17 Bell and von Neumann

In 1932, the distinguished mathematician John von Neumann wrote ‘The Mathematical Foundations of Quantum Mechanics’[i], which is famous for the elegant mathematical formalism it introduced and for its proof of the non-existence of hidden variables. In 1964, John S. Bell is credited with discovering a ‘flaw’ in this proof, and with correcting it.

The phrase ‘hidden variables’ is another of those in modern theoretical physics that is imprecisely used and inconsistently defined. One approach to it is as follows. We might infer that light is a wave from its remarkably wave-like properties, but we have historically only been able to observe its effect on absorption by a molecule on photographic paper or in our eye. Its wave nature, if it exists, is supposedly hidden from us and so would be a hidden variable in any theory.

The term can also be taken to refer to limitations in the key mathematical equation used in quantum mechanics, the Schrödinger equation, a piece of mathematics that has proven exceptionally resistant to physical interpretation. That equation is so beautiful in the eyes of mathematical physicists, and so effective, that it is tempting to believe that it tells the entire story of sub-microscopic physics. Given the considerable practical limitations on experimental observation in this area, it is clearly speculative to go beyond what can be measured and what can reliably be found in the successful mathematics.

The most famous example of such speculation is the ‘pilot-wave theory’ of David Bohm[ii]. Bohm suggested that the particles of particle physics were real and were guided by waves, and that this offered a practical and realistic explanation for what was otherwise physically inexplicable, specifically that particles display interference effects.

We have seen that quantum theory is highly reliable in its mathematics but that everything else is problematic. Bohm had introduced a more physical description than what had gone before, and his more-physical particle and associated wave were considered ‘hidden variables’ because they could not be verified by observation and were not previously identified in the mathematics. The dominant view in quantum mechanics is that anything that is not an ‘observable’, or that is not specifically covered or predicted by this mathematics, or by the earlier ‘heuristic’ suggestions of Einstein[iii], cannot properly be considered to exist, and Bohm’s theory has not found general favour.

The combined proofs of von Neumann and Bell are a major impediment to any attempt to restructure and reinterpret modern physics in a manner that is realist and deterministic. There is a huge literature on this subject, in both mathematical form and in discussion, and both are highly detailed and highly complex. Remarkably, there are simple errors and omissions in these supposed proofs that render them ineffective.

We know, from the work of Bell, and now generally accepted, that von Neumann’s proof was flawed. This is because he had made a number of initial assumptions on which the proof was constructed, and one of these was unwarranted. This means that the conclusions cannot be relied upon.

In 1966, Bell wrote[iv]:

‘Thus the formal proof of von Neumann does not justify his informal conclusion … [that] “the present system of quantum mechanics would have to be objectively false in order that another description of the elementary process than the statistical one be possible.”’

Respected mathematician Irving Segal, who extended and improved von Neumann’s work in this area, refers to ‘the journalistically excellent but mathematically heuristic treatment of light theory in von Neumann’s 1932 book’.[v]

In subsequent pages, Bell went on to discuss the work of Jauch & Piron, 1963, and of Gleason, 1957, and concludes that they, too, fail to fully rule out hidden variables. These conclusions are accepted and we are therefore dependent on the proof supplied by John Bell.

‘Bell’s theorem’[vi] is essentially a thought experiment that reasons that a universe with an underlying mechanism would behave, in particular sets of circumstances, in a different way to that described in quantum mechanics in the equation of Schrödinger. All such ‘hidden variable’ mechanisms could then be demonstrated not to work, through a single set of observations. Alain Aspect[vii] famously carried out a version of the experiment described by Bell, with an outcome that supported the quantum mechanical prediction.

Note that, despite his crucial and fairly obvious error, von Neumann’s proof was accepted in physics for thirty years. We might ask whether there is a similarly overlooked flaw in the work of Bell.

In ‘Bell’s theorem’ we find such an assumption. This paper is very different to the mathematically and conceptually dense work of von Neumann, being short and very accessible. It is therefore easy to confirm that John Bell’s paper is actually a concise comparison between the predictions of the mathematics of Schrödinger on the one hand and a simple particle model on the other. Bell describes the emission of two physical, realist particles, with linked properties, and it is this option, and this option only, that has been ruled out by the theorem of Bell and the associated experimental work of Alain Aspect and others.

What Bell and Aspect together have shown is that a simple model with some of the limitations of real particles cannot explain experimental observation, while the mathematics of Schrödinger does so very well. Bell’s widely accepted dismissal of the original reasoning of von Neumann, and the limitations of his replacement argument, leave the door open to the possibility of a realist, wave-based theory.

It is of interest that part of von Neumann’s ‘proof’ was that models that went beyond Schrödinger, as currently interpreted, and incorporated other variables could not, in principle, be consistent. It is nevertheless accepted that Bohmian theory is both realist and consistent. Bell was an adherent of the Bohm approach, and remained so. Bell was also a critic of Bohr, one of the architects of the Copenhagen Interpretation, and described some of his arguments as ‘incoherent’.

It seems inconceivable that there should be such blatant flaws in these two iconic and crucially important ‘proofs’. We might consider that, even for most physicists, von Neumann’s work is impenetrably long and mathematical. More crucially, both von Neumann and Bell were confirming what everyone already felt they knew – that light is a particle and the metaphysical interpretations that attach to that recognition. This is, after all, what has been taught, albeit without consistency or clarity, as fundamental and mainstream knowledge in physics for a very long time.

While there are more wave-like and less penetrable versions of Bell’s theorem to which we must eventually turn[viii], more important in the view of this author is that the entire edifice of physics theory is constructed on concepts and logic that are each problematic. We have seen important examples already and there will be more. This means that there is a psychological premium in physics of avoiding examining the flawed and conflicted foundations of the subject and instead, like some fundamentalist religion, being entirely focussed on moving relentlessly forward.

Bell’s theorem is remarkable for being accessible. It very clearly addresses particles as its only realist option. Putting this together with Bohm confirms that it is the pilot wave or carrier wave in the latter that allows a realist model to work. As we have seen, light as a wave would require a background medium, and while that has been considered as ruled out since the time of Maxwell, we have identified the errors in that analysis.

There are still a great many problems to be addressed, not least the Schrödinger equation, which has successfully resisted mechanistic analysis since the time of its inception. We might note that waves contain frequency, phase and amplitude data, and hence carry more information than particles. If the Schrödinger equation is indeed a wave equation, as its normal appellation implies, then it is not surprising that the Schrödinger equation accounts for more varied behaviour than Bell’s particle alternative. Could this also be the reason why a q-bit (chapter 15) is understood to carry more information than a binary element?

Curiously, we will not find that the Schrödinger equation is a wave equation at all!


[i] John von Neumann, Mathematische Grundlagen der Quanten-Mechanik, Springer-Verlag, Berlin 1932. The English translation is: Princeton UP, NJ, 1955

[ii] David Joseph Bohm (1917 – 1992), Quantum Theory. New York: Dover, 1989; original publication 1951.

[iii] We have seen that the particle nature of light, though not directly observable, is nevertheless taken as real and is not considered a hidden variable.

[iv] On the Problem of Hidden Variables in Quantum Mechanics – John S Bell, in: Reviews of Modern Physics, Volume 38, Number 3; July 1966; 447-452, page 449

[v] Irving E. Segal, The mathematical implications of fundamental physical principles; pages 137-150 of The Legacy of John von Neumann (Volume 50, Proceedings of Symposia in Pure Mathematics, American Mathematical Society, 1990). The quote is from page 156.

[vi] Bell, J. S., Physics 1, 195 (1964).

Also: John Stewart Bell, Speakable and unspeakable in quantum mechanics – collected papers in quantum mechanics – (1987, Cambridge). There is a newer and more extensive version of this book.

[vii] Aspect, A, Trois tests experimentaux des inegalites de Bell par mesure de correlation de polarisation de photons, Ph.D. thesis, Universite de Paris-Sud, Centre D’Orsay (1983).

The late Caroline H. Thomson translated part of this into English. It is available on the web. She makes serious criticisms of Aspect’s use of statistics, his ‘subtraction of accidentals’.

[viii] See chapter 30.

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