"1 + 1 = 2" A Step in the Wrong Direction? by Jens Koeplinger and John Shuster

Ok, I see where I made a mistake in my attempt at a symbolic form of your generalized action W. Sorry for that, I was writing from memory and should better not do that.

Regarding other work on split-octonions and Maxwell's equations, I'm referencing several from my paper on your octonion variance sieve. There are approaches from direct coordinate products of split-octonions, as well as, spinors on Dirac-like equations on split-octonions. The work from S de Leo should also be quoted (I'll add the de Leo references in the forthcoming revision from the above exchange with Joy). Let me know if you're interested in more details about the different approaches.

Hi Joy - looks like our messages crossed over.

Yes I agree with everything you wrote, and thanks for your quick answer! I won't be that quick with any results, you bet ...

Regarding your critics, sometimes I wonder whether you really need to be fighting all the battles you're in; but then again, I'm cherry-picking from your work so there's not much I'm qualified to say elsewhere.

Octonions are the simple and beautiful structure, fully agreed. We'll do our best not to break it. Here I'm referring back to the work with John, that is subject of the essay :), to eventually find a true octonionic exponential a^b where both a and b are octonion. Such an exponential would have a huge space of automorphisms, with all kinds of subspaces. Hopefully that will lend itself to a well-motivated introduction of observer time.

Thanks again, Jens

PS: I just wrote "find a true octonionic exponential a^b where both a and b are octonion". That was a bit sloppy in reference to the essay. The essay only envisions an arithmetic on the E8 lattice, which is formed by the integral octonions (except for a scaling factor).

Interesting. Thanks for leaving your note as well. In RxCxHxO the "R" is more or less fashion since you could simply rescale the universe to fit any such R :) - but using it as a discrete qualifier gives you a reflection symmetry, I understand.

If SU(3)xSU(2)xU(1) would really be the symmetry group of the Standard Model, in one shape or another, then I would think your S0*S1*S3*S7 or Dixon's/Furey's algebraic RxCxHxO are the way to go. So there's a good chance that you're on the right track. Personally I'm not convinced that this is the right ansatz; which is of course entirely my problem. I acknowledge SU(2)xU(1) from the electroweak force, of course, and SU(3) in itself from the Strong Force. But can they be really unified using conventional Field Theory? In my feeble understanding of the Higgs mechanism in the weak force, and Yang-Mills instantons in the Strong Force, this appears as if there is another "outer" symmetry (which could be as simple as a U(1) rotation or even a reflection) that isn't just a gauge group symmetry, but instead something that acts on the very base manifold that the Field Theory appears built over: Transitioning between Minkowskian and Euclidean spacetime geometry. With this concern, I literally don't want to start learning Field Theory properly, which leaves me poorly educated and essentially without formal help or substantial argument in physics.

So ... I wish you good luck! You've got a chance, no doubt. Best wishes, Jens

Hello Tom - sorry I haven't gotten around to leaving a note on your essay! I'll also have a look at the attachment that you point out.

As Joy also cautioned, my thought of looking for time in his model may not be a valid one. Yes, he does write about spin orientations { n_1, ..., n_4 } at that section of his paper, and he constructs a set of four 7-vectors that have locally Euclidean metric on the 7-sphere. But only because that part of it resembles the locally Euclidean metric of our familiar 3-space doesn't mean that you could interpret it as some form of observer space, and find an independent observer time such that we have a description of the quantum system in a frame of reference in the sense of special relativity. Joy hints at a hypersurface to S7 to get this done, which would leave pure octonions. That would be ugly, as we all agree.

But then again, coming from the other side and supposing that Joy's model is a special case in that all four spins are modeled at the exact same observer time, for all observers (as there is no observer time in Joy's model whatsoever), then that necessitates that all four spins are at the same point in space. Rather than modeling time as an extraneous construct, through hypersurfaces or other bolt-on constructs, I would be interested in twisting those four spins against one another. With that I would start with only two spins, model them as Joy does, and then "pull them apart". As you know, once you create a distance between any two objects, in the sense of special relativity, you must define your frame of reference that then gives you a dynamic but distinct observer time in that reference frame. What may appear as four co-located spins { n_1, ..., n_4 } in Joy's ansatz would become the extreme case of total co-location; where pulling these spins apart would appear not as pretty, as the result of being forced to define an observer frame of reference. While the formulas would start to look ugly, it would be conceptually simple.

It might not work. Worth a try, though, I think.

Best wishes, Jens

Dear Caohoàng,

Thank you for leaving your thought provoking note. You suggest that it might be lack of confidence that caused us not to boldly sketch a new theory on the E8 lattice. To me it is more a reality check that assures me that most all suggestions for new physics that we can come up with, at any time, are wrong. This assurance comes from the mere number of possibilities out there. In order to keep this essay both entertaining for a wide audience, but also to make a strong point nevertheless, we focused on what it most important to us: To sketch a sense of naturalness, beauty, and simplicity as motivating some of our current and future work elsewhere. We did that at the expense of actually proposing a model, granted, but isn't it amazing with how few and simple assumptions you necessarily arrive at the E8 lattice? Repeat, invert, closure; and self-duality of the space under addition and multiplication. Lattices in 1, 2, 4, and 8 dimensions satisfy these very simple assumptions; and only in these dimensions. The E8 lattice is of course an immensely complicated construct if you attempt to understand its properties and automorphisms - something I will never succeed in fully. But conceptually it is *that* simple. Pure math at its finest, we feel, and that's what we want to convey to the reader.

Best wishes, Jens

Dear Benjamin - I'm humbled by your note. Sorry for not replying earlier, I was out of town with my family. Just to let you know, I came across your essay on 30 August, loved it as well, told John about it, and gave it the top rating as well - whew :) You should have good chances of winning a prize, and hopefully you earn FQXi membership! I do have a question regarding your causal metric hypothesis, but let me first respond to your note.

Regarding Riemann surfaces, as you know of course they work great when modelling physical systems where the configuration space is locally Euclidean. Physicists also believe they're great for locally pseudo-Euclidean spaces; all conventional description of fundamental physical law is built on such after all. When using lattices as configuration space, in contrast, we need new mathematical tools to do analysis: Lattices are made from countably infinite objects, and when comparing lattices against one another you need to embed them into some kind of encompassing space. Taking the E8 lattice, a natural embedding would be the eight dimensional Euclidean space over the reals, R8, and you could then shift and rotate instances of such lattice against one another in R8 and do math. You know all of that, of course, I am just summarizing. The whole thought on lattices came last year when I began studying a mathematical concept developed by Prof Mark Burgin (UCLA), which he calls "hypernumbers and extrafunctions". He generalizes the concept of "number" to infinite sequences, and defines rules for comparison, arithmetic, differentiation, and integration. Using his concept, I played with making an exponential function where "1 ^ (1/n)" would have all those "n" points on the unit circle as their solution set where the point taken to its "n-th" power would be 1. Such solution set of "n" points should, in turn, be a single "number". Burgin's hypernumbers do the trick, since he allows alternating or even divergent sequences to be understood as a single number. The set of convergence points for the sequence is what he calls the "spectrum" of a number, and with that you can model a single number where the spectrum of "1 ^ (1/n)" indeed are a set of "n" points on the unit circle. The amazing result - to me at least - is that for "n --> infinity" the spectrum of "1 ^ (1/n)" becomes the topologically closed unit circle in the complexes!

Currently I am writing a paper about this, but I am very slow in math so don't expect things to turn up quickly. All of the actual material is scattered out across my little online group (e.g. the 10th topic post, "Burgin 10", contains a summary http://tech.dir.groups.yahoo.com/group/hypercomplex/message/1101 but also note the follow-ups with corrections ... I really do need to write this into a self-contained, intelligible paper, sorry for not having anything better at hand right now). Being able to make such a generalized exponentiation of an expression (a/b) ^ (c/d) where a, b, c, d are positive nonzero integers, this gave us a lot of hope since lattices in any finite dimension are made from countably infinite points as well; and Burgin sequences can always be enlarged by more (countably infinite) subsequences without leaving their realm. With that, Burgin hypernumbers appear powerful enough for modeling generalized octonionic arithmetic on the E8 lattice. That gave us the needed motivation that such a thing could even exist. So, here's what'll happen next: I do my homework and write a little paper on that, put it in the arXiv, contact Prof Burgin and other mathematicians for comment, and then go from there. Estimated time to completion: Whenever it's ready ... this is "open-source research" after all, contributors and claims of ownership welcome :)

You mention "categorification", which is an interesting field in mathematics to be looked at for use in physics, as you wrote. John and I had looked at category theory a bit a couple of years ago, but found that it may not be a good fit. There is an associativity requirement on morphisms that seems too restrictive for what we want to do, and in turn when dropping associativity from categories you end up with even less mathematical structure. Categories are already so general, so wide in what all they could encompass; we didn't want to explore weakening its definitions even more. So we opted for working on specific examples of number systems, rather than attempting to understand what more general framework they would fall into. One of our works, "W space", can indeed be restated in compact form as a primitive 2-category. Here's a preprint version of our work: http://www.jenskoeplinger.com/P/PaperShusterKoepl_WSpace.pdf -- If we would describe it as a 2-category, then the paper would just be one page! So obviously, categories can be helpful. There was another reason for making such a chatty paper, namely that we connected to rather colorful prior research and felt we needed to spend the extra time to put things on sound feet first, by themselves. A second number system that we defined, "PQ space", cannot be described as a category ( http://www.jenskoeplinger.com/P/PaperShusterKoepl-PQSpace.pdf ) since the functor on the morphisms "" and "x" would not be associative. That ended our interest in category theory, for now at least.

Regarding your causal metric hypothesis, let me think that through and then post on your essay thread. Thanks again for writing! And best wishes, Jens ( & John)

Dear Eckard, thank you for leaving your note, and for referring to your essay and the work of Spalt. I'll have a look. Of course, until proven right or wrong, everything is opinion. Best wishes, Jens