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

" ... Nature had to evolve the possibility for measurable spaces to exist somehow - in order to create detailed structures and complexity - and that topology is the only way to go."

Exactly my argument that dimensions are self organized and self limiting -- leading to an organic continuation of mathematics with physical reality and the way we measure it.

Joy's measurement framework is rich with new research potential, as any revolutionary idea ought to be.

I'd like to continue here..

I'm starting a new thread because this is a shift of topic. In 'NCG 2000' Alain Connes loudly proclaimed "Noncommutative measure spaces evolve with time!" which is a profound insight, central to his program. PC Kainen, in 'Octonion Physics,' extended this, adding that instead of being a problem the non-associative property of the Octonions is actually a blessing and assures that geometry will be emergent. In a sense; the dynamism present in the Quaternion and Octonion algebras is what drives or enables the further evolution of form and structure, perhaps up to and certainly beyond the most elementary forms. As I said; the Reals just sit there, because they do not admit dynamism, and are useless in the continuation of computation without properties of C, H, and O.

One might assert that Connes statement could be reversed to say that if we are in a space where dynamism is allowed and time evolution is displayed, it MUST be a non-commutative measure space, rather than a simple Euclidean one. That is; perhaps the evolution of form is impossible in spaces that are strictly limited by a lower-dimensional bound, and admit no higher-dimensional components, because the properties of higher-dimensional spaces like the quaternionic and octonionic space are essential to assure the continued evolution of possibilities. In my view; the sequentially evolutive aspect of the octonions clearly makes Rick correct, in that they like to drive the process or tell us (and Nature) what to do next.

More later,

Jonathan

In case you missed it..

I was suggesting that the beginning of Cosmological TIME came with the appearance of Octonionic space. This is because it is the simplest (fewest rules) algebra that flexibly admits sequential evolution. If it is an essential property that continuation or possibilities must be preserved for the universe of form to come into existence, then sequential evolution - through procedural stages - is a must, and the octonions clearly afford and preserve this property. According to Hurwitz, Frobenius, and Bott - we find that only R, C, H, and O satisfy certain essential conditions of regularity - which Nature apparently observes.

Ergo; if the universe exists and evolves with time, where successive states are necessarily dependent on prior results, we are living in octonionic space. So perhaps the new really big questions are "Does the universe actually exist?" and "Does physical reality evolve with time?"

Have Fun,

Jonathan

Joy,

You (unnecessarily) assign lambda as the orientation choice for the algebraic basis used, then sum over lambda_k with the requirement on Nature of a 50-50 statistical choice of +1 and -1. Like many before you, you conflate basis element and coefficient definitions when you talk about beta_x and -beta_x. You may be able to properly represent both possible orientations of quaternion bases by doing this, but when you multiply algebraic elements in one basis by algebraic elements in the opposite orientation basis, you break the rules of quaternion algebra for rather than three non-scalar bases, you now have six. If instead you assign lambda as a coefficient with 50-50 chance of +1 and -1, and maintain a consistent singular basis set, everything works out as you intended it to, and the basis set orientation becomes a free choice made once for the full run. The math works out to the same conclusion for either orientation choice.

This is the essence of Nature's clue to us regarding mathematical (algebraic) physics and physical reality. Orientation is a singular choice made by the physicist and must be consistently maintained throughout, and Nature demands the math had better not care which choice(s) is(are) made. For all those who fret over "extra" dimensions, this is a clue for you also. There are detectable forces with of course singular reaction directions that are mathematically defined as a non-scalar algebraic product of non-scalar items. Orientation is fully in play in this case, yet the result must have a singular reaction direction independent of the orientation choice. Think about the equivalence of rest frame electric field with moving frame magnetic field for the force on a charged particle, as well as the central scalar * non-scalar force on a charge in the presence of and electric field and realize you cannot deal with this mathematically without algebraic structure with more than 4 degrees of freedom. 4D space-time deals with this with the second rank field tensor, yet can't provide the required degrees of freedom to cover gravitation AND charge central forces in the same expression without the additional structure of intrinsic space-time curvature. A 4D space-time, up ranked tensors, intrinsic curvature, Minkowski split signature are all unnecessary if the fundamental basis space is upped to 8 dimensions and its algebra ruling ALL of physics is Octonion. Smell the coffee, wake up and drink it, then re-read my 2012 FQXi essay.

Joy, you have done great work because you realized reality is proximal to the division algebras. You have just extrapolated beyond where you needed with unnecessary basis orientation connections and you really don't need the non-division associative 8D algebra you allude to which shares the used quaternion sub-algebra with the real deal, the non-associative Octonion Algebra.

Rick

"You (unnecessarily) assign lambda as the orientation choice for the algebraic basis used, then sum over lambda_k with the requirement on Nature of a 50-50 statistical choice of +1 and -1."

Come on, Rick. The variables are nonlinear input to a continuous function. The experimenter is forbidden to assign initial condition. That's the whole problem with Bell's choice of measure space.

"The math works out to the same conclusion for either orientation choice."

The math does not work, however, for a function continuous from the initial condition. One would get only a flat line for A, flat line for B.

Tom

Aw shucks,

Rick makes some good points, and deserves some acknowledgment for what he gets right. It is absolutely accurate to state that some of the constructions used by Joy are cast in the Quaternion case, because it captures the most salient features, and that this works because S3 and S7 share certain essential properties - topologically speaking. It is clear that the case stated with S7 is more general, but might be harder to sell because people are scared off by Octonion algebra.

Beyond this; I think that the dynamism I talk about in the comment below is an essential feature of the Octonion algebra that makes it indispensable, for explaining the Physics involved, rather than just a convenient representational schema. On the other hand; I think that realizable geometry is absolutely necessary to Nature, for creating persistent structures. The Octonions describe the actions and motions possible on S7, and without the curious properties of that algebra, some portions of Joy's argument (especially as involves GHZ states) fall apart.

I think the geometry and topology involved does constitute a deeper truth, but this does not make the powerful algebra of the octonions any less essential to understanding those remarkable properties.

Regards,

Jonathan

I should probably add..

It is obvious to me that without some of the dynamic properties of the Octonions, it is likely not possible for nature to evolve topological spaces at all, but their existence (following the reasoning of Kainen) assures that this geometry must evolve or emerge. Again; the hierarchy of spaces smooth > topological > measurable, which Connes talks about (also in NCG 2000) involves the application of rules with progressively stricter conditions, as Fred was talking about with algebras above.

Spelling this out; the Associative rule in Algebra is pertaining to the subject of Surfaces or Topological distinctions - because it deals with interiority/exteriority, in determining what is or must be inside of what. The applications of parentheses can be equated with the creation of a topological boundary or distinction, in that each may be seen to be a container or separator from those elements that lie outside the boundary.

In my view; this process of acquiring topology flows naturally, once S7 or the Octonions come to be, or are well-defined as a system. I don't see how one could get topology from a blank slate without the strongly directed evolution of a number type like the octonions.

All the Best,

Jonathan

" ... in my view algebra provides merely a convenient representation of the deeper truth, which, for me, is topology---in particular, the topologies of S^3 and S^7 (I do not care much about S^1)."

Right on! Whatever misunderstanding that I or anyone has about this measurement framework, let it not be that the model isn't analytical.

Joy,

Referenced document eqs (16) and (17) seem to be multiplying as I stated. Perhaps my cursory look missed something.

Rick

" ... where the relative orientation lambda is now assumed to be a random variable, with 50/50 chance of being +1 or - 1 at the moment of creation of the singlet pair of spinning particles."

Seems clear to me that A and B are dichotomous variables. Classical probability.

Jonathan,

Thank you. That was a serious direct answer. The ensuing conversations were enjoyable to read.

James Putnam

P.S. The enjoyment of reading new on-subject messages about Joy's accomplishment has returned for now.

Tom,

Me: "You (unnecessarily) assign lambda as the orientation choice for the algebraic basis used, then sum over lambda_k with the requirement on Nature of a 50-50 statistical choice of +1 and -1."

You: "Come on, Rick. The variables are nonlinear input to a continuous function. The experimenter is forbidden to assign initial condition. That's the whole problem with Bell's choice of measure space."

You quoting Joy later in same thread: " ... where the relative orientation lambda is now assumed to be a random variable, with 50/50 chance of being +1 or - 1 at the moment of creation of the singlet pair of spinning particles."

Come on me??

I have yet to see Joy weigh in on your connecting his work to your personal view of continuous functions as implied here. Maybe this is because he does not have any better idea than I do about just what you are implying. Personally I have no idea what you are talking about, and I believe I have a good appreciation for continuous functions.

Perhaps you could elaborate.

Rick

Jonathan,

""Does physical reality evolve with time?""

Or is time simply an effect and measure of the process of evolution?

Is this process truly a vector of evolving configurations, or is it one state that evolves?

Is blocktime necessary, or is it real dynamism?

Regards,

John M

If I may offer,

I know that a crucial point Joy makes in his book is the anomalous choice of a binary measurement space by Bell, implying a codomain equivalent to S0, where the unit 0-sphere contains only the points -1 and +1, a disconnected space. A key observation, to understand Joy's work, is that a disconnected measurement space is unduly restrictive, or unrealistic.

Indeed; J.B. Pors et al., in their paper on Shannon dimensionality, show experimentally (using a circular phase plate) that a broader range is available than can be represented by a binary choice. They clearly demonstrate that the actual codomain of quantum correlations is at least as 'big' as S1, or subdivisions thereof, but do not set an upper bound on its actual range.

Joy's assertion that the actual codomain is S3 provides both a continuous range and a simply connected topology. This choice can explain a broad variety of phenomena which have been attributed to non-local entanglement. But the big piece is that the measurement space is continuous, not disconnected.

Regards,

Jonathan

Thanks Florin, for the great picture/quote.

And no John I don't think Blocktime captures the essence of time's mode of transit. The dynamism I speak of is both deterministic and open-ended, so I am really mostly in agreement that it is the evolution of form and structure which creates the flow we see as the progress of time, rather than seeing time as something linear or space-like.

I had an interesting discussion with Huw Price about this question at FFP10, championing my view of Entropy as depicting the spreading of energy or substance into a medium - rather than the common view that entropy represents disorder. His talk focused partly on the utility of the Blocktime view for relativists, and I pointed out some drawbacks (or undesirable implications) in our conversation.

On the other hand, maybe if it was an 8-dimensional block... In that case, though; since the Octonions rule, time's evolution would still be a process of becoming and not a straight line.

All the Best,

Jonathan

Hi Jonathan,

How do you make the connection from the paper that "They clearly demonstrate that the actual codomain of quantum correlations is at least as 'big' as S1,..."? Your last paragraph is mostly right on the money except the codomain is S7. EPR-Bohm only requires S3 which is part of S7.

Best,

Fred

To those familiar with Joy's work, both those in support and those in agreement. I am a novice who is unfamiliar with the computer programs used to do the simulations credited with proving or disproving Bell's theorem, but I have a few innocent, well meaning, even if naive questions to ask:

Do the Lines and Planes contain the same number of pixels?

Are the Planes made up of Lines?

If yes, where did the Plane get its 2-dimension from if the Line has only 1-dimension?

If no, what then is the Plane made of?

Does a short line ___ have the same number of pixels as a long one _________?

Is it right for Digital (discrete) assumptions to be used for simulation and be unacknowledged in giving a conclusive answer to a Continuous debate?

I raise these questions not to derogate from anyone's work. Indeed, I am made to understand by Jonathan that Joy's work is meant to bring back reality to the Quantum domain. A worthwhile task. But we must not sweep anything under the carpet if the truth and the whole truth must be unveiled by this laudable task.

Akinbo