A Topological Structure in the Set of Classical Free Radiation Electromagnetic Fields

Abstract

The aim of this work is to proceed with the development of a model of topological electromagnetism in empty space, proposed by one of us some time ago and based on the existence of a topological structure associated with the radiation fields in standard Maxwell's theory. This structure consists in pairs of complex scalar fields, say $ϕ$ and $θ$, that can be interpreted as maps $ϕ,θ: S^3\mapsto S^2$, the level lines of which are orthogonal to one another, where $S^3$ is the compactified physical 3-space $R^3$, with only one point at infinity, and $S^2$ is the 2-sphere identified with the complete complex plane. These maps were discovered and studied in 1931 by the German mathematician H. Hopf, who showed that the set of all of them can be ordered in homotopy classes, labeled by the so called Hopf index, equal to $γ=\pm 1,\,\pm 2,\,\cdots ,\, \pm k,...$ but without $γ=0$. In the model presented here and at the level of the scalars $ϕ$ and $θ$, the equations of motion are highly nonlinear; however there is a transformation of variables that converts exactly these equations (not by truncation!) into the linear Maxwell's ones for the magnetic and electric fields $\B$ and $\E$.