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

We present a unified framework that fully represents electromagnetic potentials, fields, and sources in vacuum, based on a generalization of the classical Hertz-potential formalism. In this construction, φ, A, E, B, ρ, and J are systematically derived from a single vector wavefield Γ(x, t) (called the Γ-potential), which is structurally aligned with the classical electric Hertz potential but has a 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^3C_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 solution-generating transformation that is naturally enabled by the linearity of the representation. This transformation, termed the Γ-transformation, provides a systematic method to derive new, physically distinct electromagnetic configurations from existing ones. It generally leads to changes in potentials, fields, and their underlying sources. Traditional gauge transformations, which alter potentials while preserving fields and sources, constitute a special, restricted subclass of Γ-transformations. An illustrative example is also provided to demonstrate possible applications of the framework.