Classical Spheres, Division Algebras, and the Illusion of Quantum Non-locality:

Richard Gill

PS. So the numbers "2", "1/2" have been treated as free parameters and tweaked to get us really close to the cosine! Congratulations! A victory for the programming language R, I would say, and proof of Chantal Roth's and my respective skills.

Caroline Thompson would be really proud to see this. She knew that her chaotic spinning ball model could be tweaked, by introducing "unsharp" membership of the circular caps, to get as close as you like (for any practical purposes) to the cosine. And at last, this has been done, to a fantastic degree of accuracy.

http://freespace.virgin.net/ch.thompson1/Papers/The%20Record/TheRecord.htm

Joy Christian

So let us put some perspective on my version of Richard Gill's S^2 version of Michel Fodje's original simulation of my 3-sphere, or SU(2), model for the EPR-Bohm correlation.

As we can see from the details discussed in the above paper, my analytical model predicts the (negative) cosine correlation exactly. What is more, in Bell's local-realistic framework (discussed in his 1964 paper) we are completely free to choose whatever initial or complete state of the system we like. In the present representation of the 3-sphere this initial state is represented by the pair (e_o, theta_o), which are simply four numbers. These numbers depend on the system under consideration. They depend on the symmetry of the physical situation. For example, for the GHZ or Hardy case the initial state would be nothing like these four numbers.

Now have a look at the derivation of this state within my 3-sphere model. As we can see from just above the box of eq.(10), I had made a simple choice for the initial state by choosing the function f(theta_o) defined in eq.(7). This choice simply specifies the magnitude of the sum of the two initial quaternions, p_o and q_o, and thereby also specifies the initial state (e_o, theta_o). In Michel Fodje's simulation the choice I made for f(theta_o) seemed necessary and sufficient to produce the correct correlation. It now appears that when one zooms-in with greater precision, the initial state is in fact what has now been chosen in the latest simulation.

I am grateful to Richard Gill for insisting on greater precision for the simulation, which helped me discover a more accurate choice for the initial state (e_o, theta_o) in the EPR-Bohm case. I am also grateful to Chantal Roth for encouraging me to learn R and thus investigate Richard Gill's simulation myself. The result of my investigations is not devoid of beauty.

Richard Gill

I re-ran the latest simulation http://rpubs.com/chenopodium/joychristian at larger sample size still (now 10^7) and ask you to take a look at a different spot on the curve ... say at around 60 degrees.

http://rpubs.com/gill1109/ChaoticUnsharpBall

Getting closer but not quite there yet.

Amusingly, there is another interpretation of the simulation model: this is Caroline Thompson's chaotic spinning ball model with a kind of unsharp observation of the circular caps - the circular caps don't have a fixed radius, but a random radius S, and presently it has the following probability distribution: S = (sin(Theta)^1.32)/3.16 where Theta ~ Unif(0, pi/2).

Conjecture 1: there does exist a radius distribution such that the cosine is reproduce exactly.

Conjecture 2: there are no constants a, b such that S = (sin(Theta)^a)/b where Theta ~ Unif(0, pi/2) reproduces the cosine exactly.

I do agree that these investigations are not devoid of beauty, and they are not devoid of mathematical interest either.

Thomas Ray

"This choice simply specifies the magnitude of the sum of the two initial quaternions, p_o and q_o, and thereby also specifies the initial state (e_o, theta_o)."

It seems to me that this would also satisfy general covariance independent of scale, as required of a coordinate free framework with active diffeomorphism (the topoloigcal structure).

Everything just fits. It's nice, Joy.

Richard Gill

Sorry, forgot to add this link

http://freespace.virgin.net/ch.thompson1/Papers/The%20Record/TheRecord.htm

Richard Gill

Tom: you said "Everything just fits". No, everything doesn't quite fit, yet. I think we need a new idea to make it fit exactly. See my two conjectures.

Joy Christian

"So let us put some perspective on my version of Richard Gill's S^2 version of Michel Fodje's original simulation of my 3-sphere, or SU(2), model for the EPR-Bohm correlation.

As we can see from the details discussed in the above paper, my analytical model predicts the (negative) cosine correlation exactly."

Thomas Ray

I'll think about your conjectures, Richard. In fact, I think (1) closely corresponds to a conjecture I made in January 2012, "For every observation of a classical state, there exists at least one quantum state." Actually, my notebook reflects it as a theorem following from the lemma "Joy Christian's framework for quantum pair correlations implies geometric uniformization for 3 manifolds," for which the proof is yet incomplete. I plan to return to it. Obviously these conjectures are waiting to be formalized.

My point was, that Einstein identified "up to diffeomorphism" in general relativity as the best that one could do with the mathematical methods of his day. I truly believe that Joy Christian has completed the quest for analytical continuation of the physical space, with an active diffeomorphism.

By the way, I bought and have only started reading Kevin Brown's book Reflections on Relativity from which the link was taken, after having read bits and pieces of his Mathpages.com writings for a few years. Just 25 pages in, I think it's an incredibly good read.

Tom

Joy Christian

Tom,

You are absolutely right. As you say, "everything just fits."

The Monte Carlo accuracy of Richard Gill's simulation is about 0.0001.

The slight deviation in my plot he is now making fuss about is much smaller than 0.0001.

Joy

Richard Gill

The "slight deviation" is a whole lot bigger than 0.0001

But it should be possible to make it go away altogether. But you'll need to tweak more than two parameters to do that. An interesting mathematical challenge. Glad this fits to one of Tom's ideas.

Richard Gill

Here is computational proof that we are done: clearly, a continuous convex combination of the blue curves, which are Christian-Roth models with fixed theta_0, and simultaneously Caroline Thompson models with fixed R, can be found which yields the black curve (cosine), exactly.

http://rpubs.com/gill1109/ChaoticUnsharpBall3

Questions: can the required "mixing distribution" be specified in compact analytical form?

Can Joy Christian *derive* it from his S^3 model?

Richard Gill

Another question: can Fred use Mathematica either to *prove* my "pretty safe conjecture" or to *derive* the mixture distribution? That's what's needed to give independent proof that a simulation of Joy's model, on the lines of Minkwe/Roth/Gill, can reproduce *exactly* the cosine function (in the limit, as numerical and statistical precision increase indefinitely). By "independent" I mean independent of Christian's derivation from his S^3 model.

[deleted]

Continuity isn't a new idea. It has just never been adapted to event by event computation in a way that guarantees a closed judgment on the curve without arbitrary boundary conditions.

There's a quote from Riemann in the last link I posted: "In all physical theories, partial differential equations constitute the only verifiable basis. These facts, established by induction, must also hold a priori. True basic laws can only hold in the small and must be formulated as partial differential equations."

Leibniz held a similar view, as did Hermann Weyl. And of course, Einstein.

The reason for the "fit" using Joy's technique is that the framework is coordinate free and independent of scale. The dependence on initial condition obviates dependence on measurement settings and any boundary conditions that those settings imply.

Quantum theorists who want to extend Bell's theorem to cosmological status (and thereby relegate relativity to a "locally real" limit) are assuming nonlocality to "prove" nonlocality -- in other words, the assumption that t = T = 1 gives time a special status that it does not have in relativity, because "all physics is local" in the context of spacetime continuity. The circularity of quantum cosmological reasoning is actually in favor of relativity; the universe is a local place, finite in space and unbounded in time.

Because Joy's framework is coordinate free and scale independent, my lemma, "For every classical measurement, there is at least one quantum state," holds in the small and the large both physically and mathematically. In fact, one can see its application today, without proof, in methods of computing that exploit the phenomenon of quantum discord. The next step is to simply drop the assumption of entanglement, just as science once discarded phlogiston as essential to fire. Joy has shown us what's behind the illusion of entanglement.

Tom

Thomas Ray

Lost my log-in. Last was mine.

Thomas Ray

Sorry, I meant to link Zeeya Merali's article on quantum discord.

Richard Gill

Tom: we do not yet have the "fit". We still have a small *mis-fit*. We are waiting for the final step, whether it is an existence theorem or just convincing statistical and computatational evidence.

But I think it is close to hand.

Thomas Ray

Richard, we have plenty of existence theorems on the role of particle states in a continuum, most arising from applications of the central limit theorem and the law of large numbers.

As M.D. Penrose notes (5.2, open problems) however, "The theory of piecewise deterministic Markov processes [15;35] could be relevant here. In a more general extension, one might hope to allow piecewise deterministic motion of a particle. In this case, this motion is typically able to take the particle from one patch of space into a different patch of space (this feature does not arise in [58]), which is likely to make matters more complicated."

The complication is, that piecewise determinism (all stochastic models) requires an averaging of ergodic motion that blurs the distinction between the true continuum and the average path of a particle in an ensemble. Joy's framework does not rely on piecewise determinism; it relies on pairwise correlation of points of the continuum described by a particular topological structure.

If one forgets the structure -- and I am just as amazed as Joy is, that referees can reject the framework without ever once referring to the parallelized 3 sphere -- there is no basis for correlation. All correlations are therefore accepted as random, and classical measurement results are interpreted as the product of quantum entanglement. Christian's framework is one of minimal assumptions: initial condition and topological structure; nothing superfluous, like entanglement or ergodicity. That's why everything "fits" into a continuous measurement function independent of scale, boundary conditions, coordinate systems or any form of probability measure. It harks back to Einstein's original idea of "particle" as a wave state; correlation of particle pairs from one patch of space into another is continuous.

Tom

[deleted]

Tom,

Perhaps you could explain why you think S^3 is scale independent. Seems to me it is about as scale restrictive as you can get, since the norms of all member quaternions must be exactly 1 and not some other number.

If you please could you also explain why you feel Joy's framework is "coordinate free". You seem to equate this to Hestenes' GA and not to quaternion and octonion analysis, which I think I have done a good job at demonstrating is alive and well.

You also repetitively bring up "continuous functions". Do you think mathematical descriptions of reality must be free of singularities? Why?

More important to the subject at hand, how is any of this constructive in proving Joy is on the correct path?

Rick

Thomas Ray

Rick,

"Perhaps you could explain why you think S^3 is scale independent. Seems to me it is about as scale restrictive as you can get, since the norms of all member quaternions must be exactly 1 and not some other number."

I didn't say S^3 is scale independent. I said Joy's measurement framework -- whose domain is parallelized S^3 -- is scale independent. And it is -- for there are no real measurement values outside this space; it's the complete space that we live in.

Quaternion norms have nothing to do with that issue.

"If you please could you also explain why you feel Joy's framework is 'coordinate free'. You seem to equate this to Hestenes' GA and not to quaternion and octonion analysis, which I think I have done a good job at demonstrating is alive and well."

GA is based in vector analysis, too. Coordinate free geometry, however, doesn't find it necesary to define a coordinate system to combine vectors and scalars. We need that freedom in a relativistic model, because reversibility (correlation of points of a continuous measurement function)depends on it -- in Joy's framework, nonzero topological torsion is the key. I can't see that your model is relativistic, given your emphasis on normed vectors.

"You also repetitively bring up 'continuous functions'. Do you think mathematical descriptions of reality must be free of singularities?"

Yes.

"Why?"

See the Riemann quote I posted earlier.

"More important to the subject at hand, how is any of this constructive in proving Joy is on the correct path?"

I don't know that it is. I only report what I have personally found, in the years that I have studied the subject.

Tom

Rick Lockyer

Strange answers Tom, your post at top of this thread:

"This choice simply specifies the magnitude of the sum of the two initial quaternions, p_o and q_o, and thereby also specifies the initial state (e_o, theta_o)."

It seems to me that this would also satisfy general covariance independent of scale, as required of a coordinate free framework with active diffeomorphism (the topoloigcal structure).

Everything just fits. It's nice, Joy.

Nothing to do with quaternions and norms?? Perhaps you could explain or modify your reply. You did read Joy's justification of the qualifier that was modified with some success didn't you? I did: norm 1 quaternions, and *this* is at the heart of his framework.

Rick

Fred Diether

Rick asked of Tom, "More important to the subject at hand, how is any of this constructive in proving Joy is on the correct path?"

Tom doesn't need to prove that Joy is on the correct path. Joy has already done that himself. And it is not really very complicated. Space has unique spinor properties.

Best,

Fred

Thomas Ray

"Perhaps you could explain or modify your reply."

I could, but I won't.

I've been quite clear that I think the normed algebras are necessary though not sufficient to express evolution of the state vector over time-distance scales. Were this not true, quantum mechanics would be a complete theory. The heart of Joy's framework is nonvanishing topological torsion. As Fred explained, spinor properties.

Thomas Ray

Here's what I'm talking about, Rick, from your own essay forum of 2012. Answering a post to Jonathan Dickau, you ended with the comment, "It all comes down to the structure constants p_ijk."

To which Joy made a short reply, "Or structure variables p_ijk(x) in my variable torsion picture, at every point x in S^7."

Your views topologically make the difference between passive diffeomorphism (yours) and active diffeomorphism (Joy's).

Jonathan Dickau

I think Rick raises some good points...

The viability of Joy's framework absolutely does rest on the properties and effectiveness of the quaternions and octonions. Having parallelized S^3 in an embedding space of S^7 topology makes such a linkage unavoidable - nor would you want to avoid it. As the introductory blurb on the top of this page states; no less than Michael Atiyah heads the list of those for whom the quaternion and octonion algebras appear central to Physics.

The thing is; there are a number of interesting formulations that arise from the fact that these algebras (and their attendant geometry) are of crucial importance to Math and therefore must have relevance to Physics - perhaps representing the core embodiment of the order and symmetry found therein. So maybe Michael Atiyah, Baez and Huerta, Joy Christian, Geoffrey Dixon, Rick Lockyer, and others like Tony Smith - can all have a field day with this, now that some of the more subtle generalizations have been worked out in detail.

I guess we live in exciting times.

All the Best,

Jonathan

Jonathan Dickau

I guess I should add..

It appears fair to say that Joy's model is a theory of quaternionic space, conformal or compatible with theories of octonionic gravity. So in essence; Rick is correct.

However; I agree with almost all of the comments Tom made in reply to Rick, because I feel that the overlooked element of topological torsion is essential to Joy's model. The thing is, coming to grips with the idea that we live in a simply connected space with a twist in it takes some getting used to. As Joy says again and again; people are hung up on seeing space as R3 - or flat Euclidean extendedness. Anyhow; I see how the algebraic and geometric or topological ideas come as a package, or are connected in a non-trivial way.

Regards,

Jonathan

Joy Christian

Hi Jonathan,

Thanks for your comments.

You may be interested to know that Richard Gill has started a new thread in Fred's new forum about quaternions and octonions, partly to better understand my work: http://www.sciphysicsforums.com/spfbb1/viewtopic.php?f=6&t=19.

Best,

Joy

Jonathan Dickau

Excellent!

I'm happy to see it. I have that page open in another tab right now, but I want to mention here that I've been thinking a full-on simulation would require some calculations to be done in the quaternion domain, but I didn't know how to bring that up. Now I won't have to.

Regards,

Jonathan

Joy Christian

Absolutely, Jonathan.

All the simulations to date are flatland simulations (i.e., R^3-based simulations). In the full-on S^3-based simulation statistics must be applied to all elements of the graded basis (i.e., both to scalars as well as to bivectors). That is why I fought so hard to defend my one-page paper of 2011. There you can see that the argument works because I applied statistics to all elements of the graded basis even-handedly.

Best,

Joy

Richard Gill

Joy wrote "All the simulations to date are flatland simulations (i.e., R^3-based simulations). In the full-on S^3-based simulation statistics must be applied to all elements of the graded basis (i.e., both to scalars as well as to bivectors). That is why I fought so hard to defend my one-page paper of 2011."

Precisely. That is why all the simulations are defective. Whether or not Bell's theorem applies to physical reality is one thing - but it does sure as hell apply to an event-based local realist computer simulation of a Bell-CHSH delayed choice experiment under the usual constraints (no "non-detections", outcomes binary, etc etc). Such a computer simulation model is doomed irrevocably to live in flatland, whatever kind of fancy maths computation you use to generate particles, compute measurement outcomes, and so on.

That is what I have been saying all along.

Joy Christian

You do have quite a knack of reading everything I write in a completely wrong way.

Bell made a stupid mistake in his argument. Get over it!

Thomas Ray

Quaternions and octonions describe how to calculate real measurement values; they do not describe the shape of space, which is what determines the measurement outcomes in Joy's framework.

As Richard correctly notes in the sci.physics.foundations forum, "Two copies of quaternions give you bi-quaternions, not octonions. The former is a torus but the latter is a sphere. Hence the difference in the multiplication tables."

The torus is mutltiply connected. The sphere is simply connected. One can get quantum correlations assuming a toroidal space; however, one must also bring essential assumptions of randomness and nonlocality -- along with collapse of the wave function for the observed quantum mechanical unitarity. In the simply connected space, just as Joy has exhaustively explained, the "4 sphere's worth of 3 spheres" or complete S^7 space which has the Lebesgue measure limit S^2 X S^2 on the S^3 manifold, is the natural physical space for quantum correlations, with minimal assumptions. S^7 is the space in which octonions live, not the space that octonions created. The physical generalization of the simply connected Euclidean S^n space is only possible, because the division algebras (admitting parallelizability) are limited to S^1, S^3, S^7, i.e., the even numbered dimensions > 0 and < 8. We live, therefore, in 7 1 dimensions of spacetime with manifestly discrete real measures in 3 1 dimensions; all of this space is demonstrably local and independent of scale, because topological torsion is nonvanishing at every scale.

In my opinion, until one understands the completeness of Joy's measurement domain, it is distracting to get down in the weeds over mathematical models and calculating techniques. I think that's been the problem all along.

Tom

Richard Gill

Joy wrote "You do have quite a knack of reading everything I write in a completely wrong way. Bell made a stupid mistake in his argument".

Since John Bell is dead one can accuse him of making a *stupid* mistake. As you know, I still think that Joy Christian made a mistake, and that John Bell didn't.

So yes, anyone who reads what I write, should bear in mind that I am pretty sure that it was not Bell who made the mistake. This difference of opinion between Joy and myself is obviously a motivation for my activity here. But I sincerely hope that my mathematical and computational contributions to the analysis and implementation of various simulation models are "value free".

Joy Christian

John S. Bell was, and is, one of my heroes. But he did make at least one stupid and fatal mistake in his argument, if not several. This is not a matter of opinion. Anyone can see his stupid mistake by simply reading about it on my blog. The mistake is right there in the very first equation of his famous paper, as I explain on this page of my blog.

Rick Lockyer

Tom wrote:

"Here's what I'm talking about, Rick, from your own essay forum of 2012. Answering a post to Jonathan Dickau, you ended with the comment, "It all comes down to the structure constants p_ijk."

To which Joy made a short reply, "Or structure variables p_ijk(x) in my variable torsion picture, at every point x in S^7."

Your views topologically make the difference between passive diffeomorphism (yours) and active diffeomorphism (Joy's). "

The concept of "structure variables p_ijk(x) .... at every point in S^7" is nonsense. You will no longer have a connected space in which you can do anything useful. I have said this before, and I will say it again for surely it has not sunk in. You cannot combine algebraic elements from different algebras in a single mathematical expression, nor can you combine mathematical expressions that have algebraic elements from different algebras. This includes related algebras where the distinction is a chiral choice. This is Algebra 101.

Diffeomorphism?? You can't even define differentiation if the algebra changes within the coordinate neighborhood.

Rick

Joy Christian

Rick,

You wrote: "The concept of "structure variables p_ijk(x) .... at every point in S^7" is nonsense."

I am afraid you are grossly mistaken. Please check out references [21], [40], and [41] of this paper. You will find in these references that the concept of structure variables at every point of S^7 is neither new nor problematic. In fact it is quite enlightening and refreshing.

Joy

Thomas Ray

"Diffeomorphism?? You can't even define differentiation if the algebra changes within the coordinate neighborhood."

That doesn't happen, "up to diffeomorphism," which was Einstein's limit. Active diffeomorphism, though, in which measurement values depend on the structure of the space, allows analytical continuation, as Joy pointed out. The measure space is complete.

pjackson

Richard,

"..you are making clever use of the conspiracy loophole." Not in the way assumed. I seem to have found a ten lane overpass which does indeed "circumvent" Bell's theorem. I agree the theorem is mathematically correct, but I identify an underlying physical mechanism which it does not anticipate or model

Did you read my essay? Or analyse the 99.999% data Aspect discarded (from the only proper time-resolved pair experiment) because his source was apparently not 'rotationally invariant' and there was no theory the data fitted? Doing so has exposed the cause and derived the theory to fit.

The simple physical 'classroom experiment' is now refined, and the cause explained with more clarity in a draft paper. QM and SR converge to unity. Do you have time to take a quick look?

Florin Moldoveanu

Since simulations and programming are the topics of the day, I want to show a paper on Evolution Programming.Attachment #1: EvolProgramming.pdf

Richard Gill

Wow! Nice to see the Zipf law coming up.

(But isn't this a bit off topic?)

Fred Diether

Richard said, "It can be mathematically proven that we *don't* go from Bell's straight line to the cosine curve." in relation to what I said.

OK, go for it. Let's see the rigorous mathematical proof that what I am saying above is false. I would advise you not to waste your time on it since Joy has already extremely mathematically and rigorously proven that it is true here and with this paper here from a different direction.

New graph of 50 Million trials

Richard Gill

Dear Fred

Joy has already accepted the fact. You seem not to have noticed that he's changed the simulation model. He's also working with R, nowadays. I am not going to waste my time writing out a mathematical proof of something that Joy and I both agree on.

I suggest you install and learn R yourself. Both are pretty easy jobs.

Actually we are still only very close, still not yet spot-on. The problem is, what distribution to take for theta_0? At present Chantal Roth and Joy Christian together have homed in empirically on a pretty good choice, which in terms of Caroline Thompson's equivalent ball model, is circular caps of radius

R = acos((sin(Phi)^1.32)/3.16), Phi ~ Unif(0, pi/2).

Minkwe (lifted to 3-D - another of my innovations which Joy has taken on board) had

R = acos((sin(Phi)^2)/2), Phi ~ Unif(0, pi/2).

I've written an R script so one can experiment with various different probability distributions for R. And in particular, try various *constant* values of R. It is pretty clear that as you vary R between pi/4 and pi/2 you get various curves, none of which look much like a cosine, but such that a continuous convex linear combination of them can be found, which is exactly equal to the cosine! (OK: this is not a theorem, it is a conjecture, but it is a bloody good one, pardon my French!).

See:

http://rpubs.com/gill1109/ChaoticUnsharpBall2

The code is taken by copy-paste from Chantal and Joy's latest production. The only thing I have changed is the probability distribution of theta_0

Richard Gill

Peter,

I did not read your essay nor analyse the data Aspect discarded. However Caroline Thompson did analyse that data very thoroughly, and I am do not expect that you will have anything new to say beyond what she already said. In exquisite clarity at

http://freespace.virgin.net/ch.thompson1/Papers/The%20Record/TheRecord.htm

Do I have time to take a quick look at your simple class-room experiment? Tell me *where* I can look first, and then I will take a very quick glance, and then I will tell you if I'm interested to look further. But I doubt that you can tell me anything I didn't know already on this subject. Simple class-room experiments already exist. I already use them in class.

Nothing much new has happened since Pearle (1970).

Though I would like to mention the identification of the coincidence loophole by Larsson and Gill in "Bell's inequality and the coincidence-time loophole", Europhysics Letters, vol 67, pp. 707-713 (2004), and the resolution of the memory loophole by Gill in "Accardi contra Bell (cum mundi): The Impossible Coupling", pp. 133-154 in: Mathematical Statistics and Applications: Festschrift for Constance van Eeden. Eds: M. Moore, S. Froda and C. Léger. IMS Lecture Notes - Monograph Series, Volume 42 (2003).

https://arxiv.org/abs/quant-ph/0312035

http://arxiv.org/abs/quant-ph/0110137

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