Maximum Determinacy III: Light, Charge and Gauge

This paper builds light and charge on the lattice of Papers I and II, where objects hopacross joins at most one join per pair per tick. A pair whose first member is a phase carrieswhole-number charge; with the author’s ruling that every pair an object carries steps atone join event, the hop carries the join’s phase, so U(1) transport, gauge invariance andGauss’s law follow. Light is content that hops like any other, so it reaches c∗; it is the spin-1walk on F = E + iB, run over two ticks in mirrored order, whose half-angle quaternionis exactly q0 = cos qx cos qy cos qz, ˜n = (sin qx cos qy cos qz, sin qy cos qz, sin qz), with an exactlocal canonical potential and Gauss’s law exact with stencil ˜n. Its maximal speed is c∗,reached along the axes, and its long-wavelength speed is c∗ in every direction. KeepingGauss’s law exact constrains how light sees charge: charge on the cell admits no local coupling,and among sharings of smallest reach the only one symmetric about the cell is the cornersharing Qj cos2(qj/2), which has exact local couplings at every momentum. The potentialis taken non-compact. Semiclassical runs of the coupled law give a Coulomb force. Thepaper also carries the two-pair locks and the dispersion they give, the generator problem, theconditional route to Yang–Mills law-form and the Wilson functional, and the photon’s edgeconditions, which give Fresnel’s coefficients, Brewster’s angle and frustrated total internalreflection. Every result carries its grade; the premises are collected in Paper VI.