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

Thanks for pointing out your book - I must admit that I'm familiar with some aspect of your work (though by far not all of it). When you describe correlations on the 7-sphere, in octonion space, I will look in your work for how you propose to recover spacetime. The avenue where your work is intriguing to me goes as follows: Correlations on the 7-sphere describe the complete quantum mechanical state of a system. Electromagnetism uses a certain four dimensional subspace that just so happened to be parametrizable by four independent variables. When performing distance measurements between two such sets of four-vectors (four-parameters), these parameters exhibit Minkowskian metric. In my hypothesis (which still is aligned with yours, I believe) human bias assumes such a four-parameter space as foundational and describes the remaining non-electromagnetic forces as deviations thereof. In that thought, certain properties of canonical description of physical law become artifacts (here I am starting to speculate): The Higgs mechanism in the Weak Force or Yang-Mills instantons in the Strong Force, both of which assume existence of more than one vacuum metric. While my understanding of these mechanisms is very poor, I am nevertheless not satisfied an assumption that physical reality could be built on 4-spaces with both Minkowskian and Euclidean metric. I do know that it is not valid to attempt a microscopic understanding of a field theory this way; but I am not satisfied with this, either.

It could be that nature is that way, and I am simply an unsatisfied person. But if my hypothesis were to be true, then - starting with quantum systems on the 7-sphere - I would assume that by keeping some parameters unchanged but rotating others, such rotation would be transitioning through the entirety of physical forces that may exist in reality. Maintaining the anthropocentric bias and assuming electromagnetism as foundational, these forces would then appear as gravity, weak, and strong force we know today.

In 2006/2007 I gave it a first attempt using complex octonions and pairwise multiplication of a differential operator and a wave function. I don't think anymore that it can be made to work into a reasonable quantum theory, and it uses complexified octonions ("conic sedenions") and not on pure octonions. Let me point it out nevertheless and describe in the next paragraph how it could become quantum systems on the 7-sphere, in pure octonion space: http://www.jenskoeplinger.com/P/Paper-Koepl-2006-7v1.pdf . There, a one-parameter angle alpha transitions the Minkowskian Dirac equation into a 4D Euclidean counterpart and back. The nice part of this work is that the resulting force, next to electromagnetism, indeed describes gravity (in the weak-field limit it becomes linearized gravity, as it should). The bad part, of course, is that I've given up trying to make it into a working quantum theory.

However, your work becomes relevant to me if an octonionic exponentiation a^b would exist, where both a and b are general octonions. In that case, the same reasoning from my 2006/2007 papers could be applied to such generalized exponentiation, in which case I would also have a quantum theory that I'm happy with (chapter 3 here: http://arxiv.org/abs/0910.3347 ). Pairwise multiplication c*d and exponentiation c^d have the curious property in the quaternions and octonions, in that these morphisms intersect when using only basis elements: Take e.g. c = i_1 and d = i_2, both imaginary basis units of the quaternions (and octonions). The product c*d is defined as i_3. The exponential c^d is (i_1)^(i_2) which you can define as exp(i_2 * ln(i_1)) which also becomes i_3. All the reasoning from my old 2006/2007 papers would then become valid, only this time on the octonions and the 7-sphere alone from my 2011 paper, with no need to complexify or other ugliness. I would be comfortable in defending such a quantum theory --- Problem is: No such arithmetic for a^b exists today where both a and b may be generalized octonions.

So, you see, I am very much tuned in to your work that describes quantum systems on the 7-sphere. If my speculation is right, then your work proves existence and completeness of such quantum theory. Can we find an arithmetic that works the way I need it to evaluate my hypothesis? There are some indications that such arithmetic would need to be built on the E8 lattice.

Now simply strip out all the parts that we're not sure about, or don't yet, then you essentially arrive at our essay. :)

Fun times? You bet! Jens

Interesting - thank you so much for the direct reference to your paper. There is one particular interpretation of your work that I'm focused on (and again, please excuse my tunnel vision). I see how you embed the four spin three-vectors from an EPR ansatz into equatorial 6-spheres of your general 7-sphere configuration space. You don't yet handle time in the paper/equations referenced, please correct me if I'm wrong. Glancing at your equations (113) through (116) it seems that you have at least one independent degree of freedom left; which is enough for an observer time dimension. You mention GHZ states as candidates for the entanglement scenario you're describing. A more simple proposal could be bound quark states: Ignoring quark/anti-quark states, the smallest bound quark states observed in nature have three constituents. The quark/anti-quark symmetry (if so) could then be driven by an additional time parameter. In your model, it seems straightforward to measure distances between the various n_1, n_2, n_3, n_4: The distances are simply from a 3-dimensional Euclidean metric. I'm not sure how time would enter such a system, and how to do (special) relativity in your model. ... That would be something worth tinkering around for me, to take one of your spins (say, n_1), and try to introduce a time parameter somehow to describe it in relativistic form. Have you done such a thing yet?

Regarding your question why I'm looking at exponentiation: It is purely a proposal at this point. With a modified Born rule ( http://arxiv.org/abs/0910.3347 sections 3.2 and 3.3 ) that requires invariance of the conventional eigenvalue relation (del Psi = m Psi) under changes of octonion algebras, I'm happy with crafting a quantum theory that I would be able to defend as "similar, but simpler" as compared to canonical quantum theory. Theoretical reductionism is key for me, but in order to achieve this I would need a generalized octonion exponential. Here, "octonion exponential" is a placeholder term for a new kind of morphism, to be found, between two octonions. This new morphism must reduce, in the special case of a complex number subspace, to complex number exponentiation. Therefore the terminology choice "exponentiation". If nature were one dimensional, and could be described in such a complex number subspace, and if you were ask for the influence of 1/r potentials on a test particle, then such quantum theory proposal results in what would be the Dirac equation in one dimension (seciton 3.3, equation 3.1). So my ansatz isn't necessarily wrong. Whether anything of this works at all for the description of nature, though, hinges on the existence of such octonion exponential.

Thanks for pointing to Michael Goodband's essay, I will read it soon. Sorry if some of this may take a little while; I'm currently transitioning to a new job, which is consuming a lot of my attention right now as I am leaving my current employer and prepare for what's next.

Best wishes, Jens

You must be the Joker from Batman ...

Thank you, Thomas! Will do. Octonions are around since many generations, we should not need to be in a hurry even if they seem to become more fashionable again.

Rick - you wrote: "I am a bit puzzled by both you and Michael Goodband talking Octonion Algebra, S7 and a split signature (as in Minkowski metric spaces) all in the same breath." - I've double-checked the little note I left on Michael Goodband's discussion. All I am doing is refer him to the work of others that I believe is related ... so I don't know what of my writings you're referring to.

Then you wrote: "You do not get S7 with split Octonions that are not even a division algebra." - Sure you can "get S7 with split Octonions", e.g. when supplying Lorentz boosts as in Gogberashvili http://arxiv.org/abs/0808.2496 . So when Goodman proposes that he's working with some decomposition of S7 into S3xS4 or S3x(S3xS1), then that's not a problem to me a priori, on this high level at least. Whether he can actually make it to work is a whole different story, and I'm not competent to comment on that.

You conclude that same paragraph with a statement about me: "I can see where you are coming from since it is good politics [...]" - I'm lost, what are you driving at?

Overall, I am confused how you are - on one side - advertising your approach to recover Maxwell electromagnetism (amongst other claims, of course) from octonions, but - on the other side - criticize others for "talking Octonion Algebra, S7 and a split signature (as in Minkowski metric spaces) all in the same breath". In my analysis of your octonion variance sieve I remained on the mathematical side, without going into physics. The physics part is yours to defend, though I did have a peek. I do understand that you are defining an octonionic action

W = invariant F * j

(i.e., from a generalized octonionic force and flux) where W is built from an octonion differential operator

Del := { d/dx0, ..., d/dx7 }

and an octonion potential

V := { V_0, ..., V_7 }

such that (if I got it right) you define W as:

W = Del ( Del V V Del ) ( Del V V Del )

Here, the notation V Del means applying Del on V but using the commuted octonion multiplication rule. It is a product of three terms, but since the 2nd and 3rd term are the same we don't have to worry about nonassociativity. In turn, this almost instantly proves that it is an algebraic invariant under your octonion variance sieve.

Your notation of this is different, with lots of indices; but I think my notation is faithful (or at least essentially right). Your achievement then follows from recovering Maxwell electromagnetism (and more; again, which is for you to defend) from that. This alone is a notable achievement! What confuses me is your criticism of others attempting to recover Maxwell electromagnetism, algebraic split-signatures, or Minkowski metric through other means. Just like yourself, some of these other approaches also start with octonions, and just as you are proposing an ad-hoc action W as I sketched above, these other approaches propose ad-hoc constructions as well.

I meant Michael Goodband, sorry for the typo.

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