Von Neumann's Impossibility Proof: Mathematics in the Service of Rhetorics

TL;DR

This paper reevaluates von Neumann's 1932 proof against hidden variables in quantum mechanics, clarifying its actual claims and historical misunderstandings, and discusses its implications for the structure of hidden-variable theories.

Contribution

It provides a more accurate historical account of von Neumann's proof, clarifies its actual argument, and challenges the common narrative of its immediate disproof by Bell.

Findings

Von Neumann did not claim to disprove all hidden-variable theories.

Hidden-variable theories must be contextual and deviate from quantum operator representation.

Historical misunderstandings have obscured von Neumann's actual argument.

Abstract

According to what has become a standard history of quantum mechanics, in 1932 von Neumann persuaded the physics community that hidden variables are impossible as a matter of principle, after which leading proponents of the Copenhagen interpretation put the situation to good use by arguing that the completeness of quantum mechanics was undeniable. This state of affairs lasted, so the story continues, until Bell in 1966 exposed von Neumann's proof as obviously wrong. The realization that von Neumann's proof was fallacious then rehabilitated hidden variables and made serious foundational research possible again. It is often added in recent accounts that von Neumann's error had been spotted almost immediately by Grete Hermann, but that her discovery was of no effect due to the dominant Copenhagen Zeitgeist. We shall attempt to tell a story that is more historically accurate and less ideologically charged. Most importantly, von Neumann never claimed to have shown the impossibility of hidden variables tout court, but argued that hidden-variable theories must possess a structure that deviates fundamentally from that of quantum mechanics. Both Hermann and Bell appear to have missed this point, moreover, both raised unjustified technical objections to the proof. Von Neumann's argument was basically that hidden-variables schemes must violate the "quantum principle" that physical quantities are to be represented by operators in a Hilbert space. As a consequence, hidden-variables schemes, though possible in principle, necessarily exhibit a certain kind of contextuality.