Essay Abstract

Next year we will mark the 170th anniversary of John T. Graves' discovery of Octonion Algebra. Since its discovery Octonion Algebra has for the most part languished in relative obscurity. Almost everyone that studies Octonion Algebra revels in the beauty of its algebraic structure, yet precious few have come to believe it has any connection to physical reality. Why is this? Historically, Quaternion Algebra H got a bum rap by tinkerers that sadly assumed the forms for gradient, divergence and curl were individually fundamental, instead of structured sub-components of a more fundamental H. This hindered due consideration for Octonion Algebra throughout the 20th century. Internally, there are many fundamental assumptions about Octonion Algebra that are counter-factual. Externally, physicists have been jaded by the success of alternate mathematical systems, none of which can be demonstrated as necessary, at best only sufficient for their degree of success. The bedrock foundation for a Theory of Everything should be an Algebra of Everything. I submit it is Octonion Algebra.

Author Bio

I received a B.S. in Physics from Stanford University in 1973. I was convinced at that time what I wanted to learn was not part of any graduate program in Physics. I had marketable engineering skills and lived in Silicon Valley, so I opted for an engineering career instead of continuing with formal education. I never stopped thinking about foundational Physics. The essay material presented is the result of decades of private research.

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Hello Rick,

I too feel that octonions are a good candidate for an algebra of everything. In fact; I'm inclined to believe they are the 'actual' or 'true' numbers, by being the most general case, where reals, complex numbers, and quaternions are special cases.

I look forward to reading your essay, which I have already downloaded and have open in another tab. I give some favorable attention to the octonions in my own contest essay "Cherished Assumptions and the Progress of Physics." I hope you do well in the contest.

Regards,

Jonathan

Hello again, Rick;

Very interesting on first read through, but I'll need to re-read some parts several times to understand fully. I like the proof that 480 possible tables reduce to only 16. I guess that's 8 each for left hand and right hand. Geoff Dixon claims on his web-site that only 4 are in common usage, where his and Cederwell's plus and minus conventions are cited. But doesn't Okubo do things differently? Anyhow; it was a nice job.

I especially liked the part at the end where Electrodynamics comes out in its complete form, with no fudging for the SET. As you say; when the octonion formulation gives us terms that are unobserved bits, it doesn't necessarily mean that they are useless. Rather; it means we have yet to find out what they tell us.

All the Best,

Jonathan

    Rick,

    Interesting essay. You might be interested in some of the connections between quaternions, octonions, Hopf fibrations, and quantum information theory. Unfortunately I don't have a completed paper on this, but there are some partial treatments you can find on the web. Some people also believe that quantum information theory is more deeply connected to quantum gravity than it's usually given credit for; I mention this briefly near the end of my essay

    [link:fqxi.org/community/forum/topic/1386]On the Foundational Assumptions of Modern Physics[link]

    Take care,

    Ben Dribus

      Hi Jonathan,

      Thanks for taking a look. I did look at yours when it came out, and found it very interesting. As you know from our past interactions, we agree on much but not everything, notably dimensional stability. For me if Octonions ever were appropriate, they will always be so. Liked your paper with Ray: In Defense of Octonions.

      Dixon likes to generate different Octonion Algebras with arithmetic rules on the indexes. Problem is it produces different triplet sets that disguise the fundamental structure embodied by the Quaternion subalgebra triplet chiral choices only visible if the same seven triplets are used. This is critical to algebraic invariance, which is a fundamental law of physical reality from my point of view. Common use of less than all 16 is precisely the problem that has held back some smart people that have looked long and hard on Octonion Algebra.

      Interesting that as shown in the endnotes, the non-observable variants come in three product term sets. Kind of quirky, or is that quarky?

      Good luck to you too.

      Rick

      Ben,

      Thanks for taking a look. The connection between Quaternions and Octonions is at the heart of the concepts behind the essay. The choices for the Quaternion subalgebra chirality that maintain a normed alternative composition algebra for O defines the full range of O variability. If you notice I used up every bit of 9 pages, and had a tough time shoe-horning it in. I could have said much more about many aspects of R to C to H to O without the length limitations.

      I agree with you on Hopf fibrations. My current intellectual diet has a high fiber content. I did read your essay and liked many portions of it. I think it would be difficult to cover Electrodynamics without including the notion of time within the manifold. It works for 4D and equivalently within an Octonion framework as I presented in my essay. Have you had any success leaving it out without implicitly having it in?

      Good luck with your essay.

      Rick

      Dear Rick Lockyer,

      I enjoyed your essay immensely, beginning with your observation of the essentially religious aspect of our assumptions. I too believe General Relativity and Quantum Mechanics are not fundamental, and that Electrodynamics and Gravitation should be united.

      I have read all of the material on your website, but it was a year ago and that material hardly sticks in one's mind. I wish that you could write more explanatory material. For example, the notion that "divergence, gradient, and curl are not standalone forms" seems an ideal topic to expand on.

      Since the weak field approximation to GR has the form of Maxwell's equations, I've used this in my current essay and would appreciate any comment on the feasibility of reformulating it in terms of Octonions. In particular, because the energy of gravito-magnetic fields have mass, the fields interact with themselves, in Yang-Mills fashion. Do Octonions handle this aspect of gravity?

      Your essay certainly goes on my re-read list.

      Edwin Eugene Klingman

        Ben,

        Rereading my response to you, I was distracted by a young house guest my wife and I just received that I was really being a bad host for by being on the computer at all last night. I do not know much about quantum information theory but it would seem to be going the route of quantized spaces. The algebra and analytic tools do not seem to need this, a continuum seems to fit just fine. If I am missing your point please let me know.

        Rick

        I omitted one reference I meant to include. It is "A History of Vector Analysis" by Michael J. Crowe. This book gives the story of the "bum rap" I allude to in the abstract.

        Rick

          Hi Edwin,

          I read your essay when it came out just because I know you from your posts and previous interactions we have had. I wanted to know your perspective on wave functions even though the subject does not resonate with me.

          I am in progress on a book about my work, with a fair amount of content not on the website. This has allowed me to expound more on the philosophy motivating the mathematics as well as providing more detail. I particularly like the Sedenion chapter where I extend the Boolean triplet generators from 1-7 to 1-15, and employ them on basic quads (Octonion seven minus Quaternion triplets one at a time) to show the ways to roll out Sedenions in valid and not so valid Octonion subalgebras, and exactly where the 168 terms in N(A*B) - N(A)N(B) come from. I think you will find the book up to your desire to see more explanation.

          As for the gravito-electromagnetic fields, all there can be is presented, both in the field algebraic elements and the dynamics of force-work and conservation. The big question is what are the other rotational fields, and how do they fit in to nature. I expect them to be the glue so to speak. The optimal coordinate system will not be the rectilinear native u in the essay. It will likely be some curvilinear system that pops the symmentries.

          Rick

          Rick,

          I look forward to your book. Let us know when it's available.

          Another thing I would like to see in more detail is the algebraic 'equivalent' of calculus. As I recall, derivatives are essentially 'delta'-elements and integrals are sums of such. But I would really like examples and explanations that assume a good knowledge of calculus and a minimal knowledge of Octonions.

          Edwin Eugene Klingman

          Edwin,

          The concept is not an algebraic equivalent for calculus, it is algebra working in harmony with calculus. I do not know about newer texts, but if you look up "Mathematical Methods for Physicists" by Arfkin, in chapter one on vector analysis, he mentions an integral definition for gradient, divergence and curl as limits for a volume with the point of application an interior point going to zero of the ratio of a surface integral divided by enclosed volume integral. The surface integral is over the differential surface normal vector respectively multiplied by a scalar function, an inner product with a vector function, and cross product with a vector function; for gradient, divergence and curl. He uses this to demonstrate for example spherical-polar representations of these three forms.

          I look at this not as an alternate description for n dimensional differentiation, but instead its fundamental definition. Algebra comes into play because multiplication is its dominion. The multiplication on the differential surface normal is an algebraic expression, and if you are working with Quaternions, the three forms of scalar multiplication, scalar result vector -vector products and vector result vector - vector products are all covered by a single operation, the Quaternion product of two algebraic elements, here a 4D differential surface normal and a 4D function. If you were to leg out the Quaternion Ensemble Derivative for a transformation between rectilinear native coordinates and spatial spherical-polar coordinates, you will find proper representations of spherical-polar gradient, divergence and curl, which you may individually isolate with the resultant basis element products. Do it again, you get the second order forms. We all know what they are, so there are no mysteries on whether or not the result is correct as some may argue if the work was done in Octonion 8D space.

          There still is the notion of a difference, not simply between two arbitrary points but instead over the full (n-1) dimensional surface, but also over the full set of algebraic products between the surface normal and function to differentiate in order to come up with something transformable. The limit is as the volume approaches 0, arbitrarily close but never touching the point at which we wish to define the differentiation. So there is always a definable surface and a difference between functional values at the point of application and values in a coordinate neighborhood defined by the surface.

          This is the genesis of the Ensemble Derivative.

          Hope this helps.

          Rick

          Hi Rick,

          Thanks for the above comment. I have Arfkin and will review him as you recommend and will give some thought to this comment. I am hoping that the next few days will halt the ever-growing list of essays and allow me to focus on the ones that most interest me (which includes yours.)

          Best,

          Edwin Eugene Klingman

          Rick,

          I attach a paper I published earlier this year. It discusses octonions and E_8 within the setting of computing states of a black hole. I have been less concerned with trying to employ it directly, but am trying to come to some understanding on how O might naturally occur.

          My current essay is also directed in part this way. This leads to an argument for quantum states as modular or a part of the Eisenstein series and θ functions. The E_8 lattice is computed with the Jacobi θ functions. In the context of the Eistenstein series these form so called Mock θ functions. You can read some of the comments I make on my blog page for details that lie outside my essay, which connect more with these issues than my actual essay.

          Cheers LCAttachment #1: 1_Crowell_EJTP_counting_states_in_ST.pdf

            Thanks Rick,

            I appreciate the universality factor for octonions, and I agree. Though we may appear to live in a lower dimensional space, octonions are a fundamental reality. Yes, algebraic invariance is the crucial property to be preserved or conserved, indeed. That's what makes all the nice symmetries possible. I'll think on the 16 distinct variations all being important question.

            Regards,

            Jonathan

            Rick,

            Thanks for the responses. I can certainly sympathize with the difficulty of the nine-page limit! Regarding quantum information theory, I wasn't referring to quantum gravity or the fundamental scale in that context, but merely pointing out a currently "fashionable" field that someone with your knowledge of the special algebras could contribute to. I'm a mathematician, and I always appreciate when someone takes notice of "obscure" structures or concepts that deserve more attention. Take care,

            Ben

            Lawrence,

            If you did actually read my essay, you would have gotten my opinion on your question about coming "to some understanding on how O might naturally occur". While this might not cover your immediate concerns narrowly related to your perspective on things, it fundamentally answers the question. O provides mandated structure that I show in the essay covers Electrodynamics soup to nuts as only a subset of the formalism. The remainder is explicitly provided, and IMHO explains the remainder of physical reality.

            None the less, you probably should read my essay if you haven't. You might change your mind on believing GR is what needs to be unified with QM. If there is an Octonion tie in with QM, you will have a better shot at unifying "Octonion Relativity".

            Who knows? You might even have a life changing experience reading it. Perhaps you will have a change of faith and come to realize the path to an understanding of the quantum nature of things is down here on earth, and not in the cosmos.

            Rick

            Hi Ben,

            Having a day job that has involved RF communications for a couple of decades, I have come to learn a thing or three about information theory, the works of Shannon, and error correcting codes. I have read only a small amount about conservation of "quantum information" and also about "quantum error correcting codes". I presume one is on the cosmological scale and the other on the Plank scale, right or wrong. Anyway, I have to pick my shots with the limited time I have, and this seems on the extremes of a tree limb that already can't support its own weight.

            I am very interested in the quantum character of Nature, I just do not feel it is appropriately covered by today's quantum theories. I take a more pedestrian view, believing it will naturally occur from a bottom up analysis rather than the long chain of assumptions current theory suffers.

            Rick

            I have given your essay a read through, which is to say that I have not focused on details and depth. I always at first read a paper that way. You have constructed a differential geometry which expresses a gauge theory according to octonion algebra.

            To be honest I see the octonions as a representation of E_8 or the E_8 lattice and its extended role in the Leech lattice and quantum error correction codes.

            Cheers LC