Boole vs. Bell - a mathematicians perspective?

[harrylin]

This is a spin-off from thread https://www.physicsforums.com/showthread.php?t=499002 , which discusses a new paper on Boole vs. Bell.

The question is if Bell's Theorem correctly applies Boolean logic, or if it may for example be invalid due to a probabilistic incompatibility of random Variables.

The mathematical criticism is not limited to a single paper or even a single author, but the following seems to be a good one for starters:

http://www.mdpi.com/1099-4300/10/2/19/ (open access)
"Bell-Boole Inequality: Nonlocality or Probabilistic Incompatibility of Random Variables?"

Khrennikov appears to be an expert in this field:
http://w3.msi.vxu.se/Personer/akhmasda/CV.htm

As most people here will be unfamiliar with Bell's Theorem, I should perhaps first provide links to a few simple summaries on Internet:

http://michaelnielsen.org/blog/why-the-world-needs-quantum-mechanics/
http://faculty.luther.edu/~macdonal/Spooky.pdf

And here's a link to Bell's original paper:
http://www.drchinese.com/David/Bell_Compact.pdf

OK then, back to the mathematical criticism on which the QM group gave little feedback. As for myself, I am neither strong in QM nor in generalisations of probability theory.

To summarize the Khrennikov paper:

"Is it possible to construct the joint probability distribution [..]?"
The answer is no, if Bell's inequalities are violated.

"The joint probability distribution does not exist just because
those observables could not be measured simultaneously."

"Eberhard[..] operated with statistical data obtained from three different experimental contexts, C1 , C2 , C3 , in such a way as [if] it was obtained on the basis of a single context. He took results belonging to one experimental setup and add[ed] or substract[ed] them from results belonging to another experimental setup. These are not proper manipulations from the viewpoint of statistics. One never performs algebraic mixing of data obtained for [a] totally different sample."

The paper concludes:

"nonexistence of a single probability space does not imply that the realistic description (a map λ→a(λ)) is impossible to construct."

Is perhaps anyone here who can find a flaw in the mathematical criticism? Or who can elaborate on its pertinence?

You may also be interested to comment on the new paper by De Raedt et al, which is the subject of discussion in the thread mentioned at the top of this post. That paper gives a more lengthy summary of the different objections, together with an introduction to Boolean inequalities.

Regards,
Harald

 

[mmwave]

Dear Harald,

Thank you for bringing this paper to our attention. it is always important to critically examine and analyze any new findings or theories, even those that have been widely accepted in the scientific community.

After reading the paper by Khrennikov, it is clear that he raises some valid points about the assumptions and limitations of Bell's theorem. In particular, his argument that the joint probability distribution cannot be constructed if Bell's inequalities are violated is a strong one. This raises questions about the validity of using Bell's theorem to prove non-locality in quantum mechanics.

Furthermore, Khrennikov's criticism of Eberhard's statistical data manipulation is also valid. It is important for scientists to be careful and precise in their data analysis, and any improper manipulation of data can lead to incorrect conclusions.

I believe that this mathematical criticism is relevant and should be carefully considered by the scientific community. It is always important to question and challenge established theories, as this is how scientific progress is made.

As for the new paper by De Raedt et al, I think it is a valuable contribution to the discussion on Bell's theorem and its limitations. It brings together different objections and presents them in a clear and comprehensive manner. I encourage further discussion and analysis of this paper and its implications for our understanding of quantum mechanics.

Thank you again for bringing this important topic to our attention.