Submitted by Renata Lopes (renatasil82@gmail.com) on 2016-12-22T13:25:11Z No. of bitstreams: 1 vahidnikoofard.pdf: 794368 bytes, checksum: 26feb8d15d2757f79f8d7dfb69b610c2 (MD5) / Approved for entry into archive by Diamantino Mayra (mayra.diamantino@ufjf.edu.br) on 2017-01-31T10:20:21Z (GMT) No. of bitstreams: 1 vahidnikoofard.pdf: 794368 bytes, checksum: 26feb8d15d2757f79f8d7dfb69b610c2 (MD5) / Made available in DSpace on 2017-01-31T10:20:21Z (GMT). No. of bitstreams: 1 vahidnikoofard.pdf: 794368 bytes, checksum: 26feb8d15d2757f79f8d7dfb69b610c2 (MD5) Previous issue date: 2015-06-30 / CAPES - Coordenação de Aperfeiçoamento de Pessoal de Nível Superior / Esta tese é composta por assuntos distintos entre si de teorias quânticas de campos onde alguns deles são descritos em espaços não-comutativos (NC). Em primeiro lugar, analisamos a dinâmica de uma partícula livre sobre uma 2-esfera e através da dinâmica das suas equações de movimento, obtivemos as perturbações NCs neste espaço de fase. Este modelo sugere uma origem para o Zitterbewegung do elétron. Depois disso, consideramos umaversãoNCdasegundaleideNewtonparaestemodelo, quefoiobtidocomestecenário geométrico aplicado a este modelo. Em seguida, discutimos um formalismo alternativo relacionado à não-comutatividade chamado DFR onde o parâmetro NC é considerado uma coordenada e demonstramos exatamente que ela tem obrigatoriamente um momento conjugado neste espaço de fase DFR, diferentemente do que alguns autores da atual literatura sobre DFR afirmam. No próximo assunto, usando o formalismo de solda que, em poucaspalavras,colocapartículascomquiralidadesopostasnomesmomultipleto,soldamos algumas versões NCs de modelos bem conhecidos como modelos de Schwinger quirais e modelos (anti) auto duais no espaço-tempo de Minkowski estendido. Em outro assunto estudado aqui, também construímos a versão NC do modelo de Jackiw-Pi com um grupo de calibre arbitrário e usamos o mapeamento bem conhecido de Seiberg-Witten para obter este modelo NC em termos de variáveis comutativos. Finalmente, utilizamos o formalismo de campos e anticampos (ou método BV) para construir a ação de Batalin-Vilkovisky (BV) do modelo Jackiw-Pi estendido e após o prEntendiocedimento de fixação de calibre chegamos a uma ação completa, pronta para quantização. / This thesis is composed of distinct aspects of quantum field theories where some of them are described in noncommutative (NC) spaces. Firstly, we have analyzed the dynamics of a free particle over a 2-sphere and through the dynamics of the equations of motion we have derived its NC perturbations in the phase-space. This model suggests an origin for Zitterbewegung feature of the electron. After that we have considered the NC version of Newton’s second law for this model, which was obtained with the geometricalscenarioappliedtothismodel. Thenwehavediscussedtheso-calledDoplicher– Fredenhagen–Roberts (DFR) alternative formalism concerning noncommutativity where the NC parameter has a coordinate role and we showed exactly that it has a conjugated momentum in the DFR phase-space, differently of what some authors of the current DFR-literature claims. In the next issue, using the soldering formalism which, in few words, put opposite chiral particles in the same multiplet, we have soldered some NC versions of well known models like the chiral Schwinger model and (anti)self dual models in the extended Minkowski spacetime. Changing the subject, we have constructed the NC spacetime version of Jackiw-Pi model with an arbitrary gauge group and we used the well known Seiberg-Witten map to obtain the NC model expressed in terms of commutative variables. Finally, we have used the field-antifield (or BV method) formalism to construct the Batalin-Vilkovisky (BV) action of the extended Jackiw-Pi model and after the gauge fixing procedure we have arrived at a quantized-ready action for this model.

We initiate the study of a new class of semigroup C*-algebras arising from number-theoretic considerations; namely, we generalize the construction of Cuntz, Deninger, and Laca by considering the left regular C*-algebras of ax+b-semigroups from actions of congruence monoids on rings of algebraic integers in number fields. Our motivation for considering actions of congruence monoids comes from class field theory and work on Bost–Connes type systems. We give two presentations and a groupoid model for these algebras, and establish a faithfulness criterion for their representations. We then explicitly compute the primitive ideal space, give a semigroup crossed product description of the boundary quotient, and prove that the construction is functorial in the appropriate sense. These C*-algebras carry canonical time evolutions, so that our construction also produces a new class of C*-dynamical systems. We classify the KMS (equilibrium) states for this canonical time evolution, and show that there are several phase transitions whose complexity depends on properties of a generalized ideal class group. We compute the type of all high temperature KMS states, and consider several related C*-dynamical systems. / Graduate

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Cette thèse s'intéresse à des familles de fonctions zêta spectrales (séries de Dirichlet) qui peuvent être associées à certaines algèbres d'opérateurs sur des espaces de Hilbert. Dans ce mémoire, la principale question étudiée sur ces fonctions zêta est l'existence d'un prolongement méromorphe à partir d'un demi-plan ouvert du plan complexe au plan complexe tout entier. Généralisant une idée de Nigel Higson, on propose dans la partie I, une méthode pour prouver l'existence de ce prolongement méromorphe pour certains fonction zêta spectrales. Cette méthode s’effectue dans le cadre d'algèbres d'opérateurs différentiels généralisés et elle s'appuie sur une suite de réduction. Le théorème principal donne, sous certaines conditions, l'existence d'un prolongement méromorphe, une localisation des pôles dans les supports de suites arithmétiques et une borne supérieure pour l'ordre de ces pôles. Dans la partie II, on reformule la méthode de la partie I dans le contexte et avec le vocabulaire des triplets spectraux de Connes et Moscovici. Dans la troisième partie, on donne une application pour des fonctions zêta associées à des opérateurs de type Laplace sur des variétés lisses, compactes et sans bord. Cet exemple a été initialement traité par Nigel Higson avec cette approche en 2006. Une deuxième application traite de fonctions zêta associées au tore non commutatif. Dans la partie IV, on utilise le calcul pseudodifférentiel associé à des algèbres de Lie nilpotentes et développé par Dominique Manchon, pour construire de nouveaux triplets spectraux. Dans la partie V se trouve la principale application de la méthode exposée dans ce mémoire. On prouve l'existence du prolongement méromorphe pour des fonctions zêta provenant de représentations de Kirillov d'une classe d'algèbre de Lie nilpotentes. / The thesis is about a families of zeta functions (Dirichlet series) that may be associated to certain algebras of Hilbert space operators. In this thesis, the main question in studying these zeta functions is to establish their meromorphic continuation from a half-plane in the complex plane to the full plane.Following an idea of Nigel Higson, we develop, in part I, a method for proving the existence of a meromorphic continuation for some spectral zeta functions. The method is based on algebras of generalized differential operators. The more important tool is the reduction sequence. The main theorem states, under some conditions, the existence of a meromorphic continuation, a localization of the poles in supports of arithmetic sequences and an upper bound of their order. A formulation of the method into the framework of Connes and Moscovici, the regular spectral triples, setting in part II. In the third part, we give an application for zeta functions associate to a Laplace-type operator on a smooth, closed manifold. This example was initially treated in this way by Nigel Higson in 2006. We give another application for zeta functions associate to the noncommutative torus. In part IV, using the work of Dominique Manchon on algebras of pseudodifferential operators associated to unitary representations of nilpotent Lie group, we construct new spectral triples. In part V, set the main application of the method. We applicate the reduction method for some algebras of generalized differential operators, arising from a Kirillov representation of a class of nilpotent Lie algebras.

In this thesis, we develop a framework that generalizes the previously known notions of theta-summable Fredholm modules to the setting of locally convex dg algebras. By introducing an additional action of the Clifford algebra, we may treat the even and odd cases simultaneously. In particular, we recover the theory developed by Güneysu/Ludewig and extend the definition of odd theta-summable Fredholm modules to the differential graded category. We then construct a Chern character, which serves as a differential graded refinement of the JLO cocycle, and prove that it has all the expected analytical and homological properties. As an application, we prove an odd noncommutative index theorem relating the spectral flow of a theta-summable Fredholm module to the pairing of the Chern character with the odd Bismut-Chern character in entire (differential graded) cyclic homology, thereby extending results obtained by Güneysu/Cacciatori and Getzler.