Topological electromagnetism

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Abstract
There is a topological structure in the set of the electromagnetic radiation fields (with E.B=0) in vacuum. A subset of them, called the admissible fields, are associated with maps S3(S2) and can be classified in homotopy classes labelled by the value of the corresponding Hopf indexes, which are topological constants of the motion. Moreover, any radiation field can be obtained by patching together admissible fields and is therefore locally equal to one of them. There is, however, an important difference from the global point of view, since the admissible fields obey the topological quantum conditions that the magnetic and the electric helicities are equal to integer numbers n and m times an action constant a which must be introduced because of dimensional reasons, that is integral A B d3r=na, integral C.E d3r=ma, where B and E are the magnetic and electric fields and Del *A=B, Del *C=E. A topological mechanism for the quantization of the electric charge operates in the set of the admissible fields, in such a way that the electric flux through any closed surface around a point charge is always equal to a times an integer number n', equal to the degree of a map S2(S2) corresponding to the existence of a fundamental charge with value q0a/4 pi . It is argued that results of this kind could help reaching a better understanding of quantum physics.

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