Division Algebras and Four Space TrajectoriesL. Edgar Otto 04-30-11 Eau Claire, WI
I give you first a sort of ley rant on theories of the foundations. I suppose the quality of one's playing in a chess game depends on the quality of your opponent- in this mood it is unfortunate the intellectual challenge that inspired this defense of Baez et al comes from what some conclude is not a strong enough army to command respect and civility in undertaking science war that may lead to breakthroughs and leaps.
I present this informally to show the links that come to mind- all of this impeded by our languages and notations- but I assume some of the methods have been around long enough to be common knowledge by those in the discipline- how we derive the Lorentz transformations for example.
Since the Lorentz "boosts" are associative, those who, although standing on an immutable truth over the infinite range (perhaps excluding zero) and hyperbolic spaces, (these as identity matrices from my researches in a "quasic" ordering and zeros considered at rest or otherwise, those who consider this work of the last century to be the end all and grounds for all future developments in foundational physics including what we mean by spins and rotations in spacetime, This property then limited by that property in the division algebra limited in notions to quaternions, and the applications limited to special relativity only in a discrete quantum way in which case the concept of octonions is an irrelevant background for hoped for super symmetries, that or seen as anathema let alone a useful generalizations of the opaque that may describe the universe even beyond such algebras.
In some was we are not concerned only with even or odd functions over quasic (wave particle) space but also the natural consideration of the Lorentz treatments and absolute values of invariant spacetime shifts as vertical to the upper and lower quadrants - this the basis for deriving the transformations of such a spacetime group theory. This pattern seems, on the face of it, most evident in the evolving structure of life, provided fractal like (that is the assumption of a quasicontinuum) spaces grasp the limits of symmetry and asymmetry and other hierarchical scales and values that tries to understand an infinite regress.
Considering the order of the matrix idea leading to Lorentz transforms it makes sense in quantum theory to rotate them 90 degrees to interpret say Feynman diagrams so rotated in particle descriptions and transfers, decays. It is the 45 degrees of the absolute angles of light rays that makes the quasic plane on this dimensional level relate it as if square, and what sort of determinants and linearity involved in these fixed and general vector objects fill the square 8 space with hypersphere objects. It also may follow, if the reductionists applying reduction to this idea take these algebras and general numbers seriously- that different laws across the universe seem to be there in some cases, the inside of quasars, the shift of neutrino masses and other anomalies to their system (one that btw it is futile and false to assume it can be dismantled without consideration of its deeper axioms, or the thought of a variation of the light velocity, this is perceived as a threat that can only be resolved by a more comprehensive foundational theory.
We try of course to resolve the issues of interest, fact or myth, of our time- after all communication is a human need and is important even with some sacrifice of certitude. In these issues of simple Trinity as a model to include, perhaps the Christian tradition would understand this better or at least a Western one, for the physics, for science, as well as religion it seems these issues of triality in what factions people belong to has cause more core in depth conflict by ideas alone than most any other issue of philosophy from religion.
Apparently, the use of simple graphs and geometry, after all how do we define say a Cayley number or graph? Seems to be suspect and rejected at the heart of new theory, but this is how it was for a very long time for the status of simple geometry as taught.
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Although the lecture (thru Kea's blog) concerned string theory on the atomic and then Planck level was deep into its origins and continuity of the growth of ideas from Newton to Quantum- as a creative method the lecturer asked what if someone from the future came and told Newton that his trajectory ideas were wrong or could be expanded into a more quantum view- should we not image the same state of things for a better idea that really does not undermine the old physics, just expands upon them? We certainly need more of these intuitive methods, to think about the foundations of the foundations. What do we gain by uplifting an idea, such as that of the surfaces into projection of volumes holographically, as an item of unquestioned worship- especially if we cite all the time the specially chosen saints of that discipline as if no others worthy to commune with the ultimate spirit of physics? Is that not some form of Nationalism? Some inversion that beyond the healthy egoism is a nasty self-centered egotism where both are not really defined?
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Let me say a couple of things, technically, the all important fact that it is shown that in the Graceo-Latin squares there are no solutions for 2 and 6, this shows up when we make squares of things (in four space for after all we cannot extend the matrices as Cramer? does beyond three space) and it follows that there are known ones for at least ten elements to combine. Why are these global things of simple topology and counting not considered intrinsic to physical theories? Why is this the case in reality anyway as a property of space? What is the two then three and a little more in a world were our notions and notations have to deal with 2 x 6 or 12 things over space. (the consideration of these led to the 9 cubes drawn above).
The previous table (that included a Maroi title) amounts to a Keaese-to-PeSla translator where we try to resolve these models of triality which are really a sort of compliments as maybe we should expect from the new freedoms of higher space and the use of defining numbers like the Mersenne for the essential composites and thus the factoring or division of things. As the illustration states I rather sorted these axes and labels informally and not their mirrors for example- nor have I formally shown, as Kea did, these transformations in spirit do apply to the multiplication say in the case of quark mirrors. But my doing that also shows a certain ease when in parts of combinatorics the order does not seem to matter. Yet it is clear with privileged centering in a global setting artfully assumed, the properties of division algebras asymmetrically and with the associahedra and so on are useful areas to study and explore- certainly it would enhance string ideas. While we may think of such space as M theory without strings, it is true also that there are still those who think, from a recent bygone era, we can have string theory without these higher levels of number theory as spaces-string elements not explaining so many things disembodied from any idea of space and time itself.
To what extent, I wonder, is the density as defined by say the polygons and polyhedra a density as so defined by the expansion or contraction of wide space.
I remark also that while it is not looked at too deeply, the tachyon idea is alive and well or at least in the background, that to keep in mind in our explorations as either a useful principle or one that if not there explains so much as to what is there. But again, teleology is still more philosophy than science.
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Ironically, I owe Lubos for an idea today for on his article:
http://motls.blogspot.com/2011/04/john-baez-octonions-and-string-theory.html?utm_source=feedburner&utm_medium=feed&utm_campaign=Feed%3A+LuboMotlsReferenceFrame+%28Lubos+Motl%27s+reference+frame%29
he shows this diagram which speaks for itself- that is shows such work from the alternative views of Baez et al- but as far as I can tell this drawing comes from Tony Smith. From my view, and in struggling to see things a little alien to my views that involve the twists and turns and braids and so on of complex numbers this way of representing octonions and division space, a seven sided figure that is broken down into triangles speaks volumes to me- a way perhaps to make another set of such cubes for three space. Again we would benefit from more formal explorations or at least from those who might choose to help with the work, surely they to find something new also in the new physics,- well, a theoretical brain is a terrible thing to waste, would from an existentialist view our detractors not vanish but come on board that while we breathe the wheel is in motion and a conversion to the cause if it a better and objective direction, is to be hoped for.
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