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)).
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.