Grete Hermann in 1935 claimed that the proof contains a flaw, and John Bell in 1966 came up with a similar criticism.
Who is right?
Quantum mechanics is about waves: can we model quantum mechanics with "non-wavelike" phenomena?
A wave packet which models the position ans the momentum of a particle, is a typical quantum mechanical object.
Can we somehow build a wave packet from "non-wavelike" phenomena?
Intuitively, it is not easy. We can build a wave packet as a sum of two wave packets, but how could we build a wave packet from, say, ten real numbers?
Suppose that we have a sample of 1,000 classical particles, each of which has a definite position and momentum. This sample can approximately model a wave packet, but not exactly.
We have not yet checked the details of the proof by von Neumann. We believe that von Neumann aimed to show that, under certain assumptions, one cannot build a wave from a (finite) number of non-wavelike subsystems.
Criticism by Hermann and Bell: a digital computer
Grete Hermann and John Bell criticized the proof by von Neumann. They said that it might still be possible to simulate wavelike behavior with hidden variables.
Suppose that we have a digital computer which models and calculates a wave packet. Certainly, we can, at least approximately, simulate the wave packet in the computer. A digital computer is not a "wavelike" object. We proved that one can, approximately, simulate a wave with non-wavelike objects.
Could it be that a computer could simulate a wave packet exactly?
The random number generator is a problem. A pure digital computer must contain a deterministic pseudo-random number generator. This implies that the behavior of the computer program does differ from an idealized quantum system, and that we, in principle, can observe that the output of the program is not like that of a true quantum system. We conclude that a digital computer cannot simulate a quantum system.
Maybe this is what von Neumann proved?
The de Broglie-Bohm model contains a "pilot wave"
The model of Louis de Broglie and David Bohm (1952) contains a global "pilot wave". The particle is like a small boat which sails on the pilot wave.
We know that the model can reproduce the standard quantum mechanical results, like the interference pattern in a double-slit experiment.
The model does contain a wave. It is not surprising that the model can describe wave phenomena.
But the model cannot simulate the "true" random behavior of quantum mechanics? If we initialize the particles to certain definite values, then the system certainly will not behave in a truly random way. It is just like in the case of a digital computer.
Bell's inequality (1964)
The famous inequality of John Bell (1964) shows that one cannot model the results of the Einstein-Podolsky-Rosen experiment with a "local hidden variable" theory. The state of the two particles at distant locations cannot be simulated with a model where each particle possesses a determined unique state.
The state of the two particles is "entangled". It cannot be split into two determined parts.
The result by Bell refers to locality, while the result of von Neumann does not. It is now clear that von Neumann did not prove Bell's theorem in 1932.
Did von Neumann prove a triviality?
How possibly could a short argument about expectation values rule out hidden variable models?
It has to be magic – or a triviality.
Suppose that we have a true random generator which outputs either -1 or 1 at random. Then the expectation value for its first output is 0.
But if it is a pseudo-random generator, which uses a complicated mathematical formula to generate a sequence of pseudo-random numbers, then the expectation value of the first output is -1 or 1 – not 0.
Did John von Neumann prove this triviality?
Tom Harper (2023) has posted a video and a paper where he analyzes arguments by Louis Caruana (1995):
and Mermin and Schack (2018):
According to Caruana, von Neumann proves the following trivial fact:
- If quantum systems A and B are truly indistinguishable, then there cannot exist any hidden variables whose state could differ between A and B.
The criticism by John Bell (1966)
"Dispersionless", informally, means that a system is a classical particle, or some other classical, not quantum, object. It cannot be a quantum object in a superposition state.
Suppose that we have a quantum system which consists of dispersionless subcomponents. In a sense, the quantum system then is determined by hidden variables. An example: a tense plastic string behaves much like a wavelike object (quantum), but at a low level we can decompose it into atoms, which might be non-wavelike.
In section III of the paper, John Bell writes that one should not require the additivity of expectation values of different measurements for dispersionless subcomponents. Bell's statement is suspicious. The parameters of a classical particle can be measured arbitrarily accurately, without disturbing the system. That is, all measurements "commute", in the terminology of quantum mechanics. Then the expectation value of
measured position / meter
+ measured momentum / kg m/s
is the sum of the expectation values of each summand!
If we have an arbitrary hidden variable model which outputs values for measurements, then the question what is the expectation value of
position + momentum
is not well defined. Let the hidden variable model be a computer program which outputs a number when we type either "position" or "momentum". What does the "expectation value" of position + momentum mean? We cannot type that string as the input to the computer. Should we first type "position" and then "momentum", and take the sum?
Conclusions
John von Neumann proved a triviality: something like this:
- You cannot simulate a genuine random number generator with a pseudo-random number generator, because for a pseudo-random generator you can, with a mathematical formula, predict the output of the generator!
Von Neumann used his operator algebra for the proof. The algebra is somewhat complicated, and people have had hard time evaluating the impact and relevance of the proof.
Hidden variable models are not trivial. A trivial result will not illuminate the problem much.
Our own contribution in this blog is noting that since quantum mechanics is about waves, then simplifying a system into a few real numbers is not going to work, in most cases.
This implies the famous result of John Bell: if you assume that the "wave" describing particles A and B has collapsed (i.e., the wave has been reduced into a few real numbers), then you will not be able to produce the phenomena of quantum mechanics. Letting a wave collapse often destroys information – but that information is needed to reproduce quantum phenomena.
To produce the interference pattern in the double-slit experiment, you need a wave. A classical particle which is characterized by a few real numbers (its velocity components), will not do.
Since the de Broglie-Bohm model includes the pilot wave, it can produce the interference pattern with hidden variables, deterministically.
We conclude that the critics of the von Neumann proof were right: the proof is not relevant for hidden variable models.
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