In the following dissertation, we explore the applicability of Yangian symmetry to various integrable models, in particular, in relation with S-matrices. One of the main themes in this dissertation is that, after a careful study of the mathematics of the symmetry algebras one finds that in an integrable model, one can directly reconstruct S-matrices just from the algebra. It has been known for a long time that S-matrices in integrable models are fixed by symmetry. However, Lie algebra symmetry, the Yang-Baxter equation, crossing and unitarity, which are what constrains the S-matrix in integrable models, are often taken to be separate, independent properties of the S-matrix. Here, we construct scattering matrices purely from the Yangian, showing that the Yangian is the right algebraic object to unify all required symmetries of many integrable models. In particular, we reconstruct the S-matrix of the principal chiral field, and, up to a CDD factor, of other integrable field theories with su(n) symmetry. Furthermore, we study the AdS/CFT correspondence, which is also believed to be integrable in the planar limit. We reconstruct the S-matrices at weak and at strong coupling from the Yangian or its classical limit. This version of the thesis includes minor corrections following the viva on 17 September 2010.

In this thesis we use a time-dependent tight-binding model metal evolving under semiclassical Ehrenfest dynamics to explore the effects of electron-ion energy exchange on radiation damage phenomena. By incorporating an explicit model of quantum mechanical electrons coupled to a set of classical ions, our model correctly reproduces the interaction of excited ions with cooler electrons and captures phenomena absent in classical molecular dynamics simulations and in much-used analytical models. With our simple model we have been able to simulate large numbers of radiation damage cascades. We have directly explored the electronic excitations stimulated in such cascades and have found them to be well characterized by an elevated electronic temperature. We have also analysed the effect of these excitations in weakening the bonding interactions in our model metal, and the effect of these weakened interactions on the evolution of replacement collision sequences. By separating out components of the Hellmann-Feynman forces exerted by the electrons on the ions, we have identi ed the non-adiabatic force, resulting from the finite response time of the electrons to ionic motion and responsible for the accumulating electronic excitations. Based on simplifying physical arguments we have derived a temporallyand spatially-local expression for this force suitable for incorporation within a classical MD code at very low computational cost. Data from our simulations show that our new expression for the non-adiabatic force captures much of the microscopic detail of the direction and magnitude of the force. We find that it significantly outperforms commonly used viscous damping models of ion-electron energy transfer. At higher energies, our simulations of ion channelling reveal a new resonant enhancement of the electronic charge on the channelling ion and corresponding effects on the stopping force. We explain these phenomena with reference to the detailed atomic and electronic structure of our model.

In this thesis, certain exact results in supersymmetric gauge theories are discussed. In these theories, holomorphic gauge invariant operators play a central role in understanding the structure of the space of solutions to vacuum equations, known as the moduli space. We focus on a technique to count such operators with various quantum numbers. The counting can be done by computing a partition function, known as the Hilbert series, which counts all holomorphic gauge invariant operators carrying a speci ed set of global U(1) charges. The Hilbert series can be computed exactly for various gauge theories. In Part I of this thesis, we compute the Hilbert series of four dimensional N = 1 supersymmetric QCD with classical gauge groups. In part II, we count chiral operators on the one instanton moduli space on R4 and study the hypermultiplet moduli spaces of a large class of N = 2 supersymmetric gauge theories in four dimensions. We demonstrate that the Hilbert series not only contains information about the spectrum of operators in the theory, but it also carries geometrical properties of the moduli space, e.g. the dimension. It is also an indicator of whether the moduli space is Calabi-Yau. Moreover, Hilbert series can be used as a primary tool to test various dualities in gauge theories and in string theory.

We consider a number of aspects of the problem of defining time observables in quantum theory. Time observables are interesting quantities in quantum theory because they often cannot be associated with self-adjoint operators. Their definition therefore touches on foundational issues in quantum theory. Various operational approaches to defining time observables have been proposed in the past. Two of the most common are those based on pulsed measurements in the form of strings of projection operators and continuous measurements in the form of complex potentials. One of the major achievements of this thesis is to prove that these two operational approaches are equivalent. However operational approaches are somewhat unsatisfying by themselves. To provide a definition of time observables which is not linked to a particular measurement scheme we employ the decoherent, or consistent, histories approach to quantum theory. We focus on the arrival time, one particular example of a time observable, and we use the relationship between pulsed and continuous measurements to relate the decoherent histories approach to one based on complex potentials. This lets us compute the arrival time probability distribution in decoherent histories and we show that it agrees with semiclassical expectations in the right limit. We do this both for a free particle and for a particle coupled to an environment. Finally, we consider how the results discussed in this thesis relate to those derived by coupling a particle to a model clock. We show that for a general class of clock models the probabilities thus measured can be simply related to the ideal ones computed via decoherent histories.

Topological solitons - are of broad interest in physics. They are objects with localised energy and stability ensured by their topological properties. It is possible to create them during phase transitions which break some sym- metry in a frustrated system. They are ubiquitous in condensed matter, ranging from monopole excitations in spin ices to vortices in superconduc- tors. In such situations, their behaviour has been extensively studied. Less well understood and yet equally interesting are the symmetry-breaking phase transitions that could produce topological defects is the early universe. Grand unified theories generically admit the creation of cosmic strings and monopoles, amongst other objects. There is no reason to expect that the behaviour of such objects should be classical or, indeed, supersymmetric, so to fully understand the behaviour of these theories it is necessary to study the quantum properties of the associated topological defects. Unfortunately, the standard analytical tools for studying quantum field theory - including perturbation theory - do not work so well when applied to topological defects. Motivated by this realisation, this thesis presents numerical techniques for the study of topological solitons in quantum field theory. Calculations are carried out nonperturbatively within the framework of lattice Monte Carlo simulations. Methods are demonstrated which use correlation functions to study the mass, interaction form factors, dispersion relations and excitations of quantum topological solitons. Results are compared to exact expressions obtained from integrability, and to previous work using less sophisticated numerical techniques. The techniques developed are applied to the prototypical kink soliton and to the 't Hooft-Polyakov monopole.

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Einstein's Theory of General Relativity has proven remarkably successful at modelling a wide range of gravitational phenomena. Amongst some of the novel features in this description is the existence of black holes; regions of space-time where gravity is so strong that light cannot escape. The properties of black holes have been extensively studied within General Relativity, culminating in the result that the few known space-times are the only allowed stationary black hole solutions in four dimensions. In the past half century, research has focused on how to unify the distinct theories of gravity and quantum mechanics. A common theme amongst several strong candidates is that space-time, the backdrop for gravity, is fundamentally higher dimensional. In these theories, the structure of black hole solutions is relatively unknown and expected to be much richer; finding such solutions is, however, a very hard task. In this thesis, we introduce new numerical methods to study higher dimensional black holes. The methods, based on refinements of existing work and the novel application of standard techniques, are then used to study a number of black hole space-times. Namely the structure of black holes on a Kaluza-Klein background, and rotating Kerr black holes. We demonstrate that these algorithms can be applied in a wide class of situations and yield good quality results with comparative ease. New results are presented in both cases studied. We examine the predicted merger between non-uniform black strings and localised black holes on a Kaluza-Klein background. We find evidence for a new type of non-uniform black string with one Euclidean negative mode and lower entropy than the uniform strings. We discover a window of localised black holes with one Euclidean negative mode but positive specific heat. We also look at the local structure of the merger point and find consistency with Kol's cone prediction.

As the frontiers of physics steadily progress into the 21st century we should bear in mind that the conceptual edifice of 20th-century physics has at its foundations two mutually incompatible theories; quantum mechanics and Einstein’s general theory of relativity. While general relativity refuses to succumb to quantum rule, black holes are raising quandaries that strike at the very heart of quantum theory. M-theory is a compelling candidate theory of quantum gravity. Living in eleven dimensions it encompasses and connects the five possible 10-dimensional superstring theories. However, Mtheory is fundamentally non-perturbative and consequently remains largely mysterious, offering up only disparate corners of its full structure. The physics of black holes has occupied centre stage in uncovering its non-perturbative structure. The dawn of the 21st-century has also played witness to the birth of the information age and with it the world of quantum information science. At its heart lies the phenomenon of quantum entanglement. Entanglement has applications in the emerging technologies of quantum computing and quantum cryptography, and has been used to realize quantum teleportation experimentally. The longest standing open problem in quantum information is the proper characterisation of multipartite entanglement. It is of utmost importance from both a foundational and a technological perspective. In 2006 the entropy formula for a particular 8-charge black hole appearing in M-theory was found to be given by the ’hyperdeterminant’, a quantity introduced by the mathematician Cayley in 1845. Remarkably, the hyperdeterminant also measures the degree of tripartite entanglement shared by three qubits, the basic units of quantum information. It turned out that the different possible types of three-qubit entanglement corresponded directly to the different possible subclasses of this particular black hole. This initial observation provided a link relating various black holes and quantum information systems. Since then, we have been examining this two-way dictionary between black holes and qubits and have used our knowledge of M-theory to discover new things about multipartite entanglement and quantum information theory and, vice-versa, to garner new insights into black holes and M-theory. There is now a growing dictionary, which translates a variety of phenomena in one language to those in the other. Developing these fascinating relationships, exploiting them to better understand both M-theory and quantum entanglement is the goal of this thesis. In particular, we adopt the elegant mathematics of octonions, Jordan algebras and the Freudenthal triple system as our guiding framework. In the course of this investigation we will see how these fascinating algebraic structures can be used to quantify entanglement and define new black hole dualities.

The emerging field of nano-electro-mechanical systems (NEMS), in which the single mode of a nanomechanical oscillator plays the role of an active device, is receiving much attention due to its technological importance. The characteristic component that gives the name to these devices is an oscillator of nanometer size coupled to the electrons on the dot that transfer electrons one-by-one between a source and a drain lead. From a fundamental point of view, it is important to understand the interplay between the electronic transport and the nanomechanical motion of the oscillator quantum mechanically. This thesis contains the description and analysis of the dynamics of a nanomechanical oscillator coupled to a resonant tunnel junction (RTJ) and its realization as a shuttle device. The models we consider describe both the mechanical and electrical degrees of freedom quantum mechanically; Firstly, a RTJ coupled to a nanomechanical oscillator. Secondly, we report a first complete quantum mechanical analysis of a charge shuttle. We introduce a new non-perturbative quantum mechanical description for the strong interaction of both the electrical and the mechanical object, which is beyond the existing experiments. We describe a nonequilibrium Green’s function formalism: a well suited technique to treat this kind of far from equilibrium systems, which can deal with very small to very large applied bias. The numerical implementation of these models are discussed in detail, and the transient and the steady state behavior of the system is also considered here for the quantum dynamics of the oscillator as a function of time. This will provide useful insight for the design of experiments aimed at studying the quantum behavior of an oscillator.

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