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.

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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...

    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!

    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

    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 :)

    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

      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? ;)

      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 !

      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 [math]S^3[/math], with the complex phase being the [math]S^1[/math] fibre over the [math]S^2[/math] base space, while two qubits yield an [math]S^3[/math] fibre over an [math]S^4[/math] base, and three qubits an [math]S^7[/math] fibre over the [math]S^8[/math] 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.

      OK, so apparently, that doesn't work as inline math... Sorry for uglifying your thread!

      Hopf fibrations: yet another resonant bit of maths associated with these dimensions. I'm afraid I don't know anything about their relationship to entanglement theory. Sounds interesting, but like Sherlock Holmes, who did not know the moon revolved around the earth, that fact being no help in his work as a detective, I am shockingly ignorant about a great deal of contemporary physics ideas.

      By the way, these sphere fibrations can be extended to lattice theory. I wrote a paper on that back in the age of dinosaurs, but included the idea in the windmill book too. As to those hypothetical denizens of another universe who may extol the virtues of different maths ideas ... I'm not sure I like them very much, or their universe. :)

      As to messing up the thread, my first attempt to use some of the suggested tags failed miserably. I applaud you willingness to tackle it at all.

      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

      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

      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.

      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

        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

        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 ВҐ be the probability I actually am not a robot; then ВҐ

        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

        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 valuable time on Dynamic Universe Model also and give your some of the valuable & esteemed guidance

        Some of the Main foundational points of Dynamic Universe Model :

        -No Isotropy

        -No Homogeneity

        -No Space-time continuum

        -Non-uniform density of matter, universe is lumpy

        -No singularities

        -No collisions between bodies

        -No blackholes

        -No warm holes

        -No Bigbang

        -No repulsion between distant Galaxies

        -Non-empty Universe

        -No imaginary or negative time axis

        -No imaginary X, Y, Z axes

        -No differential and Integral Equations mathematically

        -No General Relativity and Model does not reduce to GR on any condition

        -No Creation of matter like Bigbang or steady-state models

        -No many mini Bigbangs

        -No Missing Mass / Dark matter

        -No Dark energy

        -No Bigbang generated CMB detected

        -No Multi-verses

        Here:

        -Accelerating Expanding universe with 33% Blue shifted Galaxies

        -Newton's Gravitation law works everywhere in the same way

        -All bodies dynamically moving

        -All bodies move in dynamic Equilibrium

        -Closed universe model no light or bodies will go away from universe

        -Single Universe no baby universes

        -Time is linear as observed on earth, moving forward only

        -Independent x,y,z coordinate axes and Time axis no interdependencies between axes..

        -UGF (Universal Gravitational Force) calculated on every point-mass

        -Tensors (Linear) used for giving UNIQUE solutions for each time step

        -Uses everyday physics as achievable by engineering

        -21000 linear equations are used in an Excel sheet

        -Computerized calculations uses 16 decimal digit accuracy

        -Data mining and data warehousing techniques are used for data extraction from large amounts of data.

        - Many predictions of Dynamic Universe Model came true....Have a look at

        http://vaksdynamicuniversemodel.blogspot.in/p/blog-page_15.html

        I request you to please have a look at my essay also, and give some of your esteemed criticism for your information........

        Dynamic Universe Model says that the energy in the form of electromagnetic radiation passing grazingly near any gravitating mass changes its in frequency and finally will convert into neutrinos (mass). We all know that there is no experiment or quest in this direction. Energy conversion happens from mass to energy with the famous E=mC2, the other side of this conversion was not thought off. This is a new fundamental prediction by Dynamic Universe Model, a foundational quest in the area of Astrophysics and Cosmology.

        In accordance with Dynamic Universe Model frequency shift happens on both the sides of spectrum when any electromagnetic radiation passes grazingly near gravitating mass. With this new verification, we will open a new frontier that will unlock a way for formation of the basis for continual Nucleosynthesis (continuous formation of elements) in our Universe. Amount of frequency shift will depend on relative velocity difference. All the papers of author can be downloaded from "http://vaksdynamicuniversemodel.blogspot.in/ "

        I request you to please post your reply in my essay also, so that I can get an intimation that you replied

        Best

        =snp