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

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".

:) 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

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

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

(this was meant as a reply to Rick's earlier post)

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 )

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 ...