O principal objetivo do presente trabalho é expor o procedimento de quantização de teorias de Yang-Mills, através do método de Faddeev-Popov, no formalismo de 1a Ordem, e investigar num primeiro momento sua equivalência (clássica e quântica) ao formalismo usual (2a Ordem) e algumas de suas aplicações, principalmente no cálculo de correções quânticas. Para isso, ideias gerais a respeito do processo de quantização via formalismo de Faddeev-Popov foram expostas, e posteriormente utilizadas no processo de quantização de teorias de Yang-Mills no formalismo de 1a Ordem. Apresenta-se também as ideias gerais relativas ao método de regularização dimensional utilizado no cálculo de correções quânticas à nível de 1-loop para a teoria de Yang-Mills no formalismo de 1a ordem, utilizando-se, para isso, computação simbólica. Foi demonstrado que via formalismo de 1a Ordem, a estrutura ultravioleta encontrada no propagador do bóson de gauge é consistente com a renormalizabilidade da teoria. Embora tenhamos diferenças quanto a estrutura das interações neste novo formalismo, a estrutura das divergências ultravioletas continua a mesma do formalismo usual. / The main objective of the present work is to expose the quantization procedure of Yang- Mills theories in first order formalism, by Faddeev Popov\'s method. We want to investigate the classical and quantum equivalence between first and second order formalism, and look and analyze the differences in practical calculations of quantum corrections. Therefore, the general ideas about quantizantion by Faddeev-Popov\'s method was exposed, and used later in first order theory. It is also presented in this work, the main ideas concerning to dimensional regularization used in quantum corrections calculations at one-loop order for Yang-Mills theories, using for that, symbolic computation. It has been shown that upon using the first order formalism, the ultraviolet structre found in gauge boson propagator is also consistent to the theory\'s renormalizability. Although we have differences concerning to interactions structures in this new formalism, the ultraviolet structures from usual formalism is also found in it.

In this thesis we review some aspects of Yang-Baxter deformations of semi-symmetric space sigma models. We start by giving a short review of the sigma model description of superstrings and then we offer a self contained introduction to the Yang-Baxter deformation technique. We then show how to obtain an integrable deformation of the hybrid sigma model. Also, we show that the gravity dual of beta-deformed ABJM theory can be obtained as a Yang-Baxter deformation. This is done by selecting a convenient combination of Cartan generators in order to construct an Abelian r-matrix satisfying the classical Yang-Baxter equation. / Nesta tese revisamos alguns aspectos das deformações de Yang-Baxter de modelos sigma em espaços semi-simétricos. Damos uma breve revisão do modelo sigma de supercordas e, em seguida, oferecemos uma introdução ao método de deformação de Yang-Baxter. Em seguida, mostramos como obter uma deformação integrável do modelo sigma híbrido. Além disso, mostramos que o dual gravitacional da teoria ABJM beta-deformada pode ser obtida como uma deformação de Yang-Baxter. Isso é feito selecionando-se uma combinação conveniente de geradores de Cartan para construir uma matriz r Abeliana satisfazendo a equação clássica de Yang-Baxter.

Les facteurs de forme et les fonctions de corrélation déterminent les quantités dynamiques mesurables associées aux modèles de théorie des champs et de mécanique statistique. Dans le cas de modèles intégrables en dimension 2, au-delà des propriétés du spectre ou de la fonction de partition, un des grands défis actuels concerne le calcul exact des facteurs de forme et des fonctions de corrélation.Le but de cette thèse est de développer une approche permettant de résoudre ce problème dans le cadre de la méthode de séparation des variables quantique de Skyanin. Cette méthode généralise au cas quantique et pour des systèmes avec un grand nombre de degrés de liberté la méthode de Hamilton-Jacobi en mécanique analytique. Le Hamiltonien est exprimé avec des opérateurs séparés, son spectre et ses états propres caractérisés par un système d'équations de Baxter résultant des structures algébriques de Yang-Baxter, caractéristiques de l'intégrabilité de ces modèles.Cette thèse a permis, pour les modèles de sine-Gordon (théorie des champs quantique) et de Potts chiral (modèle de physique statistique), le calcul des produits scalaires entre états propres du Hamiltonien, la résolution du problème inverse, i. e. l'expression des opérateurs du modèle en termes des variables séparées, ainsi que le calcul en termes de déterminants des facteurs de forme, i. e. des éléments de matrice des opérateurs locaux du modèle dans la base propre du Hamiltonien, ce qui constitue un pas important vers le calcul des fonctions de corrélation de ces modèles. / Form factors and correlation functions determine the measurable dynamic quantities that are associated with field theories and statistical physics models. In the case of 2-dimensional integrable models, one of the main challenges beyond spectrum properties and partition function is the exact computation of form factors and correlation functions.The aim of this thesis is to develop an approach in the framework of Sklyanin's separation of variables to address this problem. This framework generalizes to the quantum case and for systems with many degrees of freedom the Hamilton-Jacobi method from analytical mechanics. The Hamiltonian is expressed in terms of separated operators, its spectrum and eigenvectors are characterized by a system of Baxter equations. These Baxter equations are a consequence of Yang-Baxter relations that are characteristic of these models being integrable.The result of this thesis is, in the case of the sine-Gordon model (quantum field theory) and of the chiral Potts model (statistical physics model), the computation of scalar products of Hamiltonian eigenstates, the resolution of the inverse problem (expressing the model operators in terms of separated variables) and the computation in terms of determinant of form factors (the matrix elements of the model local operators in the Hamiltonian eigenbasis), which is an important step towards the computation of the correlation functions of these models.

Σκοπός της παρούσας διατριβής είναι η κατασκευή και μελέτη συνολοθεωρητικών λύσεων της κβαντικής εξίσωσης Yang-Baxter (απεικονίσεις Yang-Baxter) και η συσχέτισή τους με την ολοκληρωσιμότητα διακριτών δυναμικών συστημάτων. Οι κατασκευές απεικονίσεων Yang-Baxter που προτείνονται προέρχονται από την αναπαραγοντοποίηση ισχυρών ζευγών Lax εξαρτώμενων από μια φασματική παράμετρο. Οι αντίστοιχοι πίνακες Lax προκύπτουν από την συμπλεκτική εμφύλλωση διωνυμικών πινάκων εφοδιασμένων με μια κατάλληλη δομή Poisson (αγκύλη Sklyanin). Στην περίπτωση των 2x2 πινάκων Lax, οι αντίστοιχες απεικονίσεις είναι συμπλεκτικές, τετράρητες και ταξινομούνται με βάση τον μεγιστοβάθμιο όρο του πίνακα Lax ως προς την ισοδυναμία απεικονίσεων Yang-Baxter. Εκφυλισμένες απεικονίσεις Yang-Baxter, οι οποίες σχετίζονται με γνωστές ολοκληρώσιμες εξισώσεις, προκύπτουν από όρια των τετράρητων (μη-εκφυλισμένων). Η σύνδεση μεταξύ απεικονίσεων Yang-Baxter και ολοκληρωσιμότητας επιτυγχάνεται θεωρώντας περιοδικά προβλήματα αρχικών τιμών σε δισδιάστατα πλέγματα. Σε κάθε απεικόνιση Yang-Baxter αντιστοιχεί μια οικογένεια αντιμεταθετικών απεικονίσεων μεταφοράς στο πλέγμα (transfer maps) που διατηρούν αναλλοίωτο το φάσμα του μονόδρομου πίνακά τους. Η αγκύλη Sklyanin εξασφαλίζει την ενέλιξη των ολοκληρωμάτων που προκύπτουν από το φάσμα του μονόδρομου πίνακα. Κατά αυτόν τον τρόπο από τις συμπλεκτικές απεικονίσεις Yang-Baxter που κατασκευάσαμε παράγονται ολοκληρώσιμες απεικονίσεις μεταφοράς. Τέλος, η μελέτη μας επεκτείνεται σε συστήματα πεπλεγμένων απεικονίσεων Yang-Baxter (entwining Yang-Baxter maps) . / The purpose of this thesis is the construction and the study of set theoretical solutions of the quantum Yang-Baxter equation (Yang-Baxter maps) and the connection with the integrability of discrete integrable systems. The constructions that we present are derived from the re-factorization of strong Lax pairs depending on a spectral parameter. The corresponding Lax matrices are obtained from the symplectic foliation of binomial matrices equipped with an appropriate Poisson bracket (Sklyanin bracket). In the case of 2x2 binomial Lax matrices, the corresponding maps are symplectic, quadrirational and can be classified with respect to the Yang-Baxter equivalence. Degenerate Yang-baxter maps constructed as limits of the quadrirational maps, are connected to known integrable equations. The connection between Yang-Baxter maps and integrability is achieved by considering periodic initial value problems on two dimensional lattices. For any Yang-Baxter map that admits a Lax matrix, there is a family of commuting transfer maps which preserve the spectrum of their monodromy matrix. The Skllyanin bracket ensures that the integrals obtained from the spectrum of the monodromy matrix are in involution. In this way, integrable transfer maps are generated from the symplectic Yang-Baxter maps that we constructed. Finally, our study is extended for systems of entwining Yang-Baxter maps.

Made available in DSpace on 2016-06-02T20:16:47Z (GMT). No. of bitstreams: 1 3474.pdf: 452458 bytes, checksum: 7857dc28822e6d45234586ddd2b5e98e (MD5) Previous issue date: 2011-03-02 / Universidade Federal de Minas Gerais / In this work we study the solutions of the Yang-Baxter equation associated to nineteen vertex models invariant by the parity-time symmetry from the perspective of algebraic geometry. We determine the form of the algebraic curves constraining the respective Boltzmann weights and found that they possess a universal structure. This allows us to classify the integrable manifolds in four different families reproducing three known models besides uncovering a novel nineteen vertex model in a unified way. The introduction of the spectral parameter on the weights is made via the parameterization of the fundamental algebraic curve which is a conic. The diagonalization of the transfer matrix of the new vertex model and its thermodynamic limit properties are discussed. We point out a connection between the form of the main curve and the nature of the excitations of the corresponding spin-1 chains. / Nesta dissertação estudamos as possíveis soluções da equação de Yang-Baxter para modelos de dezenove vértices invariantes por simetria de paridade e reversão temporal do ponto de vista da geometria algébrica. Determinamos a forma das curvas algébricas que vinculam os respectivos pesos de Boltzmann e descobrimos que suas estruturas são universais. Com tal observação foi possível classificar, de uma maneira unificada, as variedades algébricas integráveis em quatro diferentes famílias, três delas já conhecidas e uma delas correspondendo a um novo modelo de dezenove vértices. A introdução de um parâmetro espectral nos pesos de Boltzmann é feita através da parametrização da curva algébrica fundamental, que é uma crônica. A diagonalização da matriz de transferência do novo modelo de vértices bem como suas propriedades no limite termodinâmico são discutidas. Mencionamos ainda uma curiosa conexão entre a forma da curva principal e a natureza das excitações das Hamiltonianas de spin-1 associadas aos modelos de vértices.

Les principaux outils pour la compréhension du comportement macroscopique desystèmes quantiques à partir de leur description microscopique sont la déterminationdu spectre du Hamiltonien associé et le calcul des fonctions de corrélation. Cettethèse se place dans le cadre du développement d’un tel programme de recherche afind’étudier des systèmes intégrables quantiques avec des conditions aux bordsintégrables générales, le but à long terme étant la description exacte d’une physiquequantique hors équilibre.Plus spécifiquement, nous avons analysé la classe des systèmes intégrablesquantiques sur réseau associés aux représentations cycliques de l’algèbre de réflexionà 6-vertex, avec comme exemples les modèles de sine Gordon et de Potts chiral avecconditions aux bords intégrables.Une large partie du travail a été consacrée au développement de la méthode deséparation quantique des variables pour résoudre le problème spectral de la matricede transfert de ces modèles avec conditions de bords intégrables les plus générales,en étendant l’idée des transformations de jauge de Baxter à ces algèbres de réflexion.Nous avons caractérisé complètement le spectre de la matrice de transfert (valeurspropres et vecteurs propres) en termes des solutions d’un système discret d’équationspolynomiales et d’une façon équivalente en termes des solutions, dans une certaineclasse de fonctions, d’une équation de type Baxter fonctionnelle. Cela permet de fairele lien dans certains cas particuliers avec la méthode de l’anstaz de Bethe algébriquequi ne permet pas d’étudier ces modèles en toute généralité.Nous avons ensuite construit des familles de nouveaux Hamiltoniens locaux avecconditions aux bords intégrables qui commutent avec la matrice de transfert. Pour cefaire nous avons défini une hiérarchie de nouvelles équations de réflexion mélangeantdifférentes représentations de l’algèbre quantique à 6-vertex et utilisant entre autres,la matrice R fondamentale cyclique. / The main theoretical tools to understand the macroscopic behaviour of quantumsystems from their microscopic description are the determination of theirHamiltonian spectrum and the computation of their correlation functions. This thesistakes place in the development of such a research program to study quantumintegrable models with general integrable boundary conditions, the long-range goalbeing to be able to exactly describe out of equilibrium physics.More specifically, we have analysed the class of integrable quantum models on thelattice associated to cyclic representations of the 6-vertex reflection algebra,including as particular cases the lattice sine- Gordon model at root of unity and thechiral Potts model with general integrable boundaries.A large part of the work has been devoted to the development of the quantumseparation of variables method to solve the spectral problem for these models withgeneral integrable boundary conditions, by generalising the Baxter’s gaugetransformations to these cyclic reflection algebras.We have completely characterised the transfer matrix spectrum (both eigenvaluesand eigenstates) in terms of the set of solutions to a discrete system of polynomialequations and equivalently as the set of solutions, in a given class of functions, to aBaxter like functional equation. This last point allows in particular cases to make alink with the Algebraic Bethe Ansatz approach, which in general, cannot be used forthe study of these models.We have then constructed families of new local Hamiltonians with integrableboundaries commuting with the above transfer matrix. To that end, we have defined ahierarchy of new mixed reflection equations, involving different representations ofthe 6-vertex algebra and using, among others, the fundamental R-matrix.

Made available in DSpace on 2014-07-29T16:02:20Z (GMT). No. of bitstreams: 1 Rosane Gomes Pereira.pdf: 525845 bytes, checksum: 76abe0b472d0e4b44a4d3197912958d3 (MD5) Previous issue date: 2012-03-09 / In this work, we study eigenvalues of polyharmonic operators on compact Riemannian manifolds with boundary (possibly empty). Here, we bring in a universal inequality for the eigenvalues of the polyharmonic operator on compact domains in an Euclidean space Rn. This inequality controls the kth eigenvalue by the lower eigenvalues, independently of the particular geometry of the domain. Besides, a inequality we present covers the important Yang inequality on eigenvalues of the Dirichlet Laplacian. Finally, we introduce universal inequalities for eigenvalues of polyharmonic operator on compact domains in a unit n-sphere Sn. NOTE: Programs do not copy or copy errors with certain symbols, formulas, formatting, etc ..., n of Rn and Sn are overwritten. View all content by clicking pdf - dissertation at the bottom of the screen. / Neste trabalho, estudamos autovalores do operador poli-harmônico em variedades Riemannianas compactas com fronteira ( possivelmente vazia ). Aqui, apresentamos uma desigualdade universal para os autovalores do operador poliharmônico em domínios compactos no Espaço Euclidiano Rn. Esta desigualdade controla o k-ésimo autovalor pelos autovalores menores, independentemente da geometria particular do domínio. Além disso, a desigualdade que apresentamos cobre a importante desigualdade de Yang em autovalores do Laplaciano de Dirichlet. Finalmente, apresentamos desigualdades universais para autovalores do operador poli-harmônico em domínios compactos na esfera unitária n- dimensional Sn. OBS: Programas não copiam ou copiam com erros certos símbolos, fórmulas, formatações etc..., o n de Rn e Sn está sobrescrito. Visualize todo conteúdo clicando pdf - dissertação na parte de baixo da tela.

In this thesis we review some aspects of Yang-Baxter deformations of semi-symmetric space sigma models. We start by giving a short review of the sigma model description of superstrings and then we offer a self contained introduction to the Yang-Baxter deformation technique. We then show how to obtain an integrable deformation of the hybrid sigma model. Also, we show that the gravity dual of beta-deformed ABJM theory can be obtained as a Yang-Baxter deformation. This is done by selecting a convenient combination of Cartan generators in order to construct an Abelian r-matrix satisfying the classical Yang-Baxter equation. / Nesta tese revisamos alguns aspectos das deformações de Yang-Baxter de modelos sigma em espaços semi-simétricos. Damos uma breve revisão do modelo sigma de supercordas e, em seguida, oferecemos uma introdução ao método de deformação de Yang-Baxter. Em seguida, mostramos como obter uma deformação integrável do modelo sigma híbrido. Além disso, mostramos que o dual gravitacional da teoria ABJM beta-deformada pode ser obtida como uma deformação de Yang-Baxter. Isso é feito selecionando-se uma combinação conveniente de geradores de Cartan para construir uma matriz r Abeliana satisfazendo a equação clássica de Yang-Baxter.

CoordenaÃÃo de AperfeiÃoamento de Pessoal de NÃvel Superior / Pesquisa sobre Bioenergia Oriental e suas possÃveis relaÃÃes com os cursos superiores de EducaÃÃo FÃsica. A tese busca a inclusÃo de conceitos relacionados à visÃo oriental da Bioenergia nos currÃculos dos cursos superiores de EducaÃÃo FÃsica, buscando contribuir com a percepÃÃo de corpo/mente como indissociÃveis. A introduÃÃo apresenta um breve percurso pessoal de envolvimento do autor com o tema. O capÃtulo I fornece esclarecimentos acerca das diferenÃas iniciais sobre Bioenergia Ocidental e Oriental, para entÃo ingressar em anÃlise epistemolÃgica ocidental. Esta anÃlise trata do nascimento da ciÃncia grega, problemas da medievalidade, do mÃtodo cartesiano com Ãnfase no divÃrcio entre corpo e mente, e de alguns sÃrios problemas ainda nÃo resolvidos: os problemas da hipÃtese, do controle das variÃveis e da induÃÃo. O capÃtulo se finda com possibilidades curriculares da EducaÃÃo FÃsica. O capÃtulo II apresenta a epistemologia oriental, levantando as noÃÃes sobre a ciÃncia do Extremo Oriente, e abordando os conceitos de Tao, Yin e Yang, para posteriormente ingressar no cerne da Bioenergia Oriental: o conceito de Ki. A partir do Ki, aborda-se o teste manual do cÃrculo de energia, que à uma forma direta de analisar diminuiÃÃes locais de Ki; explica-se a Teoria dos Meridianos e suas possibilidades de atuaÃÃo prÃtica, visto que a mesma serve de base para diversas terapias, como a acupuntura e shiatsu; e apresenta-se as relaÃÃes da Bioenergia com a respiraÃÃo, como as realizadas na arte-marcial Hapkido, uma respiraÃÃo que estimula o Ki, aquecendo e energizando o organismo. Posteriormente sÃo apresentados os estudos realizados com a energia Hado. Por fim, apresenta dificuldades epistemolÃgicas e algumas possibilidades para a EducaÃÃo FÃsica em busca desta conexÃo perdida entre corpo-mente.

Neste trabalho estabelecemos o Teorema Central do Limite para o modelo O(N) de Heisenberg hierárquico na criticalidade via equação a derivadas parciais no limite N -> infinito. Por simplicidade consideramos apenas o caso d = 4, sendo o teorema também válido para d > 4. Pelo estudo de uma dada equação a derivadas parciais (EDP) determinamos a temperatura inversa crítica do modelo esférico hierárquico contínuo para um d > 2 qualquer, havendo conexão entre criticalidade e o ponto fixo da EDP. Por meio de uma análise geométrica da trajetória crítica obtemos informações sobre a dinâmica e distribuição dos zeros de Lee-Yang. / In this work we stablish the Central Limit Theorem for the hierarchical O(N) Heisenberg model at criticality via partial differential equation in the limit N -> infinity. For simplicity we only treat the d = 4 case but the theorem is still valid for d > 4. By studying a given partial differential equation (PDE) we determine for any d > 2 the critical inverse temperature of the continuum hierarchical spherical model, and we show a connection between criticality and the fixed point of PDE. By means of a geometric analysis of the critical trajectory we obtain some informations about Lee-Yang zeros´s dynamics and distribution.