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Understanding the dynamics of strongly coupled gauge theories is one of the greatest challenges in modern theoretical physics. A new hope in attacking this problem was brought by the surprising discovery of integrability in a special four-dimensional gauge theory { the N = 4 supersymmetric Yang-Mills theory (SYM) in the limit of large number of colors. Quantum integrability manifests itself as a powerful hidden symmetry which allows to explore the theory far beyond the conventional perturbative regime, and may even lead to its exact solution. Integrability should also shed light on the striking gauge/string duality, which holographically relates N = 4 SYM with a string theory in curved geometry. In this thesis we focus on one of the key quantities in the N = 4 SYM theory { its spectrum of conformal dimensions, which correspond to string state energies. The study of integrability has culminated in reformulation of the spectral problem as a compact set of Riemann-Hilbert type equations known as the Quantum Spectral Curve (QSC). We demonstrate the power of this framework by applying it to study the spectrum in a wide variety of settings. The new methods which we present allow to explore previously unreachable regimes. We rst discuss an all-loop solution in a near-BPS limit, leading also to new strong coupling predictions. Next we describe an ecient numerical algorithm which allows to compute the nite-coupling spectrum with nearly unlimited precision (e.g. 60 digits in some important cases). We also present a universal analytic iterative method, which in particular allows to solve a longstanding open problem related to the BFKL limit in which N = 4 SYM develops close links with QCD. Finally we propose the extension of the QSC to the deformed case corresponding to a cusped Wilson line, uncovering new algebraic features of the construction. This allows to systematically study the generalized quark-antiquark potential and generate numerous new results.

We investigate the validity of the collective coordinate approximation to the scattering of two solitons in several classes of (1+1) dimensional field theory models. First we consider the collision of solitons in the integrable NLS model and compare the results of the collective coordinate approximation with results obtained using a full numerical simulation. We find that the approximation is accurate when the solitons are some distance apart and is reasonably good during their interaction. We then consider a modification of the NLS model with a deformation parameter which changes the integrability properties of the model, either completely or partially (the model becomes quasi-integrable). As the collective coordinate approximation does not allow for the radiation of energy out of a system we pay particular attention to how the approximation fares when the model is quasi-integrable and therefore has asymptotically conserved charges (i.e. charges Q(t) for which Q(t -> -infinity)=Q(t -> infinity)). We find that the approximation accurately reproduces the physical properties of the solitons, and even their anomalous charges, for a large range of initial values. The only time the approximation is not totally reliable is for the scatterings when the solitons come very close together (within one width of each other). To determine whether these results hold in a model with topological solitons we then consider a modified sine-Gordon model. The deformation preserves the topology of the model but changes the integrability properties in a similar way to the modified NLS model. In this model we find that the approximation is accurate when the model is either integrable or quasi-integrable, but the accuracy was much reduced when the model was completely non-integrable. To further explore this link between the accuracy of the collective coordinate approximation in a modified sine-Gordon model and the integrability properties of the system, we then consider soliton scattering in a double sine-Gordon model. The double sine-Gordon model allows us to vary between two integrable sine-Gordon models, and when the model is not integrable it still possesses the additional symmetries necessary for quasi-integrability. We find that for all values of our deformation parameters the approximation is accurate and that, as expected, the anomalous charges are asymptotically conserved.

This thesis is concerned mainly with the accidental degeneracy of the n-dimensional anisotropic harmonic oscillator. The general concept of accidental degeneracy is defined, and a summary is given of some of the explanations for it. A description is presented of Mackey's theory of group representations for general semi-direct product groups. This is followed by a method for the explicit application of the theory to the cases which arise in this thesis.

This work is a study of the nature of the motion of a molecule near the surface of a solid. The dynamics of some simple models are analysed in detail but these models are too simple to represent a scattering experiment. So the conclusions are qualitative. A method of presenting the global dynamics of a conservative classical system is described. This method is used to present the motion of a rigid rotator colliding with a plane passive surface. Part of the motion on repulsive pair-potentials is interpreted using perturbations from an averaging approximation and part by comparing the motion with that of a dumbell striking a hard wall. Where the potential has a deep well or there is an abrupt change in the force on the molecule the motion of the rotator has no perturbation part. If the motion of a dumbell striking a hard wall is averaged over the initial orientation of the dumbell the transition probabilities are controlled by the bounds due to conserved variables. Some ideas of ergodic theory are considered to find how the strongly-coupled motion might be described. These ideas have bean developed for bound systems and so do not give a unique description of the scattering. Although it is not possible to characterise any system the motion of a dumbell striking a hard wall may be described simply. A quantum formalism to calculate transition probabilities for a dumbell or an ellipsoid striking a plane hard wall is developed. The variations with total energy of these probabilities are interpreted using the classical motion and properties of the quantum equations.

A number of numerical schemes are employed in order to gam insight into the stability of the infinite swept attachment line boundary layer. The basic flow is taken to be of the Hiemenz class, i.e. a two-dimensional stagnation line flow with an added crossflow giving rise to a constant thickness boundary layer along the attachment line. For the perturbation flow quantities the assumption is made that the chordwise velocity components is linearly dependent 011 the chordwise coordinate x in 2-D, while the spanwise and normal velocity components are taken to be independent of x. In all numerical schemes, a second-order-accurate finite-differencing scheme is tlsf'd in the normal to the wall direction, a pseudo-spectral approach is emplo)wl in the other directions; temporally, an implicit Crank-Nicolson scheme is used and the resulting system of equations is solved using an initial-value-problem approach. Extensive use of the efficient Fast Fourier Transform (FFT) algorithm has been made in order to transform the solution betw( > en real and Fourier-transform spaces; the FFT was chosen because of the substantial savings in computing-cost which result from its implementation. The linear two-dimensional results (i.e. results obtained for the perturbation flow quantities when small amplitude two-dimensional perturbation waves are introduced into the basic flow) of previous investigations were accurately reproduced using this initial-value-prohlem approach, thus departing from normal-mode analyses which has invariably been the tool for earlier work. A time-periodic scheme is also employed, producing identical results to those of the time-marching approach in the case of the time-periodic forcings. The two-dimensional work was extended to study non-linear effects in the perturbation (Le. non-small perturbation of the basic flow), although the cost of this study generally proved prohibitively high. However, some (very expensive) nonlinear results obtained suggest that the grid used for the linear calculations (which resulted in very large code sizes) has to be further refined in the spanwise direction in order to account for the growth of the non-linear terms. This result is, as expected, Reynolds number dependent and should be taken into consideration when an investigation of the subcritical regime is undertaken. The small-time behaviour of the flow, i.e. the flow development shortly after small-amplitude perturbations have been introduced into the basic flow, has been studied analytically using the matchf'd asymptot.ics expansions method. The boundary layer is taken to consist of two flow regions, an 'inner' and an 'outer' one. The parameter upon which perturbation qnantities are expanded is in both cases the normal coordinate, taken to be small in the inner region and of order one in the outer region. The stability of the flow is studied at stat.ions off the attachment-line, when three-dimensional perturbations are introduced into the Hiemenz flow; in this case assumptions analogous to those of parallel-flow are used, namely that there exist two chordwise length-scales, a slow one corresponding to the slow boundary layer acceleration and a fast one upon which perturbation quantities are dependent. An important result obtained is that, at a station off the attachment line, the critical for the onset of instability Reynolds numher is an order of magnitude smaller than the corresponding two-dimensional critical Reynolds number at the attachment line. Difficulties were experienced when trying to determine a lower branch of the neutral loop in spanwise wavenumber - Reynolds number space; a lower branch was not found for wavenumbers as low as the accuracy of the numerical method employed, a result suggesting that a lower branch of the neutral curve may not exist.

A pair of orthonormal bases are mutually unbiased (MU) if the inner products across all their elements have equal magnitude. In quantum mechanics, these bases represent observables that are complementary, i.e. a measurement of one observable implies maximal uncertainty about the possible outcome of a subsequent measurement of a second observable. MU bases have attracted interest in recent years because their properties seem to depend dramatically on the dimension d of the quantum system in hand. If the dimension is given by a prime or prime-power, the state space is known to accommodate a complete set of d+1 MU bases. However, for composite dimensions, such as d = 6, 10, 12, ..., complete sets seem to be absent and it is not understood why. In this thesis we carry out a comprehensive study of MU product bases in dimension six. In particular, we construct all MU bases in dimension six consisting of product states only. The exhaustive classification leads to several non-existence results. We also present a new construction of complex Hadamard matrices of composite order, which is a consequence of our work on MU product bases. Based on this construction we obtain several new isolated Hadamard matrices of Butson-type.

This thesis concerns how symmetries impinge on quantum mechanical measurements, and preclude certain self adjoint operators from representing observable quantities. After developing the requisite mathematical machinery and aspects of quantum measurement theory necessary for our analysis, we proceed to critically review the literature surrounding the remarkable theorem of Wigner, Araki and Yanase (WAY) that prohibits accurate and repeatable measurements of any observable not commuting with an additive conserved quantity, as well as discussing the conditions under which approximate measurements with approximate degrees of repeatability can be achieved. We strengthen the original statement of the WAY theorem and generalise it to the case of position measurements obeying momentum conservation, leading to a solution of a long-standing problem of Stein and Shimony. A superselection rule appearing as the existence of an observable which commutes with all others gives rise to a stronger restriction than the WAY theorem, yielding self adjoint operators which do not represent observable quantities. We analyse various perspectives on superselection rules, aiming to clarify different viewpoints appearing in the literature since the inception of the topic in 1952. We exploit an explicit description of relative phase observables which have been lacking in other contributions, delineating conditions under which relative and (prohibited) absolute phases become statistically close. By providing simple models we are able to mimic a number of attempts to overcome superselection rules, in order to highlight the generic features of such attempts. We show that the statistical proximity of absolute and relative quantities arises only when there is a highly localised phase reference, and that the superselection rule compatible relative phase factors between certain superpositions takes on the appearence of a forbidden relative phase factor in this limit. However, we argue that these relative phase factors can be determined fully within the confines of a superselection rule.

In this thesis, we discuss the natural dynamics of spin chains with regards to their potential role as information carriers in quantum computers and also standalone uses in a quantum computational context. We discuss a range of spin chain devices, expanding existing results on linear chains and simple branched devices to more complex geometries and circular devices. Over the course of this work, we analyse requirements and feasibility of perfect state transfer using the natural dynamics of spin chains and present an extensive investigation into the effects of a number of perturbations on the dynamics of these devices. This includes both fabrication defects and other sources of perturbations. We also present other potential uses of spin chains, including state storage, and introduce an original protocol for the generation of cluster state ladders using only a single linear spin chain.

Quantum information theory promises many advances in science and technology. This thesis presents three different results in quantum information theory. The first result addresses the theoretical foundations of quantum metrology. It is now well known that quantum-enhanced metrology promises improved sensitivity in parameter estimation over classical measurement procedures. The Heisenberg limit is considered to be the ultimate limit in quantum metrology imposed by the laws of quantum mechanics. It sets a lower bound on how precisely a physical quantity can be measured given a certain amount of resources in any possible measurement. Recently, however, several measurement procedures have been proposed in which the Heisenberg limit seemed to be surpassed. This led to an extensive debate over the question how the sensitivity scales with the physical resources such as the average photon number and the computational resources such as the number of queries that are used in estimation procedures. Here, we reconcile the physical definition of the relevant resources used in parameter estimation with the information-theoretical scaling in terms of the query complexity of a quantum network. This leads to a novel and ultimate Heisenberg limit that applies to all conceivable measurement procedures. Our approach to quantum metrology not only resolves the mentioned paradoxical situations, but also strengths the connection between physics and computer science. A clear connection between physics and computer science is also present in other results. The second result reveals a close relationship between quantum metrology and the Deutsch-Jozsa algorithm over continuous-variable quantum systems. The Deutsch-Jozsa algorithm, being one of the first quantum algorithms, embodies the remarkable computational capabilities offered by quantum information processing. Here, we develop a general procedure, characterized by two parameters, that unifies parameter estimation and the Deutsch-Jozsa algorithm. Depending on which parameter we keep constant, the procedure implements either the parameter estimation protocol or the Deutsch-Jozsa algorithm. The procedure estimates a value of an unknown parameter with Heisenberg-limited precision or solves the Deutsch-Jozsa problem in a single run without the use of any entanglement. The third result illustrates how physical principles that govern interaction of light and matter can be efficiently employed to create a computational resource for a (one-way) quantum computer. More specifically, we demonstrate theoretically a scheme based on atomic ensembles and the dipole blockade mechanism for generation of the so-called cluster states in a single step. The entangling protocol requires nearly identical single-photon sources, one ultra-cold ensemble per physical qubit, and regular photo detectors. This procedure is significantly more efficient than any known robust probabilistic entangling operation.