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Towards Self Dual Loop Quantum Gravity Jibril Ben Achour

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Towards Self Dual Loop Quantum Gravity Jibril Ben Achour Towards self dual Loop Quantum Gravity Jibril Ben Achour To cite this version: Jibril Ben Achour. Towards self dual Loop Quantum Gravity. Physics [physics]. Université Paris 7 Denis Diderot, 2015. English. tel-01396791 HAL Id: tel-01396791 https://tel.archives-ouvertes.fr/tel-01396791 Submitted on 15 Nov 2016 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. Universite´ Paris Diderot (Paris 7) Sorbonne Paris Cite´ ED 560 - STEP'UP - \Sciences de la Terre de l'Environnement et Physique de l'Univers de Paris" Th`ese de Doctorat Physique theorique Towards self dual Loop Quantum Gravity pr´esent´eepar Jibril Ben Achour pour l'obtention des titres de Docteur de l'Universite´ Paris Diderot (Paris 7) Sorbonne Paris Cite´ Th`esedirig´eepar Dr Eric Huguet / Dr Karim Noui Laboratoire AstroParticule et Cosmologie soutenue publiquement le 30 septembre 2015 devant le jury compos´ede : Dr Eric Huguet Directeur de th`ese Dr Karim Noui Co - Directeur de th`ese Dr Etera Livine Rapporteur arXiv:1511.07332v1 [gr-qc] 23 Nov 2015 Pr Hanno Sahlmann Rapporteur Dr Renaud Parentani Examinateur Pr Alejandro Perez Examinateur Pr Pierre Binetruy Pr´esident 3 \ Ce qui est plus certain, c'est que nos notions habituelles d'espace et de temps, m^emeas- sez profond´ement remani´eespar la th´eoriede la relativit´e,ne sont pas exactement appropri´ees `ala description des ph´enom`enesatomiques. [...] En v´erit´e, les notions d'espace et de temps tir´eesde notre exp´eriencequotidienne ne sont valables que pour les ph´enom`enes`agrandes ´echelles. Il faudrait y substituer, comme notions fondamentales valables en micro-physique, d'autres conceptions qui conduiraient `aretrouver asymptotiquement, quand on repasse des ph´enom`enes´el´ementaires aux ph´enom`enesobservables `anotre ´echelle, les notions habituelles d'espace et de temps. Est-il besoin de dire que c'est l`aune t^ache bien difficile ? On peut m^emese demander si elle est possible, si nous pourrons jamais arriver `a´eliminer `a ce point ce qui constitue le cadre m^emede notre vie courante. Mais l'histoire de la science montre l'extr^emefertilit´ede l'esprit humain et il ne faut pas d´esesp´erer.Cependant, tant que nous ne serons pas parvenus `a´elargirnos concepts dans le sens indiqu´e`al'instant, nous de- vrons nous ´evertuer `afaire entrer, plus ou moins gauchement, les ph´enom`enesmicroscopiques dans le cadre de l'espace et du temps et nous aurons le sentiment p´eniblede vouloir enfermer un joyau dans un ´ecrin qui n'est pas fait pour lui." Louis De Broglie - La Physique Nouvelle et les Quantas (1936) Acknowledgements Mes premiers remerciements vont tout d'abord `aEric Huguet et Jacques Renaud, pour m'avoir fait confiance en stage, puis offert la possibilit´ed'entreprendre cette th`eseau sein du laboratoire APC, m'ouvrant ainsi les portes aux joies de la physique th´eorique,et plus globalement de la recherche scientifique. Je garderai un tr`esbon souvenir de leur bonne humeur, leur disponibilit´eet des apr`esmidi pass´eesdevant le tableau `aleurs cot´es. Je tiens `aremercier tout particuli`erement Karim Noui, pour m'avoir fait partager sa passion pour la recherche, ses id´eeset ses questions, et pour m'avoir fait d´ecouvrirle champ de recherche de la gravit´equantique `aboucles. J'ai particuli`erement appr´eci´esa p´edagogie, sa patience sans limites et sa rigueur scientifique. Ce fut un privil`egeet un tr`esgrand plaisir d'apprendre `ases c^ot´es. Je souhaite aussi remercier mes rapporteurs, Hanno Sahlmann et Etera Livine, pour le temps qu'ils ont pris pour la lecture de ce manuscrit et pour leurs retours et suggestions qui m'ont permis de clarifier plusieurs points. Je remercie Marc Geiller pour ses conseils (hautement recommand´es!),son hospitalit´eet sa bonne humeur. Ce fut un tr`esgrand plaisir de faire mes premiers pas en physique th´eorique dans le bureau 424A en compagnie de Florian, Marc, Alexis ... et de poursuivre aux c^ot´es de Maxime et Alexis. Un grand merci `avous pour toutes ces discussions, de physique ou autre, et pour les doux craquages de fin de journee. Une pens´eeva ´evidemment `atous les compagnons de route, Julien, Ileyk, Georges, Pierre, Vivien, Julian, Ben, Maica, Romain, Lea et `atoute l'´equipe QUPUC plus g´en´eralement ! Une pens´eetoute particuli`ereva aux tortues de feu, sans qui rien n'aurait pu arriver. Je tiens `aremercier Julien Grain, Antonin Coutant, Daniele Pranzetti et Boris Bolliet pour les bons moments pass´esen conference, et pour la d´ecouverte de Berlin ... et tous les amis, connaissances et rencontres inattendues de Paris qui ont fait de ces trois ann´eesune exp´eriencedes plus riche ! Finalement, une pens´eetoute particuli`ereva `ama famille, pour leur ind´efectiblesoutien tout au long de cette aventure et depuis si longtemps, un grand merci pour tout. Abstract In this PhD thesis, we introduced a new strategy to investigate the kinematical and physical predictions of self dual Loop Quantum Gravity (LQG) and by-passed the old problem of implementing quantum mechanically the so called reality conditions inherent to the self dual phase space. Since our first motivations come from black holes thermodynamics, we first review, in the third chapter, the loop quantization of spherically isolated horizon based on the SU(2) Ashtekar-Barbero variables leading to the micro-canonical entropy. This approach is based on the effective model of a Chern-Simons connection coupled to point-like particles living on the horizon. Then we present the so called \gas of punctures" model for the isolated horizon, which provides the framework to go beyond the micro-canonical ensemble and compute the canonical and grand-canonical entropy. In the context of this model, we investigate how the assumption of a Bose-Einstein statistic for the punctures impacts the semi-classical result and underline a relation between the condensation phenomenon and the presence of purely logarithmic quantum corrections to the entropy. The presentation of this model enable us to point the different limits and drawbacks of the SU(2) loop quantization of spherically isolated horizon. First, one needs to proceed to an unnatural fine tuning on the Immirzi parameter to obtain the right semi-classical limit in the micro-canonical ensemble. Moreover, this quantization does not predict a holographic behavior for the degeneracy of the hole, as one could expect. Therefore, if one wants to avoid to precedent fine tuning on γ, one is led to assume the so called holographic hypothesis to obtain the right semi classical limit in the context of the gas of puncture model. The fourth chapter is devoted to studying to what extend the loop quantization based on the self dual variables could cure those problems. Obviously, since no one knows how to quantize the self dual Ashtekar phase space of General Relativity, because of the so called reality conditions, we are led to introduce a new strategy, based on an analytic continuation of the degeneracy from γ 2 R to γ = ±i. We review in details the construction of the procedure, and present the results. The self dual degeneracy turns out to be naturally holographic, supplemented with some power law corrections which conspired, at the semi classical limit, to provide the expected logarithmic quantum corrections to the entropy. At the leading term, we recover the Bekenstein-Hawking area law. The discrete real area spectrum is turned into a continuous real area spectrum, even if we are now working with γ = ±i. Finally, we recognize that our procedure is a well known map which send the Casimir and the character of the SU(2) compact group into the Casimir and character of the continuous representations of the SU(1; 1) non compact group. A detailed discussion on the status of our procedure is 8 provided at the end of the chapter. The fifth chapter is devoted to understanding more precisely the interplay between the disappearance of γ in the physical predictions of the quantum theory, the appearance of the SU(1; 1) group and the relation with the self dual variables. To to so, we introduce a new toy model of three dimensional gravity admitting a Barbero-immirzi like parameter and SL(2; C) as its symmetry group. The canonical analysis of the toy model in two different gauges, one selecting the compact group SU(2), the second one selecting the non compact SU(1; 1) group, allows us to conclude that at least in three dimensional gravity, the presence of γ in the SU(2) phase space is a pure gauge artifact. Finally, the loop quantization of the two phases spaces results in two different kinematical area spectrums, which turn out to be related through our analytic continuation procedure. At the end, we show that it is possible to reformulated the SU(2) phase space in terms of new one, based on a complex SL(2; C) connection, supplemented with some reality constraints. Once solved, the reality conditions reduces the complex SL(2; C) phase space to the SU(1; 1) phase space as expected. This chapter allows us to exhibit an interesting mechanism concerning the disappearance of the Immirzi parameter in the predictions of three dimensional loop quantum gravity.
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