It is well known that the standard canonical uncertainty relation does not apply to the angular variable ? and its conjugate LZ. That is, the relation ? ø ? L Z > h/2 is false. The break down of the result has to do with difference in topology between the line and the circle. It is thus desirable to generalize the standard uncertainty relation topologically and find satisfactory results for the non-Euclidean spaces. This problem is intimately related to the issue of finding a consistent definition for quantum mechanics on "curved spaces". Just as the Heisenberg uncertainty relation was pivotal in understanding the basic structure of standard quantum mechanics, a solution to this problem should shine some light onto the proper conduct of quantum mechanics on general topological spaces. In this study we explore in detail how the standard uncertainty relation may breakdown. We also address the importance of topological considerations in quantum mechanics in general - we shall show how a change in topological character can change the nature of the quantum mechanics for a system and how the consideration of the topology of a system can greatly organize the solution of a problem and in some cases even be necessary for a. full understanding of the problem. We then discuss the derivation of satisfactory uncertainty relations for the compact, homogeneous spaces of the circle, the n-torus and the n-sphere. Finally, we draw out any implications to the issue of properly defining quantum mechanics on the non- Euclidean spaces.

Utilizando a representação funcional de schrodinger e um ansatz gaussiano simples, obtemos um conjunto de equações finitas para a matéria e gravitação em espaços homogêneos e inomogêneos. / Using the formalism of functional Schrödinger representation and a simple Ansatz, we obtain a set of finite equations to the matter and gravitation in homogeneous and inhomogeneous spaces.

Utilizando a representação funcional de schrodinger e um ansatz gaussiano simples, obtemos um conjunto de equações finitas para a matéria e gravitação em espaços homogêneos e inomogêneos. / Using the formalism of functional Schrödinger representation and a simple Ansatz, we obtain a set of finite equations to the matter and gravitation in homogeneous and inhomogeneous spaces.

My dissertation focuses on the existence of metrics of non-positive curvature for the simplicial complexes constructed recently by Tom Brady and Daan Krammer for the braid groups and other Artin groups of finite type. In particular, for each Artin group of finite type, there is a recently constructed finite simplicial Eilenberg-Mac Lane space known as its Brady-Krammer complex. The Brady-Krammer complexes are highly symmetric objects. Prior work on the relationship between the Brady-Krammer complexes and the theory of CAT(0)spaces has produced some positive results in low-dimensions. More specifically, the Brady-Krammer complexes of dimension at most 3 have been shown to support piecewise Euclidean metrics of non-positive curvature. Similarly, the 4dimensional Brady-Krammer complexes of type A4 and type B4 also support such metrics. In every instance, the metrics assigned respect all of the symmetries alluded to above. The main results of my dissertation show that this pattern does not extend to the Brady-Krammer complexes of type F4 and D4. These are the first negative results known about the curvature of these Brady-Krammer complexes. The proofs of my main theorems involve a combination of combinatorial results and computer calculations. These negative results are particularly striking since Ruth Charney, John Meier and Kim Whittlesey have shown that a particular complex closely related to each Brady-Krammer complex admits an asymmetric metric satisfying a weak version of non-positive curvature. Thus, one corollary of my results is that the weak asymmetric version of a CAT(0) metric (initially defined by Mladen Bestvina) is strictly weaker than the traditional version.

My dissertation focuses on the existence of metrics of non-positive curvature for the simplicial complexes constructed recently by Tom Brady and Daan Krammer for the braid groups and other Artin groups of finite type. In particular, for each Artin group of finite type, there is a recently constructed finite simplicial Eilenberg-Mac Lane space known as its Brady-Krammer complex. The Brady-Krammer complexes are highly symmetric objects. Prior work on the relationship between the Brady-Krammer complexes and the theory of CAT(0)spaces has produced some positive results in low-dimensions. More specifically, the Brady-Krammer complexes of dimension at most 3 have been shown to support piecewise Euclidean metrics of non-positive curvature. Similarly, the 4dimensional Brady-Krammer complexes of type A4 and type B4 also support such metrics. In every instance, the metrics assigned respect all of the symmetries alluded to above. The main results of my dissertation show that this pattern does not extend to the Brady-Krammer complexes of type F4 and D4. These are the first negative results known about the curvature of these Brady-Krammer complexes. The proofs of my main theorems involve a combination of combinatorial results and computer calculations. These negative results are particularly striking since Ruth Charney, John Meier and Kim Whittlesey have shown that a particular complex closely related to each Brady-Krammer complex admits an asymmetric metric satisfying a weak version of non-positive curvature. Thus, one corollary of my results is that the weak asymmetric version of a CAT(0) metric (initially defined by Mladen Bestvina) is strictly weaker than the traditional version.

Orientadores: Ricardo Antonio Mosna, Guillermo Gerardo Cabrera Oyarzun / Dissertação (mestrado) - Universidade Estadual de Campinas, Instituto de Física Gleb Wataghin / Made available in DSpace on 2018-08-26T12:57:50Z (GMT). No. of bitstreams: 1 Souza_IbereOliveiraKuntzde_M.pdf: 20917156 bytes, checksum: bb95eeb451fb153542f18a4c8e165528 (MD5) Previous issue date: 2015 / Resumo: Nesse trabalho estudamos configurações geométricas de um cristal líquido bidimensional sobre substratos curvos. Em particular, estamos interessados na fase esmética-A desses materiais, em que as suas moléculas são organizadas em camadas. Isso é interessante pois grande parte das propriedades de um cristal líquido, como as propriedades ópticas e elásticas, é afetada pela curvatura do seu substrato. Diferentemente dos esméticos no plano euclidiano, em superfícies curvas a presença de curvatura gaussiana dá origem a defeitos topológicos e grain boundaries na estrutura dos esméticos. Mostrarei essa interação entre curvatura e defeitos topológicos em algumas superfícies no limite em que a contribuição à energia devido a compressão das camadas é muito maior do que as contribuições provenientes de outros tipos de deformação. Nesse regime, o estado de menor energia é obtido quando as camadas esméticas são igualmente espaçadas. Isso faz com que o vetor diretor siga as geodésicas da superfície, o que leva a uma interessante analogia entre esméticos e óptica geométrica. Além disso, é bem conhecido na comunidade de óptica que lentes planas de índice de refração não-uniformes podem ser tratadas como superfícies curvas, cujas geodésicas se propagam da mesma forma que a luz se propaga na lente. Com isso, pode-se fabricar, em princípio, superfícies com propriedades ópticas específicas e, dessa forma, construir texturas esméticas com diferentes defeitos e singularidades a partir da extensa literatura conhecida de lentes / Abstract: We study geometrical configurations of liquid crystals defined on curved bidimensional substrates. We are particularly interested in the smectics-A phase, whose molecules are organized in layers. This is an interesting problem since many of the liquid crystal characteristics, such as its optical and elastic properties, are affected by the curvature of its substrate. Differently from the planar case, in curved surfaces the presence of Gaussian curvature induces topological defects and grain boundaries in the smectic structure. We will illustrate this interplay between curvature and topological defects for different surfaces in the limit where the energy contribution due to the compression of the layers is much larger than the contributions from other types of deformations. At this regime, the ground state is obtained when the smectic layers are uniformly spaced. In this case the normals to the layers follows geodesics of the surface. This leads to an interesting analogy between smectics and geometric optics. Moreover, it is well known in the optics community that flat lenses with nonuniform refractive index can be treated as curved surfaces, where their geodesics propagate in the same way that light propagates in the lens. Therefore, one can manufacture, in principle, surfaces with specific optical properties and construct smectic textures with different topological defects and singularities by using the extensive literature of known lenses / Mestrado / Física / Mestre em Física

Cette thèse concerne l'étude des propriétés de champs scalaires classiques et quantiques en présence d'un environnement inhomogène et/ou dépendant du temps. Nous nous concentrerons sur des modèles pouvant être décrits, fondamentalement ou de manière effective, par un espace-temps courbe contenant un horizon des événements. Nous verrons en particulier comment une correspondance mathématique, provenant d'une symétrie de Lorentz effective à basse énergie, permet de relier les comportements des ondes dans un cadre non relativiste à la physique des trous noirs, quelles en sont les limites et dans quelle mesure les résultats ainsi obtenus sont og analogues fg à leurs pendants gravitationnels. Après un premier chapitre d'introduction rappelant quelques bases de relativité générale puis une dérivation de la radiation de Hawking et de la correspondance avec des systèmes non relativistes, je présenterai le détail de quatre travaux effectués durant ma thèse. Les autres articles écrits dans ce cadre sont résumés dans le dernier chapitre, précédant une conclusion générale. Mes collaborateurs et moi nous sommes concentrés sur trois aspects du comportement des champs près de l'analogue d'un horizon des événements dans des modèles avec une symétrie de Lorentz effective à basse énergie. Le premier concerne les effets non linéaires, cruciaux pour comprendre l'évolution de la radiation de Hawking ainsi que pour les réalisations expérimentales mais auparavant peu étudiés. Nous montrerons comment ceux-ci déterminent les possibles comportements aux temps longs pour des systèmes stables ou instables. Le second aspect a trait aux effets linéaires et quantiques, en particulier la radiation de Hawking elle-même, son devenir lorsque l'horizon est continûment effacé, ainsi que les diverses instabilités à même de survenir dans différents modèles. Enfin, nous avons participé à l'élaboration, à l'analyse et à l'étude d’expériences dites de og gravité analogue fg dans des condensats de Bose-Einstein et des systèmes hydrodynamiques ou acoustiques, dont je rapporte les principaux résultats. / The present thesis deals with some properties of classical and quantum scalar fields in an inhomogeneous and/or time-dependent background, focusing on models where the latter can be described as a curved space-time with an event horizon. While naturally formulated in a gravitational context, such models extend to many physical systems with an effective Lorentz invariance at low energy. We shall see how this effective symmetry allows one to relate the behavior of perturbations in these systems to black-hole physics, what are its limitations, and in which sense results thus obtained are “analogous” to their general relativistic counterparts. The first chapter serves as a general introduction. A few notions from Einstein's theory of gravity are introduced and a derivation of Hawking radiation is sketched. The correspondence with low-energy systems is then explained through three important examples. The next four chapters each details one of the works completed during this thesis, updated and slightly reorganized to account for new developments which occurred after their publication. The other articles I contributed to are summarized in the last chapter, before the general conclusion. My collaborators and I focused on three aspects of the behavior of fields close to the (analogue) event horizon in models with an effective low-energy Lorentz symmetry. The first one concerns nonlinear effects, which had been given little attention in view of their crucial importance for understanding the evolution in time of Hawking radiation as well as for experimental realizations. We showed in particular how they determine the late-time behavior in stable and unstable configurations. The second aspect concerns linear and quantum effects. We studied the Hawking radiation itself in several models and what replaces it when continuously erasing the horizon. We also characterized and classified the different types of linear instabilities which can occur. Finally, we contributed to the design and analysis of “analogue gravity” experiments in Bose-Einstein condensates, hydrodynamic flows, and acoustic setups, of which I report the main results.

O objeto de estudo desta dissertação é o efeito de criação de partículas pela curvatura sob o escopo de uma teoria de espalhamento, discutindo quando que a interpretação a partir de uma matriz S é tangível e obtendo sua expressão nesses casos. O capítulo de introdução aborda superficialmente conceitos de relatividade geral e de teoria quântica de campos em espaços planos e curvos, necessários para a construção da matriz S. O conteúdo deste capítulo segue as apresentações feitas por Wald, Parker e Birrell em geral, tendo como guia as obras de Bar, Wald e Hawking no que se trata especificamente de relatividade geral, e de Penrose e Rindler no que se trata da estrutura espinorial. A construção da matriz S se dá no capítulo 2, tendo como guia o trabalho de Wald. O capítulo 3 apresenta exemplos que permitem a contextualização da criação de partículas em casos específicos de espaços-tempos em expansão. Este estudo nos permite verificar que as condições que precisam ser satisfeitas em um espaço-tempo globalmente hiperbólico e assintoticamente estacionário para que a formulação da matriz S possa ser feita são que as teorias no passado e futuro distantes devem ser unitariamente equivalentes, que a relação entre as regiões se dá através de transformações de Bogolyubov dadas por operadores limitados definidos em toda a parte e que tais operadores satisfaçam a condição de Hilbert-Schmidt. Nestes casos obtemos uma expressão para a matriz $S$ que descreve a criação de partículas pela curvatura do espaço-tempo para o campo de Klein-Gordon e de Dirac, além de outras relações úteis, como número médio de partículas criadas e probabilidade de se encontrar partículas em determinado modo, o que permite uma analogia com a radiação de corpo negro, passo fundamental para se entender fenômenos de grande interesse na física, como a radiação de Hawking e a criação de partículas no período inflacionário. / This master\'s thesis deals with the effect of particle creation by the curvature of space-time according to the point of view of scattering theory, discussing when such interpretation is possible by means of an S-matrix and obtaining its expression in those cases. The first chapter treats, superficially, some concepts of general relativity and quantum field theory in plane and curved space-times that are imperative to understand the construction of the S-matrix. The subject of this chapter is covered in the work of Wald, Parker, and Birrell, and follows closely the work of Bar, Wald and Hawking, when treats concepts specifically from general relativity, and from Penrose and Rindler, when talking about the spinor structure of space-time. The construction of the S-matrix is made in the second chapter, along the lines of the work of Wald. The third chapter presents some examples that bring some light on the creation of particles in specific cases of expanding space-times. This study let us verify that an S-matrix formulation is tenable, on globally hyperbolic asymptotic stationary curved space-times, if both quantum theories in the distant past and distant future are unitary equivalent, the relation of both regions is made by Bogolyubov transformations by means of everywhere defined bounded operators and that those operators satisfy the Hilbert-Schmidt condition. In those cases we derive the expression of the S-matrix for the Klein-Gordon and Dirac fields. Also we obtain the number of particles created and the probability of find particles in a particular mode, with let one make an analogy with the black body radiation, which is a fundamental step in the direction of understanding interesting phenomena in quantum field theory in curved space-times, like the Hawking radiation and particle creation in the early universe.

The Vlasov-Poisson system is most commonly used to model the movement of charged particles in a plasma or of stars in a galaxy. It consists of a kinetic equation known as the Vlasov equation coupled with a force determined by the Poisson equation. The system in Euclidean space is well-known and has been extensively studied under various assumptions. In this paper, we derive the Vlasov-Poisson equations assuming the particles exist only on the 2-sphere, then take an in-depth look at particles which initially lie along a great circle of the sphere. We show that any great circle is an invariant set of the equations of motion and prove that the total energy, number of particles, and entropy of the system are conserved for circular initial distributions. / Graduate

Submitted by Vasti Diniz (vastijpa@hotmail.com) on 2017-09-18T14:40:44Z No. of bitstreams: 1 arquivototal.pdf: 5994670 bytes, checksum: 2dd7a3577c0cc6ee44dd8eb7435dfaaf (MD5) / Made available in DSpace on 2017-09-18T14:40:44Z (GMT). No. of bitstreams: 1 arquivototal.pdf: 5994670 bytes, checksum: 2dd7a3577c0cc6ee44dd8eb7435dfaaf (MD5) Previous issue date: 2015-02-26 / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior - CAPES / In this thesis we study the dynamics of charge carriers, and the electronic properties, of the C60 fullerene molecule. Characterizing it by a geometric bias. In inertial reference systems and when we have your material under rotation content. Initially we discussed the scientific advent of carbon allotropes, and the importance of modelling its derivates at low energies. We show that at low energies, the graphene - the two-dimensional carbon allotrope form - can be described for a non-massive theory of free fermions. At a second moment, we extended the nonmassive free fermions theory for the C60 molecule. Assuming the hexagonal graphene network can be entered in fullerene when we introduce topological defects. A brief study of topological defects in condensed matter was done. And soon after, we made a description these defects via a non-Euclidean geometry. Showing how the charge carriers in the network see the defects like gauge fields. Then we began to expose the results of this thesis. First we assume the fullerene by a two-dimensional spherical metric with defects, containing a fictitious t’Hooft-Polyakov monopole in its center. TheC60 is still subjected to the action of an Aharonov-Bohm flux arising of a magnetic wire running through its poles. So we get the spectrum, and the prediction of a persistent current in the molecule. Finally we return to the analysis of the molecule, now with your content of matter under rotation. For this, we studied a metric Gödel-type with spherical symmetry. We discussed the problem of causality and obtain the spectrum and the persistent current in terms of the vorticity (W) of spacetime. / Nesta tese estudamos a dinâmica de portadores de carga, e as propriedades eletrônicas, na molécula de fulerenoC60. Caracterizando-a por um viés geométrico. Tanto em sistemas de referência inercial, como quando temos seu conteúdo de matéria sob rotação. Inicialmente abordamos o advento científico das formas alotrópicas do carbono e a importância da modelagem a baixas energias dos seus derivados. Onde mostramos que no limite de baixas energias, o grafeno - que trata-se da forma alótropica bidimensional do carbono - pode ser descrito por uma teoria de férmions livres sem massa. Num segundo momento estendemos a teoria de férmions não massivos para a molécula de C60. Assumindo que a rede hexagonal do grafeno pode inscrever o C60 ao introduzirmos alguns defeitos topológicos. Um breve estudo sobre os defeitos topológicos na matéria condensada foi feito. Onde, logo em seguida, partimos para uma descrição de tais defeitos via uma geometria não-euclidiana. Mostrando como os portadores de carga no meio enxergam os defeitos como campos de gauge. Em seguida começamos a expor os resultados desta tese. Primeiramente assumimos tratar o fulereno por uma métrica de uma esfera bidimensional com defeitos, e contendo um monopolo de t’Hooft-Polyakov fictício em seu centro. O C60 é ainda submetido a ação de um fluxo de Aharonov-Bohm advindo de uma corda magnética quiral transpassando seus polos. Obtemos assim o espectro e a predição de uma corrente persistente na molécula. Por fim retomamos a análise da molécula, agora com seu conteúdo de matéria sob rotação. Para isso assumimos tratar o fulereno por uma métrica do tipo Gödel com simetria esférica. Discutimos o problema da causalidade e obtemos espectro e corrente persistente em termos da vorticidade (W) do espaço-tempo.