Loop Quantum Gravity: From Theory to Applications

Overview

The thesis builds loop quantum gravity step by step: general relativity is first rewritten as a gauge theory, then quantized, and space acquires a discrete structure at the Planck scale. Two applications follow, the entropy of black holes and a covariant formulation of the dynamics.

Key results

General relativity recast in Ashtekar–Barbero variables as an $\mathrm{SU}(2)$ gauge theory, subject to three constraints: Gauss, diffeomorphism and Hamiltonian.
Area and volume are quantized. A surface crossed by spin-network links has area $A = 8\pi\gamma\ell_{\mathrm P}^2 \sum_p \sqrt{j_p(j_p+1)}$ , and the smallest nonzero area is $a_0 = 4\sqrt3\,\pi\gamma\,\ell_{\mathrm P}^2 \simeq 6\,\ell_{\mathrm P}^2$ .
Counting the punctures of a black-hole horizon by spin-network links recovers the Bekenstein–Hawking entropy $S = A/4\ell_{\mathrm P}^2$ when the Barbero–Immirzi parameter takes the value $\gamma_0 = 0.274\,067\ldots$
The dynamics: Thiemann's regularization of the Hamiltonian constraint and its ambiguities, and the spin-foam models, from Ponzano–Regge in three dimensions to EPRL in four.

[Image: Quantum geometry: the nodes of a spin network carry quanta of volume, and its links carry quanta of area (Chapter 3).]

Quantum geometry: the nodes of a spin network carry quanta of volume, and its links carry quanta of area (Chapter 3).

About this thesis

Master's thesis in theoretical physics (Master in Physics), Department of Physics, Faculty of Exact Sciences, Frères Mentouri University Constantine 1, Algeria. Supervised by Prof. N. Mebarki and defended on 3 September 2020 before a committee chaired by Prof. H. Aissaoui, with Prof. A. Benslama as examiner. The full text is given below; the typeset thesis can also be downloaded as a PDF.

Abstract

This thesis provides a systematic introduction to loop quantum gravity, a non-perturbative and background-independent approach to the quantization of general relativity. We start from the tetrad and spin-connection formulation of general relativity and from the Holst action, which introduces the Barbero–Immirzi parameter $\gamma$ . The Hamiltonian analysis, carried out first in the ADM formalism and then in terms of the Ashtekar–Barbero variables, recasts general relativity as an ${\mathrm{SU}}(2)$ gauge theory subject to three sets of constraints: the Gauss, diffeomorphism and Hamiltonian constraints. The quantization relies on the holonomy–flux algebra: we construct the kinematical Hilbert space, the spin-network basis and the area and volume operators, whose discrete spectra express a granular structure of geometry at the Planck scale. We then present Thiemann's regularization of the Hamiltonian constraint together with its ambiguities. Two applications are developed. The first is black-hole entropy: from the local first law of Frodden, Ghosh and Perez and a statistical counting of the punctures of the horizon by spin-network links, we recover the Bekenstein–Hawking law $S = A/4{\ell_{\mathrm{P}}}^2$ for the value $\gamma_0 \simeq 0.274$ of the Barbero–Immirzi parameter, and we discuss the dependence of this result on the counting scheme. The second is the covariant spin-foam formulation: after a derivation of Feynman's path integral, we present the Ponzano–Regge model of three-dimensional gravity, Ooguri's model for four-dimensional BF theory and the EPRL model, obtained by imposing the simplicity constraints. We conclude with a discussion of the main open problems: the dynamics, the semiclassical limit and the contact with observation.

Keywords: loop quantum gravity, Ashtekar–Barbero variables, Barbero–Immirzi parameter, spin networks, area and volume spectra, black-hole entropy, spin foams.