We review the famous no-hidden-variables theorem in von Neumann’s 1932 book on the mathematical foundations of quantum mechanics (Mathematische Grundlagen der Quantenmechanik, Springer, Berlin, 1932 ). We describe the notorious gap in von Neumann’s argument, pointed out by Hermann (Abhandlungen der Fries’schen Schule 6:75–152, 1935 ) and, more famously, by Bell (Rev Modern Phys 38:447–452, 1966 ). We disagree with recent papers claiming that Hermann and Bell failed to understand what von Neumann was actually doing.
We review the famous no-hidden-variables theorem in von Neumann’s 1932 book on the mathematical foundations of quantum mechanics (Mathematische Grundlagen der Quantenmechanik, Springer, Berlin, 1932 ). We describe the notorious gap in von Neumann’s argument, pointed out by Hermann (Abhandlungen der Fries’schen Schule 6:75–152, 1935 ) and, more famously, by Bell (Rev Modern Phys 38:447–452, 1966 ). We disagree with recent papers claiming that Hermann and Bell failed to understand what von Neumann was actually doing.
It is shown here that the ``simple test of quantumness for a single system'' of Alicki and Van Ryn has exactly the same relation to the discussion of the problem of describing the quantum system via a classical probabilistic scheme (that is in terms of hidden variables or within a realistic theory) as the von Neumann theorem [Mathematische Grundlagen der Quantenmechanik (Springer, Berlin, 1932); Mathematical Foundations of Quantum Mechanics (Princeton University, Princeton, NJ, 1955)]. The latter one was shown by Bell [Rev. Mod. Phys. 38, 447 (1966)] to stem from an assumption that the hidden variable values for a sum of two noncommuting observables have to be, for each individual system, equal to sums of eigenvalues of the two operators. One cannot find a justification for such an assumption to hold for noncommeasurable variables. On the positive side, the criterion may be useful in rejecting models which are based on stochastic classical fields. Nevertheless, the example used by the authors has a classical optical realization.
Introduction In his Mathematical Foundations of Quantum Mechanics , John von Neumann presented a proof that the quantum statistics cannot be recovered from probability distributions over “hidden” deterministic states that assign definite premeasurement values to all physical quantities. Von Neumann's proof has been dismissed as “silly” by John Bell and by David Mermin, who writes: Many generations of graduate students who might have been tempted to try to construct hidden-variables theories were beaten into submission by the claim that von Neumann, 1932, had proved that it could not be done. A few years later (see Jammer, 1974, p. 273) Grete Hermann, 1935, pointed out a glaring deficiency in the argument, but she seems to have been entirely ignored. Everybody continued to cite the von Neumann proof. A third of a century passed before John Bell, 1966, rediscovered the fact that von Neumann's nohidden-variables proof was based on an assumption that can only be described as silly—so silly, in fact, that one is led to wonder whether the proof was ever studied by either the students or those who appealed to it to rescue them from speculative adventures.
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 interpretation (namely, that hidden variables are incompatible). I show that this reading of von Neumann’s theorem is obvious once one recalls several contextual 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 essentially completed already in his 1927 papers, at least concerning the status of hidden variables, and this would have been obvious to the audience for those papers. Thus, the theorem’s statement and proof were only necessary when the material was presented for a general mathematical audience, i.e., in his 1932 Mathematical Foundations of Quantum Mechanics . 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.
This article reviews the history of J. von Neumann’s analysis of hidden variables in quantum mechanics and the subsequent analysis by others. In his book The Mathematical Foundations of Quantum Mechanics, published in 1932, von Neumann performed an analysis of the consequences of introducing hidden parameters (hidden variables) into quantum mechanics. He arrived at two principal conclusions: first, hidden variables cannot be incorporated into the existing theory of quantum mechanics without major modifications, and second, if they did exist, the theory would have already failed in situations where it has been successfully applied. This analysis has been taken as an “incorrect proof” against the existence of hidden variables, possibly due to a mistranslation of the German word prufen. von Neumann’s so-called proof isn’t even wrong as such a proof does not exist, but it is an examination of the limitations imposed by internal consistency of the Hilbert space formulation of the theory. One of the earliest attempts to eliminate uncertainty, by D. Bohm, requires a major modification of quantum mechanics (observables are not represented by Hermitian operators), which supports von Neumann’s first principal conclusion. However, testing the Bohm theory requires constructing a physically impossible initial state. As such, the theory has no experimental consequences, so W. Pauli referred to it as an “uncashable check”. As there are no observable consequences, the Bohm theory is possibly a counterexample to von Neumann’s second conclusion that hidden variables in particular would have already led to a failure of the theory.
It is shown that, although correct mathematically, the celebrated 1932 theorem of von Neumann which is often interpreted as proving the impossibility of the existence of "hidden variables" in Quantum Mechanics, is in fact based on an assumption which is physically not reasonable. Apart from that, the alleged conclusion of von Neumann proving the impossibility of the existence of "hidden variables" was already set aside in 1952 by the counterexample of the possibility of a physical theory, such as given by what is usually called the "Bohmian Mechanics". Similar arguments apply to other two well known mathematical theorems, namely, of Gleason, and of Kochen and Specker, which have often been seen as equally proving the impossibility of the existence of "hidden variables" in Quantum Mechanics.
In his book \textit{The Mathematical Foundations of Quantum Mechanics}, published in 1932, J. von Neumann performed an analysis of the consequences of introducing hidden parameters (hidden variables) into quantum mechanics. He showed that hidden variables cannot be incorporated into the existing theory of quantum mechanics without major modifications, and concluded that if they did exist, the theory would have already failed in situations where it has been successfully applied. von Neumann left open the possibility that the theory is not complete, and his analysis for internal consistency is the best that can be done for a self-referenced logical system (Gödel's theorem). This analysis had been taken as an ``incorrect proof" against the existence of hidden variables. von Neumann's so-called proof isn't even wrong as such a proof does not exist. One of the earliest attempts at a hidden variable theory was by D. Bohm, and because there were no experimental consequences, W. Pauli referred to it as an ``uncashable check." To our knowledge, a successful hidden variable extension to quantum mechanics with testable consequences has not yet been produced, suggesting that von Neumann's analysis is worthy of rehabilitation, which we attempt to provide in a straightforward manner.