Hi Tom

Thanks for your kind words earlier.

On the subject of dealing with particles, without the particles, I have a question: does my condition that the hidden domain of Joy's analysis be enclosed by a S2 surface give a form of the holographic principle?

For the hidden domain containing topological monopoles, the S2 surface - with a radius small enough to render it hidden - can be labelled with the spin and particle charges within; for a singlet state this labelling includes a non-trivial spin and particle symmetry orientation, but as hidden variables. The S2 surface can be labelled with all the values of its contents for the case where those contents are topological monopoles and so the properties of the hidden contents of the domain are in effect projected onto the S2 surface.

Joy's Clifford algebra formulation proceeds on the basis of the hidden variable being an orientation - although not explicitly of such an S2, it is compatible with this interpretation. With this view, Joy's framework connects experimental correlation results with the topological properties of the S2 surface. In which case, experimental results viewed through the perspective of Joy's framework would determine the properties of the S2 surface, and so apply constraints upon the possible theory describing the contents of the hidden domain. Given the uniqueness conditions I find for my theory, I have an obvious suspicion about the answer.

I would note that the purely geometric view only gives topological monopoles/anti-monopoles (space S0), but no wave property. The replacement of S0 by the S1 fibre-bundle of wave(S1)-particle(S0) duality for the particle content of the hidden domain marks the transition which is normally associated with the appearance of QT. So the theory of the hidden domain would have to explain the origin of this replacement. In my work this comes from a replacement of the integers with the reals, which has underlying it issues about the "geometry" of (infinitely recursive) functions over the counting numbers compared to the reals. As you know, this is a separate issue from the S7 space of the observable functions, but surely it impacts the nature of the functions themselves?

Michael

Joy's question to me in this thread probably would have been better suited for my essay blog, but since it was posted here I will try to answer it here. I have only done a small amount of thinking about algebra : topology -> homeomorphic topology : algebra in terms of algebra algebra if this is what you are getting at, and had previously read about exotic spheres. On the question of whether or not one could demonstrate the diffeomorphism issue with exotic spheres in a purely algebraic fashion, I have neither answer nor informed opinion or guess. On the question of how it fits in to my "rather algebraic perspective", I can't say it does.

Understand that I do not think all of physical reality is on the unit 7-sphere. Within my perspective I fully expect the p coefficients for the native Octonion position algebraic element p_i e_i to freely range from +oo to - oo. This is not to say I think your representation as well as Michael's can't be so restricted, because I actually agree they may. But this does not require me to be similarly restricted, nor does it preclude my approach from having some topological agreement at some level of abstract representation.

I am not looking to morph coffee cups into donuts. My Ensemble Derivative indicates the proper transformation characteristics for an Octonion analytic form within a single topological space. The 1/J scaling emphasizes the requirement for the Jacobian of any suitable transformation to be non-zero hence reversible hence a diffeomorphism. It works, as demonstrated by demanding the Lorentz transformation for EM.

Octonion Algebra demands the form of the Ensemble Derivative and the Law of Algebraic Invariance as stated. The need to have algebraic invariance within covariant differential equations on potential functions describing observables produces equations of algebraic constraint. It dictates the form for the mathematical expression of all stresses, strains that must integrate to zero to describe stable stuff, be it electrons, protons, neutrons, atoms, photons, etc. I have provided the explicit form for all of this, and nothing is inserted "by hand". Is there enough there to "lead us all the way" as you ask? I think there is enough to produce a family of solutions for the potential functions that would give us a clear picture of just what reality is. Achieving these solutions I think will be a non-trivial task. I loathe solving differential equations, and really wish people with better skills would jump in.

One question to all: if you do not plan on "doing the math" in both homeomorphic topologies, does it matter if they are not diffeomorphic? I wonder if this precludes some meaningful connection.

Does this answer things satisfactorily?

Rick

Hello Michael,

" ... does my condition that the hidden domain of Joy's analysis be enclosed by a S2 surface give a form of the holographic principle?"

My opinion? -- Yes. And I base it on first principles alone: simple connectivity, orientability, geometric projection. As I note in my essay, the strength of quantum correlations orients complementary quantum properties on a line from the horizon, to a middle value. Because the middle value sweeps the entire horizon, holographic projection is assured.

Making this mathematically rigorous will, I am confident, depend heavily on the non-trivial topological properties of S^3 (as Joy has frequently emphasized). Of the four spacetime dimensions,because S^2 is the middle value of the parallelizable S^0, S^1, S^3, S^7. we should expect a continuous projection on the S^2 manifold between S^1 and S^3.

"I would note that the purely geometric view only gives topological monopoles/anti-monopoles (space S0), but no wave property."

Yep. That's why probabilistic functions on the interval {- OO, OO} cannot provide a space of complete measurement functions.

"The replacement of S0 by the S1 fibre-bundle of wave(S1)-particle(S0) duality for the particle content of the hidden domain marks the transition which is normally associated with the appearance of QT. So the theory of the hidden domain would have to explain the origin of this replacement."

And would have to include a continuous wave function, rather than a probability function.

"In my work this comes from a replacement of the integers with the reals, which has underlying it issues about the "geometry" of (infinitely recursive) functions over the counting numbers compared to the reals."

Right on, brother. It's what motivated my NECSI ICCS 2006 paper, getting a well ordered counting function without Zorn's lemma (axiom of choice). Eliminates a host of evils.

"As you know, this is a separate issue from the S7 space of the observable functions, but surely it impacts the nature of the functions themselves?"

You bet it does. It makes orientability work in every simply connected space of S^n. Seeing the limit of S^n physical space in Joy's research (corroborated by Hestenes's spacetime algebra and Rick's octonion space) was a major piece of the puzzle for me. Then getting the S^3 X S^7 ---> S^10 mapping from your research explains discrete particle events without having to assume discrete particles. I like!

All best,

Tom

Hi Michael and everyone,

In another blog I said to Joy, "You remark that "one can view a 7-sphere as a 4-sphere of 3-spheres" and elsewhere "...a 7-sphere can be thought of as a six sphere worth of circles." Then I asked, "Can this 6-sphere then be divided into two 3-spheres...?"

Joy replied: "You ask: "Can this 6-sphere ... be divided into two 3-spheres worth of circles?" The answer is no, because 6-sphere worth of circles is not the same as two 3-spheres worth of circles. It makes no sense to replace the 6-sphere base with two 3-spheres base. No such fibration of the 7-sphere exists.

But in this current blog Michael says:

"...the group spaces are S3, S3, S1 and fit into S7. [where...] The spin space S3 has a spatial origin, whereas colour space S3 and electromagnetism S1 have an origin in the particle symmetry space S7... [but] the issue I have is with the composite view: whether the differential manifolds of the particle symmetry spaces (S3, S3, S1) spaces can form S7. At best they could only give the topological S7 and *not* the differential manifold S7, which is ...required for Joy's parallelisation condition. [...] So for component spaces S3, S3, S1 - whatever their origin - this demands that from the perspective of observable functions these spaces *must* form the topological space S7, and then the parallelisation condition of Joy demands that it *must* be the differential manifold S7"

Now my interpretation of the two spheres (S3, S3) was not the same as Michael's, but I fail to see what the physical interpretation ["whatever their origin"] has to do with it. Why did it "make no sense to replace the 6-sphere base with two 3-spheres base" in the other blog, but now seems to make sense that "(S3, S3, S1) spaces can form S7" in this blog. I'm sure I'm missing something, but what is it?

Edwin Eugene Klingman

    Hi Edwin,

    Michael is simply exploring whether there are any connections between our respective approaches. He is simply thinking out loud to get input from us.

    But the answer to your question is simple. Michael is talking about the *total* or *bundle* space S7, which can be decomposed as S3 x S3 x S1 locally (i.e., at some point of S7).

    You, on the other hand, were asking me about dividing the *base* space into S3 x S3. But for that to be possible there must exist a fibration of S7 of the form

    S1 --> S7 --> S3 x S3.

    But no such fibration of S7 can exist because S3 x S3 has a hole in it (i.e., it is not simply-connected). What *does* exist is a fibration of the form

    S1 --> S7 --> S6.

    Joy

    Hi Joy,

    You say, "Michael is talking about the *total* or *bundle* space S7, which can be decomposed as S3 x S3 x S1 locally (i.e., at some point of S7). You, on the other hand, were asking me about dividing the *base* space into S3 x S3. But for that to be possible there must exist a fibration of S7 of the form S1 --> S7 --> S3 x S3."

    Actually, I was not asking about dividing the *base* space into S3 x S3. What I was trying to ask was whether S7 could be viewed as two S3's connected somehow by an S1. This seems to me equivalent to Michael's (S3, S3, S1) spaces forming S7.

    There seems to be a lot of misunderstanding all the way around. There are many well-informed and bright people on this blog, but I don't think we're all playing on the same field. Nevertheless it's quite interesting to see this tennis ball being slammed between different courts.

    Edwin Eugene Klingman

    Hi Rick,

    Thank you for your detailed reply. I appreciate that.

    You kind of addressed my question. Let me summarize what I have gathered from your answer.

    You wrote: "On the question of whether or not one could demonstrate the diffeomorphism issue with exotic spheres in a purely algebraic fashion, I have neither answer nor informed opinion or guess. On the question of how it fits into my "rather algebraic perspective", I can't say it does."

    Well, for what it's worth, my guess is that one cannot address the diffeomorphism versus homeomorphism issue of exotic spheres in a purely algebraic fashion. Therefore I am not surprised that it does not fit into your algebraic perspective.

    Having said that, I do appreciate the power and versatility of your approach---your results speak for themselves! But I think it leaves out important, global aspects of the 7-sphere, such as Milnor's startling discovery. And that is what I was deriving at. The issue, for me, is not about homeomorphism versus diffeomorphism. The issue is about local versus global. And your approach is at best silent on that issue.

    Joy

    Hi Edwin,

    You wrote: "What I was trying to ask was whether S7 could be viewed as two S3's connected somehow by an S1."

    Yes, but only locally (in a local neighborhood). It is not possible to decompose S7 globally in that manner.

    Joy

    Typo: In the last para I meant "driving at" not "deriving at."

    Hi all,

    Local-global differences are at the heart of this algebra-geometry issue: algebra can see the local, but it requires geometry to see the global. I stumbled across a quote by David Hestenes and Garret Sobczyk which seems to capture it:

    "Geometry without algebra is dumb! - Algebra without geometry is blind!"

    Edwin's point in the thread below is about the struggle with the local-global difference, where conceiving S3 as locally being S2*S1 but globally a sphere and not a torus is doable, but S7 is just that more difficult.

    Equivalence of my work with Joy's framework would be strictly local as formulated, because it is GR with curvature. But as all the relevant experiments would similarly be local, this is not really an issue. I have a critical global condition of the electroweak vacuum being a map from S7 to S3 in the product space S3*S7, which is globally a torus - a hole has to physically inserted into the "unified" S10 to get the global change from a sphere to this torus. It is this mapping S7 to S3 which picks out a S3 from the S4 base-space of S7 (the coordinate parameterisation given in my essay notes indicates that this S3 is not just the S4 equator). The identification of the S7 subspaces defined by the electroweak vacuum map with group spaces gives the form of the *local* theory in the "broken" S3*S7 phase - note physical spaces in the global theory become group spaces in the local theory. As Lie group spaces are parallelisable, it is this identification (common to Kaluza-Klein-style dimensional compactification theories) which gives the parallelised spheres (spin S3, colour S3, electromagnetism S1) in the *local* form of the theory. The possibility of a flat S3 cosmology with teleparallel gravity, suggests that the product space S3*S7 being a flat torus with a global electroweak map from S7 to S3 may also be possible. This would require the map to be a precondition, something which unifying S3*S7 into S10 by a topological transition removes - hence its appeal to me. There are also GR issues which are resolved by the same S10 unification.

    Michael

    Hi Edwin,

    There is room for confusion here, especially because there are a number of different spaces and a missing bit. The 3 spheres of my particle symmetry spaces - colour S3, isospin S3, hypercharge S1 - form S7 because that is where they came from in the first place. The electroweak vacuum (a map from S7 to the closed universe S3) breaks the isospin symmetry (S3), leaving intact symmetries: spin S3, colour S3, electromagnetism S1, which *do not* form S7 despite appearing to be the subspaces of S7.

    The simple comparison of these spaces with Joy's S7 condition isn't valid as it isn't comparing like with like. Joy's correlation condition applies to the codomain of functions giving observables, not to particle symmetry spaces. It comes from the EPR-Bell definitions of locality and completeness, and Joy's parallelisation - the conclusion is that the codomain can only be S3 or S7. This condition is independent of what the particle symmetries actually are, but should apply to any number of any correlated particles anyway.

    For colourless particles like electrons and photons my S3 colour space is irrelevant and can be ignored. This gives the symmetry spaces S3, S1 for the particles considered in Joy's correlation analysis, where the S7 condition applies for 3 particles - but the S3, S1 don't form S7 either. Yet Joy obtains the correct correlation result, and these *are* the symmetry spaces - the missing bit is how to square these 2 facts with each other (note that this applies to Joy's framework irrespective of anything of mine). My contention is that whatever the mechanism, it will also work for the case when the colour space S3 is included - as it is still an S3 like the spin space - but not if the colour space isn't S3.

    For the particles in question being topological monopoles arising from a broken S7 particle symmetry space, there is a connection through maps of S3 and S7 to a S2: topological monopoles in my case; the surface enclosing a finite hidden domain in Joy's. With the same collection of spheres, the same underlying geometric algebra and consequently a related specification of uniqueness, it doesn't just seem like co-incidence to me.

    Michael

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    • Post #161

    Michael,

    "Geometry without algebra is dumb" is on the money. The second half: "Algebra without geometry is blind" certainly works for algebra applied to physical reality but not for algebra in total. Your conclusion "algebra can see the local but it requires geometry to see the global" is somewhat counter-factual if you admit geometry generally (more than the "dumb") requires algebraic expressions for its representation, since the two cannot then be cleanly bifurcated.

    I think we could come up with a similar quote using mathematics and physics instead of algebra and geometry. And just what is this algebra - geometry controversy anyway? I hope you are not including me in it based on my detailed analysis of Octonion Algebra and attempts to use it to define a better mathematical basis for Physics. Joy's response here that my approach does not address "the global" suffers from too narrow a view on just what this means and just what I am trying to accomplish.

    Mathematics per se, physics, topology in many forms, algebra, geometry, and group theory etc. are all interlocking components of physical reality. I see no justification for pitting one against another. Anyone needing to be dismissive of any should re-evaluate their position, for it should be interpreted as a well lit billboard sized warning sign stating "Dead End Ahead".

    Rick

    Michael,

    Here is a better quote, from Michael Atiyah:

    "Algebra is the offer made by the devil to the mathematician. The devil says: I will give you this powerful machine, it will answer any question you like. All you need to do is give me your soul: give up geometry and you will have this marvelous machine."

    Sorry, Rick. I couldn't resist (I know; I am being frivolous).

    Tom, I can both drive and derive at the same time!

    Joy

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    • Post #164

    Joy,

    With the response I am getting in this essay contest, I am sure many yelled out "Blasphemer. Stone him, stone him" as soon as they got to the word "Octonion" in the abstract, no need to read further. I am happy Michael is doing as well as he is, but he should be doing better.

    Rick

    I had not heard that quote of Atiyah, and I absolutely love it! It fits to a "t" Wheeler's information-theoretic view, imbedded in our classical world experience of continuous functions.

    Tom

    Joy,

    That's a wonderful quote! I have to admit that I was a reluctant octonion, but not accepting one's own conclusions would just be bad form.

    Coming from a QFT view, my natural way to accommodate QT not being fundamental is to just accept it as an approximate descriptive form with illusions, and not try to interpret them literally. Your experience seems to indicate an attachment by some to the illusions being literal. Since to me it is just about description, I accept that there may be another way. Proponents of geometric algebra have to my mind clearly demonstrated that in some sense it is the natural way to describe space, as the descriptive form is free from the illusions - such as in your work.

    Tom,

    Before you get too excited about the idea of dealing with particles without having particles, a hidden domain enclosed by a small radius S2 looks somewhat like a particle itself, or a type of multi-particle containing any number of correlated particles. It is also only a non-relativistic view that doesn't include particle reactions that change particle numbers ... yet?

    What the octonion view of such a 2D surface (homeomorphic to S2) does give you, is an easy explanation of colour confinement. With coloured particles being topological monopoles with unbroken symmetry group (Spin(3)*U(1))/Z_3 the same topological unwinding arguments as for topological defects in field theory can be applied, with conclusions:

    1) a monopole/anti-monopole pair are connected by a colour string

    2) the Z_3 means that the colour field of 3 coloured monopoles with no net colour charge can be unwound outside of the S2 domain

    Energy minimisation then implies the S2 is shrunk to the smallest size possible, giving colour confinement in classical physics - because the monopoles come from S7. This adds to the suspicion that the conclusion is going to be, in the words of Clinton: it's the octonions, stupid ;-)

    Michael

    Michael,

    I too have been a reluctant octonion. "Extra dimensions" were anathema to me until my own investigations in the origins of quantum correlations persuaded me otherwise. There is no way to reproduce all of the observed quantum correlations local-realistically without embracing the octonionic 7-sphere fully. All other attempts to understand quantum theory will eventually hit a brick wall. In this context my attitude towards the octonion algebra is very much the same as Rick's.

    On the other hand, you are right to think that the proponents of geometric algebra have got it right. The credit for their insight, however, must go to Hermann Grassmann first before anyone else (cf. the attached slides).

    Joy

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    • Post #169

    Joy,

    Sounds like you want to have your cake and eat it too: non-associative Octonion Algebra and associative Geometric Algebra.

    Rick