Local origins of quantum correlations rooted in geometric algebra

[Joy Christian]

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This paper is now on the arXiv: https://doi.org/10.48550/arXiv.2205.11372.

In it, I review the past 15 years of my work, spelling out local, realistic, and deterministic foundations of quantum correlations, without resorting to superdeterminism, retrocausality, or any other conspiracy loophole.

The framework presented in the paper assumes that Bell's formal argument against such a possibility is wrong. I have discussed the reason why it is wrong in Section 2.4 of my Reply paper for RSOS (which is currently undergoing peer-review at the journal): http://dx.doi.org/10.13140/RG.2.2.34887.37286.
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[FrediFizzx]

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This paper is now on the arXiv: https://doi.org/10.48550/arXiv.2205.11372.

In it, I review the past 15 years of my work, spelling out local, realistic, and deterministic foundations of quantum correlations, without resorting to superdeterminism, retrocausality, or any other conspiracy loophole.

The framework presented in the paper assumes that Bell's formal argument against such a possibility is wrong. I have discussed the reason why it is wrong in Section 2.4 of my Reply paper for RSOS (which is currently undergoing peer-review at the journal): http://dx.doi.org/10.13140/RG.2.2.34887.37286.

Besides that there are now 3 different local models that give the -a.b prediction with no hidden variable. A common cause... yes, the singlet spin vector. But no actual hidden variable. 3-Sphere topology is all that is required. Bell's hidden variable program falls apart.
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[FrediFizzx]

I did some improvements to the Notebook Archive file. I got rid of the Qcoordinates thing and put in the limits on the A and B functions. If it looks OK, I will try to update the Mathematica Notebook Archive.





The only comment I would have is that the local origins of quantum correlations is rooted in 3-sphere topology instead of GA.
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[Joy Christian]

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Looks fine to me. Feel free to update the Mathematica Notebook archive if the link or DOI is not changed by updating.

Yes, "rooted in 3-sphere topology" is more accurate than "rooted in geometric algebra." GA is just a convenient tool.
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[FrediFizzx]

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Looks fine to me. Feel free to update the Mathematica Notebook archive if the link or DOI is not changed by updating.

Yes, "rooted in 3-sphere topology" is more accurate than "rooted in geometric algebra." GA is just a convenient tool.

Thanks. Well actually, do you want the DOI to be the arXiv version now or stay to the RG one? Yeah, the main trick is for the URL link in the references to stay the same. I'll work on the Notebook Archive on Tuesday since Monday is a holiday here and they probably won't do anything until then.
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[Joy Christian]

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Looks fine to me. Feel free to update the Mathematica Notebook archive if the link or DOI is not changed by updating.

Yes, "rooted in 3-sphere topology" is more accurate than "rooted in geometric algebra." GA is just a convenient tool.

Thanks. Well actually, do you want the DOI to be the arXiv version now or stay to the RG one? Yeah, the main trick is for the URL link in the references to stay the same. I'll work on the Notebook Archive on Tuesday since Monday is a holiday here and they probably won't do anything until then.

If you are able to update without breaking the URL link in the references I have put in my paper, then please use the DOI of the arXiv version, which is this:

https://doi.org/10.48550/arXiv.2205.11372.
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[FrediFizzx]

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Looks fine to me. Feel free to update the Mathematica Notebook archive if the link or DOI is not changed by updating.

Yes, "rooted in 3-sphere topology" is more accurate than "rooted in geometric algebra." GA is just a convenient tool.

Actually to be more correct, all quantum correlations are rooted in 7-sphere topology. We have 3 different local models without hidden variables for the EPR-Bohm scenario that predict -a.b using 3-sphere topology. That brings up the question; will the 7-sphere model still work if lambda is removed? I guess I will dust off the simulations for that to check it out.
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[FrediFizzx]

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Looks fine to me. Feel free to update the Mathematica Notebook archive if the link or DOI is not changed by updating.

Yes, "rooted in 3-sphere topology" is more accurate than "rooted in geometric algebra." GA is just a convenient tool.

Actually to be more correct, all quantum correlations are rooted in 7-sphere topology. We have 3 different local models without hidden variables for the EPR-Bohm scenario that predict -a.b using 3-sphere topology. That brings up the question; will the 7-sphere model still work if lambda is removed? I guess I will dust off the simulations for that to check it out.

Yes, the 2D 7-sphere model works with no hidden variable. I'll post the simulation tomorrow after I clean it up some. Perhaps the r_0 method will work also for the 7-sphere model. It is probably a doozy!
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[FrediFizzx]

Here are the files for the 7-sphere simulation no hidden variables. Large files.

sims/joyGA_7-sphere4Pabcd-noHV.pdf
sims/joyGA_7-sphere4Pabcd-noHV.nb

I'm using the "y" component of the singlet spin vectors to toggle the order in the product calculation. I'm going to try to do a 3D-2D version of this so that the "z" component could be used from a 3D spin vector.

Joy, perhaps you should do an analytical version to make sure it is working correctly?
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[FrediFizzx]

Yes, it works 3D2D with the spin vectors 3D and detectors 2D. I took out the cross correlations to concentrate on the main correlation.





And we have the "z" component direction of spin vector s11 reversing the product order in the calculation. However, we probably should figure out how to involve spin vector s33. Plus, I notice that I need to do some more clean up on the annotations.
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[FrediFizzx]

I think it is OK to have just one of the spin vectors reversing the product order since they are all linked together.


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[FrediFizzx]

Here is the 7-Sphere 2 particle version with no hidden variable.




Now there are 5 different local models with no hidden variables that give the quantum mechanical predictions! The Bell fanatics are weeping.
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[FrediFizzx]

I was able to simplify the 7-sphere 2 particle version a bit.




At the end you can see a typical interaction between detector A and particle spin, Ls1. We have a scalar, 6 bivectors, and a pseudoscalar.
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[FrediFizzx]

Well, here is a doozy for you! The correlations works for 2 particles with 7D vectors!




I guess this is just regular 7-spheres.
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[gill1109]

Joy, you define ||.|| to be a non-negative real number. Yet you give an example with ||X|| = sqrt(1 + epsilon) and ||Y|| = sqrt(1 - epsilon), where epsilon is not a real number.

[Joy Christian]

Joy, you define ||.|| to be a non-negative real number. Yet you give an example with ||X|| = sqrt(1 + epsilon) and ||Y|| = sqrt(1 - epsilon), where epsilon is not a real number.

I did not choose X = 1 + epsilon and Y = 1 - epsilon to allege a counterexample. You and Lasenby made those ad hoc and incorrect choices, so don't throw them back at me. Such two-dimensional objects play no role whatsoever in my local-realistic framework. They are entirely your mistakes, not mine.
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[gill1109]

Joy, you define ||.|| to be a non-negative real number. Yet you give an example with ||X|| = sqrt(1 + epsilon) and ||Y|| = sqrt(1 - epsilon), where epsilon is not a real number.

I did not choose X = 1 + epsilon and Y = 1 - epsilon to allege a counterexample. You and Lasenby made those ad hoc and incorrect choices, so don't throw them back at me. Such two-dimensional objects play no role whatsoever in my local-realistic framework. They are entirely your mistakes, not mine.
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Those two objects are elements of your 8-dimensional algebra. I didn’t “invent” them. I found them. They exist according to the axioms which you have chosen to adopt.

[Joy Christian]

Joy, you define ||.|| to be a non-negative real number. Yet you give an example with ||X|| = sqrt(1 + epsilon) and ||Y|| = sqrt(1 - epsilon), where epsilon is not a real number.

I did not choose X = 1 + epsilon and Y = 1 - epsilon to allege a counterexample. You and Lasenby made those ad hoc and incorrect choices, so don't throw them back at me. Such two-dimensional objects play no role whatsoever in my local-realistic framework. They are entirely your mistakes, not mine.
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Those two objects are elements of your 8-dimensional algebra. I didn’t “invent” them. I found them. They exist according to the axioms which you have chosen to adopt.

Even if those ad hoc choices of X and Y are made, they do not contradict the norm relation ||XY|| = ||X|| ||Y|| as you claimed. That is the point you and Lasenby have missed. You both are grasping for straws. The norm relation holds for all elements of the algebra considered in my paper, without exception.
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[gill1109]

I did not choose X = 1 + epsilon and Y = 1 - epsilon to allege a counterexample. You and Lasenby made those ad hoc and incorrect choices, so don't throw them back at me. Such two-dimensional objects play no role whatsoever in my local-realistic framework. They are entirely your mistakes, not mine.
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Those two objects are elements of your 8-dimensional algebra. I didn’t “invent” them. I found them. They exist according to the axioms which you have chosen to adopt.

Even if those ad hoc choices of X and Y are made, they do not contradict the norm relation ||XY|| = ||X|| ||Y|| as you claimed. That is the point you and Lasenby have missed. You both are grasping for straws. The norm relation holds for all elements of the algebra considered in my paper, without exception.
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I look forward to reading your argument for this assertion. And good luck with your “Reply” in connection with the RSOS paper. I’m looking forward to seeing it in print.

[Joy Christian]

And good luck with your “Reply” in connection with the RSOS paper. I’m looking forward to seeing it in print.

My invited reply was published by RSOS in November 2022: https://doi.org/10.1098/rsos.220147
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