Quantum mechanics from one geometry: the Schrödinger equation, the Born rule's weight, spin and Bell correlations derived in Complex Mechanics
Year: 2026
Type: preprint
Abstract: Textbook quantum mechanics postulates its state space, its observables, a commutation relation, a Hamiltonian for each problem, the Born rule and a projection on measurement, and introduces spin as a separate two-component structure. Complex Mechanics (ℂM) keeps none of these as a law. It starts from one container—real time and three complex dimensions carrying the Kähler triad (g, J, ω) as one object—and a base of eight items stated once: the container with its triad; null matter; matter as sections of the square-root bundle K^(1/2) that superpose; two demands on the field, that its conservation law be that of the count (the fraction of runs that land in a region) and that its modulus stand in one relation to the count's density for every state, its phase each particle's action; and three named posits—a particle's two alignments, its moment along an axis or against it; where the runs start; and the frame in which a pair's joint outcome is resolved. Beyond the base the paper takes in what its Table 1 lists: two named constructions for pairs; one named premise, that the magnetic moment reads the angular momentum of the J-rotation, the J-action; three readings, one that an apparatus reads the visible sector only and two in the bound electron's moment; and results, statements and inputs of the framework, two of them numbers solved once on a measurement. On that footing the imaginary unit is the complex structure J, and a field's phase is the angle of its rotation, the J-angle. A superposition of null phases, the phases of free null matter, obeys i∂ψ/∂t = √(m² − ∇²) ψ, whose leading term, with the phase's rate at rest removed, is the free Schrödinger equation; a pull enters as an input. The Born rule's weight is derived: |ψ|² is the only density the paths carry. Where the runs start stays a posit, an initial condition; in the one case computed, a particle in a box, the paths carry a start off the density onto it at each cell size read, never point by point. Spin-½'s half rate is that of K^(1/2), whose sections change sign under one full turn of the J-angle; with a named construction for pairs it gives the exchange sign −1, and with the moment law's premise and the lepton's unit of J-action, g = 2. A particle's two spin states are its two alignments, and its Stern–Gerlach odds are cos²(θ_field/2) at the angle θ_field between its moment and the field. The theory is deterministic, with hidden variables: each particle's velocity is, at the leading order, the gradient of its own J-angle over m, the velocity law of de Broglie's and Bohm's theory, here one step from the base's reading of the J-angle as each particle's action. Where quantum mechanics gives the odds, the path decides which particle goes where: the particles that start in the top cos²(θ_field/2) of a Stern–Gerlach beam land in the + lobe and the rest in the − lobe, and behind a double slit the paths bend with the state and land as the fringes. The opposite-alignment pair's correlation is −cos Δ at the angle Δ between the two analyzers, every pair state's CHSH value is at most 2√2, and nothing signals. At the order computed no prediction differs from quantum mechanics, and the one place a difference could live is where the runs start. Told with these: the moment beyond g = 2 is not yet computed, and the bound electron's leading-order moment stands +2.0 of the measurement's uncertainty from hydrogen's and about 400 and 1060 of theirs from two hydrogenlike ions'. Light as a quantized field and systems of many particles, from helium to condensates, are not yet derived. A companion to the framework paper Complex Mechanics, version 4.30 (doi:10.5281/zenodo.19349816). The script qm_repro.py, posted with this record, recomputes 110 of the paper's printed values from its printed equations.
Author: Orrick, Brian G
Language: English
Subfield: Atomic and Molecular Physics, and Optics
Field: Physics and Astronomy
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7225825218
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