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This tliesis first discusses braneworld models, we explain how the bulk geometry in codimension 2 scenarios restricts braneworld fields in a way inconsistent with observation. We then show how generalising Einstein's equations to include Gauss-Bonnet terms avoids this problem and as an example we successfully reproduce the Priedmann-Robertson-Walker cosmology familiar in Einstein gravity. The work on braneworlds concludes with a detailed perturbation analysis of a simple conical space-time in Gauss-Bonnet gravity, non-trivially we find the standard four-dimensional Lichnerowicz equation on the brane even though the calculation is performed in six dimensions. Next, motivated by the microscopic description of black hole thermodynamics, we discuss Gubser and Mitra's conjectured relationship between classical and thermodynamic stability including a review of numerical and theoretical evidence for it. We then give an argument using a recently discovered ansatz for non-uniform smeared p-brane solutions that the conjecture fails in the generality in which it is proposed. The thesis emphasises the underlying relationship between world volume field theory and bulk gravity from a geometrical point of view throughout.

This thesis presents research into the properties and features of the complex sine- Gordon theory. The CSG theory is a dimensional integrable held theory that admits soliton solutions which carry a Noether charge due to the U(I) invariance of the theory. Integrable CSG defects and boundaries are constructed and interactions between solitons, defects and boundaries are analysed at the classical and quantum level. The introduction of defects into the theory is facilitated by a new Backlund transformation involving two parameters. Defect conditions, constructed so they maintain the integrability of the theory and found to be exactly the BT, are used to sew two CSG theories together. How solitons interact with the defect is investigated, in particular whether as in the SG theory solitons can be absorbed and emitted by the defect. The classical time-delay and phase-shift are calculated for soliton-defect and particle-defect scattering. Using the CSG defect to dress the Dirichlet boundary a new CSG boundary theory is produced. Its integrability is checked by the explicit construction of conserved charges. The various interactions between solitons and the boundary are analysed, compared and contrasted with the defect theory. Finally aspects of the quantum CSG boundary theory are examined, culminating in a conjecture for the quantum reflection matrix for a Q = -1-1 soliton reflecting from an unexcited boundary. Reflection and boundary bootstrap procedures are used to generate the general reflection matrix for any charged soliton reflecting from any excited boundary

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Optimal generalised quantum measurements are important for quantum information applications in both photonic and solid state systems. However, until now, the implementations of such measurements have been optical. Entanglement is also a very important resource in quantum communication and information processing. However, highdimensional entangled states and corresponding Bell-inequality violations are challenging to detect and demonstrate experimentally. This thesis focuses on these two aspects of signal detection. A cavity quantum electrodynamics (QED) scheme to realise an optimised quantum measurement demonstrating the superadditivity of quantum channel capacity is proposed and analysed. The measurement is shown to be feasible using atoms in a cavity QED setup even in the presence of rather high levels of experimental errors. This is interesting because cavity QED realisations could potentially be more easily scaled to increase quantum coding gain. Experimental unambiguous discrimination between non-orthogonal states is also carried out for the first time in the solid state using the nuclear spin of a nitrogen atom associated with a defect in bulk diamond—an important step for implementations of solid-state quantum computing. This thesis presents a method for verifying entanglement dimension using only Bell inequality test measurements. It also shows experimental results demonstrating genuine eleven-dimensional two-photon orbital angular momentum (OAM) entanglement and violations of generalised Bell inequalities up to dimension twelve. The demonstrated highdimensional entanglement is potentially useful for closing the detection loophole in Belltest experiments and for real-world large-alphabet quantum-cryptography applications.

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A method of analytic renormalisation is developed (in PART I of the thesis) to define the three point time ordered product of massless fields of exponential type as a strictly localisable distribution in the Jaffe Class. The uniqueness property, known for the two point T-product, is verified for the three point T-product for a special choice of finite renormalisation. It is characterised by minimum singularity on the 'light cone' (the Lehraann-Pohlmeyer 'ansatz'); there are no delta function type singularities concentrated on the point x₁ = x₂ = x₃. A model of a massive neutral pseudovector field, W_μ, coupled to a non-conserved fermion current, j_μ = ψγ_μγ₅ψ, is considered (in PART II of the thesis). The generalised Stuckelberg formalism is used to convert the above non-renormalisable coupling into a conventionally renormalisable interaction, together with a non-polynomial strictly localisable interaction which can be treated by the methods developed in PART I of this thesis; (A_μ, B) are the Stuckelberg components of the W_μ field, and the B is taken to be a massless pseudoscalar field giving, thus, rise to massless 'superpropagators'. The renormalisation of the model theory is effected with the help of generalised Ward-Takahashi identities by adding suitable gauge invariant counterterms in the original interaction Lagrangian to cancel out the infinities of the theory. Thus the complete theory becomes renormalisable.

After introductory remarks on the question of colour confinement in non-Abelian gauge theories of the strong interactions, Chapter 1 presents a brief review of the infrared (IR) phenomena in QED, with a comparative discussion of the non-Abelian case. In Chapter 2, the singularities arising in a purely massless process (Compton scattering in massless QED) are studied in perturbation theory. The IR and mass singularities from soft and hard bremsstrahlung are computed, and the cancellation implied by the Kinoshita-Lee-Nauenberg theorem is verified. The appearance of mass singularities in the renormalization process is discussed in the context of dimensional regularization. The first sections of Chapter 3 are devoted to the examination of general features of the IR divergence cancellation in the Yang-Mills theory. By making use of the Ward identities for on-shell amplitudes, soft virtual corrections and emission of soft and hard Yang-Mills quanta (gluons) are analyzed. In Section 3.4, the interest of considering a purely massless process in order to bring out characteristic features of the non-Abelian theory is emphasized, and gluon scattering in a colourless external potential is discussed Soft and hard gluon emission corrections are computed, and the inclusive transition rate is found to be free of singularities, to lowest non-trivial order in perturbation theory. Thus we extend previous investigations restricted to massive fermions and soft gluon emission. Our treatment includes non-leading singularities in the order considered and questions of gauge independence. Renormalization group-type equations which have been proposed, and which are presumed to sum up the singularities in perturbation theory, are described in Chapter 4. From the invariance of the S-matrix under the renormalization group, equations of the Callan-Symanzik type involving variations with respect to the IR cut-off are derived, and their significance is discussed. Questions related to the massive fermion decoupling theorem, zero-mass renormalization and dimensional regularization are also discussed.

The way in which the macroscopic description of the interaction of the electromagnetic field with matter is related to the microscopic viewpoint has been known for many years. Two distinct steps are necessary to establish the connection. Firstly one has to develop a descriptive scheme that accounts for the dynamical behaviour of microscopic systems qua matter under the influence of the field which involves the construction of a model of matter, and secondly there is the problem of calculating the averages over a large number of microsystems of the quantities predicted by the dynamical scheme. The latter problem is essentially one of statistical mechanics. This thesis is concerned with the first aspect of the problem, that is the construction of a dynamical scheme to describe the interaction of the electromagnetic field with matter which we view as being composed of spatially separated aggregates of point charges called 'molecular systems'. The theory is illustrated with some examples of the type of calculations that are possible within this scheme. We make no attempt to discuss the associated problems in statistical mechanics. The conventional theory of the interaction of radiation with molecules has been standardized for many years. There are however some unsatisfactory features about it which very largely stem from the problems associated with the invariance of the hamiltonian under gauge transformations and the consequent freedom of choice of gauge. As shown by Power and Zienau (Phil. Trans. Roy. Soc. 1959, <strong>A251</strong>, 427) there exists a unique unitary transformation which eliminates the difficulties and casts the hamiltonian into a form in which the molecular multipoles are explicitly displayed. They also showed the extent to which earlier versions of the multipole hamiltonian were inexact about the definition of the fields involved. Power and Zienau however considered only dipolar and quadrupolar terms explicitly and it is desirable to extend the theory to encompass multipoles of arbitrary order. Moreover a proper account of the functional dependence of the canonical variables implied by the existence of the gauge transformations seems to be lacking in the case of molecular quantum electrodynamics. The first part of the thesis therefore is concerned with the problem of deriving in a consistent non-relativistic approximation the modified hamiltonian for general electric and magnetic multipolar interactions with the electromagnetic field. We start from the microscopic Maxwell-Lorentz theory of charged particles which we develop in a form that is in close analogy to the macroscopic theory of dielectrics. We then postulate a lagrangian from which the field and particle equations of motion may be recovered under suitable assumptions using the calculus of variations. This procedure however forces us to introduce the electromagnetic field 4-potential and thus there are more dynamical variables than degrees of freedom. To proceed to the hamiltonian therefore requires a discussion of the singular (degenerate) nature of the lagrangian and we use the theory first developed by Dirac (Can. J. Math. 1950, <strong>2</strong>, 147) since this is the most direct way of performing calculations with lagrangians that exhibit degeneracy. Essentially the procedure consists of eliminating the dynamical variables that are not required in the account of the dynamical behaviour of the system, and when this is done the originally singular canonical scheme becomes regular as required. With a suitable lagrangian and the use of the Coulomb gauge condition to define a specific Lie algebra for the dynamical variables, the hamiltonian that finally emerges is the required generalization of the multipole hamiltonian. The importance of the Coulomb gauge condition is that it enables us to identify the canonical momentum with the transverse electric field strength operator to within a constant. The gauge condition is therefore an integral part of the theory. Having started from a non-relativistic classical lagrangian, the spin interaction terms are naturally absent from the hamiltonian and must be added, if required, in an ad hoc fashion. The second section of the thesis is concerned with the use of the theory within the context of perturbation theory based on an expansion of the resolvent operator of the complete hamiltonian. In Chapter 3 we give a detailed account of the calculation of the optical rotation angle of a molecule in terms of the properties of its constituent groups which we assume are electronically isolated. For simplicity at this stage the interaction potential is limited to the dipole approximation. Diagrammatic perturbation theory is used to simplify the calculation as much as possible. The final results of the calculation are in general agreement with those in the literature but the method seems to give a particularly clear account of the physical processes involved. In Chapter 4 we consider the complete multipole hamiltonian and examine the properties of its matrix elements in order to assess the feasibility of extending calculations in the dipole approximation to include higher multipoles. Some interesting new integral representations of the interaction terms are developed which may be useful in calculations. We conclude that there is no difficulty in generalizing calculations if the process in question involves real photons, but that in general 'virtual' photon processes may only be dealt with if a cut-off for the field energy is introduced. We note that if the theory is to be consistent with our original restriction to the non-relativistic case then a cut-off is required but we do not analyse such a theory. Finally in the Discussion, we attempt to assess our formulation of the multipole hamiltonian in relation to other versions and find agreement only with Power (introductory Quantum Electrodynamics, 1964) though of course our final hamiltonian is not limited to quadrupolar terms. By extending the analysis of the integral representation of the electric multipole term in the hamiltonian, which we developed in Chapter 4, we are able to give a fully quantum mechanical justification of the unitary transformation operator employed by Power and Zienau (1959) and Power (1964). Our argument follows from the recognition that the phase of probability amplitudes may be altered in the presence of an electromagnetic field ('Bohm-Aharonov' effect). In short the multipole hamiltonian is related to the conventional hamiltonian by a unitary operator that changes the phase of the state vectors of the two descriptions by an amount that corresponds to formally moving every particle of a molecular aggregate to the centre of mass of the aggregate (or vice versa to go from the multipole description to the conventional one) in the presence of an electromagnetic field described by a given vector potential. We conclude by discussing some of the difficulties inherent in the current theory of molecular quantum electrodynamics which appear to be of a fundamental nature since they arise from our manner of describing charged particles.