We construct a 4d Lorentzian spin foam model capable of describing both spacelike and timelike surfaces. To do so we use a coherent state approach inspired by the Riemannian FK model. Using the coherent state method we reproduce the results of the EPRL model for Euclidean tetrahedra and extend the model to include Lorentzian tetrahedra. The coherent states of spacelike/timelike triangles are found to correspond to elements of the discrete/continuous series of SU(1,1). It is found that the area spectrum of both spacelike and timelike surfaces is quantized. A path integral for the quantum theory is defined as a product of vertex amplitudes. The states corresponding to timelike triangles are constructed in a basis diagonalised with respect to a noncompact generator. A derivation of the matrix elements of the generators of SL(2,C) in this basis is provided.

We construct a 4d Lorentzian spin foam model capable of describing both spacelike and timelike surfaces. To do so we use a coherent state approach inspired by the Riemannian FK model. Using the coherent state method we reproduce the results of the EPRL model for Euclidean tetrahedra and extend the model to include Lorentzian tetrahedra. The coherent states of spacelike/timelike triangles are found to correspond to elements of the discrete/continuous series of SU(1,1). It is found that the area spectrum of both spacelike and timelike surfaces is quantized. A path integral for the quantum theory is defined as a product of vertex amplitudes. The states corresponding to timelike triangles are constructed in a basis diagonalised with respect to a noncompact generator. A derivation of the matrix elements of the generators of SL(2,C) in this basis is provided.

The spin-foam is a covariant path-integral style approaching to the quantization of the gravity. There exist several spin-foam models of which the most successful one is the Engle-Pereira-Rovelli-Levine/Freidel-Krasnov (EPRL-FK) model. Using the EPRLFK model people are able to calculate the transition amplitude and the n-point functions of 4D geometry (both Euclidean and Lorentzian) surrounding by a given triangulated 3D geometry. The semi-classical limit of the EPRL-FK amplitude reproduces discrete classical gravity under certain assumptions, which shows that the EPRLFK model can be understood as UV completion of general relativity. On the other hand, it is very hard to dene a continuum limit and couple a cosmological constant to the EPRL-FK model. In this dissertation, we addressed the problems about continuum limit and coupling a cosmological constant to the EPRL-FK model. Followed by chapter one as a brief introduction of the loop quantum gravity and EPRL-FK model, chapter two introduces our work about demonstrating (for the first time) that smooth curved spacetime geometries satisfying Einstein equation can emerge from discrete spin-foam models under an appropriate low energy limit, which corresponds to a semi-classical continuum limit of spin-foam models. In chapter three, we bring in the cosmological constant into the spin-foam model by coupling the SL(2, C) Chern-Simons action with the EPRL action, and find that the quantum simplicity constraint is realized as the 2d surface defect in SL(2, C)Chern-Simons theory in the construction of spin-foam amplitudes. In chapter four, we present a way to describe the twisted geometry with cosmological constant whose corresponding quantum states can forms the Hilbert space of the loop quantum gravity with cosmological constant. In chapter five, we introduced a new definition of the graviton propagator, and calculate its semi-classical limit in the contents of spin-foam model with the cosmological constant. Finally the chapter six will be a outlook for my future work. / Includes bibliography. / Dissertation (Ph.D.)--Florida Atlantic University, 2019. / FAU Electronic Theses and Dissertations Collection

Dans cette thèse, nous étudions l’implémentation des contraintes de simplicité dans le nouveau modèle de mousses de spin d’Engle-Pereira-Rovelli-Livine, ainsi que les fonctions de corrélation à deux points de ce modèle. Nous définissons d’une manière simple l’espace de Hilbert limite de la théorie, puis montrons directement que toutes les contraintes s’annulent faiblement sur cet espace. Nous observons que la solution générale a cette contrainte (imposée faiblement) dépend d’un nombre quantique, en plus de ceux de la gravitation quantique a boucles. Nous généralisons également cette construction pour la version de Kaminski-Kisielowski-Lewandowski, ou la mousse n’est pas duale à une triangulation. Nous montrons que cette théorie peut aussi être obtenue comme une théorie BF satisfaisant la contrainte de simplicité, cette fois discrétisée sur un 2-complexe cellulaire oriente. Enfin, nous calculons la fonction de corrélation a deux points du modèle de mousses de spin Engle-Pereira-Rovelli-Livine avec la signature lorentzienne, et nous montrons que la fonction a deux points que nous obtenons correspond dans une certaine limite a celle obtenue a partir du calcul de Regge lorentzien. / In this thesis we study the implementation of simplicity constraints that defines the recent Engle-Pereira-Rovelli-Livine spinfoam model and two-point correlation functions of this model. We define in a simple way the boundary Hilbert space of the theory; then show directly that all constraints vanish on this space in a weak sense. We point out that the general solution to this constraint (imposed weakly) depends on a quantum number in addition to those of loop quantum gravity. We also generalize this construction to Kami´nski-Kisielowski-Lewandowski version where the foam is not dual to a triangulation. We show that this theory can still be obtained as a constrained BF theory satisfying the simplicity constraint, now discretized on a general oriented 2-cell complex. Finally, we calculate the twopoint correlation function of the Engle-Pereira-Rovelli-Livine spinfoam model in the Lorentzian signature, and show the two-point function we obtain exactly matches the one obtained from Lorentzian Regge calculus in some limit.

Une étude d'un assemblage symétrique de trois 4-simplex en géométrie classique, de Regge et quantique. Nous étudions les propriétés géométriques et surtout la présence de courbure. Nous montrons que les géométries classique et de Regge de l'assemblage ont une courbure qui évolue en fonction de ses paramètres de bordure. Pour la géométrie quantique, une version euclidienne du modèle EPRL est utilisé avec une valeur pratique du paramètre Barbero-Immirzi pour définir l'amplitude de transition de l'ensemble et de ses composants. Un code C ++ est conçu pour calculer les amplitudes et étudier numériquement la géométrie quantique. Nous montrons qu'une géométrie classique, avec une courbure, émerge déjà à bas spin. Nous reconnaissons également l'apparition de configurations dégénérées et de leurs effets sur la géométrie attendue. / A study of symmetrical assembly of three euclidean 4-simplices in classical, Regge and quantum geometry. We study the geometric properties and especially the presence of curvature. We show that classical and Regge geometry of the assembly have curvature which evolves in function of its boundary parameters. For the quantum geometry, a euclidean version of EPRL model is used with a convenient value of the Barbero-Immirzi parameter to define the transition amplitude of the assembly and its components. A C++ code is design for compute the amplitudes and study numerically the quantum geometry. We show that a classical geometry, with curvature, emerges already at low spin. We also recognize the appearance of the degenerate configurations and their effects on the expected geometry.

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.