Starting from a mathematical basis where one analyses and developing different techniques in how to solve and represent different kinds of integrals with diagrams. Representing the integrals as n-valent vertices and introducing propagators is a great tool that helps with the book-keeping of the solutions and will sometimes do the calculations redundant. The symmetries of diagrams are analysed and how one extracts the symmetry factors from looking at a diagram by using some fairly simple combinatorics and cleverness. Introducing the probability amplitude and do some analysis of the path integral formulation is the step into physics. Discussing experiments as the double-slit experiment and deriving the Schrödinger equation from the generating functional. Looking at diagrams in Quantum Mechanics and Quantum Field Theory will explore the use of the crucial understanding of our generators for the diagrams. This thesis makes use of the so called generating functional almost throughout and to connect the first discussed mathematics to real physical theories is the aim.

The study of scattering amplitudes beyond one loop is necessary for precision phenomenology for the Large Hadron Collider and may also provide deeper insights into the theoretical foundations of quantum field theory. In this thesis we develop a new method for computing two-loop amplitudes, based on unitarity rather than Feynman diagrams. In this approach, the two-loop amplitude is first expanded in a linearly independent basis of integrals. The process dependence thereby resides in the coefficients of the integrals. These expansion coefficients are then the object of calculation. Our main results include explicit formulas for a subset of the integral coefficients, expressing them as products of tree-level amplitudes integrated over specific contours in the complex plane. We give a general selection principle for determining these contours. This principle is then applied to obtain the coefficients of integrals with the topology of a double box. We show that, for four-particle scattering, each double-box integral in the two-loop basis is associated with a uniquely defined complex contour, referred to as its master contour. We provide a classification of the solutions to setting all propagators of the general double-box integral on-shell. Depending on the number of external momenta at the vertices of the graph, these solutions are given as a chain of pointwise intersecting Riemann spheres, or a torus. This classification is needed to define master contours for amplitudes with arbitrary multiplicities. We point out that a basis of two-loop integrals with as many infrared finite elements as possible allows substantial technical simplications, in terms of obtaining the coefficients of the integrals, as well as for the analytic evaluation of the integrals themselves. We compute two such integrals at four points, obtaining remarkably compact expressions. Finally, we provide a check on a recently developed recursion relation for the all-loop integrand of the amplitudes of N=4 supersymmetric Yang-Mills theory, examining the two-loop six-gluon MHV amplitude and finding agreement. The validity of the approach to two-loop amplitudes developed in this thesis extends to all four-dimensional gauge theories, in particular QCD. The approach is suited for obtaining compact analytical expressions as well as for numerical implementations.

Made available in DSpace on 2017-07-21T19:53:47Z (GMT). No. of bitstreams: 1 Elisangela Meira dos Santos.pdf: 1220706 bytes, checksum: 1387090400d1c0df8cb0bb5c88039b69 (MD5) Previous issue date: 2016-12-15 / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior / In this work we review the diferent distinct aspects associated with quantum dynamics in graphs. The approach used is that of the Green's functions, dened in the feld of energy. In this technique the Green's functions are obtained through the sum of all possible classical paths connecting the initial and Final points, the style of Feynman path integrals. Local quantum efects are included through the use of quantum amplitudes for reection and transmission are defined in each of the vertices of the graph. We show that in order to get the exact Green function can use the adjacency matrix of the graph, which sets the links between the vertices of the graph. This approach ensures that all paths have been accounted for and has the advantage of providing a rating only for the various families of paths that can be defined in a given graph. An application of the method to study resonances in graphs is also presented. / Neste trabalho revisamos os diferentes aspectos distintos associados a dinâmica quântica em grafos. A abordagem utilizada è aquela das funções de Green de únidas no domínio da energia. Nessa técnica as funções de Green são obtidas através da soma de todos os possíveis caminhos clássicos, ao estilo das integrais de caminho de Feynman. Os efeitos quânticos locais são incluídos através da utilização das amplitudes quânticas de reflexão e transmissão de únidas em cada um dos vértices do grafo. Mostramos que para obtermos a função de Green exata podemos utilizar a matriz de adjacência do grafo, a qual define as ligações existentes entre os vértices do grafo. Essa abordagem garante que todos os caminhos foram contabilizados e tem a vantagem de fornecer uma classificação única para as diferentes famílias de caminhos que podem ser de únidas em um dado grafo. Uma aplicação do método para o estudo ressonâncias em grafos é também apresentada.

Este trabalho apresenta um modelo matemático linear para um sistema hidráulico proposto para funcionar como um amortecedor para pequenas amplitudes de deslocamento, da ordem de décimos de milímetros. As dimensões para o sistema hidráulico foram adotadas e parâmetros foram determinados. Foram escolhidos dois conjuntos de parâmetros dimensionais e dois valores para o bulk modulus equivalente, o que permitiu fazer o estudo de quatro situações. A verificação das características dinâmicas do sistema foi efetuada utilizando o conceito de função descritiva. Um programa em Matlab/Simulink com um filtro de Fourier foi implementado para obtenção das curvas da resposta em freqüência. A eficiência deste programa computacional foi comprovada através de sua aplicação a sistemas cujas respostas em freqüência são conhecidas. As curvas das respostas em freqüência do sistema hidráulico e do amortecedor ideal foram comparadas. Considerando tolerância de \'+ OU -\' 1 grau para a fase e de \'+ OU -\' 1 db para magnitude, o sistema hidráulico simulado, com bulk modulus equivalente de 100000 psi teve comportamento similar ao amortecedor ideal na faixa de freqüência de 0,1 a 150 rad/seg. No caso do bulk modulus de 30000 psi, a faixa foi estendida, isto é, de 0,1 a 200 rad/seg. / This work presents a linear mathematical model for a proposed hydraulic system intended to work as a damper for small displacement, of the order of decimals of millimeters. The dimensions of the hydraulic system were adopted and parameters were determined. Two sets of the dimensions and two values for the equivalent bulk modulus were chosen and that allowed the study of four situations. The system dynamic properties were verified using the concept of describing function. A Matlab/Simulink program applying the concept of a Fourier filter was carried out in order to obtain the frequency response curves. The efficiency of this program was verified through its application in systems which frequency responses are known. The systems and the ideal damper frequency response curves were compared. Considering tolerance of \'+ OU -\' 1 degree for the phase and \'+ OU -\' 1 db for magnitude, the simulated hydraulic system with an equivalent bulk modulus of 100000 psi showed to be similar to an ideal damper, for the frequency range from 0.1 to 150 rad/sec. For the equivalent bulk modulus of 30000 psi, the frequency range was extended from 0.1 to 200 rad/sec.

Este trabalho apresenta um modelo matemático linear para um sistema hidráulico proposto para funcionar como um amortecedor para pequenas amplitudes de deslocamento, da ordem de décimos de milímetros. As dimensões para o sistema hidráulico foram adotadas e parâmetros foram determinados. Foram escolhidos dois conjuntos de parâmetros dimensionais e dois valores para o bulk modulus equivalente, o que permitiu fazer o estudo de quatro situações. A verificação das características dinâmicas do sistema foi efetuada utilizando o conceito de função descritiva. Um programa em Matlab/Simulink com um filtro de Fourier foi implementado para obtenção das curvas da resposta em freqüência. A eficiência deste programa computacional foi comprovada através de sua aplicação a sistemas cujas respostas em freqüência são conhecidas. As curvas das respostas em freqüência do sistema hidráulico e do amortecedor ideal foram comparadas. Considerando tolerância de \'+ OU -\' 1 grau para a fase e de \'+ OU -\' 1 db para magnitude, o sistema hidráulico simulado, com bulk modulus equivalente de 100000 psi teve comportamento similar ao amortecedor ideal na faixa de freqüência de 0,1 a 150 rad/seg. No caso do bulk modulus de 30000 psi, a faixa foi estendida, isto é, de 0,1 a 200 rad/seg. / This work presents a linear mathematical model for a proposed hydraulic system intended to work as a damper for small displacement, of the order of decimals of millimeters. The dimensions of the hydraulic system were adopted and parameters were determined. Two sets of the dimensions and two values for the equivalent bulk modulus were chosen and that allowed the study of four situations. The system dynamic properties were verified using the concept of describing function. A Matlab/Simulink program applying the concept of a Fourier filter was carried out in order to obtain the frequency response curves. The efficiency of this program was verified through its application in systems which frequency responses are known. The systems and the ideal damper frequency response curves were compared. Considering tolerance of \'+ OU -\' 1 degree for the phase and \'+ OU -\' 1 db for magnitude, the simulated hydraulic system with an equivalent bulk modulus of 100000 psi showed to be similar to an ideal damper, for the frequency range from 0.1 to 150 rad/sec. For the equivalent bulk modulus of 30000 psi, the frequency range was extended from 0.1 to 200 rad/sec.

Purpose To determine the test-retest reproducibility of accommodation measurements gathered in an unselected sample of primary school children. Methods Monocular and binocular amplitudes of accommodation (AA) were collected by five different Testers using the push-up method in an unselected sample of school children (n=137, age: 8.1±2.1-years). Testing was conducted on three occasions (average testing interval: 8-days) in 91.2% of the children. Results The median AA was 19.1D, the variation due to the identity of the Tester was 3.1D (p<0.001) and the within-subject variation (which takes the variation due to Tester identity into account) was 5.2D. Around 75-79% of children exhibited monocular AAs-12D when tested on the first occasion, but more than 90% exhibited an AA-12D when subsequently tested. Around 74-80% of those with an AA<12D on the first occasion had values-12D on subsequent testing even though no treatment had been undertaken. Poorer initial AA measurements were less likely to improve on repeat testing. Conclusions Our results reveal substantial intra-individual variation in AA measurements, raising questions about the usefulness of this test in children aged 4-12-years. We suggest that AA assessment may prove most useful in children in this age range as a pass/fail check for substantially reduced AA, for example, where the AA is <12D. Our sample would suggest that the prevalence of persistently reduced AA may be around 3.2% when tested under binocular conditions and 4-6.4% when tested monocularly.

The double-copy construction enables the calculation of scattering amplitudes in theories of gravity by combining amplitudes from gauge theories. It relies on obtaining numerators that obey a duality between color and numerator factors, called color-kinematics duality. This construction is reviewed, along with the spinor-helicity formalism for onshell states and supersymmetry in amplitudes. Using generalized unitarity, a one-loop amplitude is verified from literature for a N = 2 theory obeying color-kinematics duality. This amplitude, along with a one-loop amplitude for a N = 0 theory are combined with the double copy in order to produce one-loop amplitudes from homogeneous supergravities. The one-loop divergence is studied with the methods of counterterm analysis, that is, operators necessary to cancel the on-shell matrix element of the divergence are identified for the amplitudes studied. It is interesting to note that all vectors produced from the double copy behave in the same way, that is, have the same divergence, for the four special cases of the magical supergravities. Furthermore, one of the counterterms vanishes for these four special cases, which is likely related to the enhanced symmetry that these theories posses.

The laser Raman spectra of tetrachlorocyclopropene in the liquid and solid phases have been recorded and vibrational assignments are presented. These results along with root mean squared vibrational amplitudes from electron diffraction data have been employed in a normal coordinate analysis in which a 19 parameter potential function is refined. This potential function, originally expressed in terms of compliance constants, is then used to derive the corresponding conventional and relaxed force constants.

This thesis integrated geology, geophysics, and petroleum engineering data to build a detailed reservoir characterization models for three gas pay sands in the Grand Isle 33 & 43 fields, offshore Louisiana. The reservoirs are Late Miocene in age and include the upper (PM), middle (QH), and lower (RD) sands. The reservoir models address the stratigraphy of the upper (PM) sand and help delineate the lower (RD) reservoir. In addition, this research addresses the partially depleted QH-2 reservoir compartment. The detailed models were constructed by integrating seismic, well log, and production data. These detailed models can help locate recoverable oil and gas that has been left behind. The upper PM model further delineated that the PM sand has several areas that are shaled-out effectively creating a flow barrier within reservoir compartments. Due to the barrier in the PM-1 reservoir compartment, an area of potentially recoverable hydrocarbons remains. In Grand Isle 33, the middle QH sand was partially depleted in the QH-2 reservoir compartment by a series of development wells. Bottom hole pressure data from wells in Grand Isle 32 & 33 reveal that the two QH fault compartments are in communication across a leaking fault. Production wells in the QH-1 compartment produced reserves from the QH-2 compartment. The lower RD sand model helped further delineate the reservoir in the RD-2 compartment and show that this compartment has been depleted. The RD model also shows the possible presence of remaining recoverable hydrocarbons in the RD-1 compartment. It is estimated that about 6.7 billion cubic feet of gas might remain within this reservoir waiting to be recovered. A seismic amplitude anomaly response from the QH and RD sands is interpreted to be a lithologic indicator rather than the presence of hydrocarbons. Amplitude response from the PM level appears to be below the resolution of the seismic data. A synthetic seismogram model was generated to represent the PM and surrounding sands. This model shows that by increasing the frequency of the seismic data from 20 Hz to a dominant frequency of 30 Hz that the PM and surrounding sands could be seismically resolvable. Also the PM-1 compartment has possible recoverable hydrocarbons of 1.5 billion cubic feet of gas remaining.

Quantum Field Theory (QFT) provides the essential background for formulating the standard model of elementary particles and, moreover, practically all other theories attempting to explore the physical laws of nature at the sub-atomic level. One of the main observables in QFT are the scattering amplitudes, physical quantities which encode the information of the scattering process of particles. Accordingly, having authentic, well-defined and feasible prescriptions for the calculations of amplitudes is of huge importance to theoretical physicists. Actual calculations show that the text-book prescription, the Feynman method, besides in general being very cumbersome also hides some of the beautiful mathematical features of amplitudes. The last decade has seen tremendous efforts and achievements to improve such calculations, particularly in supersymmetric gauge theories, which have also led to better understanding of QFT itself. Among the known physically and mathematically interesting quantum field theories is perturbative gravity and its supersymmetric version, N=8 supergravity- much less understood than gauge theory. Following the developments in gauge theory, this dissertation mainly aims at exploring scattering amplitudes in gravity as a quantum field theory, using the modern approaches to QFT. The goal is not only to improve our understanding of gravity amplitudes by applying part of the known modern methods of calculations to it but also to introduce and develop new ones.