Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)


SIGMA 7 (2011), 083, 10 pages      arXiv:1103.4057      http://dx.doi.org/10.3842/SIGMA.2011.083
Contribution to the Special Issue “Loop Quantum Gravity and Cosmology”

A Lorentz-Covariant Connection for Canonical Gravity

Marc Geiller a, Marc Lachièze-Rey a, Karim Noui b and Francesco Sardelli b
a) Laboratoire APC, Université Paris Diderot Paris 7, 75013 Paris, France
b) LMPT, Université Francois Rabelais, Parc de Grandmont, 37200 Tours, France

Received May 27, 2011, in final form August 20, 2011; Published online August 24, 2011

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.

Key words: canonical gravity; first order gravity; Lorentz-invariance; second class constraints.

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