Impossibilty of hidden variables (Bohm, 1951)

[atyy]

Bohm writes in his 1951 book "Quantum Theory" (p623): "We conclude then that no theory of mechanically determined hidden variables can lead to all the results of quantum theory". He bases his argument on the uncertainty principle. Presumably the argument is not correct, since Bohm himself later provided the first known hidden variable account of non-relativistic quantum theory?

 

[jfizzix]

If I had to guess (so take it for what it's worth), I think Bohm might be saying that non-contextual hidden variables cannot describe quantum theory, but that other local hidden variables might do so.

A non-contextual hidden variable theory is one where the hidden variables are objectively determined, and independent of one's measurement strategy. These models have been disproven in part by Gleason's theorem.

However, there are models of hidden variables where the hidden variables are determined in the context of one's measurement strategy. In other words, if you include the measurement device in the description of your measurements, a model of hidden variables exists that could describe the measurement outcomes. I expect that this is what Bohm provided.

 

[atyy]

Yes, I wonder whether Bohm knew about the non-contextuality qualification when writing his 1951 statement. Or did he mean it without qualification, and somehow later realized he had made a mistake, when he discovered Bohmian Mechanics?

 

[jfizzix]

I think it was only later that the term "contextuality" was used explicitly, but I couldn't say exactly when it started to become a common term. The earliest mention of it I know of is Abner Shimony's 1984 paper:
"Contextual Hidden Variables Theories and Bell's Inequalities".

Bohm may have been referring to the same thing by another name, but I couldn't say for sure.

 

[bhobba]

Yes, I wonder whether Bohm knew about the non-contextuality qualification when writing his 1951 statement. Or did he mean it without qualification, and somehow later realized he had made a mistake, when he discovered Bohmian Mechanics?

The whole thing is tied up with and incorrect proof given by the highly influential (and with good reason - he was one of the greatest to ever live - in many peoples top ten of all time) mathematician Von-Neumann in his book - Mathematical Foundations Of QM. Its not that the proof itself is incorrect - as you would expect from a mathematician of his stature - but the assumption that went into it (it was an assumption on the addition of statistical averages that didn't apply to hidden variables because they can be non-contextual as was later sorted out by Gleason with his famous theorem).

There were people like Grete Hermann that spotted the error - but were ignored:
http://arxiv.org/ftp/arxiv/papers/0812/0812.3986.pdf

Its a bit of a sad history really.

Thanks
Bill

 

[atyy]

There were people like Grete Hermann that spotted the error - but were ignored:
http://arxiv.org/ftp/arxiv/papers/0812/0812.3986.pdf

Here my interest in asking the question is to what extent the error got into the textbooks. For example. Landau and Lifshitz do almost make the error, but maybe not because what they say is that position and momentum cannot simultaneously exist at all times - which is in general true (with some exceptions).

Another famous textbook by Messiah correctly says that hidden variables cannot be ruled out, but he will go with Copenhagen since it is simpler and no experiments distinguish the two interpretations at that time. Messiah's book was published in 1958, so I do also wonder why he made the correct statement. Did he

1) like Grete Hermann (and others like Einstein, according to anecdotes) correctly reject von Neumann's proof because of the hidden assumption of non-contextuality?
2) incorrectly accept the EPR argument that quantum mechanics must be incomplete?
3) correctly accept the possibility of hidden variables because he knew about Bohmian Mechanics which Bohm discovered in 1952 after Bohm's textbook but before Messiah's?

The version of Landau and Lifshitz I have access to is the 3rd English edition, 1977. It looks like their first English edition is 1958, the same year as Messiah's. I don't know when the original Russian text was published.

Its a bit of a sad history really.

Well, despite his error, it still shows von Neumann's enormous good taste in actually trying to investigate the problem, and do it by a theorem.

I guess the sad part is Grete Hermann being ignored. It would be very interesting to know how Bohm came to the wrong conclusion in his book of 1951, and then managed to realize his error by 1952.

 

[TrickyDicky]

So to be clear, both Gleason's and Kochen-Specker theorems only rule out non-contextual hidden variables theories?

 

[bhobba]

So to be clear, both Gleason's and Kochen-Specker theorems only rule out non-contextual hidden variables theories?

Kochen-Specker is a simple corollary to Gleason, but Gleason is notoriously difficult to prove so a direct proof was devised.
http://kof.physto.se/cond_mat_page/theses/helena-master.pdf

What Gleason say is if you assume non-contextuality then Borns rule follows (yes I know some other assumptions like the strong principle of superposition is required but that the main one) - no escaping it and you can't have properties having definite values at all times - specifically with Born's Rule you can't define a 0 and 1 only map on the Hilbert space. However hidden variable theories do not have to be non-contextual.

Thanks
Bill

 

[OCR]

A bit more about von Neumann's proof...

In 2010, Jeffrey Bub published an argument that Bell (and, thus, also Hermann) had misconstrued von Neumann's proof, claiming that it does not attempt to prove the absolute impossibility of hidden variables, and that it is actually not flawed, after all.

von Neumann’s ‘No Hidden Variables’ Proof: A Re-Appraisal...
http://arxiv.org/pdf/1006.0499.pdf

Grete Hermann... http://en.wikipedia.org/wiki/Grete_Hermann

 

[bhobba]

I know Von Neumanns proof from studying his text.

IMHO there was no misconstruing. He gave an operational definition of expected outcomes that easily showed it must be additive. However it didn't apply to hidden variables as a number of counter examples published by Bell and others showed - indeed Bohmian Mechanics is a counter example.

Thanks
Bill

 

[TrickyDicky]

Bohm writes in his 1951 book "Quantum Theory" (p623): "We conclude then that no theory of mechanically determined hidden variables can lead to all the results of quantum theory". He bases his argument on the uncertainty principle. Presumably the argument is not correct, since Bohm himself later provided the first known hidden variable account of non-relativistic quantum theory?

He wrote "mechanically determined hidden variables". Is it posible that even though the formal distinction between contextual and non-contextual had not been made at the time he was actually referring to non-contextual theories with that qualifier?
I mean if by "mechanically determined" he was referring to classical type theories, which were a few years later excluded by Bell's theorem, one can wonder if there is an equivalence between local hidden variables and non-contextual hidden variables theories.

 

[atyy]

He wrote "mechanically determined hidden variables". Is it posible that even though the formal distinction between contextual and non-contextual had not been made at the time he was actually referring to non-contextual theories with that qualifier?

I thought that just meant "deterministic" in the sense that dBB is deterministic with all variability being put in the initial conditions.

 

[bohm2]

Doesn't "machanical" just mean action by "contact mechancs" (or local causality). .So when he writes "no theory of mechanically determined hidden variables can lead to all the results of quantum theory", doesn't he just mean that no local hidden variables can lead to all the results of QT?

 

[TrickyDicky]

Doesn't "machanical" just mean action by "contact mechancs" (or local causality). .So when he writes "no theory of mechanically determined hidden variables can lead to all the results of quantum theory", doesn't he just mean that no local hidden variables can lead to all the results of QT?

That was basically my point, as I commented in the second part of my post.

 

[TrickyDicky]

I thought that just meant "deterministic" in the sense that dBB is deterministic with all variability being put in the initial conditions.

I don't think that could be the case, not in any dBB sense. He would have been ruling out his own interpretation which l'm pretty sure he already had in mind(remember dBB is a modification of de Broglie's own 1920s pilot wave theory) by 1951.

 

[Demystifier]

I have read the relevant parts of the Bohm's 1951 book again and concluded that his arguments against hidden variables were totally incoherent.
(And I suspect that in 1952 and later he would agree with that qualification.)

 

[atyy]

I don't think that could be the case, not in any dBB sense. He would have been ruling out his own interpretation which l'm pretty sure he already had in mind(remember dBB is a modification of de Broglie's own 1920s pilot wave theory) by 1951.

But de Broglie I think rejected his own theory, which is why the first solution of the measurement problem is (to my knowledge) due to Bohm. In his 1952 paper, Bohm does cite de Broglie, and also mentions problems that others (including de Broglie) pointed out with the de Broglie theory, but then goes on to say that he (Bohm) will show that all the problems are not problems.

However, the book is 1951 and Bohmian Mechanics is 1952, so that seems rather close. Did he make a breakthrough between 1952 and 1952, or was the book finished somewhat earlier than its date of publication, as is usually the case?

I have read the relevant parts of the Bohm's 1951 book again and concluded that his arguments against hidden variables were totally incoherent.
(And I suspect that in 1952 and later he would agree with that qualification.)

That's what I suspect too. Is there any history as to how he came to realize his mistake? I mean, from my point of view, the 1952 paper is a big breakthrough. Bell clearly thought it was, and as late as the early 1960s, we have Feynman's wonderful lectures making the same mistake that particle trajectories are not possible. Although the book and paper are so close in publication date that it seems he must have had some inkling of the 1952 development by 1951, it seems such a big breakthrough that I cannot imagine he would have written such wrong or at best ambiguous and misleading statements in his book.

 

[TrickyDicky]

I have read the relevant parts of the Bohm's 1951 book again and concluded that his arguments against hidden variables were totally incoherent.
(And I suspect that in 1952 and later he would agree with that qualification.)

That might well be the case, but we only have the sentence in the OP to judge here. That Bohm was incoherent about QM right until the very moment he saw the light and published his dBB interpretation is a possibility, it is hard to say , I can't really contribute to that debate due to my ignorance about BM and Bohm's writings.

 

[atyy]

I found a number of interesting stories on the history of BM online. They are from sources with good reputations, but I don't know the evidence behind these particular bits.

"David Bohm wrote a quantum mechanics book and also gave a proof that hidden variables theory were impossible. Einstein pointed out a flaw in the argument. Bohm responded with Bohmian mechanics, a hidden variables theory that agrees perfectly with quantum mechanical predictions. His theory was not well-received. David Bohm moved on." http://www.mathematik.uni-muenchen.de/~bohmmech/BohmHome/aboutus_members.html

"It was Einstein who explained to David Bohm at Princeton in 1951 why orthodox quantum mechanics is inacceptable. Therefore and because of his earlier studies of "hidden" variables, Einstein must be counted as a grandfather of Bohmian mechanics. However, he did not like Bohmian mechanics and did not support Bohm's proposal in 1952, probably because Bohmian Mechanics is nonlocal and "too simple"." http://www.mathematik.uni-muenchen.de/~bohmmech/BohmHome/whatisbm_history.html

So Einstein read the 1951 book, told Bohm the argument was flawed (hopefully not using EPR, since that argument is wrong), and that lead Bohm to BM?

 

[bohm2]

Not sure what it means except for some type of "wholeness/holism", but this is what Bohm's co-worker Hiley wrote:

The very term, “Bohmian Mechanics”, lies at the root of much confusion concerning the direction that Bohm’s own thinking took after he first published his two seminal papers in 1952. While I quite understand the wish to give credit to Bohm for his pioneering work, the linking of Bohm’s name with the term ‘mechanics’ has led many to believe that Bohm himself was motivated to find a classical order based on a deterministic mechanics from which the quantum formalism would emerge. That was never his intention. Indeed the content of his book “Quantum Theory” published in 1951, which gives an exhaustive account of the orthodox view of the theory, already sows the seeds of how radical a change Bohm thinks is needed in order to begin to understand the structure that underlies the quantum formalism. In that book he sees the need to go beyond mechanical ideas. In the section headed ‘The need for a nonmechanical description’, he writes,

...the entire universe must, in a very accurate level, be regarded as a single indivisible unit in which separate parts appear as idealisations permissible only on a classical level of accuracy of the description. This means that the view of the world as being analogous to a huge machine, the predominant view from the sixteenth to nineteenth century, is now shown to be only approximately correct. The underlying structure of matter, however, is not mechanical [7].​

In a footnote to this quote he writes “This means that the term ‘quantum mechanics’ is very much a misnomer. It should, perhaps, be called ‘quantum nonmechanics’.”

In Hiley, "Some Remarks on the Evolution of Bohm's. Proposals for an Alternative to Standard Quantum.", 2010.

Anyway, this seems more philosophy than physics because I don't understand what "non-mechanical", means in physical terms either than going against local causality (e.g. non-locality).

 

[atyy]

Here is a late interview with Bohm. It pretty much seems that like what Demystifier said, the arguments in the 1951 book are incoherent, and Bohm himself felt that after a number of discussion with Einstein about his book.

http://www.aip.org/history/ohilist/4513.html
Interview with Dr. David Bohm
By Lillian Hoddeson
At the home of the Bohms, Edgware, London
May 8, 1981

"Well, I had several conversations with Einstein. After writing this book on quantum mechanics, which I wrote to try to understand it (based on my graduate course), I sent a copy to various scientists including Einstein. He wanted to discuss it with me, and we discussed it. He felt that the book was as good as you could present the ordinary point-of-view, but he still didn’t accept it. So we discussed it for a while, and meanwhile I myself had been feeling that it wasn’t all that clear, and that therefore these two things together made me feel that the interpretation of quantum mechanics was not satisfactory. So I began to think about it, and I produced another interpretation, which came out in two papers in Phys. Rev, in 1952, two papers, using a particle and a wave, the causal interpretation I called it. And I discussed all those things with Einstein; we also had correspondence afterwards when I was in Brazil."

 

[TrickyDicky]

Here is a late interview with Bohm. It pretty much seems that like what Demystifier said, the arguments in the 1951 book are incoherent, and Bohm himself felt that after a number of discussion with Einstein

Maybe from the current mainstream view it seems incoherent but the quotes Bohm2 posted from Hiley suggests a different interpretation, he was not looking for a deterministic theory. Perhaps the main mistake with hidden variables theories is to frame them in terms of determinism- indeterminism as there are option outside that false dichotomy.

 

[atyy]

Maybe from the current mainstream view it seems incoherent but the quotes Bohm2 posted from Hiley suggests a different interpretation, he was not looking for a deterministic theory. Perhaps the main mistake with hidden variables theories is to frame them in terms of determinism- indeterminism as there are option outside that false dichotomy.

From the point of view of realistic theories, determinism-indeterminism is a false dichotomy - but if we give up realism (whatever that means) then it is not so clear. So the real question is realism, so that determinism-indeterminism can indeed be a false dichotomy.

What is realism? Realism just means that the theory completely obeys classical probability, as given by Kolmogorov's axioms. The achievement of Bohmian Mechanics is to show that quantum mechanics can be embedded in a theory that obeys Kolmogorov's axioms. In contrast, in a minimal Copenhagen-like interpretation, quantum mechanics is an interface between the quantum state which does not obey Kolmogorov's axioms, and measurement outcomes which do obey Kolmogorov's axioms.

 

[TrickyDicky]

From the point of view of realistic theories, determinism-indeterminism is a false dichotomy - but if we give up realism (whatever that means) then it is not so clear. So the real question is realism, so that determinism-indeterminism can indeed be a false dichotomy.

What is realism? Realism just means that the theory completely obeys classical probability, as given by Kolmogorov's axioms. The achievement of Bohmian Mechanics is to show that quantum mechanics can be embedded in a theory that obeys Kolmogorov's axioms. In contrast, in a minimal Copenhagen-like interpretation, quantum mechanics is an interface between the quantum state which does not obey Kolmogorov's axioms, and measurement outcomes which do obey Kolmogorov's axioms.

In the end we judge theories by how well they predict measurement outcomes, so by the Born rule the physical interpretation of QM is realist according to your definition. BM shows that you can keep that to be the case even if one takes seriously the wave function, quite a feat.

 

[Demystifier]

What is realism? Realism just means that the theory completely obeys classical probability, as given by Kolmogorov's axioms.

This is quite an unusual definition of realism. Have you seen that definition somewhere else, or is it your own definition?

 

[atyy]

This is quite an unusual definition of realism. Have you seen that definition somewhere else, or is it your own definition?

No, I haven't seen it anywhere else. The usual definition of realism is that things should exist independently of any observer. What I was thinking (now maybe I'm not sure this is right) is that in quantum mechanics the pure states are not extremal points of a simplex. In Kolmogorov's axioms, there should be a space of elementary events and a sigma algebra of combinations of events to which probability can be consistently assigned. I have assumed the sigma algebra over the elementary events gives rise to a state space that is a simplex (but I am not sure if that is necessarily true), and that that is the reason quantum mechanics does not fit into classical probability, or as bhobba likes to say - improper mixtures are not "ignorance interpretable". Then I have further assumed that there is no classical probability theory that will have an observer problem as quantum mechanics has. This is all rather informal, so it'd be interesting to know whether it's really right or not.

 

[TrickyDicky]

No, I haven't seen it anywhere else. The usual definition of realism is that things should exist independently of any observer. What I was thinking (now maybe I'm not sure this is right) is that in quantum mechanics the pure states are not extremal points of a simplex. In Kolmogorov's axioms, there should be a space of elementary events and a sigma algebra of combinations of events to which probability can be consistently assigned. I have assumed the sigma algebra over the elementary events gives rise to a state space that is a simplex (but I am not sure if that is necessarily true), and that that is the reason quantum mechanics does not fit into classical probability, or as bhobba likes to say - improper mixtures are not "ignorance interpretable". Then I have further assumed that there is no classical probability theory that will have an observer problem as quantum mechanics has. This is all rather informal, so it'd be interesting to know whether it's really right or not.

I don't think it works, not for Bohmian mechanics anyway where the guiding wave and the trajectories are not observable so it is not realist in many senses. Realism is a really ambiguous and complex concept anyway, that almost anyone working with it has a different idea of what it is, so I'm very skeptical it can help to clarify anything, unless one uses some form of naive realism, which is the basis of empirical sciences so everyone is sme way or another obliged to follow. In its more radical form could be what Heisenberg seeked as a basis of the new theory in his seminal 1925 paper starting QM, in his words:"A basis founded exclusively upon relationships between quantities which in principle are observable". Sadly it took less than a year for the founders to give up that program, when Schrödinger came along with the wave function and introduced the state vector, something far from the exclusive quantities Heisenberg was seeking.

 

[write4u]

Not sure what it means except for some type of "wholeness/holism", but this is what Bohm's co-worker Hiley wrote:In Hiley, "Some Remarks on the Evolution of Bohm's. Proposals for an Alternative to Standard Quantum.", 2010.

Anyway, this seems more philosophy than physics because I don't understand what "non-mechanical", means in physical terms either than going against local causality (e.g. non-locality).

Question: are waves mechanical functions, and if they are, can they differ from physical mechanics. I am thinking of the different behaviors of particle/wave duality?

 

[bhobba]

Question: are waves mechanical functions, and if they are, can they differ from physical mechanics. I am thinking of the different behaviors of particle/wave duality?

Can you perhaps elucidate what you mean by mechanical function?

In BM its a potential.

Thanks
Bill

 

[DrChinese]

The usual definition of realism is that things should exist independently of any observer.

Nice wording.