Essay Abstract

This text demonstrates that the complex i can be combined with a Hamilton style quaternion to produce a 5-D mathematical structure. Essentially, the complex plane is combined with an arbitrary unit vector. The complex i is shown to anti-commute with the unit vectors i, j, and k. The resulting geometry is shown to be an extension of Hamilton's quaternions based upon the complex plane rather than real numbers. This new geometric structure is presented in Figure 1 and Equations 3 through 3.3. This configuration makes it possible to calculate the diameter of the proton at rest with the estimated value being 1.668 x 10-15 meter. This is within the accepted measured range of the proton diameter at 1.755(102) x 10-15 meter as given by the NIST, and it is very close to the proton diameter at 1.68174(78) x 10-15 meter as measured at the Paul Scherrer Institute in 2010 by using muonic hydrogen.

Author Bio

I am a retired chemical engineer. I graduated from Texas A&M University with BS and MS degrees many years ago. I am interested in quaternions and wave functions.

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Thank you for taking a look at my essay.

The essay topic seems to me to be very challenging. The fact that it took so long to post the first set of essays seems to support this belief. Regrettably, the only contribution that I can make concerns the "arrow of time". My essay presents a different way of thinking about the "arrow of time". I argue that it is the complex i that has direction rather than time and that time is simply a scalar.

You might ask "Since they are presented together as i*c*t, what difference does this make?" The difference is that if time is the dimension then the result is a single 4-D spacetime. But if the complex i is the dimension then the complex plane can be combined with 3-D space to produce a 5-D space with a continuous group of 4-D subsets. I think this extra degree of freedom can allow QM and GR to be combined into a single entity.

For an inertial reference frame, the only physical dimension that is needed is the direction of motion. This model seems to say this and thereby allows for the other two spatial dimensions to be used for GR and QM. It is not clear to me which should be scalar and which should be the complex i.

Paul Dirac once commented that spin is the square root of geometry. I think this 5-D geometry is consistent with his thoughts on the subject. The reason is that the square root needs a direction to point just as the complex i needed a direction to point.

My hope with this essay is that someone who is knowledgable in Physics might recognize some similarity between the presented structures and known Physics. This might give me a clue as to how to proceed.

Best Regards and Good Luck to All,

Gary Simpson

    Dear Gary,

    As I have thoughtfully pointed out in my brilliant essay, SCORE ONE FOR SIMPLICITY, the real Universe consists only of one unified visible infinite surface occurring in one infinite dimension, that am always illuminated by infinite non-surface light. Invisible complexity has never existed, so it would be physically impossible for invisible complexity to combine with any other sort of invisible miasmatic state.

    Joe Fisher, Realist

    5 days later

    Hi Gary ,

    Congratulations for your papper about geometrical algebras.It is very relevant these algebras and these diamaters and others Tools.Thanks for sharing and good luck in this contest.You are going to have a prize in logic.You merit it.

    Regards.

      Steve,

      Thanks for taking a look and reading my essay. I think the geometric algebras will eventually explain everything to us. What is most interesting to me about what I have presented is that under some circumstances, an 8-D octonion can simplify to a 5-D space. I think this is very significant. It is necessary to take all of the concepts together as a single entity to understand fully what is happening with these algebras.

      Best Regards,

      Gary Simpson

      Hi Gary,

      You are welcome.I liked your method in playing with octonions and this 5D.I like also these geometrical algebras now.I try to formalise my theory of spherisation with 3D quant and cosm sphères with the spherical geometrical algebras and my equation about matter and energy E=mc²+ml² It is a big work Gary I search the good method.I like the quaternions, and others with the 3 vectors and the scalar and this and that, but we consider points for these algebras.If now we insert the spherical volumes and the motions more the geometrical algebras, that become relevant.Even for computing we can correlate with the bloch sphere for qbit.The strings are a beautiful tool for computing also.The convergences can appear after all.good luck in this contest.:)

      14 days later

      Gary Simpson,

      Your finding that the mass ratio of proton to electron is related to 'pi' is interesting; the observed deviation is attributed to absolute motion. In my opinion, the deviation may be due to quantum- continuous disparity.

      For example, make a circle using small spheres. What is its effective circumference? A continuous line passing through the outermost points, or a continuous line passing through the centers of the spheres, or something in between. Here I think the Feynman diagrams may help, because these are used in the case of wave- particle duality, which is a case of quantum- continuous disparity. I regard the Feynman diagrams as approximation techniques. However, I have not used them till now.

      As my approach is Newtonian, I take that the integration of grainy (quantized) matter follows "symmetrical spherical packing", and so 'pi' has a crucial role in deciding the masses of proton and neutron. The mass of neutron is slightly grater than 1838 electrons. This is not an arbitrary number; it represents the minimum number of spheres required to make a nearly spherical structure having perfect symmetry, when the spheres come in pairs (like electron-positron pairs). Here also, 'pi' decides the number. The surprise is that Fine Structure constant remains inbuilt in such a structure.

      So the equation you obtained for proton radius cannot be a mere coincidence; it has something buried inside (however, in my model proton and neutron have sizes proportional to that of electron).

      Jose P Koshy

        Jose,

        Thank you for commenting on my essay. I was beginning to feel a bit ignored. It is clear to me that you read and understood.

        The coincidental value 6pi^5 is what set me upon a long journey. To the best of my knowledge, it was first noted by Friedrich Lenz in the April 1951 issue of Physical Review. I independently discovered it in April 2012 and I have been trying to make sense of it since that time.

        You mention an alternative method of explaining the deviation of the empirically observed value of Mp/Me from 6pi^5. Your thinking seems to be very similar to that of Don Hotson. He authored three papers in the fringe web journal "Infinite Energy". The first two of those essays are excellent reading and very thought provoking. Regrettably, he has passed away. I would expect you could find his work using a Google search.

        My present thinking is that the pi^5 term is the result of the 5 dimensions and that the coefficient 6 is the sum result of a scalar one value for each of the three real dimensions vector i, vector j, and vector k, plus a scalar one value for each of the complex dimensions (complex i)x(vector i), (complex i)x(vector j), and (complex i)x(vector k).

        Best Regards and Good Luck,

        Gary Simpson

        Hello to both of you?

        All these reasonings are relevant.Do you know if we could find the correct spherical volumes and the correct serie with the good probablistic method.The serie and its finite number and the volumes are the keys in fact logically.I read the works of Feynman about électrons.It is a wonderful discussion.The fine structure constant is relevant also.I study a little this constant and its meaning.Feynamn in all case was incredible, when we read his papper and his words ,it is so deep and general.He was an incredible thinker.Do you remember that he said these words " one day we shall see all the truth and we shall say all oh my god but how is it possible that we have not seen a thing so simple before ?" sometimes the complexity returns to simplicity :)

        Best

        Steve,

        Thanks for the continued interest. Perhaps you will submit an essay?

        Best regards,

        Gary Simpson

        You are welcome Gary,

        My English is not good and I have difficulties to resume, furthermore I have not a program of maths.Best Gary good luck also.

        Hi Gary,

        I like your work involving complex numbers in relation to particle physics. I too believe particles can be constructed or modeled with a field of complex number vectors, though I think 3 1 dimensions is sufficient. I have such a model for electrons/positrons here:

        http://gpcpublishing.com/index.php?journal=gjp&page=article&op=view&path%5B%5D=367

        It would be good if you could explain how your work fits into the essay topic of wandering towards a goal resulting in goal oriented structures in the Universe. Clearly the formation of particles is the most essential step in this process (as I have discussed in my essay) so it would be good to explain this in the essay.

        Overall: nice work and a well written essay. Good luck!!

        Best Regards,

        Declan Traill

          Declan,

          As I state in my essay, I address the "arrow of time". Essentially, I eliminate it and replace it with scalar time that operates upon the complex plane. This simplifies the methods whereby any system may evolve with respect to time.

          Both GR and QM are 4-D models. They share the same physical space. However, they treat time differently. The time in GR is not the same as the time in QM. Therefore, there is no 4-D model that can combine them. The least number of dimensions tat can do that is 5 which is what I present.

          The fact that I have accurately calculated the size of the proton should be considered a very strong piece of circumstantial evidence in support of a 5-D model.

          Best Regards,

          Gary Simpson

          Hi Gary,

          Can you explain what is different about time on GR and QM?

          It seems to me that formulating GR in terms of a scalar field where space is fixed and the speed of light and rate of time are variable (by the same amount - thus giving the impression to any observer that the speed of light doesn't change) fits in nicely with QM, and in the same space 3D particles can be modeled that agree with QM.

          What would the fifth dimension be?

          Best Regards,

          Declan Traill

          Declan,

          Please read my essay slowly and carefully. It is not light reading. Figure 1 presents the 5-D geometry. It does so in a way that can be construed as 3-D geometry for an inertial reference frame.

          In both GR and SR, time and space are presented together as a 4-vector. The 4'th dimension is presented as i*c*t. Essentially, i*c*t is added to a space vector and the result becomes Minkowski space-time. GR is presented taught using tensors. Thus far, I have not learned to use tensors competently.

          Time in QM is more simple. It is presented as a scalar rather than a pseudo-vector component of space time. Space in QM is also treated differently ... more like a scalar than a vector. The Schrodinger Equation presents time coupled with the complex i and it presents the space variables x, y, and z presented as scalars rather than as vectors containing the unit vectors i, j, and k.

          Wikipedia is a wonderful resource. I give Wikipedia a modest amount of financial support. But it is not a substitute for other sources.

          BTW, Einstein himself specifically rejected the idea of a scalar field. I mention this near the end of the essay. Space-time is not a vector plus a scalar as in a quaternion. Space-time is a vector plus a "time-vector" based upon the speed of light. That is not the time of QM.

          Best Regards and Good Luck,

          Gary Simpson

          I see you have an essay here. I remember an essay last contest with similar content, it dealt with quaternions. I presume you are the same fellow. I will have to read it tomorrow.

          As for spelling, I often write from and form and similar errors, drop s from plural words and so forth. It is just a glitch.

          Cheers LC

          I just gave your paper an 8. The reason I did not give a 10 is that I think things go awry at the end. Up until equation 4.2 and page 8 things look very interesting. It is at the end when you write that the time is a scalar and not GR or SR and so this departs from GR. Actually this is a form of Clifford algebra CL_{3,2}(C) ~ SO(3,2) based on the interval

          s^2 = i^2 + ω^2 - (\bf i}^2 - (\bf j}^2 - (\bf k}^2,

          where for one of the "time" components constant (say i) this defines the anti-de Sitter spacetime AdS_5 ~ SO(3,2)/SO(3,1). This is general relativity! Spacetime can be described according to the spinor variables A and b with the interval

          s^2 = a·b + axb, x = times,

          so the scalar a·b plays the role of time and the "vector part" axb is spatial part.

          The equations 3.3 and 3.3.1 are a "biquaternion" version of the electromagnetic or Yang-Mills field tensor. The biquaterion form comes from the 2x2 matrices A_0 and A_i etc in eqn 3.3. I worked out a couple of years ago how the derivative

          d(f{\bf i} + g{\bf j} + h{\bf k}))

          ----------------------------------

          d(x{\bf i} + y{\bf j} + z{\bf k}))

          leads to the field tensor. This approach leads to a bi-quaterionic form that I think is connected to the Petrov-Penrose tensor.

          So there is a lot of good stuff here. Given that you are not trained to be a math physicists it is pretty remarkable.

          LC

            Hi Gary,

            I don't recall seeing the complex 'i' in GR for the time dimension, it is just c*t from my experience and thus a scalar time. Have you introduced the i in your work? I know it is in the Schrodinger wave equation and thus features in standing waves (i.e. for particles), but I don't see why the same notion of time cannot be used in an understanding of GR and QM (from a WSM standing wave perspective of the Universe).

            Regards,

            Declan Traill

              Lawrence,

              Many thanks for taking the time to read and comment on my essay. In the previous essay contest, I had an entry that dealt with using quaternions as a basis for Calculus. There were several other essays that dealt with quaternions also. Perhaps mine is the essay that you remember.

              Can you recommend a suitable textbook(s) for Clifford Algebra and de Sitter Space? Perhaps the time has come for me to learn these things. Keep in mind that I am an engineer by education and engineers do not study things like Group Theory ... perhaps there are some prerequisites necessary prior to Clifford and de Sitter ...

              Do you think that 6*pi^5 and the resulting proton diameter are coincidence, or could they have a physical significance? Or is it simply wrong?

              The concept of scalar time that I am presenting is a little different from what you have interpreted, but I did not really make it very clear. I am proposing that there is a "master" scalar time that is multiplied by Euler's Equation. This produces two time-like values. The product with the cosine is hopefully the time used by SR & GR. The product with isine is hopefully the time used by QM. I then interpret all of this geometrically by having the "master" scalar time operate on the complex plane. That is where the 2x2 matrices become essential and the bi-quaternion form of Hamilton's work presents itself.

              Best Regards and Good Luck,

              Gary Simpson

              Declan,

              Your statement is basically like saying that you do not see the unit vectors i, j, and k in Euclidian Geometry. It might be true, but it also misses the point. Minkowski space-time is based upon 4-vectors and that means i*c*t. The complex i is there whether or not you see it presented in writing. This is a prime example of how Wikipedia can unintentionally mislead people.

              Best Regards and Good Luck,

              Gary Simpson