Zugl.: Regensburg, Univ., Diss., 2005

Techn. University, Diss., 2005--Darmstadt.

Made available in DSpace on 2016-05-17T16:51:03Z (GMT). No. of bitstreams: 0 Previous issue date: 2009-05-22. Added 1 bitstream(s) on 2016-05-17T16:54:32Z : No. of bitstreams: 1 000857245.pdf: 574030 bytes, checksum: 8966f79c533ef00580f3bb1b8e5482f6 (MD5) / Este trabalho é uma revisão do método de integração em dimensão negativa como uma ferramenta poderosa no cálculo das correções radiativas presentes na teoria quântica de campos perturbativa. Este método é aplicável no contexto da regularização dimensional e permite obter soluções exatas de integrais de Feynman onde tanto o parâmetro de dimensão como os expoentes dos propagadores estão generalizados. As soluções apresentam-se na forma de combinações lineares de funções hipergeométricas cujos domínios de convergência estãoo relacionados com a estrutura analíica da integral de Feynman. Cada solução definida por seu domínio de convergência está conectada com as outras através de continuações analíticas. Além de apresentar e discutir o algoritmo geral do método com detalhe, mostram-se aplicações concretas a integrais escalares de um e dois loops e à renormalização da eletrodinâmica quântica (QED) a um loop / This work is a review of the Negative Dimension Integration Method as a powerful tool for the computation of the radiative corrections present in Quantum Field Perturbation Theory. This method is applicable in the context of Dimensional Regularization and it provides exact solutions for Feynman integrals with both dimensional parameter and propagator exponents generalized. These solutions are presentedintheformoflinearcombinationsofhypergeometricfunctionswhosedomains of convergence are related to the analytic structure of the Feynman Integral. Each solution is connected to the others trough analytic continuations. Besides presenting and discussing the general algorithm of the method in a detailed way, we offer concrete applications to scalar one-loop and two-loop integrals as well as to the one-loop renormalizationofQuantumElectrodynamics (QED)

Made available in DSpace on 2014-06-11T19:32:10Z (GMT). No. of bitstreams: 0 Previous issue date: 2003-04Bitstream added on 2014-06-13T19:21:27Z : No. of bitstreams: 1 santos_es_dr_ift.pdf: 492820 bytes, checksum: dc248eeca0c392b9f7972bb0ff08c78d (MD5) / Usando a técnica conhecida como método de integração em dimensão negativa (NDIM), calculamos as funções de 2 e 3-pontos em 1-loop e analisamos seus casos particulares de maior interesse físico. Generalizamos esta aplicação do método para o caso de n-pontos em 1-loop, obtendo uma fórmula geral dada em termos de uma função hipergeométrica generalizada. Mostramos também que esta fórmula geral pode ser obtida pela parametrização de Feynman o que aponta para uma equivalência entre estas duas abordagens bem como com o análogo via representação de Mellin-Barnes. A integral escalar associada ao diagrama 2-loop mater foi calculada usando a decomposição de integração por partes seguida do NDIM. A integral escalar não massiva em 2-loop com n-inserções de auto-energia também foi calculada. / Abstracts: Employing the technique Known as Negative Dimensional Integration Method (NDIM), we calculate he two- and three-point functions at one loop level and analyse their particular caes of greatest physical interest. We generalize this to the case of n-point functions at one loop, obtaining a general formula given in terms of a generalized hypergeometric function. We also show that this general formula can be obtained via Feynman parametrization showing that there is an equivalence between the two approaches as well as with the analogous method via Mellin-Barnes representation. The scalar integral associated with the two-loop master diagram is calculated using partial integration followed by NDIM, and massless scalar integral at two loops with n self-energy insertions is also calculated using NDIM.

Orientador: Alfedo Takashi Suzuki / Banca: Bruto Max Pimentel Escobar / Banca: Jorge Henrique de Oliveira Sales / Resumo: Este trabalho é uma revisão do método de integração em dimensão negativa como uma ferramenta poderosa no cálculo das correções radiativas presentes na teoria quântica de campos perturbativa. Este método é aplicável no contexto da regularização dimensional e permite obter soluções exatas de integrais de Feynman onde tanto o parâmetro de dimensão como os expoentes dos propagadores estão generalizados. As soluções apresentam-se na forma de combinações lineares de funções hipergeométricas cujos domínios de convergência estãoo relacionados com a estrutura analíica da integral de Feynman. Cada solução definida por seu domínio de convergência está conectada com as outras através de continuações analíticas. Além de apresentar e discutir o algoritmo geral do método com detalhe, mostram-se aplicações concretas a integrais escalares de um e dois loops e à renormalização da eletrodinâmica quântica (QED) a um loop / Abstract: This work is a review of the Negative Dimension Integration Method as a powerful tool for the computation of the radiative corrections present in Quantum Field Perturbation Theory. This method is applicable in the context of Dimensional Regularization and it provides exact solutions for Feynman integrals with both dimensional parameter and propagator exponents generalized. These solutions are presentedintheformoflinearcombinationsofhypergeometricfunctionswhosedomains of convergence are related to the analytic structure of the Feynman Integral. Each solution is connected to the others trough analytic continuations. Besides presenting and discussing the general algorithm of the method in a detailed way, we offer concrete applications to scalar one-loop and two-loop integrals as well as to the one-loop renormalizationofQuantumElectrodynamics (QED) / Mestre

archives@tulane.edu / In this work, five algorithms of negative dimensional integration (NDIM) are compared in several examples of Feynman diagram calculations, and the resulting solutions are compared. The methods used are the Ricotta method without parametrization, the Ricotta method with Schwinger parametrization, the Suzuki method, the Anastasiou method, and the method of brackets. It is found that for one-loop diagrams, the method of brackets gives the same solution as the other methods, but without requiring analytic continuation of the gamma factors in the solution. For multi-loop diagrams, the method of brackets gives solutions in a simpler form than the other methods, and often gives fewer possible solutions as well. In addition to its use in the evaluation of Feynman diagrams, the method of brackets is also useful when extended to the evaluation of definite integrals over the positive real numbers. This extended method of brackets is applied to several examples of definite integrals, and the five NDIM methods are also used to evaluate these examples when possible. In particular, it is shown that the method of brackets is the only method of NDIM which may be extended to the evaluation of a large class of definite integrals over the positive real numbers. / 1 / Kristina E. VanDusen

archives@tulane.edu / The Feynman diagrams have become a highly valued tool for complex calculations and understanding the physics of elementary particles within the framework of quantum field theory. In this thesis, we present an overview on constructing and utilizing Feynman diagrams in quantum field theory along with an overview of quantum field theory itself. We begin with a review of prerequisite topics then progress to discussing symmetries using Lie groups, algebras, and representation theory. We then use the representations of the Lorentz group to derive the fields in a classical context then proceed with quantization to create the corresponding quantum fields while providing a thorough analysis of each quantum field. Path integrals are constructed for each quantum field by deriving their propagators then the formulas for scattering are derived in the context of quantum field theory. Quantum symmetries are briefly explored with the intention of quantizing classical results such as Noether's theorem. Then we construct interacting quantum field theories and introduce the Feynman diagrams and Feynman rules for different interaction theories and provide examples and applications of the Feynman diagrams. The physics behind the diagrams is carefully analyzed and interpreted. Finally, we conclude this thesis with a summary of what we have covered along with possible routes of study after mastering the contents of this thesis that will lead to current research topics. / 1 / Junhyup Sung

El estudio de la mecánica cuántica en espacios de configuración no triviales dista mucho de estar agotado y constituye un problema de amplio interés actualmente. Por ejemplo, no existe acuerdo sobre cuál es la ecuación de Schrödinger adecuada que contemple la dependencia con respecto a la curvatura espacial de la variedad, es decir el equivalente a una ecuación de Schrödinger para casos de curvatura distinta de cero, la cual en el límite reproduzca la cuántica usual. En esta tesis se estudian métodos de cuantización inspirados en las integrales de Feynman para espacios de configuración que generalizan el euclidiano. En el caso de grupos de Lie con una métrica bi-invariante, se construye un propagador infinitesimal por medio de la integración en el álgebra de Lie del grupo vía el mapa exponencial. Se obtiene una ecuación de Schrödinger modificada que incluye un potencial correspondiente a la curvatura escalar de la variedad. También se estudian métodos de cuantización holomorfa como el desarrollado por B. C. Hall [57, 59, 61, 62], se los relaciona con la transformada de Segal-Bargmann y se los conecta con integrales de Feynman, lo cual nos permite obtener resultados originales. Se define un propagador infinitesimal que genera la evolución cuántica. La medida de integración usada surge de la solución fundamental de la ecuación del calor en la complexificación de la variedad. En el caso de variedades riemannianas conexas orientables de curvatura cero (euclidean space form) se muestra que existe un isomorfismo natural entre el espacio de Hilbert de funciones de cuadrado integrable en el espacio de configuración y el espacio de funciones holomorfas de cuadrado integrable en el espacio fase. Los productos escalares son definidos con una medida dada por la solución fundamental de la ecuación del calor en cada espacio. Este espacio de funciones holomorfas en el espacio fase resulta ser un espacio de Hilbert con núcleo reproductor (reproducing kernel Hilbert space). Haciendo uso de la existencia de un núcleo reproductor se obtiene el isomorfismo mencionado y una integral de Feynman que coincide con las expresiones conocidas para el caso euclidiano, ver [27, 139]. En particular, las euclidean space forms de dimensión 3 orientables compactas presentan especial interés en cosmología, dado que permiten modelar la parte espacial de los llamados modelos de universo plano [34]. Ver el trabajo más reciente de J. Levin et al., en donde se busca desarrollar un modelo cosmológico plausible usando euclidean space forms orientables y compactas de dimensión 3 de acuerdo con los resutados de observaciones del fondo de radiación cósmico [98, 99, 100, 97]. / The study of quantum mechanics on nontrivial configuration spaces is far from being exhausted and it is a topic of current wide interest. For instance, there is no agreement on which is the appropriate Schrödinger equation that considers the dependence on the spatial curvature of the manifold, i. e. the equivalent of a Schrödinger equation for cases of non-zero curvature, which in the limit, reproduces the usual quantum mechanics. In this thesis, quantization methods inspired by Feynman integrals for confi- guration spaces, which generalize the Euclidean case, are studied. In the case of Lie groups with a bi-invariant metric, an infinitesimal propagator is constructed by integrating in the Lie algebra of the group via the exponential map. A modified Schrödinger equation is obtained, which includes a potential corresponding to the scalar curvature of the manifold. Also, holomorphic quantization methods are studied following Hall [57, 59, 61, 62], specifically in association with the Segal-Bargmann transform and the connection with Feynman integrals, which allows us to obtain original results. An infinitesimal propagator is defined, in order to obtain the quantum evolution. The measure of integration used arises from the fundamental solution of the heat equation in the complexification of the manifold. In the case of an orientable connected compact at Riemannian manifold (euclidean space form) it is shown that there is a natural isomorphism between the Hilbert space of square integrable complex functions on the configuration space and the space of square integrable holomorphic functions on the phase space. The scalar products are defined with a measure given by the fundamental solution of the heat equation on each space. This space of holomorphic functions on the phase space turns out to be a reproducing kernel Hilbert space. Taking advantage of the existence of a reproducing kernel, the above mentioned isomorphism and a path integral are obtained, the latter of which coincides with the known expressions in the euclidean case, see [27, 139]. In particular, the 3-dimensional orientable compact euclidean space forms present a particular interest for cosmology, since they could model the spatial part of the flat-universe models [34]. See the most recent works of J. Levin et al. [98, 99, 100, 97], which seek to develop a plausible cosmological model using orientable compact euclidean space forms of dimension 3 in agreement with results of observations made on the cosmic microwave background radiation.

A computer program that symbolically generates and evaluates all Feynman diagrams required for scattering amplitude for exclusive processes is tested, corrected, extended, and brought to operational status. The sensitivity of perturbative QCD predictions for the nucleon form factors, ψ → pp̅, and 𝛾𝛾 → pp̅, to the theoretical uncertainties of the nucleon wave function and the form of the running coupling constant is investigated. A new prediction for the cross-section for 𝛾𝛾 → Δ++ Δ̅++ with sum-rule wave functions is presented. As a product of the development of the computer program, the quark amplitudes for meson-baryon scattering are obtained. Integrations of the quark amplitudes over wave functions are carried out by cutting off singularities. The numerical reliability of the integration and its sensitivity to the cut-off’s and the choice of wave function are investigated. / Ph. D. / incomplete_metadata

The path integral formulation of quantum mechanics generalizes the action principle of classical mechanics. The Feynman path integral is, roughly speaking, a sum over all possible paths that a particle can take between fixed endpoints, where each path contributes to the sum by a phase factor involving the action for the path. The resulting sum gives the probability amplitude of propagation between the two endpoints, a quantity called the propagator. Solutions of the Feynman path integral formula exist, however, only for a small number of simple systems, and modifications need to be made when dealing with more complicated systems involving singular potentials, including the Coulomb potential. We derive a generalized path integral formula, that can be used in these cases, for a quantity called the pseudo-propagator from which we obtain the fixed-energy amplitude, related to the propagator by a Fourier transform. The new path integral formula is then successfully solved for the Hydrogen atom in two and three dimensions, and we obtain integral representations for the fixed-energy amplitude. / Vägintegral-formuleringen av kvantmekanik generaliserar minsta-verkanprincipen från klassisk mekanik. Feynmans vägintegral kan ses som en summa över alla möjliga vägar en partikel kan ta mellan två givna ändpunkter A och B, där varje väg bidrar till summan med en fasfaktor innehållande den klassiska verkan för vägen. Den resulterande summan ger propagatorn, sannolikhetsamplituden att partikeln går från A till B. Feynmans vägintegral är dock bara lösbar för ett fåtal simpla system, och modifikationer behöver göras när det gäller mer komplexa system vars potentialer innehåller singulariteter, såsom Coulomb--potentialen. Vi härleder en generaliserad vägintegral-formel som kan användas i dessa fall, för en pseudo-propagator, från vilken vi erhåller fix-energi-amplituden som är relaterad till propagatorn via en Fourier-transform. Den nya vägintegral-formeln löses sedan med framgång för väteatomen i två och tre dimensioner, och vi erhåller integral-representationer för fix-energi-amplituden.