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Hilbert-Style Axiomatic Completion: On von Neumann and Hidden Variables in Quantum Mechanics

Mitsch, Chris (2022) Hilbert-Style Axiomatic Completion: On von Neumann and Hidden Variables in Quantum Mechanics. [Preprint]

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Abstract

In this paper I provide a detailed history of von Neumann’s “No Hidden Variables” theorem, and I argue it is a demonstration that his axiomatization mathematically captures a salient feature of the statistical transformation theory (namely, that hidden variables are incompatible). I show that this reading of von Neumann’s theorem is obvious once one recalls several factors of his work. First, his axiomatization was what I call a Hilbert-style axiomatic completion; indeed, it developed from work initiated by Hilbert (and Nordheim). Second, it was responsive to specific mathematical and theoretical problems faced by Dirac and Jordan’s statistical transformation theory (then called ‘quantum mechanics’). Third, the axiomatization was completed across his 1927 papers and 1932 book when he identified the basic assumptions underwriting quantum mechanics, showed that these suffice for deriving the trace rule, and showed that the trace rule is incompatible with hidden variables. With this reading in mind, his claim that quantum mechanics was in “compelling logical contradiction with causality” appears as a straightforward consequence of his theorem. I conclude by reassessing the theorem’s broader historical and scientific significance.


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Item Type: Preprint
Creators:
CreatorsEmailORCID
Mitsch, Chriscmitsch@uci.edu0000-0003-2502-4343
Additional Information: accepted at Studies in History and Philosophy of Science
Keywords: von Neumann, quantum mechanics, hidden variables, Bohm, Bell, Hilbert, axiomatization, no-go
Subjects: Specific Sciences > Mathematics > Methodology
General Issues > History of Science Case Studies
Specific Sciences > Physics > Quantum Mechanics
Depositing User: Chris Mitsch
Date Deposited: 30 Jun 2022 17:12
Last Modified: 30 Jun 2022 17:12
Item ID: 20825
Subjects: Specific Sciences > Mathematics > Methodology
General Issues > History of Science Case Studies
Specific Sciences > Physics > Quantum Mechanics
Date: June 2022
URI: http://philsci-archive.pitt.edu/id/eprint/20825

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