In this thesis, we study the "two-point approximation" for highly non-local effective actions, in the particular case of the Chiral Soliton Model of the nucleon. The nucleon in this model is regarded as being made of three valence quarks bound together by a meson field in a soliton form. Mesons are treated in mean field theory and the vacuum energy due to one-quark loops is included. The theory is defined with a finite cut-off in momentum space, consistent with an effective theory for the low-energy description of the strong interactions. We use the two-point approximation to calculate the vacuum correction to the chiral soliton energy for a variety of soliton profile functions, investigating the effect of different regularisation schemes. Results are little influenced by the choice of the cut-off, and are within 20% of exact calculations, done with the full inclusion of the Dirac sea. We then perform a dynamical calculation of the chiral soliton by including sea-quark effects self-consistently in the two-point approximation. We find a typical 20% (or less) deviation in the soliton energy from exact calculations. We apply a further "pole" approximation which leads to a significant algebraic simplification in the self-consistent equations. We show, in particular, that a simple numerical fit of the pole form to the two-point cut-off function yields essentially indistinguishable results from the latter. We finally calculate some static nucleon observables in the two-point approximation and find general agreement with exact calculations. In view of the results obtained, we may hope that the pole form of the twopoint approximation may prove to be a generally useful approach to similar problems involving highly non-local actions.

This dissertation begins with a review of Calabi-Yau manifolds and their moduli spaces, flux compactification largely tailored to the case of type IIb supergravity, and Coleman-De Luccia vacuum decay. The three chapters that follow present the results of novel research conducted as a graduate student. Our first project is concerned with bubble collisions in single scalar field theories with multiple vacua. Lorentz boosted solitons traveling in one spatial dimension are used as a proxy to the colliding 3-dimensional spherical bubble walls. Recent work found that at sufficiently high impact velocities collisions between such bubble vacua are governed by "free passage" dynamics in which field interactions can be ignored during the collision, providing a systematic process for populating local minima without quantum nucleation. We focus on the time period that follows the bubble collision and provide evidence that, for certain potentials, interactions can drive significant deviations from the free passage bubble profile, thwarting the production of a new patch with different field value. However, for simple polynomial potentials a fine-tuning of vacuum locations is required to reverse the free passage kick enough that the field in the collision region returns to the original bubble vacuum. Hence we deem classical transitions mediated by free passage robust. Our second project continues with soliton collisions in the limit of relativistic impact velocity, but with the new feature of nontrivial field space curvature. We establish a simple geometrical interpretation of such collisions in terms of a double family of field profiles whose tangent vector fields stand in mutual parallel transport. This provides a generalization of the well-known limit in flat field space (free passage). We investigate the limits of this approximation and illustrate our analytical results with numerical simulations. In our third and final project we investigate the distribution of field theories that arise from the low energy limit of flux vacua built on type IIb string theory compactified on the mirror quintic. For a large collection of these models, we numerically determine the distribution of Taylor coefficients in a polynomial expansion of each model's scalar potential to fourth order. We provide an analytic explanation of the proncounced hierarchies exhibited by the random sample of masses and couplings generated numerically. The analytic argument is based on the structure of masses in no scale supergravity and the divergence of the Yukawa coupling at the conifold point in the moduli space of the mirror quintic. Our results cast the superpotential vev as a random element whose capacity to cloud structure vanishes as the conifold is approached.

published_or_final_version / Mechanical Engineering / Master / Master of Philosophy

Nonlinear Dirac equations exhibiting soliton phenomena are studied. Conditions are derived for the existence of solitons and an analysis of their stability is presented. New results are obtained for models previously considered in the literature. A particular model is studied for which all stationary states are localized in a finite domain and have positive energy but indefinite charge. The electromagnetic field is introduced by minimal coupling and it is shown that the discrete nature of the electric charge, and of the angular momentum, follow from a many-body stability principle. This principle also implies the de Broglie frequency relation, and furnishes an expression for the fine structure constant. The resulting charged soliton is tentatively identified with the electron.

No description available.

[abstract] In the process of nucleation, the decay of a metastable state is initiated by the formation of a spatially localised region called a nucleus of critical size. In many realistic situations nucleation is initiated at an impurity or defect; such as a dust particle, an irregularity in a sample, or a crack in the wall of a container. The aim of this thesis is to identify and understand the fundamental changes different types of defect make to nucleation by studying a one-dimensional continuum model used to describe solitons. A well established theory due to Langer is extended to calculate the rate of decay of a metastable state due to the nucleation of solitons at defects. Results are used to find the rate of thermally activated magnetisation reversal for a ferromagnetic nanowire with defects in the uniaxial anisotropy. Defects which are narrower than the soliton width (point-like defects) and wider than the soliton width (step defects) are both modelled. An attractive defect breaks the translational symmetry of a soliton and leads to pinning. The pinning of solitons is found to reduce the activation energy required for nucleation, reduce the critical field above which a metastable state becomes unstable, alter the mechanism by which a metastable state decays, and modify the prefactor for the rate of decay. Changes to the prefactor are interpreted in terms of entropy and the dynamics of metastable decay when a defect is present.