Submitted by Emanuel Varela Cardoso (emanuel.varela@ufcg.edu.br) on 2018-10-09T18:55:52Z No. of bitstreams: 1 ARTHUR GONÇALVES CAVALCANTI – DISSERTAÇÃO (PPGFísica) 2016.pdf: 845402 bytes, checksum: dbdfb2a26834c477a45e9e735fa670d3 (MD5) / Made available in DSpace on 2018-10-09T18:55:52Z (GMT). No. of bitstreams: 1 ARTHUR GONÇALVES CAVALCANTI – DISSERTAÇÃO (PPGFísica) 2016.pdf: 845402 bytes, checksum: dbdfb2a26834c477a45e9e735fa670d3 (MD5) Previous issue date: 2016-02 / Capes / A teoria da relatividade geral prevê soluções tipo buraco negro, as quais são caracterizadas pela existência de um horizonte de eventos. Como exemplo, podemos citar a métrica obtida por Bãnados-Teitelboim-Zanelli (BTZ), que é uma solução da gravitação em (2+1)- dimensões, em que se considera uma constante cosmológica negativa. Nos últimos anos, buracos negros não-comutativos têm sido investigados na literatura por muitos autores. Em particular, a métrica BTZ não-comutativa foi obtida considerando-se a equivalência, que existe em três dimensões, entre gravitação e a teoria de Chern-Simons, que e uma teoria quântica de campos topológica em três dimensões, e usando-se o mapeamento de Seiberg-Witter com a solução em (2+1)-dimensões. A presença de divergências na teoria quântica de campos leva a considerar a possibilidade de modificar o princípio da incerteza de Heisemberg, introduzindo uma escala de comprimento fundamental, e esta modificação geram correções nas propriedades termodinâmica de buracos negros. Um dos efeitos associados as soluções tipo buraco negro, independente da dimensão do espaço-tempo, e a emissão térmica (Radiação Hawking), a qual e vista como um processo de tunelamento devido as flutuações do vácuo que acontece na região próxima ao horizonte de eventos. Neste trabalho, com o objetivo de investigar as correções devido a não comutatividade e ao princípio da incerteza generalizado, consideramos a métrica BTZ não-comutativa. Para tanto, usamos o formalismo de tunelamento via método de Hamilton-Jacobi. / The general relativity theory predicts black hole type solutions, which are characterized by the existence of an event horizon. As an example, the metric obtained by Ba~nados- Teitelboim-Zanelli (BTZ), which is a soluton of the gravitation in (2 + 1)-dimensions in what is considered a negative cosmological constant. In recent years, noncommutative black holes have been investigated by many authors in the literature. In particular, the BTZ metric non-commutative was obtained considering the equivalent, which exists in three dimensions, between gravitation and Chern-Simons theory, which is a quantum theory topological elds in three dimensions, and using it mapping Seiberg-Witter with the solution of (2 + 1)-dimensions. The presence of divergences in quantum eld theory leads to consider the possibility of modifying the principle of Heisenberg uncertainty by introducing a fundamental length scale, and this modi cation generate corrections to the thermodynamic properties of black holes. One of the e ects associated with the black hole type solutions, regardless of the space-time dimension is the thermal emission (Hawking radiation), which is seen as a process of tunneling due to vacuum uctuations that happens in the region near the event horizon . In this work, in order to investigate the corrections due to noncommutativity and the principle of widespread uncertainty, we consider the metric BTZ noncommutative. For this, we use tunneling formalism via Hamilton-Jacobi method.

The work is divided into three main parts. In a first time (chapter 1) we define a “BTZ” black hole in anti de Sitter space in any dimension. That will be done by means of group theoretical and symmetric spaces considerations. A physical “good domain” is identified as an open orbit of a subgroup of the isometry group of anti de Sitter. Then (chapter 2) we show that the open orbit is in fact isomorphic to a group (we introduce the notion of globally group type manifold) for which a quantization exists. The quantization of the black hole is performed and its Dirac operator is computed. The third part (appendix A and B) exposes some previously known results. Appendix A is given in a pedagogical purpose: it exposes generalities about deformation quantization and careful examples with SL(2,R), and split extensions of Heisenberg algebras. Appendix B is devoted to some classical results about homogeneous spaces and Iwasawa decompositions. Explicit decompositions are given for every algebra that will be used in the thesis. It serves to make the whole text more self contained and to fix notations. Basics of quantization by group action are given in appendix A.4. One more chapter is inserted (chapter 3). It contains two small results which have no true interest by themselves but which raise questions and call for further development. We discuss a product on the half-plane or, equivalently, on the Iwasawa subgroup of SL(2,R), due to A. Unterberger. We show that the quantization by group action machinery can be applied to this product in order to deform the dual of the Lie algebra of that Iwasawa subgroup. Although this result seems promising, we show by two examples that the product is not universal in the sense that even the product of compactly supported functions cannot be defined on AdS2 by the quantization induced by Unterberger's product. Then we show that the Iwasawa subgroup of SO(2,n) (i.e. the group which defines the singularity) is a symplectic split extension of the Iwasawa subgroup of SU(1,1) by the Iwasawa subgroup of SU(1,n). A quantization of the two latter groups being known, a quantization of SO(2,n) is in principle possible using an extension lemma. Properties of this product and the resulting quantization of AdSl were not investigated because we found a more economical way to quantize AdS4 .

In this thesis we study the time evolution of Rényi and entanglement entropies of thermal states in Conformal Field Theory (CFT). These quantities are usually hard to compute but Ryu-Takayanagi (RT) and Hubeny-Rangamani-Takayanagi (HRT) proposals allow us to find the same quantities using calculations in general relativity. We will introduce main concepts of holography, quantum information and conformal field theory that will be used to derive the results of this thesis. In the first part of the thesis, we explicitly compute entanglement entropy of the rotating BTZ black hole by directly applying HRT proposal and finding lengths of spacelike geodesics. Rényi entropy of thermal state perturbed by a local quantum quench is computed by mapping correlators on two glued cylinders to the plane for field theory containing a single free boson and for 2d CFTs in the large c limit. We consider Thermofield Double State (TFD) which is an entangled state in direct product of two 2D CFTs. It is conjectured to be holographically equivalent to the eternal BTZ black hole. TFD state is perturbed by a local quench in one CFT and mutual information between two intervals in two CFTs is computed. We find when mutual information vanishes and interpret this as scrambling time, i.e. time scale required for the system to thermalize. This field theory result is modelled with a massive free falling particle in the BTZ black hole. We have computed the back-reaction of the particle on the metric of BTZ and used RT proposal to find holographic entanglement entropy. Finally, we generalize this calculation to the case of rotating BTZ with inner and outer horizons. It is dual to the CFT with different temperatures for left and right moving modes. We calculate mutual information and scrambling time and find exact agreement between results in the gravity and those in the CFT.

We study a free quantum scalar field in the BTZ spacetime as a model of the AdS/CFT correspondence for black holes, and show the essential steps in computing Bogolyubov coefficients between modes on either side of the wormhole. As background, we review the BTZ geometry in standard, Kruskal and Poincaré coordinates, holographic renormalisation of the dual field theory and canonical quantisation in curved spacetime. / Vi studerar ett fritt skalärt kvantfält i BTZ-rumtiden som en modell av AdS/CFT-dualiteten för svarta hål och visar huvudstegen i beräkningen av Bogolyubov-koefficienter mellan moder på olika sidor av maskhålet. Som bakgrund redogör vi för BTZ-geometrin i standard-, Kruskal- och Poincarékoordinater, holografisk renormering av den duala fältteorin och kanonisk kvantisering i krökt rumtid.

We study constraints imposed on a general unitary two-dimensional conformal field theory by modular invariance. We begin with a review of previous bounds on the conformal dimension Delta1 of the lowest primary operator assuming unitarity, a discrete spectrum, modular invariance, cL, cR > 1, and no extended chiral algebra. We then obtain bounds on the conformal dimensions Delta2, Delta3 using no additional assumptions. We also show that in order to find a bound for Delta4 or higher Deltan, we need to assume a larger minimum value for ctot that grows logarithmically with n. We next extend the previous results to remove the requirement that our two-dimensional conformal field theories have no extended chiral algebra. We then show that modular invariance also implies an upper bound on the total number of states of positive energy less than ctot=24 (or equivalently, states of conformal dimension between ctot=24 and ctot=12), in terms of the number of negative energy states. Finally, we consider the case where the CFT has a gravitational dual and investigate the gravitational interpretation of our results. Using the AdS3/CFT2 correspondence, we obtain an upper bound on the lightest few massive excitations (both with and without the constraint of no chiral primary operators) in a theory of 3D matter and gravity with Lambda < 0. We show our results are consistent with facts and expectations about the spectrum of BTZ black holes in 2+1 gravity. We then discuss the upper and lower bounds on number of states and primary operators in the dual gravitational theory, focusing on the case of AdS3 pure gravity.

Les matériaux ferroélectriques non-linéaires à faibles pertes diélectriques présentent une permittivité ajustable sous champ électrique qui les destine à des applications de composants agiles hyperfréquences pour la microélectronique et les télécommunications (condensateurs et résonateurs accordables, déphaseurs, ...). Les performances requises - pertes diélectriques faibles, permittivité modérée, accordabilité maximale, stabilité thermique jusqu'aux hyperfréquences - imposent d'améliorer le matériau de base Ba0.6Sr0.4TiO3 (BST). Les céramiques élaborées à partir de BST et d'une phase à faibles pertes non-ferroélectrique telle MgO, MgTiO3, mais aussi les compositions inédites avec SnO2 et BaSnO3, sont caractérisées par des analyses physico-chimiques et électriques. Afin de réduire de manière significative les pertes diélectriques, de nouveaux composites à architecture contrôlée (coeur/écorce notamment BST@SiO2) sont développés par des procédés innovants de synthèse en chimie douce. Les essais de faisabilité et leurs propriétés électriques inédites ouvrent la voie à de nombreuses possibilités de maîtrise de l'agencement des phases dans les matériaux composites ferroélectriques.