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

        Hello again, Rick;

        I just left a comment on Lawrence's page that bears mentioning here. He said that 'octonions are really a system of quaternions' (7 of them) which relates to a statement I made in the paper "In Defense of Octonions" with Ray Munroe. I wrote to Lawrence that while octonions could be represented that way, they do have to be resolved in an orderly fashion, and it's not the same as saying O is really H x 7.

        In a paper I'm working on now; I suggest that working with octonion algebra is similar to assembling a watch. "Every layer or sub-assembly must mesh correctly, and then the layers must fit together in the correct relationship, for the watch to function. The same metaphor aptly describes what is required to do multiplication and division with octonions, as you must perform seven ordered groups of three operations in sequence."

        Is this an apt characterization? My guess is that Lawrence's approach would treat the component quaternions in the same way that Physics folks normally treat grad, div, and curl - as independent or fundamental quantities, where in reality (or as you demonstrate) they are structured components of the quaternions. I suggested his statement might be made true if octonions are treated as an ordered or nested system of seven quaternions. Is this essentially correct? Do we also need an extra scalar value, for the Real component?

        Enlighten us.

        all the best,

        Jonathan

          Jonathan, Tom, Edwin,Brendan and friends, you are bad strategists meriting simply to ba analyzed by the laws. Sort your members Mr Tegmark and MR Aguire.

          They have not theior place on this platfrom.They decrease the velocity of evolution of fQXi and its credibility must be universal.Sort these pseudos.

          They have simply a strategy for the maney, they are just frusterated vanituious and envious. They must be sorted. They delete in correlation of their strategy of discriminations.In fact they fear that I arrive at New York, so they try with the discriminations. I have faith me, them no !Don't compare a thing which cannot be compared.

          Mr Tegamrk and Mr Aguire, don't be troubled by their strategy and their words. These people have simply a heart without faith and reason. They are not scientists, but business man. And you ibm, forget also these persons !with the soa and its superimposings. Be rational and universal.

          Steve

          Jonathan,

          The Octonions are at the end of the R to C to H to O chain, and their discovery order as well as the doubling process both follow simple to more complex direction. I prefer to think of the Octonions as the most fundamental since there is no higher dimension normed division algebra, and each of the more simpler division algebras are all O subalgebras, meaning their basis set is a subset of the O parent, and the subset forms a proper algebra all by itself. This means it abides by the three rules of algebraic element addition, multiplication by a scalar, and algebraic element multiplication closed for the subset of basis elements.

          Geoffrey Dixon mentioned in his latest book his lack of enthusiasm over the doubling process, and I must say I fully agree with his position. This process has been the genesis of the "made from" mentality. As he mentions, you can double through the division algebras, but you can also double O to the Sedenions, which are not a division algebra, so the doubling does not conserve this very important characteristic. I further contend it has also led to a one O algebra mentality because many missed the fact doubling a commutative algebra (R to C or C to H) is not the same as doubling a non-commutative algebra as with H to O. The subalgebra perspective works end to end since R can have no subalgebra since it has only a single basis element.

          Sedenion algebra defines 35 perfectly valid H subalgebras, and 15 Octonion-like subalgebras of which only 8 can be made into normed composition algebras. The latter is why Sedenions can't be made into a division algebra.

          C has but one choice for definition of basis element multiplication, so no variability impact on the definition of H. But H does have 2 choices analogous to 3D right handed and left handed vector products. The subalgebra connection for H from O must accommodate these 2 choices, which is why there are 16 choices for proper O Algebra. The 16 choices, that is the full variability in O definition, is determined by handedness choices for the 7 H subalgebras that leave O a normed composition algebra. If as you question, O was simply Hx7, there would be 128 valid O for all possible H choices, but there are not.

          The proper H selections are less important from the H perspective than they are from the O perspective, since things only get interesting when we make O algebraic element products. Then one "H" in a way multiples another "H", and the ins and outs of this sets the algebraic variance and invariance characteristics of fundamental importance.

          Kind of a long answer, but hopefully you will find it satisfactory.

          Rick

          Weak reasoning ! :)

          Edwin, Lawrence, Jonathan,Joy,Tom,Rick and friends.

          It is weak, even the strategies are weak, it is easy to find the holes. 0 really.And me alone, integre, transparent, without tool,just with rational sciences. Between us ,it is ironical no? In all case me I laugh. Because even like that I have teached you so many things ahahah incredible no?

          And they insist furthermore with the compactification and the geomatrical algebras. If you you understand the 0, the 1, the infinity, the numbers, complexs, naturals, reals, R C O H or this or that.If you you understand the finite groups and the real infinity. Me really I am the queen of England you know. If you pondered intresting between us, ok, but no, even like that you insist on your stupidities for I don't know me.Probably a problem of vanity or a kind of play just due to your unconsciousness.In fact,you are not really skillings.Because If I learn here on fqxi , it is not with your team you know.

          In fact you are not general, here is the probelm.And even your details are not good.So you imagine my pity , you can delete, betrween us, you understand, isn't it ? your hate increases, logic, your strategies are just a simple bad play of a kind of super team , but in fact it is a team who makes pity.really. You are even ready for all in fact. You are bad persons simply. When I see all this story since the begining. It is incredible in fact with your false universalism and false patriotism, I am a better american that you furthermore because me I am a real universalist loving his fellowman. A real christian. And you have made all this just because you are vanitous , envious and full of hate against people who critics universally. It is a sad team with bad tools, and bad strategies. You are not universal. Fqxi merits more than this kind of comportment. The integrity, universal is essential.

          How can you have this kind of comportment in fact.What is your heart to make that. I pray for you, I have pity. All my pc is checked.All my platforms where I discuss.Linkedin,xing,fqxi also,facebook,....it is really bizare.

          I forgive always. It is sad this story in fact. How people can make that ? the world is sick, if already the imrpotant systems of foundamental sciences are bizare.Where are we going? It is bizare simply.