- Ce sujet est vide.
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Messages
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Je mets ça ici car c’est immense.
La comparaison entre le sexe et les appels clients SFR est un gentil morceau d’écriture et je suis sérieux.
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*Un génial morceau
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« L’idéal étant la masturbation sur rien. Qui ne tienne que par le style »
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« Moi je veux qu’à la fin d’une baise, le seule vainqueur ce soir la République »
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La poésie
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Quel génie ce mec !
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C’est un ancien, le tout dernier est sorti récemment (davavad.fr). On tend de plus en plus vers le cinéma expérimental. Pas mal de passages brillants !
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Oui mais je suis peut-être moins fan du nouveau que ce qui vient d’être proposé là.
J’ai le sentiment qu’Augustin est réellement rongé par la dépression, au point que si ça a pu nourrir son humour, ça commence à le desservir.-
Certains trucs de l’interview semblent sincères, notamment le coming-out, l’absence de sexe et la dépression. Je trouve que ça rend tout le reste (les escargots, Disneyland, SFR) encore plus savoureux.
Je suis moins fan de leurs formats longs, comme leur dernier film. Y a un truc qui coince un peu, peut-être aussi que la dynamique duo a fait son temps?-
Vois leur dernier spectacle, DAVA 10 et tu comprendras que le duo fonctionne toujours à plein.
URDCUC, format long, est leur chef-d’œuvre, et même si Tybalt est un peu en-dessous, ça reste très bon.
Quant à la sincérité de « certains trucs », je n’y crois pas une seconde, Sacha et Augustin étant de purs personnages. A une exception près, et pas récente, ils n’ont jamais donné d’interview sérieuse ; personne ne sait, sauf leurs proches évidemment, qui ils sont vraiment. C’est ça qui les rend unique.-
J’ai déjà vu URDUC et je me suis un peu ennuyé. J’ai aussi été les voir y a un an, j’ai trouvé Augustin plus drôle que Sacha (de loin). Mais tant mieux si ça t’a plus en tout cas.
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Et par ailleurs sur l’homosexualité, le sexe, le refoulement, l’attraction pour les heteros, c’est tellement précis que je doute qu’il ait inventé la sienne de toute pièces.
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N’oublions pas qu’il est interviewé par son copain gay Marc ( le stagiaire de « vos gueules les mouettes ») et que lui et Sacha ont toujours joué les adeptes de toutes sortes de fantaisies sexuelles; dans les géniales émissions » vos gueules les mouettes » le trio Marc Sacha Augustin ne se refuse rien à ce niveau, même les trucs les plus retors.
Dans le podcast VGLM c’est plus souvent Marc qui joue le dépressif mais chaque membre du trio a prétendu à un moment avoir perdu son père ou avoir été victime de sévices; n’avoir plus goût à la vie; être antisocial…
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J’avais pas du tout connaissance de ce truc, j’écoute ça dès que possible! Il m’avait semblé qu’en général ils allaient effectivement plutôt toujours vers le plus trashos (le fameux extrait Saint-Arnoult/sérophilie du Wan-Dang Doodle IV), tandis que cet extrait est plus en retenue quoi…
Ce truc sur la perte du père comme « moment 100% hetero », c’est vraiment parfait
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Je suis d’accord que URDCUC est génial. Mon préféré est peut-être Chombo Loco. J’ai dû les voir trois fois chacun. Et concernant Augustin, il faut bien dire qu’il est en effet le génie du duo. Même si Sacha, dans le dernier, est vraiment très bon.
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« quand on se rapproche sur des points communs on peut jamais être sûr que ce soient pas nos défauts ; c’est comme si tu grattait ton eczéma avec l’ongle incarné de l’autre «
Il lache des bombes et les transforme en bombes à eau – désarmant de douceur et de cruauté
C’est aussi assez dingue mais la façon dont il expose l’impossibilité du rapport sexuel
– déjà je comprends mieux
– en plus ça me fait rire-
« Sans amour ni mort, il ne reste que les cartes fidélités »
Je me demande à quel point cette manière de faire de l’esprit est ironique. À la fois c’est génial et en même temps, comme souvent, il surjoue un type de discours pour montrer sa vacuité. Ici, ce serait un artiste dépressif qui se prend trop pour Wilde?
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Que c’est drôle putain…Je ne sais pas pourquoi, sa voix, son ton trainant et dépressif m’a fait penser à ce merveilleux personnage de fou, Patrick le schizophrène blagueur, dans un reportage devenu culte.
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C’est dingue je repensais à ce mec hier en rentrant du boulot ! J’avais envie de revoir ce reportage mais je ne me souvenais plus de son nom. Merci Mathieu. Il est génial ce mec.
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Il faut dire aussi que Patrick est un artiste peintre habité. Et pour l’anecdote j’étais tombée sur lui sur Adopte un mec, il y a quelques années.
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De rien de rien, il y a aussi une deuxième partie où il rentre vivre chez son père, qui est parfois un peu dépassé et impuissant face à la folie de son fils. Mais ce type m’émeut beaucoup, j’adore sa voix, et il a effectivement des talents de peintre !
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Je viens de regarder, merci pour la découverte
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J’adore le moment où Patrick veut chanter Johnny à son psy qui s’enfuit presque en courant
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Oui, c’est le meilleur moment.
« La chanson la plus obscène », le psychiatre il est parti en courant, lui qui a pour objectif que tout le monde se tienne bien, et Patrick entonne Que je t’aime.
Encore une fois le réel dépasse la fiction.
Le coup du pyjama c’est incroyable.
Le dispositif de contrôle est très puissant, et on a l’impression que personne n’à le coeur à rire, à commencer par le service de la clinique, mais que Patrick, étant « fou » arrive à ne pas être complètement écrasé par l’atmosphère pesante de la clinique et garde son sourire et sa joie. Faut être un peu fou pour avoir la force de tenir face à tous ces inspecteurs.
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Pour rebondir sur sa voix et son visage, dans une autre vie il aurait été acteur de films français avec Louis Garrel et Vincent Macaigne.
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Le potentiel de ce type est impressionnant. Quel dommage qu’il ait ces accès de furie.
On aimerait que le personnel joue un peu plus le jeu de sa puissance comique, mais d’une certaine façon je les comprend, 5j/7 dans un environnement pareil, pas sûr qu’on ait beaucoup envie de rire.
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J’ai essayé de regarder mais les psychiatres, je peux pas, c’est vraiment trop des sales fdp.
Je rêve d’un enfer spécial réservé à cette corporation de nazis dans lequel ils sont condamnés à être électrocutés, marginalisés, rendus infirmes à coup de molécules diverses, mis sous camisoles, traités comme des chiens toute la journée.
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C’est vrai qu’Augustin a un sacré génie et je le préférais bien à Sacha quand j’ai découvert le duo. Mais avec son podcast ciné et ce genre de formats solo je ne peux plus les départager dans leur puissance humoristique
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je l’aime bien aussi là :
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Je pourrais regarder Sacha à la fashion week pendant des heures, c’est fascinant.
Tu m’as fait découvrir les interview dans la rue de Sacha.
Autre vidéo assez géniale, que j’imagine bien connue.
J’aime assez son approche, où il est certes déstabilisant, mais il ne prend jamais les gens par la violence ou le ricanement. Il laisse le temps de parler et de réfléchir et prend le plus gros de la gêne sur lui.
Une homme généreux.
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Tensorial methods and renormalization
in
Group Field Theories
Doctoral thesis in physics, presented by
Sylvain Carrozza
Defended on September 19th, 2013, in front of the jury
Pr. Renaud Parentani Jury president
Pr. Bianca Dittrich Referee
Dr. Razvan Gurau Referee
Pr. Carlo Rovelli Jury member
Pr. Daniele Oriti Supervisor
Pr. Vincent Rivasseau SupervisorAbstract:
In this thesis, we study the structure of Group Field Theories (GFTs) from the point of view of renormalization theory. Such quantum field theories are found in approaches to quantum gravity related to Loop
Quantum Gravity (LQG) on the one hand, and to matrix models and tensor models on the other hand. They
model quantum space-time, in the sense that their Feynman amplitudes label triangulations, which can be
understood as transition amplitudes between LQG spin network states. The question of renormalizability is
crucial if one wants to establish interesting GFTs as well-defined (perturbative) quantum field theories, and
in a second step connect them to known infrared gravitational physics. Relying on recently developed tensorial tools, this thesis explores the GFT formalism in two complementary directions. First, new results on the
large cut-off expansion of the colored Boulatov-Ooguri models allow to explore further a non-perturbative
regime in which infinitely many degrees of freedom contribute. The second set of results provide a new
rigorous framework for the renormalization of so-called Tensorial GFTs (TGFTs) with gauge invariance
condition. In particular, a non-trivial 3d TGFT with gauge group SU(2) is proven just-renormalizable at
the perturbative level, hence opening the way to applications of the formalism to (3d Euclidean) quantum
gravity.
Key-words: quantum gravity, loop quantum gravity, spin foam, group field theory, tensor models, renormalization, lattice gauge theory.
Résumé :
Cette thèse présente une étude détaillée de la structure de théories appelées GFT (« Group Field Theory »
en anglais), à travers le prisme de la renormalisation. Ce sont des théories des champs issues de divers
travaux en gravité quantique, parmi lesquels la gravité quantique à boucles et les modèles de matrices ou
de tenseurs. Elles sont interprétées comme des modèles d’espaces-temps quantiques, dans le sens où elles
génèrent des amplitudes de Feynman indexées par des triangulations, qui interpolent les états spatiaux de
la gravité quantique à boucles. Afin d’établir ces modèles comme des théories des champs rigoureusement
définies, puis de comprendre leurs conséquences dans l’infrarouge, il est primordial de comprendre leur
renormalisation. C’est à cette tâche que cette thèse s’attèle, grâce à des méthodes tensorielles développées
récemment, et dans deux directions complémentaires. Premièrement, de nouveaux résultats sur l’expansion
asymptotique (en le cut-off) des modèles colorés de Boulatov-Ooguri sont démontrés, donnant accès à un
régime non-perturbatif dans lequel une infinité de degrés de liberté contribue. Secondement, un formalisme
général pour la renormalisation des GFTs dites tensorielles (TGFTs) et avec invariance de jauge est mis au
point. Parmi ces théories, une TGFT en trois dimensions et basée sur le groupe de jauge SU(2) se révèle
être juste renormalisable, ce qui ouvre la voie à l’application de ce formalisme à la gravité quantique.
Mots-clés: gravité quantique, gravité quantique à boucles, mousse de spin, group field theory, modèles
tensoriels, renormalisation, théorie de jauge sur réseau.
Thèse préparée au sein de l’Ecole Doctorale de Physique de la Région Parisienne (ED 107), dans le
Laboratoire de Physique Théorique d’Orsay (UMR 8627), Bât. 210, Université Paris-Sud 11, 91405 Orsay
Cedex; et en cotutelle avec le Max Planck Institute for Gravitational Physics (Albert Einstein Institute),
Am Mühlenberg 1, 14476 Golm, Allemagne, dans le cadre de l’International Max Planck Research School
(IMPRS).
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ii
Acknowledgments
First of all, I would like to thank my two supervisors, Daniele Oriti and Vincent Rivasseau.
Obviously, the results exposed in this thesis could not be achieved without their constant
implication, guidance and help. They introduced me to numerous physical concepts and
mathematical tools, with pedagogy and patience. Remarkably, their teachings and advices
were always complementary to each other, something I attribute to their open-mindedness
and which I greatly benefited from. I particularly appreciated the trusting relationship we
had from the beginning. It was thrilling, and to me the right balance between supervision
and freedom.
I feel honoured by the presence of Bianca Dittrich, Razvan Gurau, Renaud Parentani and
Carlo Rovelli in the jury, who kindly accepted to examine my work. Many thanks to Bianca
and Razvan especially, for their careful reading of this manuscript and their comments.
I would like to thank the people I met at the AEI and at the LPT, who contributed to
making these three years very enjoyable. The Berlin quantum gravity group being almost
uncountable, I will only mention the people I had the chance to directly collaborate with:
Aristide Baratin, Francesco Caravelli, James Ryan, Matti Raasakka and Matteo Smerlak.
It is quite difficult to keep track of all the events which, one way or another, conspired
to pushing me into physics and writing this thesis. It is easier to remember and thank the
people who triggered these long forgotten events.
First and foremost, my parents, who raised me with dedication and love, turning the
ignorant toddler I once was into a curious young adult. Most of what I am today takes its
roots at home, and has been profoundly influenced by my younger siblings: Manon, Julia,
Pauline and Thomas. My family at large, going under the name of Carrozza, Dislaire, Fontès,
Mécréant, Minden, Ravoux, or Ticchi, has always been very present and supportive, which
I want to acknowledge here.
The good old chaps, Sylvain Aubry, Vincent Bonnin and Florian Gaudin-Delrieu, deeply
influenced my high school years, and hence the way I think today. Meeting them in different
corners of Europe during the three years of this PhD was very precious and refreshing.
My friends from the ENS times played a major role in the recent years, both at the
scientific and human levels. In this respect I would especially like to thank Antonin Coutant,
Marc Geiller, and Baptiste Darbois-Texier: Antonin and Marc, for endless discussions about
theoretical physics and quantum gravity, which undoubtedly shaped my thinking over the
years; Baptiste for his truly unbelievable stories about real-world physics experiments; and
the three of them for their generosity and friendship, in Paris, Berlin or elsewhere.
Finally, I measure how lucky I am to have Tamara by my sides, who always supported
me with unconditional love. I found the necessary happiness and energy to achieve this PhD
thesis in the dreamed life we had together in Berlin.
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Wir sollen heiter Raum um Raum durchschreiten,
An keinem wie an einer Heimat hängen,
Der Weltgeist will nicht fesseln uns und engen,
Er will uns Stuf ’ um Stufe heben, weiten.
Hermann Hesse, Stufen, in Das Glasperlenspiel, 1943.
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Contents
1 Motivations and scope of the present work 1
1.1 Why a quantum theory of gravity cannot be dispensed with . . . . . . . . . 1
1.2 Quantum gravity and quantization . . . . . . . . . . . . . . . . . . . . . . . 5
1.3 On scales and renormalization with or without background . . . . . . . . . . 7
1.4 Purpose and plan of the thesis . . . . . . . . . . . . . . . . . . . . . . . . . . 10
2 Two paths to Group Field Theories 13
2.1 Group Field Theories and quantum General Relativity . . . . . . . . . . . . 13
2.1.1 Loop Quantum Gravity . . . . . . . . . . . . . . . . . . . . . . . . . 13
2.1.2 Spin Foams . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
2.1.3 Summing over Spin Foams . . . . . . . . . . . . . . . . . . . . . . . . 23
2.1.4 Towards well-defined quantum field theories of Spin Networks . . . . 25
2.2 Group Field Theories and random discrete geometries . . . . . . . . . . . . . 29
2.2.1 Matrix models and random surfaces . . . . . . . . . . . . . . . . . . . 29
2.2.2 Higher dimensional generalizations . . . . . . . . . . . . . . . . . . . 34
2.2.3 Bringing discrete geometry in . . . . . . . . . . . . . . . . . . . . . . 35
2.3 A research direction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
3 Colors and tensor invariance 41
3.1 Colored Group Field Theories . . . . . . . . . . . . . . . . . . . . . . . . . . 42
3.1.1 Combinatorial and topological motivations . . . . . . . . . . . . . . . 42
3.1.2 Motivation from discrete diffeomorphisms . . . . . . . . . . . . . . . 44
3.2 Colored tensor models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
3.2.1 Models and amplitudes . . . . . . . . . . . . . . . . . . . . . . . . . . 45
3.2.2 Degree and existence of the large N expansion . . . . . . . . . . . . . 46
3.2.3 The world of melons . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
3.3 Tensor invariance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
3.3.1 From colored simplices to tensor invariant interactions . . . . . . . . 49
3.3.2 Generalization to GFTs . . . . . . . . . . . . . . . . . . . . . . . . . 50
4 Large N expansion in topological Group Field Theories 51
4.1 Colored Boulatov model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
4.1.1 Vertex variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
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viii CONTENTS
4.1.2 Regularization and general scaling bounds . . . . . . . . . . . . . . . 63
4.1.3 Topological singularities . . . . . . . . . . . . . . . . . . . . . . . . . 67
4.1.4 Domination of melons . . . . . . . . . . . . . . . . . . . . . . . . . . 71
4.2 Colored Ooguri model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
4.2.1 Edge variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
4.2.2 Regularization and general scaling bounds . . . . . . . . . . . . . . . 84
4.2.3 Topological singularities . . . . . . . . . . . . . . . . . . . . . . . . . 88
4.2.4 Domination of melons . . . . . . . . . . . . . . . . . . . . . . . . . . 92
5 Renormalization of Tensorial Group Field Theories: generalities 97
5.1 Preliminaries: renormalization of local field theories . . . . . . . . . . . . . . 97
5.1.1 Locality, scales and divergences . . . . . . . . . . . . . . . . . . . . . 97
5.1.2 Perturbative renormalization through a multiscale decomposition . . 99
5.2 Locality and propagation in GFT . . . . . . . . . . . . . . . . . . . . . . . . 103
5.2.1 Simplicial and tensorial interactions . . . . . . . . . . . . . . . . . . . 104
5.2.2 Constraints and propagation . . . . . . . . . . . . . . . . . . . . . . . 105
5.3 A class of models with closure constraint . . . . . . . . . . . . . . . . . . . . 107
5.3.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107
5.3.2 Graph-theoretic and combinatorial tools . . . . . . . . . . . . . . . . 110
5.4 Multiscale expansion and power-counting . . . . . . . . . . . . . . . . . . . . 116
5.4.1 Multiscale decomposition . . . . . . . . . . . . . . . . . . . . . . . . . 116
5.4.2 Propagator bounds . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117
5.4.3 Abelian power-counting . . . . . . . . . . . . . . . . . . . . . . . . . 118
5.5 Classification of just-renormalizable models . . . . . . . . . . . . . . . . . . . 121
5.5.1 Analysis of the Abelian divergence degree . . . . . . . . . . . . . . . 121
5.5.2 Just-renormalizable models . . . . . . . . . . . . . . . . . . . . . . . 126
5.5.3 Properties of melonic subgraphs . . . . . . . . . . . . . . . . . . . . . 127
6 Super-renormalizable U(1) models in four dimensions 135
6.1 Divergent subgraphs and Wick ordering . . . . . . . . . . . . . . . . . . . . . 135
6.1.1 A bound on the divergence degree . . . . . . . . . . . . . . . . . . . . 136
6.1.2 Classification of divergences . . . . . . . . . . . . . . . . . . . . . . . 137
6.1.3 Localization operators . . . . . . . . . . . . . . . . . . . . . . . . . . 139
6.1.4 Melordering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140
6.1.5 Vacuum submelonic counter-terms . . . . . . . . . . . . . . . . . . . 143
6.2 Finiteness of the renormalized series . . . . . . . . . . . . . . . . . . . . . . . 144
6.2.1 Classification of forests . . . . . . . . . . . . . . . . . . . . . . . . . . 144
6.2.2 Power-counting of renormalized amplitudes . . . . . . . . . . . . . . . 146
6.2.3 Sum over scale attributions . . . . . . . . . . . . . . . . . . . . . . . 148
6.3 Example: Wick-ordering of a ϕ
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interaction . . . . . . . . . . . . . . . . . . 149
CONTENTS ix
7 Just-renormalizable SU(2) model in three dimensions 153
7.1 The model and its divergences . . . . . . . . . . . . . . . . . . . . . . . . . . 153
7.1.1 Regularization and counter-terms . . . . . . . . . . . . . . . . . . . . 153
7.1.2 List of divergent subgraphs . . . . . . . . . . . . . . . . . . . . . . . . 156
7.2 Non-Abelian multiscale expansion . . . . . . . . . . . . . . . . . . . . . . . . 158
7.2.1 Power-counting theorem . . . . . . . . . . . . . . . . . . . . . . . . . 158
7.2.2 Contraction of high melonic subgraphs . . . . . . . . . . . . . . . . . 161
7.3 Perturbative renormalizability . . . . . . . . . . . . . . . . . . . . . . . . . . 167
7.3.1 Effective and renormalized expansions . . . . . . . . . . . . . . . . . 169
7.3.2 Classification of forests . . . . . . . . . . . . . . . . . . . . . . . . . . 174
7.3.3 Convergent power-counting for renormalized amplitudes . . . . . . . . 178
7.3.4 Sum over scale attributions . . . . . . . . . . . . . . . . . . . . . . . 179
7.4 Renormalization group flow . . . . . . . . . . . . . . . . . . . . . . . . . . . 181
7.4.1 Approximation scheme . . . . . . . . . . . . . . . . . . . . . . . . . . 182
7.4.2 Truncated equations for the counter-terms . . . . . . . . . . . . . . . 184
7.4.3 Physical coupling constants: towards asymptotic freedom . . . . . . . 188
7.4.4 Mass and consistency of the assumptions . . . . . . . . . . . . . . . . 190
8 Conclusions and perspectives 193
8.1 The 1/N expansion in colored GFTs . . . . . . . . . . . . . . . . . . . . . . 193
8.1.1 Achievements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193
8.1.2 Discussion and outlook . . . . . . . . . . . . . . . . . . . . . . . . . . 194
8.2 Renormalization of TGFTs . . . . . . . . . . . . . . . . . . . . . . . . . . . . 196
8.2.1 Achievements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 196
8.2.2 Discussion and outlook . . . . . . . . . . . . . . . . . . . . . . . . . . 197
A Technical appendix 201
A.1 Heat Kernel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201
A.2 Proof of heat kernel bounds . . . . . . . . . . . . . . . . . . . . . . . . . . . 202
Bibliography 217
x CONTENTS
Chapter 1
Motivations and scope of the present
work
Nous sommes en 50 avant Jésus-Christ. Toute la Gaule est
occupée par les Romains… Toute? Non! Un village peuplé
d’irréductibles Gaulois résiste encore et toujours à l’envahisseur.
Et la vie n’est pas facile pour les garnisons de légionnaires romains des camps retranchés de Babaorum, Aquarium, Laudanum
et Petibonum. . .
René Goscinny and Albert Uderzo, Astérix le Gaulois
1.1 Why a quantum theory of gravity cannot be dispensed with
A consistent quantum theory of gravity is mainly called for by a conceptual clash between the
two major achievements of physicists of the XXth century. On the one hand, the realization
by Einstein that classical space-time is a dynamical entity correctly described by General
Relativity (GR), and on the other the advent of Quantum Mechanics (QM). The equivalence
principle, upon which GR is built, leads to the interpretation of gravitational phenomena
as pure geometric effects: the trajectories of test particles are geodesics in a curved fourdimensional manifold, space-time, whose geometric properties are encoded in a Lorentzian
metric tensor, which is nothing but the gravitational field [1]. Importantly, the identification
of the gravitational force to the metric properties of space-time entails the dynamical nature
of the latter. Indeed, gravity being sourced by masses and energy, space-time cannot remain
as a fixed arena into which physical processes happen, as was the case since Newton. With
Einstein, space-time becomes a physical system per se, whose precise structure is the result of
a subtle interaction with the other physical systems it contains. At the conceptual level, this
is arguably the main message of GR, and the precise interplay between the curved geometry
of space-time and matter fields is encoded into Einstein’s equations [2]. The second aspect
of the physics revolution which took place in the early XXth century revealed a wealth
of new phenomena in the microscopic world, and the dissolution of most of the classical
1
2 Chapter 1 : Motivations and scope of the present work
Newtonian picture at such scales: the disappearance of the notion of trajectory, unpredictable
outcomes of experiments, statistical predictions highly dependent on the experimental setup
[3]… At the mathematical level, QM brings along an entirely new arsenal of technical tools:
physical states are turned into vectors living in a Hilbert space, which replaces the phase
space of classical physics, and observables become Hermitian operators acting on physical
states. However, the conception of space-time on which QM relies remains deeply rooted in
Newtonian physics: the Schrödinger equation is a partial differential equation with respect
to fixed and physical space-time coordinates. For this reason, Special Relativity could be
proven compatible with these new rules of the game, thanks to the Quantum Field Theory
(QFT) formalism. The main difficulties in going from non-relativistic to relativistic quantum
theory boiled down to the incorporation of the Lorentz symmetry, which also acts on timelike directions. Achieving the same reconciliation with the lessons of GR is (and has been
proven to be) extraordinarily more difficult. The reason is that as soon as one contemplates
the idea of making the geometry of space-time both dynamical and quantum, one looses
in one stroke the fixed arena onto which the quantum foundations sit, and the Newtonian
determinism which allows to unambiguously link space-time dynamics to its content. The
randomness introduced by quantum measurements seems incompatible with the definition of
a single global state for space-time and matter (e.g. a solution of a set of partial differential
equations). And without a non-dynamical background, there is no unambiguous ’here’ where
quantum ensembles can be prepared, nor a ’there’ where measurements can be performed
and their statistical properties checked. In a word, by requiring background independence to
conform to Einstein’s ideas about gravity, one also suppresses the only remaining Newtonian
shelter where quantum probabilities can safely be interpreted. This is probably the most
puzzling aspect of modern physics, and calls for a resolution.
But, one could ask, do we necessarily need to make gravity quantum? Cannot we live
with the fact that matter is described by quantum fields propagating on a dynamical but
classical geometry? A short answer would be to reject the dichotomous understanding of
the world that would result from such a combination of a priori contradictory ideas. On
the other hand, one cannot deny that space-time is a very peculiar physical system, which
one might argue, could very well keep a singular status as the only fundamentally classical
entity. However, very nice and general arguments, put forward by Unruh in [4], make this
position untenable (at least literally). Let us recapitulate the main ideas of this article here.
In order to have the Einstein equations
Gµν = 8πGTµν (1.1)
as a classical limit of the matter sector, one possibility would be to interpret the righthand side as a quantum average hTˆ
µνi of some quantum operator representing the energymomentum tensor of matter fields. The problems with such a theory pointed out in [4] are
two-fold. First, quantum measurements would introduce discontinuities in the expectation
value of Tˆ
µν, and in turn spoil its conservation. Second, and as illustrated with a gravitational
version of Schrödinger’s cat gedanken experiment, such a coupling of gravity to a statistical
average of matter states would introduce slow variations of the gravitational field caused
1.1 Why a quantum theory of gravity cannot be dispensed with 3
by yet unobserved and undetermined matter states. Another idea explored by Unruh to
make sense of (1.1) in such a way that the left-hand side is classical, and the right-hand side
quantum, is through an eigenvalue equation of the type
8πGTˆ
µν|ψi = Gµν|ψi. (1.2)
The main issue here is that the definition of the operator Tˆ
µν would have to depend nonlinearly on the classical metric, and hence on the ’eigenvalue’ Gµν. From the point of view
of quantum theory, this of course does not make any sense.
Now that some conceptual motivations for the search for a quantum description of the
gravitational field have been recalled (and which are also the author’s personal main motivations to work in this field), one should make a bit more precise what one means by ’a
quantum theory of gravitation’ or ’quantum gravity’. We will adopt the kind of ambitious
though minimalistic position promoted in Loop Quantum Gravity (LQG) [5–7]. Minimalistic because the question of the unification of all forces at high energies is recognized as not
necessarily connected to quantum gravity, and therefore left unaddressed. But ambitious in
the sense that one is not looking for a theory of quantum perturbations of the gravitational
degrees of freedom around some background solution of GR, since this would be of little
help as far as the conceptual issues aforementioned are concerned. Indeed, and as is for
instance very well explained in [8,9], from the point of view of GR, there is no canonical way
of splitting the metric of space-time into a background (for instance a Minkowski metric,
but not necessarily so) plus fluctuations. Therefore giving a proper quantum description
of the latter fluctuations, that is finding a renormalizable theory of gravitons on a given
background, cannot fulfill the ultimate goal of reconciling GR with QM. On top of that, one
would need to show that the specification of the background is a kind of gauge choice, which
does not affect physical predictions. Therefore, one would like to insist on the fact that even
if such a theory was renormalizable, the challenge of making Einstein’s gravity fully quantum
and dynamical would remain almost untouched. This already suggests that introducing the
background in the first place is unnecessary. Since it turns out that the quantum theory of
perturbative quantum GR around a Minkowski background is not renormalizable [10], we
can even go one step further: the presence of a background might not only be unnecessary
but also problematic. The present thesis is in such a line of thought, which aims at taking the background independence of GR seriously, and use it as a guiding thread towards
its quantum version [11]. In this perspective, we would call ’quantum theory of gravity’ a
quantum theory without any space-time background, which would reduce to GR in some
(classical) limit.
A second set of ideas which are often invoked to justify the need for a theory of quantum
gravity concerns the presence of singularities in GR, and is therefore a bit more linked to
phenomenology, be it through cosmology close to the Big Bang or the question of the fate
of black holes at the end of Hawking’s evaporation. It is indeed tempting to draw a parallel
between the question of classical singularities in GR and some of the greatest successes of
the quantum formalism, such as for example the explanation of the stability of atoms or the
4 Chapter 1 : Motivations and scope of the present work
resolution of the UV divergence in the theory of black-body radiation. We do not want to
elaborate on these questions, but only point out that even if very suggestive and fascinating
proposals exist [12–14], there is as far as we know no definitive argument claiming that the
cumbersome genericity of singularities in GR has to be resolved in quantum gravity. This
is for us a secondary motivation to venture into such a quest, though a very important one.
While a quantum theory of gravity must by definition make QM and GR compatible, it only
might explain the nature of singularities in GR. Still, it would be of paramount relevance
if this second point were indeed realized, since it would open the door to a handful of new
phenomena and possible experimental signatures to look for.
Another set of ideas we consider important but we do not plan to address further in
this thesis are related to the non-renormalizability of perturbative quantum gravity. As a
quantum field theory on Minkowski space-time, the quantum theory of gravitons based on
GR can only be considered as an effective field theory [15, 16], which breaks down at the
Planck scale. Such a picture is therefore necessarily incomplete as a fundamental theory, as it
was to be expected, but does not provide any clear clue about how it should be completed.
At this point, two attitudes can be adopted. Either assume that one should first look
for a renormalizable perturbative theory of quantum gravity, from which the background
independent aspects will be addressed in a second stage; or, focus straight away on the
background independent features which are so central to the very question of quantum
gravity. Since we do not want to assume any a priori connection between the UV completion
of perturbative quantum general relativity and full-fledged quantum gravity, as is for instance
investigated in the asymptotic safety program [17, 18], the results of this thesis will be
presented in a mindset in line with the second attitude. Of course, any successful fundamental
quantum theory of gravity will have to provide a deeper understanding of the two-loops
divergences of quantum GR, and certainly any program which would fail to do so could not
be considered complete [19].
The purpose of the last two points was to justify to some extent the technical character
of this PhD thesis, and its apparent disconnection with many of the modern fundamental
theories which are experimentally verified. While it is perfectly legitimate to look for a
reconciliation of QM and GR into the details of what we know about matter, space and
time, we want to advocate here a hopefully complementary strategy, which aims at finding a
general theoretical framework encompassing them both at a general and conceptual level. At
this stage, we would for example be highly satisfied with a consistent definition of quantum
geometry whose degrees of freedom and dynamics would reduce to that of vacuum GR in
some limit; even if such a theory did not resolve classical singularities, nor it would provide
us with a renormalizable theory of gravitons.
1.2 Quantum gravity and quantization 5
1.2 Quantum gravity and quantization
Now that we reinstated the necessity of finding a consistent quantum formulation of gravitational physics, we would like to make some comments about the different general strategies
which are at our disposal to achieve such a goal. In particular, would a quantization of
general relativity (or a modification thereof) provide the answer?
The most conservative strategy is the quantization program of classical GR pioneered by
Bryce DeWitt [20], either through Dirac’s general canonical quantization procedure [21, 22]
or with covariant methods [23]. Modern incarnations of these early ideas can be found in
canonical loop quantum gravity and its tentative covariant formulation through spin foam
models [6, 7, 9]. While the Ashtekar formulation of GR [24, 25] allowed dramatic progress
with respect to DeWitt’s formal definitions, based on the usual metric formulation of Einstein’s theory, very challenging questions remain open as regards the dynamical aspects of
the theory. In particular, many ambiguities appear in the definition of the so-called scalar
constraint of canonical LQG, and therefore in the implementation of four-dimensional diffeomorphism invariance, which is arguably the core purpose of quantum gravity. There are
therefore two key aspects of the canonical quantization program that we would like to keep
in mind: first, the formulation of classical GR being used as a starting point (in metric
or Ashtekar variables), or equivalently the choice of fundamental degrees of freedom (the
metric tensor or a tetrad field), has a great influence on the quantization; and second, the
subtleties associated to space-time diffeomorphism invariance have so far plagued such attempts with numerous ambiguities, which prevent the quantization procedure from being
completed. The first point speaks in favor of loop variables in quantum gravity, while the
second might indicate an intrinsic limitation of the canonical approach.
A second, less conservative but more risky, type of quantization program consists in
discarding GR as a classical starting point, and instead postulating radically new degrees of
freedom. This is for example the case in string theory, where a classical theory of strings
moving in some background space-time is the starting point of the quantization procedure.
Such an approach is to some extent supported by the non-renormalizability of perturbative
quantum GR, interpreted as a signal of the presence of new degrees of freedom at the Planck
scale. Similar interpretations in similar situations already proved successful in the past, for
instance with the four-fermion theory of Fermi, whose non-renormalizability was cured by
the introduction of new gauge bosons, and gave rise to the renormalizable Weinberg-Salam
theory. In the case of gravity, and because of the unease with the perturbative strategy
mentioned before, we do not wish to give too much credit to such arguments. However, it is
necessary to keep in mind that the degrees of freedom we have access to in the low-energy
classical theory (GR) are not necessarily the ones to be quantized.
Finally, a third idea which is gaining increasing support in the recent years is to question
the very idea of quantizing gravity, at least stricto sensu. Rather, one should more generally
look for a quantum theory, with possibly non-metric degrees of freedom, from which classical
6 Chapter 1 : Motivations and scope of the present work
geometry and its dynamics would emerge. Such a scenario has been hinted at from within
GR itself, through the thermal properties of black holes and space-time in general. For
instance in [26], Jacobson suggested to interpret the Einstein equations as equations of
states at thermal equilibrium. In this picture, space-time dynamics would only emerge
in the thermodynamic limit of a more fundamental theory, with degrees of freedom yet
to be discovered. This is even more radical that what is proposed in string theory, but
also consistent with background independence in principle: there is no need to assume
the existence of a (continuous) background space-time in this picture, and contrarily so,
the finiteness of black hole entropy can be interpreted as suggestive of the existence of an
underlying discrete structure. Such ideas have close links with condensed matter theory,
which explains for example macroscopic properties of solids from the statistical properties
of their quantum microscopic building blocks, and in particular with the theory of quantum
fluids and Bose-Einstein condensates [27, 28]. Of course, the two outstanding issues are
that no experiments to directly probe the Planck scale are available in the near future, and
emergence has to be implemented in a fully background independent manner.
After this detour, one can come back to the main motivations of this thesis, loop quantum
gravity and spin foams, and remark that even there, the notion of emergence seems to have a
role to play. Indeed, the key prediction of canonical loop quantum gravity is undoubtedly the
discreteness of areas and volumes at the kinematical level [29], and this already entails some
kind of emergence of continuum space-time. In this picture, continuous space-time cannot
be defined all the way down to the Planck scale, where the discrete nature of the spectra of
geometric operators starts to be relevant. This presents a remarkable qualitative agreement
with Jacobson’s proposal, and in particular all the thermal aspects of black holes explored
in LQG derive from this fundamental result [30]. But there are other discrete features in
LQG and spin foams, possibly related to emergence, which need to be addressed. Even if
canonical LQG is a continuum theory, the Hilbert space it is based upon is constructed in
an inductive way, from states (the spin-network functionals) labeled by discrete quantities
(graphs with spin labels). We can say that each such state describes a continuous quantum
geometry with a finite number of degrees of freedom, and that the infinite number of possible
excitations associated to genuine continuous geometries is to be found in large superpositions
of these elementary states, in states associated to infinitely large graphs, or both. In practice,
only spin-network states on very small graphs can be investigated analytically, the limit
of infinitely large graphs being out of reach, and their superpositions even more so. This
indicates that in its current state, LQG can also be considered a theory of discrete geometries,
despite the fact that it is primarily a quantization of GR. From this point of view, continuous
classical space-time would only be recovered through a continuum limit. This is even more
supported by the covariant spin foam perspective, where the discrete aspects of spin networks
are enhanced rather than tamed. The discrete structure spin foam models are based upon,
2-complexes, acquire a double interpretation, as Feynman graphs labeling the transitions
between spin network states on the one hand, and as discretizations of space-time akin to
lattice gauge theory on the other hand. Contrary to the canonical picture, this second
interpretation cannot be avoided, at least in practice, since all the current spin foam models
1.3 On scales and renormalization with or without background 7
for four-dimensional gravity are constructed in a way to enforce a notion of (quantum)
discrete geometry in a cellular complex dual to the foam. Therefore, in our opinion, at this
stage of the development of the theory, it seems legitimate to view LQG and spin foam
models as quantum theories of discrete gravity. And if so, addressing the question of their
continuum limit is of primary importance.
Moreover, we tend to see a connection between: a) the ambiguities appearing in the
definition of the dynamics of canonical LQG, b) the fact that the relevance of a quantization
of GR can be questioned in a strong way, and c) the problem of the continuum in the
covariant version of loop quantum gravity. Altogether, these three points can be taken as a
motivation for a strategy where quantization and emergence both have to play their role. It
is indeed possible, and probably desirable, that some of the fine details of the dynamics of
spin networks are irrelevant to the large scale effects one would like to predict and study. In
the best case scenario, the different versions of the scalar constraint of LQG would fall in a
same universality class as far as the recovery of continuous space-time and its dynamics is
concerned. This would translate, in the covariant picture, as a set of spin foam models with
small variations in the way discrete geometry is encoded, but having a same continuum limit.
The crucial question to address in this perspective is that of the existence, and in a second
stage the universality of such a limit, in the sense of determining exactly which aspects (if
any) of the dynamics of spin networks are key to the emergence of space-time as we know
it. The fact that these same spin networks were initially thought of as quantum states of
continuous geometries should not prevent us from exploring other avenues, in which the
continuum only emerge in the presence of a very large number of discrete building blocks.
This PhD thesis has been prepared with the scenario just hinted at in mind, but we should
warn the reader that it is in no way conclusive in this respect. Moreover, we think and we
hope that the technical results and tools which are accounted for in this manuscript are
general enough to be useful to researchers in the field who do not share such point of views.
The reason is that, in order to study universality in quantum gravity, and ultimately find
the right balance between strict quantization procedures and emergence, one first needs to
develop a theory of renormalization in this background independent setting, which precisely
allows to consistently erase information and degrees of freedom. This thesis is a contribution
to this last point, in the Group Field Theory (GFT) formulation of spin foam models.
1.3 On scales and renormalization with or without background
The very idea of extending the theory of renormalization to quantum gravity may look odd
at first sight. The absence of any background seems indeed to preclude the existence of any
physical scale with respect to which the renormalization group flow should be defined. A
few remarks are therefore in order, about the different notions of scales which are available
in quantum field theories and general relativity, and the general assumption we will make
throughout this thesis in order to extend such notions to background independent theories.
8 Chapter 1 : Motivations and scope of the present work
Let us start with relativistic quantum field theories, which support the standard model of
particle physics, as well as perturbative quantum gravity around a Minkowski background.
The key ingredient entering the definition of these theories is the flat background metric,
which provides a notion of locality and global Poincaré invariance. The latter allows in
particular to classify all possible interactions once a field content (with its own set of internal symmetries) has been agreed on [31]. More interesting, this same Poincaré invariance,
combined with locality and the idea of renormalization [32–34], imposes further restrictions
on the number of independent couplings one should work with. When the theory is (perturbatively) non-renormalizable, it is consistent only if an infinite set of interactions is taken
into account, and therefore loses any predictive power (at least at some scale). When it
is on the contrary renormalizable, one can work with a finite set of interactions, though
arbitrarily large in the case of a super-renormalizable theory. For fundamental interactions,
the most interesting case is that of a just-renormalizable theory, such as QED or QCD, for
which a finite set of interactions is uniquely specified by the renormalizability criterion. In
all of these theories, what is meant by ’scale’ is of course an energy scale, in the sense of
special relativity. However, renormalization and quantum field theory are general enough to
accommodate various notions of scales, as for example non-relativistic energy, and apply to a
large variety of phenomena for which Poincaré invariance is completely irrelevant. A wealth
of examples of this kind can be found in condensed matter physics, and in the study of phase
transitions. The common feature of all these models is that they describe regimes in which
a huge number of (classical or quantum) degrees of freedom are present, and where their
contributions can be efficiently organized according to some order parameter, the ’scale’. As
we know well from thermodynamics and statistical mechanics, it is in this case desirable to
simplify the problem by assuming instead an infinite set of degrees of freedom, and adopt
a coarse-grained description in which degrees of freedom are collectively analyzed. Quantum field theory and renormalization are precisely a general set of techniques allowing to
efficiently organize such analyzes. Therefore, what makes renormalizable quantum field theories so useful in fundamental physics is not Poincaré invariance in itself, but the fact that
it implies the existence of an infinite reservoir of degrees of freedom in the deep UV.
We now turn to general relativity. The absence of Poincaré symmetry, or any analogous
notion of space-time global symmetries prevents the existence of a general notion of energy.
Except for special solutions of Einstein’s equations, there is no way to assign an unambiguous
notion of localized energy to the modes of the gravitational field1
. The two situations in which
special relativistic notions of energy-momentum do generalize are in the presence of a global
Killing symmetry, or for asymptotically flat space-times. In the first case, it is possible to
translate the fact that the energy-momentum tensor T
µν is divergence free into both local
and integral conservation equations for an energy-momentum vector P
µ ≡ T
µνKν, where Kν
1We can for instance quote Straumann [35]:
This has been disturbing to many people, but one simply has to get used to this fact. There is
no « energy-momentum tensor for the gravitational field ».
1.3 On scales and renormalization with or without background 9
is the Killing field. In the second case, only a partial generalization is available, in the form of
integral conservation equations for energy and momentum at spatial infinity. One therefore
already loses the possibility of localizing energy and momentum in this second situation,
since they are only defined for extended regions with boundaries in the approximately flat
asymptotic region. In any case, both generalizations rely on global properties of specific
solutions to Einstein’s equations which cannot be available in a background independent
formulation of quantum gravity. We therefore have to conclude that, since energy scales
associated to the gravitational field are at best solution-dependent, and in general not even
defined in GR, a renormalization group analysis of background independent quantum gravity
cannot be based on space-time related notions of scales.
This last point was to be expected on quite general grounds. From the point of view
of quantization à la Feynman for example, all the solutions to Einstein’s equations (and in
principle even more general ’off-shell’ geometries) are on the same footing, as they need to
be summed over in a path-integral (modulo boundary conditions). We cannot expect to
be able to organize such a path-integral according to scales defined internally to each of
these geometries. But even if one takes the emergent point of view seriously, GR suggests
that the order parameter with respect to which a renormalization group analysis should be
launched cannot depend on a space-time notion of energy. This point of view should be taken
more and more seriously as we move towards an increasingly background independent notion
of emergence, in the sense of looking for a unique mechanism which would be responsible
for the emergence of a large class of solutions of GR, if not all of them. In particular, as
soon as such a class is not restricted to space-times with global Killing symmetries or with
asymptotically flat spatial infinities, there seems to be no room for the usual notion of energy
in a renormalization analysis of quantum gravity.
However, it should already be understood at this stage that the absence of any background
space-time in quantum gravity, and therefore of any natural physical scales, does not prevent
us from using the quantum field theory and renormalization formalisms. As was already
mentioned, the notion of scale prevailing in renormalization theory is more the number of
degrees of freedom available in a region of the parameter space, rather than a proper notion of
energy. Likewise, if quantum fields do need a fixed background structure to live in, this needs
not be interpreted as space-time. As we will see, this is precisely how GFTs are constructed,
as quantum field theories defined on (internal) symmetry groups rather than space-time
manifolds. More generally, the working assumption of this thesis will be that a notion of scale
and renormalization group flow can be defined before1
space-time notions become available,
and studied with quantum field theory techniques, as for example advocated in [36,37]. The
only background notions one is allowed to use in such a program must also be present in
the background of GR. The dimension of space-time, the local Lorentz symmetry, and the
diffeomorphism groups are among them, but they do not support any obvious notion of
scale. Rather, we will postulate that the ’number of degrees of freedom’ continues to be a
1Obviously, this ’before’ does not refer to time, but rather to the abstract notion of scale which is assumed
to take over when no space-time structure is available anymore.
10 Chapter 1 : Motivations and scope of the present work
relevant order parameter in the models we will consider, that is in the absence of space-time.
This rather abstract scale will come with canonical definitions of UV and IR sectors. They
should by no means be understood as their space-time related counter-parts, and be naively
related to respectively small and large distance regimes. Instead, the UV sector will simply
be the corner of parameter space responsible for divergences, or equivalently where ’most’
of the degrees of freedom sit. A natural renormalization group flow will be defined, which
will allow to average out the contributions of the degrees of freedom, from higher to lower
scales. The only strong conceptual assumption we will make in this respect is that such an
abstract definition of renormalization is physical and can be used to describe the emergence
of space-time structures. However, at this general level of discussion, we would like to convey
the idea that such a strong assumption is in a sense also minimal. Indeed, if one wants to
be able to speak of emergence of space and time, one also needs at least one new parameter
which is neither time nor space. We simply call this order parameter ’scale’, and identify it
with one of the central features of quantum field theory: the renormalization group. It is
in our view the most direct route towards new physics in the absence of space and time, as
quantum gravity seems to require.
1.4 Purpose and plan of the thesis
We are well aware of the fact that the previous motivations cannot be taken for granted.
They can be contested in various ways, and also lack a great deal of precision. The reader
should see them as a guiding thread towards making full sense of the emergence of spacetime from background independent physics, rather than definitive statements embraced by
the author. From now on, we will refrain from venturing into more conceptual discussions,
and mostly leave the specific examples worked out in this thesis speak for themselves, hoping
that they will do so in favor of the general ideas outlined before.
The rest of the thesis is organized as follows. In chapter 2, we will start by recalling
the two main ways of understanding the construction of GFT models. One takes its root
in the quantization program for quantum gravity, in the form of loop quantum gravity and
spin foam models. In this line of thoughts, GFTs are generating functionals for spin foam
amplitudes, in the same way as quantum field theories are generating functionals for Feynman
amplitudes. In this sense, they complete the definition of spin foam models by assigning
canonical weights to the different foams contributing to a same transition between boundary
states (spin networks). Moreover, a quantum field theory formalism is expected to provide
easier access to non-perturbative regimes, and hence to the continuum. For example, classical
equations of motion can be used as a way to change vacuum [38], or to study condensed
phases of the theory [39]. Of course, this specific completion of the definition of spin foam
models relies on a certeain confidence in the quantum field theory formalism. Alternative but
hopefully complementary approaches exist, such as coarse-graining methods imported from
condensed matter physics and quantum information theory [40–42]. Though, if one decides
to stick to quantum field theory weights, it seems natural to also bring renormalization-
Heureusement pour toi le ridicule ne tue pas.
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Je ne connaissais pas ces itw, merci !
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Tensorial methods and renormalization
in
Group Field Theories
Doctoral thesis in physics, presentted by
Sylvain Carrozza
Defended on September 19th, 2013, in front of the jury
Pr. Renaud Parentani Jury president
Pr. Bianca Dittrich Referee
Dr. Razvan Gurau Referee
Pr. Carlo Rovelli Jury member
Pr. Daniele Oriti Supervisor
Pr. Vincent Rivasseau SupervisorAbstract:
In this thesis, we study the structure of Group Field Theories (GFTs) from the point of view of renormalization theory. Such quantum field theories are found in approaches to quantum gravity related to Loop
Quantum Gravity (LQG) on the one hand, and to matrix models and tensor models on the other hand. They
model quantum space-time, in the sense that their Feynman amplitudes label triangulations, which can be
understood as transition amplitudes between LQG spin network states. The question of renormalizability is
crucial if one wants to establish interesting GFTs as well-defined (perturbative) quantum field theories, and
in a second step connect them to known infrared gravitational physics. Relying on recently developed tensorial tools, this thesis explores the GFT formalism in two complementary directions. First, new results on the
large cut-off expansion of the colored Boulatov-Ooguri models allow to explore further a non-perturbative
regime in which infinitely many degrees of freedom contribute. The second set of results provide a new
rigorous framework for the renormalization of so-called Tensorial GFTs (TGFTs) with gauge invariance
condition. In particular, a non-trivial 3d TGFT with gauge group SU(2) is proven just-renormalizable at
the perturbative level, hence opening the way to applications of the formalism to (3d Euclidean) quantum
gravity.
Key-words: quantum gravity, loop quantum gravity, spin foam, group field theory, tensor models, renormalization, lattice gauge theory.
Résumé :
Cette thèse présente une étude détaillée de la structure de théories appelées GFT (« Group Field Theory »
en anglais), à travers le prisme de la renormalisation. Ce sont des théories des champs issues de divers
travaux en gravité quantique, parmi lesquels la gravité quantique à boucles et les modèles de matrices ou
de tenseurs. Elles sont interprétées comme des modèles d’espaces-temps quantiques, dans le sens où elles
génèrent des amplitudes de Feynman indexées par des triangulations, qui interpolent les états spatiaux de
la gravité quantique à boucles. Afin d’établir ces modèles comme des théories des champs rigoureusement
définies, puis de comprendre leurs conséquences dans l’infrarouge, il est primordial de comprendre leur
renormalisation. C’est à cette tâche que cette thèse s’attèle, grâce à des méthodes tensorielles développées
récemment, et dans deux directions complémentaires. Premièrement, de nouveaux résultats sur l’expansion
asymptotique (en le cut-off) des modèles colorés de Boulatov-Ooguri sont démontrés, donnant accès à un
régime non-perturbatif dans lequel une infinité de degrés de liberté contribue. Secondement, un formalisme
général pour la renormalisation des GFTs dites tensorielles (TGFTs) et avec invariance de jauge est mis au
point. Parmi ces théories, une TGFT en trois dimensions et basée sur le groupe de jauge SU(2) se révèle
être juste renormalisable, ce qui ouvre la voie à l’application de ce formalisme à la gravité quantique.
Mots-clés: gravité quantique, gravité quantique à boucles, mousse de spin, group field theory, modèles
tensoriels, renormalisation, théorie de jauge sur réseau.
Thèse préparée au sein de l’Ecole Doctorale de Physique de la Région Parisienne (ED 107), dans le
Laboratoire de Physique Théorique d’Orsay (UMR 8627), Bât. 210, Université Paris-Sud 11, 91405 Orsay
Cedex; et en cotutelle avec le Max Planck Institute for Gravitational Physics (Albert Einstein Institute),
Am Mühlenberg 1, 14476 Golm, Allemagne, dans le cadre de l’International Max Planck Research School
(IMPRS).
i
ii
Acknowledgments
First of all, I would like to thank my two supervisors, Daniele Oriti and Vincent Rivasseau.
Obviously, the results exposed in this thesis could not be achieved without their constant
implication, guidance and help. They introduced me to numerous physical concepts and
mathematical tools, with pedagogy and patience. Remarkably, their teachings and advices
were always complementary to each other, something I attribute to their open-mindedness
and which I greatly benefited from. I particularly appreciated the trusting relationship we
had from the beginning. It was thrilling, and to me the right balance between supervision
and freedom.
I feel honoured by the presence of Bianca Dittrich, Razvan Gurau, Renaud Parentani and
Carlo Rovelli in the jury, who kindly accepted to examine my work. Many thanks to Bianca
and Razvan especially, for their careful reading of this manuscript and their comments.
I would like to thank the people I met at the AEI and at the LPT, who contributed to
making these three years very enjoyable. The Berlin quantum gravity group being almost
uncountable, I will only mention the people I had the chance to directly collaborate with:
Aristide Baratin, Francesco Caravelli, James Ryan, Matti Raasakka and Matteo Smerlak.
It is quite difficult to keep track of all the events which, one way or another, conspired
to pushing me into physics and writing this thesis. It is easier to remember and thank the
people who triggered these long forgotten events.
First and foremost, my parents, who raised me with dedication and love, turning the
ignorant toddler I once was into a curious young adult. Most of what I am today takes its
roots at home, and has been profoundly influenced by my younger siblings: Manon, Julia,
Pauline and Thomas. My family at large, going under the name of Carrozza, Dislaire, Fontès,
Mécréant, Minden, Ravoux, or Ticchi, has always been very present and supportive, which
I want to acknowledge here.
The good old chaps, Sylvain Aubry, Vincent Bonnin and Florian Gaudin-Delrieu, deeply
influenced my high school years, and hence the way I think today. Meeting them in different
corners of Europe during the three years of this PhD was very precious and refreshing.
My friends from the ENS times played a major role in the recent years, both at the
scientific and human levels. In this respect I would especially like to thank Antonin Coutant,
Marc Geiller, and Baptiste Darbois-Texier: Antonin and Marc, for endless discussions about
theoretical physics and quantum gravity, which undoubtedly shaped my thinking over the
years; Baptiste for his truly unbelievable stories about real-world physics experiments; and
the three of them for their generosity and friendship, in Paris, Berlin or elsewhere.
Finally, I measure how lucky I am to have Tamara by my sides, who always supported
me with unconditional love. I found the necessary happiness and energy to achieve this PhD
thesis in the dreamed life we had together in Berlin.
iii
iv
Wir sollen heiter Raum um Raum durchschreiten,
An keinem wie an einer Heimat hängen,
Der Weltgeist will nicht fesseln uns und engen,
Er will uns Stuf ’ um Stufe heben, weiten.
Hermann Hesse, Stufen, in Das Glasperlenspiel, 1943.
v
vi
Contents
1 Motivations and scope of the present work 1
1.1 Why a quantum theory of gravity cannot be dispensed with . . . . . . . . . 1
1.2 Quantum gravity and quantization . . . . . . . . . . . . . . . . . . . . . . . 5
1.3 On scales and renormalization with or without background . . . . . . . . . . 7
1.4 Purpose and plan of the thesis . . . . . . . . . . . . . . . . . . . . . . . . . . 10
2 Two paths to Group Field Theories 13
2.1 Group Field Theories and quantum General Relativity . . . . . . . . . . . . 13
2.1.1 Loop Quantum Gravity . . . . . . . . . . . . . . . . . . . . . . . . . 13
2.1.2 Spin Foams . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
2.1.3 Summing over Spin Foams . . . . . . . . . . . . . . . . . . . . . . . . 23
2.1.4 Towards well-defined quantum field theories of Spin Networks . . . . 25
2.2 Group Field Theories and random discrete geometries . . . . . . . . . . . . . 29
2.2.1 Matrix models and random surfaces . . . . . . . . . . . . . . . . . . . 29
2.2.2 Higher dimensional generalizations . . . . . . . . . . . . . . . . . . . 34
2.2.3 Bringing discrete geometry in . . . . . . . . . . . . . . . . . . . . . . 35
2.3 A research direction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
3 Colors and tensor invariance 41
3.1 Colored Group Field Theories . . . . . . . . . . . . . . . . . . . . . . . . . . 42
3.1.1 Combinatorial and topological motivations . . . . . . . . . . . . . . . 42
3.1.2 Motivation from discrete diffeomorphisms . . . . . . . . . . . . . . . 44
3.2 Colored tensor models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
3.2.1 Models and amplitudes . . . . . . . . . . . . . . . . . . . . . . . . . . 45
3.2.2 Degree and existence of the large N expansion . . . . . . . . . . . . . 46
3.2.3 The world of melons . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
3.3 Tensor invariance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
3.3.1 From colored simplices to tensor invariant interactions . . . . . . . . 49
3.3.2 Generalization to GFTs . . . . . . . . . . . . . . . . . . . . . . . . . 50
4 Large N expansion in topological Group Field Theories 51
4.1 Colored Boulatov model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
4.1.1 Vertex variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
vii
viii CONTENTS
4.1.2 Regularization and general scaling bounds . . . . . . . . . . . . . . . 63
4.1.3 Topological singularities . . . . . . . . . . . . . . . . . . . . . . . . . 67
4.1.4 Domination of melons . . . . . . . . . . . . . . . . . . . . . . . . . . 71
4.2 Colored Ooguri model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
4.2.1 Edge variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
4.2.2 Regularization and general scaling bounds . . . . . . . . . . . . . . . 84
4.2.3 Topological singularities . . . . . . . . . . . . . . . . . . . . . . . . . 88
4.2.4 Domination of melons . . . . . . . . . . . . . . . . . . . . . . . . . . 92
5 Renormalization of Tensorial Group Field Theories: generalities 97
5.1 Preliminaries: renormalization of local field theories . . . . . . . . . . . . . . 97
5.1.1 Locality, scales and divergences . . . . . . . . . . . . . . . . . . . . . 97
5.1.2 Perturbative renormalization through a multiscale decomposition . . 99
5.2 Locality and propagation in GFT . . . . . . . . . . . . . . . . . . . . . . . . 103
5.2.1 Simplicial and tensorial interactions . . . . . . . . . . . . . . . . . . . 104
5.2.2 Constraints and propagation . . . . . . . . . . . . . . . . . . . . . . . 105
5.3 A class of models with closure constraint . . . . . . . . . . . . . . . . . . . . 107
5.3.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107
5.3.2 Graph-theoretic and combinatorial tools . . . . . . . . . . . . . . . . 110
5.4 Multiscale expansion and power-counting . . . . . . . . . . . . . . . . . . . . 116
5.4.1 Multiscale decomposition . . . . . . . . . . . . . . . . . . . . . . . . . 116
5.4.2 Propagator bounds . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117
5.4.3 Abelian power-counting . . . . . . . . . . . . . . . . . . . . . . . . . 118
5.5 Classification of just-renormalizable models . . . . . . . . . . . . . . . . . . . 121
5.5.1 Analysis of the Abelian divergence degree . . . . . . . . . . . . . . . 121
5.5.2 Just-renormalizable models . . . . . . . . . . . . . . . . . . . . . . . 126
5.5.3 Properties of melonic subgraphs . . . . . . . . . . . . . . . . . . . . . 127
6 Super-renormalizable U(1) models in four dimensions 135
6.1 Divergent subgraphs and Wick ordering . . . . . . . . . . . . . . . . . . . . . 135
6.1.1 A bound on the divergence degree . . . . . . . . . . . . . . . . . . . . 136
6.1.2 Classification of divergences . . . . . . . . . . . . . . . . . . . . . . . 137
6.1.3 Localization operators . . . . . . . . . . . . . . . . . . . . . . . . . . 139
6.1.4 Melordering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140
6.1.5 Vacuum submelonic counter-terms . . . . . . . . . . . . . . . . . . . 143
6.2 Finiteness of the renormalized series . . . . . . . . . . . . . . . . . . . . . . . 144
6.2.1 Classification of forests . . . . . . . . . . . . . . . . . . . . . . . . . . 144
6.2.2 Power-counting of renormalized amplitudes . . . . . . . . . . . . . . . 146
6.2.3 Sum over scale attributions . . . . . . . . . . . . . . . . . . . . . . . 148
6.3 Example: Wick-ordering of a ϕ
6
interaction . . . . . . . . . . . . . . . . . . 149
CONTENTS ix
7 Just-renormalizable SU(2) model in three dimensions 153
7.1 The model and its divergences . . . . . . . . . . . . . . . . . . . . . . . . . . 153
7.1.1 Regularization and counter-terms . . . . . . . . . . . . . . . . . . . . 153
7.1.2 List of divergent subgraphs . . . . . . . . . . . . . . . . . . . . . . . . 156
7.2 Non-Abelian multiscale expansion . . . . . . . . . . . . . . . . . . . . . . . . 158
7.2.1 Power-counting theorem . . . . . . . . . . . . . . . . . . . . . . . . . 158
7.2.2 Contraction of high melonic subgraphs . . . . . . . . . . . . . . . . . 161
7.3 Perturbative renormalizability . . . . . . . . . . . . . . . . . . . . . . . . . . 167
7.3.1 Effective and renormalized expansions . . . . . . . . . . . . . . . . . 169
7.3.2 Classification of forests . . . . . . . . . . . . . . . . . . . . . . . . . . 174
7.3.3 Convergent power-counting for renormalized amplitudes . . . . . . . . 178
7.3.4 Sum over scale attributions . . . . . . . . . . . . . . . . . . . . . . . 179
7.4 Renormalization group flow . . . . . . . . . . . . . . . . . . . . . . . . . . . 181
7.4.1 Approximation scheme . . . . . . . . . . . . . . . . . . . . . . . . . . 182
7.4.2 Truncated equations for the counter-terms . . . . . . . . . . . . . . . 184
7.4.3 Physical coupling constants: towards asymptotic freedom . . . . . . . 188
7.4.4 Mass and consistency of the assumptions . . . . . . . . . . . . . . . . 190
8 Conclusions and perspectives 193
8.1 The 1/N expansion in colored GFTs . . . . . . . . . . . . . . . . . . . . . . 193
8.1.1 Achievements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193
8.1.2 Discussion and outlook . . . . . . . . . . . . . . . . . . . . . . . . . . 194
8.2 Renormalization of TGFTs . . . . . . . . . . . . . . . . . . . . . . . . . . . . 196
8.2.1 Achievements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 196
8.2.2 Discussion and outlook . . . . . . . . . . . . . . . . . . . . . . . . . . 197
A Technical appendix 201
A.1 Heat Kernel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201
A.2 Proof of heat kernel bounds . . . . . . . . . . . . . . . . . . . . . . . . . . . 202
Bibliography 217
x CONTENTS
Chapter 1
Motivations and scope of the present
work
Nous sommes en 50 avant Jésus-Christ. Toute la Gaule est
occupée par les Romains… Toute? Non! Un village peuplé
d’irréductibles Gaulois résiste encore et toujours à l’envahisseur.
Et la vie n’est pas facile pour les garnisons de légionnaires romains des camps retranchés de Babaorum, Aquarium, Laudanum
et Petibonum. . .
René Goscinny and Albert Uderzo, Astérix le Gaulois
1.1 Why a quantum theory of gravity cannot be dispensed with
A consistent quantum theory of gravity is mainly called for by a conceptual clash between the
two major achievements of physicists of the XXth century. On the one hand, the realization
by Einstein that classical space-time is a dynamical entity correctly described by General
Relativity (GR), and on the other the advent of Quantum Mechanics (QM). The equivalence
principle, upon which GR is built, leads to the interpretation of gravitational phenomena
as pure geometric effects: the trajectories of test particles are geodesics in a curved fourdimensional manifold, space-time, whose geometric properties are encoded in a Lorentzian
metric tensor, which is nothing but the gravitational field [1]. Importantly, the identification
of the gravitational force to the metric properties of space-time entails the dynamical nature
of the latter. Indeed, gravity being sourced by masses and energy, space-time cannot remain
as a fixed arena into which physical processes happen, as was the case since Newton. With
Einstein, space-time becomes a physical system per se, whose precise structure is the result of
a subtle interaction with the other physical systems it contains. At the conceptual level, this
is arguably the main message of GR, and the precise interplay between the curved geometry
of space-time and matter fields is encoded into Einstein’s equations [2]. The second aspect
of the physics revolution which took place in the early XXth century revealed a wealth
of new phenomena in the microscopic world, and the dissolution of most of the classical
1
2 Chapter 1 : Motivations and scope of the present work
Newtonian picture at such scales: the disappearance of the notion of trajectory, unpredictable
outcomes of experiments, statistical predictions highly dependent on the experimental setup
[3]… At the mathematical level, QM brings along an entirely new arsenal of technical tools:
physical states are turned into vectors living in a Hilbert space, which replaces the phase
space of classical physics, and observables become Hermitian operators acting on physical
states. However, the conception of space-time on which QM relies remains deeply rooted in
Newtonian physics: the Schrödinger equation is a partial differential equation with respect
to fixed and physical space-time coordinates. For this reason, Special Relativity could be
proven compatible with these new rules of the game, thanks to the Quantum Field Theory
(QFT) formalism. The main difficulties in going from non-relativistic to relativistic quantum
theory boiled down to the incorporation of the Lorentz symmetry, which also acts on timelike directions. Achieving the same reconciliation with the lessons of GR is (and has been
proven to be) extraordinarily more difficult. The reason is that as soon as one contemplates
the idea of making the geometry of space-time both dynamical and quantum, one looses
in one stroke the fixed arena onto which the quantum foundations sit, and the Newtonian
determinism which allows to unambiguously link space-time dynamics to its content. The
randomness introduced by quantum measurements seems incompatible with the definition of
a single global state for space-time and matter (e.g. a solution of a set of partial differential
equations). And without a non-dynamical background, there is no unambiguous ’here’ where
quantum ensembles can be prepared, nor a ’there’ where measurements can be performed
and their statistical properties checked. In a word, by requiring background independence to
conform to Einstein’s ideas about gravity, one also suppresses the only remaining Newtonian
shelter where quantum probabilities can safely be interpreted. This is probably the most
puzzling aspect of modern physics, and calls for a resolution.
But, one could ask, do we necessarily need to make gravity quantum? Cannot we live
with the fact that matter is described by quantum fields propagating on a dynamical but
classical geometry? A short answer would be to reject the dichotomous understanding of
the world that would result from such a combination of a priori contradictory ideas. On
the other hand, one cannot deny that space-time is a very peculiar physical system, which
one might argue, could very well keep a singular status as the only fundamentally classical
entity. However, very nice and general arguments, put forward by Unruh in [4], make this
position untenable (at least literally). Let us recapitulate the main ideas of this article here.
In order to have the Einstein equations
Gµν = 8πGTµν (1.1)
as a classical limit of the matter sector, one possibility would be to interpret the righthand side as a quantum average hTˆ
µνi of some quantum operator representing the energymomentum tensor of matter fields. The problems with such a theory pointed out in [4] are
two-fold. First, quantum measurements would introduce discontinuities in the expectation
value of Tˆ
µν, and in turn spoil its conservation. Second, and as illustrated with a gravitational
version of Schrödinger’s cat gedanken experiment, such a coupling of gravity to a statistical
average of matter states would introduce slow variations of the gravitational field caused
1.1 Why a quantum theory of gravity cannot be dispensed with 3
by yet unobserved and undetermined matter states. Another idea explored by Unruh to
make sense of (1.1) in such a way that the left-hand side is classical, and the right-hand side
quantum, is through an eigenvalue equation of the type
8πGTˆ
µν|ψi = Gµν|ψi. (1.2)
The main issue here is that the definition of the operator Tˆ
µν would have to depend nonlinearly on the classical metric, and hence on the ’eigenvalue’ Gµν. From the point of view
of quantum theory, this of course does not make any sense.
Now that some conceptual motivations for the search for a quantum description of the
gravitational field have been recalled (and which are also the author’s personal main motivations to work in this field), one should make a bit more precise what one means by ’a
quantum theory of gravitation’ or ’quantum gravity’. We will adopt the kind of ambitious
though minimalistic position promoted in Loop Quantum Gravity (LQG) [5–7]. Minimalistic because the question of the unification of all forces at high energies is recognized as not
necessarily connected to quantum gravity, and therefore left unaddressed. But ambitious in
the sense that one is not looking for a theory of quantum perturbations of the gravitational
degrees of freedom around some background solution of GR, since this would be of little
help as far as the conceptual issues aforementioned are concerned. Indeed, and as is for
instance very well explained in [8,9], from the point of view of GR, there is no canonical way
of splitting the metric of space-time into a background (for instance a Minkowski metric,
but not necessarily so) plus fluctuations. Therefore giving a proper quantum description
of the latter fluctuations, that is finding a renormalizable theory of gravitons on a given
background, cannot fulfill the ultimate goal of reconciling GR with QM. On top of that, one
would need to show that the specification of the background is a kind of gauge choice, which
does not affect physical predictions. Therefore, one would like to insist on the fact that even
if such a theory was renormalizable, the challenge of making Einstein’s gravity fully quantum
and dynamical would remain almost untouched. This already suggests that introducing the
background in the first place is unnecessary. Since it turns out that the quantum theory of
perturbative quantum GR around a Minkowski background is not renormalizable [10], we
can even go one step further: the presence of a background might not only be unnecessary
but also problematic. The present thesis is in such a line of thought, which aims at taking the background independence of GR seriously, and use it as a guiding thread towards
its quantum version [11]. In this perspective, we would call ’quantum theory of gravity’ a
quantum theory without any space-time background, which would reduce to GR in some
(classical) limit.
A second set of ideas which are often invoked to justify the need for a theory of quantum
gravity concerns the presence of singularities in GR, and is therefore a bit more linked to
phenomenology, be it through cosmology close to the Big Bang or the question of the fate
of black holes at the end of Hawking’s evaporation. It is indeed tempting to draw a parallel
between the question of classical singularities in GR and some of the greatest successes of
the quantum formalism, such as for example the explanation of the stability of atoms or the
4 Chapter 1 : Motivations and scope of the present work
resolution of the UV divergence in the theory of black-body radiation. We do not want to
elaborate on these questions, but only point out that even if very suggestive and fascinating
proposals exist [12–14], there is as far as we know no definitive argument claiming that the
cumbersome genericity of singularities in GR has to be resolved in quantum gravity. This
is for us a secondary motivation to venture into such a quest, though a very important one.
While a quantum theory of gravity must by definition make QM and GR compatible, it only
might explain the nature of singularities in GR. Still, it would be of paramount relevance
if this second point were indeed realized, since it would open the door to a handful of new
phenomena and possible experimental signatures to look for.
Another set of ideas we consider important but we do not plan to address further in
this thesis are related to the non-renormalizability of perturbative quantum gravity. As a
quantum field theory on Minkowski space-time, the quantum theory of gravitons based on
GR can only be considered as an effective field theory [15, 16], which breaks down at the
Planck scale. Such a picture is therefore necessarily incomplete as a fundamental theory, as it
was to be expected, but does not provide any clear clue about how it should be completed.
At this point, two attitudes can be adopted. Either assume that one should first look
for a renormalizable perturbative theory of quantum gravity, from which the background
independent aspects will be addressed in a second stage; or, focus straight away on the
background independent features which are so central to the very question of quantum
gravity. Since we do not want to assume any a priori connection between the UV completion
of perturbative quantum general relativity and full-fledged quantum gravity, as is for instance
investigated in the asymptotic safety program [17, 18], the results of this thesis will be
presented in a mindset in line with the second attitude. Of course, any successful fundamental
quantum theory of gravity will have to provide a deeper understanding of the two-loops
divergences of quantum GR, and certainly any program which would fail to do so could not
be considered complete [19].
The purpose of the last two points was to justify to some extent the technical character
of this PhD thesis, and its apparent disconnection with many of the modern fundamental
theories which are experimentally verified. While it is perfectly legitimate to look for a
reconciliation of QM and GR into the details of what we know about matter, space and
time, we want to advocate here a hopefully complementary strategy, which aims at finding a
general theoretical framework encompassing them both at a general and conceptual level. At
this stage, we would for example be highly satisfied with a consistent definition of quantum
geometry whose degrees of freedom and dynamics would reduce to that of vacuum GR in
some limit; even if such a theory did not resolve classical singularities, nor it would provide
us with a renormalizable theory of gravitons.
1.2 Quantum gravity and quantization 5
1.2 Quantum gravity and quantization
Now that we reinstated the necessity of finding a consistent quantum formulation of gravitational physics, we would like to make some comments about the different general strategies
which are at our disposal to achieve such a goal. In particular, would a quantization of
general relativity (or a modification thereof) provide the answer?
The most conservative strategy is the quantization program of classical GR pioneered by
Bryce DeWitt [20], either through Dirac’s general canonical quantization procedure [21, 22]
or with covariant methods [23]. Modern incarnations of these early ideas can be found in
canonical loop quantum gravity and its tentative covariant formulation through spin foam
models [6, 7, 9]. While the Ashtekar formulation of GR [24, 25] allowed dramatic progress
with respect to DeWitt’s formal definitions, based on the usual metric formulation of Einstein’s theory, very challenging questions remain open as regards the dynamical aspects of
the theory. In particular, many ambiguities appear in the definition of the so-called scalar
constraint of canonical LQG, and therefore in the implementation of four-dimensional diffeomorphism invariance, which is arguably the core purpose of quantum gravity. There are
therefore two key aspects of the canonical quantization program that we would like to keep
in mind: first, the formulation of classical GR being used as a starting point (in metric
or Ashtekar variables), or equivalently the choice of fundamental degrees of freedom (the
metric tensor or a tetrad field), has a great influence on the quantization; and second, the
subtleties associated to space-time diffeomorphism invariance have so far plagued such attempts with numerous ambiguities, which prevent the quantization procedure from being
completed. The first point speaks in favor of loop variables in quantum gravity, while the
second might indicate an intrinsic limitation of the canonical approach.
A second, less conservative but more risky, type of quantization program consists in
discarding GR as a classical starting point, and instead postulating radically new degrees of
freedom. This is for example the case in string theory, where a classical theory of strings
moving in some background space-time is the starting point of the quantization procedure.
Such an approach is to some extent supported by the non-renormalizability of perturbative
quantum GR, interpreted as a signal of the presence of new degrees of freedom at the Planck
scale. Similar interpretations in similar situations already proved successful in the past, for
instance with the four-fermion theory of Fermi, whose non-renormalizability was cured by
the introduction of new gauge bosons, and gave rise to the renormalizable Weinberg-Salam
theory. In the case of gravity, and because of the unease with the perturbative strategy
mentioned before, we do not wish to give too much credit to such arguments. However, it is
necessary to keep in mind that the degrees of freedom we have access to in the low-energy
classical theory (GR) are not necessarily the ones to be quantized.
Finally, a third idea which is gaining increasing support in the recent years is to question
the very idea of quantizing gravity, at least stricto sensu. Rather, one should more generally
look for a quantum theory, with possibly non-metric degrees of freedom, from which classical
6 Chapter 1 : Motivations and scope of the present work
geometry and its dynamics would emerge. Such a scenario has been hinted at from within
GR itself, through the thermal properties of black holes and space-time in general. For
instance in [26], Jacobson suggested to interpret the Einstein equations as equations of
states at thermal equilibrium. In this picture, space-time dynamics would only emerge
in the thermodynamic limit of a more fundamental theory, with degrees of freedom yet
to be discovered. This is even more radical that what is proposed in string theory, but
also consistent with background independence in principle: there is no need to assume
the existence of a (continuous) background space-time in this picture, and contrarily so,
the finiteness of black hole entropy can be interpreted as suggestive of the existence of an
underlying discrete structure. Such ideas have close links with condensed matter theory,
which explains for example macroscopic properties of solids from the statistical properties
of their quantum microscopic building blocks, and in particular with the theory of quantum
fluids and Bose-Einstein condensates [27, 28]. Of course, the two outstanding issues are
that no experiments to directly probe the Planck scale are available in the near future, and
emergence has to be implemented in a fully background independent manner.
After this detour, one can come back to the main motivations of this thesis, loop quantum
gravity and spin foams, and remark that even there, the notion of emergence seems to have a
role to play. Indeed, the key prediction of canonical loop quantum gravity is undoubtedly the
discreteness of areas and volumes at the kinematical level [29], and this already entails some
kind of emergence of continuum space-time. In this picture, continuous space-time cannot
be defined all the way down to the Planck scale, where the discrete nature of the spectra of
geometric operators starts to be relevant. This presents a remarkable qualitative agreement
with Jacobson’s proposal, and in particular all the thermal aspects of black holes explored
in LQG derive from this fundamental result [30]. But there are other discrete features in
LQG and spin foams, possibly related to emergence, which need to be addressed. Even if
canonical LQG is a continuum theory, the Hilbert space it is based upon is constructed in
an inductive way, from states (the spin-network functionals) labeled by discrete quantities
(graphs with spin labels). We can say that each such state describes a continuous quantum
geometry with a finite number of degrees of freedom, and that the infinite number of possible
excitations associated to genuine continuous geometries is to be found in large superpositions
of these elementary states, in states associated to infinitely large graphs, or both. In practice,
only spin-network states on very small graphs can be investigated analytically, the limit
of infinitely large graphs being out of reach, and their superpositions even more so. This
indicates that in its current state, LQG can also be considered a theory of discrete geometries,
despite the fact that it is primarily a quantization of GR. From this point of view, continuous
classical space-time would only be recovered through a continuum limit. This is even more
supported by the covariant spin foam perspective, where the discrete aspects of spin networks
are enhanced rather than tamed. The discrete structure spin foam models are based upon,
2-complexes, acquire a double interpretation, as Feynman graphs labeling the transitions
between spin network states on the one hand, and as discretizations of space-time akin to
lattice gauge theory on the other hand. Contrary to the canonical picture, this second
interpretation cannot be avoided, at least in practice, since all the current spin foam models
1.3 On scales and renormalization with or without background 7
for four-dimensional gravity are constructed in a way to enforce a notion of (quantum)
discrete geometry in a cellular complex dual to the foam. Therefore, in our opinion, at this
stage of the development of the theory, it seems legitimate to view LQG and spin foam
models as quantum theories of discrete gravity. And if so, addressing the question of their
continuum limit is of primary importance.
Moreover, we tend to see a connection between: a) the ambiguities appearing in the
definition of the dynamics of canonical LQG, b) the fact that the relevance of a quantization
of GR can be questioned in a strong way, and c) the problem of the continuum in the
covariant version of loop quantum gravity. Altogether, these three points can be taken as a
motivation for a strategy where quantization and emergence both have to play their role. It
is indeed possible, and probably desirable, that some of the fine details of the dynamics of
spin networks are irrelevant to the large scale effects one would like to predict and study. In
the best case scenario, the different versions of the scalar constraint of LQG would fall in a
same universality class as far as the recovery of continuous space-time and its dynamics is
concerned. This would translate, in the covariant picture, as a set of spin foam models with
small variations in the way discrete geometry is encoded, but having a same continuum limit.
The crucial question to address in this perspective is that of the existence, and in a second
stage the universality of such a limit, in the sense of determining exactly which aspects (if
any) of the dynamics of spin networks are key to the emergence of space-time as we know
it. The fact that these same spin networks were initially thought of as quantum states of
continuous geometries should not prevent us from exploring other avenues, in which the
continuum only emerge in the presence of a very large number of discrete building blocks.
This PhD thesis has been prepared with the scenario just hinted at in mind, but we should
warn the reader that it is in no way conclusive in this respect. Moreover, we think and we
hope that the technical results and tools which are accounted for in this manuscript are
general enough to be useful to researchers in the field who do not share such point of views.
The reason is that, in order to study universality in quantum gravity, and ultimately find
the right balance between strict quantization procedures and emergence, one first needs to
develop a theory of renormalization in this background independent setting, which precisely
allows to consistently erase information and degrees of freedom. This thesis is a contribution
to this last point, in the Group Field Theory (GFT) formulation of spin foam models.
1.3 On scales and renormalization with or without background
The very idea of extending the theory of renormalization to quantum gravity may look odd
at first sight. The absence of any background seems indeed to preclude the existence of any
physical scale with respect to which the renormalization group flow should be defined. A
few remarks are therefore in order, about the different notions of scales which are available
in quantum field theories and general relativity, and the general assumption we will make
throughout this thesis in order to extend such notions to background independent theories.
8 Chapter 1 : Motivations and scope of the present work
Let us start with relativistic quantum field theories, which support the standard model of
particle physics, as well as perturbative quantum gravity around a Minkowski background.
The key ingredient entering the definition of these theories is the flat background metric,
which provides a notion of locality and global Poincaré invariance. The latter allows in
particular to classify all possible interactions once a field content (with its own set of internal symmetries) has been agreed on [31]. More interesting, this same Poincaré invariance,
combined with locality and the idea of renormalization [32–34], imposes further restrictions
on the number of independent couplings one should work with. When the theory is (perturbatively) non-renormalizable, it is consistent only if an infinite set of interactions is taken
into account, and therefore loses any predictive power (at least at some scale). When it
is on the contrary renormalizable, one can work with a finite set of interactions, though
arbitrarily large in the case of a super-renormalizable theory. For fundamental interactions,
the most interesting case is that of a just-renormalizable theory, such as QED or QCD, for
which a finite set of interactions is uniquely specified by the renormalizability criterion. In
all of these theories, what is meant by ’scale’ is of course an energy scale, in the sense of
special relativity. However, renormalization and quantum field theory are general enough to
accommodate various notions of scales, as for example non-relativistic energy, and apply to a
large variety of phenomena for which Poincaré invariance is completely irrelevant. A wealth
of examples of this kind can be found in condensed matter physics, and in the study of phase
transitions. The common feature of all these models is that they describe regimes in which
a huge number of (classical or quantum) degrees of freedom are present, and where their
contributions can be efficiently organized according to some order parameter, the ’scale’. As
we know well from thermodynamics and statistical mechanics, it is in this case desirable to
simplify the problem by assuming instead an infinite set of degrees of freedom, and adopt
a coarse-grained description in which degrees of freedom are collectively analyzed. Quantum field theory and renormalization are precisely a general set of techniques allowing to
efficiently organize such analyzes. Therefore, what makes renormalizable quantum field theories so useful in fundamental physics is not Poincaré invariance in itself, but the fact that
it implies the existence of an infinite reservoir of degrees of freedom in the deep UV.
We now turn to general relativity. The absence of Poincaré symmetry, or any analogous
notion of space-time global symmetries prevents the existence of a general notion of energy.
Except for special solutions of Einstein’s equations, there is no way to assign an unambiguous
notion of localized energy to the modes of the gravitational field1
. The two situations in which
special relativistic notions of energy-momentum do generalize are in the presence of a global
Killing symmetry, or for asymptotically flat space-times. In the first case, it is possible to
translate the fact that the energy-momentum tensor T
µν is divergence free into both local
and integral conservation equations for an energy-momentum vector P
µ ≡ T
µνKν, where Kν
1We can for instance quote Straumann [35]:
This has been disturbing to many people, but one simply has to get used to this fact. There is
no « energy-momentum tensor for the gravitational field ».
1.3 On scales and renormalization with or without background 9
is the Killing field. In the second case, only a partial generalization is available, in the form of
integral conservation equations for energy and momentum at spatial infinity. One therefore
already loses the possibility of localizing energy and momentum in this second situation,
since they are only defined for extended regions with boundaries in the approximately flat
asymptotic region. In any case, both generalizations rely on global properties of specific
solutions to Einstein’s equations which cannot be available in a background independent
formulation of quantum gravity. We therefore have to conclude that, since energy scales
associated to the gravitational field are at best solution-dependent, and in general not even
defined in GR, a renormalization group analysis of background independent quantum gravity
cannot be based on space-time related notions of scales.
This last point was to be expected on quite general grounds. From the point of view
of quantization à la Feynman for example, all the solutions to Einstein’s equations (and in
principle even more general ’off-shell’ geometries) are on the same footing, as they need to
be summed over in a path-integral (modulo boundary conditions). We cannot expect to
be able to organize such a path-integral according to scales defined internally to each of
these geometries. But even if one takes the emergent point of view seriously, GR suggests
that the order parameter with respect to which a renormalization group analysis should be
launched cannot depend on a space-time notion of energy. This point of view should be taken
more and more seriously as we move towards an increasingly background independent notion
of emergence, in the sense of looking for a unique mechanism which would be responsible
for the emergence of a large class of solutions of GR, if not all of them. In particular, as
soon as such a class is not restricted to space-times with global Killing symmetries or with
asymptotically flat spatial infinities, there seems to be no room for the usual notion of energy
in a renormalization analysis of quantum gravity.
However, it should already be understood at this stage that the absence of any background
space-time in quantum gravity, and therefore of any natural physical scales, does not prevent
us from using the quantum field theory and renormalization formalisms. As was already
mentioned, the notion of scale prevailing in renormalization theory is more the number of
degrees of freedom available in a region of the parameter space, rather than a proper notion of
energy. Likewise, if quantum fields do need a fixed background structure to live in, this needs
not be interpreted as space-time. As we will see, this is precisely how GFTs are constructed,
as quantum field theories defined on (internal) symmetry groups rather than space-time
manifolds. More generally, the working assumption of this thesis will be that a notion of scale
and renormalization group flow can be defined before1
space-time notions become available,
and studied with quantum field theory techniques, as for example advocated in [36,37]. The
only background notions one is allowed to use in such a program must also be present in
the background of GR. The dimension of space-time, the local Lorentz symmetry, and the
diffeomorphism groups are among them, but they do not support any obvious notion of
scale. Rather, we will postulate that the ’number of degrees of freedom’ continues to be a
1Obviously, this ’before’ does not refer to time, but rather to the abstract notion of scale which is assumed
to take over when no space-time structure is available anymore.
10 Chapter 1 : Motivations and scope of the present work
relevant order parameter in the models we will consider, that is in the absence of space-time.
This rather abstract scale will come with canonical definitions of UV and IR sectors. They
should by no means be understood as their space-time related counter-parts, and be naively
related to respectively small and large distance regimes. Instead, the UV sector will simply
be the corner of parameter space responsible for divergences, or equivalently where ’most’
of the degrees of freedom sit. A natural renormalization group flow will be defined, which
will allow to average out the contributions of the degrees of freedom, from higher to lower
scales. The only strong conceptual assumption we will make in this respect is that such an
abstract definition of renormalization is physical and can be used to describe the emergence
of space-time structures. However, at this general level of discussion, we would like to convey
the idea that such a strong assumption is in a sense also minimal. Indeed, if one wants to
be able to speak of emergence of space and time, one also needs at least one new parameter
which is neither time nor space. We simply call this order parameter ’scale’, and identify it
with one of the central features of quantum field theory: the renormalization group. It is
in our view the most direct route towards new physics in the absence of space and time, as
quantum gravity seems to require.
1.4 Purpose and plan of the thesis
We are well aware of the fact that the previous motivations cannot be taken for granted.
They can be contested in various ways, and also lack a great deal of precision. The reader
should see them as a guiding thread towards making full sense of the emergence of spacetime from background independent physics, rather than definitive statements embraced by
the author. From now on, we will refrain from venturing into more conceptual discussions,
and mostly leave the specific examples worked out in this thesis speak for themselves, hoping
that they will do so in favor of the general ideas outlined before.
The rest of the thesis is organized as follows. In chapter 2, we will start by recalling
the two main ways of understanding the construction of GFT models. One takes its root
in the quantization program for quantum gravity, in the form of loop quantum gravity and
spin foam models. In this line of thoughts, GFTs are generating functionals for spin foam
amplitudes, in the sam way as quantum field theories are generating functionals for Feynman
amplitudes. In this sense, they complete the definition of spin foam models by assigning
canonical weights to the different foams contributing to a same transition between boundary
states (spin networks). Moreover, a quantum field theory formalism is expected to provide
easier access to non-perturbative regimes, and hence to the continuum. For example, classical
equations of motion can be used as a way to change vacuum [38], or to study condensed
phases of the theory [39]. Of course, this specific completion of the definition of spin foam
models relies on a certain confidence in the quantum field theory formalism. Alternative but
hopefully complementary approaches exist, such as coarse-graining methods imported from
condensed matter physics and quantum information theory [40–42]. Though, if one decides
to stick to quantum field theory weights, it seems natural to also bring renormalization
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