Your paper is a fascinating way to look at geometric algebra. I am thinking this is a way to represent higher or hypercomplex systems like quaternions without appealing to additional basis elements. The theory remains on the Hopf fibration between S^1 --- > S^3 --- > S^2 instead of on the next level S^3 --- > S^7 --- > S^4. It appears one is able to capture much algebraic geometry and Clifford Cl(3,1) structure this way. I have read it through a couple of times and am reading it much closer on the third reading.

Cheers LC

Dear Dr. Crowell,

I am happy that you find my text interesting. You are right, the main idea was to only work with explicitly defined planes in 3D, the planes playing the role of complex planes, eliminating in this way formal algebraic using of the imaginary unit. The simplest case was considered corresponding to qubit quantum states, two-dimensional complex vectors. Generalization to higher dimensional cases should be elaborated which may correspond to multiple object quantum states. I believe you realize that we are at the very beginning of a wonderful trip.

Sincerely,

A. Soiguine.

    Alexander,

    Many thanks for an excellent read. You have given me several new insights. I had not previously thought of the Clifford basis vectors as representing a plane. In retrospect, it is clear now since every vector has a plane that is normal to it. Also, I had not previously seen any advantage to Clifford over Hamilton. You have made me reconsider this. Setting i = (e2)(e3) still seems odd but at least I now understand the reason for it.

    I wonder what you think of the following ... Euler's Equation is (e^i*theta) = cos(theta) i*sin(theta). It is possible to express the dot product of two vectors by using the cosine of the angle between them. It is possible to express the cross product of two vectors by using the sine of the angle between them. Therefore, if there are two arbitrary vectors that are restricted to the j-k plane, it is possible to construct Euler's Equation using these two vectors, and the complex i then does not appear in the right-hand side of Euler's Equation. Is this a simplified version of your Equation 3.2?

    Beginning with your Equation 6.1, I see a connection between your paper and Dr. Klingman's paper since division by the absolute value normalizes the value to either plus or minus one. The un-numbered equation at the top of page 8 is also similar to part of Dr. Klingman's work.

    The two step rotation is a little hard for me to visualize although I generally understand your meaning. A sketch would be very nice, but with so much happening it might simply be confusing.

    Best Regards and Good Luck,

    Gary Simpson

    In what I can see the B_3 and the elements β_i with the i = sqrt{-1} in effect emulate quaternions. It seems to me that you have quaternions in a different guise. In doing it this way instead of working in four dimensions you are working in 2 dimension with the B_3 group. This is an interesting way to proceed.

    Cheers LC

    6 days later

    Dear Dr. Simpson,

    Did not reply because did not have time to read the Dr. Klingman's work. Have not read yet.

    Sincerely,

    A. Soiguine.

    a month later

    Dear Professor Soiguine,

    I thought that your engrossing essay was exceptionally well written and I do hope that it fares well in the competition.

    I think Newton was wrong about abstract gravity; Einstein was wrong about abstract space/time, and Hawking was wrong about the explosive capability of NOTHING.

    All I ask is that you give my essay WHY THE REAL UNIVERSE IS NOT MATHEMATICAL a fair reading and that you allow me to answer any objections you may leave in my comment box about it.

    Joe Fisher

    Alex,

    I think the importance and validity of your essay may be counterpoised by appearing a bit off topic to some, and will also be a little lost on most. The preponderance of mathematics isn't encouraged in this essay format and I think it's a shame you didn't spell out the great implications.

    Those may explain why it's not where I think it should be, at the head of the field. I suppose I would because, as you've seen before, the foundations of our work are entirely compatible. Indeed I hope you may look at this short video including closely related matters as well as reading my essay.

    3 plane OAM gives a 'time dependent' cosmic redshift and resolves CP violations etc.

    I wish you well in the contest.

    Peter

    Dear Peter,

    Thank you for the your comments. The main reason of disadvantages you are mentioning is that original text was 20 pages. I shrink it to 12 (official requirement) but soon was informed that just 9 allowed (why?). So many important explanations disappeared. I definitely need to look for a place to publish much bigger text with proposed applications, etc.

    Alex.

      6 days later

      Dear Alexander,

      The idea of «Geometric evolution» is very deep and heuristic. And if to expand this idea from "origin of geometry" in the spirit of E.Gusserl ("Origin of Geometry"), then from "beginning of the Universe" (taking into account "fundamental constants" of the Nature) to the Universe as whole in the spirit of V. Nalimov ("The self-aware Universe") ? I think that then will be revealed the deep ontological structural connection Mathematics (consciousness as an absolute vector) and Physics (direction of the evolution of Nature and states of matter).

      Kind regards,

      Vladimir

      Dear Vladimir,

      Thank you for the comment. Actually, I am just trying to create quantum mechanical framework comfortable, convenient for my research purposes which are mainly about topological quantum computing. I believe your expertise in structural connections between math and physics is much deeper than mine. I am not looking so far, at least yet.

      Thank you again,

      Sasha

        Thank you very much, Alexander, for your concrete and rapid response. I hope that in the "topological quantum computing" concept "ontological (structural) memory" take its rightful place. My high score. I wish you success in research and contest.

        Yours sincerely,

        Vladimir

        10 days later

        Alex,

        It's now clear what you've had to do. It's a shame you couldn't give it more time, and many will consider it not in line with the competition guidance. I on the other hand do not, but do perceive the great value of your work, and will score it appropriately.

        best of luck.

        I do hope you'll make time to read and score mine as I think it may contribute to the direction you're heading, and your comments will anyway be valued.

        Best of luck

        Peter

        4 days later

        Nice paper, Alex.

        The Hopf fibration may be a key to elementary particle topology.

        .....David

        Thank you so much David.

        Maybe you remember the story began more than 20 years ago when I was still in Russia and had read some your works on Geometric Algebra. I wrote then a small book on the subject and sent you a copy. You kindly answered confessing that Russian language was a bit problem for you.

        This essay is just the beginning. I am working on some further interesting ideas. You are absolutely right about importance of Hopf fibration. All will be about even geometric subalgebras, their fibrations to complex number systems, tangent bundles, connections, etc.

        Thank you again.

        Alex.

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