Thèse (doctorat)--Université Louis Pasteur, Strasbourg, 1992. / "Cette thèse se compose de cinq mémoires originaux, deux articles ..., et deux courtes notes de probabilités non commutatives"--4e de couv. Notes bibliogr.

Nous étudions des fluctuations qui sont omniprésentes dans des collisions entre particules aux hautes énergies. Ces fluctuations peuvent être de nature classique ou quantique et nous allons considérer ces deux cas. D'abord, nous étudions les fluctuations quantiques qui sont présentes dans des collisions entre protons. Celles-ci sont calculables en théorie quantique des champs, et nous allons nous concentrer sur une certaine classe de diagrammes dans ce cadre. Dans un second temps nous allons étudier des fluctuations qui sont présentes dans des collisions entre particules plus lourdes que le proton. Celles-ci sont décrites par les lois quantiques de la nature qui donnent les positions des nucléons dans le noyau, ou bien des fluctuations classiques, d'origine thermique, qui affectent l'évolution hydrodynamique du milieu produit dans une collision. Les fluctuations dans des collisions entre protons peuvent être calculées analytiquement jusqu'à un certain ordre en théorie quantique des champs. Nous allons nous concentrer sur des diagrammes à une boucle, d'une topologie donnée. Ces diagrammes aux boucles donnent des intégrales, qui typiquement sont difficiles à calculer. Nous allons démontrer comment des outils des mathématiques modernes peuvent être utilisés pour faciliter leur évaluation. En particulier, nous allons étudier des relations entre des coupures d'un diagramme, la discontinuité à travers d'un branchement et le coproduit. Nous allons démontrer comment l'intégrale originale peut être reconstruit à partir de l'information contenue dans le coproduit. Nous nous attendons à ce que ces méthodes seront utiles pour le calcul des diagrammes avec des topologies plus difficiles et ainsi aident au calcul des nouvelles amplitudes de diffusion. A la fin, nous étudions les deux types de fluctuations qui ont lieu dans des collisions entre ions lourds. Celles-ci sont liées soit à l'état initial de la matière, soit à l'état intermédiaire produit dans une telle collision. Les fluctuations de l'état initial ont été mesurées expérimentalement, et on voit qu'elles donnent lieu à des non-Gaussianités dans le spectre final de particules. Nous allons démontrer comment ces non-Gaussianités peuvent être comprises comme des positions et des énergies d'interaction aléatoires des 'sources' dans les noyaux entrant en collision. En plus, nous étudions le bruit hydrodynamique dans le milieu produit juste après une collision. Le comportement de ce milieu est celui d'un fluide à basse viscosité. / We study fluctuations that are omnipresent in high-energy particle collisions. These fluctuations can be either of either classical or quantum origin and we will study both. Firstly, we consider the type of quantum fluctuations that arise in proton-proton collisions. These are computable perturbatively in quantum field theory and we will focus on a specific class of diagrams in this set-up. Secondly, we will consider the fluctuations that are present in collisions between nuclei that can be heavier than protons. These are the quantum laws of nature that describe the positions of nucleons within a nucleus, but also the hydrodynamic fluctuations of classical, thermal origin that affect the evolution of the medium produced in heavy-ion collisions. The fluctuations arising in proton-proton collisions can be computed analytically up to a certain order in perturbative quantum field theory. We will focus on one-loop diagrams of a fixed topology. Loop diagrams give rise to integrals that typically are hard to evaluate. We show how modern mathematical methods can be used to ease their computation. We will study the relations among unitarity cuts of a diagram, the discontinuity across the corresponding branch cut and the coproduct. We show how the original integral corresponding to a given diagram can be reconstructed from the information contained in the coproduct. We expect that these methods can be applied to solve more complicated topologies and help in the computation of new amplitudes in the future. Finally, we study the two types of fluctuations arising in heavy-ion collisions. These are related either to the initial state or the intermediate state of matter produced in such collisions. The initial state fluctuations are experimentally observed to give rise to non-Gaussianities in the final-state spectra. We show how these non-Gaussianities can be explained by the random position and interaction energy of `sources' in the colliding nuclei. Furthermore, we investigate the effect of hydrodynamical noise in the evolution of the medium produced just after a collision. This medium behaves like a fluid with a very low viscosity, and so the corresponding evolution is hydrodynamical.

O aumento pelo interesse na teoria de integração estocástica é, basicamente, consequência da acirrada competição para entender, desenvolver e aplicar a matemática subjacente ao mercado mobiliário. Neste trabalho desenvolvemos, de maneira didática e visando aplicações, tal teoria. Para tanto, começamos apresentando um desenvolvimento cuidadoso da teoria dos martingais e dos principais resultados de medida e probabilidade relacionados. Depois apresentamos de maneira formal a teoria de integração estocástica com respeito aos semi-martingais contínuos. Finalizamos com um tratamento das principais aplicações dessa teoria como a fórmula de Itô, uma introdução às equações diferenciais estocásticas e a fórmula de Feynman-Kac. Apresentamos também, em um apêndice, a teoria de mudança de medida e o teorema de Girsanov. Tentamos durante o trabalho apresentar exemplos relacionados com finanças e ilustrar a importância do movimento Browniano. / The increasing interest in the theory of Stochastic Integration is due mainly to the competitive pressure to understand, develop and apply the underlying mathematics of security markets. In this work, we attempt to develop part of the theory in a didactical approach and focused toward some particular applications. For this purpose, we begin by introducing a thorough development of Martingale theory and the main related results on Measure and Probability theory. We then present in a formal way the Stochastic Integration Theory with respect to continuous Semimartingales. Subsequentially, we show some of the theory\'s main applications, such as Itô\'s formula, an introduction to the theory of Stochastic Differential Equations and Feynman-Kac\'s formula. We also present in the appendix Girsanov\'s theorem and a construction of Brownian motion. During the development of this text we endeavored to enrich it by including examples relevant to finance and emphasizing the importance of the ubiquitous Brownian motion.

O aumento pelo interesse na teoria de integração estocástica é, basicamente, consequência da acirrada competição para entender, desenvolver e aplicar a matemática subjacente ao mercado mobiliário. Neste trabalho desenvolvemos, de maneira didática e visando aplicações, tal teoria. Para tanto, começamos apresentando um desenvolvimento cuidadoso da teoria dos martingais e dos principais resultados de medida e probabilidade relacionados. Depois apresentamos de maneira formal a teoria de integração estocástica com respeito aos semi-martingais contínuos. Finalizamos com um tratamento das principais aplicações dessa teoria como a fórmula de Itô, uma introdução às equações diferenciais estocásticas e a fórmula de Feynman-Kac. Apresentamos também, em um apêndice, a teoria de mudança de medida e o teorema de Girsanov. Tentamos durante o trabalho apresentar exemplos relacionados com finanças e ilustrar a importância do movimento Browniano. / The increasing interest in the theory of Stochastic Integration is due mainly to the competitive pressure to understand, develop and apply the underlying mathematics of security markets. In this work, we attempt to develop part of the theory in a didactical approach and focused toward some particular applications. For this purpose, we begin by introducing a thorough development of Martingale theory and the main related results on Measure and Probability theory. We then present in a formal way the Stochastic Integration Theory with respect to continuous Semimartingales. Subsequentially, we show some of the theory\'s main applications, such as Itô\'s formula, an introduction to the theory of Stochastic Differential Equations and Feynman-Kac\'s formula. We also present in the appendix Girsanov\'s theorem and a construction of Brownian motion. During the development of this text we endeavored to enrich it by including examples relevant to finance and emphasizing the importance of the ubiquitous Brownian motion.

Submitted by Rosivalda Pereira (mrs.pereira@ufma.br) on 2017-08-23T18:24:30Z No. of bitstreams: 1 RoemirMoreira.pdf: 689688 bytes, checksum: 82f6a2259d4db9b3772731a4618fdc96 (MD5) / Made available in DSpace on 2017-08-23T18:24:30Z (GMT). No. of bitstreams: 1 RoemirMoreira.pdf: 689688 bytes, checksum: 82f6a2259d4db9b3772731a4618fdc96 (MD5) Previous issue date: 2011-09-30 / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES) / The gauge sector of the Standard Model Extended (MPE) has been investigated in many respects in recent years, discussing the effects of Lorentz symmetry violation in physical systems and the limitations on the magnitude of the parameters of violation. This work revisited some planar theories derived from the dimensional reduction of the gauge sector of the MPE, and perform an original contribution: the dimensional reduction of the CPT-even and nonbirefringent gauge sector of the MPE, composed of nine components. The resulting planar theory includes a gauge sector and scalar sector (which has an nonsual kinetic term), coupled together by a 3-vector Cα Lorentz violation. Both sectors, gauge and scale, are affected by the six components of a symmetric tensor violates Lorentz, kµρ. The energy-momentum tensor is explicitly calculated, revealing that the energy of the gauge and scalar sectors is stable for small values of the parameters of violation. The equations of motion for the electric and magnetic elds, as well as the potentials, are written and analyzed in the steady state. Then employ the method of Green to get the stationary solutions of classical electrodynamics to rst order in the parameters of violation. It is observed that the coe cients of Lorentz violation does not alter the asymptotic behavior of the elds, but does not induce an angular dependence observed in the planar theory of Maxwell. The dispersion relation is exactly computed and is compatible with a theory does not birefringent, and demonstrating that the theory is stable, but in general, not causal. Finally, we calculate the Feynman propagator for the gauge elds and scalar theory of planar, accurately, using a set of 11 projectors that form a closed algebra. We use the expression of the Feynman propagator to analyze the consistency of the theory regarding its stability, causality and unitarity. / O setor de gauge do Modelo Padrão Estendido (MPE) tem sido investigado em muitos aspectos nos últimos anos, discutindo os efeitos da violação da simetria de Lorentz em sistemas físicos e as limitações da magnitude dos parâmetros de violação. Neste trabalho, rediscutimos algumas teorias planares obtidas a partir da redução dimensional do setor de gauge do MPE, e realizamos uma contribuição original: a redução dimensional do setor de gauge CPT-par e não-birrefringente do MPE, composto por nove componentes. A resultante teoria planar abarca um setor de gauge e um setor escalar (dotado de um termo cinético não usual), acoplados entre si por um 3-vetor Cα de violação de Lorentz (LV). Ambos os setores, de gauge e escalar, são afetados pelas seis componentes de um tensor simétrico violador de Lorentz, kµρ. O tensor de energia-momento É explicitamente calculado, revelando que a energia dos setores de gauge e escalar são estáveis para pequenos valores dos parâmetros de violação. As equações de movimento para os campos elétrico e magnético, assim como para os potenciais, são escritas e analisadas no regime estacionário. Empregamos então o método de Green para obter as soluções clássicas estacionárias desta eletrodinâmica em primeira ordem nos parâmetros de violação. É observado que os coefi cientes de violação de Lorentz não alteram o comportamento assintótico dos campos, mas induzem uma dependência angular não observada na teoria planar de Maxwell. A relação de dispersão é exatamente computada, sendo compatível com uma teoria não birrefringente, e demonstrando que a teoria é estável, mas, em geral, não causal. Por fim, calculamos o propagador de Feynman para os campos de gauge e escalar desta teoria planar, de forma exata, usando um conjunto de 11 projetores que formam uma álgebra fechada. Usamos a expressão do propagador de Feynman para analisar a consistência da teoria no que concerne a sua estabilidade, causalidade e unitariedade.

Les calculs de précision dans les théories de jauge jouent un rôle très important pour l’étude de la physique du Modèle Standard et au-delà dans les super-collisionneurs de particules comme le LHC, TeVatron et ILC. Par conséquent, il est extrêmement important de fournir des outils du calcul d’amplitudes à une boucle stables, rapides, efficaces et hautement automatisés. Cette thèse a pour but de développer la bibliothèque d’intégrales Golem95. Cette bibliothèque est un programme écrit en Fortran95, qui contient tous les ingrédients nécessaires pour calculer une intégrale scalaire ou tensorielle à une boucle avec jusqu’à six pattes externes. Golem95 utilise une méthode traditionnelle de réduction (réduction à la Golem) qui réduit les facteurs de forme en des intégrales de base redondantes qui peuvent être scalaires (sans paramètres de Feynman au numérateur) ou tensorielles (avec des paramètres de Feynman au numérateur); ce formalisme permet d’éviter les problèmes de l’instabilité numérique engendrés par des singularités factices dues à l’annulation des déterminants de Gram. En plus, cette bibliothèque peut être interfacée avec des programmes du calcul automatique basés sur les méthodes d’unitarité comme GoSam par exemple. Les versions antérieures de Golem95 ont été conçues pour le calcul des amplitudes sans masses internes. Le but de ce travail de thèse est de généraliser cette bibliothèque pour les configurations les plus générales (les masses complexes sont incluses), et de fournir un calcul numériquement stable dans les régions problématique en donnant une représentation intégrale unidimensionnelle stable pour chaque intégrale de base de Golem95. / Higher order corrections in gauge theories play a crucial role in studying physics within the standard model and beyond at TeV colliders, like LHC, TeVatron and ILC. Therefore, it is of extreme importance to provide tools for next-to-leading order amplitude computation which are fast, stable, efficient and highly automatized. This thesis aims at developing the library of integrals Golem95. This library is a program written in Fortran95, it contains all the necessary ingredients to calculate any one-loop scalar or tensorial integral with up to six external legs. Golem95 uses the traditional reduction method (Golem reduction) to reduce the form factors into redundant basic integrals, which can be scalar (without Feynman parameters in the numerator) or tensorial (with Feynman parameter in the numerator); this formalism allows us to avoid the problems of numerical instabilities generated by the spurious singularities induced by the vanishing of the Gram determinants. In addition, this library can be interfaced with automatic programs of NLO calculation based on the unitarity inspired reduction methods as GoSam for example. Earlierversions of Golem95 were designed for the calculation of amplitudes without internal masses. The purpose of this thesis is to extend this library for more general configurations (complex masses are supported); and to provide numerically stable calculation in the problematic regions (det(G) → 0), by providing a stable one-dimensional integral representation for each Golem95 basic integral.

In dieser Dissertation erweitern wir die Ideen des Yangian-Bootstrap-Algorithmus auf Feynman-Diagramme mit massiven Teilchen. Ausgehend von der massiven dual-konformen Symmetrie der N = 4 Super-Yang-Mills Theorie auf dem Coulomb-Zweig konstruieren wir einen Satz von bilokalen Yangian Level-Eins Generatoren und zeigen, dass sie eine unendliche Anzahl von planaren ein- und zwei-Schleifen-Diagrammen vernichten. Wir beschreiben außerdem wie der dual-konforme Level-Eins Impuls-Operator auf eine massive Verallgemeinerung des gewöhnlichen spezial-konformen Generators im Impulsraum abgebildet wird. Als nächstes wenden wir den Yangian-Bootstrap-Algorithmus mit großem Erfolg auf eine Reihe von massiven Ein-Schleifen-Diagrammen mit verallgemeinerten Propagatorexponenten und in beliebiger Anzahl von Raumdimensionen an. Im Spezialfall der dual-konformen Integrale, deren Propagatorexponenten sich zur Raumdimension addieren, finden wir neue sehr einfache Darstellungen durch hypergeometrische Funktionen, die eine natürliche Verallgemeinerung für Diagramme mit beliebig vielen äußeren Punkten erlauben. Außerdem diskutieren wir Aspekte des Yangian-Bootstrap-Algorithmus in Minkowski-Raumzeit am Beispiel des masselosen Box-Integrals. Wir zeigen, dass dessen Yangian-Symmetrie gemeinsam mit seinen diskreten Permutationssymmetrien das Box-Integrals bis auf 12 unbestimmte Konstanten komplett festlegt. Schließlich schlagen wir vor, dass das Auftreten von Yangian-Symmetrie in massiven Fischnetz-Diagrammen mit deren Rolle als Ein-Spur-Streuamplituden in einer massiven Fischnetz-Theorie zusammenhängen könnte. In Analogie mit der masselosen Fischnetz-Theorie zeigen wir, wie diese Theorie als Deformation der N = 4 Super-Yang-Mills Theorie auf dem Coulomb-Zweig definiert werden kann. Wir diskutieren eine bestimmte Klasse von planaren Grenzfällen, in der die off-shell Streuamplituden der Theorie eine massive dual-konforme Symmetrie sowie Yangian-Symmetrie aufweisen. / In this dissertation, we extend the ideas of the Yangian bootstrap algorithm to massive Feynman diagrams. Based on the massive dual-conformal symmetry of Coulomb branch N = 4 super-Yang-Mills theory, we construct a set of bi-local Yangian level-one generators and show that they annihilate infinite classes of massive planar Feynman integrals at one and two loops. We also describe how the dual-conformal level-one momentum generator maps to a massive deformation of the ordinary momentum space special conformal generator. We then apply the Yangian bootstrap to a set of massive one-loop integrals with generalised propagator powers and in an arbitrary number of space dimensions to great success. In the special case of dual-conformal integrals, whose propagator powers sum to the space dimension, we find very simple novel hypergeometric structures, suggesting a natural generalisation to diagrams with an arbitrary number of external points. In the particular case of the massless box integral we also discuss elements of the Yangian bootstrap in Minkowski space. We show that its Yangian and discrete permutation symmetries constrain it up to 12 undetermined constants. We then derive the values of these constants via analytic continuation from the box integral in the Euclidean region. Finally, we provide evidence that the appearance of Yangian symmetry for massive fishnet diagrams is related to their role as colour-ordered scattering amplitudes in a massive fishnet theory. We show how to construct this theory from Coulomb branch N = 4 super-Yang-Mills theory, paralleling the original construction of the massless fishnet theory. We discuss how a particular class of planar limits leads to the emergence of massive dual-conformal symmetry as well as massive Yangian symmetry for the theory’s off-shell scattering amplitudes.

In dieser Dissertation untersuchen wir unterschiedliche Aspekte im Zusammenhang mit Korrelationsfunktionen in der Fischnetz-Theorie. Zunächst betrachten wir einen der einfachsten Korrelatoren der Fischnetz Theorie, das konforme Box-Integral, in Minkowski Signatur. Während dieses Integral in Euklidischer Signatur eine konforme Symmetrie aufweist, wird diese Symmetrie in Minkowski-Raumzeit subtil gebrochen. Wir beschreiben die Brechung der konformen Symmetrie quantitativ, indem wir die funktionale Form des Box-Integrals in allen kinematischen Regionen untersuchen. Ausserdem untersuchen wir das Ausmass zu dem das Box integral durch seine Yangian-Symmetrie festgelegt ist. Als nächstes widmen wir uns den Basso-Dixon-Graphen, die ebenfalls konforme Vier-Punkt-Integrale sind und Verallgemeinerungen des Box-Integrals zu höheren Schleifenordnungen darstellen. Wir leiten die Yangian-Ward-Identitäten ab, die diese Klasse von Integralen erfüllen. Die Ward-Identitäten sind einhomogene Erweiterungen der partiellen Differentialgleichungen, die im homogenen Fall durch Appell-Hypergeometrische Funktionen gelöst werden. Die Ward-Identitäten können natürlicherweise auf eine Ein-Parameter-Familie von D-dimensionalen Integralen erweitert werden, die Korrelatoren in der verallgemeinerten Fischnetz-Theorie von Kazakov und Olivucci darstellen. Schliesslich untersuchen wir den Dilatationsoperator in einem Drei-Skalar-Sektor der Fischnetztheorie, der auch als Eklektisches Modell bezeichnet wird. In diesem Sektor der Dilatationsoperator nimmt nicht--diagonalisierbare Form an. Das führt dazu, dass die Zwei-Punkt-Korrelationsfunktionen eine logarithmische Abhängigkeit von der Raumzeitseparierung der Operatoren annimmt. Unter Zuhilfenahme von kombinatorischen Argumenten führen wir eine generierende Funktion ein, die das Jordan-Block-Spektrum eines verwandten Modells, der hypereklektischen Spinkette, vollständig charakterisiert. / We study various aspects of correlation functions in fishnet theory. We begin with the study of the simplest correlator in theory theory, represented by the conformal box integral, in Minkowski space. While this integral is conformally invariant in Euclidean space, this symmetry is subtly broken in Minkowski space. We quantify the extent to which conformal symmetry is broken by analysing the functional form of the box in each kinematic region. We propose a new method to calculate the box integral directly in Minkowski space, by introducing a family of configurations with two points at infinity. Furthermore, we investigate the extent to which the box integral is constrained by Yangian symmetry. We constrain the functional form of the box integral in all kinematic regions up to twelve undetermined constants, which we fix by three separate analytic continuations from the Euclidean region. Next, we study the Basso-Dixon graphs, which represent higher-loop versions of the box integral. We derive and study Yangian Ward identities for this class of integrals. These take the form of inhomogeneous extensions of the partial differential equations defining the Appell hypergeometric functions. The Ward identities naturally generalise to a one-parameter family of D dimensional integrals representing correlators in a generalised fishnet theory. Finally, we study the dilatation operator in a particular three scalar sector of the fishnet theory, which has been dubbed the eclectic model. This dilatation operator is non-diagonalisable in this sector. This leads to logarithmic spacetime dependence in the corresponding two-point functions. Using combinatorial arguments, we introduce a generating function which fully characterises the Jordan block spectrum of a related model: the hypereclectic spin chain. This function is found by purely combinatorial means and can be expressed in terms of the q-binomial coefficient.

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