Dissertations / Theses on the topic 'Quantum theories as models of complexity'

Despite a general recognition of the importance of complex systems, there is a dearth of general models capable of describing their dynamics. This is attributed to a complexity scale; the models are attempting to describe systems at different parts of the scale and are hence not compatible. We require new models capable of describing complex behaviour at different points of the complexity scale. This work identifies, and proceeds to examine systems at the high end of the complexity scale, those which have not to date been well understood by our current modelling methodology. It is shown that many such models exhibit what might be termed contextual dependency, and that it is precisely this feature which is not well understood by our current modelling methodology. A particular problem is discussed; our apparent inability to generate systems which display high end complexity, exhibited by for example the general failure of strong ALife. A new model, Process Physics, that has been developed at Flinders University is discussed, and arguments are presented that it exhibits high end complexity. The features of this model that lead to its displaying such behaviour are discussed, and the generalisation of this model to a broader range of complex systems is attempted.

Thesis: S.M., Massachusetts Institute of Technology, Department of Electrical Engineering and Computer Science, 2015.
This electronic version was submitted by the student author. The certified thesis is available in the Institute Archives and Special Collections.
Cataloged from student-submitted PDF version of thesis.
Includes bibliographical references (pages 141-145).
While physical theories attempt to break down the observed structure and behavior of possibly large and complex systems to short descriptive axioms, the perspective of a computer scientist is to start with simple and believable set of rules to discover their large scale behaviors. Computer science and physics, however, can be combined into a new framework, wherein structures can be compared with each other according to scalar observables like mass and temperature, and also complexity at the same time. For example, similar to saying that one object is heavier than the other, we can discuss which system is more complex. According to this point of view, a more complex system can be interpreted as the one which can be programmed to encode and simulate the behavior of the others within its own degrees of freedom. Within this framework, at the most fundamental layer, physical systems are related to languages. In this thesis, I try to exemplify this point of view through an analysis of certain quantum theories in two dimensional space-time. In simple words, these models are the quantum analogues of elastic scattering of colored balls moving on a line. The models are closely related to each other in both relativistic and non-relativistic regimes. Physical examples that motivate this are the factorized scattering matrix of quantum field theory, and the repulsive delta interactions of quantum mechanics, in 1+1 dimensions. In classical mechanics, when two hard balls collide, they bounce off and remain in the same order. However, in the quantum setting, during a collision, either the balls bounce off, or otherwise they tunnel through each other, and exchange their configurations. Each event occurs with a certain probability. As a result, moving balls are put in a superposition of being in different color configurations. Thereby, if we consider n distinct balls, the state space is according to their n! possible arrangements, and their collisions act as quantum transpositions. Quantum transpositions can then be viewed as local quantum gates. I therefore consider the general Hilbert space of permutations, and study the space of unitary operators that can be generated by the local permuting gates. I first show that all of the discussed quantum theories can be programmed into an idealized model, the quantum ball permuting model, and then I will try to pin down the language of this model within the already known complexity classes. The main approach is to consider a series of models, as the variations of the ball scattering problem, and then to relate them to each other, using tools of computational complexity and quantum complexity theory. I find that the computational complexity of the ball permuting model depends on the initial superposition of the balls. More precisely, if the balls start out from the identity permutation, the model can be simulated in a one clean qubit, which is believed to be strictly weaker than the standard model of quantum computing. Given this upper-bound on the ball permuting model, the result is that the model of ball scatterings can be simulated within a one clean qubit, if they start out from an identity permutation. Furthermore, I will demonstrate that if special superpositions are allowed in the initial state, then the ball permuting model can efficiently simulate and sample from the output distribution of standard quantum computers. Next, I show how to use intermediate demolition ball detections to simulate the ball permuting model nondeterministically. According to this result, using post-selection on the outcome of these measurements, one obtains the original ball permuting model. Therefore, the post-selected analogue of ball scattering model can efficiently simulate standard quantum computers, when arbitrary initial superpositions are allowed. In the end, I formalize a quantum computer based on ball collisions and intermediate ball detections, and then I prove that the possibility of efficient simulation of this model on a classical computer is ruled out, unless the polynomial hierarchy collapses to its third level.
by Saeed Mehraban.
S.M.

Thesis: Ph. D., Massachusetts Institute of Technology, Department of Mathematics, 2019
Cataloged from PDF version of thesis.
Includes bibliographical references (pages 355-372).
Restricted models of quantum computation are mathematical models which describe quantum computers that have limited access to certain resources. Well-known examples of such models include the boson sampling model, extended Clifford circuits, and instantaneous quantum polynomial-time circuits. While unlikely to be universal for quantum computation, several of these models appear to be able to outperform classical computers at certain computational tasks, such as sampling from certain probability distributions. Understanding which of these models are capable of performing such tasks and characterizing the classical simulation complexity of these models--i.e. how hard it is to simulate these models on a classical computer--are some of the central questions we address in this thesis. Our first contribution is a classification of various extended Clifford circuits according to their classical simulation complexity.
Among these circuits are the conjugated Clifford circuits, which we prove cannot be efficiently classically simulated up to multiplicative or additive error, under certain plausible conjectures in computational complexity theory. Our second contribution is an estimate of the number of qubits needed in various restricted quantum computation models in order for them to be able to demonstrate quantum computational supremacy. Our estimate is obtained by fine-graining existing hardness results for these restricted models. Our third contribution is a new alternative proof of the Gottesman-Knill theorem, which states that Clifford circuits can be efficiently simulated by a classical computer. Our proof uses the sum-over-paths technique and establishes a correspondence between quantum circuits and a class of exponential sums. Our final contribution is a theorem characterizing the operations that can be efficiently simulated using a particular rebit simulator.
An application of this result is a generalization of the Gottesman-Knill theorem that allows for the efficient classical simulation of certain nonlinear operations.
"Funding support from the National Science Scholarship awarded by the Agency for Science, Technology and Research (A*STAR), Singapore, as well as the Enabling Practical-scale Quantum Computing (EPiQC) Expedition, an NSF expedition in computing"--Page 6.
by Dax Enshan Koh.
Ph. D.
Ph.D. Massachusetts Institute of Technology, Department of Mathematics

This thesis is concerned with the Schrödinger representation of quantum field theory. We describe techniques for solving the Schrödinger equation which supplement the standard techniques of field theory. Our aim is to develop these to the point where they can readily be used to address problems of current interest. To this end, we study realistic models such as gauge theories coupled to dynamical fermions. For maximal generality we consider particles of all physical spins, in various dimensions, and eventually, curved spacetimes. We begin by considering Gaussian fields, and proceed to a detailed study of the Schwinger model, which is, amongst other things, a useful model for (3+1) dimensional gauge theory. One of the most important developments of recent years is a conjecture by Mal-dacena which relates supergravity and string/M-theory on anti-de-Sitter spacetimes to conformal field theories on their boundaries. This correspondence has a natural interpretation in the Schrödinger representation, so we solve the Schrödinger equation for fields of arbitrary spin in anti-de-Sitter spacetimes, and use this to investigate the conjectured correspondence. Our main result is to calculate the Weyl anomalies arising from supergravity fields, which, summed over the supermultiplets of type JIB supergravity compactified on AdS(_s) x S(^5) correctly matches the anomaly calculated in the conjecturally dual N = 4 SU{N) super-Yang-Mills theory. This is one of the few existing pieces of evidence for Maldacena's conjecture beyond leading order in TV.

In this thesis we have employed the holographic duality to study the non-perturbative regime of a one-parameter family of theories with multi-scale dynamics. Normally, this (super)string theory motivated duality identifies gauge theories in flat space with string theories in a certain curved spacetime. Its relevance roots in its ability to relate the strongly-coupled regime of gauge theories with classical gravity governed by Einstein's equations. In the Introduction of the thesis, we have reviewed the main string theory ingredients that we used throughout the thesis and revisited some of the previous results which are the starting point of out study. In Chapter 2, we gather the explicit form of the supergravity solutions whose dual are the gauge theories of interest. These are three-dimensional gauge theories. Generically, they share the same physics at high energies, given by Yang-Mills and Chern-Simons interactions. Remarkably, when the low energy regime of the theories is studied, a rich variety of non-perturbative phenomenology is discovered. In particular, for a generic value of the parameter distinguishing the theories, they develop a mass gap in their spectrum. However, the two theories which are obtained at the limiting values of the parameter are special. On the one hand, the theory flows towards an infrarred fixed point, dual to a Conformal Field Theory. On the other hand, a confining theory is obtained, in the sense that the potential felt between quarks grows linearly with the distance between them for large separations. All these phenomena, together with the computation of the spectrum of spin-0 and spin-2 states, are studied in Chapter 3. The fact that in this system the Renormalisation Group flow can pass close to a Conformal Field Theory motivated the search of a light dilaton in the spectrum. But such light state was not found, the reason for that being that the values of the source and the vacuum expectation value that prevented the flow from finishing at the fixed point where of the same order. On top of that, in this Chapter some entanglement entropy measures were studied. This last investigation was motivated by the fact that in the literature some quantities extracted from such magnitudes where proposed as a probe for confinement. Our results show that, when these quantities are considered in our system, they are not able to discriminate between confining and non-confining gapped theories. Not only did we consider the theories at zero temperature case, but we also studied thermal states by constructing numerical black brane solutions in the gravity side. Black branes are very much like black holes, with the peculiarity that their surface extends in non-compact directions. Such solutions are discussed in Chapter 4. As a result, we understood their phase diagram, exhibiting a rich structure endowed with first and second order phase transitions, as well as a triple point where three phases coexist and a critical point where the second order phase transition takes place. Intrigued by the effect that the proximity of a Conformal Field Theory could have in the Renormalisation Group flow of a field theory, in Chapter 5 we carried out a study on complex Conformal Field Theories. We proposed their holographic dual, and analysed some of their properties in the strongly-coupled case. Finally, in Chapter 6, we studied transport coefficients in holographic theories which model Quantum Chromodynamics. We concluded that the holographic results are quite different from the ones obtained using perturbative techniques. These studies could have phenomenological consequences and find their application in astrophysical observations concerning neutron stars.
En esta tesis hemos utilizado la dualidad holográfica para entender el régimen no perturbativo de una familia uni-paramétrica de teorías con múltiples escalas. Primeramente, hemos repasado los ingredientes esenciales que necesitamos de teoría de cuerdas. A la vez, hemos introducimos algunos resultados previos que son el punto de partida de nuestras investigaciones. Tras dicha introducción, se recogen todas las soluciones de supergravedad duales a las teorías en tres dimensiones que estudiamos. Genéricamente, comparten la misma física a altas energías pero a bajas energías muestran una rica fenomenología. En particular, desarrollan un salto de masa en su espectro. Curiosamente, las teorías correspondientes a tomar los valores límites del parámetro son especiales. En un caso, la teoría fluye a una teoría de campos conforme. En el otro se obtiene una teoría confinante, con potencial lineal entre quarks. También se calcula el espectro de estados con espín 0 y espín 2. Además, se analizan diferentes medidas de entrelazamiento cuántico que en nuestro caso no son capaces de discriminar entre teorías con confinamiento y teorías con un salto de masa. Esto contrasta con algunas propuestas que se encuentran en la literatura. Adicionalmente hemos construido numéricamente soluciones de branas negras, que describen estados térmicos de las teorías. Hemos descubierto un diagrama de fases muy rico, con transiciones de fase de primer y segundo orden, junto a un punto crítico y un punto triple. Interesados por el efecto que una teoría conforme de campos pudiera tener si es cercana al flujo del grupo de renormalización de otra teoría, en el Capítulo 5 nos adentramos en el estudio de teorías conformes de campos complejas, dando su el dual holográfico. Finalmente, se calculan coeficientes de transporte en teorías holográficas que modelan Cromodinámica Quántica y que podrían tener consecuencias fenomenológicas en observaciones referentes a estrellas de neutrones.

Diese Arbeit befasst sich mit Quantenfeldtheorien auf nicht-kommutativen Räumen. Solche Modelle treten im Zusammenhang mit der Stringtheorie und mit der Quantengravitation auf. Ihre nicht-störungstheoretische Behandlung ist üblicherweise schwierig. Hier untersuchen wir jedoch drei nicht-kommutative Quantenfeldtheorien nicht-perturbativ, indem wir die Wirkungsfunktionale in eine äquivalente Matrixformulierung übersetzen. In der Matrixdarstellung kann die jeweilige Theorie dann numerisch behandelt werden. Als erstes betrachten wir ein regularisiertes skalares Modell auf der nicht-kommutativen Ebene und untersuchen den Kontinuumslimes bei festgehaltener Nicht-Kommutativität. Dies wird auch als Doppelskalierungslimes bezeichnet. Insbesondere untersuchen wir das Verhalten der gestreiften Phase. Wir finden keinerlei Hinweise auf die Existenz dieser Phase im Doppelskalierungslimes. Im Anschluss daran betrachten wir eine vier-dimensionale U(1) Eichtheorie. Hierbei sind zwei der räumlichen Richtungen nicht-kommutativ. Wir untersuchen sowohl die Phasenstruktur als auch den Doppelskalierungslimes. Es stellt sich heraus, dass neben den Phasen starker und schwacher Kopplung eine weitere Phase existiert, die gebrochene Phase. Dann bestätigen wir die Existenz eines endlichen Doppelskalierungslimes, und damit die Renormierbarkeit der Theorie. Weiterhin untersuchen wir die Dispersionsrelation des Photons. In der Phase mit schwacher Kopplung stimmen unsere Ergebnisse mit störungstheoretischen Berechnungen überein, die eine Infrarot-Instabilität vorhersagen. Andererseits finden wir in der gebrochenen Phase die Dispersionsrelation, die einem masselosen Teilchen entspricht. Als dritte Theorie betrachten wir ein einfaches, in seiner Kontinuumsform supersymmetrisches Modell, welches auf der "Fuzzy Sphere" formuliert wird. Hier wechselwirken neutrale skalare Bosonen mit Majorana-Fermionen. Wir untersuchen die Phasenstruktur dieses Modells, wobei wir drei unterschiedliche Phasen finden.
This work deals with three quantum field theories on spaces with noncommuting position operators. Noncommutative models occur in the study of string theories and quantum gravity. They usually elude treatment beyond the perturbative level. Due to the technique of dimensional reduction, however, we are able to investigate these theories nonperturbatively. This entails translating the action functionals into a matrix language, which is suitable for numerical simulations. First we explore a scalar model on a noncommutative plane. We investigate the continuum limit at fixed noncommutativity, which is known as the double scaling limit. Here we focus especially on the fate of the striped phase, a phase peculiar to the noncommutative version of the regularized scalar model. We find no evidence for its existence in the double scaling limit. Next we examine the U(1) gauge theory on a four-dimensional spacetime, where two spatial directions are noncommutative. We examine the phase structure and find a new phase with a spontaneously broken translation symmetry. In addition we demonstrate the existence of a finite double scaling limit which confirms the renormalizability of the theory. Furthermore we investigate the dispersion relation of the photon. In the weak coupling phase our results are consistent with an infrared instability predicted by perturbation theory. If the translational symmetry is broken, however, we find a dispersion relation corresponding to a massless particle. Finally, we investigate a supersymmetric theory on the fuzzy sphere, which features scalar neutral bosons and Majorana fermions. The supersymmetry is exact in the limit of infinitely large matrices. We investigate the phase structure of the model and find three distinct phases. Summarizing, we study noncommutative field theories beyond perturbation theory. Moreover, we simulate a supersymmetric theory on the fuzzy sphere, which might provide an alternative to attempted lattice formulations.

Thesis (Ph. D.)--University of California, San Diego, 2009.
Title from first page of PDF file (viewed September 17, 2009). Available via ProQuest Digital Dissertations. Vita. Includes bibliographical references (p. 444-468).

In questo lavoro di tesi abbiamo analizzato alcuni modelli conformi con perturbazioni integrabili, in particolare il modello di Ising tri-critico e i successivi modelli minimali. Abbiamo costruito un protocollo che realizza questi modelli in un regime fuori dall'equilibrio termodinamico. Questo sistema è stato ottenuto connettendo due sistemi semi-infiniti termalizzati a due diverse temperature. In tempi e spazi grandi ci si aspetta che questo sistema evolva verso uno stato stazionario indipendente dal tempo. Le quantità fisiche di nostro interesse sono le correnti stazionarie generate in tale situazione. Per studiare questo sistema abbiamo utilizzato strumenti di integrabilità come il Bethe ansatz termodinamico, concetti di idrodinamica generalizzata e l'insieme di Gibbs generalizzato. Finora questo schema è stato formulato per le teorie di campo con un'interazione tra le particelle data da una matrice S diagonale, ovvero per i modelli con lo spettro di quasi-particelle prive di gradi di libertà interni. In questa tesi abbiamo proposto un'estensione di questo metodo a un modello dotato di uno spettro contenente quasi-particelle organizzate in multipletti di simmetrie e quindi dotate di gradi di libertà interni detti magnoni con processi d'urto descritti da matrici S non diagonali. Abbiamo quindi risolto numericamente le equazioni differenziali che descrivono il sistema di non equilibrio e abbiamo discusso questi risultati.

Cette thèse se scinde en deux volets, avec vue sur la renormalisation de théorie quantique des champs.Le premier volet traite de trois modèles tensoriels en trois dimensions, un quartique fermionique de rang 3 et deux sextiques bosonique, de rangs 3 et 5. On se base sur l'expansion melonique à grand N des théories tensorielles. Pour le premier modèle, invariant sous le groupe U(N)³, on calcule le flot du groupe de renormalisation des deux couplages meloniques et on dresse le diagramme des phases du vide de la théorie, en étudiant sa reformulation par un champ intermédiaire matriciel diagonalisable. Observant une brisure spontanée de la symétrie discrète chirale, la comparaison avec le modèle de Gross-Neveu tri-dimensionel est faite. Au-delà de la phase symétrique U(N)³ sans masse, on note aussi une phase massive de même symétrie et une autre où la symétrie est brisée vers U(N²) x U(N/2) x U(N/2). Un modèle matriciel de symétrie U(N) x U(N²), présentant les mêmes caractéristiques, est aussi considéré.Dans les deux autres modèles tensoriels, de groupes de symétrie U(N)³ et O(N)⁵, un couplage non-melonique (la ``roue") adjoint d'une puissance de N optimale nous conduit à une expansion melonique généralisée. Les termes cinétiques sont pris de courte ou longue portée et on étudie, à grand N, perturbativement les différents groupes de renormalisation des couplages d'ordre 6, jusqu'à quatre boucles. Tandis que le modèle de rang 5 ne présente pas de point fixe non-trivial, celui de rang 3 possède deux points fixes non-triviaux réels de type Wilson-Fisher dans le cas à courte portée et une ligne de points fixes dans l'autre. On obtient enfin les dimensions conformes réelles des opérateurs primaires bilinéaires en le champ fondamental.Le second volet établit les premiers résultats de renormalisation constructive multi-échelle pour un modèle scalaire quartique sur des arbres de Galton-Watson critiques, avec un terme cinétique à longue portée. Au point critique, l'émergence d'une spine infinie fournit un espace de dimension effective 4/3 sur lequel calculer des fonctions de corrélations moyennées. Cela formalise la notion de théorie des champs sur une géométrie aléatoire. Nous utilisons dans notre approche des bornes probabilistes sur le noyau de la chaleur dans un graphe aléatoire. On esquisse pour terminer l'extension du formalisme à des fermions et à une spine compactifiée
This thesis divides into two parts, focusing on the renormalization of quantum field theories. The first part considers three tensor models in three dimensions, a fermionic quartic with tensors of rank-3 and two bosonic sextic, of ranks 3 and 5. We rely upon the large-N melonic expansion of tensor models. For the first model, invariant under U(N)³, we compute the renormalization group flow of the two melonic couplings and establish the vacuum phase diagram, from a reformulation with a diagonalizable matrix intermediate field. Noting a spontaneous symmetry breaking of the discrete chiral symmetry, the comparison with the three-dimensional Gross-Neveu model is made. Beyond the massless U(N)³ symmetric phase, we also observe a massive phase of same symmetry and another where the symmetry breaks into U(N²) x U(N/2) x U(N/2). A matrix model invariant under U(N) x U(N²), sharing the same properties, is also studied.For the two other tensor models, with symmetry groups U(N)³ and O(N)⁵, a non-melonic coupling (the ``wheel") with an optimal scaling in N drives us to a generalized melonic expansion. The kinetic terms are taken of short and long range, and we analyze perturbatively, at large-N, the renormalization group flows of the sextic couplings up to four loops. While the rank-5 model doesn't present any non-trivial fixed point, that of rank 3 displays two real non-trivial Wilson-Fisher fixed points in the short-range case and a line of fixed points in the other. We finally obtain the real conformal dimensions of the primary operators bilinear in the fundamental field.In the second part, we establish the first results of constructive multi-scale renormalization for a quartic scalar field on critical Galton-Watson trees, with a long-range kinetic term. At the critical point, an emergent infinite spine provides a space of effective dimension 4/3 on which to compute averaged correlation fonctions. This approach formalizes the notion of a quantum field theory on a random geometry. We use known probabilistic bounds on the heat-kernel on a random graph. At the end, we sketch the extension of the formalism to fermions and to a compactified spine

Cette thèse présente une étude détaillée de la structure de théories appelées GFT ("Group Field Theory" en anglais),à travers le prisme de la renormalisation. Ce sont des théories des champs issues de divers travaux en gravité quantique, parmi lesquels la gravité quantique à boucles et les modèles de matrices ou de tenseurs. Elles sont interprétées comme desmodèles d'espaces-temps quantiques, dans le sens où elles génèrent des amplitudes de Feynman indexées par des triangulations,qui interpolent les états spatiaux de la gravité quantique à boucles. Afin d'établir ces modèles comme des théories deschamps rigoureusement définies, puis de comprendre leurs conséquences dans l'infrarouge, il est primordial de comprendre leur renormalisation. C'est à cette tâche que cette thèse s'attèle, grâce à des méthodes tensorielles développées récemment,et dans deux directions complémentaires. Premièrement, de nouveaux résultats sur l'expansion asymptotique (en le cut-off) des modèles colorés de Boulatov-Ooguri sont démontrés, donnant accès à un régime non-perturbatif dans lequel une infinité de degrés de liberté contribue. Secondement, un formalisme général pour la renormalisation des GFTs dites tensorielles (TGFTs) et avec invariance de jauge est mis au point. Parmi ces théories, une TGFT en trois dimensions et basée sur le groupe de jauge SU(2) se révèle être juste renormalisable, ce qui ouvre la voie à l'application de ce formalisme à la gravité quantique
In this thesis, we study the structure of Group Field Theories (GFTs) from the point of view of renormalization theory.Such quantum field theories are found in approaches to quantum gravity related to Loop Quantum Gravity (LQG) on the one hand,and to matrix models and tensor models on the other hand. They model quantum space-time, in the sense that their Feynman amplitudes label triangulations, which can be understood as transition amplitudes between LQG spin network states. The question of renormalizability is crucial if one wants to establish interesting GFTs as well-defined (perturbative) quantum field theories, and in a second step connect them to known infrared gravitational physics. Relying on recently developed tensorial tools, this thesis explores the GFT formalism in two complementary directions. First, new results on the large cut-off expansion of the colored Boulatov-Ooguri models allow to explore further a non-perturbative regime in which infinitely many degrees of freedom contribute. The second set of results provide a new rigorous framework for the renormalization of so-called Tensorial GFTs (TGFTs) with gauge invariance condition. In particular, a non-trivial 3d TGFT with gauge group SU(2) is proven just-renormalizable at the perturbative level, hence opening the way to applications of the formalism to (3d Euclidean) quantum gravity

The holographic duality between gauge theories and string theories has opened a new door to access the strongly coupled regime of quantum field theories and offers, at the same time, a completely new way to understand the elusive nature of quantum gravity and the non-perturbative regime of string theory. After almost two decades of research, the current status of the correspondence is that of a solid conjecture that has passed a great number of nontrivial tests, to the point that it is generally believed to be true. The present thesis includes a collection of four papers published in peer-reviewed scientific journals, all of them in the context of the AdS/CFT correspondence and with a particular focus on studying gauge theories by inserting heavy external probes, following prescribed trajectories and transforming under various representations of the gauge group. Each of these works reports a little step forward in the development of new strategies for capturing correc- tions beyond the leading order as well as in using exact results available in quantum field theory in order to derive exact expressions for other relevant observables and new non-trivial string theory predictions. In chapters 2 and 3 we use the AdS/CFT correspondence in order to compute several observables of N = 4 SU (N ) super Yang-Mills theory related with the presence of an infinitely heavy particle transforming in the k-symmetric or the k-antisymmetric representations of the gauge group and following particular trajectories. This is achieved by means of adding certain D-brane probes with electric fluxes turned on and reaching the boundary of AdS on the very trajectories followed by the dual particles. For the antisymmetric case we consider D5-branes reaching the boundary at arbitrary time-like trajectories, while for the symmetric case, we consider a D3-brane fully embedded in AdS5 that reaches the boundary at either a straight line or a hyperbola. This generalizes previous computations that used fundamental strings, which are claimed to be dual to infinitely heavy point particles transforming in the fundamental. Besides the intrinsic interest of these generalizations, our main motivation in studying them is that, as it happens in the computation of certain Wilson loops, the results obtained with D3-branes give an all- orders series of corrections in 1/N to the leading order result for the fundamental representation obtained by means of fundamental strings. It is important to remark, one more time, that we can not really extrapolate up to k = 1, since this is beyond the regime of validity of the supergravity approximation. Therefore, it is not justified a priori to set k = 1 in our results. Nevertheless, when compared with the exact results available, we find that the D3-brane computation reproduces the correct result in the large N , λ limit and with k = 1. This better than expected performance suggests the exciting possibility that certain D3-branes with electric fluxes might capture correctly all the 1/N corrections, but it is fair to say that we still lack of a precise string-theoretic argument to prove this.
Durant les darreres dues dècades ha aparegut un nou paradigma que permet reformular completament certes teories quàntiques de camps i ens aporta una nova eina que ens permet realitzar càlculs analítics en règims fins ara inaccessibles. Aquest nou paradigma sorgeix del descobriment d’una correspondència o dualitat exacta entre dues teories aparentment molt diferents. Per una banda de la dualitat tenim certes teories quàntiques de camps, com per exemple les denominades teories de Yang-Mills, similars a les teories del Model Estàndard. Aquestes descriuen partícules interactuant en un espai pla d-dimensional sense gravetat. A l’altra banda de la dualitat trobem teories que inclouen la gravetat, com ara la Teoria de la Relativitat General d’Einstein o les seves generalitzacions en el marc de la Teoria de Cordes. Aquestes teories de gravetat estan definides sobre espais de dimensió més alta que d, i és per això que aquesta correspondència rep sovint l’adjectiu de “hologràfica”. Depenent del context, aquesta rep el nom de dualitat gauge/gravetat, dualitat gauge/corda o AdS/CFT (acrònim anglès per la correspondència particular entre teoria de cordes a espais d’Anti-de Sitter i teories de camps conformes). Fins ara, una de les correspondències més ben estudiades i que comprenem millor (i sobre la qual es centra la present tesi) és la dualitat entre la teoria quatre-dimensional N = 4 super Yang-Mills amb grup de gauge SU (N ) i teoria de cordes tipus IIB en un espai deu-dimensional AdS5 × S5 . Aquesta tesi presenta una recopilació de quatre articles publicats en revistes científiques d’alt impacte, tots ells en el camp de la correspondència AdS/CFT i centrats en l’estudi de teories gauge supersimètriques mitjançant la inserció de partícules de prova infinitament massives, seguint trajectòries determinades i transformant sota diverses representacions del grup de gauge. Cadascun d’aquests treballs aporta un pas endavant en el desenvolupament de noves estratègies per calcular correccions més enllà del primer ordre així com en l’ús de resultats exactes accessibles a la Teoria Quàntica de Camps per tal de derivar expressions exactes d’altres observables rellevants de la teoria i realitzar prediccions de Teoria de Cordes.

Les comportements critiques des systèmes de mécanique statistique en 2 dimensions ou de mécanique quantique en 1+1 dimensions, ainsi que certains aspects des systèmes sans interactions en 2+1 dimensions, sont efficacement décrits par les méthodes de la théorie des champs conforme et de l'intégrabilité, dont le développement a été spectaculaire au cours des 40 dernières années. Plusieurs problèmes résistent cependant toujours à une compréhension exacte, parmi lesquels celui de la transition entre plateaux dans l'Effet Hall Quantique Entier. La raison principale en est que de tels problèmes sont généralement associés à des théories non unitaires, ou théories conformes logarithmiques, dont la classification se révèle être d'une grande difficulté mathématique. Se tournant vers la recherche de modèles discrets (chaînes de spins, modèles sur réseau), dans l'espoir en particulier d'en trouver des représentations en termes de modèles exactement solubles (intégrables), on se heurte à la deuxième difficulté représentée par le fait que les théories associées sont la plupart du temps non compactes, ou en d'autres termes qu'elles donnent lieu à un continuum d'exposants critiques. En effet, le lien entre modèles discrets et théories des champs non compactes est à ce jour loin d'être compris, en particulier il a longtemps été cru que de telles théories ne pouvaient pas émerger comme limites continues de modèles discrets construits à partir d'un ensemble compact de degrés de libertés, par ailleurs les seuls qui donnent a accès à une construction systématique de solutions exactes.Dans cette thèse, on montre que le monde des modèles discrets compacts ayant une limite continue non compacte est en fait beaucoup plus grand que ce que les quelques exemples connus jusqu'ici auraient pu laisser suspecter. Plus précisément, on y présente une solution exacte par ansatz de Bethe d'une famille infinie de modèles(les modèles $a_n^{(2)}$, ainsi que quelques résultats sur les modèles $b_n^{(1)}$, où il est observé que tous ces modèles sont décrits dans un certain régime par des théories conformes non compactes. Parmi ces modèles, certains jouent un rôle important dans la description de phénomènes physiques, parmi lesquels la description de polymères en deux dimensions avec des interactions attractives et des modèles de boucles impliqués dans l'étude de modèles de Potts couplés ou dans une tentative de description de la transition entre plateaux dans l'Effet Hall par un modèle géométrique compact.On montre que l'existence insoupçonnéede limite continues non compacts pour de tels modèles peut avoir d'importantes conséquences pratiques, par exemple dans l'estimation numérique d'exposants critiques ou dans le résultats de simulations de Monte Carlo. Nos résultats sont appliqués à une meilleure compréhension de la transition theta décrivant l'effondrement des polymères en deux dimensions, et des perspectives pour une potentielle compréhension de la transition entre plateaux en termes de modèles sur réseaux sont présentées
The critical points of statistical mechanical systems in 2 dimensions or quantum mechanical systems in 1+1 dimensions (this also includes non interacting systems in 2+1 dimensions) are effciently tackled by the exact methods of conformal fieldtheory (CFT) and integrability, which have witnessed a spectacular progress during the past 40 years. Several problems have however escaped an exact understanding so far, among which the plateau transition in the Integer Quantum Hall Effect,the main reason for this being that such problems are usually associated with non unitary, logarithmic conformal field theories, the tentative classification of which leading to formidable mathematical dificulties. Turning to a lattice approach, andin particular to the quest for integrable, exactly sovable representatives of these problems, one hits the second dificulty that the associated CFTs are usually of the non compact type, or in other terms that they involve a continuum of criticalexponents. The connection between non compact field theories and lattice models or spin chains is indeed not very clear, and in particular it has long been believed that the former could not arise as the continuum limit of discrete models built out of acompact set of degrees of freedom, which are the only ones allowing for a systematic construction of exact solutions.In this thesis, we show that the world of compact lattice models/spin chains with a non compact continuum limit is much bigger than what could be expected from the few particular examples known up to this date. More precisely we propose an exact Bethe ansatz solution of an infinite family of models (the so-called $a_n^{(2)}$ models, as well as some results on the $b_n^{(1)}$ models), and show that all of these models allow for a regime described by a non compact CFT. Such models include cases ofgreat physical relevance, among which a model for two-dimensional polymers with attractive interactions and loop models involved in the description of coupled Potts models or in a tentative description of the quantum Hall plateau transition by somecompact geometrical truncation. We show that the existence of an unsuspected non compact continuum limit for such models can have dramatic practical effects, for instance on the output of numerical determination of the critical exponents or ofMonte-Carlo simulations. We put our results to use for a better understanding of the controversial theta transition describing the collapse of polymers in two dimensions, and draw perspectives on a possible understanding of the quantum Hall plateautransition by the lattice approach

Thesis (MA)--Stellenbosch University, 2003.
ENGLISH ABSTRACT: The advent of the computer age has seen many fundamental changes in the economics. The ease with which organisations can store and transmit information in unprecedented quantities and speeds has changed the face of the economy as well as the way in which organisations conduct their day to day operations. Information has become the primary resource for organisational competitiveness and this has seen an increasing drive for efficient information generation and management in an economy that is interconnected on a global scale. The demand for better information management practices is driven by the realisation that the global economy is susceptible to sudden and unpredictable changes that can potentially have global consequences. The more information organisations have at their disposal, the better their chances are of remaining competitive and relevant in the global economy. The informational economy confronts organisations with two very significant problems, the first is information overload due to the sheer volume of information that is available to them. The second problem is that despite the volume of available information organisations still are not privy to all the information that is required to lessen the impact of uncertainty that is so characteristic of the global economy. Organisations therefore always run the' risk of becoming irrelevant if they do not change constantly. This drive for continuous change and the dependence on information has led some organisational theorists and economists to compare the global economy and organisations to nonlinear systems found in nature. Examples of nonlinear systems are living organisms, ecologies and solar systems. All of these systems are characterised by high levels of interconnectedness and interdependence among individual units within a shared environment, which they co-create. Nonlinear systems are of particular interest to organisational theorists because these systems process information about the environment to adapt in an unpredictable way to unpredictable changes. Such systems are incredibly resilient because they are able to learn and adapt to different conditions. Another notable aspect of nonlinear systems is the clear structured and complex organisation that they exhibit in the absence of centralised control mechanisms. Every unit has the liberty to experiment with new designs and from the success of individual units an organised and stable system emerges with a strong link between the success of individuals and the whole system. The order that exists within nonlinear systems is known as self-organisation because it is not superimposed but emerges instead in a spontaneous manner. Nonlinear systems are therefore more than just the sum of their parts. The notion of nonlinear systems and self-organisation has seen authors such as Stacey, Wheatley and Senge develop new ideas about organisational development, leadership and organisational strategic thinking. Their ideas are based on what is popularly known as 'The New Science'. These ideas attempt to encourage organisations realise that the global economy functions as a nonlinear system and that organisations stand a better chance of success if they learn to understand the principles of nonlinear systems and to utilise the inherent creative and organising characteristics of such systems.
AFRIKAANSE OPSOMMING: Die aanvang van die rekenaar era het verskeie fundamentele veranderinge in ekonomie mee gebring. Die gemak en snelheid waarmee organisasies informasie kan stoor en versprei is ongekend en het terselfde tyd die voorkoms van die ekonomie verander asook die wyse waarop organisasies op 'n daaglikse basis funksioneer. Informasie het die belangrikste hulpbron geword vir organisasies in terme van kompetering en dit het 'n groter dryfkrag vir doeltreffende informasie ontginning en bestuur mee gebring in 'n ekonomie wat op 'n wereldwye skaal in mekaar gevleg is. Die aanvraag vir beter informasie bestuur praktyke word gedryf deur die wete dat die wereld ekonomie vatbaar is vir skielike en onvoorspelbare veranderinge wat potensieel 'n wereldwye impak kan he. Hoe meer informasie organisasies tot hul beskikking het hoe beter is hul kans om relevant en kompeterend te bly in die wereld ekonomie. Die informasie ekonomie konfronteer organisasies met twee fundamentele probleme. Die eerste gevaar is dat organisasies oorlaai kan word met informasie as gevolg van die absolute volume van beskikbare informasie. Die tweede probleem spruit voort uit die feit dat ten spyte van die beskikbare informasie, lei organisasies steeds aan 'n gebrek aan algehele informasie, organisasies kan dus nooit toegang he tot al die informasie wat benodig word om die impak te verminder van die onsekerheid wat so kenmerkend is van die wereld ekonomie. Organisasies loop dus altyd die gevaar om irrelevant te raak as hulle nie konstant aanpas by nuwe omstandighede nie. Hierdie soeke na konstante verandering en die afhanklikheid op informasie het verskeie organisasie teoretici en ekonome daartoe gelei om 'n vergelyking te tref tussen die wereld ekonomie en organisasies aan die een kant en nie-Iiniere sisteme wat in die natuur voorkom. Voorbeelde van sulke sisteme sluit lewende organismes, ekostelsels en sterre stelsels in. Die komponente van al hierdie sisteme is op 'n komplekse wyse inmekaar geweef en interafhanklik op mekaar binne die raamwerk van gemeenskaplike omgewing waarvoor hierdie komponente mede verantwoordelik is. Nie-liniere sisteme is van besondere belang vir organisasie teoretici omdat die betrokke sisteme informasie verwerk aangaande hul omgewing om op 'n onvoorspelbare wyse aan te pas by onvoorspelbare veranderinge in die omgewing. Sulke sisteme is uitsonderlik standvastig deurdat hulle kan leer en aanpas by verskillende omstandighede. Nog 'n merkbare aspek van sulke sisteme is die duidelik gestruktureerde en komplekse organisasie wat bestaan ten spyte van 'n algehele gebrek aan gesentraliseerde beheer meganismes. Elke komponent is vry om met 'n nuwe ontwerp te eksperimenteer en vanuit die sukses van die komponente spruit die sukses van die sisteem. Die organisasie wat sigbaar is in nie-liniere sisteme staan bekend as self-organisasie omdat dit nie voortspruit uit 'n sentrale beheer meganisme nie maar instede spontaan onstaan as 'n gevolg van die aksies van komponente. Nie-Iiniere sisteme het die potensiaal om meer te kan wees as die somtotaal van hul komponente. Die beginsel van nie-liniere sisteme en selforganisasie het skrywers soos Stacey, Wheatley en Senge daartoe gelei om nuwe idees te ontwikkel rakende organisasie ontwikkeling, leierskap en strategiese beplanning in organisasies. Hierdie idees is gegrond in wat algemeen bekend staan as 'The New Science'. Die idees van hierdie skrywers is gemik daarop om organisasies aan te moedig om raak te sien dat die wereld ekonomie soos 'n nie-liniere sisteem funksioneer en dat organisasies as sulks 'n beter kans staan om sukses te behaal as hulle sou leer om die beginsels van nie-liniere sisteme te begryp en die inherente kreatiewe en organiserings eienskappe van sulke sisteme uit te buit.

This thesis covers research in two disjoint research areas: The ontological model program (formerly hidden-variables program) for quantum theory has a long and noble tradition in the quantum foundations literature. By postulating a physical reality beyond the quantum state, we gain intuition on quantum phenomena and also come to understand constraints on realist interpretations of quantum theory. Bell's theorem tells us that such an underlying reality must be non-local, while the Kochen-Specker contextuality theorem abuses the classical notion that measurement should simply reveal pre-existing properties of reality. Recent research programs suggest that it is beneficial to view the quantum state as representing purely information. We show that the only current model which does this in a satisfactory manner is unable to reproduce all the statistics of quantum measurements. A recent generalization of the notion of contextuality has allowed for proofs of contextuality which differ from the original Kochen-Specker notion. We add a new result which shows that measurements in a model where the quantum state represents information must be contextual. Additionally, we refine the generalized notion of contextuality into strong and weak forms in order to parse the relationship between new and old results. Entanglement resource theory is a highly successful investigation of the usefulness of entanglement for information processing tasks. In this thesis we apply the ideas from entanglement resource theory to another resource: state distinguishability. We show analogies between distinguishability resource theory and entanglement resource theory. In particular, the analogy includes: measures which are monotonic under a class of transformations; units of a resource; and bounds on measures in terms of the amount of the unit resource needed to form states and the amount of unit resource that can be extracted from states. We show that the pairs of states which can be reversibly converted into \emph{classical states} are exactly the pairs of simultaneously diagonalizable states. Lastly, we characterize the trace-distance distinguishability of formation on a qubit system.