Defining the electromagnetic potentials

Abstract

This paper offers a critical examination of the classical method of introducing the elecrtromagnetic potentials in a nonrelativistic context. It applies the nonretarded Helmholtz theorem to show that the usual method is ambiguous, with that ambiguity being reflected in the gauge transformation equations. This ambiguity can be removed by carefully invoking the Helmholtz decomposition theorem, but the process exposes the inadequacy of the conventional procedure. It then shows how the retarded Helmholtz theorem can be used to motivate rigorous definitions of the vector and scalar potentials and the field variables based upon the current density as the single electrical source variable. The method depends upon the assumption of causality, hence is restricted to classical field theory. It uses the powerful theory of operators as the main analysis tool.