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The hidden variables program in quantum mechanics has been largely discredited by two powerful theorems, namely those of Bell and Kochen/Specker. Nonetheless, this program retains a certain philosophical appeal ("God does not play dice" and all that jazz) so I won't bother to motivate my interest in the topic.

More specifically, I am investigating recent efforts to construct alternative frameworks for QM in which hidden variables are possible, thereby making the theory deterministic/realistic, etc. In particular, there are two very intriguing papers, one by William Boos and the other by Robert Van Wesep, which make use of set theoretical tools to create (plausible?) hidden variable theories:

  • William Boos (1996) claims that random ultrafilters can provide a realization of the hidden variable program.

  • Robert Van Wesep (2006) argues that the hidden variable program is entirely characterized by generic filters and uses forcing techniques on the algebra of quantum propositions.

Interestingly, both authors use related techniques (ultrafilters & forcing) which perhaps indicates that they are on to something... However, the papers are very technical and I do not fully understand their results; sadly, I could not locate any reviews of either paper online (which is surprising to me, considering how intriguing these papers are).

So my question is the following: has anyone read these papers, and if so, could you please comment on them?

(Although the question ultimately relates to physics, I feel that the highly mathematical nature of the methods used in these papers (and their beauty!) should appeal to the audience of Math Overflow, and indeed, I hope that someone here has already perused them...)

Thank you!

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  • $\begingroup$ (Fixed a tag. Quantum algebra refers to a different, although related, branch of mathematical research, and not to quantum mechanics proper.) $\endgroup$
    – Qiaochu Yuan
    Commented Jun 30, 2010 at 16:50
  • $\begingroup$ FYI, you may also be (if you are not already) interested in t'Hooft's work. $\endgroup$
    – Steve Huntsman
    Commented Jun 30, 2010 at 17:00
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    $\begingroup$ Thank you for fixing the tag. The papers are concerned with algebras of quantum propositions (the famous lattice structure of Hilbert space) and, having been engrossed in this topic for a while now, that's what I thought "quantum algebra" referred to... $\endgroup$
    – Alex Lupsasca
    Commented Jun 30, 2010 at 18:06
  • $\begingroup$ @Steve: I'm not familiar with t'Hooft's work, and he has 227 publications on his website. Are there papers in particular which you'd suggest? $\endgroup$
    – Alex Lupsasca
    Commented Jun 30, 2010 at 18:08
  • $\begingroup$ @Alex: here's a description of some recent work of his: technologyreview.com/blog/arxiv/24044 $\endgroup$
    – Steve Huntsman
    Commented Jun 30, 2010 at 19:30

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There are also papers from the 80's by Itamar Pitowsky on hidden variables models using nonmeasurable sets.

http://edelstein.huji.ac.il/staff/pitowsky/ (papers #1 and #4 for example)

For the Kochen-Specker theorems there is a long line of later results but see especially the Conway-Kochen "Free Will Theorem", available on arxiv (quant-ph).

I don't think physicists take seriously the idea that set theory is (or even might be) relevant to quantum mechanics. On the mathematical side the set-theoretic models can be seen as demonstrations that certain types of hidden variable models are logically consistent, and so cannot be ruled out without additional assumptions, such as Lebesgue measurability, that are implicit in ordinary physics reasoning.

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    $\begingroup$ Thank you for your answer. I was already aware of Pitowsky's results, which were later generalized by Stanley Gudder. Also, in his book "Quantum Probability", Gudder makes use of an alternative notion of probability (namely, generalized probability spaces) to construct hidden variable theories without necessarily giving up Lebesgue measurability. David Malament has an excellent review of the topic: uploading.com/files/edit/d6md96mc $\endgroup$
    – Alex Lupsasca
    Commented Jun 30, 2010 at 17:53
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    $\begingroup$ Also, Malament published a wonderful critique of Pitowsky's papers: links.jstor.org/… $\endgroup$
    – Alex Lupsasca
    Commented Jun 30, 2010 at 17:54

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