Von Neumann’s impossibility proof: Mathematics in the service of rhetorics

摘要

AbstractAccording 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 CopenhagenZeitgeist.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 variablestout 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.As we shall illustrate, early reactions to Bohm’s theory are in agreement with this account. Leading physicists pointed out that Bohm’s theory has the strange feature that pre-existing particle properties do not generally reveal themselves in measurements, in accordance with von Neumann’s result. They did not conclude that the “impossible was done” and that von Neumann had been shown wrong.Highlights?Presents a balanced history of von Neumann’s no-hidden variables proof.?Analyzes the meaning of the proof.?Analyzes the contributions of Grete Hermann and John Bell.?Corrects misunderstandings in the literature.]]>