A Unified Representation and Transformation of Electromagnetic Configurations Based on Generalized Hertz Potentials

Abstract

We present a unified framework that fully represents electromagnetic potentials, fields, and sources in vacuum, based on a reinterpretation of the classical Hertz-potential formalism. In this construction, $\phi$, $A$, $E$, $B$, $\rho$, and $J$ are systematically derived from a single vector wavefield $\Gamma(x, t)$ (called the "$\Gamma$-potential"), which is structurally aligned with the classical electric Hertz potential but of broader scope. A surjective mapping is established from such wavefields to all electromagnetic configurations in vacuum (that are sufficiently regular). This mapping induces a well-defined algebraic correspondence between the solution space of Maxwell's equations and the linear space of $C_t^{3} C_x^{3}$ vector wavefields (modulo the relevant symmetries), thereby enabling a framework for structural analysis of electromagnetic fields via their associated wavefields. Gauge freedom and the Lorenz gauge are naturally preserved; charge conservation and Maxwell's equations are inherently encoded in this representation. Building on this framework, we also introduce a transformation that provides a systematic method for generating new electromagnetic solutions from known ones. This transformation, called the "$\Gamma$-transformation", generalizes classical gauge transformations and may facilitate the exploration of hidden structures and symmetries in the solution space of Maxwell's equations.