Thesis (MSc)--Stellenbosch University, 2014. / ENGLISH ABSTRACT: The tools of perturbative Quantum Field Theory are by now a standard part of the theoretical physicist's arsenal. In this thesis we investigate the spectral method, an approach that uses tools from the quantum theory of scattering to calculate the O(~) corrections to fields. Speci cally, we investigate whether the approach can be extended to deal with two interacting scalar elds with a mass gap in (1+1) dimensions. To achieve this we need to verify the analyticity of the appropriate Jost function. All the machinery to do so is introduced during the course of the thesis. This includes the eld theoretic formalism which describes such a system, and the derivation of a number of di erential equations from which the density of states can be constructed. The numerical method is also outlined. Concrete results are presented to verify that the approach reproduces known results. Arguments related to Levinson's theorem are then presented that suggest that the Jost function is indeed analytic, with some caveats. / AFRIKAANSE OPSOMMING: Die gereedskap van perturbatiewe Kwantumveld-teorie is teen di e tyd al `n standaard-deel van die teoretiese sikus se werkskis. In hierdie tesis ondersoek ons die spektrale metode, `n benadering wat gereedskap van die kwantumteorie van verstrooing gebruik om die O(~) korreksie van velde te bereken. Ons ondersoek spesi ek of die metode uitgebrei kan word om toepaslik te wees in die geval van twee wisselwerkende skalaarvelde met `n massa-gaping in (1+1) dimensies. Vir hierdie doel moet ons bepaal of die toepaslike Jost-funksie analities is of nie. Al die masjinerie om dit te bepaal word deur die loop van hierdie tesis ingevoer. Dit sluit in die veld-teorietiese formalisme wat so `n stelsel beskryf, en die a eiding van `n aantal di erensiaal vergelykings wat gebruik kan word om die digtheid van toestande te konstrueer. Die numeriese metode word ook beskryf. Konkrete resultate wat bevestig dat die metode die korrekte antwoorde in `n analities bekende geval weergee, word verskaf. Argumente wat verband hou met Levinson se stelling word gebruik om te bevestig dat die Jost-funksie inderdaad analities is, met sekere voorbehoude.

Thesis (PhD (Physics))--University of Stellenbosch, 2010. / ENGLISH ABSTRACT: The aim of this thesis is threefold. Firstly, I give an overview of non-commutative quan- tum mechanics and build up a description of non-commutative piecewise constant poten- tial wells in this context. Secondly, I look at some of the stationary properties of a finite non-commutative well using the mathematical tools laid out in the first part. Lastly, I in- vestigate how non-commutativity affects the tunneling rate through a barrier. Throughout this work I give the normal commutative descriptions and results for comparsion. / AFRIKAANSE OPSOMMING: Die doel van hierdie tesis is drievoudig. Eerstens gee ek ’n oorsig van niekommutatiewe kwantummeganika en bou daarmee ’n beskrywing van niekommutatiewe deelswyskon- stante potensiaal putte op. Tweedens kyk ek na ’n paar van die stasionˆere eienskappe van ’n eindige niekommutatiewe potensiaal put deur die wiskunde te gebruik wat in die eerste deel uiteengesit is. Laastens ondersoek ek hoe niekommutatiwiteit die spoed van tonneling deur ’n potensiaal wal be¨ınvloed. Dwarsdeur die hierdie hele tesis gee ek die normale kommutatiewe beskrywings en resultate vir maklike vergelyking.

In thinking about quantum causality one would like to approach rigorous QFT from outside the perspective of QFT, which one expects to recover only in a specific physical domain of quantum gravity. This thesis considers issues in causality using Category Theory, and their application to field theoretic observables. It appears that an abstract categorical Machian principle of duality for a ribbon graph calculus has the potential to incorporate the recent calculation of particle rest masses by Brannen, as well as the Bilson-Thompson characterisation of the particles of the Standard Model. This thesis shows how Veneziano n point functions may be recovered in such a framework, using cohomological techniques inspired by twistor theory and recent MHV techniques. This distinct approach fits into a rich framework of higher operads, leaving room for a generalisation to other physical amplitudes. The utility of operads raises the question of a categorical description for the underlying physical logic. We need to consider quantum analogues of a topos. Grothendieck's concept of a topos is a genuine extension of the notion of a space that incorporates a logic internal to itself. Conventional quantum logic has yet to be put into a form of equal utility, although its logic has been formulated in category theoretic terms. Axioms for a quantum topos are given in this thesis, in terms of braided monoidal categories. The associated logic is analysed and, in particular, elements of linear vector space logic are shown to be recovered. The usefulness of doing so for ordinary quantum computation was made apparent recently by Coecke et al. Vector spaces underly every notion of algebra, and a new perspective on it is therefore useful. The concept of state vector is also readdressed in the language of tricategories.

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We determine the dynamical structure factors of two gapped correlated electron systems, namely the Ising model in a strong transverse field and the two-leg spin-1/2 Heisenberg ladder in the limit of strong rung coupling. We consider the low-temperature limit, employing a variety of analytical and numerical techniques. The coherent modes of single-particle excitations, which are delta functions at zero temperature, are shown to broaden asymmetrically in energy with increasing temperature. Firstly, we apply a low-temperature “resummation” inspired by the Dyson equation to a linked-cluster expansion of the two-leg Heisenberg ladder. We include matrix elements to second order in the interaction between states containing up to two particles. A low-frequency response similar to the “Villain mode” is also observed. Next, we apply a cumulant expansion technique to the transverse field Ising model. We resolve the issue of negative spectral weight caused by double pole in the leading self-energy diagram by including a resummation of terms obtained from the six-point function, demonstrating that the perturbation series in 2n-spin correlation functions can be extended to higher orders. The result generalises to higher dimensions and the analytic calculation is compared to a numerical Pade approximant. We outline the extension of this method to the strong coupling ladder. Finally, we compare the previous results to numerical data obtained by full diagonalisation of finite chains and numerical evaluation of the Pfaffian, a method specific to the transverse field Ising chain. The latter method is used for a phenomenological study of the asymmetric broadening as well as an evaluation of fitting functions for the broadened lineshapes.

We discuss a method of classifying 2-dimensional invertible topological quantum field theories (TQFTs) whose domain surface categories allow non-orientable cobordisms. These are known as Klein TQFTs. To this end we study the 1+1 dimensional open-closed unoriented cobordism category K, whose objects are compact 1-manifolds and whose morphisms are compact (not necessarily orientable) cobordisms up to homeomorphism. We are able to compute the fundamental group of its classifying space BK and, by way of this result, derive an infinite loop splitting of BK, a classification of functors K → Z, and a classification of 2-dimensional open-closed invertible Klein TQFTs. Analogous results are obtained for the two subcategories of K whose objects are closed or have boundary respectively, including classifications of both closed and open invertible Klein TQFTs. The results obtained throughout the paper are generalisations of previous results by Tillmann [Til96] and Douglas [Dou00] regarding the 1+1 dimensional closed and open-closed oriented cobordism categories. Finally we consider how our results should be interpreted in terms of the known classification of 2-dimensional TQFTs in terms of Frobenius algebras.

Einstein’s equations are derived by following Jacobson’s thermodynamic method. It is seen that a family of possible field equations exist which satisfy the thermodynamic argument. Modified theories of gravity are addressed as possible candidates for replacing dark matter as an explanation for anomalous cosmological phenomena. Many of the proposed modified theories are not powerful enough to explain the currently observed phenomena and are rejected as viable theories of gravity. A surviving candidate, TeVeS, is further analyzed under the aforementioned thermodynamic argument to check for its consistency with thermodynamics.

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.

Doctor of Philosophy / Department of Mathematics / Louis Crane / In this thesis we explore the possibilities of obtaining Topological Quantum Field Theories using cobordisms with corners to break further down in the structure of manifolds of a given dimension. The algebraic data obtained is described in the language of higher category theory, more precisely in its cubical approach which we explore here as well. Interesting connections are proposed to some important objects in Physics: the representations of the Poincaré group. Finally we will describe in great details the topological tools needed to describe the categories of cobordisms with corners and give some conjectures on their nature.