Edwin,

Many thanks for reading my essay. I am planning to post one more paper to viXra.org to show the inversion of the matrix that I present in this essay. Then I plan to spend as much time as needed to learn GA and Dr. Hestenes' work. The extra reference will prove useful I am sure. I also think of the complex i as an operator.

I think there is a connection between the consciousness field that you propose in your essay and the scalar field that I propose in mine. It might be a way to add the observer to the picture.

Do you think that the 6*pi^5 observations are coincidence, or might they contain some truth? Or is the proton size calculation simply wrong?

Best Regards and Good Luck,

Gary Simpson

Gary,

I worked this up a couple of years ago. It is not difficult to understand and is related to your work. It also has content in your equations 3.

Your paper is not at all related to my paper here, but if you are interested you might find it interesting. It has little bearing on quaternions in a direct was. If you are interested I can send a paper I published which does illustrate a Clifford algebraic format for the equivalency of spacetime geometry and the Tsirelson bound of quantum mechanics.

Cheers LCAttachment #1: 2_quaternion_notes.pdf

Lawrence,

Many thanks for the post. Yes, please send a copy of the file that you mention to the email address on the cover sheet of my essay.

I've studied the note already and part of it is helpful to me. However, part of it is not consistent with Hamilton's definitions. You state that ij = jk = ki = ijk = -1. That is not what Hamilton stated. He stated that i^2 = j^2 = k^2 = -1 = ijk. As a result, ij = k, jk = i, and ik = -j. These identities impose handedness onto the system. We agree regarding anti-commutation of the unit vectors.

BTW, does anti-commutation of the complex i with the unit vectors seem reasonable to you?

Best Regards and Good Luck,

Gary Simpson

Here is the paper I published a few months ago. I intend to publish an extended version of this in something like Annals of Physics.

You are right in that ij = k, jk = i and ki = j, it is cyclic. I wrote wrongly there, even though I know otherwise. As for commutation, you do have ij = k and ji = -k. So this looks like commutators, not anti-commutators. One could potentially of course look at Jordan products or a graded quaternion system with

E^i = e^i + iσ^_{ab}(\bar θ^aψ_b + θ^a\bar ψ_b) + F_{ab}\bar θ^bθ^a,

which would be a sort of supersymmetric version of this.Attachment #1: LawrenceCrowell_V7N13.pdf

Dear Garry Simpson

Thank you for nice reply , which made me to go into deep thinking.... Best wishes....

I am reproducing the reply for the question you asked on my essay...

see there for the attachments....

Thank you very much for studying my paper so thoroughly and giving esteemed questions. I am just giving two reported cases of Galaxies / Clusters of Galaxies which are being generated after Bigbang

[35] Rakos, Schombert, and Odell in their paper 'The Age of Cluster Galaxies from Continuum Colors' Astrophys.J., 677 , 1019, DOI: 10.1086/533513, e-Print: arXiv:0801.3665 [astro-ph] | PDF arXiv:0801.3665v1 [astro-ph] 23 Jan 2008

[36] C. PAPOVICH et el, CANDELS OBSERVATIONS OF THE STRUCTURAL PROPERTIES OF CLUSTER GALAXIES AT Z=1.62, https://arxiv.org/pdf/1110.3794v2.pdf

See the CANDLES web pages also for simple language explanations.

There are many other papers and websites also if want them I will give them,

By the way, see the attachments to this post, to see these files for your quick reference...

Best Regards

SNP. Gupta

Hi Gary. It occurs to me that the Higgs field is supposed to be a product of complex and quaternion fields, so perhaps the multiplication in your first equation is relevant there.

I have become comfortable with the idea of 3 temporal dimensions from working with Fourier analysis of plane waves, but your essay reveals another possibility - complex time. Quite an interesting idea. Had to chuckle at your comment about nature being devious, having asked myself that a few times.

But I have to question the hypothesis of an absolute speed relating to the Mp/Me ratio in the way you suggest. If the cosmic microwave background defines the rest frame, the speed of the solar system is about 0.0012c, which is about 1/5 the absolute speed, 0.006136c, required to produce the Mp/Me ratio. A practical researcher does not ignore hunches based on numerical coincidence, but sometimes that's all it is. In any case, failure of this final speculation would not affect the rest of the work.

By the way, I think I first read about three time dimensions in Milo Wolff's book about space resonance and matter waves which seems to reflect some aspects of the Higgs mechanism - the book with Milo (presumably) and his motorcycle on the cover. His tripod website is still there. Sad to hear of his passing.

Anyway, nice work! - cw

    Colin,

    Thanks for reading and commenting upon my essay. If my memory is correct, you are familiar with quaternions and such. Hopefully this was an easy read for you.

    Also, thanks for the heads up regarding the Higgs. That is exactly the sort of clue that I was hoping to get by posting this essay. I'm not sure where it will lead me but I will study the idea.

    I think nature is more devious than we can even imagine ... I'm glad you liked the humor:-)

    I was not aware that the cosmic background radiation implied a velocity of 0.0012 c. This is another good clue. There are certain types of average whereby a value is divided by the number of degrees of freedom that a system possesses. Since I argue for 5 dimensions, then it is possible that both arguments are true. I will need to refresh my memory on this. I seem to remember it from statistics. In any event, as you note, the mathematical structure does not depend upon the hypothesis presented as Equation 2.

    You can definitely get three time dimensions simply by multiplying the complex i by the three spatial dimensions. But that implies to me that the complex i has the dimension and that time is simply a scalar that operates on the complex i.

    Thanks again.

    Best Regards and Good Luck,

    Gary Simpson

    Dear Gary Simson,

    But previously also proton diameter was calculated, did that method also used the 5 axes grid ...? I dont know how they calculated earlier. Hope you can show some light on that....

    Best Regards...

    =snp.gupta

    Gupta,

    The calculation that I present is based upon a simplified 5-D model.

    To the best of my knowledge, QED can calculate the size of the proton by solving a system of simultaneous differential equations. I have read that it is an extremely difficult task. I do not have the ability to determine the proton size using QED.

    Best Regards,

    Gary Simpson

    Dear Sir,

    You have correctly said that this topic is about mindless mathematics and that time is not a dimension. But do extra-dimensions exist? We are hearing about it for over a century. But it has never been found. In fact, the term dimension is still to be unambiguously and scientifically defined. Quaternions are a complex number of the form w xi yj zk, where w, x, y, z are real numbers and i, j, k are imaginary units that satisfy certain conditions. Is it a physical description? If yes, why cannot complex numbers be used in computer programming? If i stands for square root of -1, what does j, k stand for? This type of mindless mathematics are being addressed in this contest. Kindly clarify.

    Regards,

    basudeba

      Basudeba,

      Thank you for reading and commenting upon my essay. I will answer your questions to the best of my ability.

      You agree that time is not a dimension and you question the existence of extra dimensions. You ask for proof of extra dimensions. I have presented an accurate calculation of the size of the proton based upon a model that uses 5 dimensions. I argue that the accuracy of that calculation is circumstantial evidence in support of extra dimensions.

      There are two i's in my essay. One is the complex i. The other is the unit vector i. They are not the same although they both satisfy i^2 = -1.

      The thing that is puzzling you also puzzled me for a long time. Regarding quaternions, you state that i, j, and k are imaginary units. You believe this because they satisfy i^2 = j^2 = k^2 = -1. This thinking is incomplete. The unit vectors i, j, and k represent the x, y, and z axes respectively. The unit vectors make it possible for the axes themselves to be a part of computations. The unit vectors and the complex i can also be operators. For example, ij = k and the application of the complex i twice causes direction to reverse.

      As an example, consider two simple addition problems. Let x = 1 and let y = 1. It follows that x y = 2. This is a simple scalar result. Now let x = i and let y = j. It follows that x y = i j. This is a vector from the origin of length sqrt(2) at an angle midway between the x and y axes.

      Is this a physical description? Yes. Quaternions and complex numbers are used in computer programming. Quaternions produce rotations in video games.

      Essentially, you must train your mind to think of the unit vectors as the axes used in geometry.

      What my essay does, is combine physical space, in the form of an arbitrary unit vector, with the complex plane used by QM and GR. For the case of an inertial reference frame, it is very easy to visualize that combination as a 3-D universe.

      I hope that clarifies your questions.

      Best Regards and Good Luck,

      Gary Simpson

      Fun stuff Gary!

      Your work appears to have tie-ins with recent work by Stephen Adler. Here he talks about complex-valued spacetime foam..

      arXiv:1401.0353

      Here he suggests we need to test for quaternionic values in QM experiments.

      arXiv:1604.04950

      Also notable is recent work by Hyun Seok Yang asserting that non-commutative spacetime is inherently emergent. See these papers for starters.

      arXiv:1504.00464

      arXiv:1610.00011

      I already know I want to give you a high score, but I also know that elevating you now will make you a target. So I'll wait until your rating dips down a bit, before I rate your essay, and then boost it back up.

      All the Best,

      Jonathan

      Jonathan,

      Many thanks for reading my essay and for the numerous references for further study. That will probably take me awhile.

      Dr. Klingman also chooses to vote late. You are both wise men.

      Best Regards and Good Luck,

      Gary Simpson

      Gary,

      Your work starts with the wave function and ends with a very nice acknowledgement to the late Dr. Milo Wolff, founder of the Wave Structure of Matter. In between, your understanding of math to explain the proton is what this FQXi contest is all about - well done and good luck to you.

      A quick note about Dr. Wolff for those who are not familiar with his work. Dr. Wolff started a revolution for those that are working on a theory of matter that can be explained by wave energy. There's a lot of work remaining to prove this theory, but it has been an inspiration to some, and hopefully a call to action to others to explain the mysteries of the universe.

      Jeff

        Jeff,

        Many thanks for reading my essay. Yes, Dr. Wolff was a major influence in my thinking. He is sorely missed.

        Best Regards and Good Luck,

        Gary Simpson

        Dear Sir,

        You say: "The unit vectors i, j, and k represent the x, y, and z axes respectively. The unit vectors make it possible for the axes themselves to be a part of computations". In that case why complicate things by adding terms i, j, and k? The x, y, and z axes could have been sufficient by treating them as unit vectors. After all, vectors are different only because they have movement (energy) and direction. The axes provide direction. The axes have no meaning without something to represent. We also use mobile coordinates. Thus, what is the justification of adding i, j, and k? Further, x, y, and z are real, whereas i, j, and k satisfy i^2 = -1, which means complex. Why should we use complex numbers at all? They do not have physical presence. Anything that has no physicality cannot be a part of physics. Your statement that you have tried to "combine physical space, in the form of an arbitrary unit vector, with the complex plane..." presupposes that both do exist physically. Is there any proof in its support? Can you give examples?

        Your statement: "Let x = 1 and let y = 1. It follows that x y = 2. This is a simple scalar result. Now let x = i and let y = j. It follows that x y = i j. This is a vector from the origin of length sqrt(2) at an angle midway between the x and y axes" only conforms our views. Addition is linear accumulation, which is possible between similars. Here x and y have the same value and belong to one class. But x = i and let y = j shows that they belong to two different classes. You cannot add 5 oranges and 3 apples. You can add them only as fruits. We have submitted an essay to physically explain 10 dimensions. Unlike your 5 -D inferred space, we have shown direct correlation, where we have used the same logic as you have shown here.

        Regards,

        basudeba

        Basudeba,

        Let me see if I understand this ... you don't think I am correct to argue for 5 dimensions, but you have submitted an essay that uses 10 dimensions? That is puzzling to say the least. BTW, I do not see an essay associated with your name.

        I will simply restate my evidence. The calculation of proton size is based upon a 5-D model. The resulting value is within the accepted NIST range. If you wish to challenge this calculation, then please do so. Perhaps your 10-D model makes predictions or calculations?

        You continue to misrepresent the unit vectors as complex numbers. They are not. They represent physical space. You will not understand Hamilton until you abandon that thinking.

        Actually, I can add apples and oranges. A quaternion can contain 4 separate types of things. That is one way of interpreting the scalar, i, j, and k. Although treating them as something non-spatial would destroy identities such as ij = k. It would also destroy the i^2 = j^2 = k^2 = -1 identities.

        I look forward to your essay.

        Best Regards and Good Luck,

        Gary Simpson

        Dear Sir,

        We have started showing the inadequacies of mathematical physics and argued for physical mathematics. You are silent on this aspect. Similarly, we took 8 pages to refute modern notion of extra-dimensions, which has not been physically found even after a century. This includes your notion of dimension also. Kindly comment on that. How long you will chase a mirage? In case you have any other notion of dimension, kindly educate us.

        Our 10 dimensions are different from your 5 dimensions and are physical. But can you show your 5 dimensions physically.

        A quaternion is a complex number of the form w xi yj zk, where w, x, y, z are real numbers and i, j, k are imaginary units that satisfy certain conditions. Firstly, complex numbers itself have no physical representation. All text book representations are manipulated. The concept of complex numbers was introduced by Euler, when he tried to solve the equation x^2 1 = 0. But is there any physical system where this equation is fulfilled? The answer is no. If we rewrite the equation as x^2 = -1, still then it will not lead to mathematics or physics because squaring is done with not only the numbers, but also signs. Two negative signs square up to positive as per the mathematical rules. Then how can x^2 be equal to -1? Further if i denotes square root of -1, what about the equations x^2 2 = 0, x^2 3 = 0, x^2 5 = 0, x^2 7 = 0, etc.? Why do not we invent suitable terms to explain square roots of -2,-3,-5,-7, etc? In that case, the required symbols will be infinite and doing mathematics will be impossible. For this reason, complex numbers cannot be used in computer programming. Some people say complex numbers include real numbers and more. In that case, dream should be used instead of observation, because dream includes what we observe and more. May be for that reason modern scientists are including dream with observation to formulate theories based on extra-dimensions, strings, foams, chamelions, axions, gravitons, sparticles, bare mass, bare charge, dark matter, dark energy, expansion of the universe (even though it is not observed in less than galactic scales and we observe blue-shift), inflation, etc. But their dream costs humanity huge costs, which could otherwise have been enough to eradicate poverty from the world. Should dreams get primacy over real sufferings of the world?

        Hope you will educate us on these issues.

        Regards,

        basudeba

        Hi again Gary. Not sure if this turns out to be useful, or even makes sense, but it jumped out because there is a chance it might relate to the factor of 1/5 which seemed to be missing from the speed in the CMB frame in your final conjecture.

        I was just reading Jonathan Dickau's essay and was intrigued when he mentioned that the greatest hypersphere is produced in 5 dimensions. I don't know whether your 5-d construction is a hypersphere, but he refers to a Wolfram page Ball which has a formula V=S/n (Eq.2) relating the volume and surface area of a unit hypersphere to the dimension n. For n=5, the surface would be 4-dimensional. So it looks like there could possibly be a factor of 5 floating around in the math going between 5 dimensions and 4 dimensions.

        I am still working on my essay involving the derivative of a conjugate quaternion, which I started fiddling with after reading your Calculus 2.0 essay last year. -cw

          Colin,

          I am flattered and delighted that my previous essay gave you something to think about. Supporting work for that essay and this essay are posted to viXra.org. The paper names are "Quaternion Dynamics Part 1 and Part 2". They can be found here:

          http://vixra.org/author/gary_d_simpson

          I'm not sure yet what to think about the statement that you mention is Jonathan's essay. He speaks of the hyper-volume of a hyper-sphere being maximized for n = 5. I was not aware of this when I hypothesized a 5-D geometry. He also uses that as a rationale for dimensions becoming small for n > 5.

          I look forward to reading your essay ... you better hurry, the deadline is approaching.

          Best Regards and Good Luck,

          Gary Simpson