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In this Thesis, we present some direct quantitative tests for AdS/CFT correspondence using the newly discovered “integrability”. We shall begin by explaining the ideas of algebraic Bethe ansatz and scattering matrix in both gauge and string theories, and discussions how they enable us to extract the spectra of the scaling dimensions of the gauge invariant operators or the energies of the dual string states. In chapter 3, we explicitly apply the Bethe ansatz techniques in thermodynamically limit to the so-called <i>β</i>-deformation of ℵ = 4 SYM, and extract the one-loop anomalous dimensions for its long gauge invariant operators; we also construct the corresponding string solutions in the dual background, and show their energies precisely match with the gauge theory results. In the second part of Thesis, we consider a new asymptotic limit in both gauge and string theories, in such limit the elementary excitations are known as “magnons”; we shall also switch our focus to the building element of Bethe Ansatz, the scattering matrix between magnons. In chapter 4, we apply the magnon scattering matrix and explain how additional stable bound states can appear; both elementary magnons and their bound states have exact expressions for the dispersion relations. The classical string configuration dual to magnon bound state in gauge theory is constructed in chapter 5, where the connection between string sigma model on <i>R x S</i>³ and integrable complex sine-Gordon model is exploited. In chapter 6, the scattering matrix between the magnon bound states is considered via bootstrap method, and in the semiclassical limit, the result coincides with the scattering matrix between complex sine-Gordon solitons. This provides the direct verification for the proposed all-loop magnon scattering matrix and the lowest order “dressing factor”. In chapter 7, by applying the extended residual symmetry algebra psu(2|2)² x ℝ³, we classify all possible magnon bound states in terms of the constituent fields in ℵ = 4 SYM and justify the exactness of their dispersion relation.

Methods for improving the size of accurate Kohn-Sham (KS) DFT calculations have often focused on embedding the KS calculation within a region of fixed electronic density. This thesis examines the potential for a similar type of approach within Car-Parrinello, <i>ab-initio</i>, Molecular Dynamics (CPMD). A frozen electron density (FED) model is incorporated into a KS-DFT based simulations of liquid water. Interactions between FED and KS species are determined using total energy expressions taken from the study of embedded systems. The initial step during this development was a study of the FED electron gas model proposed by Gordon and Kim for rare gas systems, using a modified version of the DL_POLY MD code. The adapted code was applied to investigate various options for extending this scheme to inter-molecular interactions in liquid water, including assessment of the suitability of a number of kinetic energy (KE) functionals. The optimisation of electron density for a FED model of water was explored semi-empirically, based on adaptation of the fixed molecular electron density to the condensed phase environment. Consistent with experience from force field methods, enhancement of the molecular dipole moment proved to be necessary to reproduce the properties of the liquid. Using the TF, PW01 and LLP KE functionals, models giving good agreement with experimental results were obtained. These each carried a dipole moment of 2.95 D, the same as has been observed by recent <i>ab-initio</i> MD studies based on fully self-consistent KS methods. Finally the CPMD code was adapted to incorporate these models into mixed KS/FED simulations, in which both molecular types occupied the same simulation cell. All MD runs showed energetic stability and good overall structure. However, more acceptable reproduction of the inter-molecular interactions of a fully KS-DFT calculation were obtained with the PW91 and LLP based models, than with that based on the TF functional. This is in agreement with similar calculations reported for the gas phase dimer.

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This dissertation is concerned with a new formulation of fermionic quantum field theory in classical (electromagnetic or gravitational) backgrounds, which uses methods analogous to those used in conventional multiparticle quantum mechanics. Emphasis is placed on the states of the system, described in terms of Slater determinants, rather than on the field operator, <I>ψ</I>(<I>x</I>). The vacuum state 'at time <I>τ</I>', defined as the Slater determinant of a basis for the span of the negative spectrum of the 'first quantized' Hamiltonian <I><I>H</I></I><SUB>1</SUB> (<I>τ</I>), provides a concrete realisation of the Dirac Sea. By using the concept of 'radar time', I propose a generalisation of the concept of 'hypersurface of simultaneity', which can be applied to an arbitrarily moving observer in curved spacetime. This is used to provide a consistent particle interpretation for this observer, which depends <I>only</I> on the choice of observer and the background present, not on the choice of coordinates, the choice of gauge (in electromagnetic backgrounds) or the detailed construction of the observer's particle detector. It is also the first definition that does not rely on the spacetime possessing any convenient symmetries. I show that in the cases of a uniformly accelerating observer in flat space (Unruh effect), and a comoving observer in an exponentially inflating universe, my definition reduces to previously accepted definitions. Although this definition is necessarily non-local (no local definition of particle could possibly be consistent with the Unruh effect) I demonstrate with a simple example that this non-locality is only significant on scales of the order of the Compton wavelength λ<I><SUB>c</SUB> </I>= <I>h/mc<SUB> </SUB></I>of the particle concerned. The general S-matrix element of the theory is derived in terms of time-dependent Bogoliubov coefficients, demonstrating that this follows directly from the definition of inner product between Slater determinants. The process of 'Hermitian extension', inherited directly from conventional multiparticle quantum mechanics, allows second quantized operators to be defined without appealing to a complete set of orthonormal modes, and provides an extremely straightforward derivation of the general expectation value of the theory. Applications of the formalism to pair creation in spatially uniform electric fields, and to the treatment of discrete symmetries, are presented.

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