Akinbo,

I am pleased that this dialog has been of benefit to you. Continuing it is no burden to me although I perhaps have little left to add.

Now that I have examples of what you mean by an extended line I can give a few small thoughts. Your line exists in the physical vacuum. The properties of your line and whether or not it can be cut are controlled by the properties of the vacuum. Time is a component of your line. In my opinion, your lines are cut into past and future by a particle at here and now. I apologize if this sounds like some kind of philosophical gibberish but I don't have better words to express the concept.

Also, since you are thinking in terms of the extremely small, you should think of your particles as wave-functions. You should also consider the implication of the Heisenberg Uncertainty Principle. Physicists have avoided your dilemma by simple stating that you cannot know where something is if you know its momentum or that you cannot know the momentum of a particle if you know its location. Essentially they argue that your question cannot be answered.

Best Regards,

Gary Simpson

Lawrence,

Many thanks ... it looks very similar indeed. I am curious, is this new material or is this old material. Since I am an engineer rather than a mathematician, I do not know what is known ... if that makes sense to you.

I think there is some disagreement however ... you indicate that ij = jk = ki = ijk = -1. This is not what Hamilton states. Hamilton states I^2 = j^2 = k^2 = ijk = -1. Therefore, ij = k; jk = i; and ki = j.

Also, it is not necessary to use the product rule to get the differential with respect to a quaternion although what you present is certainly correct. Part of what I wanted to show was that the quaternion functions should be viewed as a system rather than as four separate problems. The trick then is to solve the system in one step.

If you want to see something really cool, take a look at the quaternion exponential function. It is in the paper at this URL:

http://vixra.org/abs/1412.0257

Jump down to pages 13-15

Thanks for the feedback.

Best Regards and Good Luck,

Gary Simpson

Sujatha,

Thank you for reading and considering my essay. You are correct. I have a long way to go. But a journey of 1000 miles begins with a single footstep ... or something like that. I'm not real good with quotations.

Best Regards and Good Luck,

Gary Simpson

Comment

This essay is really quite clever, although the math is somewhat difficult. The quaternion nature of 4-space is very nicely and clearly laid out and the device of imagining Newton using Hamilton's quaternion algebra to presage relativity is alluring. But Hamilton was a smart guy too...maybe even smarter than Newton. So why didn't Hamilton come to relativity?

It just seems such a shame that no one has been able to link the quaternion algebra of relativity with the quaternion algebra of quantum spin. We really should have a quantum gravity...

2.0, entertaining

2.0, well written

1.0, understandable

2.5, relevance to theme

7.5

    Steve,

    Thank you for having a read and giving me some feedback. You are most kind.

    You are very correct regarding Hamilton ... he was a smart guy ... a child prodigy from some accounts. Unfortunately, he died prior to the publication of his textbook on quaternions. His son took the responsibility of getting the work published. He worked on the problem until the end of his life. No one can ask more. RIP.

    Maxwell originally formulated his work using quaternions but from 1890 - 1895 there was a serious conflict in mathematics between advocates of quaternions and those who favored tensors and four vectors. Quaternions lost and were almost completely forgotten.

    I have seen a few things on the internet that indicate that Einstein and Dirac both knew of quaternions. Why they did not attempt to use them, I do not know.

    Regarding using them to link QM with gravity ... I have a few ideas but there is much work to do. As I noted in my essay, the inverse of the square of the distance occurs very naturally.

    Best Regards and Good Luck,

    Gary Simpson

    Hi Gary,

    I posted this elsewhere in conversation and I thought I would share this with you to add to our previous conversation.

    Here is what Roger Penrose has to say in his book, The Emperor's New Mind, p.113... "The system of real numbers has the property for example, that between any two of them, no matter how close, there lies a third. It is not at all clear that physical distances or times can realistically be said to have this property. If we continue to divide up the physical distance between two points, we should eventually reach scales so small that the very concept of distance, in the ordinary sense, could cease to have meaning. It is anticipated that at the 'quantum gravity' scale (...10-35m), this would indeed be the case.

    I think this may help clear up what is meant by dividing a distance. Hence, my asking that assuming, without conceding that the system of real numbers applies to distance, how can a distance be divided if there is always a third element between two elements and going by geometrical considerations these elements are uncuttable into parts?

    Regards,

    Akinbo

    Akinbo,

    Thanks for the continued dialog.

    For someone to say that a distance does not have meaning ... that itself is a meaningless statement to me. The value he gives is 10^-35 meter. I assume that he is referencing the Planck Length. That is shorter than the wavelength of any known radiation. The wavelength of a high energy gamma ray is roughly 10^-12 meter (1 Pico meter). The distance that he references is probably closer to the length of a matter wave associated with most or all of the visible universe. So you would need the energy of the entire visible universe to probe something that small. This is where people start talking about extra dimensions and such things ...

    I don't agree that the elements that you describe cannot be cut. I have shown you above how to cut a line at any desired location by removing the point at that location. I do not concede that the point cannot be displaced due to the presence of something else already being at the new location.

    I hold my hands in front of me. There is a line of length 12 inches that has an endpoint on each hand. I move them towards each other. They are now 6 inches apart. I have divided the space. The previous line is still there but now there is also a line that is 6 inches in length and it occupies the same space as the longer line. We must be smarter than Zeno. Use an interval around the point where the cut is to be made instead of making a cut between two points.

    Take a look at the attached .pdf file ... I gave it your name. I have presented the non-limit forms of the derivatives of a few polynomials. You can see that they will each simplify to the correct expression for the derivative of each function. The minimum distance does not appear in the derivative expression for a line but it does appear in expressions that are quadratic or higher.

    The complicated looking expressions come from the Binomial Theorem. Factorials are used to determine the values for the various coefficients of the polynomials.

    So calculus will work with the limit as the infinitesimal goes to zero or if the infinitesimal goes to some arbitrarily small value. If the minimum allowed distance is not zero then if the distance involved is on the order of the size of the smallest distance then an effect should be observable.

    It should also be possible to repeat this exercise for integrals. I will do so at some future time.

    Feel free to contact me at my aol.com email address. It is listed on the coversheet of my paper.

    Best Regards and Good Luck,

    Gary SimpsonAttachment #1: Akinbo_Ojo.pdf

    Dear Gary,

    You are certainly very good in mathematics and its use. We may not fully agree on some aspects but no matter, as it helps both sides fine-tune their model. In brief, some of the areas of divergence I itemize are:

    "The distance that he references is probably closer to the length of a matter wave associated with most or all of the visible universe. So you would need the energy of the entire visible universe to probe something that small."

    If E = hf = hc/λ, the associated energy or mass (given E = mc^2) is about 10^9. That is not the energy of the entire visible universe. Probably, about 10^60th of it.

    ..."I have shown you above how to cut a line at any desired location by removing the point at that location. I do not concede that the point cannot be displaced"

    That is okay. But it means a point can be physically removed or displaced to a pre-existing location, which is as well a point if I get your meaning. Mathematically, I agree you have shown me. But physically, hmmm... If action-reaction is what causes displacement to another location according to Newton's law, then there is the problem how something without mass can be so displaced since it cannot provide a reaction. But I will let that ride.

    "I hold my hands in front of me. There is a line of length 12 inches that has an endpoint on each hand. I move them towards each other. They are now 6 inches apart. I have divided the space. The previous line is still there but now there is also a line that is 6 inches in length and it occupies the same space as the longer line."

    Well, that is one way to look at it. In my humble opinion, the space between your hands was not divided. A part of it equal to 6 inches was destroyed. While in the line outside your hands, an equivalent amount was simultaneously created. In my model therefore it may not be correct to say "the previous line is still there" or that the line of 6 inches occupies the same space as that of 12 inches. This may lead to absurdity, i.e. 6 inches = 12 inches.

    "Use an interval around the point where the cut is to be made instead of making a cut between two points"

    What is 'an interval'? Is it some distance that does not consist of any points? Does an 'interval' exist in physical reality or only in mathematics? The use of 'around' connotes it is a place. Can there be a place without a point, either mathematically or in physics?

    "So calculus will work with the limit as the infinitesimal goes to zero or if the infinitesimal goes to some arbitrarily small value (i.e. not zero). If the minimum allowed distance is not zero then if the distance involved is on the order of the size of the smallest distance then an effect should be observable."

    This got me twisting my neck this way and that, trying to dodge zero and get hit by zero at the same time. It is this sort of argument that made some humorously call these quantity names. Berkeley calls them "ghosts of departed quantities", Cantor "cholera-bacilli" infecting mathematics, Russel as "unnecessary, erroneous and self-contradictory", all quoted from the . There is no doubt that the quantities are very, very useful in spite of seeming to be logically dubious. Of course, one can say since it works, who cares? They may be right in some sense.

    Regards,

    Akinbo

    Sorry for being so late, but I have not been on FQXi much the last week. You might want to look at Soiguine's paper in this contest. It is rather complicated, but it works with the geometric algebra of Hestenes and Clifford algebras. The quaternion product is a Clifford algebra.

    Cheers LC

    Akinbo,

    I agree that the dialog is useful. Thank you.

    Planck Length ... oops. You are correct. My bad. I was using hyperbole and did not perform a calculation. A wavelength of 10^-35 meter has an energy of 1.986 x 10^11 joules. This is 1.240 x 10^29 eV. The LHC at CERN is 7 x 10^12 eV. So it would require something 15 to 16 orders of magnitude (roughly) more powerful than the LHC to probe something that small.

    This amount of energy is equivalent to a mass of 2.206 x 10^-6 kg. This is the mass equivalent of 1.3 x 10^21 protons. The distance itself is 1.755 x 10^20 times smaller than the diameter of the proton.

    I think we can safely assume that questions concerning distances of that scale will not be physically tested in the near future ... perhaps even never.

    The idea of motion as being equivalent to the creation and destruction of space is very interesting ... it could reasonably be true ... but how would you empirically show such a thing?

    If you are trying to understand if it is possible to divide or partition the vacuum, I think that our understanding of the vacuum is not sufficient to answer your question. The only clues that I know concern the number of fermions that can occupy a given space or the number of bosons that can occupy a given space. The strange behavior of the Stern-Gerlach experiment is also a clue.

    In the sense that I use the word "interval", it is strictly a mathematical concept. If there is truly a minimum length, then the concept of interval could never be less than two such lengths.

    Best Regards and Good Luck,

    Gary Simpson

      10 days later

      Gary,

      "This text demonstrates that how we think about both Mathematics and Physics can be influenced by the mathematical tools that are available to us." Do math tools come first? Einstein needed clarity for his theory of general relativity, thus utilizing new ventures into Riemannian geometry. Did Math come second here? Or should we say that achievement is built on the foundations of both?

      These are questions that you aptly discuss in your essay.

      Jim

        Gary,

        I also think that it was Euler who reduced Newton's law of motion down to F = ma. Before that no one could understand what Newton was talking about.

        You write in a precise and clear manner. I learned something from this essay.

          Jim,

          Many thanks for taking the time to read my essay. I think that usually the mathematics comes first. Then when science finds an application for a new concept in mathematics, the mathematicians return to that concept and expand upon it some more. It seems like physics has gotten ahead of math now though.

          When Hamilton developed quaternions, I think he was thinking about both math and physics. He knew of developments in electro-magnetism and hoped to incorporate them using anti-commutation.

          Best Regards and Good Luck,

          Gary Simpson

          Efthimios,

          Thanks for reading my essay. Hopefully you gained something useful.

          I think that Hamilton's methods have not been satisfactorily applied. My objective with this essay was to present several ideas. I wanted to show that ordinary Calculus could be applied to quaternion functions. That allows four times as much information to be expressed by a given number of symbols. I wanted to show that the resulting kinematics can produce a curved path. That hints at how to treat gravity. And I wanted to show that the Lorentz Transform could be extended into a time quaternion whose vector portion resides in 3-D space. That hints at how to remain consistent with Special Relativity and it offers the possibility of eliminating time as a fourth dimension.

          Best Regards and Good Luck,

          Gary Simpson

          9 days later

          Gary,

          Your quaternion derivative is very similar to the Gateaux derivative for quaternions according to the Wikipedia article on "quaternionic analysis". If you were unaware of Gateaux, you are to be congratulated for your insight in finding a directionally dependent quaternion derivative. In your case, the direction of the quaternion itself is a natural choice.

          The difference is that you take the limit (eq.6) of a ratio, while the Gateaux derivative does not include the denominator. It looks like the direction quaternion in the denominator has to be divided out separately. The wiki article gives the example of q^2.

          I have been playing around trying to find another way to produce dq*/dq with no luck yet.

          Best regards,

          Colin

            Colin,

            I was not aware of Gateaux. It looks like you have become quite interested in quaternions and possible methods associated with them. Excellent. This will be a long and difficult battle.

            Best Regards and Good Luck,

            Gary Simpson

            Dear Gary,

            I just post a reply to your comment on my paper.

            Friday I'll have time to approach your essay before replying here on your own forum.

            Best regards and good look

            Peter

            Gary,

            I enjoyed reading your essay, and felt grateful that you made me think about "the topic" in the context of the history of science. Obviously, this approach ought to inspire at least some of the answers.

            Your method of involving the human element (including the fact that things would have progressed differently had certain individuals known about the work of others - perhaps even more so if not coeval) supports the narrative that the connection between physics and math should not be viewed in isolation from the people actually "doing" the two disciplines. I was not sure if one should interpret this observation to mean that you view mathematics as something that people "develop," rather than something that would have been always out there (somewhere, somehow) even if no human ever existed. Taken to its extreme, this interpretation could imply that the connection between physics and math "resides" in the nature of humanity. It appeared safer not to draw such a conclusion without your blessing.

            In any case, your essay is good work, and deserves a good rating.

            I also wish you Good Luck.

            En

            P.S. I replied to your comment on "my page."