in loop quantum gravity, identifying a family of Lorentz connections and analyzing their implications for the area spectrum and spacetime invariance.

Contribution

It introduces a two-parameter family of Lorentz connections and shows how only one preserves 4d diffeomorphism invariance, affecting the area spectrum and the role of the Immirzi parameter.

Findings

Existence of a two-parameter family of Lorentz connections.

Only one connection preserves 4d diffeomorphism invariance.

Area spectrum is independent of the Immirzi parameter.

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.