Essay Abstract

Fundamental questions in physics can be asked anytime, anywhere. Often they arise at the interface of physics, mathematics, and philosophy - where scrapping conversation turns into testable hypothesis. This essay explores the idea that the primitive act of counting "1, 2, 3 ..." makes an implicit assumption that ultimately causes some of the challenges faced in quantum mechanics today. A hypothesis for what could be done differently is developed during a humorous, yet serious, conversation among a physics student, a math student, an ex-philosophy student, and a city councilor. Beginning with a physics student's ill-fated attempt at bargaining for a lower price, the essay touches upon beauty in numbers and nature; repetition, inversion, and algebraic closure in mathematics; and observability in quantum mechanics. A surprising property of the complex numbers will be shown to indicate incompleteness or inadequacy in regard to resolving certain questions in quantum mechanics. A new kind of number and arithmetic may be needed, and a proposal for such is sketched using the E8 lattice.

Author Bio

Jens Koeplinger received a "Diplom" (M. Sci.) in physics at the University of Heidelberg, Germany, in 1999. When not exploring possibilities in physical mathematics, he is working daytime hours as IT Systems Analyst for AT&T, and off-hours developing mobile apps at "Dirty Little Cyborg". John A Shuster earned an A.B. in math and economics (physics minor) from Kenyon College (OH) in 1971, then did graduate work in operations research at the University of Rochester (NY). He is a retired Systems Analyst who enjoys grandchildren, travel, writing, and exploring new math systems.

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  • Post #2

Dear Messrs Koeplinger and Shuster,

Despite the dismal fact that I do not know an awful lot about mathematics, I loved reading your exceptionally well written, humorous, yet perceptively cogent essay, and I do hope it garners one of the prizes. I hesitate to mention what my picayune quibble with the essay might be lest you might question my motive for bringing it up, but I will risk elaborating on it if you do not mind. In my essay Sequence Consequence, I concentrate on reality. I believe that one real Universe having one real appearance can be perpetually occurring in a real here for a real now in one real dimension once. Real stuff has always to be in one real dimension. I think that if there were three abstract spatial dimensions, it would be difficult if not impossible to determine how abstract stuff was distributed. Would heavy abstract stuff helpingly remain in dimension A, moderate abstract stuff stay in Dimension B, and light abstract stuff linger in dimension C. I prefer to think that only 1 of anything could only ever exist once. Unfortunately, the most confounding illogical code seems to be the numeric representation of numbers. For instance, a single line is used to depict each of the numbers from zero to nine. The number 0 could visibly equal the number1if only the number of lines used to construct both numbers was considered. Does the space inside of the 0 have a value? Is that spatial value greater, equal, or less than the space between the 0 and the 1? There has been a standardization of the measured speed of light. Why has there never been a set standard for the presentation of numbers?

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    • Post #3

    Jens,

    Your essay reminds me of the many times over the past decade that I have been trying to communicate with "System International (SI) loyalists" that their base units are not suitable for "scientific units of measure." One of my communicants even stated "they are totally anthropocentric, arbitrary, and non-natural base units, from the POV of physical law."

    My paper had not been published when I had that one particular communications, he had a draft, but I provided the individual with the IEEE citation after the paper was published. I have his comments on file, but his constant exposure to SI units has completely immunized him to even thinking the SI base units are unsuitable for scientific purposes.

    IEEE paper titled, "A methodology to define physical constants using mathematical constants"

    IEEE Methdology

    or postprint

    Postprint Methodology

    IEEE no longer allows authors to post the published version anywhere.

    E8 based QM theory isn't complete, but it presents a structure that contemporary QM and string theory does not have.

      Dear Joe,

      Thank you for your kind words! I'm glad you enjoyed the essay, we sure had a lot of fun writing it. Relating nature's observed geometric dimensionality and magnitudes to abstract defined algebraic dimensionality and numbers is one of the big riddles to be solved, we feel as well. Thanks for sharing your thoughts towards the numbers 0 and 1, and for referring to your essay.

      Jens

      Hello Frank, thank you for pointing out your research. Aside from formal publication, do you have a reference to freely available material that would give the reader here an overview of your thoughts? You must admit that accusing the IEEE as having an "anthropocentric" bias is a bit odd, given that the customers of an Engineering society are humans after all. It's like accusing Barnes & Noble of selling books. Re "E8 based QM", I wish I knew what that is ... Best wishes, Jens

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      • Post #6

      Another essay on considering the context of numbers as opposed to just considering their value as a number is at

      http://fqxi.org/community/forum/topic/1375

      Thanks.

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        • Post #7

        Jens, John,

        I did not state the IEEE has an anthropocentric bias. I don't know if the person I quoted is an IEEE member, that used the term anthropocentric, but I do know he is an electrical engineer.

        The IEEE publication I cited was the culmination of over ten years of trying to get the concept published. It was rejected by several publications before I submitted to an IEEE publication, and it was rejected. I rewrote the introduction and submitted the paper to another IEEE publication and it was accepted.

        Presenting a physical law in the form of two right triangles is not taught in the text books.

        The methodology in the IEEE paper disposes of anthropocentric bias in how base units of measure should be derived. Physicists are trying to derive physical laws that govern the characteristics of the universe, and the use of man-defined base units does not help. My topic, 1294, discusses the multi-century assumption that SI units are suitable for scientific units. Even the BIPM admits they are not based upon fundamental physical constants; they don't know how to correct it. My emails to various BIPM officials have never been answered.

        The BIPM is a bureaucracy that exists for the purpose of preserving artifacts that represent purely anthropocentric base units of measure. SI units are fine for commerce. Bureaucracies do not take any action that will eliminate their existence.

        To give the reader information of what you're advertising, your article is titled: "Thought Experiments in the Abstract Field of the Mathematics of Infinities Produce Experimental Artifacts Suggesting That Their Use in the Real-World Science of Physics Should Be Reexamined".

        11 days later

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        • Post #9

        Ok - thank you for clarifying. It would still be good to have a public overview of your work somewhere. It could be as simple as a personal web page or so. I'd be glad to have a look. Thanks, Jens

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        • Post #10

        Another dream where we wake up just before the answer is given.

          :) Well then I suppose we better get to work and find the answer while awake!

          Seriously, going beyond discussing a physical assumptions that may be wrong, to actually provide a fully working answer, that would be truly amazing. We decided against writing about published ideas that are developed further. With that, the essay sides in favor of inspiration, but at the expense of presenting a working model. Those familiar with my line of research know of course where the work with John is heading: From the 4D Euclidean quantum gravity model that needed a geometry, to the octonionic background geometry that needed a quantum theory, to the nonassociative quantum theory in one dimension that needs math yet to be determined in order to go higher-dimensional. "Just" about a year ago did I learn about a technique that lets us do exponentiation and differentiation on one-dimensional lattices. In a very optimistic estimate, this essay is a half-way point for our work of making this a reality on the E8 lattice. Next to formal publications ( http://www.jenskoeplinger.com/P ) we're working in a glass house ( http://groups.yahoo.com/group/hypercomplex/ ). Open-source research, so to speak - contributors welcome, to the least we appreciate if you post to our group if you've done related work, or work inspired by us.

          Hope this helps describing where we come from in a bit more detail! Thanks, Jens

          Hi Jens,

          I liked your essay, but it did leave me wanting more explanation. I think due to the length limitations the story line took up too much space. Knowing you a bit, I am quite sure you could have provided more content. I do appreciate your intent using the method you did, and had mused myself about presenting my essay within a story line with the theme Crazy Uncle O's Magical Mystery Tour of Physical Reality. The wife talked me out of it. Just as well, without the prop I had to leave out quite a bit of content I wanted to put in to be able to shoe horn it into 9 pages.

          I look forward to announcements on your blog http://groups.yahoo.com/group/hypercomplex/ about further developments. Keep up the good work.

          Regards,

          Rick

            Hello Rick - thank you for leaving your note. John and I noticed your essay as well. You understand that I am disappointed about seeing no mention, favorable or otherwise, of my analysis of your work ( arxiv:1103.4748 ). It is of course your choice on what to write about, and what to ignore. Jens

            Jens,

            Trust me when I say I did not intentionally fail to mention your paper, there just was no space available. I chose to discus algebraic invariance in terms of what I called Iso(). I was remiss not to include your paper in the Reference section, and have posted such in my essay blog. This also was not an intentional act, I just tried to give references related to the essay content. It is easy to leave things out of the references, like your omission of this in the very same arxiv paper. No worries, it is all good.

            Rick

            Ok, I appreciate your note. You wrote at some point about your vision: "Algebra, analysis, topology and groups are interlocking parts. The most fundamental is the algebra, for it sets the tone for the remainder." Your 'octonion variance sieve' can indeed be expressed elegantly using derivation algebras. Those exhibit properties similar to what one would expect from arithmetic. There are a couple of formal bugs in my paper on this part of your work, in its current version on the arXiv at least; but since it has attracted no feedback whatsoever I'm somewhat demotivated towards fixing them. Maybe that explains my negativity.... I do believe that your octonion variance sieve works, and that - for differential equations - there are solution spaces that don't simply collapse into the quaternion case. Best wishes, Jens

              Hello Jens,

              Please do not be "demotivated" towards fixing the bugs in your arXiv paper. It has been more comprehensible to me than Rick's own writings and explanations (because of my own limitations as a physicist rather than a mathematician, and because of my associative, Clifford-algebraic perspective derived from the works of Hestenes and Lounesto). So, please, do revise your arXiv paper if necessary because it has been useful at least to me. In particular, I would be interested in understanding how the solution spaces for some differential equations do not collapse into the quaternion case. This is not what I would expect from my topological perspective of the octonionic 7-sphere.

              Best,

              Joy

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              • Post #18

              Hi Jens

              After reading your essay i would like to send my observation

              http://vixra.org/abs/0907.0014

              Hello Joy - thank you so much for posting!

              Regarding the bugs, it's good to see your interest. I'll try to get to them soon. Essentially the problem is that I'm treating polynomial functions and algebras as if they were the same thing. E.g., on the right-hand side of (5.7), a set of functions { f[N], u, v } is of course not contained in the quaternions. Oops - that doesn't make sense. What I meant to write was that the multiplication rules used in the polynomials f are quaternion, therefore making the f[N] quaternionic polynomial functions.

              Then regarding where the approach collapses into the quaternions, I admit that my work is incomplete in that I only state in (5.9) that

              der( Df ) contained in H

              does not necessarily require that the polynomial f is quaternionic. In order to be complete it needs to be shown exactly where f may be octonionic and whether there exist any interesting differential operators D such that (5.9) still holds *and* Df is not already quaternion. Rick is proposing such a construct for his recovery of the Maxwell equations; and I've checked his multiplication rules by hand and found no error. But that doesn't make it formal proof, of course ...

              Thank you again for your interest!

              Jens

              corr: I incorrectly quoted; here's the correct quote:

              (5.9) der( Df ) contained in der( H )