SU(2) Loop Quantum Gravity seen from Covariant Theory

TL;DR

This work presents a covariant canonical formulation of general relativity using the generalized Hilbert-Palatini action and Dirac brackets for second-class constraints, revealing a two-parameter family of Lorentz connections and a unique true spacetime connection. It shows how SU(2) LQG can be embedded within the covariant framework by selecting a covariant Ashtekar-Barbero connection, while also detailing the diffeomorphism-invariant quantization route that naturally connects to spin foam models like Barrett-Crane. A key result is that the Immirzi parameter becomes physically irrelevant when diffeomorphism invariance is preserved, and the area spectrum can be described without beta dependence in the spacetime-connection formulation. The paper thus links canonical LQG and spin foams, clarifying covariance and suggesting a consistent quantization that respects all classical symmetries, while outlining open problems for fully implementing the second-class constraints in the spacetime-connection framework.

Abstract

Covariant loop gravity comes out of the canonical analysis of the Palatini action and the use of the Dirac brackets arising from dealing with the second class constraints (``simplicity'' constraints). Within this framework, we underline a quantization ambiguity due to the existence of a family of possible Lorentz connections. We show the existence of a Lorentz connection generalizing the Ashtekar-Barbero connection and we loop-quantize the theory showing that it leads to the usual SU(2) Loop Quantum Gravity and to the area spectrum given by the SU(2) Casimir. This covariant point of view allows to analyze closely the drawbacks of the SU(2) formalism: the quantization based on the (generalized) Ashtekar-Barbero connection breaks time diffeomorphisms and physical outputs depend non-trivially on the embedding of the canonical hypersurface into the space-time manifold. On the other hand, there exists a true space-time connection, transforming properly under all diffeomorphisms. We argue that it is this connection that should be used in the definition of loop variables. However, we are still not able to complete the quantization program for this connection giving a full solution of the second class constraints at the Hilbert space level. Nevertheless, we show how a canonical quantization of the Dirac brackets at a finite number of points leads to the kinematical setting of the Barrett-Crane model, with simple spin networks and an area spectrum given by the SL(2,C) Casimir.