A particle's clock is its phase: how electromagnetic geometry forces the gauge circle and its Compton radius

Relativity takes the time along a path to be what a clock carried along it reads. Quantum mechanics gives an elementary particle exactly one clock, its phase. We ask of electromagnetism the question general relativity answered for gravity: which geometry has the particle's action as its length. For a charged particle the answer is Minkowski space with the clock form replaced by the phase form, a metric that places the electric potential in its time–time part and the vector potential in its time–space part; it is unique at first order, and its length along any path is the particle's action, so its paths of extremal length are the exact Lorentz motion. A change of gauge acts on it as a change of coordinates at first order but not beyond. We show that no local four-dimensional metric can do better: making a change of gauge a change of coordinates at second order needs a quantity that shifts by the gauge function itself, which no local function of the potentials supplies. The accumulated phase does. Taken as a coordinate, it is a closed cycle whose geometry is fixed uniquely; the particle moves around it at the speed of light, and its whole mass is that motion. With the electron as the single winding, the cycle's radius is forced to be the reduced Compton wavelength ħ/mc, and one completed turn carries the action h, in a rest form and a moving form that occupy the time–time and time–space parts of the metric. Waves on the cycle are the charged particles of quantum electrodynamics: the turning of the cycle is its U(1) gauge symmetry, the closure quantizes charge, and the length carries the mass. A spinor that carries the charge phase has g = 2; one turned by the cycle has g = 1, and measurement selects the first to one part in 10¹².