Despite remarkable progress made in the past century, which has revolutionized our understanding of the universe, there are numerous open questions left in theoretical physics. Particularly important is the fact that the theories describing the fundamental interactions of nature are incompatible. Einstein's theory of general relative describes gravity as a dynamical spacetime, which is curved by matter and whose curvature determines the motion of matter. On the other hand we have quantum field theory, in form of the standard model of particle physics, where particles interact via the remaining interactions - electromagnetic, weak and strong interaction - on a flat, static spacetime without gravity. A theory of quantum gravity is hoped to cure this incompatibility by heuristically replacing classical spacetime by quantum spacetime'. Several approaches exist attempting to define such a theory with differing underlying premises and ideas, where it is not clear which is to be preferred. Yet a minimal requirement is the compatibility with the classical theory, they attempt to generalize. Interestingly many of these models rely on discrete structures in their definition or postulate discreteness of spacetime to be fundamental. Besides the direct advantages discretisations provide, e.g. permitting numerical simulations, they come with serious caveats requiring thorough investigation: In general discretisations break fundamental diffeomorphism symmetry of gravity and are generically not unique. Both complicates establishing the connection to the classical continuum theory. The main focus of this thesis lies in the investigation of this relation for spin foam models. This is done on different levels of the discretisation / triangulation, ranging from few simplices up to the continuum limit. In the regime of very few simplices we confirm and deepen the connection of spin foam models to discrete gravity. Moreover, we discuss dynamical, e.g. diffeomorphism invariance in the discrete, to fix the ambiguities of the models. In order to satisfy these conditions, the discrete models have to be improved in a renormalisation procedure, which also allows us to study their continuum dynamics. Applied to simplified spin foam models, we uncover a rich, non--trivial fixed point structure, which we summarize in a phase diagram. Inspired by these methods, we propose a method to consistently construct the continuum theory, which comes with a unique vacuum state. / Trotz bemerkenswerter Fortschritte im vergangenen Jahrhundert, die unser Verständnis des Universums revolutioniert haben, gibt es noch zahlreiche ungeklärte Fragen in der theoretischen Physik. Besondere Bedeutung kommt der Tatsache zu, dass die Theorien, welche die fundamentalen Wechselwirkungen der Natur beschreiben, inkompatibel sind. Nach Einsteins allgemeiner Relativitätstheorie wird die Gravitation durch eine dynamische Raumzeit dargestellt, die von Materie gekrümmt wird und ihrerseits durch die Krümmung die Bewegung der Materie bestimmt. Dem gegenüber steht die Quantenfeldtheorie, die die verbliebenen Wechselwirkungen - elektromagnetische, schwache und starke Wechselwirkung - im Standardmodell der Teilchenphysik beschreibt, in dem Teilchen auf einer statischen Raumzeit -- ohne Gravitation -- miteinander interagieren. Die Hoffnung ist, dass eine Theorie der Quantengravitation diese Inkompatibilität beheben kann, indem, heuristisch, die klassische Raumzeit durch eine 'Quantenraumzeit' ersetzt wird. Es gibt zahlreiche Ansätze eine solche Theorie zu definieren, die auf unterschiedlichen Prämissen und Ideen beruhen, wobei a priori nicht klar ist, welche zu bevorzugen sind. Eine Minimalanforderung an diese Theorien ist Kompatibilität mit der klassischen Theorie, die sie verallgemeinern sollen. Interessanterweise basieren zahlreiche Modelle in ihrer Definition auf Diskretisierungen oder postulieren eine fundamentale Diskretheit der Raumzeit. Neben den unmittelbaren Vorteilen, die Diskretisierungen bieten, z.B. das Ermöglichen numerischer Simulationen, gibt es auch gravierende Nachteile, die einer ausführlichen Untersuchung bedürfen: Im Allgemeinen brechen Diskretisierungen die fundamentale Diffeomorphismensymmetrie der Gravitation und sind in der Regel nicht eindeutig definiert. Beides erschwert die Wiederherstellung der Verbindung zur klassischen, kontinuierlichen Theorie. Das Hauptaugenmerk dieser Doktorarbeit liegt darin diese Verbindung insbesondere für Spin-Schaum-Modelle (spin foam models) zu untersuchen. Dies geschieht auf sehr verschiedenen Ebenen der Diskretisierung / Triangulierung, angefangen bei wenigen Simplizes bis hin zum Kontinuumslimes. Im Regime weniger Simplizes wird die bekannte Verbindung von Spin--Schaum--Modellen zu diskreter Gravitation bestätigt und vertieft. Außerdem diskutieren wir dynamische Prinzipien, z.B. Diffeomorphismeninvarianz im Diskreten, um die Ambiguitäten der Modelle zu fixieren. Um diese Bedingungen zu erfüllen, müssen die diskreten Modelle durch Renormierungsverfahren verbessert werden, wodurch wir auch ihre Kontinuumsdynamik untersuchen können. Angewandt auf vereinfachte Spin-Schaum-Modelle finden wir eine reichhaltige, nicht-triviale Fixpunkt-Struktur, die wir in einem Phasendiagramm zusammenfassen. Inspiriert von diesen Methoden schlagen wir zu guter Letzt eine konsistente Konstruktionsmethode für die Kontinuumstheorie vor, die einen eindeutigen Vakuumszustand definiert.

La relativité générale est la théorie rendant compte de la gravitation via une déformation de l'espace-temps. Son application à l'Univers permet, dans le modèle Lambda-CDM, de bien rentre compte des observations cosmologiques. Cependant, à l'échelle de Planck, la théorie ne fonctionne plus et s'avère incohérente. Pour résoudre ce problème, il est sans doute essentiel de tenir compte des effets quantiques. Depuis près d'un siècle, concilier relativité générale et mécanique quantique est considéré comme une priorité de la physique théorique. La tâche s'avère néanmoins extraordinairement difficile et cette thèse est consacrée à l'une des pistes les plus sérieuses : la gravitation quantique à boucles. Pour aller de l'avant dans cette démarche nécessaire mais complexe, des confrontation avec des données expérimentales seraient essentielles. Nous nous sommes ainsi intéressés aux perturbations cosmologiques générées dans ce cadre. Nous avons étudié en détails les conséquences phénoménologiques des corrections de cosmologie quantique à boucles aux modes tensoriels dans un modèle d'univers en rebond. Une analyse de Fisher a été développée pour comparer ces prédictions aux éventuelles futures observations. Pour les autres modes, nous nous sommes placés dans un formalisme spécifique incluant le calcul de contre-termes permettant de prévenir l'apparition d'anomalies dans la structure de l'algèbre des contraintes. Ce formalisme a été appliqué aux cas des perturbations vectorielles puis scalaires. Les équations du mouvement invariantes de jauges permettant de calculer les spectres ont alors été dérivées. / General relativity describes gravity as a deformation of space-time. Applied to the Universe as a whole, it explains well cosmological observations in the lambda-CDM paradigm. However, at the Planck scale, the theory is not anymore self-consistent. It is most probably necessary to include quantum effects. For a century, this has been considered as one of the main challenges for theoretical physics. This is however an extremely difficult aim to reach and this thesis is devoted to one of the main proposal: Loop Quantum Gravity. To go ahead in the construction of any quantum theory of gravity, it would be most useful to compare predictions with observations. To this aim, we have studied cosmological perturbations in this framework. We have investigated into the details the phenomenological consequences of loop quantum cosmology corrections in a bouncing universe. A Fisher analysis was carried out to compare the predictions with future data. For the other modes, we have used a specific formalism to include counterterms that prevent anomalies from appearing in the algebra of constraints. This formalism was applied to vector and scalar perturbations. The gauge-invariant equations of motion leading to the calculation of measurable spectra were derived.

Orientador: Prof. Dr. Rodrigo Fresneda / Dissertação (mestrado) - Universidade Federal do ABC, Programa de Pós-Graduação em Matemática , 2016. / O modelo de Ponzano-Regge é um modelo de gravitação quântica em três dimensões. O principal objetivo deste trabalho é apresentar os fundamentos para construção desse modelo. Buscamos introduzir conceitos necessários para entendê-lo, abordando a teoria de representações de grupos de Lie compactos, tais como: redutibilidade de uma representação, representações de produto direto, e representações no espaço de funções. Tratamos especialmente do caso particular do grupo SU(2). Nesse contexto particular, apresentamos os símbolos 3j e 6j e suas propriedades. O modelo de Ponzano-Regge descreve uma geometria tridimensional discretizada, dada em termos de uma triangulação por simplexos (tetraedros, em três dimensões), de tal forma que o comprimento de cada aresta corresponde a uma representação irredutível do grupo de Lie SU(2). Estes tetraedros são descritos como símbolos 6j, cuja fórmula assintótica possibilita a passagem ao limite clássico, levando a uma expressão para a função de partição que representa uma soma sobre geometrias em três dimensões. / The Ponzano-Regge model is a quantum gravity model in three dimensions. The main goal of this work is to present the foundations for the construction of this model. We aim at introducing the necessary concepts to understand it, taking into account the theory of representations of compact Lie groups, such as: reducibility of representations, direct product representations, and representations in function spaces. We treat the particular case of the SU(2) group. In this special case, we present the 3j and 6j symbols and their properties. The Ponzano-Regge model describes a discretized 3-geometry, given in terms of a triangulation through simplices (tetrahedrons, in 3 dimensions), such that the length of each edge corresponds to an irreducible representation of the Lie group SU(2). These tetrahedrons are described as 6j symbols, whose asymptotic formula allows taking the classical limit, leading to an expression of the partition function that represents a sum over 3-geometries.

Estudamos duas classes de teorias de campos com métrica indefinida: os modelos sigma não linear não-compactos e as teorias quárticas da gravitação. Mostramos que a prova da unitariedade para os modelos sigma de simetria não compacta é a mesma que para os modelos compactos nas regiões fisicamente relevantes. Além disso, analisamos a possibilidade de ocorrer geração dinâmica de massa nos modelos sigma sob influência de temperatura finita, em várias dimensões. No que se refere às teorias quárticas da gravitação, calculamos o valor esperado no vácuo da função de dois pontos com inserção da relação de Gauss-Bonnet. Encontramos que a identidade clássica não é preservada quando empregamos a regularização dimensional, mesmo num espaço sem singularidades. Isto revela o surgimento de anomalias gravitacionais / Two classes of field theories with indefinite metric are studied: noncompact nonlinear sigma models and higher-derivative quantum gravity. We show that the proof of unitarity for noncompact sigma models is the same as the one for compact models in the physically relevant regions. Moreover, we analyze the possibility of dynamical mass generation at finite temperature in various space-time dimensions. Concerning to the higher-derivative quantum gravity, we calculate the vacuum expectation value of the two-point function with the insertion of Gauss-Bonnet Relation. In the quantum context, we also show, using dimensional regularization, that the Gauss-Bonnet relation is not satisfied due to gravitational anomalies.

nÃo hà / Neste trabalho, apresenta-se o formalismo canÃnico de quantizaÃÃoo da gravidade, tanto em sua formulaÃÃo original, para a qual a mÃtrica à a variÃvel canÃnica, quanto na de Ashtekar, onde a conexÃo autodual assume o papel de variÃvel canÃnica. Nesta Ãltima formulaÃÃo, as equaÃÃes de vÃnculo do formalismo sÃo drasticamente simplificadas, e, fazendo uso da teoria de Chern-Simons, constrÃi-se um estado que satisfaz estas equaÃÃes no vÃcuo, constituindo uma importante soluÃÃo para a equaÃÃo de Wheeler-DeWitt. O estado de Chern-Simons tambÃm tem uma representaÃÃo em loops, que recebe este nome por ser formulada em termos dos loops de Wilson.

Orientador: Patricio Anibal Letelier Sotomayor / Tese (doutorado) - Universidade Estadual de Campinas, Instituto de Matemática, Estatística e Computação Científica / Made available in DSpace on 2018-08-18T20:03:53Z (GMT). No. of bitstreams: 1 Manoel_JoaoPauloPitelli_D.pdf: 2670867 bytes, checksum: 990119329fe5abbf22d8a42384ff3e72 (MD5) Previous issue date: 2011 / Resumo: Espaços-tempo classicamente singulares serão estudados de um ponto de vista quântico. A utilização da mecânica quântica será feita de duas maneiras. A primeira consiste em encontrar a função de onda do Universo, resolvendo a equação de Wheeler-DeWitt para as variáveis canônicas do espaço-tempo. A segunda consiste em acoplar conformemente campos escalares e spinoriais ao campo gravitacional, estudando o comportamento de pacotes de ondas neste espaço-tempo curvo / Abstract: Classically singular spacetimes will be studied from a quantum mechanical point of view. The use of quantum mechanics will be handled in two different ways. The first consists in finding the wave function of the universe by solving the Wheeler-DeWitt equation for the canonical variables of spacetime. The second is through the conformal coupling of scalar and spinorial fields with the gravitational field, where we will study the behavior of wave packets in this curved spacetime / Doutorado / Matematica Aplicada / Doutor em Matemática Aplicada

Je propose la première étude quantitative des corrections radiatives du modèle EPRL en gravité quantique à mousse de spins. Ce modèle est la proposition la plus élaborée de gravité quantique Lorentzienne 4D dite 'indépendante du fond' ('background independent'). C'est une réalisation, par intégrale de chemin, de la quantification de la Relativité Générale comme somme sur les géométries. L'étude se focalise sur les propriétés et les aspects géométriques de l'analogue du graphe de self-énergie du modèle, connu comme le graphe 'melonique'. Je montre que les contributions dominantes à un tel graphe divergent beaucoup moins que celles de modèles similaires en théorie topologique des champs. De plus, je dérive en détails la dépendance des amplitudes aux données de bords, et montre que ce graphe n'induit pas une renormalisation de la fonction d'onde. Ceci est dû à des raisons reliées aux fondements du modèle. Cependant, il se trouve que l'amplitude se réduit à une telle renormalisation dans la limite de nombres quantiques élevés. Ensuite, je montre les conséquences de ces calculs sur une observable physique : la fonction à deux points de la métrique quantique. Ainsi, je montre comment l'insertion du graphe de self-énergie dans l'intérieur de la mousse de spins utilisée a des effets non-triviaux sur la fonction à deux points, modifiant ses contributions à l'ordre dominant. De façon intéressante, ces effets ne disparaissent pas dans la limite des nombres quantiques élevés. Enfin, je discute les conséquences de ces calculs pour le modèle lui-même, et je souligne et commente les traits généraux qui semblent commun à tout modèle de mousse de spins basé sur le schéma présenté ici. / I present the first quantitative study of radiative corrections within the EPRL model of quantum gravity. This model is the most advanced proposal of Lorentzian 4-dimensional background-independent quantum gravity. It is a realization of the path-integral quantization of general relativity as a sum over geometries. The present study focuses on the properties and geometrical features of the analogue of the self-energy graph within the model, often referred to as the "melon"-graph. Here, I show that the dominating contribution to such a graph is characterized by a degree of divergence much smaller than that of closely related topological quantum field theories. Moreover, I work out in detail the dependence of the amplitude from the boundary data, and find that the self-energy graph does not simply induce a wave function renormaliziation. This happens for reasons deeply related to the model foundations. However, it turns out that the amplitude reduces to a wave function renormalzation in the limit of large quantum numbers. Then, I show the consequences of this calculations on a concrete spinfoam observable: the quantum-metric two-point function. In doing this, I show how the insertion of the self-energy graph in the bulk of the (first-order) spinfoam used in the calculation, has non-trivial effects on the correlation function, modifying its leading order contributions. Most interestingly, this effects do not disappear in the limit of large quantum number. Finally, I discuss the consequences of these calculations for the model itself, and I point out and comment those general features which seem to be common to any spinfoam model based on the present model-building schemes.

Établir une théorie de gravité quantique qui décrit de manière cohérente les propriétés quantiques de la matière et de l'espace-temps est l'un des défis majeurs de la physique théorique. Malgré plusieurs décennies de recherches, de nombreux problèmes conceptuels et techniques doivent encore être résolus. L'étude de modèles simplifiés donne des idées de résolution. La première partie de la thèse traite de la gravité quantique bidimensionnelle. À deux dimensions, la gravité quantique est beaucoup mieux comprise et de nombreux calculs peuvent être faits exactement. Si la gravité quantique bidimensionnelle a été largement étudiée quand elle est couplée à de la matière conforme, le cas de la matière non-conforme était très peu connu jusque récemment. Nous calculons d'abord l'action gravitationnelle pour un champ scalaire massif sur une surface de Riemann avec bords puis pour un fermion de Majorana massif sur une variété compacte. Ce dernier cas correspond à une CFT perturbée par une perturbation conforme et est d'ordinaire étudié grâce à l'ansatz de DDK, mais les résultats sont différents. Finalement, on calcule le spectre de l'action de Mabuchi dans l'approximation du minisuperespace. La seconde partie étudie les propriétés thermales des trous noirs dans le contexte de la correspondance AdS/CFT. On construit un modèle de mécanique quantique fondé sur les principes holographiques pour simuler la dynamique des trous noirs quantiques. Ce modèle permet d'obtenir des résultats numériques exacts. / Finding a theory of quantum gravity describing in a consistent way the quantum properties of matter and spacetime geometry is one of the greatest challenges of modern theoretical physics. However after several decades of research, many conceptual and technical issues are still to be resolved. Insights on these questions can be given by simplified toy models that allow for exact computations. The first part of the thesis deals with two-dimensional quantum gravity. In two dimensions quantum gravity is much better understood and many computations can be carried out exactly. Whereas two-dimensional quantum gravity coupled to conformal matter has been widely studied and is now well understood, much less was known until recently when matter is non-conformal. First we compute the gravitational action for a massive scalar field on a Riemann surface with boundaries and then for a massive Majorana fermion on a manifold without boundary. The latter case corresponds to a CFT perturbed by a conformal perturbation and is usually tackled through the DDK ansatz, but the results do not seem to match. Finally we give a minisuperspace computation of the spectrum of the Mabuchi action, a functional that appears in the gravitational action for a massive scalar field. In the second part we focus on black hole thermal behaviour which provides a lot of insight of how a theory of quantum gravity should look like. In the context of string theory the AdS/CFT correspondence provides powerful tools for understanding the microscopic origin of black holes thermodynamics. We construct a quantum mechanical toy model based on holographic principles to study the dynamics of quantum black holes.

In this thesis we are dealing with basic methods of theoretical physics focusing on quantum theory of gravity, that are: Hamilton-Dirac formalism for singular systems, Dirac`s method of quantizing systems with constraints and its mathematical formulation - refined algebraic quantization, representation of compact groups and representation of Lorentz group. We apply these methods to find eigenstates of Lorentz group and General linear group generators. We construct a physical Hilbert space on temporal part of 3+1 decomposition of Einstein-Cartan theory. Powered by TCPDF (www.tcpdf.org)

In this thesis we offer a broad introduction into loop quantum gravity against the backdrop of the quantum gravity research as a whole. We focus on both the canonical and covariant version of the theory. In the latter version we investigate the dynamics of some simple configurations in the simplified setting of Ponzano-Regge model. We ascertain that the naïve approach to define a consistent dynamics, where the path integral's partition function is computed as a sum of amplitudes corresponding to all boundary and bulk states, fails in this case, on account of an appearance of divergences. This opens up space for the utilization of some more sophisticated methods.