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Cool! Thank you for the link and info. Will read both tomorrow with great interest.

- local
**Posts:**86**Joined:**Mon Aug 05, 2019 1:19 pm

Here is some typical output of the GAViewer code that is in the paper.

One can see that the correlation matches up perfectly to neg_adotb event by event so verifies the product calculation. And that the bivectors are vanishing in the "mean".

.

- Code: Select all
`neg_adotb = 0.574949`

theta = 234.903931

correlation = 0.574949

neg_adotb = 0.386546

theta = 247.260223

correlation = 0.386546

neg_adotb = 0.006321

theta = 90.362167

correlation = 0.006321

neg_adotb = -0.487911

theta = 60.796616

correlation = -0.487911

neg_adotb = 0.196614

theta = 101.339050

correlation = 0.196614

neg_adotb = -0.526722

theta = 58.215748

correlation = -0.526722

neg_adotb = 0.898704

theta = 153.988235

correlation = 0.898704

neg_adotb = -0.832386

theta = 33.655334

correlation = -0.832386

neg_adotb = 0.113726

theta = 96.530151

correlation = 0.113726

neg_adotb = 0.461435

theta = 242.520279

correlation = 0.461435

neg_adotb = 0.399192

theta = 246.472336

correlation = 0.399192

neg_adotb = 0.532410

theta = 122.168465

correlation = 0.532410

neg_adotb = -0.392552

theta = 66.886620

correlation = -0.392552

neg_adotb = 0.254378

theta = 104.736710

correlation = 0.254378

mean = 0.001660 + 0.004355*e2^e3 + -0.001763*e3^e1 + -0.002109*e1^e2

aveA = 0.004108

aveB = 0.000323

One can see that the correlation matches up perfectly to neg_adotb event by event so verifies the product calculation. And that the bivectors are vanishing in the "mean".

.

- FrediFizzx
- Independent Physics Researcher
**Posts:**1732**Joined:**Tue Mar 19, 2013 7:12 pm**Location:**N. California, USA

Happy Thanksgiving to all!

And thank you FrediFizzx for the great forum.

And thank you FrediFizzx for the great forum.

- local
**Posts:**86**Joined:**Mon Aug 05, 2019 1:19 pm

local wrote:Happy Thanksgiving to all!

And thank you FrediFizzx for the great forum.

You're welcome and Happy Thanksgiving to you and all.

Now, if I could just figure out the function for the sign flip, it would be a really great Thanksgiving.

.

- FrediFizzx
- Independent Physics Researcher
**Posts:**1732**Joined:**Tue Mar 19, 2013 7:12 pm**Location:**N. California, USA

Joy Christian wrote:***

Ok., here is the new paper: https://arxiv.org/abs/1911.11578.Title: Dr. Bertlmann's Socks in the Quaternionic World of Ambidextral Reality

Abstract:

In this pedagogical paper, John S. Bell's amusing example of Dr. Bertlmann's socks is reconsidered, first within a toy model of a two-dimensional one-sided world of a non-orientable Möbius strip, and then within a real world of three-dimensional quaternionic sphere, S^3, which results from an addition of a single point to R^3 at infinity. In the quaternionic world, which happens to be the spatial part of a solution of Einstein's field equations of general relativity, the singlet correlations between a pair of entangled fermions can be understood as classically as those between Dr. Bertlmann's colorful socks.

I have revised the pedagogical paper: https://arxiv.org/abs/1911.11578.

I have added an appendix proving the equivalence of the conservation of spin angular momentum and the twists in the Hopf bundle of S^3. This makes the paper more technical than I intended it to be, but the technicality (as beautiful as it is) is in the appendix and can be ignored in the first reading.

I have also added a couple of explanatory lines in the code presented in Section IV to bring the code more in line with the analytical derivation of the singlet correlations.

***

- Joy Christian
- Research Physicist
**Posts:**2187**Joined:**Wed Feb 05, 2014 4:49 am**Location:**Oxford, United Kingdom

***

Quite independently of any cosmological model, for the past twelve years I have argued for a closed S^3 model for the 3D physical space. My argument comes from a local-realistic understanding of quantum correlations. It is good to know that cosmological data also now seem to support the closed model for the physical space:

"... the Planck cosmic microwave background spectra now preferring a positive curvature at more than 99% confidence level":

https://www.nature.com/articles/s41550-019-0906-9

***

Quite independently of any cosmological model, for the past twelve years I have argued for a closed S^3 model for the 3D physical space. My argument comes from a local-realistic understanding of quantum correlations. It is good to know that cosmological data also now seem to support the closed model for the physical space:

"... the Planck cosmic microwave background spectra now preferring a positive curvature at more than 99% confidence level":

https://www.nature.com/articles/s41550-019-0906-9

***

- Joy Christian
- Research Physicist
**Posts:**2187**Joined:**Wed Feb 05, 2014 4:49 am**Location:**Oxford, United Kingdom

***

Happy New Year to everyone!

I have revised my pedagogical paper once again: https://arxiv.org/abs/1911.11578. The last couple of lines of the following paragraph, including Eq. (40), are new in this version:

Nota bene: If a null vector is like the teeth of the Cheshire Cat, then a null bivector is like the grin of the Cheshire Cat, with the Cat itself being the vector or the bivector, respectively.

But seriously, just as a null vector is a vector that has no length (or has vanishing magnitude) and no direction, a null bivector is a bivector that spans no area (or spans zero area) and has no direction. Moreover, in Geometric Algebra there is only one notion of zero for elements of all grades. In other words, there is only one notion of zero for the scalars, vectors, bivectors, trivectors, and multivectors, not separate notions of zero for each grade or a composite of grades.

More importantly, I claim that my pedagogical paper provides the best explanation to date of the observed strong correlations in strictly local, realistic, and deterministic terms.

***

Happy New Year to everyone!

I have revised my pedagogical paper once again: https://arxiv.org/abs/1911.11578. The last couple of lines of the following paragraph, including Eq. (40), are new in this version:

Nota bene: If a null vector is like the teeth of the Cheshire Cat, then a null bivector is like the grin of the Cheshire Cat, with the Cat itself being the vector or the bivector, respectively.

But seriously, just as a null vector is a vector that has no length (or has vanishing magnitude) and no direction, a null bivector is a bivector that spans no area (or spans zero area) and has no direction. Moreover, in Geometric Algebra there is only one notion of zero for elements of all grades. In other words, there is only one notion of zero for the scalars, vectors, bivectors, trivectors, and multivectors, not separate notions of zero for each grade or a composite of grades.

More importantly, I claim that my pedagogical paper provides the best explanation to date of the observed strong correlations in strictly local, realistic, and deterministic terms.

***

- Joy Christian
- Research Physicist
**Posts:**2187**Joined:**Wed Feb 05, 2014 4:49 am**Location:**Oxford, United Kingdom

***

I have revised my pedagogical paper once again to make the above comments regarding the disappearing Cheshire Cat more transparent (Reference: Alice's Adventures in Wonderland).

The revised part of the paper is a new paragraph that includes equations (54) to (62). In fact, let me reproduce the new derivation of the strong correlations to show how beautiful it is. What is calculated in the derivation of the expectation value E(a, b) below is correlations between the limiting scalar points, A = +/-1 and B = +/-1, of a quaternionic 3-sphere (i.e. S^3):

A line-by-line explanation of this derivation is given in the paragraph that includes it. There are two differences between this derivation and all of the previous derivations of the strong correlations I have presented. While all derivations are based on imposing the law of conservation of spin angular momentum by setting s1 = s2, in this derivation the two limits s1 --> a and s2 --> b are actually carried out rather than made superfluous. The consequence is that we end up with a null bivector in eq. (59) instead of a bivector over the direction a x b. Thus the Cheshire Cat disappears already in eq. (59). Only its ghostly grin remains. But, as we can see from eqs. (59) to (62), even the cat's grin is wiped out in the end, thanks to the fact that handedness of the 3-sphere is necessarily a fair coin, which is the hidden variable in the model, and therefore it must be summed over in the calculation of the expectation value E(a, b).

***

I have revised my pedagogical paper once again to make the above comments regarding the disappearing Cheshire Cat more transparent (Reference: Alice's Adventures in Wonderland).

The revised part of the paper is a new paragraph that includes equations (54) to (62). In fact, let me reproduce the new derivation of the strong correlations to show how beautiful it is. What is calculated in the derivation of the expectation value E(a, b) below is correlations between the limiting scalar points, A = +/-1 and B = +/-1, of a quaternionic 3-sphere (i.e. S^3):

A line-by-line explanation of this derivation is given in the paragraph that includes it. There are two differences between this derivation and all of the previous derivations of the strong correlations I have presented. While all derivations are based on imposing the law of conservation of spin angular momentum by setting s1 = s2, in this derivation the two limits s1 --> a and s2 --> b are actually carried out rather than made superfluous. The consequence is that we end up with a null bivector in eq. (59) instead of a bivector over the direction a x b. Thus the Cheshire Cat disappears already in eq. (59). Only its ghostly grin remains. But, as we can see from eqs. (59) to (62), even the cat's grin is wiped out in the end, thanks to the fact that handedness of the 3-sphere is necessarily a fair coin, which is the hidden variable in the model, and therefore it must be summed over in the calculation of the expectation value E(a, b).

***

- Joy Christian
- Research Physicist
**Posts:**2187**Joined:**Wed Feb 05, 2014 4:49 am**Location:**Oxford, United Kingdom

***

By the way, the entire second term in eq. (60) is a null bivector. And, just like a null vector, a null bivector is an additive identity. Therefore step (61) is not needed for the derivation of the strong correlation. In other words, eq. (60) is already equal to eq. (62). And with that, all the hullabaloo of a "sign error" by some unscrupulous critics of my work goes down the drain.

***

By the way, the entire second term in eq. (60) is a null bivector. And, just like a null vector, a null bivector is an additive identity. Therefore step (61) is not needed for the derivation of the strong correlation. In other words, eq. (60) is already equal to eq. (62). And with that, all the hullabaloo of a "sign error" by some unscrupulous critics of my work goes down the drain.

***

- Joy Christian
- Research Physicist
**Posts:**2187**Joined:**Wed Feb 05, 2014 4:49 am**Location:**Oxford, United Kingdom

Joy Christian wrote:

By the way, the entire second term in eq. (60) is a null bivector. And, just like a null vector, a null bivector is an additive identity. Therefore step (61) is not needed for the derivation of the strong correlation. In other words, eq. (60) is already equal to eq. (62). And with that, all the hullabaloo of a "sign error" by some unscrupulous critics of my work goes down the drain.

I have updated the paper on the arXiv to make the above point clear in the derivation. In the updated version 6, I have added the following two lines in the paragraph just below eq. (62):

***

- Joy Christian
- Research Physicist
**Posts:**2187**Joined:**Wed Feb 05, 2014 4:49 am**Location:**Oxford, United Kingdom

***

The story goes that Bertrand Russell, in a lecture on logic, mentioned that in the sense of material implication, a false proposition implies any proposition.

A student raised his hand and said: "In that case, given that 1 = 0, prove that you are the Pope."

Russell immediately replied:

"Add 1 to both sides of the equation: then we have 2 = 1. The set containing just me and the Pope has 2 members. But 2 = 1, so it has only 1 member; therefore, I am the Pope."

The moral of the story is that you can prove anything and claim anything about anything if you start out with an equation that is manifestly wrong in that its LHS is not equal to its RHS.

So don't forget to check your starting equation before making a fool of yourself.

Just saying.

***

The story goes that Bertrand Russell, in a lecture on logic, mentioned that in the sense of material implication, a false proposition implies any proposition.

A student raised his hand and said: "In that case, given that 1 = 0, prove that you are the Pope."

Russell immediately replied:

"Add 1 to both sides of the equation: then we have 2 = 1. The set containing just me and the Pope has 2 members. But 2 = 1, so it has only 1 member; therefore, I am the Pope."

The moral of the story is that you can prove anything and claim anything about anything if you start out with an equation that is manifestly wrong in that its LHS is not equal to its RHS.

So don't forget to check your starting equation before making a fool of yourself.

Just saying.

***

- Joy Christian
- Research Physicist
**Posts:**2187**Joined:**Wed Feb 05, 2014 4:49 am**Location:**Oxford, United Kingdom

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