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

Hi Jonathan,

"One thing Ray Munroe insisted on that I agree with is that, geometrically speaking, both the minimal case and maximal or extremal cases must be considered as bounding conditions of reality, if we are to completely make sense of things."

This is one of the things in which I disagreed with Ray. A coordinate-free analytical framework without boundary conditions is indifferent to maxima and minima. "Finite and unbounded" is a perfect description of the space of general relativity; only when we reduce it to a measure space, are we compelled to have end points. Spacetime is not so constrained a priori.

Extra dimensions are mathematical artifacts -- string theorists, I think, are largely enchanted with the power that extra degrees of freedom impart to the calculating machinery.

Einstein, himself, though, was not opposed to using extra dimensions (Kaluza-Klein had convinced him of the utility of their argument) provided that "... there are sound physical reasons to do so."

Joy has convinced me that the sound physical reasons lie in spacetime topology that (by the very definition of continuity in topology) is continuous at the extreme of torsion, which remains nonzero. This is the condition that guarantees a consistent measure space from minus infinity to plus infinity -- the same measure space that John Bell chose *without* the topological continuity. Bell's measure space is disconnected before a measurement event, and multiply connected following a series of measurement events. Joy is right that it should have seemed obvious that a measurement function continuous from the initial condition would have to be the product of a simply connected space in order to get EPR's predicted result. Except that it wasn't so obvious (to me or anyone) until Joy's subtle mathematical treatment made it so.

I always get a little uncomfortable here when we get into the properties of the division algebras in which the measurement framework is explained. It isn't the discrete measure space that accounts for the result; it is the simply connected continuous function that allows correlation of points of the parallelized 3-sphere. I look at the division algebras as a scaffold from which to build the framework, and I fully expect that when the measurement *theory* is complete, the scaffold may be removed.

All best,

Tom

Thanks, Jonathan.

" ... it is the far edge of the universe that sets the local scale of objects."

Here's the thing, though -- if the universe has no edge ("finite and unbounded") there is no global-local boundary. Joy's result instantiates that fact in a measurement schema.

"But to think about things this way turns our perceptions inside out, or makes us see the fabric of spacetime that way, when one needs to see that fabric from the outside in - to know its nature."

Agreed. Non-zero torsion in the topological framework makes inside-out, outisde-in.

All best,

Tom

Speaking of inside-out and outside-in, the examples of a Mobius band or the Dirac belt trick (both of which Joy has used in the past) give a good representation of how this physically happens. We know that a complete transit of an orientable object on a Mobius band embedded in a 3 dimension space reverses its orientation. Where? -- it isn't sufficient to say "at the twist of the band," because there is no way to determine objectively that the object has transited the twist when the observer is embedded on the band with the object. The Bell-Aspect result assumes just such a condition -- identifying the observer's orientability with that of the object.

A couple of FQXi essay competitions ago, I tried to explain reversibility in figures 3 & 4 -- such that one can see objectively, that every path of an oriented element leads to its reverse orientation in a simply connected space of continuous functions.

Tom

Wow great comments!

I partly disagree with the either/or flavors offered, though, as I think the a priori and ab initio approaches come together seamlessly. This is implied by the result presented in Torsten's essay this year, showing that simplicial and analytic constructions actually create identical structures (also recent work by the CDT folks). The big thing is that one cannot have a measurement framework that is not geometrical, so as the constructivists assert, any act of determination is inherently a construction process as well.

Fred's remarks reflect the mindset of Wolfram's New Kind of Science, and I think it is sound reasoning that Nature's explorations would tend to seek rules that afford continuation or possibilities, while threads that do not afford continued evolution would quickly die out. My most focused research over the past 3 years has come out of the simple premise that all higher learning hinges on the skill of distance estimation, which is acquired by human children around the age of 2 1/2. What is actually acquired is a sense of the dimensionality of objects and spaces.

So it is germane to our discussion to ask "How does Nature acquire a sense of dimensionality?" I think this involves the hierarchy of smooth > topological > measurable spaces that Connes talks about and as with Fred's line of reasoning about numbers - a gradual application of stricter and stricter rules (which Joy explains very well above).

All the Best,

Jonathan

Continuing..

If we posit that the ONLY cases worth examining are those where the a priori and ab initio approaches lead to the same result, then look at what remains; a pattern emerges where certain invariant structures in Math play a pivotal role. If we assume "make no preference" is the way to go, and further assert that Nature does the same, we find the following correspondences are essential.

First off; "make no preference" sounds like the Sedenions and S15, but the only decompositions yield S7, S3, S1, and S0 or O, H, C, and R algebras equivalently. If only threads that allow continuation or possibilities are allowed; what's obvious is that the Octonions are the workhorse, and do most of the driving, just as Rick has asserted all along.

So we don't need to express a preference for one type of number over another, and neither does Nature, but the Octonions are pure dynamism, while the Reals just sit there - computationally speaking. So if the criterion is only those rules that allow continuation or possibilities, the choice of the Octonions for an early creative role (shaping Natural law and the universe) is automatic. I'll stop there for now.

All the Best,

Jonathan

"The big thing is that one cannot have a measurement framework that is not geometrical, so as the constructivists assert, any act of determination is inherently a construction process as well."

That's excellent, Jonathan. It's also a way of describing the result of Brouwer, Dedekind, Weyl, et al, that all real functions of a real valued variable are continuous.

Tom

I wanted to add..

This all speaks to the question of computability, and what level of structure is necessary to making computation possible, or whether one needs a structure for computation to take place. My opinions about this are somewhat colored by long discussions with Brian Whitworth, and by a short conversation I had with Gerard 't Hooft at FFP10 where I asked the question explicitly and he replied that we don't need atoms of space, because the laws of nature do the calculating for us.

In my view; the move from smooth to topological spaces is essential for this to take place (because topology stores information efficiently), and further makes possible measurability. If gauging the measure of dimensionality is essential to higher learning, and we combine fact this with the constructivist principle that measurement = construction, we arrive at the conclusion that 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.

All the Best,

Jonathan

" ... 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

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.