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Impossibilty of hidden variables (Bohm, 1951)

  1. Feb 11, 2015 #1

    atyy

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    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?
     
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  3. Feb 11, 2015 #2

    jfizzix

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    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.
     
  4. Feb 11, 2015 #3

    atyy

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    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 realised he had made a mistake, when he discovered Bohmian Mechanics?
     
  5. Feb 11, 2015 #4

    jfizzix

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    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.
     
  6. Feb 11, 2015 #5

    bhobba

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    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
     
  7. Feb 11, 2015 #6

    atyy

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    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.

    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 realise his error by 1952.
     
    Last edited: Feb 11, 2015
  8. Feb 12, 2015 #7
    So to be clear, both Gleason's and Kochen-Specker theorems only rule out non-contextual hidden variables theories?
     
  9. Feb 12, 2015 #8

    bhobba

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    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 [Broken]

    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 cant 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
     
    Last edited by a moderator: May 7, 2017
  10. Feb 12, 2015 #9

    OCR

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    A bit more about von Neumann's proof...
    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
     
  11. Feb 13, 2015 #10

    bhobba

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    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
     
  12. Feb 13, 2015 #11
    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 refering to non-contextual theories with that qualifier?
    I mean if by "mechanically determined" he was referring to classical type theories, wich 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.
     
  13. Feb 13, 2015 #12

    atyy

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    I thought that just meant "deterministic" in the sense that dBB is deterministic with all variability being put in the initial conditions.
     
  14. Feb 13, 2015 #13
    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?
     
  15. Feb 13, 2015 #14
    That was basically my point, as I commented in the second part of my post.
     
  16. Feb 13, 2015 #15
    I don't think that could be the case, not in any dBB sense. He would have been ruling out his own interpretation wich 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.
     
  17. Feb 13, 2015 #16

    Demystifier

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    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.)
     
  18. Feb 13, 2015 #17

    atyy

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    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?

    That's what I suspect too. Is there any history as to how he came to realise 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.
     
  19. Feb 13, 2015 #18
    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.
     
  20. Feb 13, 2015 #19

    atyy

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    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?
     
  21. Feb 13, 2015 #20
    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).
     
    Last edited: Feb 13, 2015
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