Fermion-soliton interaction in 1+1 dimensions

Abstract

This thesis deals with the concept of fermion-soliton interactions. In the very beginning, we provide a historical background on how solitons were discovered and how they led to other solutions with similar features in many branches of physics. Then we study the main solitonic models in 1+1 dimensions and discuss some important results concerning these types of models, including Derrick theorem stating that there is no stable soliton solution for Lagrangians with only scalar fields in spatial dimensions above 2. Besides that, we present some solitonic models in higher dimensions, e.g. magnetic monopoles and vortices. We also study instantons which are soliton solutions on a Euclidean spacetime and compare the formalism of instantons for tunneling processes in quantum mechanics with the corresponding one in Yang-Mills theory. Moreover, we present the formalism of fermions interacting with solitons as background fields, as well as some essential mathematical tools including Stationary Phase Approximation, Grassman numbers, Path Integral formalism. We also investigate one of the most important consequences of the interaction, the so-called Casimir Energy induced in systems containing non-trivial background fields such as solitons, using the phase shift method. Finally, we study a nonlinear interaction of a fermion field with a solitonic solution called compaction and compare the results with the known limiting cases in literature.