In previous pages I have sketched the crucial role played by the 3- and 7-dimensional spheres in understanding the existence of quantum correlations. What is so special about 3 and 7 dimensions? Why is the vector cross product definable only in 3 and 7 dimensions and no other? Why are , , , and the only possible normed division algebras? Why are only the 3- and 7-dimensional spheres nontrivially parallelizable out of infinitely many possible spheres? Why is it possible to derive all quantum correlations as local-realistic correlations among the points of only the 7-sphere?
The answers to all of these questions are intimately connected to the notion of factorizability introduced by Bell within the context of his theorem. Mathematicians have long been asking: When is a product of two squares itself a square: ? If the number is factorizable, then it can be written as a product of two other numbers, , and then the above equality is seen to hold for the numbers , , and . For ordinary numbers this is easy to check. The number 8 can be factorized into a product of 2 and 4, and we then have . But what about sums of squares? A more profound equality holds, in fact, for a sum of two squares times a sum of two squares as a third sum of two squares:
There is also an identity like this one for the sums of four squares. It was first discovered by Euler, and then rediscovered and popularized by Hamilton in the 19th century through his work on quaternions. It is also known that Graves and Cayley independently discovered a similar identity for the sums of eight squares. This naturally leads to the question of whether the product of two sums of squares of different numbers can be a sum of different squares? In other words, does the following equality hold in general for any ?
As I noted on this page, it turns out that this equality holds only for = 1, 2, 4, and 8. This was proved by Hurwitz in 1898. It reveals a deep and surprising fact about the world we live in. Much of what we see around us, from elementary particles to distant galaxies, is an inevitable consequence of this simple mathematical fact. The world is the way it is because the above equality holds only for = 1, 2, 4, and 8. For example, the above identity is equivalent to the existence of a division algebra of dimension over the field of real numbers. Indeed, if we define vectors , , and in such that are functions of and determined by equation (2), then
Thus the division algebras (real), (complex), (quaternion), and (octonion) we use in much of our science are intimately related to the dimensions = 1, 2, 4, and 8. Moreover, from the equation of a unit sphere,
it is easy to see that the four parallelizable spheres , , , and correspond to = 1, 2, 4, and 8, which are the dimensions of the respective embedding spaces of these four spheres. What is not so easy to see, however, is the fact that there is a deep connection between Hurwitz’s theorem and the quantum correlations. As we saw in the previous sections, all quantum correlations are inevitable consequences of the parallelizability of the 7-sphere, which in turn is a consequence of Hurwitz’s theorem. So the innocent looking algebraic equality (2) has far reaching consequences, not only for the entire edifice of mathematics, but also for that of quantum physics:
I recently became interested in Garret Lisi’s E8 theory. There are a lot of commonalities between that and your octonion sphere model. What caught my attention is his description of spin as “gravitational charge” (referring to cartan theory), and the higgs field basically being the torsion of spacetime. Ever looked into it? I admit I have like 10% of the required math knowledge to look at it beyond his descriptions in a ted talk, so perhaps before diving into it more and start researching the stuff i need to know i was wondering if you have any thoughts on that, if at all.
Thank you for all your answers Joy.
I have met Gerret Lisi. I was at the 2011 FQXi conference where he gave a talk on his E8 theory. His work is interesting, but I haven’t spent time on it beyond listening to his talk at that conference because my focus is elsewhere. I don’t know how much Geometric Algebra Lisi uses, but David Hestenes was also present in his FQXi talk, and Lisi gave credit to Hestenes for inspiration. Sabine Hossenfelder was also at that FQXi conference and, if I recall correctly, she has made a video about Lisi’s theory, which may help you.
Dear Joy,
quick question. In the paper “Bell’s Theorem Versus Local Realism in a Quaternionic Model of Physical Space” I don’t understand how you derived eq. 25. In particular I don’t understand how exactly the angles ne and zs are assigned or what their relationship is, as if we set such angles both to pi/2 (with k=+1), or some interval around pi/2 (like 80-100°) the norm of |P+Q| becomes larger than 2, which is not allowed since on S3 this norm is bounded by 2.
Eq. 25 is not derived. It is postulated. It is an ansatz. I have made it up, because it works.
The angles ne and zs are constrained by the inequality 28, which follows from the triangle inequality 23. Thus, ne and zs can take any angular values subject to the constraint 28. As long as the inequality 28 is satisfied, the norm will not exceed the value of 2.
I see. Is there an intuitive geometric idea behind ne and zs? In your paper you mention n can be thought of as a measurement direction, z as a “reference” vector at another point of S3 (which i personally think of as either i,j or k), and e and s are related to the spin system. In particular you mention that s can be thought of as the vector dual to the bivector Is representing the spin of the combined pair, but isn’t that bivector by definition 0 since angular momentum is conserved (e1=e2)? Also I’m struggling to geometrically relate all of this to the fact that certain angles are essentially forbidden.
You are struggling because it is not an elegant approach like that based on the even subalgebra of the Geometric Algebra Cl(3, 0) or a quaternionic 3-sphere. I used that “ugly” approach in that paper to make contact with Pearle’s non-Clifford-algebraic approach [ see Ref. 34, where I cite his famous paper ], which allows numerical simulations of the singlet correlations.
No angles are forbidden by the constraint 28. What is forbidden are certain *relations* between the angles ne and zs. If these relations are admitted, then they would throw us outside the geometry of the 3-sphere, because including them would violate the triangle inequality 23, which must be respected to respect the geometry of the 3-sphere.
By “composite pair” in the paragraph above Eq. 23, I mean just “spin-1 and spin-2” regardless of the net values of their spins, not their singlet state defined in Eq. 9. The net angular momentum of the two spins in the singlet state would indeed be zero.
Dear Joy,
i studied more carefully your “Symmetric” paper. In it, you show the example that on a moebius strip, the intrinsic angle Nas is related to the extrinsic angle (Bas) by B=pi(1-cosNas). You also derived the linear correlation using B as E(a,b) = -1 + B/pi.
.
Then, I noticed that the linear correlations derived by Bell in his Toy model has the form
E(a,b) = -1 + 2Alpha/pi. Obviously there’s a very clear similarity here between the two correlations, with B = 2Alpha. Tell me if this is the correct way to think about it:
.
If we assume R3, there’s no twist, and the correlation is the linear one of Bell. If we assume S3, there’s a twist, then we need to relate Theta to the intrinsic twist angle Nas. The factor of 2 comes from the double covering of SO(3). So doing the substitution Alpha = pi/2 (1-cosNas) we obviously end with E(a,b) = -cos(Nas).
.
There’s more. The relation psi=pi(1-cos(theta)) derives from the holonomy on the Hopf Projection on S2, with psi is the phase on the S1 fibers after a full loop and theta is the polar angle on S2 where the loop is constructed. So if we make theta=pi/2, we’re at the equator, where the twist becomes a full pi angle, exactly like in the moebius strip. This justifies making the substitution psi = Nas as an intrinsic angle. Then from the relation alpha = pi/2(1-cosNas) it becomes clear that Nas is equal to alpha, that is, we can relate the angle between detectors with the angle in the twist of S1! Hence,the non-linear correlations!
.
I think this makes a lot of sense, but I wanted to ask you if my understanding is indeed correct.Months thinking about S3 left me in the clouds a bit too much.
I think you are talking about “Dr. Bertlmann’s Socks” paper, not the “Symmetric” paper.
Your understanding above is essentially correct, as long as you keep in mind that the toy example is not the real thing. The Mobius strip is not a 3-sphere. At best, it is just one of the S^2 worth of circles that make up the 3-sphere. What is more, the Mobius strip is not *orientable*. A consistent sense of handedness cannot be specified on a Mobius strip because there is a twist in it. By contrast, a 3-sphere is orientable and can be parallelized to specify a consistent sense of handedness over the entire sphere, without ever having to change the “hand.” As long as you keep in mind that the Mobius toy example only provides an analogy to help our intuition, you will be fine.
Happy Easter Joy!
Why exactly is the orientability important here? A cosine correlation can be derived regardless. I know orientation is your hidden variable Lambda, but even that drops when the two measurement functions are multiplied.
Happy Easter! We do not live in a 2-dimensional, one-sided world of a Mobius strip. As I have argued, we live in a 3-dimensional world of a 3-sphere, which differs from a flat Euclidean space R^3 only by a single mathematical point at infinity. Moreover, an ordinary unit 3-sphere defined by the equation w^2 + x^2 + y^2 + z^2 = 1 and embedded in R^4 cannot be charted by a single coordinate system without encountering a singularity somewhere, usually chosen to be at the north pole. However, a 3-sphere is parallelizable with quaternions, which renders it a flat, orientable set of unit quaternions. Meaning, unlike the ordinary 3-sphere, a quaternionic 3-sphere can be charted by a single coordinate system with a fixed handedness without any discontinuities, singularities, or fixed points. This is important because excluding a single point (say, at the north pole) would render it topologically equivalent to a flat Euclidean space R^3 — S^3 minus a point is equal to R^3. And we know that measurements within R^3 cannot reproduce strong quantum correlations, as proved by Bell’s theorem. Thus, orientability is a necessary and sufficient criterion for a 3-sphere to be able to reproduce strong quantum correlations.
Joy,
Reading through the pages of your blog I’ve seen you also used a different derivation for the triangle inequality, where you derive |cos(ae)| >= 1/2 sin^2(theta). Now this to me is a more rigorous result, as this function itself does constraint the sum of two quaternions to 2 for all angles.
.
So I’ve tried simulating it in excel, and indeed the result is the correct correlation. But I was wondering how I should interpret geometrically this constraint. I realize it essentially relates “projected” quantities (vectors instead of bivectors) to the geometry of a 3-sphere. But is there a direct correspondence to, for example, the Hopf projection? Could theta be a polar angle on the base space S2, and ae0 the angle on the fiber? Or perhaps some other interpretation?
.
Another question as well. The initial state (e0, theta) gives the final measurement result as sign(cos(ae0)). But for some angles theta this “metric” returns zero, so you say the state for which the metric is zero does not exist on S3. But of course for a given (e0, theta) whether the metric is zero or not depends on the measurement direction a, and as this varies different states are said to “not exist”. How should I interpret this geometrically, perhaps related to the projection picture? For a given theta, why are some angle relationships between e0 and a constrained? For example, you mention that at theta=pi/2 most of e0 and a are orthogonal to each other, but if we sub in the constraint this value we get |cos(ae)| = 0.5, which means ae=60°.
The approach with the constraint |cos(ae)| >= 1/2 sin^2(theta) was phenomenologically motivated. Its purpose was to produce numerical agreement with the observed correlations. It is pointless to force a geometrical interpretation on such a crude approach, or compare it with the elegant approach using a Hopf fibration of the 3-sphere.
Joy,
I don’t think it is pointless. Without such a geometrical interpretation people won’t stop accusing it to be simply a detection loophole model. It’s not enough to just say “it follows the topology and geometry of the 3 sphere” without directly creating a correspondence between them. At least, that is what I’m concerned with since I want to make it into a video, but I’ve been stuck on these details for months.
In Section III G of the following paper (open access), I discuss what is wrong with Pearle’s detection loophole model, and then make a detailed comparison with the quaternionic S^3 model:
In that section, you will see that the geometrical relation between the two models is quite involved and requires considerable effort to identify the flaw in the detection loophole argument. Most people like Gill are mathematically too incompetent to see the flaw in Pearle’s model. You, on the other hand, are one of the very few people who are making an effort to understand the S^3 model. I appreciate that because it is hard work. But, unfortunately, the geometrical and topological subtleties of the S^3 model are too difficult to explain in online discussions, such as in your effort to make a YouTube video about them.
Note also that the comparison of the two models requires a different constraint derived from the triangle inequality within S^3, not the one with the phenomenologically motivated sine function you have noted above.
Dear Joy,
one more question about the “Bertlmann” paper, which came to me after more scrutiny. In the toy model you state that the relation between the extrinsic, “cylindrical” angle on the strip and the intrinsic twist angle, seen from a cross-sectional view, is beta(ab) = pi(1-cos(nab)). But since the twist is uniformly distributed along the entire length of the strip, the actual relationship is linear, more specifically beta(ab) = 2n(ab).
.
The relation you write is an holonomy relation, but I’m not sure how you square the two together.
Angle is always defined in terms of the arc length of a circle, not a straight line. So, the relation beta(ab) = 2n(ab) you wrote down is incorrect. The correct relation between the two angles is beta(ab) = pi(1- cos(nab)).
Dear Joy,
sorry to bother you on the same issue again. I’ve been trying to find a derivation of the relation beta(ab) = pi(1- cos(nab)) for a moebius strip for months now, but i could not understand how it is done (despite your previous answer on the matter). Since this kind of formula is found when dealing with areas and curvatures, i tried looking up integrals for the same relation but for flat surfaces with torsion like the moebius strip, but they all give a linear result. Could you provide a resource to understand how to derive this relation, on the strip specifically? In your paper you simply state “for the well known properties of the moebius strip” which unfortunately wasn’t much of help for me.
I am not sure why you are wasting so much time on that toy example. It is not a model for the singlet correlations observed in the real world. The purpose of the toy example is to give us an intuitive understanding of a characteristic feature of the geometry of the quaternionic 3-sphere in the real world. I have warned in my papers not to take the toy model too seriously. In particular, the relation beta(ab) = pi(1- cos(nab)) cannot be derived for a Möbius strip. It postulates a physical characteristic of the toy world that a two-dimensional Alice and Bob are assumed to inhabit. That is why I have stressed that the relation beta(ab) = pi(1- cos(nab)) is the defining relation of the Möbius world of Alice and Bob. It has to be accepted, not derived, for the illustration to work.
Dear Joy,
the reason being that I want to understand the reasoning in your papers. A toy model is supposed to present a simplified, yet not necessarily comprehensive or even widely applicable, version of a model. If I understood the toy model completely, then I would feel more confident trying to understand the whole model, wary of the limitations of the toy model itself. The problem with just giving out mathematical relationships ad hoc without fully justifying them, even on a matter of principle, leaves more questions than answers, which defeats the point of creating a toy model. I’m not sure why you question me questioning the derivation, I just didn’t understand it. Should I be held at fault for not understanding “defining relation” as “we are just assuming this holds”?
Given that I’m much more of a visual person rather than a math one, I can now understand the relation pi(1- cos(nab)) as a torsion not uniformly distributed on the strip, centered on the midpoint between a and b. But that seems to mean the choice of a influences the possible choice of b, so by accepting that relation we are introducing non-locality in the strip (there’s no global distribution of torsion that works for all a and b). Perhaps you have objections here, but please don’t just say “ignore the toy model”. I’ve been passionately reading tons of posts on your work, by many people, and I understand most of your critics to be wrong on many points they make. Just to make an example, I immediately recognized that the “sign error” many of them talked about is in fact their misunderstanding of the idea behind the change in orientation. One thing they are not wrong about though, is that sometimes your writing style can be a bit cryptic, especially to those without years of experience on GA, topology and GR.
There is no deep issue to resolve here. By hypothesis, the relation beta(ab) = pi(1- cos(nab)) *defines* the toy geometry of the two-dimensional world Alice and Bob live in; just as, by hypothesis, we live in a three-dimensional geometry isomorphic to a quaternionic 3-sphere. The relation beta(ab) = pi(1- cos(nab)) is not a standard property of a Möbius strip. So, it cannot be derived. But there is no need for its derivation, since we are working within a hypothesised toy model. I think what you are missing here is not mathematics, but the logic behind the toy model.
Hello Joy,
I’ve come to terms with your toy model. Unfortunately, I don’t think it helps at all with anything, but that’s just my very humble opinion. After reading and reading, ive come to the conclusion that the most fundamental thing in all of this matter about Bell’s theorem, QM correlations and so is a single property of S3, that it is a non trivial bundle. The central argument for which Bell’s inequalities are considered inapplicable, here and on many posts on sciphysicsforum, is the impossibility of measuring different settings on the same particle pair. This in turn means the outcomes corresponding to the same setting cannot be factorized through different integrals, thus making deriving the classical bound impossible. Another way that this argument was put in was that the additivity of expectation values is not respected across different settings.
To me, all these arguments are really just one argument, that the distribution p(lambda) is different for different setting pairs, preventing the existence of a global joint distribution valid for all settings. This is exactly in line with some critiques by Karl Hess, and Fine’s result (whereby the existence of such a distribution = validity of the CHSH inequality).
I’ve also learned that, through yours and tsirelson’s equivalent derivation, that non commuting observables are what cause an obstruction in QM to the existence of a global joint. In particular, when viewed as a topological obstruction, no global joint = no global section on the outcome space. I’ve also learned that a non trivial bundle provides exactly such a lack of global section. This mirrors the simple fact that the same path in SO(3) can lift to two different paths on SU(2). From the point of view of R3 outcomes really do seem to depend on the distant setting (they look context dependent), but that is simply because we ignore the full specification of the state, which must include the relative points on the fibers for the same bloch vector. Likewise, we can’t know which SU(2) path was taken from a single rotation; but we can infer it through comparison of two different rotations of the same state. The response functions depend on the lift. Thus, a joint probability on the base implicitly pretends that all outcome-determining variables are functions of basepoints only; if the true ontic dependency lives in the fiber (lift) and there is no global, consistent way to choose a lift over the whole base (no global section, which is guaranteed for a jon trivial bundle), then we cannot compress the model to a single, context-independent probability on the base that reproduces all lift-dependent statistics.
I’m not sure you’ll agree with my conclusion here; but this has been the clearest way for me to make sense of all I’ve read. I also believe that this helps with the conceptual issues I’ve had with your simulations. The “detector loophole” can likewise be framed in terms of differing joints for different settings; when elements of one distribution are used for a pair of settings which uses a different distribution, some of such elements may not be present in that distribution. Thus the need to specify a full list of “initial states” for setting pairs like in your simulation.
Overall, I think I will take a break from studying these matters. On one hand, im tired of the vitriol directed at both parties in the matter, which also has made it very hard to follow through with your proposed experiment (which would settle the matter once and for all), even thogh initially i did find interested individuals; on the other hand, I feel like I don’t really have the cognitive energies to change once again the way I think about all of this, which i hope i made with sufficient precision above.
Joy, with my previous post I didn’t mean that I dont value your input in the matter. I want to make that clear since I realized it could have been interpreted that way.
I am happy to answer any genuine technical questions about my work. But only about my own work. I have wasted enough time addressing bogus claims about my model by incompetent non-physicists such as Gill and Aaronson, so I am no longer inclined to answer any issues that arise from their online, non-peer-reviewed comments. As you may know, the Royal Society has now published my detailed rebuttal to Gill’s bogus claims about my work:
This has allowed me to expose just how incorrect and incompetent all of the critics of my work have been. Consequently, I am no longer inclined to fight a proxy war with them, via you or anyone else. I have more serious work to do, and such proxy wars (however well-intended) are a huge distraction from my ongoing research.
My post above did not stem from either Gill, Aaronson or any other of your critics. It is my observations only on what your work implies; the main point i was trying to make is that mathematically, those expectation values are different integrals over different setting choices (hope we agree here). If we assume the global joint p(lambda) over outcomes exists and is the same for all settings, then mathematically for the large N limit the sum of those integrals is the same as the integral of the sum. Many posts of Michel Fodje, which you endorsed, point out that Bell’s inequalities involve dependent terms like A1B1 + A1B’1 + A’1B1 – A’1B’1, which can indeed be factorized. For independent terms, which would be the case for experiments since we can only measure two settings at a time, the correct expression is A1B1 + A2B’2 + A’3B3 – A’4B’4. But in the large N limit, if we assume those As and Bs are drawn from the same fair sample, the two expressions are indeed equivalent. Thus, my observation that since QM violates the inequality, then it means, in view of the above, that the As and Bs are NOT drawn from the same sample for the different integrals. Which means either 1) outcome dependence (A depends on B and viceversa, i.e. non locality) or 2) p(lambda) being a “function” of settings. What I’m proposing, in view of your own results, is that condition 2) does not necessarily mean superdeterminism; that is only the case if we insist on making A,B functions of the base only. But on S3, what looks like superdeterminism in S2 simply stems from the topological properties of S3, namely the lack of a global section. Functions on the whole space (like your own As and Bs) are not limited in this way. That is what I tried to get across, and what I hoped to get your feedback on. I simply tried to find a way to restate your work in a different manner.
Ok, you have now expressed your point more clearly, and it is worth thinking about.
To begin with, you are quite right to point to the lack of a global section in the fibration of S^3 as the root cause of the differences that arise in the strength of the correlations when measurement results are charted on S^2 versus S^3.
However, your suggestion that this lack of a global section should be interpreted as making the probability density p(lambda) depend on settings is problematic, because a setting-dependent function p(lambda | a, b), in fact, implies a more subtle form of non-locality. Clauser and Horne have explained this nicely in footnote 13 of their paper “Experimental consequences of objective local theories,” Phys. Rev. D., vol. 10, pp 526 – 535 (1974). So, your observation concerning the lack of a global section on S^3 does not let you escape a form of non-locality, if you want to think of strong correlations in terms of Bell inequalities.
But independently of S^3, the real reason why strong correlations violate Bell inequalities has to do with the assumption of linear additivity of expectation values, regardless of the setting-dependent or setting-independent probability density p(lambda). As is well known, linear additivity holds in quantum mechanics, making it a non-local theory. On the other hand, while we must assume setting-independent probability density p(lambda) to preserve strict locality, there is no reason to assume linear additivity of expectation values in hidden variable theories. I have explained this in considerable detail in my latest analysis of Bell’s argument: https://arxiv.org/abs/2302.09519
I recently became interested in Garret Lisi’s E8 theory. There are a lot of commonalities between that and your octonion sphere model. What caught my attention is his description of spin as “gravitational charge” (referring to cartan theory), and the higgs field basically being the torsion of spacetime. Ever looked into it? I admit I have like 10% of the required math knowledge to look at it beyond his descriptions in a ted talk, so perhaps before diving into it more and start researching the stuff i need to know i was wondering if you have any thoughts on that, if at all.
Thank you for all your answers Joy.
I have met Gerret Lisi. I was at the 2011 FQXi conference where he gave a talk on his E8 theory. His work is interesting, but I haven’t spent time on it beyond listening to his talk at that conference because my focus is elsewhere. I don’t know how much Geometric Algebra Lisi uses, but David Hestenes was also present in his FQXi talk, and Lisi gave credit to Hestenes for inspiration. Sabine Hossenfelder was also at that FQXi conference and, if I recall correctly, she has made a video about Lisi’s theory, which may help you.
Dear Joy,
quick question. In the paper “Bell’s Theorem Versus Local Realism in a Quaternionic Model of Physical Space” I don’t understand how you derived eq. 25. In particular I don’t understand how exactly the angles ne and zs are assigned or what their relationship is, as if we set such angles both to pi/2 (with k=+1), or some interval around pi/2 (like 80-100°) the norm of |P+Q| becomes larger than 2, which is not allowed since on S3 this norm is bounded by 2.
Hi Sandra,
Eq. 25 is not derived. It is postulated. It is an ansatz. I have made it up, because it works.
The angles ne and zs are constrained by the inequality 28, which follows from the triangle inequality 23. Thus, ne and zs can take any angular values subject to the constraint 28. As long as the inequality 28 is satisfied, the norm will not exceed the value of 2.
I see. Is there an intuitive geometric idea behind ne and zs? In your paper you mention n can be thought of as a measurement direction, z as a “reference” vector at another point of S3 (which i personally think of as either i,j or k), and e and s are related to the spin system. In particular you mention that s can be thought of as the vector dual to the bivector Is representing the spin of the combined pair, but isn’t that bivector by definition 0 since angular momentum is conserved (e1=e2)? Also I’m struggling to geometrically relate all of this to the fact that certain angles are essentially forbidden.
You are struggling because it is not an elegant approach like that based on the even subalgebra of the Geometric Algebra Cl(3, 0) or a quaternionic 3-sphere. I used that “ugly” approach in that paper to make contact with Pearle’s non-Clifford-algebraic approach [ see Ref. 34, where I cite his famous paper ], which allows numerical simulations of the singlet correlations.
No angles are forbidden by the constraint 28. What is forbidden are certain *relations* between the angles ne and zs. If these relations are admitted, then they would throw us outside the geometry of the 3-sphere, because including them would violate the triangle inequality 23, which must be respected to respect the geometry of the 3-sphere.
By “composite pair” in the paragraph above Eq. 23, I mean just “spin-1 and spin-2” regardless of the net values of their spins, not their singlet state defined in Eq. 9. The net angular momentum of the two spins in the singlet state would indeed be zero.
Dear Joy,
i studied more carefully your “Symmetric” paper. In it, you show the example that on a moebius strip, the intrinsic angle Nas is related to the extrinsic angle (Bas) by B=pi(1-cosNas). You also derived the linear correlation using B as E(a,b) = -1 + B/pi.
.
Then, I noticed that the linear correlations derived by Bell in his Toy model has the form
E(a,b) = -1 + 2Alpha/pi. Obviously there’s a very clear similarity here between the two correlations, with B = 2Alpha. Tell me if this is the correct way to think about it:
.
If we assume R3, there’s no twist, and the correlation is the linear one of Bell. If we assume S3, there’s a twist, then we need to relate Theta to the intrinsic twist angle Nas. The factor of 2 comes from the double covering of SO(3). So doing the substitution Alpha = pi/2 (1-cosNas) we obviously end with E(a,b) = -cos(Nas).
.
There’s more. The relation psi=pi(1-cos(theta)) derives from the holonomy on the Hopf Projection on S2, with psi is the phase on the S1 fibers after a full loop and theta is the polar angle on S2 where the loop is constructed. So if we make theta=pi/2, we’re at the equator, where the twist becomes a full pi angle, exactly like in the moebius strip. This justifies making the substitution psi = Nas as an intrinsic angle. Then from the relation alpha = pi/2(1-cosNas) it becomes clear that Nas is equal to alpha, that is, we can relate the angle between detectors with the angle in the twist of S1! Hence,the non-linear correlations!
.
I think this makes a lot of sense, but I wanted to ask you if my understanding is indeed correct.Months thinking about S3 left me in the clouds a bit too much.
I think you are talking about “Dr. Bertlmann’s Socks” paper, not the “Symmetric” paper.
Your understanding above is essentially correct, as long as you keep in mind that the toy example is not the real thing. The Mobius strip is not a 3-sphere. At best, it is just one of the S^2 worth of circles that make up the 3-sphere. What is more, the Mobius strip is not *orientable*. A consistent sense of handedness cannot be specified on a Mobius strip because there is a twist in it. By contrast, a 3-sphere is orientable and can be parallelized to specify a consistent sense of handedness over the entire sphere, without ever having to change the “hand.” As long as you keep in mind that the Mobius toy example only provides an analogy to help our intuition, you will be fine.
Happy Easter Joy!
Why exactly is the orientability important here? A cosine correlation can be derived regardless. I know orientation is your hidden variable Lambda, but even that drops when the two measurement functions are multiplied.
Happy Easter! We do not live in a 2-dimensional, one-sided world of a Mobius strip. As I have argued, we live in a 3-dimensional world of a 3-sphere, which differs from a flat Euclidean space R^3 only by a single mathematical point at infinity. Moreover, an ordinary unit 3-sphere defined by the equation w^2 + x^2 + y^2 + z^2 = 1 and embedded in R^4 cannot be charted by a single coordinate system without encountering a singularity somewhere, usually chosen to be at the north pole. However, a 3-sphere is parallelizable with quaternions, which renders it a flat, orientable set of unit quaternions. Meaning, unlike the ordinary 3-sphere, a quaternionic 3-sphere can be charted by a single coordinate system with a fixed handedness without any discontinuities, singularities, or fixed points. This is important because excluding a single point (say, at the north pole) would render it topologically equivalent to a flat Euclidean space R^3 — S^3 minus a point is equal to R^3. And we know that measurements within R^3 cannot reproduce strong quantum correlations, as proved by Bell’s theorem. Thus, orientability is a necessary and sufficient criterion for a 3-sphere to be able to reproduce strong quantum correlations.
Joy,
Reading through the pages of your blog I’ve seen you also used a different derivation for the triangle inequality, where you derive |cos(ae)| >= 1/2 sin^2(theta). Now this to me is a more rigorous result, as this function itself does constraint the sum of two quaternions to 2 for all angles.
.
So I’ve tried simulating it in excel, and indeed the result is the correct correlation. But I was wondering how I should interpret geometrically this constraint. I realize it essentially relates “projected” quantities (vectors instead of bivectors) to the geometry of a 3-sphere. But is there a direct correspondence to, for example, the Hopf projection? Could theta be a polar angle on the base space S2, and ae0 the angle on the fiber? Or perhaps some other interpretation?
.
Another question as well. The initial state (e0, theta) gives the final measurement result as sign(cos(ae0)). But for some angles theta this “metric” returns zero, so you say the state for which the metric is zero does not exist on S3. But of course for a given (e0, theta) whether the metric is zero or not depends on the measurement direction a, and as this varies different states are said to “not exist”. How should I interpret this geometrically, perhaps related to the projection picture? For a given theta, why are some angle relationships between e0 and a constrained? For example, you mention that at theta=pi/2 most of e0 and a are orthogonal to each other, but if we sub in the constraint this value we get |cos(ae)| = 0.5, which means ae=60°.
Hi Sandra,
The approach with the constraint |cos(ae)| >= 1/2 sin^2(theta) was phenomenologically motivated. Its purpose was to produce numerical agreement with the observed correlations. It is pointless to force a geometrical interpretation on such a crude approach, or compare it with the elegant approach using a Hopf fibration of the 3-sphere.
Joy,
I don’t think it is pointless. Without such a geometrical interpretation people won’t stop accusing it to be simply a detection loophole model. It’s not enough to just say “it follows the topology and geometry of the 3 sphere” without directly creating a correspondence between them. At least, that is what I’m concerned with since I want to make it into a video, but I’ve been stuck on these details for months.
In Section III G of the following paper (open access), I discuss what is wrong with Pearle’s detection loophole model, and then make a detailed comparison with the quaternionic S^3 model:
https://ieeexplore.ieee.org/document/9693502
In that section, you will see that the geometrical relation between the two models is quite involved and requires considerable effort to identify the flaw in the detection loophole argument. Most people like Gill are mathematically too incompetent to see the flaw in Pearle’s model. You, on the other hand, are one of the very few people who are making an effort to understand the S^3 model. I appreciate that because it is hard work. But, unfortunately, the geometrical and topological subtleties of the S^3 model are too difficult to explain in online discussions, such as in your effort to make a YouTube video about them.
Note also that the comparison of the two models requires a different constraint derived from the triangle inequality within S^3, not the one with the phenomenologically motivated sine function you have noted above.
Dear Joy,
one more question about the “Bertlmann” paper, which came to me after more scrutiny. In the toy model you state that the relation between the extrinsic, “cylindrical” angle on the strip and the intrinsic twist angle, seen from a cross-sectional view, is beta(ab) = pi(1-cos(nab)). But since the twist is uniformly distributed along the entire length of the strip, the actual relationship is linear, more specifically beta(ab) = 2n(ab).
.
The relation you write is an holonomy relation, but I’m not sure how you square the two together.
Angle is always defined in terms of the arc length of a circle, not a straight line. So, the relation beta(ab) = 2n(ab) you wrote down is incorrect. The correct relation between the two angles is beta(ab) = pi(1- cos(nab)).
Ooh, so it’s due to the fact that the strip is not straight but curved! Obvious in hindsight. Thank you Joy.
Dear Joy,
sorry to bother you on the same issue again. I’ve been trying to find a derivation of the relation beta(ab) = pi(1- cos(nab)) for a moebius strip for months now, but i could not understand how it is done (despite your previous answer on the matter). Since this kind of formula is found when dealing with areas and curvatures, i tried looking up integrals for the same relation but for flat surfaces with torsion like the moebius strip, but they all give a linear result. Could you provide a resource to understand how to derive this relation, on the strip specifically? In your paper you simply state “for the well known properties of the moebius strip” which unfortunately wasn’t much of help for me.
Hi Sandra,
I am not sure why you are wasting so much time on that toy example. It is not a model for the singlet correlations observed in the real world. The purpose of the toy example is to give us an intuitive understanding of a characteristic feature of the geometry of the quaternionic 3-sphere in the real world. I have warned in my papers not to take the toy model too seriously. In particular, the relation beta(ab) = pi(1- cos(nab)) cannot be derived for a Möbius strip. It postulates a physical characteristic of the toy world that a two-dimensional Alice and Bob are assumed to inhabit. That is why I have stressed that the relation beta(ab) = pi(1- cos(nab)) is the defining relation of the Möbius world of Alice and Bob. It has to be accepted, not derived, for the illustration to work.
Dear Joy,
the reason being that I want to understand the reasoning in your papers. A toy model is supposed to present a simplified, yet not necessarily comprehensive or even widely applicable, version of a model. If I understood the toy model completely, then I would feel more confident trying to understand the whole model, wary of the limitations of the toy model itself. The problem with just giving out mathematical relationships ad hoc without fully justifying them, even on a matter of principle, leaves more questions than answers, which defeats the point of creating a toy model. I’m not sure why you question me questioning the derivation, I just didn’t understand it. Should I be held at fault for not understanding “defining relation” as “we are just assuming this holds”?
Given that I’m much more of a visual person rather than a math one, I can now understand the relation pi(1- cos(nab)) as a torsion not uniformly distributed on the strip, centered on the midpoint between a and b. But that seems to mean the choice of a influences the possible choice of b, so by accepting that relation we are introducing non-locality in the strip (there’s no global distribution of torsion that works for all a and b). Perhaps you have objections here, but please don’t just say “ignore the toy model”. I’ve been passionately reading tons of posts on your work, by many people, and I understand most of your critics to be wrong on many points they make. Just to make an example, I immediately recognized that the “sign error” many of them talked about is in fact their misunderstanding of the idea behind the change in orientation. One thing they are not wrong about though, is that sometimes your writing style can be a bit cryptic, especially to those without years of experience on GA, topology and GR.
Hi Sandra,
There is no deep issue to resolve here. By hypothesis, the relation beta(ab) = pi(1- cos(nab)) *defines* the toy geometry of the two-dimensional world Alice and Bob live in; just as, by hypothesis, we live in a three-dimensional geometry isomorphic to a quaternionic 3-sphere. The relation beta(ab) = pi(1- cos(nab)) is not a standard property of a Möbius strip. So, it cannot be derived. But there is no need for its derivation, since we are working within a hypothesised toy model. I think what you are missing here is not mathematics, but the logic behind the toy model.
Hello Joy,
I’ve come to terms with your toy model. Unfortunately, I don’t think it helps at all with anything, but that’s just my very humble opinion. After reading and reading, ive come to the conclusion that the most fundamental thing in all of this matter about Bell’s theorem, QM correlations and so is a single property of S3, that it is a non trivial bundle. The central argument for which Bell’s inequalities are considered inapplicable, here and on many posts on sciphysicsforum, is the impossibility of measuring different settings on the same particle pair. This in turn means the outcomes corresponding to the same setting cannot be factorized through different integrals, thus making deriving the classical bound impossible. Another way that this argument was put in was that the additivity of expectation values is not respected across different settings.
To me, all these arguments are really just one argument, that the distribution p(lambda) is different for different setting pairs, preventing the existence of a global joint distribution valid for all settings. This is exactly in line with some critiques by Karl Hess, and Fine’s result (whereby the existence of such a distribution = validity of the CHSH inequality).
I’ve also learned that, through yours and tsirelson’s equivalent derivation, that non commuting observables are what cause an obstruction in QM to the existence of a global joint. In particular, when viewed as a topological obstruction, no global joint = no global section on the outcome space. I’ve also learned that a non trivial bundle provides exactly such a lack of global section. This mirrors the simple fact that the same path in SO(3) can lift to two different paths on SU(2). From the point of view of R3 outcomes really do seem to depend on the distant setting (they look context dependent), but that is simply because we ignore the full specification of the state, which must include the relative points on the fibers for the same bloch vector. Likewise, we can’t know which SU(2) path was taken from a single rotation; but we can infer it through comparison of two different rotations of the same state. The response functions depend on the lift. Thus, a joint probability on the base implicitly pretends that all outcome-determining variables are functions of basepoints only; if the true ontic dependency lives in the fiber (lift) and there is no global, consistent way to choose a lift over the whole base (no global section, which is guaranteed for a jon trivial bundle), then we cannot compress the model to a single, context-independent probability on the base that reproduces all lift-dependent statistics.
I’m not sure you’ll agree with my conclusion here; but this has been the clearest way for me to make sense of all I’ve read. I also believe that this helps with the conceptual issues I’ve had with your simulations. The “detector loophole” can likewise be framed in terms of differing joints for different settings; when elements of one distribution are used for a pair of settings which uses a different distribution, some of such elements may not be present in that distribution. Thus the need to specify a full list of “initial states” for setting pairs like in your simulation.
Overall, I think I will take a break from studying these matters. On one hand, im tired of the vitriol directed at both parties in the matter, which also has made it very hard to follow through with your proposed experiment (which would settle the matter once and for all), even thogh initially i did find interested individuals; on the other hand, I feel like I don’t really have the cognitive energies to change once again the way I think about all of this, which i hope i made with sufficient precision above.
Joy, with my previous post I didn’t mean that I dont value your input in the matter. I want to make that clear since I realized it could have been interpreted that way.
Hi Sandra,
Understood.
I am happy to answer any genuine technical questions about my work. But only about my own work. I have wasted enough time addressing bogus claims about my model by incompetent non-physicists such as Gill and Aaronson, so I am no longer inclined to answer any issues that arise from their online, non-peer-reviewed comments. As you may know, the Royal Society has now published my detailed rebuttal to Gill’s bogus claims about my work:
https://royalsocietypublishing.org/doi/10.1098/rsos.220147
This has allowed me to expose just how incorrect and incompetent all of the critics of my work have been. Consequently, I am no longer inclined to fight a proxy war with them, via you or anyone else. I have more serious work to do, and such proxy wars (however well-intended) are a huge distraction from my ongoing research.
My post above did not stem from either Gill, Aaronson or any other of your critics. It is my observations only on what your work implies; the main point i was trying to make is that mathematically, those expectation values are different integrals over different setting choices (hope we agree here). If we assume the global joint p(lambda) over outcomes exists and is the same for all settings, then mathematically for the large N limit the sum of those integrals is the same as the integral of the sum. Many posts of Michel Fodje, which you endorsed, point out that Bell’s inequalities involve dependent terms like A1B1 + A1B’1 + A’1B1 – A’1B’1, which can indeed be factorized. For independent terms, which would be the case for experiments since we can only measure two settings at a time, the correct expression is A1B1 + A2B’2 + A’3B3 – A’4B’4. But in the large N limit, if we assume those As and Bs are drawn from the same fair sample, the two expressions are indeed equivalent. Thus, my observation that since QM violates the inequality, then it means, in view of the above, that the As and Bs are NOT drawn from the same sample for the different integrals. Which means either 1) outcome dependence (A depends on B and viceversa, i.e. non locality) or 2) p(lambda) being a “function” of settings. What I’m proposing, in view of your own results, is that condition 2) does not necessarily mean superdeterminism; that is only the case if we insist on making A,B functions of the base only. But on S3, what looks like superdeterminism in S2 simply stems from the topological properties of S3, namely the lack of a global section. Functions on the whole space (like your own As and Bs) are not limited in this way. That is what I tried to get across, and what I hoped to get your feedback on. I simply tried to find a way to restate your work in a different manner.
Hi Sandra,
Ok, you have now expressed your point more clearly, and it is worth thinking about.
To begin with, you are quite right to point to the lack of a global section in the fibration of S^3 as the root cause of the differences that arise in the strength of the correlations when measurement results are charted on S^2 versus S^3.
However, your suggestion that this lack of a global section should be interpreted as making the probability density p(lambda) depend on settings is problematic, because a setting-dependent function p(lambda | a, b), in fact, implies a more subtle form of non-locality. Clauser and Horne have explained this nicely in footnote 13 of their paper “Experimental consequences of objective local theories,” Phys. Rev. D., vol. 10, pp 526 – 535 (1974). So, your observation concerning the lack of a global section on S^3 does not let you escape a form of non-locality, if you want to think of strong correlations in terms of Bell inequalities.
But independently of S^3, the real reason why strong correlations violate Bell inequalities has to do with the assumption of linear additivity of expectation values, regardless of the setting-dependent or setting-independent probability density p(lambda). As is well known, linear additivity holds in quantum mechanics, making it a non-local theory. On the other hand, while we must assume setting-independent probability density p(lambda) to preserve strict locality, there is no reason to assume linear additivity of expectation values in hidden variable theories. I have explained this in considerable detail in my latest analysis of Bell’s argument: https://arxiv.org/abs/2302.09519