Topology and geometry of the electromagnetic field


  • Walter L. Engl


A separation of topology and geometry which is well known in network theory may be reached in electrodynamics as well by using adequate differential forms instead of classical vectors; this, moreover, yields a better insight into the physical nature of the field quantities. Differential forms are objects which can be integrated over curves, surfaces, etc., without reference to geometrical structures. With Ē and math formula as even 1- and 2-forms, respectively, math formula and math formula as odd 1- and 2-forms, respectively, Maxwell's equations become purely topological constraints. The metric properties of space then appear only in the constituent relations which become an isomorphism mapping the even multiforms (measures of intensity) Ē, math formula onto the odd multiforms (measures of quantity) math formula, math formula. Transition from threedimensional electrodynamics to the theory of special relativity can be done in a quite natural way.