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 thingeven for finite N there are unitarily inequivalent representations of the Heisenberg commutation relations among which are physically interesting quantum phenomena, such as the AharonovBohm 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  

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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://philsciarchive.pitt.edu/id/eprint/22115 
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