Discrete Mass, Elementary Length, and a Topological Invariant as a Consequence of a Relativistic Invariant Variational Principle

Abstract

A nonlinear, Lorentz-invariant action principle is introduced. This variation principle, which deals with an "internal" angular variable $\theta$, is obtained from analogy considerations of moving Bloch walls in magnetic crystals and of elementary particles. In the one-dimensional case the resulting Euler equations can be solved by elementary functions, describing, in the rest system, a spatially extended energy distribution. The total energy, and thus the mass, has a discrete character. An elementary length appears as a parameter of this energy distribution, i.e., as a parameter of the structure. The moving structure undergoes the appropriate Lorentz contraction. An invariant of topological nature comparable to the invariant of the Möbius strip is furthermore inherent in this structure. This invariant has the symmetry properties of the elementary charge. A solution of the three-dimensional case including an internal rotation seems possible.