Indiana University-Purdue University Indianapolis (IUPUI) / I study d-bar and Dirac operators on classical and quantum domains subject to the APS boundary conditions, APS like boundary conditions, and other types of global boundary conditions. Moreover, the inverse or inverse modulo compact operators to these operators are computed. These inverses/parametrices are also shown to be bounded and are also shown to be compact, if possible. Also the index of some of the d-bar operators are computed when it doesn't have trivial index. Finally a certain type of limit statement can be said between the classical and quantum d-bar operators on specialized complex domains.

Submitted by Maria Aparecida (cidazen@gmail.com) on 2017-12-04T14:44:31Z No. of bitstreams: 1 João Andrade..pdf: 3163183 bytes, checksum: 0c7d19f31b8e570d13e85b371ea43554 (MD5) / Made available in DSpace on 2017-12-04T14:44:31Z (GMT). No. of bitstreams: 1 João Andrade..pdf: 3163183 bytes, checksum: 0c7d19f31b8e570d13e85b371ea43554 (MD5) Previous issue date: 2017-02-22 / CAPES. / For the recent years, there has been a growing interest in Lorentz-violating theories. Studies have been carried out addressing the inclusion of Lorentz-violating terms into the Standard Model (SM). This has led to the development of the Standard Model Extension (SME), which is a framework containing modifications that are power-counting renormalizable and consistent with the gauge structure of the SM. More recently, a nonminimal version of the SME was developed for the photon, neutrino, and fermion sector additionally including higher-derivative terms. One of the new properties of this nonminimal version is the lost of renormalizability. In this work, we study the main aspects of a modified Dirac theory in the nonminimal Standard-Model Extension. We focus on two types of operators namely, pseudovector and two-tensor operators. These two operators display an unusual property; they break the degeneracy of spin. This new property becomes manifest in providing two di erent dispersion relations, one for each spin projection. To solve the Dirac equation modified by those operators, we introduce a new method that was suggested by Kostelecký and Mewes in a recent research paper. This method allows to block-diagonalizing the modified Dirac equation and, thus, permits us to obtain the spinors. The objectives of the current work are as follows. First, we will review the main concepts for understanding the SME. Second, we will introduce how to extend the minimal fermion sector to the nonminimal one. Third, we will describe the method that block-diagonalizes the modified Dirac equation and we will compute the field equations. And,finally, we will get the exact dispersion relations and the spinor solutions for operators of arbitrary mass dimension. / Nos últimos anos, houve um aumento significativo no interesse em teorias que violam a simetria de Lorentz. Estudos têm sido realizados na tentativa de incluir termos que violam a simetria de Lorentz no Modelo Padrão (MP). Esta tentativa culminou no surgimento do chamado Modelo Padrão Estendido (MPE). Este modelo contempla todas as possíveis modificações que são consistentes com as propriedades já bem estabelecidas, tais como renormalizabilidade e a estrutura de gauge do MP. Mais recentemente, uma versão não-mínima do MPE foi desenvolvida para os setores dos fótons, neutrinos e para os férmions. Esta versão não-mínima caracteriza-se pela presença de altas derivadas. Uma das novas propriedades nesta versão não-mínima é a perda da renormalizabilidade. Neste trabalho, estudamos os principais aspectos da teoria de Dirac modi cada no MPE não-mínimo. Nós nos concentramos em dois tipos de operadores a saber, operadores pseudovetoriais e tensoriais. Estes dois operadores exibem uma propriedade incomum, eles quebram a degenerescência de spin. Esta nova propriedade manifesta-se, por exemplo, na presença de duas relações de dispersão diferentes, uma para cada projeção do spin. Para resolver a equação de Dirac modi cada por esses operadores, introduzimos um novo método que foi sugerido por Kostelecký e Mewes em um trabalho recente. Este método permite bloco-diagonalizar a equação de Dirac modi cada e, assim, nos fornece uma nova maneira de obter os espinores. Os objetivos do presente trabalho são os seguintes. Primeiro, iremos rever alguns conceitos essenciais para o entendimento do MPE. Segundo, apresentaremos a extens ão do setor fermiônico mínimo para o não-mínimo. Terceiro, descreveremos o método que bloco-diagonaliza a equação de Dirac modi cada e calcularemos as equações de campo. Por fim, calcularemos as relações de dispersão exatas e as soluções espinoriais para cada configuração não-mínima dos operadores citados.

The pseudospin symmetry concept is investigated by solving the Dirac equation for the exactly solvable potentials such as pseudoharmonic potential, Mie-type potential, Woods-Saxon potential and Hulth&eacute / n plus ring-shaped potential with any spin-orbit coupling term $kappa$. Nikiforov-Uvarov Method, Asymptotic Iteration Method and functional analysis method are used in the calculations. The energy eigenvalue equations of the Dirac particles are found and the corresponding radial wave functions are presented in terms of special functions. We look for the contribution of the ring-shaped potential to the energy spectra of the Dirac particles. Particular cases of the potentials are also discussed. By considering some particular cases, our results are reduced to the well-known ones presented in the literature. In addition, by taking equal mixture of scalar and vector potentials together with tensor potential, solutions of the Dirac equation are found and then the energy splitting between the two states in the pseudospin doublets is investigated. We indicate that degeneracy between members of pseudospin doublet is removed by tensor interactions. Effects of the potential parameters on the pseudospin doublet splitting are also studied. Radial nodes structure of the Dirac spinor are presented.

Ce travail de thèse s’intéresse aux corrections relativistes induites par une impulsion lumineuse ultra-brève et intense dans la matière condensée. Il s’inscrit dans la thématique nouvelle de la désaimantation ultra-rapide cohérente de systèmes ferromagnétiques induite par une impulsion laser femto-seconde [Nature 5, 515 (2009)] [1]. Un couplage de nature relativiste entre les spins et les photons a été proposé pour expliquer les résultats expérimentaux observés dans [1]. La première partie de ce travail étudie la limite non relativiste du formalisme de Dirac en présence d’un champ électromagnétique dépendant du temps. En utilisant la transformation de Foldy-Wouthuysen , le hamiltonien électronique de Dirac en présence d’un champ électromagnétique dépendant du temps est développé au cinquième ordre en 1/m. Les résultats obtenus ont permis de postuler une expression générale de l’interaction directe entre le spin et le champ électromagnétique sous la forme d’un développement en série entière. Un travail similaire est réalisé dans le cadre du problème relativiste à deux électrons en interaction coulombienne. La diagonalisation du hamiltonien de Breit au troisième ordre en 1/m fait apparaître une interaction singulière entre le spin, le champ coulombien et le champ électromagnétique externe dépendant du temps. Dans la deuxième partie, on propose un modèle classique pour modéliser une expérience de magnéto-optique non-linéaire réalisée sur des échantillons ferromagnétiques. Les prédictions théoriques des angles de rotation Faraday sont comparées aux résultats expérimentaux de la référence [1] et permettent d’ouvrir une discussion à propos des mécanismes physiques gouvernant les phénomènes magnéto-optiques observés. Le rôle joué par l’interaction spin-orbite entre les spins et le champ électrique du laser est discuté. / This thesis focuses on the relativistic corrections induced by an ultra-short and intense light pulse in condensed matter. It is part of the new theme of the coherent ultra-fast demagnetization of ferromagnetic systems induced by a femtosecond laser pulse [ Nature, 5, 515 (2009)] [1]. A relativistic coupling between spins and photons has been proposed to explain the experimental results obtained in [1]. The first part of this work focuses on the nonrelativistic limit of the Dirac’s formalism. By means of the Foldy–Wouthuysen transformation the nonrelativistic approximation of the external-electromagnetic-field Dirac equation to fifth order in powers of 1/m is obtained. Generalizing this result we postulate a general expression of the direct spin–field electronic hamiltonian valid at any order in 1/m. A similar work is performed on a two-interacting electrons system described with the Breit hamiltonian, whose the diagonalization at third order in 1/m illustrates an original coupling between the spin, the coulombian interaction and the time-dependent external electromagnetic field. In a second part, a classical model is developed for modeling ultrafast nonlinear coherent magneto-optical experiments performed on ferromagnetic thin films. Theoretical predictions of the Faraday rotation angles are compared to available experimental values and give meaningful insights about the physical mechanisms underlying the observed coherent magneto-optical phenomena. The crucial role played by the spin-orbit mechanism resulting from the direct interaction between the external electric field of the laser and the electron spins of the sample is underlined.

In this project a theoretical study of the so called gyromagnetic ratio was done by investigating classical mechanics, Dirac theory and string theory. The gyromagnetic ratio is a constant term appearing in the coupling between angular momentum and magnetic moment for a particle. A universality in quantum field theory claiming g = 2 regardless of spin is known to exist which also agrees with the found values (g = 2) of the Dirac and string theory. The proof of the aforementioned universality in quantum field theory was sketched in the project by showing that the W-boson Lagrangian is well behaved in the massless limit. Furthermore it is shown that the spin equations of motion for a particle is greatly reduced if g = 2 regardless of spin. / I det här projektet utfördes en teoretisk studie av den så kallade gyromagnetsika kvoten genom att undersöka klassisk mekanik, Dirac-teori och strängteori. Den gyromagnetiska kvoten är en konstant term som visar sig i kopplingen mellan rörelsemängsmoment och magnetiskt moment för en partikel. I kvantfältsteori existerar en universalitet som hävdar att g = 2 oberoende av spin. Denna universalitet stämmer överens med de funna värden på g från Dirac-teori och strängtoeri. Beviset för denna universalitet testades genom ett exempel där Lagrangianen för W-bosonen i kvantfältsteori visades bete sig som förväntat när massan för partikeln tilläts gå mot noll. Vidare undersöktes rörelseekvationerna för ett system helt bestämt av dess spin. I dessa kan det visas att ett universellt värde på g oavsett spin reducerar dessa ekvationer avsevärt.

No description available.

Made available in DSpace on 2015-05-14T12:14:00Z (GMT). No. of bitstreams: 1 arquivototal.pdf: 1354058 bytes, checksum: 2f549ef4aed5937520237cd0b6df944f (MD5) Previous issue date: 2011-08-08 / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior - CAPES / Since many models in physics depend on the confinement of particles in certain regions of the space-time, like Rubakov and Randall-Sundrum models, we analyze the possibility of using geometrical fields to confine particles. In doing so, we exhibit some examples of the confinement of particles by using only geometrical fields such as torsion and Weyl 1- form. In order to prepare the reader to these examples, we give a brief introduction to the Riemannian and the non-Riemannian geometries. It turned out to be impossible to avoid controversial issues such as the equation of motion of a particle, the use of the minimal coupling procedure, and the application of the variational principle for non-Riemannian geometries. However, we avoided choosing what approach was right and decided to take two completely different approaches into account, namely, Kleinert's and Hehl's ones. Kleinert claims that particles must follow autoparallel, while Hehl and others state that the equation of motion of a particle must be derived from a conservation law related to the energy-momentum tensor of the particle. As a matter of fact, there are more differences between those approaches than we have mentioned here, but we expect this thesis to clarify those differences. To be more precise, we managed to exhibit examples of confinement only for Kleinert's approach. We had dificulty finding a example of confinement to hehl's approach, however we were able to eliminate the possibility of confinement for many cases, like scale fields for example. / Levando em consideração o interesse visível que muitos modelos da física têm em manter a matéria usual confinada em uma certa região do espaço-tempo, como por exemplo o modelo de Rubakov e o de Randall-Sundrum, exibimos a possibilidade da utilização de campos com origem geométrica para realizar este confinamento. Antes, porém, preparamos o leitor com todo o aparato geométrico necessário para a compreensão do que é feito nos últimos capítulos desta tese. Tornou-se impossível fugir de questões polêmicas envolvendo geometrias mais gerais que a riemanniana, como por exemplo a polêmica sobre a equação de movimento da partícula, o uso do acoplamento mínimo e a aplicação do princípio variacional. Entretanto, tentamos adotar uma postura imparcial e fizemos a análise do confinamento seguindo duas vertentes distintas. Uma das vertentes, defendidas por Kleinert, consiste em postular que partículas seguem autoparalelas. A outra vertente, a mais comum na literatura, segue a linha de Hehl, Gasperini e outros. Nesta vertente, a equação de movimento de uma partícula não pode ser postulada, mas sim obtida a partir da lei de conservação associada ao tensor de energia-momento da partícula, pois este contém informação sobre o movimento da partícula. Há mais diferenças entre essas duas linhas do que citamos aqui, como será indicado no decorrer da tese. Para ser mais preciso, fomos capazes de exibir o confinamento apenas para a primeira vertente. No caso da segunda, dificuldades técnicas nos limitaram a somente descartar certos campos de origem geométrica como campos confinadores.

Nous étudions l’analyse asymptotique en relativité générale sous deux aspects: le peeling et le scattering (diffusion) conforme. Le peeling est construit pour les champs scalaires linéaire et non-linéaires et pour les champs de Dirac en espace-temps de Kerr (qui est non-stationnaire et à symétrie simplement axiale), généralisant les travaux de L. Mason et J-P. Nicolas (2009, 2012). La méthode des champs de vecteurs (estimations d’énergie géométriques) et la technique de compactification conforme sont développées. Elles nous permettent de formuler les définitions du peeling à tous ordres et d’obtenir les données initiales optimales qui assurent ces comportements. Une théorie de la diffusion conforme pour les équations de champs sans masse de spîn n/2 dans l’espace-temps de Minkowski est construite.En effectuant les compactifications conformes (complète et partielle), l’espace-temps est complété en ajoutant une frontière constituée de deux hypersurfaces isotropes représentant respectivement les points limites passés et futurs des géodésiques de type lumière. Le comportement asymptotique des champs s’obtient en résolvant le problème de Cauchy pour l’équation rééchelonnée et en considérant les traces des solutions sur ces bords. L’inversibilité des opérateurs de trace, qui associent le comportement asymptotique passé ou futur aux données initiales, s’obtient en résolvant le problème de Goursat sur le bord conforme. L’opérateur de diffusion conforme est alors obtenu par composition de l’opérateur de trace futur avec l’inverse de l’opérateur de trace passé. / This work explores two aspects of asymptotic analysis in general relativity: peeling and conformal scattering.On the one hand, the peeling is constructed for linear and nonlinear scalar fields as well as Dirac fields on Kerr spacetime, which is non-stationary and merely axially symmetric. This generalizes the work of L. Mason and J-P. Nicolas (2009, 2012). The vector field method (geometric energy estimates) and the conformal technique are developed. They allow us to formulate the definition of the peeling at all orders and to obtain the optimal space of initial data which guarantees these behaviours. On the other hand, a conformal scattering theory for the spin-n/2 zero rest-mass equations on Minkowski spacetime is constructed. Using the conformal compactifications (full and partial), the spacetime is completed with two null hypersurfaces representing respectively the past and future end points of null geodesics. The asymptotic behaviour of fields is then obtained by solving the Cauchy problem for the rescaled equation and considering the traces of the solutions on these hypersurfaces. The invertibility of the trace operators, that to the initial data associate the future or past asymptotic behaviours, is obtained by solving the Goursat problem on the conformal boundary. The conformal scattering operator is then obtained by composing the future trace operator with the inverse of the past trace operator.

Made available in DSpace on 2015-05-14T12:14:06Z (GMT). No. of bitstreams: 1 arquivototal.pdf: 1577961 bytes, checksum: c71d976d783495df566e0fa6baadf8ca (MD5) Previous issue date: 2009-08-06 / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior - CAPES / We start this work studying the appearance of geometric quantum phases as in the relativistic as in the non-relativistic quantum dynamics of a neutral particle with permanent magnetic and electric dipole moment which interacts with external electric and magnetic fields in the presence of linear topological defects. We describe the linear topological defects using the approach proposed by Katanaev and Volovich, where the topological defects in solids are described by line elements which are solutions of the Einstein's equations in the context of general relativity. We also analyze the in uence of non-inertial effects in the quantum dynamics of a neutral particle using two distinct reference frames for the observers: one is the Fermi-Walker reference frame and another is a rotating frame. As a result, we shall see that the difference between these two reference frames is in the presence/absence of dragging effects of the spacetime which makes its in uence on the phase shift of the wave function of the neutral particle. In the following, we shall use our study of geometric quantum phases to make an application on the Holonomic Quantum Computation, where we shall show a new approach to implement the Holonomic Quantum Computation via the interaction between the dipole moments of the neutral particle and external fields and the presence of linear topological defects. Another applications for the Holonomic Quantum Computation is based in the structure of the topological defects in graphene layers. In the presence of topological defects, a graphene layer shows two distinct phase shifts: one comes from the mix of Fermi points while the other phase shift comes from the topology of the defect. To provide a geometric description for each phase shift in the graphene layer, we use the Kaluza-Klein theory where we establish that the extra dimension describes the Fermi points in the graphene layer. Hence, we can implement the Holonomic Quantum Computation through the possibility to build cones and anticones of graphite in such way we can control the quantum uxes in graphene layers. In the last part of this work, we study the Landau quantization for neutral particles as in the relativistic dynamics and non-relativistic dynamics. In the non-relativistic dynamics, we study the Landau quantization in the presence of topological defects as in an inertial as in a non-inertial reference frame. In the relativistic quantum dynamics, we start our study with the Landau quantization in the Minkowisky considering two different gauge fields. At the end, we study the relativistic Landau quantization for neutral particles in the Cosmic Dislocation spacetime. / Neste trabalho estudamos inicialmente o surgimento de fases geometricas nas dinâmicas quânticas relativística e não-relativística de uma partícula neutra que possui momento de dipolo magnético e elétrico permanente interagindo com campos elétricos e magnéticos externos na presença de defeitos topológicos lineares. Para descrevermos defeitos topológicos lineares usamos a aproximação proposta por Katanaev e Volovich, onde defeitos lineares em sólidos são descritos por elementos de linha que são soluções das equações de Einstein no contexto da relatividade geral. Analisamos também a inuência de efeitos não-inerciais na dinâmica quântica de uma partícula neutra em dois tipos distintos de referenciais para os observadores: um é o referencial de Fermi-Walker e outro é um referencial girante. Vemos que a diferença entre dois referenciais está na presença/ausência de efeitos de arrasto do espaço-tempo que irá influenciar diretamente na mudança de fase na funçãao de onda da partícula neutra. Em seguida, usamos nosso estudo de fases geométricas para fazer aplicações na Computação Quântica Holonômica onde mostramos uma nova maneira de implementar a Computação Quântica Holonômica através da interação entre momentos de dipolo e campos externos e pela presença de defeitos topológicos lineares. Outra aplicação para a Computação Quântica Holonômica está baseada na estrutura de defeitos topológicos em um material chamado grafeno. Na presença de defeitos topológicos lineares, esse material apresenta duas fases quânticas de origens distintas: uma da mistura dos pontos de Fermi e outra da topologia do defeito. Para dar uma descrição geométrica para a origem de cada fase no grafeno usamos a Teoria de Kaluza-Klein, onde a dimensão extra sugerida por esta teoria descreve os pontos de Fermi no grafeno. Portanto, a implementação da Computação Quântica Holonômica no grafeno está baseada na possibilidade de construir cones e anticones de grafite de tal maneira que se possa controlar os fluxos quânticos no grafeno. Na última parte deste trabalho estudamos a quantização de Landau para partículas neutras tanto na dinâmica não-relativística quanto na dinâmica relativística. Na dinâmica não-relativítica, estudamos a quantização de Landau na presença de defeitos em um referecial inercial e, em seguida, em um referencial nãoo-inercial. Na dinâmica relativística, estudamos inicialmente a quantização de Landau no espaço-tempo plano em duas configurações de campos diferentes. Por fim, estudamos a quantização de Landau relativística para partículas neutras no espaço-tempo da deslocação cósmica.