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As Revealing in the Breach as in the Observance: von Neumann's Uniqueness Theorem

Earman, John (2023) As Revealing in the Breach as in the Observance: von Neumann's Uniqueness Theorem. [Preprint]

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Abstract

von Neumann's celebrated uniqueness theorem is often advertised as demonstrating the essential uniqueness of representations (or realizations) of the Heisenberg commutation relations for a finite number N of degrees of freedom by showing that all such representations are unitarily equivalent to the Schrödinger representation and, thereby, securing the equivalence of Schrödinger wave mechanics to the matrix mechanics of Born, Heisenberg, and Jordan. But the theorem proves no such thing---even for finite N there are unitarily inequivalent representations of the Heisenberg commutation relations among which are physically interesting quantum phenomena, such as the Aharonov-Bohm effect; and von Neumann's own explanation of the equivalence of Schrödinger wave mechanics to the matrix mechanics makes no use of his uniqueness theorem. There are other loopholes and ways around the uniqueness theorem; but this does not constitute a criticism of von Neumann's theorem but is rather is a nod to the genius that produced a theorem as revealing in the breach as in the observance, for the exceptions illustrate (in the words of Asao Arai) how the universe uses inequivalent irreducible representations of the canonical commutation relations to produce "characteristic" quantum effects.


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Item Type: Preprint
Creators:
CreatorsEmailORCID
Earman, Johnjearman@pitt.edu
Keywords: quantum mechanics, canonical commutation relations, unitarily inequivalent representations
Subjects: General Issues > Explanation
General Issues > History of Science Case Studies
Specific Sciences > Physics > Quantum Field Theory
Specific Sciences > Physics > Quantum Mechanics
Depositing User: John Earman
Date Deposited: 17 May 2023 13:51
Last Modified: 17 Jun 2023 11:31
Item ID: 22115
Subjects: General Issues > Explanation
General Issues > History of Science Case Studies
Specific Sciences > Physics > Quantum Field Theory
Specific Sciences > Physics > Quantum Mechanics
Date: 16 May 2023
URI: https://philsci-archive-dev.library.pitt.edu/id/eprint/22115

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