The Volumes: Geometry of Electromagnetism

This work, in three volumes, identifies the geometry of electromagnetism. Its structure is summarized in Table I: every entry is established in the volumes, and the right-hand column, the identity of a particle's Compton cycle with the U(1) gauge circle of quantum electrodynamics, seen at rest as the Compton form and in motion as the de Broglie form, is shown to be forced. Electromagnetism has a metric (Volume I). It is built on a single length, r_EM = (q/q_P)² r₀: the reduced Compton wavelength r₀ = ħ/mc, scaled by the square of the particle's charge measured against the Planck charge q_P = √(4πε₀ħc). For the elementary charge, (e/q_P)² is the fine-structure constant α, so r_EM = αr₀, which for the electron is the classical electron radius. Through this length electromagnetism maps onto the Einstein field equations in the same way gravity does through r_G = (m/m_P)² r₀ = ℓ_P²/r₀ = Gm/c², the same Compton wavelength scaled by the square of the mass measured against the Planck mass, equivalently the Planck area divided by r₀ (top row of Table I). The scalar potential Φ sits in g₀₀ and the vector potential A in g₀ᵢ; the Christoffel symbols give the Lorentz force, and Coulomb's law, written as a curvature law, gives Maxwell's equations. The quantum of action belongs to the same geometry (Volume II). One completed cycle of a particle's phase carries h = E Δt, in two forms. At rest, h = E₀ (cλ_C/c²): the Compton interval reads the clock's rate and belongs to g₀₀, beside Φ. In motion, h = E (vλ_dB/c²): the interval is the offset of simultaneity across one de Broglie wavelength, which reads the offset of the phase across space and belongs to g₀ᵢ, beside A. These are one structure, and it is forced (Volume III), with nothing added and nothing fitted. It rests on general relativity and the U(1)-coupled quantum phase alone. Asked for the geometry whose length is the charged particle's action, as general relativity asks it of gravity, the answer is Minkowski space with the particle's clock replaced by its phase: exactly the metric of electromagnetism. Beyond first order that metric's curvature no longer agrees with its own motion, and the one way out is the phase itself, a closed cycle traversed at the speed of light. With the unit of charge a single winding, the cycle is one Compton wavelength long and every turn completes h; its turning is the U(1) gauge symmetry of quantum electrodynamics. The gauge circle that quantum electrodynamics takes as an abstract symmetry is the particle's own clock.