Tensoring the field content of two super-Yang-Mills theories results in the field content of a certain supergravity theory, a procedure we call squaring. This thesis investigates how both the local and global internal symmetries enjoyed by the supergravity theory are inherited from the super-Yang-Mills factors. This is part of a much larger framework for studying a supergravity theory through its factorisation into simpler theories. The thesis begins by introducing local and global symmetries in the general context of relativistic field theory. The introduction is a short review on spacetime and internal symmetries in both gravitational and non-gravitational theories with particular focus on the supersymmetric regime. This is followed by the squaring idea and its appearance in various different contexts. After providing a unified description of all super-Yang-Mills theories over the four Normed Division Algebras R,C,H,O, the global internal supergravity symmetries are built with the help of the mathematical construction of the magic square, generalised to what we call the magic pyramid of supergravities. A physical interpretation of the formula reveals the Yang-Mills origin of the symmetries and demonstrates how simultaneous supersymmetry transformations on both factors can contribute to bosonic generators. The analysis is then extended to accommodate more exotic squarings by allowing for the coupling of matter multiplets to the super-Yang-Mills factors. Finally, the focus shifts to local internal symmetries whose linear form is derived directly from the corresponding linear factors. After a general treatment of off-shell squaring in various spacetime dimensions, the possibility of extending the construction to non-linear gravity is discussed.

We show how to evaluate divergent asymptotic series using a modified Borei resummation method. We develop and test this technique using three different perturbative expansions of the anharmonic oscillator. In the first two expansions this provides the energy eigenvalues directly; however, in the third method we tune the wavefunctions to achieve the correct large X behaviour, as first illustrated in 1. This tuning technique allows us to determine the energy eigenvalues up to an arbitrary level of accuracy with remarkable efficiency. We give numerical evidence to explain this behaviour. We also refine the modified Borei summation technique to improve its accuracy. The main sources of error are investigated with reasonable error corrections calculated. Having developed a suitable resummation technique we show how to generate a type of local expansion for vacuum, one-and two-particle states in the Schrödinger representation of quantum field theory. We also develop a large distance expansion of the ร matrix in terms of a momentum cut-off. Computer programs capable of producing the local expansions of the wavefunctionals and S matrix to an arbitrary order are generated.

R-matrix theory is applied to three-body resonances by treating the decay as two sequential two-body decays. This allows the decay to proceed through up to three different decay routes, each with an associated width. From the literature it is unclear whether these widths should be combined incoherently or coherently, especially in the case of two-neutron emission. In this work three-body R-matrix theory is applied to the 1.8 MeV 2+ resonance in He, as the interactions of the two-body subsystems are well known. This resonance can decay via two possible routes, each going through an intermediate state; (i) 6He(2+) -5He(3/2- g.s.) + n → alpha + n + n (ii) 6He(2+) → 2n(0+) + alpha → alpha + n + n Limiting cases of the theory are explored, in particular the sensitivity to the shape of the probability density function (PDF) which describes the shape of the intermediate state. A new formula is developed for the PDF which allows for the choice of optimised boundary conditions. This is particularly important where the intermediate state is a virtual state. In order to determine whether the widths from the R-matrix decay routes interfere coherently or incoherently, fully dynamical three-body hyper spherical harmonic (HH) calculations are also performed. The HH calculations produce a single value for the total width of the three-body decay. It was found that the coherent and HH widths are in agreement, but both fall short of the experimentally measured value. This is consistent with the calculations for two-proton decay found in the literature.

This investigation concerns the transition pathway of the condensation phase transition. Under certain conditions condensation is initiated by nucleation events, which are driven by fluctuations or instabilities in the vapour phase. This involves the spontaneous formation of groups of particles, which we refer to as clusters. The clusters have a highly unstable nature and exist momentarily, before breaking up. This makes them difficult to study experimentally and model mathematically, in comparison to larger more stable systems. The aim of this study is to explore the stability of these tiny molecular clusters that exist momentarily within their environment, in terms of the time taken for the cluster to lose particles (decay). To do this we employ a microscopic cluster model of n-nonane molecules in which the cluster is treated in isolation from the vapour particles that would normally surround it. The interactions between cluster particles are modelled using empirical potentials. The cluster's dynamics is modelled using deterministic molecular dynamics simulations. The simulations generate a time evolved trajectory of all the positions, velocities and forces of all the atoms in the cluster. The process of cluster decay in n-nonane clusters is modelled using a Langevin interpretation of the decay mechanism. This treatment views cluster decay as a process of single particle escape from a confining potential of mean force, driven by a particle's interactions with the surrounding cluster particles. The motion of a cluster particle is modelled using a Langevin equation, which is parameterised using the MD generated data in order to extract the decay related parameters. The decay parameters are used to evaluate an Arrhenius type equation for the kinetic decay rate. This is used to calculate the mean timescale of cluster decay for n-nonane clusters, which we refer to as the mean cluster lifetime. We compare the dynamically generated lifetimes calculated from the model to those predicted by experimental measurements, as well as classically derived lifetimes. We discover the dynamical model predicts lifetimes that compare well to experimental predictions. The cluster decay model allows us to predict cluster decay timescales without decay events actually occurring. This makes it an essential tool for systems with long decay timescales, for which decay events can not be feasibly observed through MD simulations alone. Finally, the last chapter presents recent work that has been conducted on ice cluster embryos. The ice embryos emerge during the freezing transition of supercooled water into ice I. Unlike the previous method of treating clusters in isolation from their surroundings, this study involves the treatment of ice clusters in coexistence with their environment. We utilise a molecular dynamics trajectory of supercooled water freezing into ice, which is used to identify and extract ice cluster embryos. It is evident from the MD simulations that at the initial stages of freezing the clusters are very amorphous and disordered. We investigate cluster properties such as the size distribution and molecular connectivity, and explore whether we are able to quantify the potential of mean force in order to estimate the mean lifetime of disordered ice cluster embryos.

The concern of this thesis is a performance analysis of certain Markov chain Monte Carlo algorithms. The performance of Markov chain Monte Carlo algorithms in general is determined by the rate of convergence toward stationarity of the Markov chains involved. There now exists a substantial body of work on this subject and to begin with a synopsis of “classical” techniques for bounding convergence rate is given. Such techniques are: coupling, spectral gap, conductance and canonical paths. The focus then shifts to recent improvements on these techniques followed by sample applications: the first one illustrates how the path coupling method simplifies the analysis of a Markov chain for generating random lozenge tilings. The next example highlights the gains made by average conductance over “classical” conductance for the bases-exchange walk on balanced matroids. The result for the latter example is obtained by an inductive argument and is subsequently improved by the use of so-called logarithmic Sobolev constants. Bounding the logarithmic Sobolev constant is here achieved by following a decomposition-cum-induction approach. Such decompositional approaches for analysing Markov chain convergence are another manifestation of the “divide-and-conquer” paradigm. The thesis concludes with the treatment of a novel decomposition method and its application to bounding the convergence rate of a random process on bounded Cayley trees.

This thesis is a report of work which develops the study of electromagnetic radiation by accelerating charges in the scalar quantum electrodynamic theory. We investigate aspects of this theory in flat spacetime, and in a class of conformally flat and curved spacetimes. In particular, we show that in flat spacetime, the quantum-theoretic prediction for the emission of energy by the particle, in the limit h-bar tends to 0 and to order e^2 in the coupling constant, may be shown to match the classical calculation. We also calculate the order h-bar correction to this quantity for two specific classes of problem. In the class of conformally flat and curved spacetimes, we compare the change in position due to the radiation reaction with the classical result, and we also consider some of the one-loop corrections to the theory. We show that as h-bar tends to 0, the conformally flat result and the classical result match, but that in that limit the general spacetime results differ.

This thesis consists of three parts. In the rst part we review the quantization of Yang-Mills theories and perturbative quantum gravity in curved spacetime. In the second part we calculate the Feynman propagators of the Faddeev- Popov ghosts for Yang-Mills theories and perturbative quantum gravity in the covariant gauge. In the third part we investigate the physical equivalence of covariant Wightman graviton two-point function with the physical graviton two-point function. The Feynman propagators of the Faddeev-Popov ghosts for Yang-Mills theories and perturbative quantum gravity in the covariant gauge are infrared (IR) divergent in de Sitter spacetime. We point out, that if we regularize these divergences by introducing a nite mass and take the zero mass limit at the end, then the modes responsible for these divergences will not contribute to loop diagrams in computations of time-ordered products in either Yang-Mills theories or perturbative quantum gravity. We thus nd eective Feynman propagators for ghosts in Yang-Mills theories and perturbative quantum gravity by subtracting out these divergent modes. It is known that the covariant graviton two-point function in de Sitter spacetime is infrared divergent for some choices of gauge parameters. On the other hand it is also known that there are no infrared problems for the physical graviton two-point function obtained by xing all gauge degrees of freedom, in global coordinates. We show that the covariant Wightman graviton two-point function is equivalent to the physical one in the sense that they result in the same two-point function of any local gauge-invariant quantity. Thus any infrared divergence in the Wightman graviton two-point function in de Sitter spacetime can only be an gauge artefact.

This thesis studies quantum integrable structures such as Yangians and quantum affine algebras that arise in and are inspired by the AdS/CFT duality, with a primary emphasis on the exploration of integrable boundaries deeply hidden in the duality. The main goal of this thesis is to find novel algebraic structures and methods that could lead to new horizons in the theory of quantum groups and in the exploration of boundary effects in the gauge/gravity dualities. The main thrust of this work is the exploration of the AdS/CFT worlsheet scattering theory and of integrable boundaries that manifest themselves as Dp-branes (p+1-dimensional Dirichlet submanifolds) which are a necessary part of the superstring theory. The presence of these objects breaks some of the underlying symmetries and leads to boundary scattering theory governed by coideal subalgebras of the bulk symmetry. Here the boundary scattering theory for D3-, D5- and D7-branes is considered in detail, and the underlying boundary Yangian symmetries are revealed. The AdS/CFT worldsheet scattering theory is shown to be closely related to that of the deformed Hubbard chain. This similarity allows us to apply the quantum deformed approach to the boundary scattering theory. Such treatment of the system leads to quantum affine symmetries that manifest themselves in a very elegant and compact form. In such a way the symmetries of distinct boundaries that previously seemed to be unrelated to each other emerge in a uniform and coherent form. The quantum deformed approach also helps us to better understand the phenomena of the so-called secret symmetry. It is called secret due to its peculiar feature of appearing as a level-one generator of the Yangian of the system. However it does not have a Lie algebra (level-zero) analogue. This symmetry is shown to have origins in the quantum deformed model, where it manifest itselfs as two, level-one and level-minus-one, generators of the corresponding quantum affine algebra.

In this thesis we consider a number of different aspects of dynamical locality, an axiom on locally covariant theories proposed by Fewster and Verch that is closely related to the question of whether a theory describes the same physics in all spacetimes. After some introductory material, in Chapters 3 and 4 we examine dynamical locality for the nonminimally coupled scalar field and its enlarged algebra of observables. We show that dynamical locality holds at all masses, including non-zero masses, for the nonminimally coupled scalar field theory. We also demonstrate that dynamical locality holds in the massive minimally coupled and massive conformally coupled cases for the enlarged algebra of observables, and fails to hold in the massless minimally coupled case. In Chapter 5, we discuss a number of categorical structures that can be used in the construction of classical theories that may be quantized using canonical anticommutation relations (CAR), and their subsequent quantization. We prove a number of results pertaining to dynamical locality of classical theories and their CAR-quantized counterparts. In Chapters 6 and 7, we give a simplified version of the locally covariant classical and quantum Dirac theories, using the machinery developed in Chapter 5. We also formulate for the first time versions of these theories that are entirely independent of the choice of a global reference frame for the spacetime, and depend only on an equivalence class of these frames. We demonstrate that both the simplified frame-dependent theories and the frame-independent theories are dynamically local.

In this thesis I explore using the qubus for quantum computing. The qubus is an architecture of quantum computing, where a continuous variable ancilla is used to generate operations between matter qubits. I concentrate on using the qubus for two purposes - quantum simulation, and generating cluster states. Quantum simulation is the idea of using a quantum computer to simulate a quantum system. I focus on conducting a simulation of the BCS Hamiltonian. I demonstrate how to perform the necessary two qubit operations in a controlled fashion using the qubus. In particular I demonstrate an O(N3) saving over an implementation on an NMR computer, and a factor of 2 saving over a naıve technique. I also discuss how to perform the quantum Fourier transform on the qubus quantum computer. I show that it is possible to perform the quantum Fourier transform using just, 24⌊N/2⌋ + 7N − 6, this is an O(N) saving over a naıve method. In the second part of the thesis, I move on, and consider generating cluster states using the qubus. A cluster state, is a universal resource for one-way or measurement-based computation. In one-way computation, the pre-generated, entangled resource is used to perform calculations, which only require local corrections and measurement. I demonstrate that the qubus can generate cluster states deterministically, and in a relatively short time. I discuss several techniques of cluster state generation, one of which is optimal, given the physical architecture we are using. This can generate an n × m cluster in only 3nm − 2n − 2m + 4 operations. The alternative techniques look at generating a cluster using layers or columns, allowing it to be built dynamically, while the cluster is used to perform calculations. I then move on, and discuss problems with error accumulation in the generation process.