From UD Theory to Electromagnetism: The Vector Projection

This paper derives classical electromagnetism as the vector projection of UD theory. The dynamical scalar fields U_D (the D-attribute in space, manifest as dark matter) and D_U (the U-attribute in matter, manifest as quantum fluctuations) form a coupled Klein-Gordon system. In the low-energy electromagnetic limit, the background fields U_U and D_D sit at their vacuum expectation values, and the active equations reduce to a closed system. The canonical energy-momentum tensor Tμν of this system is exactly conserved. The electromagnetic field strength Fμν is defined as the normalized Hodge dual of Tμν: Fμν = (2m_0^2)⁻¹ εμνρσ Tρσ, where m_0 = 1/(2πe^π) is the UD ground mass scale. This definition is the unique Lorentz-covariant mapping from a symmetric rank-2 tensor to an antisymmetric one, with the dimension corrected by the only fundamental mass scale of the theory. From the conservation law ∂_μ Tμν = 0, it follows directly that ∂_μ Fμν = 0. Together with the Bianchi identity ∂[λ Fμν] = 0, this yields the source-free Maxwell equations. The photon is identified as the quantum of Fμν: a propagating collective excitation of the U_D-D_U energy-momentum transfer, not a permanent elementary particle. A vector potential A^μ satisfying Fμν = ∂^μ A^ν - ∂^ν A^μ exists by the Poincaré lemma and inherits the residual U(1) gauge symmetry. Coupling to the conserved U(1) Noether current j^μ from the spinor projection (Ref. [2]) via the gauge-invariant action S = ∫ d^4x(-1/4 FμνFμν + e A_μ j^μ) yields the full Maxwell equations ∂_μ Fμν = e j^ν. A massive scalar mode with m = m_0/√2 remains as a UV signature, inducing a Yukawa correction to the Coulomb potential at the UD length scale. All derivations proceed from the UD action without introducing external gauge fields. The U(1) gauge symmetry and the masslessness of the photon are derived consequences, not postulates. Together with Ref. [1] (general relativity) and Ref. [2] (quantum mechanics), UD theory now accounts for all three fundamental interactions as projections of a single four-aspect ontology. Key Points - The electromagnetic field strength Fμν is the normalized Hodge dual of the U_D-D_U energy-momentum tensor- Fμν = (2m_0^2)⁻¹ εμνρσ Tρσ is the unique Lorentz-covariant mapping from a symmetric to an antisymmetric rank-2 tensor- ∂_μ Fμν = 0 follows directly from ∂_μ Tμν = 0, giving the source-free Maxwell equations without postulating a fundamental vector field- The photon is an emergent collective excitation of U_D-D_U energy-momentum transfer, not a permanent elementary particle- U(1) gauge symmetry emerges as a residual symmetry of the massless field strength, not as an external postulate- A massive scalar mode with m = m_0/√2 is predicted as a testable UV signature References [1] Zhu, D. "From UD Theory to General Relativity: A Complete Derivation of Gravitational Theory." Zenodo, 2026. https://doi.org/10.5281/zenodo.19903855 [2] Zhu, D. "From UD Theory to Quantum Mechanics: The Spinor Projection." Zenodo, 2026. https://doi.org/10.5281/zenodo.19911827 [3] Faraday, M. Experimental Researches in Electricity, Vol. 1. London: Bernard Quaritch, 1839. [4] Maxwell, J. C. "A dynamical theory of the electromagnetic field." Phil. Trans. R. Soc. Lond. 155, 459–512 (1865). [5] Jackson, J. D. Classical Electrodynamics, 3rd ed. New York: Wiley, 1999. [6] Williams, E. R., Faller, J. E., and Hill, H. A. "New experimental test of Coulomb's law: A laboratory upper limit on the photon rest mass." Phys. Rev. Lett. 26, 721–724 (1971).