Ce mémoire est une poursuite de l’étude de la superintégrabilité classique et quantique dans un espace euclidien de dimension deux avec une intégrale du mouvement d’ordre trois. Il est constitué d’un article. Puisque les classifications de tous les Hamiltoniens séparables en coordonnées cartésiennes et polaires sont déjà complétées, nous apportons à ce tableau l’étude de ces systèmes séparables en coordonnées paraboliques. Premièrement, nous dérivons les équations déterminantes d’un système en coordonnées paraboliques et ensuite nous résolvons les équations obtenues afin de trouver les intégrales d’ordre trois pour un potentiel qui permet la séparation en coordonnées paraboliques. Finalement, nous démontrons que toutes les intégrales d’ordre trois pour les potentiels séparables en coordonnées paraboliques dans l’espace euclidien de dimension deux sont réductibles. Dans la conclusion de l’article nous analysons les différences entre les potentiels séparables en coordonnées cartésiennes et polaires d’un côté et en coordonnées paraboliques d’une autre côté. Mots clés: intégrabilité, superintégrabilité, mécanique classique, mécanique quantique, Hamiltonien, séparation de variable, commutation. / This thesis is a contribution to the study of classical and quantum superintegrability in a two-dimensional Euclidean space involving a third order integral of motion. It consists of an article. Because the classifications of all separable hamiltonians into Cartesian and polar coordinates are already complete, we bring to this picture the study of those systems in parabolic coordinates. First, we derive the determinating equations of a system into parabolic coordinates, after which we solve the obtained equations in order to find integrals of order three for potentials, which allow the separations of variables into the parabolic coordinates. Finally, we prove that all the third order integrals for separable potentials in parabolic coordinates in the Euclidean space of dimension two are reducible. In the conclusion of this article, we analyze the differences between the separable potentials in Cartesian and polar coordinates and the separable potentials in parabolic coordinates. Keywords: integrability, superintegrability, classical mechanics, quantum mechanics, Hamiltonian, separation of variables, commutation.

Submitted by Eligio Carlos Eduardo (eligio.eduardo1.618@gmail.com) on 2018-05-23T13:49:44Z No. of bitstreams: 1 dissertacao1.pdf: 4158096 bytes, checksum: 54d269ca38f12cb406b7db6f9a37bc62 (MD5) / Approved for entry into archive by Ana Paula Santulo Custódio de Medeiros null (asantulo@rc.unesp.br) on 2018-05-23T16:58:19Z (GMT) No. of bitstreams: 1 eduardo_ec_me_rcla.pdf: 4148882 bytes, checksum: 140172be0b9d254f494210f1ed894661 (MD5) / Made available in DSpace on 2018-05-23T16:58:19Z (GMT). No. of bitstreams: 1 eduardo_ec_me_rcla.pdf: 4148882 bytes, checksum: 140172be0b9d254f494210f1ed894661 (MD5) Previous issue date: 2018-04-27 / Este trabalho aborda os aspectos teóricos e numérico da Transformada de Fourier, bem como aplicações em sistemas mecânicos periódicos. O estudo iniciou-se com uma revisão bibliográfica que abordou inicialmente aspectos básicos de equações diferenciais ordinárias, métodos numéricos e implementação computacional, o desenvolvimento teórico da Transformada de Fourier, bem como sua implementação. Realizou-se estudos baseados em simulações numéricas de três modelos físicos: o oscilador harmônico, o pêndulo e o pião simétrico. / This work deals with the theoretical and numerical aspects of the Fourier Transform, as well as applications in periodic mechanical systems. It begins with a bibliographical study about a basic review of ordinary differential equations and its numerical solution methods. We also revisit the theoretical of the Fourier Transform as well an its computational implementation. We applied this theory studed in three physical models: the harmonic oscillator, the pendulum and the symmetrical top.

Submitted by Geandra Rodrigues (geandrar@gmail.com) on 2018-10-10T14:48:12Z No. of bitstreams: 1 guilhermeenriquealemanlopez.pdf: 6728480 bytes, checksum: d96cb4f60e6df31a85d6b383dec056f1 (MD5) / Approved for entry into archive by Adriana Oliveira (adriana.oliveira@ufjf.edu.br) on 2018-10-16T13:35:18Z (GMT) No. of bitstreams: 1 guilhermeenriquealemanlopez.pdf: 6728480 bytes, checksum: d96cb4f60e6df31a85d6b383dec056f1 (MD5) / Made available in DSpace on 2018-10-16T13:35:18Z (GMT). No. of bitstreams: 1 guilhermeenriquealemanlopez.pdf: 6728480 bytes, checksum: d96cb4f60e6df31a85d6b383dec056f1 (MD5) Previous issue date: 2018-08-24 / O objetivo deste trabalho é o estudo de procedimento de quantização canônica de uma partícula em um espaço curvo e sobre uma superfície de espaço Euclidiano. Estudaremos quantização canônica em coordenadas curvilineares, e em seguida vamos adaptar no caso de espaço curvo. Fizemos comparação crítica deste formalismo com três procedimentos principais apresentados na literatura desta área. / The aim of this work is to study the canonical quantization procedure of a particle in a Riemann space, as well as on a surface embedded in Euclidean space. To this aim, we rewrite quantum mechanics of a particle in Euclidean space in curvilinear coordinates. This allows us to formulate certain quantization procedure in a Riemann space. The resulting picture is compared with three quantization proposals known in the literature.

I defend three general claims concerning inter-theoretic reduction in physics. First, the popular notion that a superseded theory in physics is generally a simple limit of the theory that supersedes it paints an oversimplified picture of reductive relations in physics. Second, where reduction specifically between two dynamical systems models of a single system is concerned, reduction requires the existence of a particular sort of function from the state space of the low-level (purportedly more accurate and encompassing) model to that of the high-level (purportedly less accurate and encompassing) model that approximately commutes, in a specific sense, with the rules of dynamical evolution prescribed by the models. The third point addresses a tension between, on the one hand, the frequent need to take into account system-specific details in providing a full derivation of the high-level theory’s success in a particular context, and, on the other hand, a desire to understand the general mechanisms and results that under- write reduction between two theories across a wide and disparate range of different systems; I suggest a reconciliation based on the use of partial proofs of reduction, designed to reveal these general mechanisms of reduction at work across a range of systems, while leaving certain gaps to be filled in on the basis of system-specific details. After discussing these points of general methodology, I go on to demonstrate their application to a number of particular inter-theory reductions in physics involving quantum theory. I consider three reductions: first, connecting classical mechanics and non-relativistic quantum mechanics; second,connecting classical electrodynamics and quantum electrodynamics; and third, connecting non-relativistic quantum mechanics and quantum electrodynamics. I approach these reductions from a realist perspective, and for this reason consider two realist interpretations of quantum theory - the Everett and Bohm theories - as potential bases for these reductions. Nevertheless, many of the technical results concerning these reductions pertain also more generally to the bare, uninterpreted formalism of quantum theory. Throughout my analysis, I make the application of the general methodological claims of the thesis explicit, so as to provide concrete illustration of their validity.

Theoretical and experimental studies on intense laser atom interaction have drawn many interests over the past few decades. In this thesis, we consider strong-field tunnel ionization to explore two different problems dealing with the ionized-electron dynamics in the presence of an infrared, high-intensity, elliptically-polarized laser pulse. In the first part, we discover the electron dynamics from a static potential, this describes the complicated field-driven dynamics by a simple time-independent problem. In the second part, we set up an analytical expression for the attoclock offset angle. We use the time-dependent Kramers-Henneberger (KH) potential, and show how some approximations within the KH potential lead to the static potential and the analytical offset angle. We elucidate good agreement of our theory with the numerical results obtained from classical equations of motion. Finally, the comparison with the available experimental data has led to an interestingly new tunnel exit-radius different from the conventional models.

<p>Optimization and Machine learning (ML) have emerged as two positively disruptive methodologies and have thus resulted in unprecedented applications in several domains of technology. In recent years, ML has forayed into physical sciences and provided promising outcomes thanks to its ability in representing and generalizing complex functions to reveal underlying relations among variables describing a system. By casting ML as an optimization task, we first focus on its application in solving quantum many-body problems. Leveraging the power of quantum computation, we develop hybrid quantum machine learning protocols and implement benchmark tests to calculate the band structures of two-dimensional materials. We also show how this method can be used to estimate the critical point for a quantum phase transition. One  hurdle in such techniques is related to parameter optimization, wherein to obtain the desired result, the parameters have to be optimized, which can be computationally intensive. For a particular class of problem and a choice of algorithm, we deduce a simple parameter setting rule. This rule is projected as a heuristic and is validated numerically for several problem instances. Finally, by venturing into thermal photonics, a framework that takes advantage of the spectral and spatial information of hyperspectral thermal images to establish a completely passive machine perception, titled HADAR is presented. A conventional deep neural network is developed that utilizes the governing equation of HADAR and its performance in semantic segmentation is demonstrated. Altogether, this report establishes the need for creative algorithms that exploit modern hardware to solve complex problems that were previously deemed unsolvable.</p>

Notre thèse étudie la relation entre les mécaniques classique et relativiste. Il est généralement supposé, à partir de l’hypothèse des petites vitesses, que la mécanique classique correspond à la mécanique relativiste dans les cas où la vitesse des objets est petite par rapport à la vitesse de la lumière. Cette position nous semble inadéquate pour la simple raison que la mécanique classique ne peut être restreinte au seule domaine des petites vitesses. Nous proposons l’hypothèse que les deux mécaniques ont une structure commune et que chacune se distingue sous certaines conditions. Pour appuyer cette hypothèse, nous proposons une axiomatisation de la mécanique suffisamment générale pour servir de structure commune aux mécaniques classique et relativiste. Cette axiomatisation comporte une théorie de la relativité qui précise comment les quantités relatives sont reliées entre elles lorsque déterminées par rapport à différents référentiels, et les lois du mouvement qui précisent comment les forces exercées sur un objet détermine son mouvement. Cette mécanique générale est déterminée à deux constantes près et c’est en déterminant la valeur de ces constantes qu’apparaît le bris de la structure commune qui génère la mécanique classique d’une part et la mécanique relativiste d’autre part. / Our thesis studies the relationship between classical and relativistic mechanics. It is generally assumed, based on the assumption of small velocities, that classical mechanics corresponds to relativistic mechanics in cases where the speed of objects is small compared to the speed of light. This position seems inadequate to us, for the simple reason that classical mechanics cannot be restricted to the realm of small velocities alone. We propose the hypothesis that the two mechanics have a common structure, and that each can be distinguished under certain conditions. To support this hypothesis, we propose an axiomatization of mechanics that is sufficiently general to serve as a common structure for both classical and relativistic mechanics. This axiomatization includes a theory of relativity that specifies how relative quantities are related to each other when determined with respect to different reference frames, and laws of motion that specify how forces exerted on an object determine its motion. This general mechanics is determined to within two constants, and it is by determining the value of these constants that the common structure that generates classical mechanics on the one hand and relativistic mechanics on the other is broken down.

Dans cette thèse nous étudions un système quantique, obtenu comme un analogue d'un système classique superintégrable perturbé au moyen de la quantification géométrique. Notre objectif est de mettre en évidence la présence des phénomènes analogues à ceux qui caractérisent la superintégrabilité classique, notamment la coexistence des mouvements réguliers et chaotiques liés aux effets des résonances ainsi que la régularité du régime non-résonant. L'analyse est effectuée par l'étude des distributions du Husimi des états quantiques sélectionnés, avec une attention particulière aux états stationnaires et à l'évolution des états cohérents. Les calculs sont effectués en utilisant les méthodes numériques et les méthodes perturbatives. Les calculs sont effectués en utilisant les méthodes numériques et les méthodes perturbatives. Bien que cette thèse devrait être considérée comme une étude préliminaire, dont l'objectif est de créer le socle des études futures, nos résultats donnent des indications intéressantes sur la dynamique quantique. Par exemple, il est démontré comment les résonancees classiques exercent une influence considérable sur le spectre du système quantique et comment il est possible, dans le comportement quantique, de trouver une trace de l'invariant adiabatique dans le régime de résonance. / The abundance, among physical models, of perturbations of superintegrable Hamiltonian systems makes the understanding of their long-term dynamics an important research topic. While from the classical standpoint the situation, at least in many important cases, is well understood through the use of Nekhoroshev stability theorem and of the adiabatic invariants theory, in the quantum framework there is, on the contrary, a lack of precise results. The purpose of this thesis is to study a perturbed superintegrable quantum system, obtained from a classical counterpart by means of geometric quantization, in order to highlight the presence of indicators of superintegrability analogues to the ones that characterize the classical system, such as the coexistence of regular motions with chaotic one, due to the effects of resonances, opposed to the regularity in the non resonant regime. The analysis is carried out by studying the Husimi distributions of chosen quantum states, with particular emphasis on stationary states and evolved coherent states. The computation are performed using both numerical methods and perturbative schemes. Although this should be considered a preliminary work, the purpose of which is to lay the fundations for future investigations, the results obtained here give interesting insights into quantum dynamics. For instance, it is shown how classical resonances exert a considerable influence on the spectrum of the quantum system and how it is possible, in the quantum behaviour, to find a trace of the classical adiabatic invariance in the resonance regime. / L'abbondanza, fra i modelli fisici, di perturbazioni di sistemi Hamiltoniani superintegrabili rende la comprensione della loro dinamica per tempi lunghi un importante argomento diricerca. Mentre dal punto di vista classico la situazione, perlomeno in molti case importanti, è ben compresa grazie all'uso del teorema di stabilità di Nekhoroshev e della teoria degli invariantiadiabatici, nel caso quantistico vi è, al contrario, una mancanza di risultati precisi. L'obiettivo di questa tesi è di studiare un sistema superintegrabile quantistico, ottenuto partendo da un corrispettivo classico tramite quantizzazione geometrica, al fine di evidenziare la presenza di indicatori di supertintegrabilità analoghi a quelliche caratterizzano il sistema classico, come la coesistenza di moti regolari e caotici, dovuta all'effetto delle risonanze, in contrapposizione con la regolarità nel regime non risonante. L'analisi è condotta studiando le distribuzioni di Husimi di stati quantistici scelti, con particolare enfasi posta sugli stati stazionari e sugli stati coerenti evoluti. I calcoli sono effettuati sia utilizzando tecniche numeriche che schemi perturbativi. Pur essendo da considerardi questo un lavoro preliminare, il cui compito è di porre le fondamenta per analisi future, i risultati qui ottenuti offrono interessanti spunti sulla dinamica quantistica. Per esempio è mostrato come le risonanze classiche abbiano un chiaro effeto sullo spettro del sistema quantistico, ed inoltre comesia possibile trovare una traccia, nel comportamento quantistico, dell'invarianza adiabatica classica nel regime risonante.

<div>We consider actions of the current Lie algebras $\gl_{n}[t]$ and $\gl_{k}[t]$ on the space $\mathfrak{P}_{kn}$ of polynomials in $kn$ anticommuting variables. The actions depend on parameters $\bar{z}=(z_{1}\lc z_{k})$ and $\bar{\alpha}=(\alpha_{1}\lc\alpha_{n})$, respectively.</div><div>We show that the images of the Bethe algebras $\mathcal{B}_{\bar{\alpha}}^{\langle n \rangle}\subset U(\gl_{n}[t])$ and $\mathcal{B}_{\bar{z}}^{\langle k \rangle}\subset U(\gl_{k}[t])$ under these actions coincide.</div><div></div><div>To prove the statement, we use the Bethe ansatz description of eigenvectors of the Bethe algebras via spaces of quasi-exponentials. We establish an explicit correspondence between the spaces of quasi-exponentials describing eigenvectors of $\mathcal{B}_{\bar{\alpha}}^{\langle n \rangle}$ and the spaces of quasi-exponentials describing eigenvectors of $\mathcal{B}_{\bar{z}}^{\langle k \rangle}$.</div><div></div><div>One particular aspect of the duality of the Bethe algebras is that the Gaudin Hamiltonians exchange with the Dynamical Hamiltonians. We study a similar relation between the trigonometric Gaudin and Dynamical Hamiltonians. In trigonometric Gaudin model, spaces of quasi-exponentials are replaced by spaces of quasi-polynomials. We establish an explicit correspondence between the spaces of quasi-polynomials describing eigenvectors of the trigonometric Gaudin Hamiltonians and the spaces of quasi-exponentials describing eigenvectors of the trigonometric Dynamical Hamiltonians.</div><div></div><div>We also establish the $(\gl_{k},\gl_{n})$-duality for the rational, trigonometric and difference versions of Knizhnik-Zamolodchikov and Dynamical equations.</div>

<p>Holographic duality as a rigorous approach to quantum gravity claims that a quantum gravitational system is exactly equal to a quantum theory without gravity in lower spacetime dimensions living on the boundary of the quantum gravitational system. The duality maps key questions about the emergence of spacetime to questions on the non-gravitational boundary system that are accessible to us theoretically and experimentally. Recently, various aspects of quantum information theory on the boundary theory have been found to be dual to the geometric aspects of the bulk theory. In this thesis, we study the exact and approximate quantum error corrections (QEC) in a general quantum system (von Neumann algebras) focused on QFT and gravity. Moreover, we study entanglement theory in the presence of conserved charges in QFT and the multiparameter multistate generalization of quantum relative entropy.</p>