A Lorentz-Covariant Connection for Canonical 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.

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Cited by in corpus (15)

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• Discrete Gravity Models and Loop Quantum Gravity: a Short Review
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• On the role of the Barbero-Immirzi parameter in discrete quantum gravity
• From Classical To Quantum Gravity: Introduction to Loop Quantum Gravity
• An elementary introduction to loop quantum gravity
• Testing the role of the Barbero-Immirzi parameter and the choice of connection in Loop Quantum Gravity
• Manifestly Lorentz-covariant variables for the phase space of general relativity
• Spontaneously broken Lorentz symmetry for Hamiltonian gravity
• Gravity as an SU(1,1) gauge theory in four dimensions
• $SU(1,1)$ Barbero-like variables derived from Holst action
• Revisiting the solution of the second-class constraints of the Holst action
• Barbero's formulation from a $BF$-type action with the Immirzi parameter
• New insights in quantum geometry