Lawrence,

Many thanks for the kind words. I am very appreciative of your enthusiasm. If I can supply someone with a new tool to use when attempting to solve some of these difficult problems then I will count myself as fortunate.

Having said that, I should mention that it would be better not to indicate how you might have rated an essay. The administrators at FQXI might construe that to be vote trading and that could be cause for disqualification.

One of my objectives is to get back to Maxwell in quaternion form. I need to spend some time working on identities and simple kinematics prior to that. I also think that it should be possible to formulate and solve a quaternion style wave equation. That should closely resemble Dirac's solution without the need for factorization using Clifford Algebras. Something that puzzles me regarding that work by Dirac is the equation ab = -ba. I understand non-commutation etc ... What seems odd to me is that the first thing that I think of is simply the cross product of two vectors. If a and b are both vectors then (a cross b) = - (b cross a) for any arbitrary vectors. Obviously, I need to study the subject some more.

I have read your essay once. I will need to read it one or two more times before being able to make any meaningful comments. I see from your e-mail address that you are no stranger to quaternions.

Best Regards and Good Luck,

Gary Simpson

The Dirac equation can be looked at as the multiplication of quaternions. The Dirac operator is a quaternion valued set of differential operators and the spinor field is also quaternion valued.

What you might be pondering is the role of forming differential forms from quaternions that are antisymmetric and quantum commutators. One can think of the 1, i, j, k as one-forms that give wedge products that are a generalization of the cross product. I think this is some question with the relationship between the Heisenberg group and quaternions. I think this relationship involves the AdS_5 spacetime.

My email address golden field quaternions refers to the 120 quaternions in the icosian group, that is half of the E8 lattice. That gets into octonions.

Cheers LC

Dear Gary D. Simpson

Your essay is an example of how learning enables identifying or finding new problems and persistence leads to proposed solutions. Moreover, the presentation of your ideas was logical and accessible, which is considerate of the reader, likely reflecting a desire that reader can share in the benefit of your hard work.

Does the following question in relation to your essay make sense? Can a space of points described by quaternions each of which isotropically scales by the same increasing (scalar) scale factor model the isotropic cosmological expansion of space?

Regards.

Bob Shour

    Dear Bob,

    Thank you much for your generous comments. You are most gracious. I try to write as simply as possible with short sentences in a linear, logical sequence with clear endpoints. Sometimes I have to leave an idea dangling so that I can merge it with something else.

    Regarding your question, I think that I understand your question and I think that the answer is yes. I'll expand on this a little. If you look at Equation 3 in my essay and you make the vectors collinear, the cross product term becomes zero. Now, you only have to worry about the scalar term. I have done the calculation using 13.8 billion light-years as the radius of the universe. This allows me to predict an expansion rate equal to 70.75 km/sec per Mpc. The observed expansion rate is 67.80 km/sec per Mpc.

    So it looks like I can get to within roughly 5% of the accepted value by using a fairly simple analysis. But I think there may be a problem. The way the cross product term is defined, it looks to me like linear velocity and angular velocity are linked (ie, they are not independent of each other). The implication is that if something is far away and moving linearly very fast then the space associated with it must be rotating fast.

    Best Regards,

    Gary Simpson

    5 days later

    Dear Gary,

    You forewarned that knowledge of quaternions and Linear Algebra would be required to fully grasp your essay. I don't have much knowledge of either but was able to decipher that your essay was trying to bring out something important from the depths using calculus. I am happy that your essay suggests a way of reconciling irreconciliables like Special relativity and the Aether.

    My reservation about calculus is how the limit or the infinitesimal can be equated with zero and in the same breadth not zero. This looks like a mathematical trick, even if a useful one. Under a magnifying lens what appears infinitesimal or tending towards zero can be seen to be physically non-zero.

    What is your take on these statements about the infinitesimal, dx

    dx ~ 0 (~ = indistinguishable from)

    neither dx = 0 nor dx тЙа 0

    dx2 = 0

    dx тЖ' 0 (тЖ' = becomes vanishingly small)

    The statement dx = 0 and dx тЙа 0 cannot both be logically true thus raising suspicion. While not denying the usefulness, suggest to equate dx with the Planck length, 10-35m . You may read my efforts and comment when you have the time.

    Regards,

    Akinbo

    Akinbo,

    Thank you for reading my essay. In truth, I think that all of the FQXI readership is easily able to follow my mathematics but I do feel obliged to give a small warning regarding Linear Algebra and quaternions. Having said that, all that is really needed is to know regarding LA is how to multiply matrices and to understand the meaning of an inverse matrix.

    I have read your essay and will be commenting shortly.

    Regarding infinitesimals in general, I am perfectly comfortable with them as mathematical entities. Regarding dx^2, I can only say that it goes to zero faster than dx.

    Infinitesimals are not really a difficulty in mathematics since they are only used to integrate or differentiate. When used for integration, you multiply by them and take a sum. So you are summing the infinitesimals to produce an area. When used to differentiate, they cancel each other out in part of the calculation and simply go to zero in another part ... consider the following:

    let y = mx

    dy/dx = (m(x delta) - mx)/((x delta) - x)

    this simplifies to

    dy/dx = m*delta/delta = m

    So when you take the limit as delta goes to zero, there is no delta in the equation. SO it is not really a problem at all. When higher order polynomials are differentiated, a similar behavior is observed except that there will be one or more delta terms. When the limit is taken, they become zero and only the first term (where there is no delta) remains.

    This brings out a more general problems of zero divided by zero, infinity divided by infinity, and zero multiplied by infinity. These are all considered to be indeterminate and L'Hopital's Rule is used to evaluate them. Essentially, the question is asked ... "which function goes to zero the fastest or which function goes to infinity the fastest". L'Hopital evaluates this by looking at the derivatives.

    So you see, dx is not really a trick.

    It is interesting that you should mention the Aether. I will confess to being a believer therein. The problem of course is how to measure something if it is not understood what that something is. It was thinking about that problem that ultimately produced the thinking that went into my essay.

    Best Regards,

    Gary Simpson

      Dear Gary,

      Can dx have a smallest possible size?

      As you pointed out, infinitesimals may not be a difficulty for mathematics, but what of for physics?

      Calculus and Cauchy's solution are used to solve Zeno's paradox, what do you think about the last step, which though indeterminate must be taken?

      Give these some thought.

      Regards,

      Akinbo

      Akinbo,

      To me, the infinitesimal can be zero. It is simply a concept. Whether or not it has physical meaning does not alter its usefulness to me as a concept.

      For Physics, it does not make any sense to me to think about distances that are smaller than the size of a proton or a neutron since those are the smallest stable particles that have a physical dimension. I'll discuss the electron further in your forum since that is where you pose the question. It is true that both the proton and neutron can be destroyed by high-energy collisions. However, it is not true that either can simply be divided into parts. To do so would simply destroy them.

      Regarding Zeno's Paradox for motion ... to me, it is not a paradox. He makes note that you must go 1/2 of the remaining distance each step but he does not mention that at constant velocity it will only take 1/2 of the time of the previous step. To me, Zeno is an example of a missed opportunity. He knows that motion is possible yet his logic gives him doubts. What he needed was a new concept (sum of infinite series) that would connect the two. But he was not able to make the mental step and humans had to wait for 2000 years for Isaac Newton to come along with Calculus. I do not give Zeno a high score ... of course, everything is obvious after someone else shows you the truth.

      The discussion between Dr. Klingman and Dr. Maudlin reminds me of Zeno. There is a deeper truth to be had there. I just hope we don't have to wait 2000 years to get a solution.

      Best Regards,

      Gary Simpson

        Gary,

        Thanks for your response over at my forum topic. I admit Calculus has achieved a lot but I don't want to let you off so easily :).

        You say, to you the infinitesimal can be zero. This runs against the common notion that the infinitesimal is a quantity almost indistinguishable from zero, but not zero, at least as is discussed HERE.

        Anyway, the reason for my post is that I have been giving Calculus more thought following your responses. It appears clear that for calculus dx can have no lowest finite value and that being the case, a line would be continuous, having an infinite number of points. Because you have also once considered motion as the creation and destruction of space, and motion as the destruction of the moving object and its creation and reappearance in the next adjacent space, it means all scenarios have at least once featured on your table.

        What I want to now know is how the continuous line in space can be cut since between any two points there is always a third by definition. Where on this line can you cut, since a point is uncuttable by definition?

        Regards,

        Akinbo

        Akinbo,

        Thanks for the continued dialog.

        I will answer your question concerning making a cut for both a physical structure such as a piece of string and for a mathematical concept such as a continuous line.

        Let us say that a piece of string is cut by a knife. The knife is harder than the string. Its atoms are bound to each other more strongly than the atoms in the string. The knife is forced into the space of the string and the bond between some of the atoms in the string is broken. The cut occurs in the space between atoms.

        Now let us consider an abstraction such as a continuous geometric line. A cut is to be made between two points but there are an infinite number of such points between any two points ... so where is the cut to be made? Let us be smarter than Zeno and introduce a new concept ....

        Let an interval near point x be defined as follows:

        (x - delta) < x < (x delta) where x and delta are both reals

        We can now make a cut at point x by taking the limit as delta approaches zero. Essentially, the line is cut into segments by removing point x. Something to remember about real numbers is that they have an infinite number of digits. So, the real number one is 1.000000000 ... ad infinitum. This is equivalent to infinite precision or to taking the limit of the above interval as delta goes to zero.

        The more interesting question is what happens to point x? Both line segments approach it but neither segment includes it. I suppose that it could be reconnected to one of the segments but not to both.

        Regards,

        Gary Simpson

          Gary,

          Yours is one of the more informed contributions hence my coming back. Your reply resembles the one that Tim Maudlin gave on the question. Mathematically correct, Yes. Physically correct with regards to an extended line? I still have doubts.

          For example, what does "interval" mean on a physical line in empty space? Does it not mean some amount of small space? Note that I am not discussing the abstract number line. I am talking of extension, of matter or empty spatial distance.

          Then talking of "taking the limit as delta approaches zero". Delta had better not reach zero lest x cease to exist. And for cutting at point x, going by Euclid's definition which I referenced, x can have no parts, so it cannot be cut. Again, x occupies some position, can it be displaced? If so, what does it leave behind on displacement? Wherever, it is displaced to, must also have an x there, can more than one x occupy a point x? I think you get a bit of the dilemma now?

          Regards,

          Akinbo

          • [deleted]

          • Post #28

          Akinbo,

          Thanks for the continued dialog. I hope that I can give you a fresh perspective on your interesting question.

          I will attempt to answer. However, I might need some clarification regarding what you mean by "physical line in empty space". The space cannot be empty if it is occupied by a physical line.

          My best interpretation of what you are asking is that you have constructed a line using small components that are interconnected. The components should not be considered to be infinitesimal. The components themselves cannot be cut or broken.

          The meaning of interval as used above is a small region near a point. When the limit is taken as delta goes to zero, the interval becomes the point. It does not cease to exist as you state. A point is an infinitesimal at a specific location. So the geometric line is cut by removing a point at the location to be cut. It is not necessary to cut the point. You simply take it out of the line.

          If you consider an interval near a physical object, then when the limit is taken as delta approaches zero, only the object remains. None of the nearby space would be included as part of the limit. Only the object itself would be included.

          The physical line that you describe is cut in the manner I previously described for a piece of string. The cut occurs in the space between the atoms. Nothing is displaced. Nothing is removed.

          The abstract geometric line is cut by removing a point. The point must be displaced but this does not mean it is displaced by something. It is not a physical entity. You simply mentally remove the point and you get two line segments.

          Are you certain that two things cannot occupy the same space? The Pauli Exclusion Principle allows two electrons to share the same orbital provided they have opposite spins. So the Helium atom has a single s orbital with two electrons. One electron has an up spin and the other has a down spin. This is a general property of fermions. Bosons are even more gregarious. An unlimited number of bosons can occupy the same space.

          Think about DNA and RNA for a moment. Sometimes atoms swap positions. Perhaps the answer to your question is that two lines cut each other with the result being that they each swap an object at the location of the cut.

          Best Regards,

          Gary Simpson

          Gary,

          Our exchange has been helpful in refining and clarifying my argument (or as it may turn out misconception although I don't think so yet).

          - In reply to your request for clarification on what I mean by "physical line in empty space"

          To make it have a precedent, let me answer thus. That line is the same line that Newton says in his first law as the path of an object moving with uniform motion. It is also the same object that Einstein's General relativity say will become distorted and curve in the presence of a gravitational body. It is also the path a light signal would traverse. It is also the line that shortens and elongates alternately when a gravitational wave passes orthogonal to it. I use these well-known examples deliberately for definition for you to decide what 'emptiness' could mean.

          - If we now proceed from this and consider whether a 'line' is just a concept without any physical reality and consider that by its ability to vary in character it is more than a concept, what is a line constituted of? Some say, it is of points, infinite or finite in number. And Euclid says unlike the number in a number line, points cannot be cut, we get to my headache of how to cut a line. If you read my exchange with Tim Maudlin, he talked about Dedekind's cut. That cut applies to a line whose elements are real numbers and are cuttable. But the elements on an extended line are not numbers, unless you think otherwise.

          - In my 2013 essay, the argument between Plato, Proclus, Aristotle, The Pythagoreans and later Leibniz on whether the point is an abstract concept or a physically real entity (monad) was discussed. Even Plato later told Aristotle that the point is not just a concept but real. Quoting from that essay: In Metaphysics, Book I, Part 9, paragraph 14, Aristotle tells us, "...Plato even used to object to this class of things as being a geometrical fiction". Instead Plato preferred that points be referred to as the 'beginning of lines' or as 'indivisible lines'.

          Best regards,

          Akinbo

          *We may take this discussion elsewhere on the FQXi site, e.g. Alternative models forum, if it becomes too much of a bother for you to engage here. Thanks.

          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