Orientador: Lucio Centrone / Dissertação (mestrado) - Universidade Estadual de Campinas, Instituto de Matemática Estatística e Computação Científica / Made available in DSpace on 2018-08-26T01:00:17Z (GMT). No. of bitstreams: 1 Manuel_AlexSandroFaria_M.pdf: 1576341 bytes, checksum: 9b70b637adbe03cbed89830126b23521 (MD5) Previous issue date: 2014 / Resumo: Esta dissertação foi escrita com a intenção de conter os seus principais pré-requisitos. Assim, inicialmente, recordaremos algumas definições básicas e alguns resultados da álgebra clássica. Então, listaremos alguns resultados clássicos da teoria de PI-álgebras, bem como alguns resultados sobre codimensões e série de Hilbert. Este último nos dará ferramentas para descrever, pelo menos parcialmente, as identidades polinomiais da álgebra de Grassmann em característica positiva (principalmente a álgebra de Grassmann unitária). No entanto, muitos dos resultados podem funcionar em característica zero. Levaremos em consideração dois casos: no primeiro, o corpo base será considerado infinito (de acordo com um artigo escrito por Giambruno e Koshlukov) enquanto que, no segundo, consideraremos que o corpo base seja finito (de acordo com um artigo escrito por Regev) / Abstract: This dissertation was written with the intent of containing its main prerequisites. So, initially, we will recall some basic definitions and some results from classical algebra. Then we will list some classical results of the theory of PI-algebras as well as the ones about codimensions and Hilbert series. The latter will give us tools to describe, at least partially, the polynomial identities of the Grassmann algebra in positive characteristic (mainly the unitary Grassmann algebra). Nevertheless, many of the results may work in characteristic zero too. We will take in consideration two cases: in the first one the ground field will be considered infinite (according to a paper written by Giambruno and Koshlukov) while in the second one we will consider the ground field to be finite (according to a paper written by Regev) / Mestrado / Matematica / Mestre em Matemática

Este trabalho é dedicado à aplicação de integrais de trajetória de Grassmann para o cálculo de operadores relevantes aos problemas da teoria quântica relativística. Primeiramente uma visão geral detalhada do método é fornecida. Então concentramos nas definições e aplicações das integrais de trajetória sobre as variáveis de Grassmann. Discutimos, em detalhe, um importante papel das integrais de trajetória de Grassmann na representação de propagadores de partículas relativísticas. Derivamos o chamado fatores de spin para tais representações, fazendo as integrações Grasmannianas. Uma contribuição completamente original foi feita aplicando tais integrais ao cálculo de operadores. Derivamos, desta forma, um conjunto de fórmulas novas para as funções de operadores das matrizes y. A aplicações de tais fórmulas são apresentadas. / This work is devoted to an application of Grassmann path integrals to operator calculus relevant to problems of relativistic quantum theory. A detailed survey of path integral method is given first. Then we concentrate ourselves on definitions and applications of path integrals over Grassmann variables. We discuss in detail an important role of Grassmann path integrals in representations of relativistic particle propagators. We derive the so called spin factors for such representations doing Grassmann integrations. A completely original contribution was made in application of such integrals to operator calculus. We have derived in such a way a set of new formulas for operator functions of y-matrices. Applications of such formulas are presented.

Este trabalho é dedicado à aplicação de integrais de trajetória de Grassmann para o cálculo de operadores relevantes aos problemas da teoria quântica relativística. Primeiramente uma visão geral detalhada do método é fornecida. Então concentramos nas definições e aplicações das integrais de trajetória sobre as variáveis de Grassmann. Discutimos, em detalhe, um importante papel das integrais de trajetória de Grassmann na representação de propagadores de partículas relativísticas. Derivamos o chamado fatores de spin para tais representações, fazendo as integrações Grasmannianas. Uma contribuição completamente original foi feita aplicando tais integrais ao cálculo de operadores. Derivamos, desta forma, um conjunto de fórmulas novas para as funções de operadores das matrizes y. A aplicações de tais fórmulas são apresentadas. / This work is devoted to an application of Grassmann path integrals to operator calculus relevant to problems of relativistic quantum theory. A detailed survey of path integral method is given first. Then we concentrate ourselves on definitions and applications of path integrals over Grassmann variables. We discuss in detail an important role of Grassmann path integrals in representations of relativistic particle propagators. We derive the so called spin factors for such representations doing Grassmann integrations. A completely original contribution was made in application of such integrals to operator calculus. We have derived in such a way a set of new formulas for operator functions of y-matrices. Applications of such formulas are presented.

Made available in DSpace on 2016-06-02T20:28:31Z (GMT). No. of bitstreams: 1 6450.pdf: 757192 bytes, checksum: 765b66ca6aed0686ecbcd10c145cefac (MD5) Previous issue date: 2014-11-28 / Financiadora de Estudos e Projetos / In this dissertation we ll present a discussion about the Representations of the Symmetric Group Sn and Alternating Group An. We ll study basics results of the Young s Theory about the representations of the Symmetric Group and discover the decomposition of the algebra FSn in simple subalgebras. After, we ll utilize this decomposition to find the decomposition of the algebra FAn in simple subalgebras. Finally, we ll use this decompositions, together with the PI Theory, for get the sequence of A-codimensions for the Grassmann Algebra (Exterior Algebra) infinitely generated. / Neste trabalho apresentamos uma discussão a respeito das Representações dos Grupos Simétrico Sn e do Grupo Alternante An. Estudaremos resultados básicos da Teoria de Young sobre as representações do grupo simétrico para encontrarmos a decomposição da álgebra de grupo FSn em subálgebras simples. Depois utilizaremos tal decomposição para encontrar a decomposição da álgebra de grupo FAn em subálgebras simples. Por fim empregaremos as informações a respeito das decomposições acima citadas, juntamente com a PI-Teoria, para obter a sequência de A-codimensões para a álgebra de Grassmann (álgebra exterior) infinitamente gerada.

Orientador: Plamen Emilov Koshlukov / Tese (doutorado) - Universidade Estadual de Campinas, Instituto de Matematica, Estatistica e Computação Cientifica / Made available in DSpace on 2018-08-12T22:59:30Z (GMT). No. of bitstreams: 1 Goncalves_DimasJose_D.pdf: 561175 bytes, checksum: 463bf9f78a417a27d1bcf83549bc65a9 (MD5) Previous issue date: 2009 / Resumo: Nesta tese estudamos identidades polinomiais em álgebras associativas. Mais precisamente, estudamos as A-identidades satisfeitas por algumas classes importantes de álgebras. O primeiro resultado principal da tese consiste em uma descrição completa das A-identidades satisfeitas pela álgebra de Grassmann sobre um corpo algebricamente fechado e de característica o. Desta maneira respondemos em afirmativo a uma conjetura devida a Henke e Regev. Em seguida estudamos as A-identidades satisfeitas pela álgebra das matrizes triangulares superiores. Obtemos uma cota inferior para o grau mínimo de uma A-identidade satisfeita por tais álgebras. Como consequência obtemos uma resposta negativa a uma outra conjetura de Henke e Regev. Além disso, descrevemos as A-identidades de grau 5, da álgebra das matrizes triangulares superiores de ordem 2, e obtemos os graus mínimos de A-identidades satisfeitas por tais álgebras de ordem 3 e 4. / Abstract: In this PhD thesis we study polynomial identities in associative algebras. More precisely we study the A-ideIltities for several important classes of algebras. The first main result of the thesis gives a complete description of the A-identities for the Grassmann algebra over an algebraically closed field of characteristic O. In this way we give a positive answer to a conjecture due to Henke and Regev. Afterwards we study A-identities for the upper triangular matrix algebras. We give a lower bound for the minimal degree of an A-identity satisfied by such algebras. As a corollary we give a negative answer to another conjecture due to Henke and Regev. Furthermore we describe the A-identities of degree 5 for the upper triangular matrices of order 2 and compute the minimal degree of an A-identity for such algebras of order 3 and 4. / Doutorado / Algebra / Doutor em Matemática

Orientador: Plamen Emilov Kochloukov / Tese (doutorado) - Universidade Estadual de Campinas, Instituto de Matemática, Estatística e Computação Científica / Made available in DSpace on 2018-08-19T21:39:41Z (GMT). No. of bitstreams: 1 Mello_ThiagoCastilhode_D.pdf: 1364753 bytes, checksum: 66955ce4a4c6b84e5c6dcc1a414f3f24 (MD5) Previous issue date: 2012 / Resumo: Nesta tese estudamos a álgebra genérica de M1;1 em dois geradores sobre um corpo infinito de característica diferente de 2. Descrevemos o centro desta álgebra e provamos que este é a soma direta do corpo com um ideal nilpotente da álgebra. Como consequência mostramos que este centro contém elementos não escalares, respondendo a uma pergunta feita por Berele. Em característica zero, estudamos também as identidades polinomiais de tal álgebra genérica e exibimos uma base finita para seu T-ideal, utilizando a descrição do seu centro e os resultados de Popov sobre as identidades de M1;1 em característica zero. Segue que tal base é formada pelos polin^omios [x1; x2][x3; x4][x5; x6], [[x1; x2][x3; x4]; x5] e s4, a identidade polinomial standard de grau 4. Por fim, utilizando ideias e resultados de Nikolaev sobre as identidades em duas variáveis de M2(K) em característica zero, mostramos que todas as identidades polinomiais em duas variáveis de M1;1 são consequências das identidades [[x1; x2]2; x1] e [x1; x2]³ / Abstract: In this thesis, we study the generic algebra of M1;1 in two generators over an infinite field of characteristic different from 2. We describe the centre of this algebra and prove that this centre is a direct sum of the field and a nilpotent ideal of the algebra. As a consequence, we show that such centre contains nonscalar elements and thus we answer a question posed by Berele. In characteristic zero we also study the identities of this generic algebra and find a finite basis for its ideal of identities using the description of its centre and the results of Popov, about the identities of M1;1 in characteristic zero. It follows that such a basis is formed by the polynomials [x1; x2][x3; x4][x5; x6], [[x1; x2][x3; x4]; x5] and by s4, the standard identity of degree four. Finally, using ideas and results of Nikolaev about the identities in two variables of M2(K) in characteristic zero, we show that the polynomial identities in two variables of M1;1 follow from [[x1; x2]2; x1] and [x1; x2]³ / Doutorado / Matematica / Doutor em Matemática

Im Rahmen des Funktionalintegralzugangs zur Quanteneichfeldtheorie wird in der vorliegenden Arbeit eine Quantisierungsprozedur in Termen eichinvarianter Felder vorgeschlagen und am Beispiel zwei- und vierdimensionaler abelscher Modelle (Thirring-Modell und QED) sowie der One-Flavour QCD konkret realisiert. Dazu wird die Algebra der aus der eichabhängigen Feldkonfiguration der zugrunde liegenden Quantenfeldtheorie gebildeten eichinvarianten Grassmann-Algebra-wertigen Differentialformen, welche die Struktur einer Z_2-graduierten Differentialalgebra trägt, näher untersucht. Danach erfolgt die Implementierung eines geeignet gewählten Satzes eichinvarianter Felder sowie bestimmter algebraischer Relationen in das Funktionalintegral, wodurch die ursprüngliche eichabhängige Feldkonfiguration ausintegriert werden kann. Diese als "Reduktion des Funktionalintegrals" bezeichnete Prozedur führt schließlich auf eine effektive bosonisierte (Quanten-) Theorie wechselwirkender eichinvarianter, und damit physikalischer Felder. Die vorgestellte Prozedur kann als allgemeines Bosonisierungsschema für Quantenfeldtheorien in beliebigen Raum-Zeit-Dimensionen angesehen werden. Die physikalische Auswertung der erhaltenen effektiven Theorien wird am Beispiel der Berechnung der chiralen Anomalie sowie bestimmter Vakuum-Erwartungswerte im Rahmen der untersuchten abelschen Modelle demonstriert. Wie sich dabei zeigt, wird man mit einer Reihe neuartiger Phänomene und Probleme konfrontiert, die bei geeigneter Behandlung tiefere Einblicke in nichtperturbative Fragestellungen erlauben. / Within the thesis a new procedure, called "reduction of the functional integral", is developed for formulating quantum field theories in terms of gauge invariant quantities (physical observable fields). It provides a new way for the construction and analysis of effective field theoretical models. Starting with a detailed mathematical analysis of the algebra of Grassmann--algebra valued gauge invariants, the procedure is applied to the two--dimensional Thirring--model, the four--dimensional spinor QED and the one--flavor QCD in four dimensions. For each of these three models an effective theory of interacting bosonic gauge invariant fields was deduced on the quantum level. Apart from this more theoretical considerations, first steps on the way to an analysis of the obtained effective models towards their application in various physical problems are performed. In the case of the two Abelian models a new approach to the bosonisation scheme and the calculation of the chiral anomaly in two and four dimensions were obtained, giving some deeper insight into the nature of the bosonisation phenomenon as well as the nature of anomalies, respectively. Moreover, the investigation of current--current expectation values shows that the suggested procedure can be viewed as a new way towards a non--perturbative formulation and understanding of quantum field theories.

L’étude réalisée dans cette thèse porte sur les phénomènes critiques classiques et quantiques. En effet, les phénomènes critiques et les transitions de phases sont devenus des sujets fondamentaux en physique statistique moderne et en théorie des champs et nous proposons dans cette thèse d’étudier certains modèles qui présentent un comportement critique, à la fois à l’équilibre et hors de l’équilibre. Dans la première partie de la thèse, certaines propriétés du modèle de dimères à deux dimensions sont étudiées. Ce modèle a été largement étudié dans les communautés de physique statistique et de mathématiques et un grand nombre d’applications en physique de la matière condensée existent. Ici, nous proposons de mettre l’accent sur des solutions exactes du modèle et d’utiliser l’invariance conforme afin d’avoir une compréhension profonde de ce modèle en présence de monomères et/ou en présence de bords. Les mêmes types d’outils sont ensuite utilisés pour explorer un autre phénomène important apparaissant dans les modèles de dimères et de chaînes de spin : le cercle arctique. Le but étant de trouver une description adéquate en termes de théorie des champs de ce phénomène, en utilisant des calculs exacts ainsi que de l’analyse asymptotique. La deuxième partie de la thèse concerne les phénomènes critiques hors de l’équilibre dans le contexte des modèles de croissance d’interfaces. Ce domaine de recherche est très important de nos jours, principalement en raison de la découverte de l’équation Kardar-Parisi-Zhang et de ses relations avec les ensembles de matrices aléatoires. La phénoménologie de ces modèles en présence des bords est analysée via des solutions exactes et des simulations numériques, on montre alors que des comportements surprenants apparaissent proches des bords / The study carried in this thesis concerns classical and quantum critical phenomena. Indeed, critical behaviors and phase transitions are fundamental topics in modern statistical physics and field theory and we propose in this thesis to study some models which exhibit such behaviors both at equilibrium and out of equilibrium. In the first part of the thesis, some properties of the two-dimensional dimer model are studied. This model has been studied extensively in the statistical physics and mathematical communities and a lot of applications in condensed matter physics exist. Here we propose to focus on exact solutions of the model and conformal invariance in order to have a deep understanding of this model in presence of monomers, and/or boundaries. The same kind of tools are then used to explore another important phenomenon appearing in dimer models and spin chains: the arctic circle. The goal was to find a proper field theoretical description of this phenomenon using exact solutions and asymptotic analysis. The second part of the thesis concerns out of equilibrium critical phenomena in the context of interface growth models. This field of research is very important nowadays, mainly because of the Kardar-Parisi-Zhang equation and its relations with random matrix ensembles. The phenomenology of these models in presence of boundaries is studied via exact solutions and numerical simulations, we show that surprising behaviors appear close to the boundaries

Im Rahmen des Funktionalintegralzugangs zur Quanteneichfeldtheorie wird in der vorliegenden Arbeit eine Quantisierungsprozedur in Termen eichinvarianter Felder vorgeschlagen und am Beispiel zwei- und vierdimensionaler abelscher Modelle (Thirring-Modell und QED) sowie der One-Flavour QCD konkret realisiert. Dazu wird die Algebra der aus der eichabhängigen Feldkonfiguration der zugrunde liegenden Quantenfeldtheorie gebildeten eichinvarianten Grassmann-Algebra-wertigen Differentialformen, welche die Struktur einer Z_2-graduierten Differentialalgebra trägt, näher untersucht. Danach erfolgt die Implementierung eines geeignet gewählten Satzes eichinvarianter Felder sowie bestimmter algebraischer Relationen in das Funktionalintegral, wodurch die ursprüngliche eichabhängige Feldkonfiguration ausintegriert werden kann. Diese als "Reduktion des Funktionalintegrals" bezeichnete Prozedur führt schließlich auf eine effektive bosonisierte (Quanten-) Theorie wechselwirkender eichinvarianter, und damit physikalischer Felder. Die vorgestellte Prozedur kann als allgemeines Bosonisierungsschema für Quantenfeldtheorien in beliebigen Raum-Zeit-Dimensionen angesehen werden. Die physikalische Auswertung der erhaltenen effektiven Theorien wird am Beispiel der Berechnung der chiralen Anomalie sowie bestimmter Vakuum-Erwartungswerte im Rahmen der untersuchten abelschen Modelle demonstriert. Wie sich dabei zeigt, wird man mit einer Reihe neuartiger Phänomene und Probleme konfrontiert, die bei geeigneter Behandlung tiefere Einblicke in nichtperturbative Fragestellungen erlauben. / Within the thesis a new procedure, called "reduction of the functional integral", is developed for formulating quantum field theories in terms of gauge invariant quantities (physical observable fields). It provides a new way for the construction and analysis of effective field theoretical models. Starting with a detailed mathematical analysis of the algebra of Grassmann--algebra valued gauge invariants, the procedure is applied to the two--dimensional Thirring--model, the four--dimensional spinor QED and the one--flavor QCD in four dimensions. For each of these three models an effective theory of interacting bosonic gauge invariant fields was deduced on the quantum level. Apart from this more theoretical considerations, first steps on the way to an analysis of the obtained effective models towards their application in various physical problems are performed. In the case of the two Abelian models a new approach to the bosonisation scheme and the calculation of the chiral anomaly in two and four dimensions were obtained, giving some deeper insight into the nature of the bosonisation phenomenon as well as the nature of anomalies, respectively. Moreover, the investigation of current--current expectation values shows that the suggested procedure can be viewed as a new way towards a non--perturbative formulation and understanding of quantum field theories.