Dans une première partie théorique, nous nous penchons sur une analyse rigoureuse des performances de l'algorithme Sequential Monte Carlo (SMC) conduisant à des résultats de type bornes L^p et inégalités de concentration. Nous abordons notamment le cas particulier des SMC associés à des schémas de température, et analysons sur ce sujet un processus à schéma adaptatif.Dans une seconde partie appliquée, nous illustrons son utilisation par la résolution de problèmes inverses concrets en électromagnétisme. Le plus important d'entre eux consiste à estimer les propriétés radioélectriques de matériaux recouvrant un objet de géométrie connue, et cela à partir de mesures de champs rétrodiffusés. Nous montrons comment l'algorithme SMC, couplé à des calculs analytiques, permet une inversion bayésienne, et fournit des estimées robustes enrichies d'estimations des incertitudes. / Sequential and Quantum Monte Carlo methods, as well as genetic type search algorithms, can be interpreted as a mean field and interacting particle approximation of Feynman-Kac models in distribution spaces. The performance of these population Monte Carlo algorithms is strongly related to the stability properties of nonlinear Feynman-Kac semigroups. In a first theoretical part, we analyze these models in terms of Dobrushin ergodic coefficients of the reference Markov transitions and the oscillations of the potential functions. Sufficient conditions for uniform concentration inequalities w.r.t. time are expressed explicitly in terms of these two quantities. We provide an original perturbation analysis that applies to annealed and adaptive FK models, yielding what seems to be the first results of this kind for these type of models. Special attention is devoted to the particular case of Boltzmann-Gibbs measures' sampling. In this context, we design an explicit way of tuning the number of Markov Chain Monte Carlo iterations with temperature schedule. We also propose and analyze an alternative interacting particle method based on an adaptive strategy to define the temperature increments. In a second, applied part, we illustrate the use of these SMC algorithms in the field of inverse problems. Mainly, the following electromagnetism (EM) inverse problem is addressed. It consists in estimating local radioelectric properties of materials recovering an object from global EM scattering measurements, at various incidences and wave frequencies. This large scale ill-posed inverse problem is explored by an intensive exploitation of an efficient 2D Maxwell solver, distributed on high performance computing machines. Applied to a large training data set, a statistical analysis reduces the problem to a simpler probabilistic metamodel, on which Bayesian inference can be performed. Considering the radioelectric properties as a hidden dynamic stochastic process, that evolves in function of the frequency, it is shown how the Sequential Monte Carlo methods can take benefit of the structure and provide local EM property estimates.

Yung Man Hong = 受歐姆阻尼振子和光學系統內的彼德曼因數及費曼圖展開 / 翁文康. / Thesis (M.Phil.)--Chinese University of Hong Kong, 2004. / Includes bibliographical references (leaves 95-99). / Text in English; abstracts in English and Chinese. / Yung Man Hong = Shou ou mu zu ni zhen zi he guang xue xi tong nei de Bideman yin shu ji Feiman tu zhan kai / Weng Wenkang. / Acknowledgement --- p.iii / Chapter 1 --- Overview --- p.1 / Chapter 1.1 --- The Langevin Equation --- p.1 / Chapter 1.2 --- Excess Noise in Lasers --- p.4 / Chapter 1.3 --- Non-orthogonality --- p.9 / Chapter 2 --- Bilinear Map and Eigenvector Expansion --- p.12 / Chapter 2.1 --- Introduction --- p.12 / Chapter 2.2 --- Mathematical Formalism --- p.14 / Chapter 2.3 --- Criticality and Divergence --- p.19 / Chapter 2.4 --- Perturbations and Cancellations --- p.25 / Chapter 3 --- Generalized Petermann Factor --- p.34 / Chapter 3.1 --- Introduction --- p.34 / Chapter 3.2 --- Petermann Factor in Optical Systems --- p.36 / Chapter 3.3 --- Generalized Petermann Factor --- p.41 / Chapter 3.4 --- Thermal Correlation Functions --- p.43 / Chapter 3.5 --- Fluctuation-Dissipation Theorem --- p.46 / Chapter 3.6 --- Weak Damping versus Near-Degeneracy --- p.49 / Chapter 4 --- Continuum Generalization --- p.56 / Chapter 4.1 --- Bilinear map --- p.56 / Chapter 4.2 --- Critical Points --- p.58 / Chapter 4.3 --- Semiclassical Laser Theory --- p.63 / Chapter 5 --- Diagrammatic Expansions --- p.71 / Chapter 5.1 --- Introduction --- p.71 / Chapter 5.2 --- Nonlinearly Coupled Oscillators --- p.72 / Chapter 5.3 --- Path Integral Method --- p.76 / Chapter 5.4 --- Feynman Diagram --- p.81 / Chapter 6 --- Conclusion --- p.87 / Chapter A --- Derivation of the Langevin equation --- p.89

In this project, we examine how integration over matrices is performed. We investigate and develop a method for calculating matrix integrals over the set of real square matrices. Matrix integrals are used for calculations in several different areas of physics and mathematics; for example quantum field theory, string theory, quantum chromodynamics, and random matrix theory. Our method consists of ways to apply perturbative Taylor expansions to the matrix integrals, reducing each term of the resulting Taylor series to a combinatorial problem using Wick's theorem, and representing the terms of the Wick sum graphically with the help of Feynman diagrams and fat graphs. We use the method in a few examples that aim to clearly demonstrate how to calculate the matrix integrals. / I detta projekt undersöker vi hur integration över matriser genomförs. Vi undersöker och utvecklar en metod för beräkning av matrisintegraler över mängden av alla reell-värda kvadratiska matriser. Matrisintegraler används för beräkningar i ett flertal olika områden inom fysik och matematik, till exempel kvantfältteori, strängteori, kvantkromodynamik och slumpmatristeori. Vår metod består av sätt att applicera perturbativa Taylorutvecklingar på matrisintegralerna, reducera varje term i den resulterande Taylorserien till ett kombinatoriellt problem med hjälp av Wicks sats, och att representera termerna i Wicksumman grafiskt med hjälp av Feynmandiagram. Vi använder metoden i några exempel som syftar till att klart demonstrera hur beräkningen av matrisintegraler går till.

Orientador: Bin Kang Cheng / Dissertação (mestrado) - Universidade Federal do Parana / Resumo: No formalismo de Feynman na mecânica estatística quântica, a função partição pode ser representada como uma integral de caminho. O método variacional proposto recentemente por Feynman e Kleinert permite transformar a integral de caminho numa integral no espaço de fase. no qual as flutuações quânticas são consideradas introduzindo potencial clássico efetivo. Este método foi testado com sucesso para potenciais suaves e para o potencial com a singularidade delta de Dirac. Nesta dissertação, nos aplicamos o método para potenciais com singularidade forte :(a) um potencial quadrático e (b) um potencial linear, ambos com uma parede rígida na origem. Para que a densidade de probabilidade seja zero na origem, introduzimos o método de Feynman-Kleinert adaptado. Primeiro obtivemos o potencial clássico efetivo analiticamente e avaliamos os potenciais clássicos efetivos, energias livres { ou funções partições ) e densidades de probabilidades numericamente. Por ultimo comparamos nossos resultados com os exatos, os de Feynman-Kleinert e os semi-clássicos. Nossos resultados são bons para osciladores com baixas frequências angulares e potenciais lineares fracos, mesmo para baixas temperaturas. Para osciladores com frequências angulares altas e potenciais lineares fortes, os resultados são válidos somente a temperaturas mais altas ( beta menor ou igual 1). / Abstract: In Feynman's approach to quantum statistical mechanics, the partition function can be represented as a path integral. A recently proposed variational method of Feynmam-Kleinert is able to transformed the path integral into an integral in phase space, in which the quantum fluctuations have been taken care of by introducing the effective classical potentential. This method has been tested with succeed for the smooth potentials and for the singular potential of delta. In this dissertation, we apply the method to the strong singular potentials: (a) a quadratic potential and (b) a linear potential both with a rigid wall at the origin. By satisfying the density of the particle be vanish at the origin, we introduce an adaptated method of Feynman-Kleinert in order to improve the method. We first obtain the effective classical potential analytictly and then evaluate effective classical potentials, free energies (or partition functions) and the densities of particle numerically. Finally we compare our results with those of exact, Feynman- Kleinert and semi- classical. Our results are good for lower angular frequency of oscillator and for weak linear potential even at lower temperatures. For higher angular frequency of oscillator and for strong linear potential, our results are valid only in higher temperature (up to beta greather than or equal to 1).

Dans cette thèse, nous nous intéressons à la renormalisation de Connes et Kreimer dans le contexe des algèbres de Hopf de graphes de Feynman spécifiés. Nous construisons une structure d'algèbre de Hopf $\mathcal{H}_\mathcal{T}$ sur l'espace des graphes de Feynman spécifié d'une théorie quantique des champs $\mathcal{T}$. Nous définissons encore un dédoublement $\wt\mathcal{D}_\mathcal{T}$ de la bigèbre de graphes de Feynman spécifiés, un produit de convolution \divideontimes et un groupe de caractères de cette algèbre de Hopf à valeurs dans une algèbre commutative qui prend en compte la dépendance en les moments extérieurs. Nous mettons en place alors la renormalisation décrite par A. Connes et D. Kreimer et la décomposition de Birkhoff pour deux schémas de renormalisation : le schéma minimal de renormalisation et le schéma de développement de Taylor. Nous rappelons la définition des intégrales de Feynman associées à un graphe. Nous montrons que ces intégrales sont holomorphes en une variable complexe D dans le cas des fonctions de Schwartz, et qu'elles s'étendent en une fonction méromorphe dans le cas des fonctions de types Feynman. Nous pouvons alors déterminer les parties finies de ces intégrales en utilisant l'algorithme BPHZ après avoir appliqué la procédure de régularisation dimensionnelle. / In this thesis, we study the renormalization of Connes-Kreimer in the contex of specified Feynman graphs Hopf algebra. We construct a Hopf algebra structure $\mathcal{H}_\mathcal{T}$ on the space of specified Feynman graphs of a quantum field theory $\mathcal{T}$. We define also a doubling procedure for the bialgebra of specified Feynman graphs, a convolution product and a group of characters of this Hopf algebra with values in some suitable commutative algebra taking momenta into account. We then implement the renormalization described by A. Connes and D. Kreimer and the Birkhoff decomposition for two renormalization schemes: the minimal subtraction scheme and the Taylor expansion scheme.We recall the definition of Feynman integrals associated with a graph. We prove that these integrals are holomorphic in a complex variable D in the case oh Schwartz functions, and that they extend in a meromorphic functions in the case of a Feynman type functions. Finally, we determine the finite parts of Feynman integrals using the BPHZ algorithm after dimensional regularization procedure.

In Kooperation mit der Joachim Herz Stiftung haben wir Unterrichtsmaterial entwickelt, das Lehrkräfte dabei unterstützt, Teilchen- und Astroteilchenphysik ins Klassenzimmer zu bringen. Die Reihe umfasst vier Bände: (1) Wechselwirkungen, Ladungen und Teilchen, (2) Forschungsmethoden, (3) Kosmische Strahlung sowie (4) Mikrokurse. Alle Hefte enthalten Texte für Schüler und Lehrkräfte, Aufgabenblätter sowie didaktische Hinweise. Das Material wurde federführend vom Netzwerk Teilchenwelt unter Leitung von Prof. Dr. Michael Kobel erarbeitet.

The Hellmann-Feynman theorem is presented together with certain allied theorems. The origin of the Hellmann-Feynman theorem in quantum physical chemistry is described. The theorem is stated with proof and with discussion of applicability and reliability. Some adaptations of the theorem to the study of the variation of zeros of special functions and orthogonal polynomials are surveyed. Possible extensions are discussed.

<p> The electron density approach in conjunction with the Hellmann-Feynman theorem is used for a systematic analysis of binding characteristics of the two isoelectronic molecular series: N₂, CO, BF, and LiF, BeO. Electron density distributions, forces and field gradients corresponding to static properties of electron densities, have been calculated from Hartree-Fock wavefunctions (obtained from the work of other authors) for these molecules. Correlation of these static properties with binding characteristics are presented. Covalent and ionic characteristics are made evident by an analysis of the density distributions, density difference maps obtained by subtracting atomic from molecular distributions, and the forces exerted on nuclei by these distributions. A discussion of the field gradients, as related to quadrupole polarizations of the electron densities, is presented and the relevance of these polarizations to the interpretation of nuclear quadrupole coupling constants is indicated. </p> <p> Dynamic properties, as reflected by the magnitude of force constants, are analyzed in terms of functionals of the one-electron density. Force constant expressions are derived from the Hellmann-Feynman theorem. Any relation of force constants to field gradients is shown to be not unique as a result of cancellation of static and dynamic electron contributions to the total force constant. The total electronic contribution is shown to arise from a relaxation of density after a displacement of a certain nucleus. Relaxation of density with respect to one nucleus but which remains localized on some other nucleus in a molecule is shown to be equivalent to a field gradient. Thus, such density is separated from other density and its contribution to the force constant is treated as a field gradient. All contributions are computed from polynomial fits of the corresponding forces calculated at a number of internuclear distances. Relaxation density maps for the remaining atomic and overlap densities centered on a specific nucleus are presented. These maps are calculated as the difference between densities of the extended and equilibrium configurations of a molecule. The relaxation densities are correlated to the magnitude of the corresponding electronic force constant components. Thus, for the first time, there is demonstrated the concrete relation between covalent and ionic characteristics of electron densities in molecules and their dynamic properties which result in the magnitude of force constants. </p> / Thesis / Doctor of Philosophy (PhD)

Die Auswertung von Mehrschleifen-Feynman-Integralen ist eine der größten Herausforderungen bei der Berechnung präziser theoretischer Vorhersagen für die am LHC gemessenen Wirkungsquerschnitte. In den vergangenen Jahren hat sich die Nutzung von Differentialgleichungen bei der Berechnung von Feynman-Integralen als sehr erfolgreich erwiesen. Es wurde dabei beobachtet, dass die von den Feynman-Integralen erfüllte Differentialgleichung oftmals in eine sogenannte kanonische Form transformiert werden kann, welche die Integration der Differentialgleichung mittels iterierter Integrale wesentlich vereinfacht. Das zentrale Ergebnis der vorliegenden Arbeit ist ein Algorithmus zur Berechnung rationaler Transformationen von Differentialgleichungen von Feynman-Integralen in eine kanonische Form. Neben der Existenz einer solchen rationalen Transformation stellt der Algorithmus keinerlei weitere Bedingungen an die Differentialgleichung. Insbesondere ist der Algorithmus auf Mehrskalenprobleme anwendbar und erlaubt eine rationale Abhängigkeit der Differentialgleichung vom dimensionalen Regulator. Bei der Anwendung des Algorithmus wird zunächst das Transformationsgesetz im dimensionalen Regulator entwickelt, um Differentialgleichungen für die Koeffizienten in der Entwicklung der Transformation herzuleiten. Diese Differentialgleichungen werden dann mit einem rationalen Ansatz für die gesuchte Transformation gelöst. Es wird zudem eine Implementation des Algorithmus in dem Mathematica Paket CANONICA vorgestellt, welches das erste veröffentlichte Programm dieser Art ist, das auf Mehrskalenprobleme anwendbar ist. CANONICAs Potential für moderne Mehrschleifenrechnungen wird anhand mehrerer nicht trivialer Mehrschleifen-Integraltopologien demonstriert. Die gezeigten Topologien hängen von bis zu drei Variablen ab und umfassen auch vormals ungelöste Topologien, die zu Korrekturen höherer Ordnung zum Wirkungsquerschnitt der Produktion einzelner Top-Quarks am LHC beitragen. / The evaluation of multi-loop Feynman integrals is one of the main challenges in the computation of precise theoretical predictions for the cross sections measured at the LHC. In recent years, the method of differential equations has proven to be a powerful tool for the computation of Feynman integrals. It has been observed that the differential equation of Feynman integrals can in many instances be transformed into a so-called canonical form, which significantly simplifies its integration in terms of iterated integrals. The main result of this thesis is an algorithm to compute rational transformations of differential equations of Feynman integrals into a canonical form. Apart from requiring the existence of such a rational transformation, the algorithm needs no further assumptions about the differential equation. In particular, it is applicable to problems depending on multiple kinematic variables and also allows for a rational dependence on the dimensional regulator. First, the transformation law is expanded in the dimensional regulator to derive differential equations for the coefficients of the transformation. Using an ansatz in terms of rational functions, these differential equations are then solved to determine the transformation. This thesis also presents an implementation of the algorithm in the Mathematica package CANONICA, which is the first publicly available program to compute transformations to a canonical form for differential equations depending on multiple variables. The main functionality and its usage are illustrated with some simple examples. Furthermore, the package is applied to state-of-the-art integral topologies appearing in recent multi-loop calculations. These topologies depend on up to three variables and include previously unknown topologies contributing to higher-order corrections to the cross section of single top-quark production at the LHC.

In dieser Dissertation geht es um Schwinger-parametrische Feynmanintegrale in der Quantenelektrodynamik. Mittels einer Vielzahl von Methoden aus der Kombinatorik und Graphentheorie wird eine signifikante Vereinfachung des Integranden erreicht. Nach einer größtenteils in sich geschlossenen Einführung zu Feynmangraphen und -integralen wird die Herleitung der Schwinger-parametrischen Darstellung aus den klassischen Impulsraumintegralen ausführlich erläutert, sowohl für skalare Theorien als auch Quantenelektrodynamik. Es stellt sich heraus, dass die Ableitungen, die benötigt werden um Integrale aus der Quantenelektrodynamik in ihrer parametrischen Version zu formulieren, neue Graphpolynome enthalten, die auf Zykeln und minimalen Schnitten (engl. "bonds") basieren. Danach wird die Tensorstruktur der Quantenelektrodynamik, bestehend aus Dirac-Matrizen und ihren Spuren, durch eine diagrammatische Interpretation ihrer Kontraktion zu ganzzahligen Faktoren reduziert. Dabei werden insbesondere gefärbte Sehnendiagramme benutzt. Dies liefert einen parametrischen Integranden, der über bestimmte Teilmengen solcher Diagramme summierte Produkte von Zykel- und Bondpolynomen enthält. Weitere Untersuchungen der im Integranden auftauchenden Polynome decken Verbindungen zu Dodgson- und Spannwaldpolynomen auf. Dies wird benutzt um eine Identität zu beweisen, mit der sehr große Summen von Sehnendiagrammen in einer kurzen Form ausgedrückt werden können. Insbesondere führt dies zu Aufhebungen, die den Integranden massiv vereinfachen. / This thesis is concerned with the study of Schwinger parametric Feynman integrals in quantum electrodynamics. Using a variety of tools from combinatorics and graph theory, significant simplification of the integrand is achieved. After a largely self-contained introduction to Feynman graphs and integrals, the derivation of the Schwinger parametric representation from the standard momentum space integrals is reviewed in full detail for both scalar theories and quantum electrodynamics. The derivatives needed to express Feynman integrals in quantum electrodynamics in their parametric version are found to contain new types of graph polynomials based on cycle and bond subgraphs. Then the tensor structure of quantum electrodynamics, products of Dirac matrices and their traces, is reduced to integer factors with a diagrammatic interpretation of their contraction. Specifically, chord diagrams with a particular colouring are used. This results in a parametric integrand that contains sums of products of cycle and bond polynomials over certain subsets of such chord diagrams. Further study of the polynomials occurring in the integrand reveals connections to other well-known graph polynomials, the Dodgson and spanning forest polynomials. This is used to prove an identity that expresses some of the very large sums over chord diagrams in a very concise form. In particular, this leads to cancellations that massively simplify the integrand.