SU(2) loop quantum gravity seen from covariant theory

By S. Alexandrov, Etera R. Livine. 2002

144 citations recorded by the source corpus.

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 spacetime 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.

Subfields: Atomic and Molecular Physics, and Optics, Nuclear and High Energy Physics, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/8709a80fb36907d421a1a7f1df0798422f04c19b

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Citations in the corpus

Loop Quantum Gravity (2003, background): “This problem has been investigated by Sergei Alexandrov, see [4, 5, 7] and references therein.”
Quantum gravity in terms of topological observables (2005, background): “quantization once the dynamics is fully taken into account or wether it can just be reabsorb into a redeﬁnition of the Newton constant. This point have already been raised at the kinematical level in [12] where a seemingly more covariant approach to loop gravity leads to a geometrical spectra indepe”
A new spin foam model for 4D gravity (2007, background): “It is not unlikely that the projected spin networks of [32] will be an important part of this relation, see also a more recent work [27] for some steps in this direction.”
The Spin-Foam Approach to Quantum Gravity (2012, background): “In this case, one gets a special kind of spin-network states that are a subclass of the projected spin networks introduced in [19, 247] in order to define an heuristic quantization of the (non-commutative and very complicated) Dirac algebra of a Lorentz connection formulation of the phase space of g”
Flipped spinfoam vertex and loop gravity (2007, unknown): “12 36 v1 [ gr -q c] 9 A ug 2 00”
Solving the simplicity constraints for spinfoam quantum gravity (2007, background): “A criticism of the spinfoam quantization is that it seems to take into account only the primary constraints [9, 12, 13, 16].”
Edge modes of gravity. Part I. Corner potentials and charges (2020, background): “This normal also plays a prominent role in extensions of LQG beyond the time gauge [102,109–111].”