Arrow of Matter: Waves and Quantization

Sometimes matter behaves like a wave. In 1923 de Broglie saw that every moving particle carries a wave, in step with a clock riding with it. Physicists then described it in steps, each with its own mathematics: complex waves, matrices, sums over paths, and negative squared lengths. In our opinion, one mathematics is enough: the Euclidean arrow numbers, with no negative lengths; Minkowski's form survives only as the scalar part of a square. Arrow of Time gave measured time a Euclidean length; the arrow of matter, an octonion of a body's time and the time content of its energy, appears to describe a single body completely. The Euler wave, a rotating arrow, gives de Broglie's wave. With Shannon's source, carrier and detector, every physical law becomes a three-body problem: three-body relativity. Multiplying a potential by the arrow gradient, on either side, gives the left and right fields. A second gradient makes Maxwell's equations one arrow equation. On the Euler wave, the Maxwell operator gives the arrow Klein–Gordon equation and, at low speed, Schrödinger's. Without matrices, Euler waves give two arrow Dirac equations, with spin one half, g = 2, and Pauli's equation. The waves carry a conserved, positive density of matter; their turn along a path is the action, and on a closed path they are quantised, with Planck's constant and no quantum postulate. Our central proposal is that one electron is one wave: its squared length is that electron's density of matter, not a probability, and probabilities are this density averaged over a turn with an unknown time. Schrödinger's cat needs no wave of its own: it is alive or dead, decided by one counter click. Two slits follow, in space and time, matching a time-slit experiment; a solenoid between them shifts the fringes with no force, as Aharonov–Bohm measurements show.