A Unified Representation and Transformation of Electromagnetic Configurations Based on Generalized Hertz Potentials
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866917343564660736 |
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| author | Yi, Ting |
| author_facet | Yi, Ting |
| contents | 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, $ϕ$, $A$, $E$, $B$, $ρ$, 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 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 "$Γ$-transformation", generalizes classical gauge transformations and may facilitate the exploration of hidden structures and symmetries in the solution space of Maxwell's equations. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2510_13875 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | A Unified Representation and Transformation of Electromagnetic Configurations Based on Generalized Hertz Potentials Yi, Ting Classical Physics 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, $ϕ$, $A$, $E$, $B$, $ρ$, 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 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 "$Γ$-transformation", generalizes classical gauge transformations and may facilitate the exploration of hidden structures and symmetries in the solution space of Maxwell's equations. |
| title | A Unified Representation and Transformation of Electromagnetic Configurations Based on Generalized Hertz Potentials |
| topic | Classical Physics |
| url | https://arxiv.org/abs/2510.13875 |