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

Hello Joy,

Good to hear of the computer simulations confirming your results! Shouldn't it also be possible to confirm this experimentally? From what I understand of your results you mathematically show quantum correlations to be dependent on the geometry of the experimental apparatus. While Bell assumes a "flat" geometry you use "spheres". Any way such design be tested confirming your predictions?

Best wishes,

Constantinos

I wanted to add to my comments above..

It is common to assume with higher dimensions, that they are somewhere 'out there,' when the reality is that this notion only works up to a point (~5.25-d) and after that adding more dimensions makes the array smaller, or more compact - at least until we reach 24-d.

So rather than being something 'out there,' higher dimensions could be 'in here' instead. This actually corresponds to the fact that the algebras and spaces are first non-commutative, and later non-associative as well. So our common notions of size and distance, and then of interiority/exteriority, become invalid. This explains how the higher-d reality can be situated 'in here' rather than 'out there' "external to the universe."

All the Best,

Jonathan

Joy,

If I had $200,000 to spare I would have gladly made that investment for science. Perhaps SciAm or other well funded sponsors may seize this opportunity to contribute to the betterment of mankind! Does anybody out there have $200,000 to sponsor the future of physics? Bill Gates? Buffett? Elliason? Anybody? If there are funds for fqxi essay contests, why not for real experiments?

Constantinos

Hi Jonathan,

"..the fact that the algebras and spaces are first non-commutative, and later non-associative as well."

I believe you have it backwards from what Nature actually did. I will say it again this time with some more explanation; In the beginning there was no math (rules), then came sedonions, then octonions, then quaternions, then complex rules, then finally the rules for real numbers. Easy to see that the order here is from no rules to the most math rules. Take a bunch of massless point-like entities and let them go in a complete void that has no math rules to start with. The only property that the void has is it is a stage for the actors. You basically end up going from "infinite" dimensional to 4 dimensions. I will explain more if necessary.

Best,

Fred

Ah So,

Yes you have it exactly right, and I understand - mostly. The point I was making was to clarify that as we move toward discussing higher-dimensions, they need not be seen as something 'out there,' apart from the universe. And accordingly; it is the absence of the rules of commutativity and associativity that mess with our ability to apply the convenient distinction between near and far, or inside and outside, respectively - so that is what makes those dimensions appear compact or within instead.

I like your construction Fred. But I would argue that what you are really talking about is the upper bound on dimensions that shrinks as more rules are applied, and that there is a lower bound at each stage as well. 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.

All the Best,

Jonathan

I should add..

I agree precisely with your main point above Fred. While people have a hard time imagining a blank slate being a higher-dimensional space, it makes sense that having no rules would allow just that. And one can certainly assert that specific arrays of dimensions would appear as rules were formulated or instated. The sedenions set the stage, or engender the possibility for further evolution, and S15 has only one fibration, yielding O, H, C, and finally R.

So yeah; the Reals are in a way an end product of a process that began with the Octonions, or arguably before that.

Regards,

Jonathan

Thanks Joy,

Glad to see you chime in, and your clarifications of course make sense of things.

Regards,

Jonathan

Yes, thanks Joy,

Since your great discovery of the origins of quantum correlations, we have to take very serious now the aspect of extra spatial dimensions over the 3 that we experience on a daily basis so what I am proposing makes logical sense to me based on the division algebras. Funny that we discovered them in basically reverse order of perhaps how they naturally happened. Of course that "happening" could have been very fast. :-) Maybe there is a clue in this somehow to make all of physics fit together.

Best,

Fred

FQXi has become a bit boring as of late with the same people saying the same things I find quite uninteresting, so I rarely take a look. Nice to see some discussion here on more interesting things.

Emergence as applied to physics is a bunch of poppycock. The only thing about physics that is emergent is understanding from ignorance, and it happens not with Nature but only between our ears. It is quite incorrect to say the reals emerge from a higher dimension algebra, especially when you say an algebra over the *field* of real numbers. For then you cannot define multiplication in the higher dimension algebra without a priori definition of multiplication over reals, in essence the algebra of real numbers.

What exactly is your point here anyway? What is gained by Nature did this, *then* it did that? I see no utility whatsoever.

Might take a look at the simulations. Perhaps it will answer my question of whether or not the algebraic operation of addition is performed between results generated from two distinct algebras, the distinction being orientation. This would be quite improper.

Rick

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