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

        Hi Rick,

        I always enjoy reading your well-crafted and cogent arguments. I appreciate the beauty of reducing 480 multiplication tables to a manageable 64 and its identity with the 8 X 8 Hadamard matrix.

        Why algebraic invariance over analytical covariance, however? Comparing your law of invariance with Lamport's Buridan's principle ("A discrete decision involving a continuous range of values cannot be made in a bounded length of time") I find that the choice of left and right, physically, is compelled by a continuous function, not an algebraic multiplicaiton rule. Nevertheless, I grok the utility of an axial-polar rotation relation between electric and magnetic fields -- as operations that result in a union of open and closed results in a 3-d coordinate system. (If you are interested, I recently posted a draft paper on my essay site that in fig. 6 pictorially shows a topological interpretation of the phenomenon, a closed external manifold connected to an open internal plane, with all external points mapped to all internal points.)

        I think the models that fall out of your, Joy Christian's, Michael Goodband's and my research may converge in a deep sense. However:

        "There is an assumption that it is impossible to define analycity within Octonion Algebra." Of course, you know that I am one of those who assume so. And I do agree with you that the tensor calculus is inadequate to the task of a closed logical judgment on wave propagation and electrodynamics in a 4-d continuum. The topological solution still most appeals to me, however -- not as a preference, but precisely because it eliminates the necessity for a preference; everything neatly follows from a free choice of topological initial condition.

        Because we are working in the same 8-dimensional space, though, one can't help thinking that our 4-dimensional measure results all originate from a common source. All these mathematical methods may turn out to be dual.

        Thanks for a great, forward-thinking essay, Rick! Best wishes in the contest.

        Tom

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          • Post #23

          Rick

          "We must look for the connection between... and physical reality"

          Indeed, any representational device must correspond with physical reality. So the question is, what underpins this mathematical system, and does that have proven correspondence with physical reality?

          To put that question in context, I would suggest that physical reality does not have three dimensions, this is just the conceptual minimum. And the number of possible dimensions in physical reality is half (because dimension relates to a direction, either way) the number of possible directions that the smallest elementary particle could travel from a given spatial point. Dimension/size being an expression of 'physical presence' which can be conceptualised in terms of spatial footprint.

          Paul

          Hi Tom,

          I do not at all put "algebraic invariance over analytic covariance". If you realize algebra and analysis are interlocking components, you see both from a more fundamental perspective. Algebraic invariance as I have defined it is a natural and simple principle that matches observation. All currents, forces, work, energy, energy flux, stresses and strains described in an Octonion framework are algebraic invariants. They are not simply such, they are the full complement of algebraic invariants available, complete. It is really difficult for me to think this is not a very loud statement the concept is a fundamental truth.

          Without the application of a suitable analysis process, we have no connection to physical reality. Algebraic invariance can be understood without this connection as pure algebra, which I feel is important since it can be understood without the added complexity of physical reality and our current uncertain mathematical cover of it. After all, we are all hopeful we can improve the math side, whether we believe Octonion Algebra is they way or not. We are ahead in the game if we can separate out components, fully understand them, and then be able to better apply them to the greater whole.

          The Ensemble Derivative is the interlocking of analytic and algebraic concepts. It works for the banal transformation of rectilinear coordinates to spherical-polar curvilinear coordinates in a Quaternion setting to the more interesting Lorentz covariance of Electrodynamics in the Octonion setting. Realizing both halves of the Electrodynamic field components transforming in an identical fashion is non-trivial. Algebraic invariance demands the algebraic basis element products for the transformed Electrodynamic field components exactly match those they rotate into. It is all there. So besides the fact that the Ensemble Derivative works, just what is your issue with it?

          Thanks for your time and consideration on this Tom, and good luck in the competition. I read yours when it came out, but had no additional comments beyond interchanges we have already had. I still can't extrapolate as well as you can. I imagine I am missing things. I will try harder.

          Rick

          Well, now that you put it that way ... :-)

          I could be persuaded. No matter -- I think we're on different pages of the same book with the same ending. Thanks again, Rick.

          Tom