Essay Abstract

The previously initialized approach is used for description and analysis of qubits, geometric phase parameters - things critical in the area of topological quantum computing. The used tool, Geometric (Clifford) Algebra is the most convenient formalism for that case. Generalizations of formal "complex plane" to an arbitrary variable plane in 3D, and of usual Hopf fibration to the map generated by an arbitrary unit value element are resulting in more profound description of qubits compared to quantum mechanical Hilbert space formalism.

Author Bio

Math/Physics Professorship, R&D in advanced simulation software. Education: St. Petersburg State University, Russia. Research work on feasibility of geometric algebra generalization of qubits for the topological quantum computing purposes, their relations to QM wave functions, unspecified variable fibration probabilities. Web-site www.soiguine.com

Download Essay PDF File

Dear Alexander Soiguine,

Your most impressive essay is long on math and short on interpretation. If I were to take a stab at interpreting it, I would use your focus on 'bivector' and 'qubits' to associate the bivector representation of the spin, which is a circulation in a plane with a fixed area but undefined shape, with the orthogonal vector usually considered as the spin axis and often interpreted as a 2-D 'qubit', since the vector can point 'up' or 'down'. The well-known identification of the imaginary i with the bivector is of course Hestenes idea of the way Schrödinger unwittingly incorporated spin in his non-relativistic equation which is usually interpreted as being 'spin less'.

You appear to have discovered some unorthodox feature of Barry curvature in the form of additional bivector elements (page 8) which, as you say, appear to represent a torsion term but could use further elaboration.

I tend to agree that "With explicitly defined variable "imaginary unit" many things become not just more informative but also much simpler." You illustrate this by generalizing the Schrödinger equation.

I would appreciate any correction of my above interpretation and I would encourage you strongly to make maximum use of the comments in your thread to flesh out a very dense presentation. This is a legitimate way of presenting information that did not fit in nine pages.

I invite you to read my essay and would welcome your comments.

Best,

Edwin Eugene Klingman

  • [deleted]

  • Post #3

I have yet to read your paper. I just got to it now. It looks very interesting. This seems connect with modular gravity and quaternion gauge theory.

LC

    Dear Dr. Klingman,

    Thank you very much for careful reading of my text.

    I appreciate your comments and agree with what you said.

    David Hestenes was the scientist who opened my eyes on Geometric Algebra. Though he has done huge job, but being not hundred percent consistent with the purpose to fully eliminate formal imaginary unit from traditional quantum mechanics he did not achieve new comprehensive logically perfect QM formalism.

    Back to my text. What, for example, was excluded was related to Clifford translations on S^3 spheres. Moving along the translation orbit (vertical tangent space component) causes synchronic rotation in horizontal tangent space plane.

    My further work will first of all be about holonomies and two point value states in the proposed geometrical algebra terms which is necessary to deal with anyons. Only practically valuable results, say demonstrating Hamiltonians that support stable anyonic states, can show that the formalism has its own value.

    Sincerely,

    Alex (aka Sasha) Soiguine

    Dear Dr. Fisher,

    Thank you for the comment.

    My text deals with something not too abstract. Imagine you are watching the rain of golden coins falling from the sky. Coins are chaotically rotating and you see them reflecting light at different random angles. This is more or less exactly what is behind the geometric algebra bivector formalism.

    Sincerely,

    Sasha Soiguine.

    Dear Professor Soignine,

    Thank you for the inspiring essay, in your view what is the description of the wave equation considering your arguments and the fact that classically it was based on wave particle duality and collapse of the wave function whenever the particle was observed. Do you think that the mathematical description formerly based on Hilbert space formalism is not reflecting the fact??

    I have written an essay before where I used new arguments regarding the interpretation of SR and shown that length contraction doesn't happen in a slow moving frame which is approximately inertial but uses linear transformation based on fiber bundle method.

    http://fqxi.org/community/forum/topic/1769

    You are also welcome to read my essay in this forum.

    Kind Regards

    Koorosh

    Dear Dr. Koorosh,

    Thank you for the message. I will carefully read your essay. Do expect enjoying it. Will write more later.

    Sincerely,

    A. Soiguine.

    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