A Lorentz-Covariant Connection for Canonical Gravity

TL;DR

This paper constructs a Lorentz-covariant connection in canonical gravity with a non-zero Barbero-Immirzi parameter, avoiding Dirac brackets and aligning with previous results, advancing Lorentz-covariant loop quantum gravity.

Contribution

It introduces a unique, commutative Lorentz-covariant connection in canonical gravity that matches previous Dirac bracket results, simplifying the phase space analysis.

Findings

Identifies a unique Lorentz-covariant connection compatible with the Holst action.

Demonstrates the connection's commutativity in the Poisson bracket.

Provides a new framework for Lorentz-covariant loop quantum gravity.

Abstract

We construct a Lorentz-covariant connection in the context of first order canonical gravity with non-vanishing Barbero-Immirzi parameter. To do so, we start with the phase space formulation derived from the canonical analysis of the Holst action in which the second class constraints have been solved explicitly. This allows us to avoid the use of Dirac brackets. In this context, we show that there is a "unique" Lorentz-covariant connection which is commutative in the sense of the Poisson bracket, and which furthermore agrees with the connection found by Alexandrov using the Dirac bracket. This result opens a new way toward the understanding of Lorentz-covariant loop quantum gravity.