Thesis (MSc (Physics))--University of Stellenbosch, 2010. / ENGLISH ABSTRACT: Due to the minimal length scale induced by non-commuting co-ordinates, it is not clear a priori what is meant by a position measurement on a non-commutative space. It was shown recently in a paper by Scholtz et al. that it is indeed possible to recover the notion of quantum mechanical position measurements consistently on the non-commutative plane. To do this, it is necessary to introduce weak (non-projective) measurements, formulated in terms of Positive Operator-Valued Measures (POVMs). In this thesis we shall demonstrate, however, that a measurement of position alone in non-commutative space cannot yield complete information about the quantum state of a particle. Indeed, the aforementioned formalism entails a description that is non-local in that it requires knowledge of all orders of positional derivatives through the star product that is used ubiquitously to map operator multiplication onto function multiplication in non-commutative systems. It will be shown that there exist several equivalent local descriptions, which are arrived at via the introduction of additional degrees of freedom. Consequently non-commutative quantum mechanical position measurements necessarily confront us with some additional structure which is necessary (in addition to position) to specify quantum states completely. The remainder of the thesis, based in part on a recent publication (\Noncommutative quantum mechanics { a perspective on structure and spatial extent", C.M. Rohwer, K.G. Zloshchastiev, L. Gouba and F.G. Scholtz, J. Phys. A: Math. Theor. 43 (2010) 345302) will involve investigations into the physical interpretation of these additional degrees of freedom. For one particular local formulation, the corresponding classical theory will be used to demonstrate that the concept of extended, structured objects emerges quite naturally and unavoidably there. This description will be shown to be equivalent to one describing a two-charge harmonically interacting composite in a strong magnetic eld found by Susskind. It will be argued through various applications that these notions also extend naturally to the quantum level, and constraints will be shown to arise there. A further local formulation will be introduced, where the natural interpretation is that of objects located at a point with a certain angular momentum about that point. This again enforces the idea of particles that are not point-like. Both local descriptions are convenient, in that they make explicit the additional structure which is encoded more subtly in the non-local description. Lastly we shall argue that the additional degrees of freedom introduced by local descriptions may also be thought of as gauge degrees of freedom in a gauge-invariant formulation of the theory. / AFRIKAANSE OPSOMMING: As gevolg van die minimum lengteskaal wat deur nie-kommuterende ko ordinate ge nduseer word is dit nie a priori duidelik wat met 'n posisiemeting op 'n nie-kommutatiewe ruimte bedoel word nie. Dit is onlangs in 'n artikel deur Scholtz et al. getoon dat dit wel op 'n nie-kommutatiewe vlak moontlik is om die begrip van kwantummeganiese posisiemetings te herwin. Vir hierdie doel benodig ons die konsep van swak (nie-projektiewe) metings wat in terme van 'n positief operator-waardige maat geformuleer word. In hierdie tesis sal ons egter toon dat 'n meting van slegs die posisie nie volledige inligting oor die kwantumtoestand van 'n deeltjie in 'n niekommutatiewe ruimte lewer nie. Ons formalisme behels 'n nie-lokale beskrywing waarbinne kennis oor alle ordes van posisieafgeleides in die sogenaamde sterproduk bevat word. Die sterproduk is 'n welbekende konstruksie waardeur operatorvermenigvuldiging op funksievermenigvuldiging afgebeeld kan word. Ons sal toon dat verskeie ekwivalente lokale beskrywings bestaan wat volg uit die invoer van bykomende vryheidsgrade. Dit beteken dat nie-kommutatiewe posisiemetings op 'n natuurlike wyse die nodigheid van bykomende strukture uitwys wat noodsaaklik is om die kwantumtoestand van 'n sisteem volledig te beskryf. Die res van die tesis, wat gedeeltelik op 'n onlangse publikasie (\Noncommutative quantum mechanics { a perspective on structure and spatial extent", C.M. Rohwer, K.G. Zloshchastiev, L. Gouba and F.G. Scholtz, J. Phys. A: Math. Theor. 43 (2010) 345302) gebaseer is, behels 'n ondersoek na die siese interpretasie van hierdie bykomende strukture. Ons sal toon dat vir 'n spesi eke lokale formulering die beeld van objekte met struktuur op 'n natuurlike wyse in die ooreenstemmende klassieke teorie na vore kom. Hierdie beskrywing is inderdaad ekwivalent aan die van Susskind wat twee gelaaide deeltjies, gekoppel deur 'n harmoniese interaksie, in 'n sterk magneetveld behels. Met behulp van verskeie toepassings sal ons toon dat hierdie interpretasie op 'n natuurlike wyse na die kwantummeganiese konteks vertaal waar sekere dwangvoorwaardes na vore kom. 'n Tweede lokale beskrywing in terme van objekte wat by 'n sekere punt met 'n vaste hoekmomentum gelokaliseer is sal ook ondersoek word. Binne hierdie konteks sal ons weer deur die begrip van addisionele struktuur gekonfronteer word. Beide lokale beskrywings is gerie ik omdat hulle hierdie bykomende strukture eksplisiet maak, terwyl dit in die nie-lokale beskrywing deur die sterproduk versteek word. Laastens sal ons toon dat die bykomende vryheidsgrade in lokale beskrywings ook as ykvryheidsgrade van 'n ykinvariante formulering van die teorie beskou kan word.

In this thesis we are concerned with the existence of infrared fixed points and the conformal window for gauge theories with fermions. We are particularly interested in those theories that are candidates for walking technicolor. We discuss the background of technicolor and the techniques relevant to a theoretical understanding of the conformal window. Following this we extend the ideas of metric confinement and causal analyticity to theories with fermions in non-fundamental representations. We use these techniques to, respectively, provide a lower bound on the lower end of the conformal window and to provide a measure of perturbativity. As well as analytic calculations we use lattice techniques to investigate two particular candidate theories for walking technicolor - SU(2) with two adjoint fermions and with six fundamental fermions. We use Schrodinger Functional techniques to investigate the running of the theory across a wide range of scales. We measure both the running of the coupling and an estimator for the fermion mass anomalous dimension, $gamma$. We find that both theories are consistent with an infrared fixed-point. However, paying particular attention to our error estimates, we are unable to absolutely confirm their existence. This is a not unexpected result for SU(2) with two adjoint fermions but is rather surprising for SU(2) with only six fundamental fermions. In the region where we are consistent with a fixed point we find $0.05<gamma<0.56$ for $SU(2)$ with two adjoint fermions and $0.135<gamma<1.03$ for $SU(2)$ with six fundamental fermions. The measurement of $gamma$ for $SU(2)$ with two adjoint fermions is the first determination of $gamma$ for any candidate theory of walking technicolor.

In the first part of this thesis we study baryonic U(1) symmetries dual to Betti multiplets in the AdS_4/CFT_3 correspondence for M2 branes at Calabi-Yau four-fold singularities. Such short multiplets originate from the Kaluza-Klein compactification of eleven-dimensional supergravity on the corresponding Sasaki-Einstein seven-manifolds. Analysis of the boundary conditions for vector fields in AdS_4 allows for a choice where wrapped M5 brane states carrying non-zero charge under such symmetries can be considered. We begin by focusing on isolated toric singularities without vanishing six-cycles, which we classify, and propose for them field theory duals. We then study in detail the cone over the well-known Sasaki-Einstein space Q^111, which is a U(1) fibration over CP^1 x CP^1 x CP^1. The boundary conditions considered are dual to a CFT where the gauge group is U(1)^2 x SU(N)^4. We find agreement between the spectrum of gauge-invariant baryonic-type operators in this theory and M5 branes wrapping five-cycles in the Q^111 space. Moreover, the physics of vacua in which these symmetries are spontaneously broken precisely matches a dual gravity analysis involving resolutions of the singularity, where we are able to match condensates of the baryonic operators, Goldstone bosons and global strings. We then study the implications of turning on a closed three-form with non-zero periods through torsion three cycles in the Sasaki-Einstein manifold. This three-form, otherwise known as torsion G-flux, non-trivially affects the supergravity dual of Higgsing, and we show that the supergravity and field theory analyses precisely match in an example based on the Sasaki-Einstein manifold Y^1,2(CP^2), which is a S^3 bundle over CP^2. We then explain how the choice of M-theory circle in the background can result in exotic renormalization group flows in the dual field theory, and study this in detail for the Sasaki-Einstein manifold Y^1,2(CP^2). We also argue more generally that theories where the resolutions have six-cycles are expected to receive non-perturbative corrections from M5 brane instantons. We give a general formula relating the instanton action to normalizable harmonic two-forms, and compute it explicitly for the Sasaki-Einstein Q^222 example, which is a Z_2 orbifold of Q^111 in which the free Z_2 quotient is along the R-symmetry U(1) fibre. The holographic interpretation of such instantons is currently unclear. In the second part of this thesis we study the breaking of baryonic symmetries in the AdS_5/CFT_4 correspondence for D3 branes at Calabi-Yau three-fold singularities. This leads, for particular vacuum expectation values, to the emergence of non-anomalous baryonic symmetries during the renormalization group flow. We identify these vacuum expectation values with critical values of the NS-NS B-field moduli in the dual supergravity backgrounds. We study in detail the C^3/Z_3 orbifold theory and the dual supergravity backgrounds that correspond to the breaking of the emerging baryonic symmetries, and identify the expected Goldstone bosons and global strings in the infra-red. In doing so we confirm the claim that the emerging symmetries are indeed non-anomalous baryonic symmetries.

We first consider four-dimensional gauge theory on twistor space, taking as a case study maximally supersymmetric Yang-Mills theory. Using a twistor action functional, we show that gauge theory scattering amplitudes are naturally computed on twistor space in a manner that is much more efficient than traditional space-time Lagrangian techniques at tree-level and beyond. In particular, by rigorously studying the Feynman rules of a gauge-fixed version of the twistor action, we arrive at the MHV formalism. This provides evidence for the naturality of computing scattering amplitudes in twistor space as well as an alternative proof of the MHV formalism itself. Next, we study other gauge theory observables in twistor space including gauge invariant local operators and Wilson loops, and discuss how to compute their expectation values with the twistor action. This enables us to provide proofs for the supersymmetric correlation function / Wilson loop correspondence as well as conjectures on mixed Wilson loop - local operator correlators at the level of the loop integrand. Furthermore, the twistorial formulation of such observables is naturally algebro-geometric; this leads to novel recursion relations for computing mixed correlators by performing BCFW-like deformations of the observables in twistor space. Finally, we apply these twistor actions to gravity. Using the on-shell equivalence between Einstein and conformal gravity in de Sitter space, we argue that the twistor action for conformal gravity should encode the tree-level graviton scattering amplitudes of Einstein's theory. We prove this in terms of generating functionals, and derive the flat space MHV amplitude as well as a recursive version of the MHV amplitude with cosmological constant. We also include some discussion of super-connections and Coulomb branch regularization on twistor space.

This thesis is mainly concerned with the study of hypersymplectic structures in gauge theory. These structures arise via applications of the hypersymplectic quotient construction to the action of the gauge group on certain spaces of connections and Higgs fields. Motivated by Kobayashi-Hitchin correspondences in the case of hyperkähler moduli spaces, we first study the relationship between hypersymplectic, complex and paracomplex quotients in the spirit of Kirwan's work relating Kähler quotients to GIT quotients. We then study dimensional reductions of the ASD equations on $mathbb R^{2,2}$. We discuss a version of twistor theory for hypersymplectic manifolds, which we use to put the ASD equations into Lax form. Next, we study Schmid's equations from the viewpoint of hypersymplectic quotients and examine the local product structure of the moduli space. Then we turn towards the integrability aspects of this system. We deduce various properties of the spectral curve associated to a solution and provide explicit solutions with cyclic symmetry. Hitchin's harmonic map equations are the split signature analogue of the self-duality equations on a Riemann surface, in which case it is known that there is a smooth hyperkähler moduli space. In the case at hand, we cannot expect to obtain a globally well-behaved moduli space. However, we are able to construct a smooth open set of solutions with small Higgs field, on which we then analyse the hypersymplectic geometry. In particular, we exhibit the local product structures and the family of complex structures. This is done by interpreting the equations as describing certain geodesics on the moduli space of unitary connections. Using this picture we relate the degeneracy locus to geodesics with conjugate endpoints. Finally, we present a split signature version of the ADHM construction for so-called split signature instantons on $S^2 imes S^2$, which can be given an interpretation as a hypersymplectic quotient.

Nesta tese estudamos possíveis extensões do modelo CPN-1 em (2+1) dimensões. Provamos que quando tomado na representação fundamental à esquerda ele é renormalizável e não possui divergências infravermelhas perigosas. O mesmo não ocorre se o campo principal . Mostramos que a inclusão de férmions, minimamente acoplados ao campo de calibre, traz alguma melhoria no comportamento das divergências infravermelhas no setor de calibre em ordem dominante em 1/N. Discutimos também a invariância de calibre no procedimento de renormalização. / In this thesis investigate possible extensions of the (2+1) dimensional CPN-1 model to the noncommutative space. Up to leading nontrivial order of 1/N, we prove that the model restricted to the left fundamental representation is renormalizable and does not have dangerous infrared divergences. By contrast, IF the pricipal Field transforms in accord with the adjoint representation, linearly divergent, nonintegrable singularities are present in the two point function of the auxiliary gauge Field and also in the leading correction to the self-energy of the Field. It is showed that the inclusion of fermionic matter, minimally coupled to the gauge Field, ameliorates this behavior by eliminating infrared divergences in the gauge sector at the leading 1/N order. Gauge invariance of the renormalization is also discussed.

Discutimos um modelo de calibre com simetria Z (N) na rede, sendo as variáveis dinâmicas definidas em faces de cubos. Mostramos a dualidade com um sistema de spins Z (N) em quatro dimensões e a autodualidade em seis dimensões para este modelo, utilizando o formalismo da matriz de transferência. Analisamos as funções de correlação invariantes por transformações de calibre, constatando os decaimentos exponenciais com o volume (para altas temperaturas e d &#8805 3) e com a área (para baixas temperaturas e d > 3). Para três dimensões, o modelo não apresenta transição de fase sendo exatamente solúvel. Estudamos também a versão U (1) do modelo e mostramos sua equivalência com uma teoria de campos clássica livre na região de baixas temperaturas / We discussus a model with a Z (N) gauge symmetry on a lattice, the dynamical variables being defined on faces of cubes. The duality with a Z (N) spin system in four dimensions and the selfduality in six dimensions is shown for this model, using the transfer matrix formalism. The gauge invariant correlation functions have been analysed and we verify their exponential decay with volume (at high temperatures and d &#8805 3) and with the área (at low temperatures and d > 3). For three dimensions, the model exhibits no phase transition, being exactly soluble. We also study a U (I) version o four model and show its equivalence with a free classical field theory in the low temperature region

Topological order is a new paradigm for quantum phases of matter developed to explain phase transitions which do not fit the symmetry breaking scheme for classifying phases of matter. They are characterized by patterns of entanglement that lead to topologically depended ground state degeneracy and anyonic excitations. One common approach for studying such phases in two-dimensional systems is through exactly solvable lattice Hamiltonian models such as quantum double models and String-Net models. The former can be understood as the Hamiltonian formulation of lattice gauge theories and, as such, it is defined by a finite gauge group. However, not much is known about topological phases in tridimensional systems. Motivated by this we develop a new class of three-dimensional exactly solvable models which go beyond quantum double models by using finite crossed modules instead of gauge groups. This approach relies on a lattice implementation of 2-gauge theory to obtain models with a richer topological structure. We construct the Hamiltonian model explicitly and provide a rigorous proof that the ground state degeneracy is a topological invariant and that the ground states can only be characterized with nonlocal order parameters. / Ordem topológica é um novo paradigma para fases quânticas da matéria desenvolvido para explicar transições de fase que não se encaixam no esquema de classificação de fases da matéria por quebra de simetria. Estas fases são caracterizadas por padrões de emaranhamento que levam a uma degenerescência de estado fundamental topológica e a excitações anyonicas. Uma abordagem comum para o estudo de tais fases em sistemas bidimensionais é através de modelos Hamiltonianos exatamente solúveis de rede como os modelos duplos quânticos e modelos de String-Nets. O primeiro pode ser entendido como a formulação Hamiltoniana de teorias de gauge na rede e, desta maneira, é definido por um group de gauge finito. Entretanto, pouco é conhecido a respeito de fases topológicas em sistemas tridimensionais. Motivado por isso nós desenvolvemos uma nova classe de modelos tridimensionais exatamente solúveis que vai alem de modelos duplos quânticos pelo uso de módulos cruzados finitos no lugar de grupos de gauge. Esta abordagem se baseia numa implementação em redes de teoria de 2-gauge para obter modelos com uma estrutura topológica mais rica. Nós construímos o modelos Hamiltoniano explicitamente e fornecemos uma demonstração rigorosa de que a degenerescência de estado fundamental é um invariante topológico e que os estados fundamentais só podem ser caracterizados por parâmetros de ordem não locais.

Submitted by Pedro Alexandre dos Santos null (retiarus@gmail.com) on 2017-03-22T23:00:09Z No. of bitstreams: 1 dissertacao-versão-final-pedro.pdf: 1127822 bytes, checksum: 02edcd6c8cccc0fb68f13733a6af5f48 (MD5) / Approved for entry into archive by Luiz Galeffi (luizgaleffi@gmail.com) on 2017-03-23T14:34:41Z (GMT) No. of bitstreams: 1 santos_pa_me_guara.pdf: 1127822 bytes, checksum: 02edcd6c8cccc0fb68f13733a6af5f48 (MD5) / Made available in DSpace on 2017-03-23T14:34:41Z (GMT). No. of bitstreams: 1 santos_pa_me_guara.pdf: 1127822 bytes, checksum: 02edcd6c8cccc0fb68f13733a6af5f48 (MD5) Previous issue date: 2017-02-20 / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES) / Neste trabalho, fez-se uma introdução aos mapas dinâmicos, um conjunto de técnicas desenvolvido por Mat- sumoto, Umezawa, entre outros colaboradores, e a prescrição i na formulação usual da teoria quântica de campos no contı́nuo para descrever quebra espontânea de simetria de calibre (QES). Esta técnica baseia-se na utilização de representações unitárias não equivalentes, para construir as diferentes fases fı́sicas observadas em sistemas que apresenta QES. Introduzido o mapa dinâmico, tentou-se adaptar esta coleção de técnicas junto da prescrição i em rede, obtendo como resultado que a primeira não é satisfatoriamente desenvolvida em redes finitas, entretanto no limite termodinâmico o resultado aparenta estar adequado. Então, visitou-se o Teorema de Elitzur para as descrições obtidas por esta ferramenta, isto é, tentou-se demonstrar o Teorema de Elitzur para modelos com termos complexos provenientes da prescrição i . Uma vez que este não forneceu as informações esperadas, aplicou-se outras discussões para investigar a possibilidade da quebra espontânea de simetria de calibre em rede. Neste texto, o ferramental adotado se baseou em uma discussão apresentada por Splittorff. / In this work, an introduction to dynamic maps, a set of techniques developed by Matsumoto, Umezawa, among other collaborators, and the iepsilon prescription in the usual formulation of the quantum field theory in the continuum to describe spontaneous breaking of gauge symmetry (QES). This technique is based on the use of non-equivalent unitary representations to construct the different physical phases observed in systems that apresented QES. Introduced dynamic map, it was started the adaptation fo this collection of techniques to- gether with the prescription i in lattice, obtaining as a result that the first one is not satisfactorily developed in finite lattice, however in the thermodynamic limit the apparent result is adequate. Then the Elitzur’s The- orem was visited for the descriptions obtained by this tool, that is, it was attempted to demonstrate Elitzur’s Theorem for models with complex terms from the prescription i . As soon as the latter did not provide the expected information, other discussions were conducted to investigate the possibility of spontaneous breaking of lattice gauge symmetry. In this text, the tooling adopted was based on a discussion presented by Splittorff.

In this thesis, we study several different renormalization group (RG) methods, including the conventional Wilson renormalization group, Monte Carlo renormalization group (MCRG), exact renormalization group (ERG, or sometimes called functional RG), and tensor renormalization group (TRG). We use the two dimensional nearest neighbor Ising model to introduce many conventional yet important concepts. We then generalize the model to Dyson's hierarchical model (HM), which has rich phase properties depending on the strength of the interaction. The partition function zeros (Fisher zeros) of the HM model in the complex temperature plane is calculated and their connection with the complex RG flows is discussed. The two lattice matching method is used to construct both the complex RG flows and calculate the discrete β functions. The motivation of calculating the discrete β functions for various HM models is to test the matching method and to show how physically relevant fixed points emerge from the complex domain. We notice that the critical exponents calculated from the HM depend on the blocking parameter b. This motivated us to analyze the connection between the discrete and continuous RG transformation. We demonstrate numerical calculations of the ERG equations. We discuss the relation between Litim and Wilson-Polchinski equation and the effect of the cut-off functions in the ERG calculation. We then apply methods developed in the spin models to more complicated and more physically relevant lattice gauge theories and lattice quantum chromodynamics (QCD) like theories. Finite size scaling (FSS) technique is used to analyze the Binder cumulant of the SU(2) lattice gauge model. We calculate the critical exponent nu and omega of the model and show that it is in the same universality class as the three dimensional Ising model. Motivated by the walking technicolor theory, we study the strongly coupled gauge theories with conformal or near conformal properties. We compare the distribution of Fisher zeros for lattice gauge models with four and twelve light fermion flavors. We also briefly discuss the scaling of the zeros and its connection with the infrared fixed point (IRFP) and the mass anomalous dimension. Conventional numerical simulations suffer from the critical slowing down at the critical region, which prevents one from simulating large system. In order to reach the continuum limit in the lattice gauge theories, one needs either large volume or clever extrapolations. TRG is a new computational method that may calculate exponentially large system and works well even at the critical region. We formulate the TRG blocking procedure for the two dimensional O(2) (or XY ) and O(3) spin models and discuss possible applications and generalizations of the method to other spin and lattice gauge models. We start the thesis with the introduction and historical background of the RG in general.