CoordenaÃÃo de AperfeiÃoamento de Pessoal de NÃvel Superior / Conselho Nacional de Desenvolvimento CientÃfico e TecnolÃgico / Neste trabalho, demonstramos teoremas estabelecendo condiÃÃes suficientes para a existÃncia de imersÃes isomÃtricas com curvatura extrÃnseca prescrita em grupos de Lie nilpotentes e solÃveis. Obtemos assim uma generalizaÃÃo do Teorema Fundamental da Teoria de Subvariedades em Rn e, em particular, obtemos resultados de imersÃo em todos os grupos tipo-Heisenberg e em todos os espaÃos de Damek-Ricci. / In this paper, we prove theorems establishing sufficient conditions to existence for isometric immersions with prescribed extrinsic curvature in two-step nilpotent Lie groups and solvmanifolds. We obtain a generalization of the Fundamental Theorem of Submanifold Theory in Rn and, in particular, we one has immersion results in the generally Heisenberg type groups and Damek-Ricci spaces.

Lattice gauge theory is a special regularization of continuum gauge theories and the numerical simulation of lattice quantum chromodynamics (QCD) remains as the only first principle method to study non-perturbative QCD at low energy. The lattice spacing a, which serves as the ultraviolet cut off, plays a significant role in determining error on any lattice simulation results. Physical results come from extrapolating a series of simulations with different values for a to a=0. Reducing the size of these errors for non-zero a improves the extrapolation and minimizes the error. In the strong coupling limit the coarse lattice spacing pushes the analysis of the finite lattice spacing error to its limit. Section 4 measures two renormalized physical observables, the neutral kaon mixing parameter BK and the Delta I=3/2 K pi pi decay amplitude A2 on a lattice with coarse lattice spacing of a ~ 1GeV and explores the a^2 scaling properties at this scale. In the weak coupling limit the lattice simulations suffer from critical slowing down where for the Monte Carlo Markov evolution the cost of generating decorrelated samples increases significantly as the lattice spacing decreases, which makes reliable error analysis on the results expensive. Among the observables the topological charge of the configurations appears to have the longest integrated autocorrelation time. Based on a previous work where a diffusion model is proposed to describe the evolution of the topological charge, section 2 extends this model to lattices with dynamical fermions using a new numerical method that captures the behavior for different Fourier modes. Section 3 describes our effort to find a practical renormalization group transformation to transform lattice QCD between two different scales, whose knowledge could ultimately leads to a multi-scale evolution algorithm that solves the problem of critical slowing down. For a particular choice of action, we have found that doubling the lattice spacing of a fine lattice yields observables that agree at the few precent level with direct simulations on the coarser lattice. Section 5 aims at speeding up the lattice simulations in the weak coupling limit from the numerical method and hardware perspective. It proposes a preconditioner for solving the Dirac equation targeting the ensemble generation phase and details its implementation on currently the fastest supercomputer in the world.

京都大学 / 0048 / 新制・課程博士 / 博士(理学) / 甲第18042号 / 理博第3920号 / 新制||理||1566(附属図書館) / 30900 / 京都大学大学院理学研究科数学・数理解析専攻 / (主査)教授 堤 誉志雄, 教授 加藤 毅, 教授 上田 哲生 / 学位規則第4条第1項該当 / Doctor of Science / Kyoto University / DGAM

No description available.

L'effet Hawking prédit, dans un espace-temps décrivant l'effondrement d'une étoile à symétrie sphérique vers un trou noir de Schwarzschild, qu'un observateur statique, situé à l'infini, observera un flux thermal de particules quantiques à la température de Hawking. La première démonstration mathématique de l'effet Hawking pour des champs quantiques libres est due à Bachelot, dont le travail sur les champs de Klein-Gordon a été ensuite étendu aux champs de Dirac, d'abord par Bachelot lui-même, puis par Melnyk. Ces travaux, placés dans le cadre d'une symétrie sphérique, ont été complétés par Häfner, qui donna une démonstration rigoureuse de l'effet Hawking pour des champs de Dirac, autour d'une étoile s'effondrant vers un trou noir de Kerr. Le but de cette thèse est d'étudier l'effet Hawking non plus dans un modèle de champs quantiques libres, où les problèmes posés se ramènent à l'étude d'équations aux dérivées partielles linéaires, mais dans un modèle de champs de Dirac en interaction. L'interaction est supposée à support compact, statique, et localisée à l'extérieur de l'étoile. Nous choisissons de traiter le cas d'un modèle jouet, dans un espace-temps de dimension 1+1, situation à laquelle on peut se ramener, au moins dans le cas libre, en utilisant la symétrie sphérique du problème. Nous étudions le comportement de champs de fermions de Dirac dans différentes situations : d'abord, pour une observable suivant l'effondrement de l'étoile ; puis pour une observable stationnaire ; enfin, pour une interaction dépendante du temps, localisée près de la surface de l'étoile. Dans chacun de ces cas, nous montrons l'existence de l'effet Hawking et donnons l'état limite correspondant. / The Hawking effect predicts that, in a space- time describing the collapse of a spherically symmetric star to a Schwarzschild black hole, a static observer at infinity sees the Unruh state as a thermal state at Hawking temperature. The first mathematical proof of the Hawking effect, in the original setting of Hawking, is due to Bachelot. His work on Klein-Gordon fields has been extended to Dirac fields, in the first place by Bachelot himself, and by Melnyk after that. Those works, placed in the setup of a spherically symmetric star, have been completed by Häfner, who gave a rigorous proof of the Hawking effect for Dirac fields, outside a star collapsing to a Kerr black hole. The aim of this thesis is to study the Hawking effect not for a model of free quantum fields, in which case the problems can be reduced to studies on linear partial differential equations, but for a model of interacting Dirac fields. The interaction will be considered as a static, compactly-supported interaction, living outside the star. We choose to study a toy model in a 1+1 dimensional space-time. Using the fact that the problem is spherically symetric, one can, at least in the free case, reduce the real problem to this toy model. We study the behavior of Dirac fermions fields in various situations : first, for an observable following the star's collapse ; then, for a static observable ; finally, for a time-dependent interaction, fixed close to the star's boundary. In each of those cases, we show the existence of the Hawking Effect and give the corresponding limit state.

Neste trabalho apresentamos um protocolo para simular, no contexto da eletrodinâmica quântica de cavidades, a equação de Dirac 2+1 D e 1+1 D para uma partícula relativística livre, de spin ½. Especificamente, tratamos dois sistemas distintos: no primeiro consideramos um átomo de quatro níveis interagindo com dois modos da cavidade e quatro campos clássicos; no segundo, consideramos um átomo de três níveis interagindo com um modo da cavidade e dois campos clássicos. O primeiro sistema foi utilizado para simular a equação de Dirac 2+1 D. Através do segundo sistema mostramos como simular a equação de Dirac 1+1 D. Com esse sistema mostramos como manipular e controlar por meio das forças de acoplamentos dos campos, os valores da velocidade da luz e a energia de repouso da partícula relativística livre de Dirac simulada. Verificamos que a dinâmica de um elétron no formalismo da mecânica quântica relativística pode ser simulada usando experimentos em Eletrodinâmica Quântica de Cavidades. Neste contexto, analisamos o movimento oscilatório inesperado de uma partícula quântica relativística livre conhecido como Zitterbewegung. / In this work we present, in the context of cavity quantum electrodynamics, a protocol for simulating Dirac equation 2+1 and 1+1 for a relativistic free particle with spin ½. Specifically, we deal with two different systems: In the first one we consider a four level atom interacting with two modes of the cavity and four classical fields; In the second system we deal consider a three level atom and interacting with one mode of the cavity and two classical fields. The first system was used to simulate a 2+1 D Dirac equation. With the second system we show how to simulate a 1+1D Dirac equation. With these systems we show how to simulate and control through the field coupling strength, the values of the velocity of light and rest energy of the simulated Dirac´s relativistic free particle. We verify that the dynamics of one electron in the formalism of relativistic quantum mechanics can be simulated using experiments in cavity quantum electrodynamics. In this context, we analyzed the unexpected but known oscillatory movement of a relativistic free quantum particle.

Neste trabalho apresentamos um protocolo para simular, no contexto da eletrodinâmica quântica de cavidades, a equação de Dirac 2+1 D e 1+1 D para uma partícula relativística livre, de spin ½. Especificamente, tratamos dois sistemas distintos: no primeiro consideramos um átomo de quatro níveis interagindo com dois modos da cavidade e quatro campos clássicos; no segundo, consideramos um átomo de três níveis interagindo com um modo da cavidade e dois campos clássicos. O primeiro sistema foi utilizado para simular a equação de Dirac 2+1 D. Através do segundo sistema mostramos como simular a equação de Dirac 1+1 D. Com esse sistema mostramos como manipular e controlar por meio das forças de acoplamentos dos campos, os valores da velocidade da luz e a energia de repouso da partícula relativística livre de Dirac simulada. Verificamos que a dinâmica de um elétron no formalismo da mecânica quântica relativística pode ser simulada usando experimentos em Eletrodinâmica Quântica de Cavidades. Neste contexto, analisamos o movimento oscilatório inesperado de uma partícula quântica relativística livre conhecido como Zitterbewegung. / In this work we present, in the context of cavity quantum electrodynamics, a protocol for simulating Dirac equation 2+1 and 1+1 for a relativistic free particle with spin ½. Specifically, we deal with two different systems: In the first one we consider a four level atom interacting with two modes of the cavity and four classical fields; In the second system we deal consider a three level atom and interacting with one mode of the cavity and two classical fields. The first system was used to simulate a 2+1 D Dirac equation. With the second system we show how to simulate a 1+1D Dirac equation. With these systems we show how to simulate and control through the field coupling strength, the values of the velocity of light and rest energy of the simulated Dirac´s relativistic free particle. We verify that the dynamics of one electron in the formalism of relativistic quantum mechanics can be simulated using experiments in cavity quantum electrodynamics. In this context, we analyzed the unexpected but known oscillatory movement of a relativistic free quantum particle.

Speculations in the signicance of retardation aects in fundamental physics, especiallythe Dirac equation, that Atiyah and Moore bring up in "A shifted view of fundamental physics" are summarized and reviewedin terms of basic undergraduate conceptions. Some remarks are further investigated and ashifted version of the Klein Gordon equation is derived.

Orientador: Prof. Dr. Marcos Roberto da Silva Tavares / Dissertação (mestrado) - Universidade Federal do ABC, Programa de Pós-Graduação em Física, 2015.

Wave phenomena appear in many fields of science such as acoustics, geophysics, and quantum mechanics. They can often be described by partial differential equations (PDEs). As PDEs typically are too difficult to solve by hand, the only option is to compute approximate solutions by implementing numerical methods on computers. Ideally, the numerical methods should produce accurate solutions at low computational cost. For wave propagation problems, high-order finite difference methods are known to be computationally cheap, but historically it has been difficult to construct stable methods. Thus, they have not been guaranteed to produce reasonable results. In this thesis we consider finite difference methods on summation-by-parts (SBP) form. To impose boundary and interface conditions we use the simultaneous approximation term (SAT) method. The SBP-SAT technique is designed such that the numerical solution mimics the energy estimates satisfied by the true solution. Hence, SBP-SAT schemes are energy-stable by construction and guaranteed to converge to the true solution of well-posed linear PDE. The SBP-SAT framework provides a means to derive high-order methods without jeopardizing stability. Thus, they overcome most of the drawbacks historically associated with finite difference methods. This thesis consists of three parts. The first part is devoted to improving existing SBP-SAT methods. In Papers I and II, we derive schemes with improved accuracy compared to standard schemes. In Paper III, we present an embedded boundary method that makes it easier to cope with complex geometries. The second part of the thesis shows how to apply the SBP-SAT method to wave propagation problems in acoustics (Paper IV) and quantum mechanics (Papers V and VI). The third part of the thesis, consisting of Paper VII, presents an efficient, fully explicit time-integration scheme well suited for locally refined meshes.