Renormalization and Coarse-graining of Loop Quantum Gravity Christoph Charles To cite this version: Christoph Charles. Renormalization and Coarse-graining of Loop Quantum Gravity. General Rela- tivity and Quantum Cosmology [gr-qc]. Université de Lyon, 2016. English. NNT : 2016LYSEN053. tel-01429887 HAL Id: tel-01429887 https://tel.archives-ouvertes.fr/tel-01429887 Submitted on 9 Jan 2017 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. Numéro National de Thèse : 2016LYSEN053 THESE de DOCTORAT DE L’UNIVERSITE DE LYON opérée par l’Ecole Normale Supérieure de Lyon Ecole Doctorale N° 52 Physique et astrophysique de Lyon Discipline : Physique Soutenue publiquement le 25/11/2016, par : Christoph CHARLES Renormalization and Coarse-graining of Loop Quantum Gravity Renormalisation et coarse-graining de la gravitation quantique à boucle Devant le jury composé de : Noui, Karim Maître de conférences Université de Tours Rapporteur Dittrich, Bianca Faculty member Perimeter Institute Rapporteur Freidel, Laurent Faculty member Perimeter Institute Président Speziale, Simone Chargé de recherche Université d’Aix-Marseille Examinateur Samtleben, Henning Professeur ENS de Lyon Examinateur Livine, Etera Directeur de recherche ENS de Lyon Directeur de thèse christoph charles RENORMALIZATIONANDCOARSE-GRAININGOF LOOP QUANTUM GRAVITY RENORMALIZATIONANDCOARSE-GRAININGOFLOOP QUANTUM GRAVITY christoph charles November 2016 Christoph Charles: Renormalization and Coarse-graining of Loop Quan- tum Gravity, , © November 2016 “Together, or not at all” — Amy Pond To my parents to whom I owe nearly everything. To my wife who has always been supportive. ABSTRACT The continuum limit of loop quantum gravity is still an open problem. Indeed, no proper dynamics in known to start with and we still lack the mathematical tools to study its would-be continuum limit. In the present PhD dissertation, we will investigate some coarse-graining methods that should become helpful in this enterprise. We concen- trate on two aspects of the theory’s coarse-graining: finding natural large scale observables on one hand and studying how the dynamics of varying graphs could be cast onto fixed graphs on the other hand. To determine large scale observables, we study the case of hyper- bolic tetrahedra and their natural description in a language close to loop quantum gravity. The surface holonomies in particular play an important role. This highlights the structure of double spin networks, which consist in a graph and its dual, which seems to also appear in works from Freidel et al. To solve the problem of varying graphs, we consider and define loopy spin networks. They encode the local curvature with loops around an effective vertex and allow to describe different graphs by hidding them in a coarse-graining process. More- over, their definition gives a natural procedure for coarse-graining allowing to relate different scales. Together, these two results constitute the foundation of a coarse- graining programme for diffeomorphism invariant theories. vii RÉSUMÉ Le problème de la limite continue de la gravitation quantique à boucle est encore ouvert. En effet, la dynamique précise n’est pas connue et nous ne disposons pas des outils nécessaires à l’étude de cette limite le cas échéant. Dans cette thèse, nous étudions quelques méthodes de coarse-graining (étude à gros grains) qui devraient contribuer à cette entreprise. Nous nous concentrons sur deux aspects du flot: la détermination d’observables naturelles à grandes échelles d’un côté et la manière de s’abstraire du problème de la dynamique à graphe variable en la projetant sur des graphes fixes de l’autre. Pour déterminer les observables aux grandes distances, nous étu- dions le cas des tétraèdres hyperboliques et leur description naturelle dans un langage proche de celui de la gravitation quantique à boucle. Les holonomies de surface en particulier jouent un rôle important. Cela dégage la structure des double spin networks constitués d’un graphe et de son dual, structure qui semble aussi apparaître dans les travaux de Freidel et al. Pour résoudre le problème des graphes variables, nous considérons et définissons les loopy spin networks. Ils encodent par des boucles la courbure locale d’un vertex effectif et permettent ainsi de décrire différents graphes en les masquant via le processus de coarse-graining. De plus, leur définition donne un procédé naturel systématique de coarse-graining pour passer d’une échelle à une autre. Ensemble, ces deux principaux résultats posent le fondement d’un programme de coarse-graining pour les théories invariantes sous dif- féomorphismes. viii PUBLICATIONS [1] Christoph Charles and Etera R. Livine. “Ashtekar-Barbero holon- omy on the hyperboloid: Immirzi parameter as a cutoff for quantum gravity.” In: D92.12 (2015), p. 124031. doi: 10.1103/ PhysRevD.92.124031. eprint: 1507.00851. [2] Christoph Charles and Etera R. Livine. “Closure constraints for hyperbolic tetrahedra.” In: 32.13 (2015), p. 135003. doi: 10. 1088/0264-9381/32/13/135003. eprint: 1501.00855. [3] Christoph Charles and Etera R. Livine. “The closure constraint for the hyperbolic tetrahedron as a Bianchi identity.” In: (2016). eprint: 1607.08359. [4] Christoph Charles and Etera R. Livine. “The Fock Space of Loopy Spin Networks for Quantum Gravity.” In: 48.8 (2016), p. 113. doi: 10.1007/s10714-016-2107-5. eprint: 1603.01117. ix ACKNOWLEDGMENTS I was told this is the hardest part to write. It is certainly most difficult not to forget someone as so many have helped in one way or another. The order of mention should not be taken to represent anything and certainly does not reflect the amount of work these people provided. First and foremost, I want to give a very special thanks to my ad- visor, doctor Etera Livine, who has mentored me, taught me and guided me during the whole thesis. Etera, it took some time to adapt to each other but in the end, I’m grateful. I’m grateful for learning to work with people with different approaches but I’m more especially grateful for all the sacrifices you made and the help you provided as a mental support, most notably in the end. You have introduced the world of research to me and taught me to love it enough to continue. I discovered quantum gravity and more specifically loop quantum gravity thanks to you. For all this, I want to say thank you. Thank you also to all of the defense committee members, doctor Laurent Freidel, doctor Henning Samtleben, doctor Bianca Dittrich, doctor Karim Noui and doctor Simone Speziale. All of you have been kind enough to answer every e-mail, to wait when I had some trouble getting all the information in time and have been flexible enough to accommodate all the difficulties. Thank you also for the help you gave when I had administrative issues. I also want to thank the administrative team of the ENS of Lyon, which is one of the most effective I have ever met. But most impor- tantly, I am always impressed by their willingness to help, their kind- ness and the support they provide. I must thank most particularly Myriam Friat who has borne with me for at least two weeks in the last run for the Ph.D. defense and helped me at every step of the process. My labmates have been of tremendous help too and it is difficult not to omit someone. I want to thank all of them for welcoming me in the first place, for all the discussions about nearly everything, but also for all the support and the encouragement they provided. Thank you to Michel Fruchart, David Lopes Cardozo, to Daniele Malpetti and to Baptiste Pezelier. A particular thank you must be addressed to Michel who have taken the trouble of listening to me talking for hours about quantum gravity just to help me clarify my ideas. His help cannot be overstated. I also want to thank Christophe Göller whose internship I had the chance to oversee. He encouraged me a lot, allowed some very interesting conversations and made the summer work a bit more xi lifeful. I want to thank him more specifically for the help he provided on some quite lengthy computations. I want to thank all my close friends who have been an incredible and inexhaustible source of courage and joy, namely, Chérif Adouama, Ivan Bannwarth, Dimitri Cobb, Siméon Giménez and Cyril Philippe. Thank you for all the discussions. Thank you Cyril for our talks on epistemology guiding my research. Thank you Dimitri for your help in math, where your knowledge is quickly exceeding mine. Thank you Ivan and Chérif for all the discussions about the Ph.D. life and your attentive ear. Thank you Siméon for the numerous conversations, the way too many jokes and the galaxy savings. I am also thinking about my flatmates though this group inter- sects with the previous one. Thank you Nathanaël Dobé and Mar- jorie Dobé, thank you Cyril Philippe, thank you Siméon Giménez, thank you Dimitri Cobb, thank you Maxime Kristanek and thank your Alexandre Drouin. Thank you for all the time playing, sharing meals and talking. Thank you for the good food and wine we shared and thank you for the less good burgers and coke we shared. I also want to thank people from church who have always been keen to ask for news and have been of great support: Gilles and Yvette Campoy, Denis and Maïlys Blum, Jonah and Amy Haddad, Alexandre and Suzanne Sarran.