Charged particles in the model of topological fermions

Abstract

The \emph{model of topological fermions} describes particles with spin quantum number $1/2$ as solutions of a nonlinear partial differential equation. Single particle solutions are stable and characterised by topological quantum numbers, they are topological \emph{solitons}. For introducing the concept the soliton solutions of the~$1+1$ dimensional sine-Gordon equation are presented first. Then we define the model in~$3+1$ dimensions and describe charged particles. We sketch the mathematic toolkit -- Lie groups and Lie algebras, using quaternions for visualizing the necessary considerations. Field and energy density of single solitons is calculated and the relation of the curvature in color space to the electric field in real space will be explained. In addition, two topological quantum numbers are introduced, characterizing spin and charge of topological fermions. The soliton fields are interpreted as extended objects in real space -- the topological fermions, compared to the first generation of leptons (e$ -$, e$ +$). It is shown that far away from the soliton centers -- in the \emph{electrodynamic limit} -- the electromagnetic interaction behaves like the Coulomb interaction. In the second part, we investigate a quasi static collision of two topological fermions by calculating the total energy of soliton fields with decreasing distance on a lattice. For this purpose, we have programmed a computer simulation. Due to size and runtime limits of the discrete lattice simulation, an additional surface energy term has to be included. This surface term is evaluated in the electrodynamic limit. To stabilize the lattice solutions, we freeze the values of the soliton-field around the soliton centers and the field at the boundaries of the lattice, initialized by precalculated values of the electric field of a normal dipole from Maxwell's theory. Comparing the lattice simulation of a single soliton with the algebraic solutions, we find an error of less than $\nicefrac{1}{2}\%$. However, the more interesting part is the comparison of the interaction of the two soliton with the calculated total energy of a particle-antiparticle\footnote{For better comparability we have effectively used the values of an electron-positron dipole.} dipole. At distances of the soliton center smaller than $10\,r_0$ with $r_0$ denoting the soliton radius the interaction starts to deviate from Coulomb's $1/r 2$-law. This behaviour, which is also known from QED, can be interpreted as \emph{running coupling}.