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FQXi FORUM

September 17, 2021

CATEGORY: FQXi Essay Contest - Spring, 2017 [back]
TOPIC: What is Fundamental? by Geoffrey Dixon [refresh]

Author Geoffrey Dixon wrote on Jan. 11, 2018 @ 17:12 GMT

Essay Abstract

Short answer: I don't know. I have some thoughts - almost opinions.

Author Bio

A disillusioned curmudgeon who, like many others of this sort, has retired to the woods of New Hampshire to write and enjoy life.

Download Essay PDF File

Jochen Szangolies wrote on Jan. 11, 2018 @ 21:49 GMT

Dear Geoffrey,

it was a pleasant surprise discovering your essay among the latest batch. I've long thought that your particular algebraic approach to physics deserves much more attention than it has gotten so far---looking at the history of this sort of thing, back to the suggestions of Gürsey and Günaydin about a connection between the strong force and octonions, there seem to be a lot of missed opportunities, research programs that came to be abandoned not because of some fundamental flaw, but simply, it seems, because of a lack of interest. However, one still sees the occasional algebraic spark light up---Christinel Stoica's entry in this contest being one of those.

My own training is in quantum information, so I've always had to struggle a little with unfamiliar formalisms when trying to understand your work, but the topic still fascinates me very much, although time constraints these days make an active interest more difficult. Still, it's good to see that you're still thinking about these things.

Anyway, on to your essay. Regarding the complex numbers being more fundamental than the reals, maybe one could make a different argument based on Cayley-Dickson: you need to introduce a fundamentally new idea (the square root of -1) to go from the reals to the complex numbers, while you only need to forget something---to coarse-grain, in some sense---to get the real numbers out of the complex field. Similarly for the quaternions and octonions. From yet another point of view, more fundamental structures contain less assumptions---so we strike the assumption of having a well-ordering and find the complex numbers among the options, we strike commutativity and get the quaternions, and we strike associativity and get the octonions.

So I think I'm ready to follow you there. But one question that must come up is, why stop there? Sure, the sedenions aren't a division algebra anymore, but why is that particular property necessary?

You answer this with the idea of resonances: certain mathematical structures are just way more fruitful, for lack of a better word, than others. Why is that the case? Well, that's just the way things work out: nothing more needs to be said.

I have some sympathy for this line of thinking. But the realm of mathematical structures is vast: is it really likely that we already have discovered the most fruitful structures? I know that (although I have forgotten much, it seems), for certain properties of interest, we can prove the uniqueness of the dimensions you mention---for instance, supersymmetry really only works in 3, 4, 6, and 10 dimensional spacetimes, which is directly related to the division algebras. (As an aside, I don't seem to recall much on supersymmetry in your works, which always struck me as somewhat surprising giving the close connection to division algebras---but nature here seems somewhat reluctant to conform to theorist's dreams, anyway.)

But do we care about these particular properties because they are of intrinsic interest, or just because we abstract them from the world around us most readily? Couldn't an entirely different physical universe exist, whose denizens marvel at the deep connections of its physics to mathematical structures whose properties seem wholly uninteresting to us? I find it hard to untangle these issues.

Anyway, I think I'll have to go brush up on my algebra...

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Jochen Szangolies replied on Jan. 11, 2018 @ 22:23 GMT

Oh, and by the way, I think it's a good thing to have thoughts that are almost opinions. These days, it seems much more in vogue to have opinions that are almost thoughts!

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Author Geoffrey Dixon replied on Jan. 12, 2018 @ 00:30 GMT

Regarding R and C being the first two in a series of algebras, I tried to stress in that first section that they were being considered there as mathematical fields, and not algebras. Analysis is done wrt fields, and even algebras are generally defined as real or complex, and in fact I treat C the algebra as being distinct from C the field in my paper on the potential for a 24-d periodicity, beyond the 8-d Bott periodicity (all can be found on 7stones).

(I utterly failed in my attempt to make R and C boldface.)

As to the division algebras, I am not a huge fan of Cayley-Dickson, as I wrote in my 2nd technical book (... Windmill Tilting). Rather I like to think that the real Ur-series, more fundamental than the series of algebras (a human construct), is the series of parallelizable spheres in dimensions 1,2,4,8 (so spheres of dimension 0,1,3,7). From these, if you're so inclined, you can get the division algebras, and from them all the classical Lie groups, and so much more.

As to the sedenions, their existence requires the Cayley-Dickson coincidence, but as I mentioned in my first boook, if you derive the division algebras using Hadamard matrices you get a next algebra quite different from the sedenions. Which is correct? Is that word, "correct", even relevant? These are human constructs, at least, in my opinion, more so than the notion of parallelizable spheres, which are true and real even without a human to think them up.

And, indeed, "the realm of mathematical structures is vast", and I do not believe "we already have discovered the most fruitful structures". I indicate as much in my papers about the number 24, which, I believe, has a very important role to play in the design of our physical reality, one I certainly do not understand (although I have some guesses in my published work).

As to supersymmetry, I am not a huge fan. My mentors in graduate school pushed me hard to be a fan, but I did not find the idea elegant or mathematically beautiful. I invented a version of symplectic Clifford algebras at one point that I found much more elegant, and connected to supersymmetry, but I did not pursue the idea very far.

You ask, "Couldn't an entirely different physical universe exist...?" I have an essay on that in my windmill book. Basically, assume such a universe exists, founded on entirely different mathematical constructs. In that universe there will still be only the four parallelizable spheres, and from these will arise the division algebras, and from these the standard symmetry ... The point is, whatever constructs you choose, and whatever reality results, our reality will be inherent and derivable from that alternate reality. But not, I think, visa versa, for our reality has already exploited the best, most beautiful, and resonant bits of mathematics.

Thanks for the good thoughts.

IMO :)

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Jochen Szangolies replied on Jan. 12, 2018 @ 15:55 GMT

I think I see what you mean about considering the field- rather than the algebra-angle; to me, it's just that the latter includes H and O, as well. But I don't really have any strong feelings on the issue.

The parallelizable spheres are indeed very interesting objects. I once was very impressed with the way they turn up in entanglement theory, with the state spaces of one, two, and three qubits essentially corresponding to the three Hopf fibrations. Thus, a single qubit is an

$S^3$

, with the complex phase being the

$S^1$

fibre over the

$S^2$

base space, while two qubits yield an

$S^3$

fibre over an

$S^4$

base, and three qubits an

$S^7$

fibre over the

$S^8$

base. What makes this whole thing (perhaps) nontrivial is the fact that these mappings yield data about the entanglement between the qubits: if the state is separable, the image of the Hopf map will lie in the complex numbers, whereas for a general state, it is quaternion-valued.

Still, I'm no longer sure if this is something significant, or just a kind of coincidence. I'll have to re-read your essay in the "Windmill Tilting"-book, but to me, it seems plausible that mathematical objects that carry no significance to us may seem just as important to denizens of another universe as parallelizable spheres are to us, whereas those don't hold any intrigue to them.

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Edwin Eugene Klingman wrote on Jan. 11, 2018 @ 23:15 GMT

Dear Geoffrey Dixon,

"I was not born with a humble-gene."

Few who participate in this contest have an over-expressed humble-gene, so you're in good company.

Let me congratulate you on a most enjoyable and interesting essay. You observe in conclusion that if a theory of everything varies too markedly from accepted dogma – upon which rests the livelihoods and reputations of the ruling elite – it will be resisted [unless monetarily profitable.] You conclude that attempts to enlarge understanding should be a labor of love, not based on an expectation of beating the rigged system. Of course that correlates with the advanced age of many of us authors.

You note that math is a powerful tool and also that "a great deal of modern theoretical physics rests on things assumed…" My essay retains the math of the Lorentz transformation, but deals with physical interpretations, specifically Einstein's space-time symmetry versus energy-time conjugation. I think you might enjoy it.

My very best regards,

Edwin Eugene Klingman

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Georgina Woodward wrote on Jan. 12, 2018 @ 12:09 GMT

Geoffrey, I like the style of your essay. I enjoyed reading it. Thank you for sharing your thoughts and opinions. Times are changing and with increasing rapidity. Perhaps there is room for a little more optimism, even though you do not feel like it. You say "We shall never be in agreement". I agree that we will never agree on everything but agreement on some things will be inevitable. (Feel free to disagree.) Kind regards Georgina

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Author Geoffrey Dixon replied on Jan. 12, 2018 @ 13:31 GMT

I might disagree, but even were I to do so, however vehemently, my indignation would quickly evaporate, and my attention be drawn to something uplifting, like puppies. But I don’t disagree. I had a brief email exchange with Noam Chomsky decades ago. I questioned his assertion that right ideas will always prevail in the end, feeling then that mainstreams learn from past mistakes, resulting is in an increasingly viscous intellectual environment. (He posited at one point that the 60s never would have happened had the world’s elites been united in opposition to change: but at the time they were not, and worldwide youthful revolution ensued.). Anyway, I have sidelined myself, but I do enjoy watching the struggle of the mainstream to come up with novel ways to justify its continued funding. The first time I witnessed this was at a particle physics colloquium at Harvard in the late 1970s. A prominent theorist, hoping to maintain funding for neutrino research, suggested that zapping the “wee sleekit cowran tim'rous beasties” through the earth could potentially help us find hidden oil reserves.

Anyway, I do not disagree. For every step backward, we take 1.00001 steps forward. Occasionally even more.

Wait ... what was the question? ;)

Brajesh Mishra wrote on Jan. 12, 2018 @ 15:33 GMT

Dear Geoffrey,

I found a few common threads in the essays submitted in this contest. They fundamentally echo a sense of 'mysteriousness' with the subject under consideration. I like the clarity of thought inbuilt in your write-up. Congratulations !

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Branko L Zivlak wrote on Jan. 15, 2018 @ 21:24 GMT

Dear Geoffrey,

Regarding: "The Boltzmann-Mach debate was a mere two lifetimes ago. Have we evolved in the intervening decades? Uh, no. We have not."

I do agree. That is the reason to return to the study of the achievements of the great physicist. In addition to the mentioned two: Newton, Boskovic, Maxwell, Planck ..

But please do not misunderstanding of Lemeitre, Habble and some modern science promoters. You wrote a really good essay.

Regards,

Branko

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Joe Fisher wrote on Jan. 16, 2018 @ 19:36 GMT

Dear Geoffrey Dixon,

You wrote: ”The sum total of mathematics at its profoundest is an explanation of why only certain mathematical objects are interesting. The sum total of physics is these objects.” All real visible objects have surface.

I have concluded from my deep research that Nature must have devised the only permanent real structure of the Universe obtainable for the real Universe existed for millions of years before man and his finite complex informational systems ever appeared on earth. The real physical Universe consists only of one single unified VISIBLE infinite surface occurring eternally in one single infinite dimension that am always illuminated mostly by finite non-surface light.

Joe Fisher, ORCID ID 0000-0003-3988-8687. Unaffiliated

post approved

John Brodix Merryman wrote on Jan. 20, 2018 @ 02:01 GMT

Professor Dixon,

You provide an enjoyable point of view. No one is truly humble, as each of us is the center of the entire universe. We are born knowing everything and spend our lives learning otherwise.

I would ask though, since I put the idea out in my entry, whether zero is the foundation of math?

The flatline from which all features and qualities expand and to which they coalesce. Otherwise it would seem maths exist as some platonic realm, rather then emergent with the features they map.

I extend that out to the proposition that empty space is the physics equivalent of zero. The vacuum that is the metric of C.

It does seem physics would prefer it to be emergent, from geometry, from the Big Bang, from time, etc, but it keeps sitting there quietly in the background and that would seem to be the quality of being foundational.

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Jonathan J. Dickau wrote on Jan. 23, 2018 @ 05:30 GMT

I like this offering a lot Geoff...

I am in broad agreement with your statements about the manner in which the primacy of Math applies. One of my past FQXi essays argued that the "Totality of Mathematics Shapes Physics" with a similar notion that the division algebras and other prominent or recurrent patterns in Math are naturally selected as relevant.

My essay has yet to post, but I will be sure to give you a high rating once it does. You aptly address the contest question, going to the heart of several questions about what is really fundamental, given certain pairings. I will probably reread once I do return to this, but you make some things very simple and direct, so there is no ambiguity or complication to speak of or complain about. This means you are also speaking at the appropriate technical level for this audience and contest.

Good luck!

Jonathan

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Jonathan J. Dickau replied on Jan. 23, 2018 @ 05:36 GMT

Once my entry does post...

You will see that I also appreciate the fact that complex numbers are closer to the source than the reals, or share in your thinking on that, and that I have some appreciation for the quaternions and octonions as well. I hope this contest gives your work some visibility.

All the Best,

Jonathan

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Author Geoffrey Dixon replied on Jan. 23, 2018 @ 11:26 GMT

Merci beaucoup, mais ...

Let ∆ be the probability of visibility. Let Ω be the probability that it matters. Let ø be the probability that bananas can be grown on the sun. Then we propose:

ø = ∆Ω.

Hmm. I really AM a curmudgeon.

And now, inorder to post this comment, I have to click the box entitled "I'm not a robot". Let ¥ be the probability I actually am not a robot; then ¥

Author Geoffrey Dixon replied on Jan. 23, 2018 @ 11:29 GMT

Damn. ¥ ≤ ø. Let's see if that posts correctly.

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Jonathan J. Dickau replied on Jan. 24, 2018 @ 04:22 GMT

O.K. Wait...,

So bananas grow on the sun? I'm just having fun with you Geoff. But my current essay has posted, if you want to take a look. One point of possible interest is how the Mandelbrot Set recreates Cartan's rolling ball analogy for G2, which I mention is also the automorphism group of the octonions.

In the meanwhile, enjoy the elevation, however brief. It is well-deserved. You address the topic head-on, and I give you kudos for cogent answers.

All the Best,

Jonathan

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Author Geoffrey Dixon replied on Jan. 26, 2018 @ 17:28 GMT

The last time I looked my banana plantation on the surface of the sun was doing quite well. Haven't you noticed the sun is a bit yellow?

Jonathan J. Dickau replied on Feb. 3, 2018 @ 00:34 GMT

Ah yes...

The bananas must be thriving.

Warm Regards,

Jonathan

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Satyavarapu Naga Parameswara Gupta wrote on Jan. 26, 2018 @ 16:46 GMT

Hi Geoffrey Dixon

It is wonderful to meet you an expert Algebra person, especially complex numbers… Very nice. A great deal of modern theoretical physics rests on complex and imaginary algebras…. Dear Geoffrey Dixon…. I want you to have a look in this paper also, where complex numbers are omitted….…..….. I highly appreciate your essay and request you please spend some of the...

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Author Geoffrey Dixon replied on Jan. 26, 2018 @ 17:25 GMT

Hello SNPG

I am retired, and even when not retired was never drawn to gravity theory, as I never felt there was anything I could contribute to such a crowded field. I am like a person who has heard a good song, and even when it is over can not get it out of my head. But I have been hearing that song in my head for over 30 years (R⊗C⊗H⊗O), and I shall likely still be hearing it on my deathbed.

I wish you luck. But think about how the world works, how it has always worked, and always will work ("always" means as long as our species is here to muck things up), and you should realize you will need much more than luck. But that's ok, as long as the work gives you joy.

Joe Fisher replied on Jan. 30, 2018 @ 20:18 GMT

Dear Fellow Essayists

This will be my final plea for fair treatment.,

FQXI is clearly seeking to find out if there is a fundamental REALITY.

Reliable evidence exists that proves that the surface of the earth was formed millions of years before man and his utterly complex finite informational systems ever appeared on that surface. It logically follows that Nature must have permanently devised the only single physical construct of earth allowable.

All objects, be they solid, liquid, or vaporous have always had a visible surface. This is because the real Universe must consist only of one single unified VISIBLE infinite surface occurring eternally in one single infinite dimension that am always illuminated mostly by finite non-surface light.

Only the truth can set you free.

Joe Fisher, Realist

post approved

Lawrence B. Crowell wrote on Feb. 2, 2018 @ 17:32 GMT

Geoffrey,

I enjoyed your essay. This is because I read your https://arxiv.org/abs/1407.4818 paper some years ago. I do have a few questions. In particular is the 128 dimensional T^2 hyperspinor space the same as the E8/SO(16) = 128? The other is that I have written notes or a pre-paper on some work with the Jordan J^3(O). This is I think more general than the Leech lattice, or embeds the Leech lattice. I was wondering if you have done any work on this and automorphism of the FS “monster group.”

I have pondered how it is that spin ½ leads to FD statistics. I have found myself thinking exactly what Feynman responded with, “I can’t do it.” It does seem plausible that because BE statistics integrates 1/(e^{-Eβ} - 1) into ζ-functions. The FD statistics 1/(e^{-Eβ} + 1) can be thought of as related to the BE with the general form 1/(e^{-Eβ} + e^{iθ}) for θ a phase angle. This is a bit like anionic statistics. It seems in a way this involves some deep relationship with the Riemann zeta function.

The motivation by mathematics can at times be compelling. I have some resonance with Dirac’s call to seek beauty. It is though not clear to me whether mathematics is more fundamental than physics. There was a time when I thought this might be the case. Then as time goes on this seemed difficult to uphold, while on the flip side it appears to be a collapse of objectivity to just assume mathematics is a sort of game or human invention. I am at a stage where I have not the faintest idea what the deep relationship between mathematics and physics is.

Cheers LC

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Author Geoffrey Dixon replied on Feb. 3, 2018 @ 01:20 GMT

I replied to this below, but may have failed to make it a reply, as opposed to a comment.

And again, I check "I'm not a robot", but I likely am.

Lawrence B. Crowell replied on Feb. 4, 2018 @ 20:31 GMT

I intend to work explicitely the quotient construction E8/SO(16). By saying E8/SO(16) = 128, as far as I know this means there is some vector space that is the 128-dim adjoint representation of --- well something. The thought has occurred to me it is U(8)xU(8), where each is 64 dimensions. The snag I can see with that is with Zamolodchikov representation of the golden quaternions, with magnitudes given by the golden mean, there would not likely be this even partition.

In fact I was going to work on this last May, but I had a death in the family. My great buddy and pal Umbriel, Umbra for sporadic group stuff and for the moon around neptune, was this big wonderful black pit bull that I adored. It might sound silly, but it took me a while to get over that.

I did a derivation quite some years ago on how Λ_{24} as derived by O^3. I used θ-functions. I think I have the derivation written in some notebook somewhere. The idea of a trilinear product is also something I have been kicking around. The Freudenthal diagonalization leads to three sets of eigenvalues. Behind this is the hyperdeterminant of Cayley and it seems to me there is a generalization of the 3-form that involves the associator. I have done a lot of work fairly recently on Morse theory with the Jordan algebra. It is work I did mostly about 2 years ago.

I have read on the higher sporadic groups, in particular Conway and SLoane Sphere Packing, lattices and groups. I have not though done any calculations on this. I figure if we can just get reasonable physics with O, E8, O^3 and J^3(O) then we might have something workable. Since these have automorphism properties on the FG monster this then will suggest a deeper layer of structure.

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Jonathan J. Dickau wrote on Feb. 3, 2018 @ 00:53 GMT

Hello again Geoff,

I wanted to thank you for the thoughtful 'play by play' review of my essay. It is amazing what one can learn seeing what you have written through someone else's eyes. I especially appreciate your catching me on the dodgy usage of the word 'likely' which has no place in academic writing, where the goal is to be crystal clear and mathematically precise.

As for the Mandelbrot Set; it was the horse I rode in on, as it were. I had a few phone conversations with Ben Mandelbrot more than 30 years ago that greatly encouraged and shaped my learning. The last time we spoke; he called me out of the blue on an Easter Sunday morning. I had worked until midnight, the night before, and decided to sleep in rather than attending a church service - but I got lucky and talked to Ben instead.

Since then; I've found out M doesn't stand alone, but connects with a number of other mathematical objects - so it was a good place to start me going in a worthwhile direction. I'm glad to have had your musings to refer to, as well, because there does appear to be a special significance to T. As I recall; the Sedenion sphere S15 only has three possible fibrations, S1, S3. & S7 - yielding the C, H, and O algebras. This would seem to indicate they are a foundational trio.

I thought the comment above was priceless, and I'm very glad to see that your banana plantation is thriving!

All the Best,

Jonathan

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Author Geoffrey Dixon wrote on Feb. 3, 2018 @ 01:16 GMT

"In particular is the 128 dimensional T^2 hyperspinor space the same as the E8/SO(16) = 128?"

I don't have any reason to believe it is or isn't. Tony Smith thinks so, but I am unwilling to lead the maths. I prefer to be led, even if led astray. At least it won't be my fault.

"Jordan J^3(O). This is I think more general than the Leech lattice, or embeds the Leech lattice."

Wilson, at U of London, and I have independently represented the Leech lattice over O³. Anyone with any interest in J^3(O) will naturally point out that there is a nice copy of O³ in J^3(O). I don't know if this is meaningful. If one could show that there was a bilinear of trilinear multiplication on the Leech lattice as a subset of J^3(O) that closed on this subset, then I personally would be hugely interested. But yes, one can stick it into J^3(O), but without some further interesting structure ...

"I was wondering if you have done any work on this and automorphism of the FS “monster group.”"

No. I of course find the monster group enticing, because it is exceptional, but I haven't got around to looking into it.

As to your paragraph 2, this leads into the quantum quagmire. If you can find two people who agree on much of anything in that quagmire, let me know. Meanwhile, the level of disagreement means to me that we are not ready to understand at a deep level.

"... it appears to be a collapse of objectivity to just assume mathematics is a sort of game or human invention. I am at a stage where I have not the faintest idea what the deep relationship between mathematics and physics is."

This is a hard one, because one has to define what one means by mathematics. I think of it as a thing that is there even in the absence of any intelligent life or any conception of consciousness. Many will likely give this idea a label named after a dead Greek and think they understand. Maybe they do. Anyway, in the presence of human intelligence we have this notion of Ur-mathematics, and we have the human symbols and formalism that is our lens onto this world. So, ignoring the invented formalism, I think of physics - I'm just making this up, but it sounds good to me - anyway, physics crystallizes on the exceptional, generative, and resonant bits of this Ur-maths, for only there is there a structure rich enough to nurture it. And to give rise to us, for what that's worth.

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Lawrence B. Crowell replied on Feb. 4, 2018 @ 20:33 GMT

Duplicate of above

I intend to work explicitely the quotient construction E8/SO(16). By saying E8/SO(16) = 128, as far as I know this means there is some vector space that is the 128-dim adjoint representation of --- well something. The thought has occurred to me it is U(8)xU(8), where each is 64 dimensions. The snag I can see with that is with Zamolodchikov representation of the golden quaternions, with magnitude given by the golden mean, there would not likely be this even partition.

In fact I was going to work on this last May, but I had a death in the family. My great buddy and pal Umbriel, Umbra for sporadic group stuff and for the moon around neptune, was this big wonderful black pit bull that I adored. It might sound silly, but it took me a while to get over that.

I did a derivation quite some years ago on how Λ_{24} as derived by O^3. I used θ-functions. I think I have the derivation written in some notebook somewhere. The idea of a trilinear product is also something I have been kicking around. The Freudenthal diagonalization leads to three sets of eigenvalues. Behind this is the hyperdeterminant of Cayley and it seems to me there is a generalization of the 3-form that involves the associator. I have done a lot of work fairly recently on Morse theory with the Jordan algebra. It is work I did mostly about 2 years ago.

I have read on the higher sporadic groups, in particular Conway and SLoane Sphere Packing, lattices and groups. I have not though done any calculations on this. I figure if we can just get reasonable physics with O, E8, O^3 and J^3(O) then we might have something workable. Since these have automorphism properties on the FG monster this then will suggest a deeper layer of structure.

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Ajay Pokhrel wrote on Feb. 3, 2018 @ 04:26 GMT

Hello Geoffrey,

I liked the way your essay is written, mainly the introductory part. I believe your essay makes much sense, no one can reach an agreement for fundamental. I also believe that mathematics is more fundamental than physics which I have written very descriptively on my essay.

I gave a good rating to your essay because I find it unique and interesting and I hope you'd enjoy mine too.

Kind Regards

Ajay Pokharel

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Cristinel Stoica wrote on Feb. 4, 2018 @ 08:38 GMT

Geoffrey Dixon

Dear Geoffrey,

I rarely feel so lucky and pleased to read something with which I agree almost completely. Your essay gave me this high satisfaction, so thank you. There may be two points where we slightly diverge, but only as a matter of preference. The first one is that I may be a bit biased towards geometry and consider complex, quaternionic, and octonionic structures as living on real vector spaces. The second one follows from the first one, since the compositions of transformations preserving structures is associative. I fully agree with the paramount role of spinors, but consequently I tend to see them as representations of Clifford algebras (a quite mainstream position among mathematicians). So my views about the Standard Model are shaped by this. Not that I would disagree with you, in fact I think you are right from another perspective. Another consequence of my view is that it kept me away from properly investing time in studying your work, although I I knew about the Dixon algebra and that you made a mathematically beautiful and physically insightful model for leptons and quarks. It's time to fix these lacunae I have and read carefully your writings. Your essay convinced me of this, even though you mentioned your work only incidentally, being focused on answering the question "what is 'fundamental'". I also just included in my paper about the Standard Model based on a Clifford algebra a mention of your model (fortunately my manuscript is still under review). I think your work deserves more attention. What I find intriguing is that, unlike Clifford algebras which are infinitely many, the Dixon algebra is one of a kind. Since I still am a bit biased against nonassociativity, I would like to ask you if you know some physical consequences of this feature of octonions. Congratulations for your excellent essay, and success!

Best wishes,

Cristi

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Author Geoffrey Dixon replied on Feb. 4, 2018 @ 11:54 GMT

Yes, the octonion algebra O is nonassociative. But for each division algebra (R,C,H,O) there are associated (and associative) algebras of actions of the algebra on itself. With R and C, because they are commutative and associative, this gives rise to nothing new. But H is noncommutative, so the algebra of all left multiplications of H on itself (H_L elements do this: xH) is distinct...

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Author Geoffrey Dixon replied on Feb. 4, 2018 @ 12:37 GMT

Hi Cristinel

Just downloaded your essay. Impressive. This contest will be long over before I could possibly dig into it deeply enough to say anything cogent.

You mentioned Furey's work using the Dixon algebra. I began working on that algebra some 40 years ago. I worked alone, and the work was far enough removed from mainstream thought that it was largely ignored. But it was, and is, right in its fundamentals.

Baez was the first to refer to R⊗C⊗H⊗O as the Dixon algebra, after noting that recent work exploiting this algebra largely ignored my decades of books and papers. Such is science, and there is no cure (as I point out in my book, A Fire in the Night).

Cristinel Stoica replied on Feb. 4, 2018 @ 16:55 GMT

Dear Geoffrey,

Thank you for the clear explanations, the properties you described are really great, in particular the dual action of quaternions (with which I am more familiar) and especially the double role of the octonion algebra. I definitely want to know more about your work and come back with more questions later. In the mean time, I hope this contest will give more visibility to your works and ideas.

Best wishes,

Cristi Stoica, Indra's net

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Member Tejinder Pal Singh wrote on Feb. 4, 2018 @ 16:09 GMT

Dear Geoffrey,

I greatly liked your essay!

I was struck in particular by the statement that Dirac even speculated that one day physics and mathematics will become one. I did not know Dirac had such a view, and I humbly mention that some considerations lead me to the same conclusion in my essay.

My thanks and regards,

Tejinder

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Jonathan J. Dickau wrote on Feb. 5, 2018 @ 04:18 GMT

You said something above greatly of note Geoff..

Regarding the octonions, you stated "The bottom line is this: the nonassociativity of O is a feature, not a bug, and an amazing feature at that, coming into play in just the perfect way." I share your enthusiasm regarding this feature of the octonions, and I likewise exalt that it comes into play in a most amazing way.

Warm Regards,

Jonathan

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peter cameron wrote on Feb. 5, 2018 @ 20:44 GMT

Hello Geoffrey,

Good metaphor, thank you. Immediately we bump heads, tho, with C more fundamental than R. Given that the goal in our two essays is to have a satisfactory model of agency in the physical world at the level of the elementary particle spectrum, I'm of the view that R is more fundamental than C. This is the position taken by the geometric algebra community of the 'Hestenes school', as so simply and lucidly presented in his 1966 book, Spacetime Algebra, which resurrected Grassman and Clifford's original geometric intrepretation and introduced it to physics.

In the geometric view one can take the vacuum wavefunction to be comprised of the eight fundamental geometric objects of the 3D Pauli algebra - one scalar, three vectors (3D space), three bivectors, and one trivector. Endowing the geometric objects with topologically appropriate fields, this becomes an agent in the physical world.

Interaction of these agents/wavefunctions can be modeled by the nonlinear geometric product, which generates a 4D Dirac algebra of flat Minkowski. Time, the dynamics, emerges from the interactions, encoded in the 4D pseudoscalars. There is no need for complex numbers, for complex algebras, for this particular legacy of Euler.

re spinors, they are likewise understood as being comprised of a scalar plus bivector, can be visualized. Reinvention of Clifford algebra by Pauli and Dirac in the matrix representation has left the community stuck with something that is too abstract. Basis vectors of geometric algebra are equivalent of matrix representation....

I admire your knowledge of group theory, a knowledge that i sorely lack, hope that the above outline of the geometric wavefunction is helpful to you in applying that knowledge to the physics.

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peter cameron wrote on Feb. 5, 2018 @ 20:51 GMT

neither your 7 stones nor your arxiv link seem to work properly. 7 stones is not found, and the pdf at arxiv has it's format incomprehensibly scrambled, is not readable.

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Author Geoffrey Dixon replied on Feb. 5, 2018 @ 21:39 GMT

http://www.7stones.com/7_new/7_Pubs.html

What OS nd browser are you using? I just tried this, and it works on my old Mac using Firefox, and on my iPhone using Chrome.

https://arxiv.org/pdf/1407.4818.pdf

Likewise this link works fine on my Mac. Methinks the problem is closer to home. Please let me know if you figure it out.

peter cameron replied on Feb. 5, 2018 @ 23:14 GMT

both fine now, dunno what that was about. Glad to see what appear to me to be fairly strong connections between your work and what Michaele and I are doing. Sure wish we had your group theory expertise, tho.

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Jonathan J. Dickau replied on Feb. 5, 2018 @ 23:58 GMT

I had the same issue...

Underline characters in the PDF links were turned into spaces, by acrobat or my browser, in the first example. And some part of the https:// got lost too. But I later got the links to work as well.

All the Best,

Jonathan

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peter cameron wrote on Feb. 6, 2018 @ 01:14 GMT

Geoffrey/Cristi/Jonathan/...

Looking at Geoffrey's comment

O = spinor space; OL = Clifford algebra.

does this mean the left handed neutrino is built into OL?

Spinor wavefunction is scalar plus bivector if i understand correctly (please explain if wrong).

Handedness comes from the bivector, of which there are three in the Pauli algebra of 3D space. However it is the two component spinor that comprises wavefunction, not just the bivector (Bohr magneton).

Seems like both the non-commutative and the non-associative properties would come from the bivector, and don't quite understand how the scalar enters into it from consideration of those two broken properties (symmetries?). Is it nothing more than just the 'gauge', not playing any role as an additional topological object (the singularity) with the bivector in the wavefunction that is somehow involved in understanding what's going on?

coming back to my opening question

O = spinor space; OL = Clifford algebra.

does this mean the left handed neutrino is built into OL?

Does this mean that the eight component Pauli wavefunction Michaele and I are working with has chiral symmetry breaking built in?

and ditto the 16 component Dirac algebra of the eight by eight geometric representation of the S-matrix generated by geometric products of Pauli wavefunctions?

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peter cameron replied on Feb. 6, 2018 @ 01:18 GMT

darn. another issue with formatting getting scrambled when working thru this interface. Trying again:

Geoffrey/Cristi/Jonathan/...

Looking at Geoffrey's comment nO = spinor space; OL = Clifford algebra.

does this mean the left handed neutrino is built into OL?

Spinor wavefunction is scalar plus bivector if i understand correctly (please explain if wrong). Handedness comes from the bivector, of which there are three in the Pauli algebra of 3D space. However it is the two component spinor that comprises wavefunction, not just the bivector (Bohr magneton).

Seems like both the non-commutative and the non-associative properties would come from the bivector, and don't quite understand how the scalar enters into it from consideration of those two broken properties (symmetries?). Is it nothing more than just the 'gauge', not playing any role as an additional topological object (the singularity) with the bivector in the wavefunction that is somehow involved in understanding what's going on?

coming back to my opening question

O = spinor space; OL = Clifford algebra.

does this mean the left handed neutrino is built into OL?

Does this mean that the eight component Pauli wavefunction Michaele and I are working with has chiral symmetry breaking built in?

and ditto the 16 component Dirac algebra of the eight by eight geometric representation of the S-matrix generated by geometric products of Pauli wavefunctions?

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peter cameron replied on Feb. 6, 2018 @ 01:20 GMT

ok giving up on this for the moment,

gonna restart my computer,

see if that helps.

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Jonathan J. Dickau replied on Feb. 6, 2018 @ 03:16 GMT

Seeing that the character n now replaces all carriage returns, I think it is a system wide problem with the FQXi forum platform software. I think they may be trying to make returns appear as an en-dash for compactness but this is ridiculous.

All the Best, JJD

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Author Geoffrey Dixon replied on Feb. 6, 2018 @ 11:20 GMT

First a test.

If this looks good, I'll respond.

Author Geoffrey Dixon replied on Feb. 6, 2018 @ 11:23 GMT

This is another test.

The yellow banana grows on the sun. O_L

GD

Anonymous replied on Feb. 6, 2018 @ 12:05 GMT

Ok, so no carriage returns. My starting position is T = R⊗C⊗H⊗O. It's just a mathematical object. If it is an essential part of any viable mathematical model of reality, then I suggest it is required because fermion fields require parallelizable spheres in some manner as yet to be determined. Lacking a clear understanding of that I start with T. Like C, H and O, individually, and P =R⊗C⊗H, T is a spinor space. P is the spinor space of P_L, which is essentially the Pauli algebra, so the associated geometry is 3-space. (Still just math.) It a very similar fashion T is the spinor space of the Clifford algebra T_L, and its associated geometry is 9-space. In my first book I used the fact that a Dirac spinor is a pair of Pauli spinors (so P², although this is actually an SU(2) doublet of Dirac spinors) to motivate basing my theory building on T², which is an SU(2) doublet of Dirac spinors for a 1,9-spacetime. T itself is not only a spinor, it is also an algebra, and its identity can be decomposed into a set of orthogonal projection operators (a la Gürsey and company at Yale in the 1970s) with respect to which the bivectors of the 1,9-Clifford algebra, which is also the Lie algebra so(1,9), gets decomposed to so(1,3) x u(1) x su(3), and the spinor T² is decomposable into a collection of su(2) doublet ordinary 1,3-Dirac spinors. Again, this is just pure mathematics, with some physics words thrown in because the maths has a kind of obvious interpretation in the physics context. Anyway, I think I won't try to duplicate all the details of two books and a number of papers in this comment, because I need breakfast. Since the neutrino was mentioned in the initial comment, I'll just add that interpreted as a basis for physics modelling, the neutrino that pops out of the maths is a Dirac neutrino, and it has the potential to have a Dirac mass. So, in conclusion: there is maths; there is physics (standard model); everything I've done is simply to show that one can use the maths as a skeleton onto which all of your preferred QFT-flesh can be attached. If the maths has something to say about the deeper quantum side of things, I am not competent to say what it is. If some other idea proves eventually to be the correct mathematical model of reality, T-maths will still have all these properties that look a lot like they ought to have something to do with our presently accepted theories of physics.

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Author Geoffrey Dixon replied on Feb. 6, 2018 @ 15:58 GMT

The above post is me. Thought I was logged in.

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Rick Lockyer wrote on Feb. 10, 2018 @ 18:42 GMT

As I said over on my essay thread, liked your essay very much. We are in agreement in many ways, but not all. For me, the progression of fundamentalness is physical reality -> a subset of mathematics-> algebraic structure. I would put the algebra that allows the paralizeable spheres as more fundamental rather than the other way around as you seem to imply. Chicken or the egg though to some degree. Then there is that pesky issue of whether or not two algebras with demonstrably different structure can be truly considered isomorphic. It is not “Greek” to me.

I have learned a lot from you over the years, along with from John Baez. Can’t thank you enough for championing division algebras. You are a legend in my mind.

I just put up a discussion on Octonion Algebraic Invariance/Variance referencing the table in my essay. Might be worth a look for you.

Rick

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Author Geoffrey Dixon replied on Feb. 10, 2018 @ 22:03 GMT

I think I'm logged in, but if not, this is GD again.

As to spheres vs algebra, years ago I had a confab with Baez in which I posited the idea that octonion maths is in a sense holographic, as most of the bits and pieces we like can be used to derive the remaining bits and pieces. That is, no given bit is any more fundamental than any other. Parallelizable spheres yield division algebras; and visa versa. As you do, I start with algebra, but in my heart I view algebra as a kind of intellectual microscope, devised by humans to better see structures that exist without us. I mean, in the total absence of any intellect in the universe, planets and stars still take roughly spherical form. However, without intellect algebra does not exist. But again, I start with algebra, because it allows me to play with things more fundamental.

You say: "Then there is that pesky issue of whether or not two algebras with demonstrably different structure can be truly considered isomorphic." I assume you are referring to the left/right octonion thing. If so, isomorphism is demonstrable. I think I even did it in my windmill book. There is only one octonion algebra, but many ways to organize it. Really, one needn't use integers to to label the units. One could use fruit, or puppies, or anime characters. Integers just allows us to see some structures that are otherwise hidden. Like my fav multiplication table with e₁ e₂ = e₄, imposing invariance with respect to cycling and doubling of integer subscripts (yielding a finite invariance group of order 21).

Anyhum, give me any two multiplication tables for O, and I'll build an isomorphism. None of the best minds in the fields (like Conway and Sloane and others) ever discuss two versions of O that are not isomorphic. There is one O. Until proven otherwise, of course.

Rick Lockyer replied on Feb. 11, 2018 @ 01:01 GMT

I don’t really want to belabor the point, but in my way of thinking, sure you can call the basis elements any names you want and index them anyway. I draw my line at calling any element -name from +name, and thinking this negation does not modify fundamental Algebraic structure. You can get away with it for Quaternions since a minus sign in front of 1 or 3 non-scalar bases is the same as a simple transposition, that implying both forms are ordered triplet multiplication rules so same basic structure.

Not so much for Octonion Algebra. You can stick a minus sign in front of an even number of non-scalar bases and demonstrate this new rule set is equivalent to a number of transpositions. This would map Right to Right and Left to Left. But if you do it on an odd number of negations, this would be the map between Right and Left, and no possible set of transpositions without sticking in a minus sign will do the deed. It is not that you CAN do the negations, you MUST if you go between Right and Left.

As far as others not believing this, I have yet to see anyone else deal with Octonion definition variation the way I do. They bought the story all Octonion Algebras are the same so haven’t bothered to look at the differences, which are extremely important. Not surprised by the bias and lack of effort checking closely. Their loss, for Algebraic Invariance and variance dealing explicitly with this variation is quite important.

Rick

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Author Geoffrey Dixon replied on Feb. 11, 2018 @ 01:37 GMT

I'm trying to interpret words in terms of old equations in my head. Inevitably it's coming out looking like something I'm familiar with. If it isn't, then I'd need to see equations and multiplication tables. Without that I can't judge. Not that it's important that I do. What's important is that you carry on. Ignore my quibbles. I don't really know if they are valid.

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Author Geoffrey Dixon wrote on Feb. 13, 2018 @ 13:19 GMT

Prediction for those serious about applying division algebras:

So, I encountered the following online ...

https://hackaday.com/2018/02/10/all-the-stuff-you-wished-
you-knew-about-fourier-transforms-but-were-afraid-to-ask/

Thi
s is an excellent description on the Fourier transform, described as taking a signal and wrapping it around a circle, the circle (S¹) being represented as the set of unit elements in C. I then did a little googling and discovered a number of papers on Fourier transforms over H and O, which means over the parallelizable spheres, S³ and S⁷, including discussions of corresponding uncertainty relations. This kind of thing is the analytical future of this field.

Rick Lockyer wrote on Feb. 15, 2018 @ 03:16 GMT

Geoffrey,

Knowing you are not big on the Cayley-Dickson doubling algorithm, I did neglect to mention another reason my Quaternion triplet enumeration algorithm is cooler than yours (no offense intended). That would be for enumerating the triplets for the sedenions. For Octonion Algebra I used the binary numbers 1 through 7 and partitioned the triplets with the bit wise exclusive or logic function. This can be extended to the sedenions by going to binary 1 through 15, of course representing directly the 15 non-scalar basis elements enumerations. Doing the exclusive or logic operation on binary 1 through 15 partitions them into 35 unique closed triplet sets, and since there are (n-1)*(n-2) = 15*14 = 210 basis products not e0 * en, en * e0 or en * en fixed product rules, and each triplet does 6 rules, 35*6 = 210, covering all 210 basis products with 35 Quaternion triplets.

For each Octonion subalgebra using this enumeration technique the seven non-scalar basis element indexes will exclusive or to a zero result, as well each Quaternion index triplet. If you take away from the seven Octonion non-scalar basis indexes any set of three which are Quaternion triplet indexes, the remaining four basis elements are called “basic quads” and their indexes will also exclusive or to zero. They are basic quads because given just them, the exclusive or of any two will be the index of one of the companion Quaternion basis triplets and all are provided 2 up with this.

This gives a method to determine how many Octonion subalgebra candidates there would be, just go through all quads of distinct indexes 1 through 15 for combinations that exclusive or to zero. A simple computer program will show there are 105 such quads. Any Octonion definition has seven Quaternion triplets each with a unique basic quad, so we would expect to see 105/7 = 15 candidate Octonion subalgebras. Each of the 35 Quaternion triplets occurs in 3 Octonion subalgebras 3*35 = 105.

The rub is you can’t make all 15 Octonion subalgebra candidates represent valid normed composition algebras, e.g. division algebras. An algebraic proof the sedenions are not a division algebra? Maybe so.

Thought you might find this interesting.

Rick

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Author Geoffrey Dixon replied on Feb. 15, 2018 @ 11:48 GMT

"Exclusive or" is this, right?

0+0 = 1+1 = 0;

1+0 = 0+1 = 1.

The same thing that I used in my Hadamard matrix extension of the octonions. Anyway, it sounds interesting, but I'm still light years away from thinking Cayley-Dickson and sedenions are interesting ... to me. As to "Quaternion triplet enumeration algorithm": I have no problem with this being cooler than mine (although I've no idea how the word "algorithm" applies to anything I do with the octonions).

Meanwhile I have a new bee in my bonnet: Fourier transforms over H and O. So little time.

Anonymous replied on Feb. 15, 2018 @ 15:33 GMT

Yes, at the bit level, the exclusive or operation is equivalent to modulo 2 addition. You can consider modulo 2 polynomials and addition instead of binary numbers and the exclusive or logic function. I have used both in the past.

Rick

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Dizhechko Boris Semyonovich wrote on Feb. 19, 2018 @ 14:18 GMT

Dear Geoffrey Dixon,

I think that the fundamental must be simple, clear and to save our thinking.

Mathematics complicates physics the abstract forms, which require it to abandon the basis on which there is a specific thinking. According to the principle of identity of space and matter Descartes, matter is space and space is matter that moves. Time is a synonym for universal total movement. Thus, space is the Foundation for fundamental theories.

You may like to look at the sky and it seems to you empty infinite space in which it moves large and small body. However, this impression is deceptive. According to the principle of identity of space and matter Descartes, space is matter that moves. When Copernicus asserted that the Earth revolves around the Sun, it had, according to Descartes, to add that along with the Earth revolves around the Sun, all the solar space. Space is what built the world. Space contains information about the development of the Universe . Take a look at my essay in which I showed how radically can change physics, if it will follow this principle. Leave your autograph. FQXi Fundamental in New Cartesian Physics by Dizhechko Boris Semyonovich

Do not allow New Cartesian Physics go away into nothingness, which can to be the theory of everything OO.

Sincerely, Dizhechko Boris Semyonovich.

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Rick Lockyer wrote on Feb. 23, 2018 @ 17:05 GMT

Hi Geoffrey,

Put my Octonion symbolic algebra JavaScript tool up on my essay thread. Don’t know what your programming skills or inclinations are, but if you wanted to better understand my Octonion mathematical physics, you could learn a lot about it just from the logged results of the included script that I verified my derivation of the Octonion conservation of energy and momentum equations with.

Getting old and needing to write things down, I do have a ~170 page PDF “book” that goes into things in great detail. Been thinking of sharing with you, John Baez and others for review. Not sure of the interest leve.

Rick

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Author Geoffrey Dixon replied on Feb. 23, 2018 @ 18:12 GMT

Hi Rick

First, let’s face it: none of us really players well with others. I’m taking a big break from all this. I’ve spent much of the last two weeks pouring over a paper I was asked to review. I liked it. I had to dredge up lots of old knowledge and see it in a new light. Now I’m pooped. No more math or physics for a while, except maybe my own.

Why not publish on CreateSpace?