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 desmodèles 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 deschamps 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. / 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.

Made available in DSpace on 2016-08-03T16:34:26Z (GMT). No. of bitstreams: 1 Rafael Erik de Menezes.pdf: 561756 bytes, checksum: f5cb063fe9c3a9d91af334a57ef99665 (MD5) Previous issue date: 2013-03-21 / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior / The population and school communities have been pushing and charging the authorities of Justice Systems and Education for a coherent and effective intervention in resolving conflicts established in the school environment, since the current model of Justice, the retributive, has remedied the situation. The Restorative Justice then arises as a new way to tackle this problem and one of its strategies is the restorative circle, characterized as a group to restore relations and conflicts. This research aims to describe and analyze the structural elements of restorative circles and phenomena of the field group, performed in the school environment to intervene in conflict situations. The sample consisted of five restorative practices involving pre-circle, and the circle after circle mediated by a facilitator and two co-facilitators. The processing of data was based on the analysis of the structural elements of restorative justice (opening and closing ceremony, Cane speech, and consensual decision-making), considering such elements equal to the basic psychoanalytic setting, since this specific case, has intended to make clear to the participants of the group which is the proposed operation of the same, and a content analysis organized from predefined categories, according to psychoanalytic concepts (resistance, acting and insight). The results showed that were established structural elements (setting) conducive to meeting participants and the predominant field strengths, which resulted in the reestablishment of good fellowship in all cases analyzed. The structural elements established for the completion of the restorative circle created a safe space where participants bind positively, even with the conflict situation. It is considered important to credit the figure of the facilitator (Psychologist) part of achieving conflict resolution. Conclude that the function continent; handling and understanding of resistors, actings and insights contributed to the group field in configure cohesion rather than disintegration. Finally, it should be added that experience has shown that children and adolescents respond very well when they are invited to participate in a restorative circle there and learn to act according to the values experienced as an educational process. / A população e as comunidades escolares vêm pressionando e cobrando as autoridades dos Sistemas de Justiça e da Educação por uma intervenção coerente e efetiva na resolução de conflitos que se estabelecem no ambiente escolar, uma vez que o atual modelo de Justiça, o retributivo, não tem sanado a situação. A Justiça restaurativa surge então como uma nova maneira para enfrentar esse problema e uma de suas estratégias é o círculo restaurativo, caracterizado como um grupo para restauração das relações e dos conflitos. Esta pesquisa visa: descrever e analisar os elementos estruturais dos círculos restaurativos e os fenômenos do campo grupal em processos restaurativos realizados no ambiente escolar para intervir em situações de conflito. A amostra foi composta de cinco práticas restaurativas que envolveram pré-círculo, círculo e o pós-círculo mediados por um facilitador e dois co-facilitadores. O tratamento dos dados se deu a partir da análise dos elementos estruturais da justiça restaurativa (cerimônia de abertura e fechamento, bastão de fala, e processo decisório consensual), considerando tais elementos iguais ao setting de base psicanalítica, visto que neste caso especifico, tem como objetivo deixar claro aos participantes do grupo qual é a proposta de funcionamento do mesmo; e de uma análise de conteúdo organizada a partir de categorias pré-definidas, segundo conceitos psicanalíticos (resistência, acting/atuação e insight/elaboração). Os resultados mostraram que foram estabelecidos elementos estruturais (setting) favorável ao encontro dos participantes e que predominaram no campo aspectos positivos, o que resultou no bom reestabelecimento do convívio em todos os casos analisados. Os elementos estruturais estabelecidos para a realização do círculo restaurativo criaram um espaço seguro onde os participantes se ligaram de modo positivo, mesmo com a situação de conflito. Considera-se importante creditar a figura do facilitador (Psicólogo) parte da realização da resolução do conflito. Conclui-se que a função continente; o manejo e compreensão das resistências, actings e dos insights contribuíram para que o campo grupal configurasse em coesão ao invés da desintegração. Finalmente, cabe acrescentar que a experiência demonstrou que as crianças e adolescentes respondem muito bem quando são convidados a participar de um círculo restaurativo e ali aprendem a agir de acordo com os valores vivenciados como em um processo educativo.

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 desmodèles 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 deschamps 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.

In einigen Ansätzen zu einer Quantentheorie der Gravitation wie Gruppenfeldtheorie und Schleifenquantengravitation zeigt sich, dass Zustände und Entwicklungen der geometrischen Freiheitsgrade auf einer diskreten Raumzeit basieren. Die dringendste Frage ist dann, wie die glatten Geometrien der Allgemeinen Relativitätstheorie, beschrieben durch geeignete geometrische Beobachtungsgrößen, aus solch diskreten Quantengeometrien im semiklassischen und Kontinuums-Limes hervorgehen. Hier nehme ich die Frage geeigneter Beobachtungsgrößen mit Fokus auf die effektive Dimension der Quantengeometrien in Angriff. Dazu gebe ich eine rein kombinatorische Beschreibung der zugrunde liegenden diskreten Strukturen. Als Nebenthema erlaubt dies eine Erweiterung der Gruppenfeldtheorie, so dass diese den kombinatorisch größeren kinematischen Zustandsraum der Schleifenquantengravitation abdeckt. Zudem führe ich einen diskreten Differentialrechnungskalkül für Felder auf solch fundamental diskreten Geometrien mit einem speziellen Augenmerk auf dem Laplace-Operator ein. Dies ermöglicht die Definition der Dimensionsobservablen für Quantengeometrien. Die Untersuchung verschiedener Klassen von Quantengeometrien zeigt allgemein, dass die spektrale Dimension stärker von der zugrunde liegenden kombinatorischen Struktur als von den Details der zusätzlichen geometrischen Daten darauf abhängt. Semiklassische Zustände in Schleifenquantengravitation approximieren die entsprechenden klassischen Geometrien gut ohne Anzeichen für stärkere Quanteneffekte. Dagegen zeigt sich im Kontext eines allgemeineren, auf analytischen Lösungen basierenden Modells für Zustände, die aus Überlagerungen einer großen Anzahl von Komplexen bestehen, ein Fluss der spektralen Dimension von der topologischen Dimension d bei kleinen Energieskalen hin zu einem reellen Wert zwischen 0 und d bei hohen Energien. Im Spezialfall 1 erlauben diese Resultate, die Quantengeometrie als effektiv fraktal aufzufassen. / In several approaches towards a quantum theory of gravity, such as group field theory and loop quantum gravity, quantum states and histories of the geometric degrees of freedom turn out to be based on discrete spacetime. The most pressing issue is then how the smooth geometries of general relativity, expressed in terms of suitable geometric observables, arise from such discrete quantum geometries in some semiclassical and continuum limit. In this thesis I tackle the question of suitable observables focusing on the effective dimension of discrete quantum geometries. For this purpose I give a purely combinatorial description of the discrete structures which these geometries have support on. As a side topic, this allows to present an extension of group field theory to cover the combinatorially larger kinematical state space of loop quantum gravity. Moreover, I introduce a discrete calculus for fields on such fundamentally discrete geometries with a particular focus on the Laplacian. This permits to define the effective-dimension observables for quantum geometries. Analysing various classes of quantum geometries, I find as a general result that the spectral dimension is more sensitive to the underlying combinatorial structure than to the details of the additional geometric data thereon. Semiclassical states in loop quantum gravity approximate the classical geometries they are peaking on rather well and there are no indications for stronger quantum effects. On the other hand, in the context of a more general model of states which are superposition over a large number of complexes, based on analytic solutions, there is a flow of the spectral dimension from the topological dimension d on low energy scales to a real number between 0 and d on high energy scales. In the particular case of 1 these results allow to understand the quantum geometry as effectively fractal.