On the dynamics of topological solitons

Abstract

This thesis investigates the dynamics of lump-like objects in non-integrable field theories, whose stability is due to topological considerations. The work concerns three different low dimensional ((1 + 1)- and (2 + l)-dimensional) systems and addresses the questions of how the topology and metric structure of physical space, the quantum mechanics of the basic field quanta and intersoliton interactions affect soliton dynamics. In chapter 2 a sine-Gordon system in discrete space, but with continuous time, is presented. This has some novel features, namely a topological lower bound on the energy of a kink and an explicit static kink which saturates this bound. Kink dynamics in this model is studied using a geodesic approximation which, on comparison with numerical simulations, is found to work well for moderately low kink speeds. At higher speeds the dynamics becomes significantly dissipative, and the approximation fails. Some of the dissipative phenomena observed are explained by means of a dispersion relation for phonons on the spatial lattice. Chapter 3 goes on to quantize the kink sector of this model. A quantum induced potential called the kink Casimir energy is computed numerically in the weak coupling approximation by quantizing the lattice phonons. The effect of this potential on classical kink dynamics is discussed. Chapter 4 presents a study of the low-energy dynamics of a CP(^1) lump on the two-sphere in the geodesic approximation. By considering the isometry group inherited from globalsymmetries of the model, the structure of the induced metric on the unit-charge moduli space is so restricted that the metric can be calculated explicitly. Some totally geodesic submanifolds are found, and the qualitative features of motion on these described. The moduli space is found to be geodesically incomplete. Finally, chapter 5 contains an analysis of long range intervortex forces in the abelian Higgs model, a massive field theory, extending a point source. approximation previously only used in massless theories. The static intervortex potential is rederived from a new viewpoint and used to model type II vortex scattering. Velocity dependent forces are then calculated, providing a model of critical vortex scattering, and leading to a conjecture for the analytic asymptotic form of the metric on the two-vortex moduli space.