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Compt. rend.

THE VON NEUMANN THEOREM CONCERNING THE ESSENTIAL INDETERMINISM OF QUANTUM MECHANICS

J. Tharrats

Published Jun 1, 1960• 0 citations
OpenAlex

Abstract

The problem of the impossibility of cancelling the dispersion of all observable magnitudes is restudied by centralizing the question in the von Neumann theorem which is presented under a mcdified form. The von Neumann solution appears doubtful principally because the demonstration is made by projectors which do not represent observable magnitudes. (tr-auth)

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The problem of the impossibility of cancelling the dispersion of all observable magnitudes is restudied by centralizing the question in the von Neumann theorem which is presented under a mcdified form. The von Neumann solution appears doubtful principally because the demonstration is made by projectors which do not represent observable magnitudes. (tr-auth)
Introduction To know the quantum mechanical state of a system implies, in general, only statistical restrictions on the results of measurements. It seems interesting to ask if this statistical element be thought of as arising, as in classical statistical mechanics, because the states in question are averages over better defined states for which individually the results would be quite determined. These hypothetical ‘dispersion free’ states would be specified not only by the quantum mechanical state vector but also by additional ‘hidden variables’ – ‘hidden’ because if states with prescribed values of these variables could actually be prepared, quantum mechanics would be observably inadequate. Whether this question is indeed interesting has been the subject of debate. The present paper does not contribute to that debate. It is addressed to those who do find the question interesting, and more particularly to those among them who believe that ‘the question concerning the existence of such hidden variables received an early and rather decisive answer in the form of von Neumann's proof on the mathematical impossibility of such variables in quantum theory.’ An attempt will be made to clarify what von Neumann and his successors actually demonstrated. This will cover, as well as von Neumann's treatment, the recent version of the argument by Jauch and Piron, and the stronger result consequent on the work of Gleason. It will be urged that these analyses leave the real question untouched.
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.
The impossibility of making quantum mechanics deterministic by adding ‘hidden variables’ was discussed by J. von Neumann in 1932. His proof of the impossibility of dispersion-free ensembles was criticized as being logically circular, by Grete Hermann soon after, and as fundamentally flawed, by John Bell in 1964. Bell’s severe criticism and an explicit counter-example are generally considered as the definitive demonstration that J. von Neumann’s proof is inadequate. I show, without using the formalism of quantum mechanics, that J. von Neumann’s central assumption of the linear additivity of the expectation values, even for incompatible (noncommuting) observables, is a necessary consequence that follows directly from the fundamental conservation laws of physical quantities, applicable universally. With this result, J. von Neumann’s result is fully resurrected. I identify the physical reason why dispersion-free ensembles cannot exist. After discussing the conflict of Bell’s counter-example with the factual mechanics, I highlight the result that the local hidden variable theories, for which the Bell’s inequalities were derived, are grossly incompatible with the fundamental conservation laws. In a scrutiny of the vital difference between the ‘causal incompleteness’ of quantum mechanics, discussed by J. von Neumann, and the ‘representational incompleteness’ discussed by Einstein, Podolsky, and Rosen, I prove the striking demarcating result that a multi-particle system that is considered causally completed can remain EPR-incomplete. I identify the wave nature of the action as the reason behind the irreducible quantum dispersion and the impossibility of dispersion-free ensembles.