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This thesis is concerned with the possible sums of perturbation series in mass- less, renormalizable field theories. It shows that, given a free choice of scheme, the limit of the sequence of approximants is arbitrary. Restricting the choice to finite schemes, in particular "zero schemes", yields a perturbatively unique limit to the sequence of approximants. An operational method for calculating perturbative expansions in the class of zero schemes is discussed. A comparison of various optimization schemes is given for a few phenomenological examples in QCD and QED.

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A dissertation presented to The Faculty of Science University of the Witwatersrand Johannesburg in ful lment of the requirements for the degree of Master of Science June 2015 / At nite N, the number of restricted Schur polynomials is greater than, or equal to the number of generalized restricted Schur polynomials. In this dissertation we study this dis- crepancy and explain its origin. We conclude that, for quiver gauge theories, in general, the generalized restricted Shur polynomials correctly account for the complete set of nite N constraints and they provide a basis, while the restricted Schur polynomials only account for a subset of the nite N constraints and are thus overcomplete. We identify several situations in which the restricted Schur polynomials do in fact account for the complete set of nite N constraints. In these situations the restricted Schur polynomials and the gen- eralized restricted Schur polynomials both provide good bases for the quiver gauge theory. Further, we demonstrate situations in which the generalized restricted Schur polynomials reduce to the restricted Schur polynomials and use these results to study the anomalous dimensions for scalar operators in ABJM theory in the SU(2) sector. The operators we consider have a classical dimension that grows as N in the large N limit. Consequently, the large N limit is not captured by summing planar diagrams { non-planar contributions have to be included. We nd that the mixing matrix at two-loop order is diagonalized using a double coset ansatz, reducing it to the Hamiltonian of a set of decoupled oscilla- tors. The spectrum of anomalous dimensions, when interpreted in the dual gravity theory, shows that the energy of the uctuations of the corresponding giant graviton is dependent on the size of the giant. The rst subleading corrections to the large N limit are also considered. These subleading corrections to the dilatation operator do not commute with the leading terms, indicating that integrability probably does not survive beyond the large N limit.

A dissertation submitted to the Faculty of Science, University of the Witwatersrand, Johannesburg, in fulfilment of requirements for the degree of Master of Science. Johannesburg, 2015. / The AdS/CFT correspondence asserts a duality between non-Abelian gauge theories and quantum theories of gravity, established by the value of the gauge coupling . Gerard t'Hooft found that the large N0 limit in non-Abelian Yang-Mills gauge theories results in a planar diagram simpli cation of the topological expansion. In this dissertation, SU(2) gauge theories are written in terms of vector models (making use of collective eld theory to obtain an expression for the Jacobian), a saddle point analysis is performed, and the large N limit taken. Initially this procedure is done for gauge theories dimensionally reduced on T4 and R T3, and then attempted for the full eld theory (without dimensional reduction). In each case this results in an expression for the non-perturbative propagator. A nite volume must be imposed to obtain a gap equation for the full eld theory; directives for possible solutions to this di culty are discussed.

A dissertation submitted to the Faculty of Science, University of the Witwatersrand, Johannesburg, in fulfilment of requirements for the degree of Master of Science. Johannesburg, 30.09.2014. / ABSTRACT In this dissertation we study the action of the one loop dilatation operator on operators with a classical dimension of order N. We consider the su(3) and su(2) sectors. The operators in the su(3) sector are constructed using three complex fields X, Y and Z, while operators in the su(2) sector are constructed from only the two complex fields Y and Z. For the operators in these sectors non-planar diagrams contribute already at the leading order in N and the planar and large N limits are distinct. Although the spectrum of anomalous dimensions in su(3) has been computed for this class of operators, previous studies have neglected certain terms which were argued to be small. After dropping these terms diagonalizing the dilatation operator reduces to diagonalizing a set of decoupled oscillators. In this dissertation we explicitly compute the terms which were neglected previously and show that diagonalizing the dilatation operator still reduces to diagonalizing a set of decoupled oscillators. In the su(2) sector the action of the one loop and the two loop dilatation operator reduces to a set of decoupled oscillators and factorizes into an action on the Z fields and an action on the Y fields. Direct computation has shown that the action on the Y fields is the same at one and two loops. In this dissertation, using the su(2) symmetry algebra as well as structural features of field theory, we give compelling evidence that the factor in the dilatation operator that acts on the Y s is given by the one loop expression, at any loop order. I hereby declare that the content of this dissertation is based on my following original works: • R. de Mello Koch, S. Graham and W. Mabanga, “Subleading corrections to the Double Coset Ansatz preserve integrability” (2013) [arXiv:1312.6230v1 [hep-th]] • R. de Mello Koch, S. Graham and I. Messamah, “Higher Loop Nonplanar Anomalous Dimensions from Symmetry” (2013) [arXiv:1312.6227v1 [hep-th]].

Nesta dissertação, estudamos implicações geradas pela quebra da simetria de Lorentz na Eletrodinâmica Quântica. Analisamos férmions interagindo com um campo eletromagnético nos contextos da mecânica quântica e ao efetuar correções radiativas. Na mecânica quântica, os termos de quebra da simetria de Lorentz foram tratados como perturbações à equação de Dirac, e seus valores esperados no vácuo foram obtidos. Nas correções radiativas, a quebra da simetria de Lorentz foi introduzida nessa interação para que o termo tipo Chern-Simons pudesse ser induzido em (3+1) dimensões. Também discutimos as consequências geradas por este termo sobre as velocidades de propagação de fótons clássicos. / In this dissertation, we study the implications generated by the Lorentz breaking symmetry in quantum electrodynamics. We analyze fermions interacting with an electromagnetic field in the contexts of quantum mechanics and we make radiative corrections. In quantum mechanics, the terms of the Lorentz breaking symmetry were treated as perturbations to the Dirac equation, and their expected values were obtained in a vacuum. In the radiative corrections, the Lorentz breaking symmetry was introduced in this interaction for the Chern-Simons like term could be induced in (3 +1) dimensions. We also discussed the consequences generated by this term on the propagation speeds of classic photons.

Track stiffness is the one of the most critical parameters of the track structure. Its evaluation is important to assess track quality, component performance, localised faults and optimise maintenance periods and activities. Keeping the track stiffness within acceptable range of values is connected with keeping the railway network in a satisfactorily performing condition, allowing thereby upgrade of its capacity (speed, load, intensity). Current railway standards are changing to define loading and stiffness requirements for improved ballasted and ballastless performance under high speed train traffic. In recent years various techniques have been used to measure track deflection which have been also used to validate numerical models to assess various problems within the railway network. Based on recent introduction of the Video Gauge for its application in the civil engineering industry this project provides the proof of effective applicability of this DIC (Digital image correlation) tool for the accurate assessment of track deflection and the calculation of track stiffness through its effective applicability in various track conditions for assessing the stiffness of various track forms including track irregularities where abrupt change in track stiffness occur such as transition zones and rail joints. Attention is given in validation of numerical modelling of the response of insulated rail joints under the passage of wheel load within the goal to improve track performance adjacent to rail joints and contribute to the sponsoring company s product offering. This project shows a means of improving the rail joint behaviour by using external structural reinforcement, and this is presented through numerical modelling validated by laboratory and field measurements. The structural response of insulated rail joints (IRJs) under the wheel vertical load passage is presented to enhance industry understanding of the effect of critical factors of IRJ response for various IRJ types that was served as a parametric FE model template for commercial studies for product optimisation.

This thesis investigates applications of the gauge/gravity duality to strongly coupled quantum field theories. After a review of the duality and of correlators and transport in quantum systems, we present our results on second-order non-conformal hydrodynamics. We derive new Kubo formulae for five second-order transport coefficients in non-conformal relativistic fluids. We then apply these Kubo formulae to a class of non-conformal holographic fluids at infinite coupling. We find strong evidence that the Haack-Yarom identity, known to relate second-order coefficients in conformal holographic fluids at infinite coupling, continues to hold in holographic fluids without conformal symmetry: Within our class of models, we prove that it still holds when leading non-conformal corrections are taken into account, and we show numerically that it is also obeyed beyond leading order. This provides further evidence that the identity may be a universal feature of strongly coupled fluids. Next, we present our results on magnetic spin impurities in strongly correlated systems. We build a holographic two-impurity Kondo model, identifying the inter-impurity interaction as double-trace deformation. Our numerical results for the phase diagram suggest a quantum phase transition between a trivial phase with uncorrelated spins and no Kondo screening, and a non-trivial phase with anti-ferromagnetic correlations and simultaneous Kondo screening. Computing the spectrum in the single-impurity case, we observe Fano resonances, which at low temperatures we identify with the Kondo resonance.