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Thesis (Ph.D.)--Massachusetts Institute of Technology, Dept. of Nuclear Engineering, 1981. / MICROFICHE COPY AVAILABLE IN ARCHIVES AND SCIENCE. / Includes bibliographical references. / by Teymour Boutros-Ghali. / Ph.D.

Strongly coupled systems are elusive and not suitable to be described by conventional perturbative approaches. However, they are ubiquitous in nature, especially in particle physics. The lattice formulation of quantum field theories provided a unique framework in which the physical content of these systems could be precisely determined. Combined with numerical techniques, the lattice formalism allowed to precisely determined physical quantities describing the thermodynamics, as well as the spectroscopy of strongly interacting theories. In this work, the lattice formulation has been employed to probe the effectiveness of a recently proposed method, which aims at determining the renormalized energy-momentum tensor in non perturbative regimes. The latter plays a fundamental role to quantitatively describe the thermodynamics and fluid-dynamics of hot, dense systems, or to characterize theories that enlarge the actual standard model. In all these aspects, only a non perturbative approach provides physically reliable results: hence a non perturbative determination of the energy momentum tensor is fundamental. The new method consists in defining suitable lattice Ward identities probed by observables built with the gradient flow. The new set of identities exhibits many interesting qualities, arising from the UV finiteness of such probes, and allows to define a numerical strategy for estimating the renormalization constants of the lattice energy-momentum tensor. In this work the method has been tested within two different quantum theories, with the purpose of understanding its effectiveness and reliability.

For the latter half of the last century renormalization methods have played an important part in tackling problems in fundamental physics and in providing a deeper understanding of systems with many interacting scales or degrees of freedom with strong coupling. The study of turbulence is no exception, and this thesis presents an investigation of renormalization techniques available in the study of the statistical theory of homogeneous and isotropic turbulence. The thesis consists of two parts which assess the two main renormalization approaches available in modeling turbulence. In particular we will be focusing on the renormalization procedures developed by McComb and others. The first part of this thesis will discuss Renormalization Group (RG) approaches to turbulence, with a focus on applications to reduce the degrees of freedom in a large-eddy simulation. The RG methods as applied to classical dynamical systems will be reviewed in the context of the Navier-Stokes equations describing fluid flow. This will be followed by introducing a functional based formalism of a conditional average first introduced by McComb, Roberts and Watt [Phys. Rev A 45, 3507 (1992)] as a tool for averaging out degrees of freedom needed in an RG calculation. This conditional average is then used in a formal RG calculation applied to the Navier-Stokes equations, originally done by McComb and Watt [Phys. Rev. A 46, 4797 (1992)], and later revised by Mc- Comb and Johnston [Physica A 292, 346 (2001)]. A correction to the summing of the time-integral detailed in the latter work is shown to introduce an extra viscous life-time term to the denominator of the increment to the renormalized viscosity and is shown to have a negligible effect in the numerical calculations. We follow this study by outlining some problems with the previous approach. In particular it is shown that a cross-term representing the interaction between high and low wavenumber modes which was neglected in the previous studies on the grounds that it does not contribute to energy dissipation, does in fact contribute significantly. A heuristic method is then put forward to include the effects of this term in the RG calculation. This leads to results which agree qualitatively with numerical calculations of eddy-viscosities. We finish this part of the thesis with an application of the RG method to the modeling of a passive scalar advected by a turbulent velocity field. The second part of this thesis will begin by reviewing Eulerian renormalized perturbation theory attempts in closing the infinite moment hierarchy introduced by averaging the Navier-Stokes equations. This is followed by presenting a new formulation of the local energy transfer theory (LET) of McComb et. al. [J. Fluid Mech. 245, 279 (1992)] which resolves some problems of previous derivations. In particular we show by the introduction of time-ordering that some previous problems with the exponential representation of the correlator can be overcome. Furthermore, we show that the singularity in the LET propagator equation cancels by way of a counter-term. We end this study by introducing a single-time Markovian closure based on LET which, unlike other Markovian closures, does not rely on any arbitrary parameters being introduced in the theory.

Estendemos para a teria de campos o método variacional de Kleinert. Este método foi primeiramente usado na mecânica quântica e fornece uma expansão em cumulantes convergente. Sua extensão para a teoria de campos não é trivial devido às divergências ultravioletas que aparecem quando a dimensão do espaço é maior que 2. Devido a estas divergências, a teoria deve ser regularizada e normalizada. Além das dificuldades usuais associadas com a renormalização, devemos decidir se calculamos o valor ótimo do parâmetro variacional antes ou depois da renormalização. Nesta tese abordamos o problema da renormalização do potencial efetivo variacional. Primeiramente, mostramos que o potencial efetivo variacional em temperatura zero coincide com o \"potencial efetivo pós-gaussiano\" introduzido por Stancu e Stevenson. Em seguida, apresentamos um esquema de renormalização que permite que renormalizemos a teoria antes de calcular o parâmetro variacional ótimo. Usando este esquema mostramos que o potencial efetivo usual, calculado em ordem 1-loop, pode ser obtido a partir do esquema variacional de Kleinert inteirando uma única vez a equação que determina o parâmetro variacional. Para o potencial efetivo em ordem 2-loops esta aproximação não é tão boa. A renormalização da teoria antes do cálculo do parâmetro variacional permite que estudemos o potencial efetivo variacional numericamente e de forma não-perturbativa, como foi feito por Kleinert para a mecânica quântica. / We have extended the Kleinert variational technique to field theory. This method was first used in quantum mechanics and provides a convergent cumulate expansion that is extremely accurate. Its extension to field theory is non-trivial because of the ultraviolet divergences that appear when the space dimension is greater than 2. Due to these divergences the theory has to be regularized and renormalized. In addition to the usual difficulties associated with renormalization, one has to decide whether one calculates the optimum value of the variational parameter before or after renormalization. In this thesis we deal with the renormalization of the variational effective potential. Firstly, we show that the zero temperature regularized variational potential coincides with the post-Gaussian effective potential introduced by Stancu and Stenvenson. Secondly, we present a renormalization scheme that enables one to renormalize the theory before calculating the optimum variational parameter. Using this scheme we show that the usual 1-loop effective potential can be obtained from the Kleinert variational scheme by interacting only once the equation that determines the variational parameter. In this sense, the 1-loop expansion is contained within the variational scheme. For the 2-loop effective potential the same approximation is not so good. The renormalization of the theory before the calculation of the variational parameter allows one to study the variational effective potential numerically and in a non-pertubative way, as it was done in quantum mechanics by Kleinert.

Sem dúvida a questão central em Dinâmica Holomorfa é aquela sobre a densidade de hiperbolicidade. Temos a seguinte conjectura devida a Pierre Fatou: No espaço das aplicações racionais de grau d o conjunto das aplicações racionais hiperbólicas neste espaço formam um subconjunto aberto e denso. Nem mesmo para a família dos polinômios quadráticos esta questão foi respondida. Para a família quadrática este problema é equivalente a mostrar a não existência de polinômios quadráticos que suportam sobre o seu conjunto de Julia um campo de linhas invariante. Devido a resultados de Jean-Christophe Yoccoz sabemos da não existência de campos de linhas invariante para polinômios quadráticos no máximo finitamente renormalizáveis. Nesta dissertação é mostrado que um polinômio quadrático infinitamente renormalizável satisfazendo certa hipótese geométrica, denominada robustez, não suporta sobre o seu Julia um campo de linhas invariante. Esta prova foi obtida por Curtis T. McMullen e publicada em [McM1]. Os avanços na teoria de renormalização e quanto ao problema da densidade de hiperbolicidade e problemas relacionados tem contado com a colaboração de inúmeros renomados matemáticos como Mikhail M. Lyubich, Artur Ávila, Mitsuhiro Shishikura, Curtis T. McMullen, Jean-Christophe Yoccoz, Sebastien van Strien, Hiroyuki Inou, dentre outros / Undoubtedly one of the central open questions in Holomorphic Dynamics is about proving the density of hyperbolicity. That question was first raised by Pierre Fatou: In the space of rational functions of degree d the set of hyperbolic rational functions form a open and dense subset. Not even for the family of quadratic polynomials this question been answered. For this particular quadratic family the problem is equivalent to showing the non-existence of quadratic polynomial with a Julia set supporting an invariant line field. Due to results by Jean-Christophe Yoccoz we already know the non-existence of invariant line fields for the quadratic polynomials that are at most finitely renormalizable. In this dissertation it is shown that an infinitely renormalizable quadratic polynomial satisfying a certain geometric hypotesis, called robustness, does not have an invariant line field supported on its Julia set. This proof was obtained by Curtis T. McMullen and published in [McM1]. Many advances on the theory of renormalization and on the problem of density of hyperbolicity have been already accomplished through the collective work of several renowned mathematicians such as Mikhail M. Lyubich, Artur Ávila, Mitsuhiro Shishikura, Curtis T. McMullen, Jean-Christophe Yoccoz, Sebastien van Strien, Hiroyuki Inou among others.

Nós estudamos a dinâmica do operador de renormalização atuando no espaço de pares (?, t), onde ? é um difeomorfismo e t ? [0, 1], interpretados como aplicações unimodais ? o qt, onde qt(x) = -2t|x|? + 2t - 1. Estabelecemos cotas complexas a priori para pares suficientemente renormalizáveis com combinatória limitada e então a utilizamos para mostrar que quando o expoente crítico ? está próximo de um número par, o operador de renormalização tem um único ponto fixo, o qual é hiperbólico e possui uma variedade estável de codimensão um que contém todos os pares infinitamente renormalizáveis / We study the dynamics of the renormalization operator acting on the space of pairs (?, t), where ? is a diffeomorphism and t ? [0, 1], interpretated as unimodal maps ? o qt, where qt(x) = -2t|x|? + 2t - 1. We prove the so called complex bounds for sufficiently renormalizable pairs with bounded combinatorics. This allows us to show that if the critical exponent ? is close to an even number then the renormalization operator has a unique fixed point. Furthermore this fixed point is hyperbolic and its codimension one stable manifold contains all infinitely renormalizable pairs

O comportamento critico de sistemas unidimensionais de spin do tipo Z(N) na ausência de campos magnéticos, é estudado sob a luz da teoria do grupo de renormalização. Os modelos são resolvidos exatamente pelo método da matriz de transferência e expressões para as funções de correlação e susceptibilidade (a campo zero) por si tio são também calculadas. As transformações do grupo de renormalização são efetuadas através de um traço parcial na função de partição, obtendo- se um conjunto de relações de recorrência que podem ser escritas de maneira simples para qualquer valor inteiro do fator de reescala espacial, mediante o uso de campos de escala convenientes. Tirando vantagem de um ponto fixo inteiramente atrativo, calculamos uma expressão para a energia livre por sitio, exata para T ¢ O. Analisamos o comportamento de nossos modelos no espaço de parâmetros, onde identificamos em particular as ~s ferro e antiferromagnéticas. O problema de correções às previsões de escala em termos de campos de escala não lineares é discutido. Aventamos também a possibilidade de calcular os auto valores da matriz de transferência através dos campos não lineares / In this work we study the criticai behaviour of one dimensional Z(N) spin systems in zero magnetic fields, using the approach of the renormalization group (RG) theory. The models are solved by the transfer matrix method and expressions for the correlation functions and zero field susceptibility per site are found. The RG transformations are carried out via a partial trace over the partition function and one obtains a set of recursion relations which, with the use of a convenient set of scaling fields, are written out in a simple manner for any integer value of the spatial rescaling factor. Using a totaly attractive fixed point we calculate an expression for the free energy per site, valid exactly for non zero values of the temperature. We analyse the behaviour of our models in the space of parameters, identifying in particular ferro and antiferromagnetic regions. The problem of corrections to scaling in terms of nonlinear scaling fields is discussed and a possibility of finding the eigen values of the transfer matrix from such fields is contemplated

The formulation of rigid supersymmetric field theories on curved space leads to a number of results on their strongly-interacting regime, crucial from both the mathematical and physical point of view, starting from Witten's topological twist of four-dimensional Yang-Mills theory. At the same time, strongly-coupled field theories may also be studied holographically via the AdS/CFT correspondence. The aim of this thesis is to study aspects of the holographic dictionary for supersymmetric theories on curved manifolds. A key aspect of the correspondence is the renormalization of gravity observables, which is realized via holographic renormalization. If the dual boundary field theory is supersymmetric, it is natural to ask whether this scheme is compatible with the rigid supersymmetry at the curved boundary. The latter requires specific geometric structures, and general arguments imply that BPS observables, such as the partition function, are invariant under certain deformations of these structures. We may then formulate a precise check of the holographic dictionary by asking whether the dual holographic observables are similarly invariant, as the free energy of the gauge theory is identified with the holographically renormalized supergravity action. In the first part of the thesis, we consider this question in N = 4 gauged supergravity in four and five dimensions for the holographic dual to the topological twists of N = 4 gauge theories on Riemannian three-manifolds and N = 2 gauge theories on Riemannian four-manifolds. We show that the renormalized on-shell action is independent of the metric on the boundary four-manifold, as required for a topological theory. We then go further, analyzing the geometry of supersymmetric bulk solutions. This allows us to show that the gravitational free energy of any smooth filling vanishes in both AdS₄/CFT₃ and AdS₅/CFT₄. In the second part of the thesis, we study the same question in minimal N = 2 gauged supergravity in four and five dimensions. In four dimensions we show that holographic renormalization precisely reproduces the expected field theory results for the dependence of the partition function on the background. Surprisingly, in five dimensions we find that no choice of standard holographic counterterms is compatible with supersymmetry, which leads us to introduce novel finite boundary terms. For a class of solutions satisfying certain topological assumptions we provide some independent tests of these new boundary terms, in particular showing that they reproduce the expected VEVs of conserved charges. We also briefly comment on the relation between these terms and boundary supercurrent anomalies.

by Siu Wing Hon. / Thesis (Ph.D.)--Chinese University of Hong Kong, 1996. / Includes bibliographical references. / by Siu Wing Hon. / Acknowledgement --- p.i / Abstract --- p.ii / Publication List --- p.iv / Chapter 1 --- Introduction --- p.1 / References --- p.6 / Chapter 2 --- Real-Space Renormalization Group (RG) Theory in Electrical Conduction --- p.9 / Chapter 2.1 --- Scale Invariance --- p.10 / Chapter 2.2 --- Critical Exponents --- p.14 / Chapter 2.3 --- Alternative View-Point of RG Theory --- p.15 / References --- p.18 / Chapter 3 --- "Weakly Nonlinear Composites: Critical Behavior, Flicker Noise and Crossover Behavior" --- p.19 / Chapter 3.1 --- Introduction --- p.19 / Chapter 3.2 --- Formalism --- p.20 / Chapter 3.3 --- Critical Exponents by RG Method --- p.22 / Chapter 3.4 --- Connection to Flicker Noise Problem and Crossover Behavior --- p.25 / Chapter 3.5 --- Discussions and Conclusions --- p.27 / References --- p.28 / Chapter 4 --- Critical Behavior of Strongly Nonlinear Composites --- p.30 / Chapter 4.1 --- Introduction --- p.30 / Chapter 4.2 --- Formalism --- p.31 / Chapter 4.3 --- Applications of RG Theory to Strongly Nonlinear Composites --- p.32 / Chapter 4.4 --- Connections with Links-Nodes-Blobs picture --- p.36 / Chapter 4.5 --- Discussions and Conclusions --- p.39 / References --- p.41 / Chapter 5 --- "Enhanced Nonlinear Response of Superconductor-Normal-conductor Composite Wires, Strips and Rods" --- p.43 / Chapter 5.1 --- Introduction --- p.43 / Chapter 5.2 --- Formalism --- p.45 / Chapter 5.3 --- Linear and Nonlinear Responses of Composite Wires --- p.46 / Chapter 5.4 --- Linear and Nonlinear Response of Composite Strips --- p.49 / Chapter 5.5 --- Linear and Nonlinear Responses of Composite Rods --- p.56 / Chapter 5.6 --- Scaling Behaviors --- p.59 / Chapter 5.7 --- Discussions and Conclusions --- p.63 / References --- p.64 / Chapter 6 --- Renormalized Effective Medium Theory for Weakly Nonlinear Composites --- p.66 / Chapter 6.1 --- Introduction --- p.66 / Chapter 6.2 --- Weakly Nonlinear Conductance Network --- p.69 / Chapter 6.3 --- Simulation --- p.70 / Chapter 6.4 --- Effective Medium Approximation --- p.76 / Chapter 6.5 --- Renormalized Effective Medium Approximation --- p.79 / Chapter 6.6 --- Discussion and Conclusions --- p.81 / References --- p.83 / Chapter 7 --- Conclusions --- p.86 / Chapter A --- Derivation of Voltage-Summation Formulas --- p.88 / Chapter B --- Effective Linear and Nonlinear Response of 2 x 2 cell --- p.92 / Chapter C --- Duality Symmetry in 2D Network --- p.97 / Chapter D --- Derivation of Effective-Medium Approximation --- p.99