Testing the role of the Barbero-Immirzi parameter and the choice of connection in Loop Quantum Gravity

Abstract

We study the role of the Barbero-Immirzi parameter $\gamma$ and the choice of connection in the construction of (a symmetry-reduced version of) loop quantum gravity. We start with the four-dimensional Lorentzian Holst action that we reduce to three dimensions in a way that preserves the presence of $\gamma$. In the time gauge, the phase space of the resulting three-dimensional theory mimics exactly that of the four-dimensional one. Its quantization can be performed, and on the kinematical Hilbert space spanned by SU(2) spin network states the spectra of geometric operators are discrete and $\gamma$-dependent. However, because of the three-dimensional nature of the theory, its SU(2) Ashtekar-Barbero Hamiltonian constraint can be traded for the flatness constraint of an sl(2,C) connection, and we show that this latter has to satisfy a linear simplicity-like condition analogous to the one used in the construction of spin foam models. The physically relevant solution to this constraint singles out the non-compact subgroup SU(1,1), which in turn leads to the disappearance of the Barbero-Immirzi parameter and to a continuous length spectrum, in agreement with what is expected from Lorentzian three-dimensional gravity.

36 pages

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• Edge modes of gravity -- II: Corner metric and Lorentz charges
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• The Thiemann Complexifier and the CVH algebra for Classical and Quantum FLRW Cosmology
• Linking loop quantum gravity quantization ambiguities with phenomenology
• Covariance in self dual inhomogeneous models of effective quantum geometry: Spherical symmetry and Gowdy systems
• Black hole spectroscopy from Loop Quantum Gravity models
• Spectra of geometric operators in three-dimensional LQG: From discrete to continuous
• Manifestly Lorentz-covariant variables for the phase space of general relativity
• Timelike twisted geometries
• Classical r-matrices for the generalised Chern-Simons formulation of 3d gravity
• Ashtekar-Barbero holonomy on the hyperboloid: Immirzi parameter as a Cut-off for Quantum Gravity
• BTZ Black Hole Entropy and the Turaev-Viro model
• $SU(1,1)$ Barbero-like variables derived from Holst action
• Towards self dual Loop Quantum Gravity
• Analytic continuation of real Loop Quantum Gravity : Lessons from black hole thermodynamics
• Barbero's formulation from a $BF$-type action with the Immirzi parameter
• Semi-classical analysis of black holes in Loop Quantum Gravity: Modelling Hawking radiation with volume fluctuations
• Loop quantum gravity and observations