Arrow of Matter: Maxwell Equations and Ohm's Law

Maxwell's original equations were written in the potentials,with time and space together, and included Ohm's law and theelectromotive force; the textbooks' equations are written inthe fields and state those two separately. Written in theEuclidean arrow numbers, the real quaternions with theircalculus, the four-dimensional form returns, and with it aseventh field component that the textbooks do not count: thescalar field, built from Maxwell's J and Ψ, which hisequations carry and set aside. It vanishes for charges at restand for steady currents whose vector potential has nodivergence, which is why the laws of Coulomb and of Biot andSavart never see it, but where matter conducts it does thework: it drives the conduction current, j_c = −c∇E₀, the arrowOhm's law, with no empirical constant of its own; it drags thedrift of each kind of carrier, so that one balance of forcesgives Ohm's law and the Hall effect together; and it decideswhere charge is conserved. Maxwell's equations become one waveequation for the potentials, the sum of the potential's twosecond derivatives; the rest are identities of the product.The force on charges is Maxwell's own electromotive force,with one new push of the scalar field. The two orders of theproduct are the two hands of the quaternions; moving the zerofrom the source to the detector changes the hand and keepsevery length: the three-body relativity of a source, a carrierand a detector, which begins with Euclid's Optics. By deBroglie's postulate the arrowmatter holds both of Einstein'sreadings of E = mc²; with the band's maximum speed in theplace of c it is the two-band model of a crystal, whose massesfollow those measured in silicon and indium antimonide with nofitted parameter, and in graphene with the measuredsquare-root law.