Question

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: http://www.springerlink.com/content/n3gr194551472536/

• 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: http://arxiv.org/abs/quant-ph/0506040

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!

PS: Boos' paper is not freely available from the publisher, but a copy exists online: http://uploading.com/files/37m88a11/boos%2Bultrafilters.pdf/

Answer 1

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.