Essay Abstract

Quaternions arise from the solution of an elementary two-dimensional polynomial, and can be visualized in terms of amplitude and phase spectra by means of a multidimensional Fourier transform. Three-dimensional quaternion configurations can be found in the structure of an octonion, including one that lies outside a plane, a tetrahedron. These configurations are formulated as polynomials and spectra of their quaternion and nonquaternion solutions are discussed in relation to the Higgs field.

Author Bio

This work does not have its origin in the study of physics, but in digital signal processing beginning twenty years ago as an attempt to form a two-dimensional spectrum estimator based on an elementary polynomial, which has solutions I eventually realized are quaternions. Interest in quantum computing and curiosity about the Bell state led to octonions, and the prospect of a 3D quaternion spectrum. I am a co-author of several papers in signal processing. I received a BMath from the University of Waterloo in 1970.

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Mr. Walker,

Please excuse me; I am just a decrepit old realist. All of the stars and planets in the sky seem to be spherical. My question is: If nature can only produce spheres, why does man persist in producing so many cubes?

    Colin,

    Knowing very little of quaternions and octonians I found your essay enlightening, particularly the excellent use of graphics. Traditionally maths is a killer for essays here so you may struggle, but I think your work has value. In fact I hope you'll read my essay and consider octonians in more dynamic terms as the multiple vector orbital angular momentum/translation underpinning mine.

    You would however find it in a very different non mathematical language, and suggesting a ('Dirac') dividing line between pure mathematics and pure nature which I believe octonians may be best prepared to bridge. (I also discussed in terms of PDL modal logic last year)

    PS, I wouldn't worry about the initial '1' score, we all got that from a troll the moment our essays appeared. You're certainly down for a better score from me.

    Best of luck

    Peter

      Joe, you can count me among the decrepit old realists. I have a hard time accepting black holes or the big bang for instance.

      As to your question, a physical analogy from nature might be the cubic structure of some crystals like salt.

      There is an animated scene in Monty Python's "Meaning of Life" showing a cartoon god figure against a background of spherical planets. The figure is contemplating a sphere in one hand and a cube in the other, finally tossing the sphere to join the rest of the planets but keeping the cube to contemplate further. Maybe we just like cubes.

      Colin

      Peter,

      I have no experience with quaternions or octonions in physical applications beyond the scope of the essay, which has time and frequency as Fourier pairs. Position and momentum are typically Fourier pairs in quantum mechanics, and I suppose angle and angular momentum would also work. Unfortunately, that's all I've got.

      By the way, I notice you are following D-Wave. The idea that a company would try to make a quantum computer was what got me interested in quantum computing a few years ago.

      Thank you for your kind comments.

      Colin

      Colin,

      I would be happier with your answer if it was not for the fact that the Eskimos and the Kalahari Desert dwellers and a lot of African native people and even the Native Americans with their teepees naturally erected round abodes. Yet every city and town that has ever been built anywhere on earth at any time has always been built of cubes.

      6 days later

      Dear Colin,

      I am glad to see you entered the current contest, I also have an entry in preparation which I could not get to sooner because I was attending a conference in Sweden. During that conference, I heard a talk by Gregor Weihs in which he tried to do an experiment suggested by Asher Peres in his paper "Proposed Test for Complex versus Quaternion Quantum Theory" which purported to distinguish between a complex vs. a quaternion formulation. In this experiment there is a particular value (designated by the letter F) which under the complex formulation comes very close to 1 but under the quaternion formulation is closer to 0. When he did the experiment he did in fact find value for F much smaller than 1, but then realized that there were certain experimental limitations (which Peres had hinted at in his paper) which conspired to lower the value, and during the talk he concluded that, because he did not know of any way to overcome these limitations, the Peres test was not really an experimental test after all.

      I briefly spoke to him after the talk and it seems that the problem is that to overcome this difficulty he needs to find out independently what value F should have under the quaternion formulation with those limitations factored in in order to be able to distinguish between the two possibilities.

      Anyway, if the quaternion formulation suggested by Peres is similar to yours, then you may wish to correspond with Weihs, as he is obviously interested in doing a test (and in fact has already attempted it) and see whether, given your detailed knowledge in this area, you might be able to find a way to overcome the problem. I hope you can.

      All the best,

      Armin

        Dear Armin,

        It is good to hear from you. I found one of Peres' papers with the title "Quaternionic Quantum Interferometry" which is substantially similar to the paper you mention, if not the same. It appears Peres was interested in a radical modification of quantum mechanics in which some complex exponentials would be replaced by unit quaternions - like QM isn't complicated enough already. The experimental technique involves the examination of nuclear scattering cross-sections. It looks like quite a difficult task to find a material with three sufficiently different scatterers required to determine whether the scattering amplitude is a complex function or quaternion. It is an interesting idea I was not familiar with, and one worth investigating in my opinion.

        In contrast, my essay deals with quaternions in a way that parallels quantum mechanics (I think) by using complex exponentials. I have yet to find an example from QM but the Higgs mechanism seems promising. It is only in the last few months that I found out about the Higgs field having quaternion structure.

        It is unfortunate that quaternions have a bad reputation among physicists. Here are a couple of zingers about quaternions taken from Wikipedia. Kelvin's comment is especially harsh considering Maxwell originally formed his equations as quaternions, and whose untimely death (from cancer I believe, not quaternions) had occurred over a decade previously.

        "Quaternions came from Hamilton after his really good work had been done; and, though beautifully ingenious, have been an unmixed evil to those who have touched them in any way, including Clerk Maxwell." - Lord Kelvin, 1892.

        "...quaternions appear to exude an air of nineteenth century decay, as a rather unsuccessful species in the struggle-for-life of mathematical ideas. Mathematicians, admittedly, still keep a warm place in their hearts for the remarkable algebraic properties of quaternions but, alas, such enthusiasm means little to the harder-headed physical scientist." - Simon L. Altmann, 1986.

        Perhaps there has been a slight thaw over the last century in physicist's attitudes toward quaternions and Clifford algebras.

        Your dimensional theory is the most promising idea for a theory of mass I have seen, and there are hints that dimensionality plays a role in the states of quaternion resonance. I look forward to your essay.

        Colin

        Dear Hai

        I was a bit worried about the large size of my essay. There is a smaller version (385 kB) here. The filename is qspecSMALL.pdf. You can download the file using the down arrow on the right hand side of the page. It has the most important figures in case you were unable to see the graphics. I hope this helps.

        Colin

        Dear Colin

        Thank you for your interesting note on my fqxi essay page - I enjoyed reading your very nicely written and illustrated paper. The artist in me enjoyed the beauty and complexity of the figures. What program did you use? I was happily surprised to find a reference to my Beautiful Universe (BU) vacuum made up of a Kepler packing of spinning nodes. As we both pointed out tetrahedral configurations emerge from such a Face-Centered-Cubic packing - or less technically apples stacked on a grocers shelf in a square grid with alternate layers staggered half a space. Your highly technical approach is beyond me, but I have always been fascinated by the story of Hamilton and his quaternions and an eponymous bridge in Dublin. In fact (BU) can be summarized as Hamiltonian transfer of energy by spinning nodes making up an FCC lattice.

        It is encouraging that you think a Higgs condensate may be related to such a structure. You might have read last year's fqxi essay by Norman Cook of nuclear dynamics also based on FCC. In fact it was Norman who inspired me to choose this particular packing from amongst the other possible ones. I will alert him to your work.

        I found it interesting that you associate matter with a 1/2 spin tetrahedron - that is exactly how I illustrated E=mc^c (matter creation from energy and vice versa) in my Beautiful Universe Theory paper. This link gives a revised pdf - in the origional I had all the nodes spin according to a left hand rule, now reversed. Beyond these qualitative conclusions I could not go much further. I am now trying to use the Basic! app on my iphone to simulate BU interactions...enough said :)

        I must add that in my theory the lattice is 'absolute' - I feel flexible spacetime is an unnecessary chimera in physics, and that time is not a dimension - but I am curious to read Mr. Shirazi's paper.

        Your work goes to to the heart of the It-Bit question, but perhaps it is too technical for many (but by no means all) of us here at fqxi - and I hope you will continue to pursue the promising thread you have taken.

        Cheers indeed

        Vladimir

        Vladimir

          Dear Vladimir

          The diagrams come from my Linux C program written using the Cairo graphics library.

          I share your view about space having an absolute nature. I have found that there are two ways to view gravitational potential energy which is what ultimately determines spatial curvature:

          1: energy is a sum (the result of addition) and space is curved.

          2: energy is a product (the result of multiplication) and space is flat

          Conventional physics has concentrated on option 1. Option 2 was the subject of my last year's essay with more gory details at this website. I have a suspicion that formulating the energy of gravitation as an exponential map could be compatible with general relativity in the sense that no arbitrary parameters need be introduced, but I have a little hope of showing this.

          Anyway, we are on the same page, so ... Cheers again,

          Colin

          Sorry to barge in at this discussion. I tend to agree with Colin. If the vacuum structure is face-centered-cubic (Kepler packing) of basic building blocks as I think we both agree it might be, then as a sphere gets smaller and smaller the surface will become 'stepped' or pixillated. Eventually you end up with just an icosahedron (?) and then a cube.

          Cheers

          Thanks Colin - I now remember our discussion of your paper last year. I have a feeling that GR is unnecessarily complex and should not be the 'standard' formulation of gravity, predictive as it might be. Einstein built GR on the basis of SR - tied up as the latter is with an observer-referenced paradigm. Is there an observer needed as a light beam curves around the sun? Without SR the equivalence of gravity with acceleration is enough to describe local linear gravitational density gradients in a flat space. Does that agree with your point (2) ? Again I could not prove that this will serve to replace all of GR but I suspect it might.

          Best,

          Vladimir

            Dear Vladimir

            My (and probably your) uneasy feeling about general relativity is that it is a top-down formulation. There is no doubt that GR is a complicated set of inter-relations, but at their foundation is one relatively simple equation which has passed every first-order test. This coupling of a simple model with predictive power is compelling in my opinion - so compelling that in the event of failing a second-order test such as LATOR proposed for the end of the decade, I would first try to finagle GR into satisfying the new observations.

            The term "frame of reference" might be more appropriate than the term "observer" which has a subjective connotation. GR has shown that it covers complicated weak-field cases which otherwise would be a hodge-podge. I think SR and the equivalence of gravity with acceleration are sufficient to model strong-field gravity in a flat space given idealized conditions of free-fall, which is in accord with my point (2) from the previous post. Perhaps this could be part of a bottom-up formulation of gravity.

            Colin

            10 days later

            Hello Colin,

            I enjoyed reading your paper. It was especially nice because quaternions and octonions are something I'm enthusiastic about. You'll see them mentioned in my essay, once it posts. But I feel like your essay wandered off topic a bit, and failed to end conclusively. I'm not sure you made your point.

            I think I probably have to read Nikkah Shirazi's paper - which I have downloaded - to understand what you mean at the end. So the relevance was not emphasized enough, and I see there clearly is some. But what you did write about, I enjoyed reading. Good luck in the contest, and enjoy interacting with the other authors.

            Have Fun,

            Jonathan

              Colin,

              If given the time and the wits to evaluate over 120 more entries, I have a month to try. My seemingly whimsical title, "It's good to be the king," is serious about our subject.

              Jim

              Hi Jonathan,

              Your observation about my essay wandering off topic is true. I learned about quaternions and the Higgs mechanism just as I was nearing the end and got completely side-tracked.

              The connection to Nikkah Shirazi's theory is a bit tenuous, but a promising one that seems obvious on looking at spectra of Pauli and Hadamard matrices. His premise is that there is 2+1 areatime existing as a superposition in our 3+1 spacetime. Areatime objects must be described "in terms of a superposition of all possible worldlines of the massive objects into which they can emerge" [from his 2012 essay].

              Now replace "massive object" by "resonance". What I noticed was that the zeroes of quaternion polynomials (those frequencies at which resonance can occur) differed in their dimensionality depending whether the underlying qubit was in a Pauli (real or 'actual') state, or in a Hadamard (superposed or 'actualizable') state. [Taking a quaternion as a qubit (u,v) with a twisted companion (-u*,v*).] While there is a clear and consistent difference in dimensionality, the sense is opposite what I expected - for example, in 3D spectra the zeroes of Pauli states require a plane while Hadamard states produce lines of zeroes (more convincing than 2D but not shown in the essay). Resonance requires inverting the polynomial, so considerations of dimensionality ought to be inverted as well. This notion of dimensionality applying to states of resonance is what I most want to show.

              I hope this gives a better idea of how my essay relates to Armin's dimensional theory. Looking forward to reading yours.

              Now if I can just keep the condensate off these Higgs goggles...

              Colin

              8 days later

              Thanks greatly for the comments Colin,

              Given the content; you might find a lot to like in the essay by Tom Ray, where he talks about fermionic condensates as a model for primordial spacetime. I hope you also get around to reading my essay, now that it has posted. As I said earlier; I mention the quaternions and octonions therein. I expect to be rating your work later today, after a brief review.

              All the Best,

              Jonathan

              Hi Colin,

              I've just sent you an e-mail about available references on quaternion Physics.

              There is a lot out there, and I've been sifting through it for a while.

              I've plenty to recommend or share.

              All the Best,

              Jonathan

              Dear Colin. Hello, and apologies if this does not apply to you. I have read and rated your essay and about 50 others. If you have not read, or did not rate my essay The Cloud of Unknowing please consider doing so. With best wishes.

              Vladimir

              Thanks for the comments on my page, Colin.

              I'm glad you enjoyed the Steinhardt lecture; I thought it was a lot of fun to watch him walk from one side of the room to the other, playing both defending and prosecuting attorney - for the theory he helped develop. I hope your research goes well, and bears interesting fruit.

              All the Best,

              Jonathan

              8 days later

              Dear Colin,

              I have down loaded your essay and soon post my comments on it. Meanwhile, please, go through my essay and post your comments.

              Regards and good luck in the contest,

              Sreenath BN.

              http://fqxi.org/community/forum/topic/1827

              4 days later

              Dear Colin,

              I very mucjoyed your essay which also touches on the existence topic from the mathematical angle. This is understandable given your Bio. Mine is more from a logician's point of view.

              I am sceptical about the Higg's mechanism which you mention. But even if correct some of these issues cannot be fully resolved without the nature of space being put to rest. A 6 rating I think is fair on your essay not being a mathematician myself and being limited in this respect.

              Best regards,

              Akinbo

              *If you would be reading my essay, read the JUDGEMENT in the blog in the case of Atomistic Enterprises Inc. vs. Plato & Ors delivered on Jul. 28, 2013 @ 11:39 GMT. In that judgement points and monads turnout to be the possible bits. Thanks

              Hello Colin,

              I like the Quarternion approach. I have a theory based on the tetrahedron that partly unifies the four forces of nature and resolves the three paradoxes of cosmogony. Further it relates the relative masses of the proton, neutron and electron to 99.999988% of prediction.

              I like how you apply your approach to the Higgs field.

              If you get the time please take a look at my essay which deals with dimensionality of observation - perhaps hinting at Black Hole existence, but perhaps suggesting that the singularity can't destroy information and that eventually they release all information.

              I note above you had issues yourself accepting black holes and I think you're right to question them.

              Best wishes,

              Antony

              Hi Colin,

              Thanks for an intriguing perspective.

              > Quaternions arise from the solution of an elementary two-dimensional polynomial, and can be visualized in terms of amplitude and phase spectra by means of a multidimensional Fourier transform.

              I share your enthusiasm for quaternions and I advocate the use of Geometric Algebra. In my essay Software Cosmos you can see how these play into both the explicate world that we measure and also an implicate space. These two spaces (one observable, one not) might be linked via just such a Fourier transform.

              > Two forms correspond to slices through the octonion vector space and are essentially two-dimensional. The third form is a tetrahedron in three dimensions.

              > The Fano plane shows the seven imaginary elements of an octonion having seven sets of quaternion cyclic ordering given by three sides of the triangle (426, 635, 514), the three altitudes (473, 671, 572), and the circle of midpoints (123).

              Sounds like this would be useful in constructing models for quantum contextuality. Michel Planat discusses such models in his essay, including the Fano plane. My comment to Michel suggested he look at k-rational points based on Q(phi), the extension of the rationals by the golden ratio. The coordinates of a tetrahedron (and many other polytopes) would qualify.

              > I can imagine the picture of existence that emerges could be analogous to a quantum computer operating coherently in the Higgs condensate.

              I hope you get a chance to check my essay, as I think my picture nicely complements your views.

              Hugh

              Dear Colin,

              We are at the end of this essay contest.

              In conclusion, at the question to know if Information is more fundamental than Matter, there is a good reason to answer that Matter is made of an amazing mixture of eInfo and eEnergy, at the same time.

              Matter is thus eInfo made with eEnergy rather than answer it is made with eEnergy and eInfo ; because eInfo is eEnergy, and the one does not go without the other one.

              eEnergy and eInfo are the two basic Principles of the eUniverse. Nothing can exist if it is not eEnergy, and any object is eInfo, and therefore eEnergy.

              And consequently our eReality is eInfo made with eEnergy. And the final verdict is : eReality is virtual, and virtuality is our fundamental eReality.

              Good luck to the winners,

              And see you soon, with good news on this topic, and the Theory of Everything.

              Amazigh H.

              I rated your essay.

              Please visit My essay.

              Colin - a truly outstanding piece of work. I'm blown away with how few people have noticed the excellence of this essay. You just filled in a critical piece of the puzzle of a mathematical problem I've had for several years. Thank you.

              Kind regards, Paul

              4 years later

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