Borns Rule from Reversible Evolution and Irreversible Outcomes

Abstract:We show that the quadratic measure associated with the Born rule need not be introduced as an independent probabilistic postulate, but follows, under the stated structural and regularity assumptions, from the compatibility between reversible linear evolution and the formation of stable records. Physical processes alternate between reversible dynamics and episodes of record formation. Quantities that remain well defined throughout such processes must therefore admit a representation that remains consistent across both regimes. Prior to record formation, alternatives combine additively at the level of complex amplitudes, whereas successive record-forming distinctions admit a multiplicative representation through refinement. Requiring compatibility between these compositional structures constrains the admissible weight assignment to the family of weights proportional to the pth power of the amplitude magnitude. The finite-dimensional specialization of Lamperti's isometry theorem then singles out p = 2 as the unique value compatible with continuous reversible linear mixing while preserving total weight. The Born rule therefore emerges as the unique weight assignment compatible with reversible linear composition, multiplicative record refinement, phase invariance, and continuous reversible invariance, without invoking probabilistic assumptions or a specific interpretation of quantum measurement.

Published version. 11 pages, 1 figure. Derivation of the Born rule from compatibility between reversible evolution and irreversible outcomes

arXiv:2604.07418 [quant-ph]

(or arXiv:2604.07418v2 [quant-ph] for this version)

https://doi.org/10.48550/arXiv.2604.07418 Focus to learn more arXiv-issued DOI via DataCite

https://doi.org/10.4236/jmp.2026.1710056 Focus to learn more DOI(s) linking to related resources

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