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Solitons and Their Symmetries: A Mathematical Analysis

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Amanda Steinhebel

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2015

This work represents a rigorous investigation of the symmetries of soliton creation. Natural symmetries are expressed mathematically as matrix Lie groups. From a pure math perspective, we present characteristics of Lie groups and their algebras. With these, we develop functions of groups and algebras that we use to define matrix Lie representations, or actions of the groups. In particular, we are interested in the spacetime symmetry of the Lorentz group SO(3,1) and the flavor symmetry SU(2).
We introduce classical field theory as a framework for the creation of particles as perturbations of a field. Of particular interest are large, stable perturbations called solitons. We find through the use of toy models of fields in 1+1 and 2+1 dimensions that solitons break the spacetime and flavor symmetries of the original action of the field, but gain their own symmetry group in the process. This new symmetry involves a mixing of the original symmetries and leads to the presence of a conserved soliton number. It is this conserved quantity that indicates that the soliton created is a stable particle. We graduate to the creation of a Yang-Mills soliton in 4+1 dimensions which lays the groundwork for the creation of a string theory soliton.