Symmetry Arguments in Electrodynamics

Symmetry principles can be applied in the study of electrodynamics to streamline the development of proofs and provide efficient problem-solving methods. It is often possible, for example, to gain information about electric/magnetic fields from the symmetries of their corresponding charge/current densities. The goal of our research is to place such principles of symmetry on a mathematically rigorous foundation. To do this, we need certain results about the relationships between divergence/curl operations and vector field transformations. In particular, it should not matter whether one transforms a vector field then calculates divergence/curl, or calculates divergence/curl then transforms the result. Though physically intuitive, proving these is rather involved, and there are some transformations for which these relationships do not hold at all. So far we have been investigating the transformations for which these relationships can be proven. Key results include successfully proving these relationships for a particular group of transformations, proving several symmetry theorems relying on them, developing a precise and consistent definition of vector field transformations, and exploring how group theory can be applied in this project to further mathematical rigor and elegance.