A Lorentz-Covariant Connection for Canonical Gravity

• Main Authors: Marc Geiller, Marc Lachièze-Rey, Karim Noui, Francesco Sardelli
• Format: Article
• Language: English
• Published: National Academy of Science of Ukraine 2011-08-01
• Series: Symmetry, Integrability and Geometry: Methods and Applications
• Subjects: canonical gravity; first order gravity; Lorentz-invariance; second class constraints
• Online Access: http://dx.doi.org/10.3842/SIGMA.2011.083

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.