In this thesis, we extend the numerical S-matrix bootstrap program to 1+1d theories with a boundary, where we bootstrap the 1-to-1 reflection matrix (R-matrix). We review the constraints that a physical R-matrix must obey, namely unitarity, analyticiy and crossing symmetry. We then carve out the allowed space of 2d R-matrices with the O(N) nonlinear sigma model and the periodic Yang Baxter solution in the bulk. We find a variety of integrable R-matrices along the boundary of the allowed space both with and without free parameters. The integrable models without a free parameter appear at vertices of the allowed space, while those with a free parameter occupy the whole boundary. We also introduce the extended analyticity constraint where we increase the domain of analyticity beyond the physical region. In some cases, the allowed space of R-matrices shrinks drastically and we observe new vertices which correspond to integrable theories. We also find a new integrable R-matrix through our numerics, which we later obtained by solving the boundary Yang--Baxter equation. Finally, we derive the dual to the extended analyticity problem and find that the formalism allows for R-matrices which do not saturate unitarity to lie on the boundary of the allowed region.

Diese Arbeit befasst sich mit Quantenfeldtheorien auf nicht-kommutativen Räumen. Solche Modelle treten im Zusammenhang mit der Stringtheorie und mit der Quantengravitation auf. Ihre nicht-störungstheoretische Behandlung ist üblicherweise schwierig. Hier untersuchen wir jedoch drei nicht-kommutative Quantenfeldtheorien nicht-perturbativ, indem wir die Wirkungsfunktionale in eine äquivalente Matrixformulierung übersetzen. In der Matrixdarstellung kann die jeweilige Theorie dann numerisch behandelt werden. Als erstes betrachten wir ein regularisiertes skalares Modell auf der nicht-kommutativen Ebene und untersuchen den Kontinuumslimes bei festgehaltener Nicht-Kommutativität. Dies wird auch als Doppelskalierungslimes bezeichnet. Insbesondere untersuchen wir das Verhalten der gestreiften Phase. Wir finden keinerlei Hinweise auf die Existenz dieser Phase im Doppelskalierungslimes. Im Anschluss daran betrachten wir eine vier-dimensionale U(1) Eichtheorie. Hierbei sind zwei der räumlichen Richtungen nicht-kommutativ. Wir untersuchen sowohl die Phasenstruktur als auch den Doppelskalierungslimes. Es stellt sich heraus, dass neben den Phasen starker und schwacher Kopplung eine weitere Phase existiert, die gebrochene Phase. Dann bestätigen wir die Existenz eines endlichen Doppelskalierungslimes, und damit die Renormierbarkeit der Theorie. Weiterhin untersuchen wir die Dispersionsrelation des Photons. In der Phase mit schwacher Kopplung stimmen unsere Ergebnisse mit störungstheoretischen Berechnungen überein, die eine Infrarot-Instabilität vorhersagen. Andererseits finden wir in der gebrochenen Phase die Dispersionsrelation, die einem masselosen Teilchen entspricht. Als dritte Theorie betrachten wir ein einfaches, in seiner Kontinuumsform supersymmetrisches Modell, welches auf der "Fuzzy Sphere" formuliert wird. Hier wechselwirken neutrale skalare Bosonen mit Majorana-Fermionen. Wir untersuchen die Phasenstruktur dieses Modells, wobei wir drei unterschiedliche Phasen finden. / This work deals with three quantum field theories on spaces with noncommuting position operators. Noncommutative models occur in the study of string theories and quantum gravity. They usually elude treatment beyond the perturbative level. Due to the technique of dimensional reduction, however, we are able to investigate these theories nonperturbatively. This entails translating the action functionals into a matrix language, which is suitable for numerical simulations. First we explore a scalar model on a noncommutative plane. We investigate the continuum limit at fixed noncommutativity, which is known as the double scaling limit. Here we focus especially on the fate of the striped phase, a phase peculiar to the noncommutative version of the regularized scalar model. We find no evidence for its existence in the double scaling limit. Next we examine the U(1) gauge theory on a four-dimensional spacetime, where two spatial directions are noncommutative. We examine the phase structure and find a new phase with a spontaneously broken translation symmetry. In addition we demonstrate the existence of a finite double scaling limit which confirms the renormalizability of the theory. Furthermore we investigate the dispersion relation of the photon. In the weak coupling phase our results are consistent with an infrared instability predicted by perturbation theory. If the translational symmetry is broken, however, we find a dispersion relation corresponding to a massless particle. Finally, we investigate a supersymmetric theory on the fuzzy sphere, which features scalar neutral bosons and Majorana fermions. The supersymmetry is exact in the limit of infinitely large matrices. We investigate the phase structure of the model and find three distinct phases. Summarizing, we study noncommutative field theories beyond perturbation theory. Moreover, we simulate a supersymmetric theory on the fuzzy sphere, which might provide an alternative to attempted lattice formulations.

La position centrale occupée par les Théories Quantiques des Champs Topologiques (TQFTs) dans l’étude de la topologie en basse dimension est due à leur structure extraordinairement riche, qui permet différentes interactions et applications à des questions de nature géométrique. Depuis leur première apparition, un grand effort a été mis dans l’extension des invariants quantiques de 3-variétés en TQFTs et en TQFT Étendues (ETQFTs). Cette thèse s’attaque à ce problème dans deux cadres généraux différents. Le premier est l’étude des invariants quantiques semi-simples de Witten, Reshetikhin et Turaev issus de catégories modulaires. Bien que les ETQFTs correspondantes étaient connues depuis un certain temps, une réalisation explicite basée sur la construction universelle de Blanchet, Habegger, Masbaum et Vogel apparaît ici pour la première fois. L’objectif est de tracer la route à suivre dans la deuxième partie de la thèse, où la même procédure est appliquée à une nouvelle famille d’invariants quantiques non semi-simples due à Costantino, Geer et Patureau. Ces invariants avaient déjà été étendus en TQFTs graduées par Blanchet, Costantino, Geer and Patureau, mais seulement pour une famille explicite d’exemples. Nous posons la première pierre en introduisant la définition de catégorie modulaire relative, un analogue non semi-simple aux catégories modulaires. Ensuite, nous affinons la construction universelle pour obtenir des ETQFTs graduées étendant à la fois les invariants quantiques de Costantino, Geer et Patureau et les TQFTs graduées de Blanchet, Costantino, Geer et Patureau dans ce cadre général / The central position held by Topological Quantum Field Theories (TQFTs) in the study of low dimensional topology is due to their extraordinarily rich structure, which allows for various interactions with and applications to questions of geometric nature. Ever since their first appearance, a great effort has been put into extending quantum invariants of 3-dimensional manifolds to TQFTs and Extended TQFTs (ETQFTs). This thesis tackles this problem in two different general frameworks. The first one is the study of the semisimple quantum invariants of Witten, Reshetikhin and Turaev issued from modular categories. Although the corresponding ETQFTs were known to exist for a while, an explicit realization based on the universal construction of Blanchet, Habegger, Masbaum and Vogel appears here for the first time. The aim is to set a golden standard for the second part of the thesis, where the same procedure is applied to a new family of non-semisimple quantum invariants due to Costantino, Geer and Patureau. These invariants had been previously extended to graded TQFTs by Blanchet, Costantino, Geer an Patureau, but only for an explicit family of examples. We lay the first stone by introducing the definition of relative modular category, a non-semisimple analogue to modular categories. Then, we refine the universal construction to obtain graded ETQFTs extending both the quantum invariants of Costantino, Geer and Patureau and the graded TQFTs of Blanchet, Costantino, Geer and Patureau in this general setting

<p>In recent years, there has been a large interest in the subject of extra dimensions in particle physics. In particular, a number of models have been suggested which provide solutions to some of the problems with the current Standard Model of particle physics, and which could be tested in the next generation of high-energy experiments. Among the most important of these models are the large extra dimensions model by Arkani-Hamed, Dimopoulos, and Dvali, the universal extra dimensions model, and models allowing right-handed neutrinos to propagate in the extra dimensions. In this thesis, we study phenomenological aspects of these three models, or simple modifications of them.</p><p> </p><p>The Arkani-Hamed-Dimopoulos-Dvali model attempts to solve the gauge hierarchy problem through a volume suppression of Newton's gravitational constant, lowering the fundamental Planck scale down to the electroweak scale. However, this solution is unsatisfactory in the sense that it introduces a new scale through the radius of the extra dimensions, which is unnaturally large compared to the electroweak scale. It has been suggested that a similar model, with a hyperbolic internal space, could provide a more satisfactory solution to the problem, and we consider the hadron collider phenomenology of such a model.</p><p> </p><p>One of the main features of the universal extra dimensions model is the existence of a potential dark matter candidate, the lightest Kaluza-Klein particle. In the so-called minimal universal extra dimensions model, the identity of this particle is well defined, but in more general models, it could change. We consider the indirect neutrino detection signals for a number of different such dark matter candidates, in a five- as well as a six-dimensional model.</p><p> </p><p>Finally, right-handed neutrinos propagating in extra dimensions could provide an alternative scenario to the seesaw mechanism for generating small masses for the left-handed neutrinos. Since extra-dimensional models are non-renormalizable, the Kaluza-Klein tower is expected to be cut off at some high-energy scale. We study a model where a Majorana neutrino at this cutoff scale is responsible for the generation of the light neutrino masses, while the lower modes of the tower could possibly be observed in the Large Hadron Collider. We investigate the bounds on the model from non-unitarity effects, as well as collider signatures of the model.</p>

This thesis is an investigation of the subject of extra dimensions in particle physics. In recent years, there has been a large interest in this subject. In particular, a number of models have been suggested that provide solutions to some of the problem with the current Standard Model of particle physics. These models typically give rise to experimental signatures around the TeV energy scale, which means that they could be tested in the next generation of high-energy experiments, such as the LHC. Among the most important of these models are the universal extra dimensions model, the large extra dimensions model by Arkani-Hamed, Dimopolous, and Dvali, and models where right-handed neutrinos propagate in the extra dimensions. In the thesis, we study phenomenological aspects of these models, or simple modifications of them. In particular, we focus on Kaluza–Klein dark matter in universal extra dimensions models, different aspects of neutrino physics in higher dimensions, and collider phenomenology of extra dimensions. In addition, we consider consequences of the enhanced renormalization group running of physical parameters in higher-dimensional models. / QC 20120427

In recent years, there has been a large interest in the subject of extra dimensions in particle physics. In particular, a number of models have been suggested which provide solutions to some of the problems with the current Standard Model of particle physics, and which could be tested in the next generation of high-energy experiments. Among the most important of these models are the large extra dimensions model by Arkani-Hamed, Dimopoulos, and Dvali, the universal extra dimensions model, and models allowing right-handed neutrinos to propagate in the extra dimensions. In this thesis, we study phenomenological aspects of these three models, or simple modifications of them.   The Arkani-Hamed-Dimopoulos-Dvali model attempts to solve the gauge hierarchy problem through a volume suppression of Newton's gravitational constant, lowering the fundamental Planck scale down to the electroweak scale. However, this solution is unsatisfactory in the sense that it introduces a new scale through the radius of the extra dimensions, which is unnaturally large compared to the electroweak scale. It has been suggested that a similar model, with a hyperbolic internal space, could provide a more satisfactory solution to the problem, and we consider the hadron collider phenomenology of such a model.   One of the main features of the universal extra dimensions model is the existence of a potential dark matter candidate, the lightest Kaluza-Klein particle. In the so-called minimal universal extra dimensions model, the identity of this particle is well defined, but in more general models, it could change. We consider the indirect neutrino detection signals for a number of different such dark matter candidates, in a five- as well as a six-dimensional model.   Finally, right-handed neutrinos propagating in extra dimensions could provide an alternative scenario to the seesaw mechanism for generating small masses for the left-handed neutrinos. Since extra-dimensional models are non-renormalizable, the Kaluza-Klein tower is expected to be cut off at some high-energy scale. We study a model where a Majorana neutrino at this cutoff scale is responsible for the generation of the light neutrino masses, while the lower modes of the tower could possibly be observed in the Large Hadron Collider. We investigate the bounds on the model from non-unitarity effects, as well as collider signatures of the model. / QC 20110324

Several different approaches to renormalization are studied. The Callan-Symanzik equation is derived and we study its beta functions. An effective potential for the Coleman-Weinberg model is studied to find that the beta function is positive and that spontaneous symmetry breaking will occur if we expand around the classical field. Lastly we renormalize a non-abelian gaugetheory to find that the beta function in QCD is negative.