**In essence, quantum correlations are a measure of torsion within our physical space**.

(see also these two papers for more details)

If you know about teleparallel gravity, then it will be easier for you to understand the above statement and the rest of my argument. Fortunately, it is not essential to know about teleparallel gravity to understand why quantum correlations are a manifestation of torsion in our physical space. It is, however, important to understand what is meant by torsion in a smooth Riemannian manifold. A simply-connected space such as defined on the previous page is said to be absolutely parallelized (in the topological sense) if its curvature tensor (which, as a tensor, is defined *locally, *at every point *p* of ) vanishes identically with respect to the Weitzenböck connection

where is the symmetric Levi-Civita connection and is the totally anti-symmetric torsion tensor. This vanishing of the curvature tensor renders the resulting parallelism on absolute and guarantees the path-independence of any parallel transported vector within it. For a non-Euclidean space such as this is possible if and only if the parallelizing torsion in the space is non-vanishing:

Thus the non-Euclidean properties of the parallelized 3-sphere are encoded entirely within its torsion rather than a curvature. And as we saw on the previous page, it is this non-vanishing torsion within that is responsible for the strong quantum correlations between measurement events and occurring within it. Mathematically the measurement functions thus take the form

with their image points being

It turns out that this local-realistic framework can be generalized to reproduce *ALL* quantum correlations, provided we take the codomain of the above functions to be an absolutely parallelized 7-sphere instead of a 3-sphere:

In fact the choice of here is both necessary and sufficient to provide a *complete local account* of *ALL* quantum correlations. It may seem *ad hoc*, but it stems from the profound relationship between the normed division algebras and the parallelizability of the unit spheres. As is well known, the only spheres that can be parallelized absolutely are , , , and , and they correspond precisely to the four division algebras: , , , and . The 7-sphere is thus homeomorphic to the set of all unit octonions, and corresponds to the most general possible normed division algebra . It is therefore fitting that it also gives rise to the existence and strength of *ALL* quantum correlations. That is to say, quantum correlations exist and exhibit the strength they exhibit because the equality

holds only for the integers , and , for any generalized numbers **x**. Among the four parallelizable spheres, however, only and are non-trivially parallelized, and thus characterized by a non-vanishing torsion. Therefore, only with the choices and for the codomain of the measurement functions (4) can we reproduce the strong quantum correlations. Moreover, can be viewed as a 4-sphere worth of 3-spheres in the language of Hopf fibration (cf. this page). Thus, in general, we only need to consider , since is already contained within it as one of its fibres.

This last observation suggestes that for general quantum correlaions the form of the measurement functions (4) has to be generalized to admit an interpediate stage (or map) by giving it the form

where is a tangent space at a point of , spanned by vectors of the form , with and (explicit constructions of how this map works in practice can be found in this paper). Thus, although the actual measurement results are occurring in the familiar 3D tangent space of , their observed values are dictated by the trivial tangent bundle structure of the absolutely parallelized 7-sphere. In this sense, then, quantum correlations are the evidence of the fact that rotational symmetries respected by our physical space are those of a parallelized 7-sphere. It is important to note here, however, that, unlike the torsion within , the non-associativity of the octonionic numbers leads to torsion within that is necessarily different at each point *p* of :

This variability of torsion gives rise to the rich variety of quantum correlations we observe in nature.

Suppose now we consider an arbitrary quantum state and a corresponding self-adjoint operator in some Hilbert space , parameterized by an arbitrary number of local contexts etc. Note that I have imposed no restrictions on the state , or on the size of the Hilbert space . In particular, can be as entangled or unentangled as one may like, and can be as large or small as one may like. The quantum mechanical expectation value of the operator in the state would then be

where is a statistical operator of unit trace representing the state. As noted, I have imposed no restrictions whatsoever on the state, the observable, or the number of local contexts. Nevertheless, it turns out that—because the octonionic division algebra remains closed under multiplication—the quantum correlation predicted by this expectation value can always be reproduced as local and realistic correlation among a set of points of a parallelized 7-sphere, by following a procedure very similar to the one discussed on the previous page. This leads us to the following awesome theorem:

**Every quantum mechanical correlation can be understood as a deterministic, local-realistic correlation among a set of points of a parallelized 7-sphere, specified by maps of the form**

The proof of this theorem can be found in this paper as well as on the pages 13-17 of my book.

**The Raison D’être of Quantum Correlations:**

The above theorem demonstrates that the discipline of absolute parallelization in the manifold of all possible measurement results is responsible for the existence and strength of all quantum correlations. More precisely, it identifies quantum correlations as evidence that the physical space we live in respects the symmetries and topologies of a parallelized 7-sphere. There are profound mathematical and conceptual reasons why the topology of the 7-sphere plays such a significant role in the manifestation of quantum correlations. Essentially it is because 7-sphere happens to be homeomorphic to the most general possible division algebra. And it is the property of division that turns out to be responsible for maintaining strict local causality in the world we live in.

To understand this reasoning better, recall that, just as a parallelized 3-sphere is a 2-sphere worth of 1-spheres but with a twist in the manifold , a parallelized 7-sphere is a 4-sphere worth of 3-spheres but with a twist in the manifold . More precisely, just as is a nontrivial fiber bundle over with Clifford parallels as its linked fibers, is also a nontrivial fiber bundle, but over , and with entire 3-dimensional spheres as its linked fibers. Now it is the twist in the bundle that forces one to forgo the commutativity of complex numbers (corresponding to the circles ) in favor of the non-commutativity of quaternions. In other words, a 3-sphere is not parallelizable by the commuting complex numbers but only by the non-commuting quaternions. And it is this non-commutativity that gives rise to the non-vanishing of the torsion in our physical space. In a similar vein, the twist in the bundle forces one to forgo the associativity of quaternions (corresponding to the fibers ) in favor of the non-associativity of octonions. In other words, a 7-sphere is not parallelizable by the associative quaternions but only by the non-associative octonions. And the reason why it can be parallelized at all is because its tangent bundle happens to be trivial:

Once parallelized by a set of unit octonions, both the 7-sphere and each of its 3-spherical fibers remain closed under multiplication. This, in turn, means that the factorizability or locality condition of Bell is automatically satisfied within a parallelized 7-sphere. The lack of associativity of octonions, however, entails that, unlike the unit 3-sphere [which is homeomorphic to the spinor group SU(2)], a 7-sphere is not a group manifold, but forms only a quasi-group. As a result, the torsion within the 7-sphere continuously varies from one point to another of the manifold. It is this variability of the parallelizing torsion within that is ultimately responsible for the diversity and non-linearity of the quantum correlations we observe in nature:

The upper bound on all possible quantum correlations is thus set by the maximum of possible torsion within the 7-sphere:

This reaffirms that **quantum correlations are a measure of torsion within our physical space**.

Dear Joy,

I’ve spent the last couple of weeks reading about your work. I definitely find it intriguing, but I’m still not understanding some technical details; I just want to ask here about one particular problem that has been bugging me.

In the papers “Restoring Local Causality and Objective Reality to the Entangled Photons” and “Failure of Bell’s Theorem and the Local Causality of the Entangled Photons” (which look to me fundamentally arguing the same things) you use the expression A (α, µ) = (− I · ã ) ( + µ · ã ). The definition of ã is not consistent within the two papers, in one it is ã=ex cos 2α + ey sin 2α while in the other it is defined as ã=ex sin 2α + ey cos 2α.

I’m having a really hard time understanding where these definitions for ã come from; it looks like an ad hoc expression to be able to obtain cos 2(α − β) in the final limit, and is not further justified in the paper. Normally, a refers to the polarization direction chosen by Alice lying in the exey plane at the angle α from ez; it can be expressed as excosα + eysinα in vector form. What does ã stand for, and how has it been derived?

All the best,

Sandra

Hi Sandra,

Thank you for your interest in my work and for putting in the effort to understand it.

The reason for the difference in the definitions of ã in the two papers is because of the different polarization states used in them — see equation (15) in both papers. The quantum state (15) used in the paper “Failure …” is EVEN under reflections whereas the quantum state (15) used in the paper “Restoring …” is ODD under reflections. Since the goal of the papers is to reproduce the correlations predicted by these quantum states, it is necessary to parameterize the 2-sphere differently without changing the requirements set by Bell for the definitions of the measurement functions A (α, µ) = (− I · ã ) ( + µ · ã ). In other words, we are free to parameterize the 2-sphere differently using different definitions of the vector ã as long as the polarization angle α is not affected and can be freely chosen by Alice to make her measurements. Note that the result A (α, µ) = +/-1 on the left-hand side of the above equation is not affected by the different definitions of ã. And that is all that matters to satisfy the requirements set by Bell.

I hope this explanation helps.

All the best,

Joy

Thank you Joy for the fast reply. I understand now the reason for the difference, but it’s still nebulous to me the physical significance of ã. Had we used any other parametrization we would not have obtained the right correlation between measurements. But in actual experiments all we can choose are the angles α and β. What physically prevents us to use a different parametrization? My question might have an obvious answer, I’m just not yet familiar with the intricacies of geometric algebra.

The question here is not about geometric algebra but about (1) what freedom Bell’s theorem allows us in the definition of the measurement function A (α, µ) = +/-1, and (2) how the coordinates are set up by the experimentalists in the Bell-test experiments involving photons?

The answer to the first question is that Bell’s theorem allows us *complete* freedom in the definition of measurement functions. Bell says that we can choose whatever functions we like, however complicated (or simple) we like. The only requirement he sets is that the left-hand side of the functions must depend *only* on the hidden variable µ and the chosen experimental setting α, nothing else. Thus you are needlessly worried about the specific choice of the expression I have made on the right-hand side in defining A (α, µ) = (− I · ã ) ( + µ · ã ).

But you may ask: Why have I made that particular choice? The answer is: To match with the coordinates and settings used by the experimenters in those specific experiments. If you make a different choice for ã, for example, then you would still be able to derive the cosine correlations, but they may not match with the settings α. You may end up with something like -cos(a, b), where a and b vectors would be your new choice of vectors. Then, for you, the task would be to match with the coordinates used by the experimentalists. You would then end up with my choice for ã.

Thank you very much again Joy. It’s a great feeling when researchers are so easily available for the general public questions. In case I have more doubts that I don’t find easily answered in your other reply papers, would it bother you if I were to ask away? Would this blog be appropriate, or would you rather other online addresses? In any case, good luck with your research program. Even if it doesn’t turn out to be the right path forward, we need new ideas like yours.

Sandra

PS: I also read “Completing the Theory of Electron with Gravitational Torsion”. I’m not sure whether it might be of any interest to you, but I once read some interesting (albeit VERY speculative) ideas about the electron structure. You might want to check “Is the electron a photon with Toroidal Topology?” by Williamson and Van der Mark. Maybe not quite toroidal, maybe more like an Hopf fibration.

You are welcome, Sandra. I will be happy to answer any reasonable questions. This blog is a good place for discussion. That way other people may also benefit from your questions.

Hi again. It wasn’t too long until I found some other thing that escapes my comprehension.

In the original one page paper “Disproof of Bell’s Theorem” and subsequent more in-depth “On the Origins of Quantum Correlations”, you differentiate between raw and standard scores. It is my understanding that these correspond respectively to actual measured values (i.e. the detector outputs, as a raw score) and the true value of the measurement (the standard score). Is this a correct intepretation? You reason this differentiation is necessary because the two measurements have different standard deviations, and this is because each score “depends

on different units and scales (or different scales of dispersion) that

may have been used (advertently or inadvertently) in the measurements of such scores.” Why is that? Is it because for each measurement the detector bivector D(n) is different?

In a more recent paper (“Bell’s Theorem Versus Local Realism in a Quaternionic

Model of Physical Space”) you calculate the correlation function using the quaternionic spin values s1 and s2 instead. This paper makes no mention of standard deviations. Is this because the derivation here corresponds to using the “true value” of the measurement, i.e. the standard score, automatically?

Thank you, Sandra, for further questions.

There are several different ways to compute correlations within S^3. In the original one-page paper, correlations are computed using Pearson’s correlation coefficient, albeit applied to bivectors within geometric algebra. You can find Pearson’s method in this Wikipedia page:

https://en.wikipedia.org/wiki/Pearson_correlation_coefficient

This method, applied within S^3, is explained in more detail after Eq. (104) in this paper:

https://arxiv.org/abs/1211.0784

In your post above, you have understood my reasoning correctly.

In the recent paper published in IEEE Access, on the other hand, the correlations are computed using the phenomenological method used by the experimentalists. This method, as you note, involves the standard scores automatically, by construction.

Theoretically, both methods are equivalent and give the same results for the correlations.

Hi again Joy.

So, after a bit more reading I came to the paper “Dr. Bertlmann’s Socks in the Quaternionic World of Ambidextral Reality”. I really liked this paper because it shows a very clear example in 2D of the basic idea. Still, I recognize that this example is not exactly a perfect analogy for the 3-sphere case. In particular, I find it hard to picture what exactly it means for our physical space to be “parallelized”, or having the topology of a 3-sphere in the first place. I was always taught that the overall intrinsic curvature of the universe reflects on parallel rays staying parallel or not as they travel, or alternatively about the angles of a big enough “triangle” exceeding 180°. The entanglement results are supposed to be caused by the topology of our space, but these are performed locally, how can the curvature, which locally is indistinguishable from flat, have any effect?

Another thing that escapes my intuition is about the Hopf fibration, and how it relates to the physical experiments. I understand how, in the words of Eric Weinstein, the map S3 to S2 effectively corresponds to little circles at each point in spacetime. These circles (their angle, 2π), in your paper, are assigned to the angle formed by the measurement vector (a or b), which is decided by Alice and Bob, and the spin bivector s. In this sense I’m picturing that each point on the sphere effectively has a phase attached to it, going from 0 to 2pi. Then, the angle between a and b is assigned to the equatorial angle of the sphere. I’m failing to picture how in my mind the “phase” cycles at each point as the equatorial angle goes from 0 (where the 2 phases, or Hopf fibers, are identical) to π (where they get a – sign), and how physically this happens as I rotate the detectors/polarizers. I’m also failing to understand what the vertical angle stands for: for the equatorial angle 0, all the points between 0<rho<π are supposed to have the same twist in the fiber; their projections though are obviously different, being larger and larger circles as rho gets smaller.

On another note, have you thought about appearing on famous physics podcasts like Brian Keating's or Kurt Jomungal? The latter I think would be delighted to have you. This would help a lot getting your ideas out there.

Hi Sandra,

The concept of parallelizability of space (or manifold) has nothing to do with Euclid’s parallel postulate, or parallel rays, or triangles exceeding 180°. It has to do with the possibility of assigning a consistent sense of handedness throughout the space. The toy example of the 2D Mobius strip is a good counter-example of this. On a Mobius strip, we cannot consistently assign handedness throughout the space. Because of the twist in the space, the handedness changes from left to right or right to left as we go around the strip. By contrast, on the 3-sphere the quaternions that constitute it assign a consistent sense of orientation or handedness throughout the sphere, without exception — i.e., without a single point left out to have a different handedness from the rest of the points. In the technical language, this means that a 3-sphere is parallelizable. And the very quaternions that constitute the 3-sphere, parallelize it.

I am afraid it is not easy to develop intuition about the geometry of the 3-sphere. Your understanding of the Hopf fibration, namely, that a circle, S1, is attached to each point of the base space S2, is correct. However, it is not easy to visualize how the phase angle of each quaternion “conspires” to produce the strong correlations we observe in the experiments. Unlike in the 2D example of the Mobius strip, for the 3-sphere it is necessary to follow the mathematics of the quaternions to understand what is going on. In this regard, perhaps my latest paper may help, because in it I work out the mathematics in more detail. See, especially, Section V of the paper:

https://arxiv.org/pdf/2204.10288.pdf

Some months ago Kurt Jomungal had approached me about appearing on his podcast to explain my model. But then he lost interest for some reason. I guess he found my model too difficult or too controversial for his podcast.

Thank you for your answer Joy.

I have used this online tool (https://philogb.github.io/page/hopf/#) to visualize 2pi rotations of a fibration. Indeed, the motion of the circle is such that any point on the fibration will find itself 180 degrees from its starting position, after a 2pi rotation on S2.

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I’m still not sure how detectors can differentiate between the two signs: might it be that it’s because they use Stern-Gerlach apparatuses? The relationships in the electromagnetic field behave eerily similar to quaternion rotations, like the current in a wire generating a magnetic field with the right hand rule. So basically we’re testing handedness with another intrinsically chiral apparatus.

.

I find your math completely correct (it’s much easier to follow with L(s) instead of lambda). What I struggle with is what it all means. Spinors are not that hard to visualize if you use belts or rocks attached to ropes. But we’re talking about electrons here; they aren’t belts nor rocks with ropes. It’s clear that there is something rotating about. Could it be space itself? I can visualize a grid moving like in this picture (https://commons.wikimedia.org/w/index.php?title=Special:ListFiles/JasonHise&ilshowall=1#/media/File:Anti-Twister.gif). After all, the only way to continuosly rotate a 3D elastic medium without breaking it (or without the elastic tension going to infinity) is exactly the spinorial motion. Would you agree?

.

It’s unfortunate Kurt never got through with interviewing you. People on the net immediately dismiss your work because they heard some “authorities” like Aaronson or because someone (cough Gill cough) said you violated some Hurwitz theorem, while it’s clear to me that you adressed those criticism perfectly well. In any case, you have my support!

.

PS: did you ever manage to perform the experiment with the exploding balls? It doesn’t sound that expensive. But again, I’m not an experimentalist.

Thank you, Sandra, for your support and interest in my work. As you can see, considerable effort is required to understand the 3-sphere model based on quaternions. So, thank you for investing your time to try to understand it.

The detectors in the 3-sphere model are modeled by bivectors D(a) and D(b). These bivectors are the counterparts of the vectors `a’ and `b’ used by Bell in his local model within R^3. The actual experiments indeed use Stern-Gerlach apparatuses to detect spins. But in a theoretical model like Bell’s, only the directions of the Stern-Gerlach apparatuses are relevant. In the 3-sphere model, the vectors `a’ and `b’ translate to bivectors D(a) and D(b). The handedness of these detector bivectors is fixed by Alice and Bob for the course of their experiment. The spin bivectors, L(s1) and L(s2) emerging from the source, whose handedness is random (i.e., not fixed by Nature), then interact with the detectors D(a) and D(b), respectively. The mathematics of bivectors then dictates that the resulting values of the spins will be either +1 or -1, as required by Bell and the Bell-test experiments. So, yes, we are measuring the spin handedness with an intrinsically chiral apparatus, because that is precisely the nature of the 3-sphere geometry.

Note also that, as with the belt or rock example, electors in a singlet state are not isolated. They come in pairs. And even though their spins are detected at spacelike separation, the electrons are correlated with each other. However, there is nothing non-local about this correlation. It is like a correlation between a pair of gloves. If you find only one glove in your pocket and it is right-handed, then you instantly know that the one you forgot at home is left-handed. In other words, the two gloves are correlated. It is the same with a pair of electrons in a singlet state. But that correlation also means that the electrons are rotating with respect to each other. So, the “belt” in this case is the preexisting correlation between the electrons, like that between the glove in your pocket and the one you forgot at home. More locally, the spinning electron L(s) is spinning with respect to the detector D(a) of fixed-handedness. Thus, the situation here is no different from that with a belt or a rock. It is just that a “belt” in this case is more abstract.

The experiment you mention would indeed be rather inexpensive. However, I am also not an experimentalist, so I can’t perform the experiment myself. Besides, it must be performed independently by some well-known experimentalists for the community to take the results seriously. Unfortunately, the negative propaganda about my work by the non-physicists you mention has so far prevented any serious experimentalists from taking an interest in my work or my proposed experiment.

Another thing I’m not sure I understand, that you might have already clarified in some other paper. In the paper you last linked me “Symmetric derivation of singlet correlations in a quaternionic 3-sphere model”, you say:

− L(s1) + L(s2) = 0 ⇐⇒ L(s1) = L(s2)

⇐⇒ s1 = s2 ≡ s.

Because of conservation of angular momentum. You use this notation in other papers as well, but shouldn’t it be L(s1) + L(s2) = 0 ⇐⇒ L(s1) = – L(s2) ? S is the spin direction, and it makes sense for both particles to have the same direction, but shouldn’t it be of opposite sign? I might be getting confused here.

.

Later in the paper you define the measurement functions

.

S3 ∋ A (a, si1) = lim s1 → µ1a {− D(a)L(s1)}

and

S3 ∋ B(b, si2) = lim s2 → µ2b {+ L(s2) D(b)}

.

I don’t quite understand why the two have a different sign inside the parenthesis. It was established L(s1) = L(s2) so the sign can’t come from L(s). D(a) and D(b) are freely chosen bivectors.

This one is easy to sort out. The sign convention for spins is chosen to respect the fact that unit bivectors square to -1 in Geometric Algebra. As shown in Figure 1 of the “Symmetric” paper you mention, the two spins are rotating in the opposite senses of each other, but about the same vector s1 = s2 = s. Now, instead of what I have chosen in the paper, let us set L(s1) + L (s2) = 0 as you have written. Then we indeed have L(s1) = -L(s2), as you say. But then the product of the two spins will give -L(s)L(s) = -(I s)(I s) = +1, because the squared bivector (I s)^2 = -1 in Geometric Algebra. Here `I’ is the standard trivector. Thus, the square of the two spin momenta gives the wrong sign. I have preferred the sign convention for spins to be consistent with the standard use of minus sign for the square of the unit bivectors in Geometric Algebra.

The signs in the definitions of the measurement functions also come from how I have defined the spins for Alice and Bob. I have denoted the spin approaching Alice’s detector D(a) to be -L(s1), and the spin approaching Bob’s detector D(b) to be +L(s2). That is where the signs in the measurement functions come from. Also, the functions A and B must be chosen so that for the perfect correlation, i.e., when a = b, the product A(a)B(a) must be equal to -1. This requirement comes from the definition of the singlet state as well as the original EPR-Bohm argument. If we drop the sign difference from the measurement functions then the requirement A(a)B(a) = -1 is not respected.

Thank you again Joy. Yes that sorted it out, it makes sense: the sign is purely a convention, and here it’s preferable to make it explicit.

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It’s wild to me your work is getting ignored like this. You know, this journey for me started around 2 years ago when I posted on a physics forum a question about bell’s theorem and how it looked like we were missing something about the detectors themselves. I still didn’t quite have clear the whole issue, but I got surprisingly close (detectors are bivectors!). I also mentioned that it did not make sense to use counterfactuals to derive the inequality. I was met with extreme dismissal, the thread was locked before I could reply, and subsequent posts with questions on unrelated things had as first answers something like “oh you’re the nut that posted about bell’s theorem this is going to be long” . I’m usually very self aware about what I think I know; especially in physics, whenever there is some talk about very complicated stuff like gauge theory, QCD, yang-mills, symmetry breaking… I always tell myself “it’s not so easy. You’re missing so much before you can even comment on these things”.

.

But when I look at your work, I can follow it. It is logically sound. I look around for criticisms, because I remind myself “it can’t be so easy!”, but all I see are people completely missing the point. The SAME people that do complicated stuff like gauge theory, yang-mills, etc. are missing the point. It’s disorienting. I’m feeling pretty confident you are right, yet I can’t shake this uncertainty about everything else. The best hidden things are those right under our noses I guess.

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Sorry for rant, and thank you again Joy.

You are welcome, Sandra.

Yes, the reaction of some members of the community to my work is surprising. It seems that Bell’s theorem has become a kind of religion for some people. They get offended if you tell them that it is based on mistaken assumptions and that you can demonstrate that. Instead of people showing interest in my work, I have encountered much abuse, knee-jerk reactions, and ostracization from some members of the Bell community. But slowly some people in the community (like yourself) have begun to take a positive interest in my work. So there is hope that, eventually, people will understand what I have been saying.