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arXiv:2107.02297

What Do We Learn by Deriving Born's Rule

James B. Hartle

Published Jul 5, 2021• 0 citations
arXivHTMLPDF

Abstract

Derivations of Borns Rule are of interest because they tell us what's connected to what in quantum mechanics if we ever need to change or modify it.
DOI
10.48550/arxiv.2107.02297

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The emergence of intrinsic probability has long been one of the most important and puzzling problems in quantum mechanics, and the law most directly related to this problem is the Born rule. For a century, there have been many attempts to derive the Born rule as a theorem rather than postulating it. However, existing derivations of the Born rule are each based on different frameworks and have attracted different criticisms. The assumptions from which they start are also highly divergent, and the connections between them have not been sufficiently studied. These possible connections are very likely to be the key to answering questions about the origin of probability in quantum mechanics. This paper focuses on proving the necessity and indispensability of the additivity assumption in the derivation of the Born rule. This supports the view that the Born rule cannot be derived solely from other non-probabilistic quantum or additional postulates. We first prove that additivity cannot be derived from two other commonly used non-probabilistic additional assumptions, non-contextuality and normalization. Then we analyze the crucial role of the additivity assumption in five important existing derivations of the Born rule. These include Gleason's Theorem, Busch's extension of Gleason's Theorem, the Deutsch-Wallace Theorem, Zurek's envariance proof, and the Finkelstein-Hartle Theorem. We show that these derivations either depend heavily on the additivity assumption or lead to obvious loopholes due to the lack of additivity. We also point out some problems arised from the lack of a non-contextuality assumption. Our results provide a novel insight into the important role of additivity assumption in quantum measurement, as well as into the origin of probability in quantum mechanics.
This package provides an educational reproducibility tool for exploring the dynamical emergence of the Born Rule based on the Unified Dynamic Cohesion Theory (UDCT). It includes:- An educational manual explaining the theoretical background, Stability Landscape, and Detector Freezing mechanism- A Python reproducibility package with visualization features (Mode A) designed for learning and verification purposes This release (v4.7.1) focuses on educational usability, including self-healing features and clear explanations connecting theory with code implementation. The package is intended for undergraduate students, self-learners, and researchers interested in the foundations of quantum mechanics and alternative interpretations of the Born Rule.