Differential Forms and Maxwell's Field: An Application of Harmonic Integrals

Abstract

Maxwell's equations are derived under the assumption of 4-dimensionality of Euclidean space from a somewhat different definition of time from that considered by Einstein, using differential forms and de Rham's theorem in the theory of harmonic integrals. It is shown that the continuity equation of the current density is an elementary consequence of the Jordan-Brouwer theorem of topology under the requirement of integrability of the field. Matter appears as a singular point of the field, and introduces various kinds of ``currents,'' in the sense of de Rham and Kodaira, according to the topological character of the domain of integration of the field. These ``currents'' describe characters of the material, and are represented not by ordinary functions, but by generalized functions in the Schwartz sense. Examples of these ``currents,'' such as electric and magnetic polarizations and the supercurrent, are given, and the origin of the fundamental difficulties with dimensions in the usual theory of electromagnetism is attributed to this fact.