Tuesday, May 3, 2011
Division Algebra over Relativistic Fields
Division Algebra over Relativistic Fields (Ottonians)
L. Edgar Otto (PeSla) 05-01-11 Eau Claire, Wisconsin
Someone quipped that we now understand the infinite, but the finite will take us a little while longer. I suggest that the deep relationship between the finite and the infinite will take a little longer still. In effect we do not understand the infinite as well as we might think even if we have become used to ideas like renormalization.
Some concepts seem to me for all practical purposes to be more of quasi-concepts. Among those more or less centered in this view is the idea of dimensionless constants, like the Reynold's constant based on Newton's second law and important for models of fluid flow. In this at some initial levels or closer to the relative micro-scale the first approximations can take time as a close description and as the perturbations expand from this into series the lesser smaller values may be discarded in their modification of the pristine case.
I suggest that perturbations in itself is a quasi concept and it is quite remarkable that in some cases like the prediction and discovery of the outer planets that this was more a stroke of luck than a certain mathematical procedure.
The scope of the unfolding of certain geometric objects like the hypercube is also quasi certain, we do not necessarily know if such folding applies to the depth or span of a wider range of dimensions, if the scale is local or extended without bounds, if it applies to the macro or micro worlds exclusively.
We have swept too much of the foundations under our rugs as to what is dust or fluid in our models, the inertia and viscosity of it in our changes, the idea of dynamical force. At some level we say that it reaches a zero or point of relative rest which does suggest a sort of vacuum energy source and even models for the structure and dynamics of the general cosmos. This too is a quasi concept at the heart of how far we can probe the structures of vacua and singularities.
We also sweep it under the rug of quantum ideas and no further going on to explain the deep source of oscillation in these geometrical transformations for the nature of particles and wave theory except from a vague metaphysical perspective. As such we imagine the complex number systems and rotations greater than the simple rigid rotations and translations in space so as to appeal to chance as also a working method of accuracy in the indeterminate that depends on a wide induction of the data. Rater than the simple fact of treating objects like classical spin, their loops and twists, and asymmetries, we interpret them in a landscape that brings them back beginning at 720 degrees, strings and the like. Some intuitively sense that this resolves our ideas of gravity, some interpret this as a vast world of super symmetries and quantum gravity.
But in the familiar scale of things, the human scale, this idea of what then is non-linearity is a quasi concept. There may be a vast and inaccessible world of some sort of curved space with few solutions, a higher relativity, but other than the ideas of time from the human scale even pushed to observations of the micro world and macro world to fine detail and definition and with more intricate machines, what relevance is left as we understand the reality of the dust and the various restrictions of its numbers and geometry, or for that matter conservation laws, energy and entropy itself perhaps laws to seem to be quasi laws and broken at least in the short view. Chiral ideas and shadow particles so described too all for the key to some particular evidence for some paradigm of physics.
Add to this that even in fluid flow, say when the ink drop expands into a torus if droped into still water, that it continues to make sub tori, this not yet explained. How is it that smoke rings of sufficient size can oscillate as if hexagons?
Our reach of computation to questions of non-linearity is also a law that can be of a greater freedom of expression as to what may be practical physicality.
I propose then a method, not necessarily the method and one that because of my time restraints will be illustrated from arbitrary labels as a standardization to which those with the time may go on to discover wider generalization or applications to the concept of particles. From that perhaps a better and general dynamics of the algebra and topology. That, in this idea of quasi-scale we decide the non-linear describes the non-euclidean geometries of one form or another, or what coordinates are the positive or negative in dimensions such that we can find descriptions of time stopped and the traditional hyperbolic invariants, from an abstract global perspective of such topological space and objects we might conclude that decoherence like perturbation is a non necessary quasi concept.
Yet can we regard a theory that does not in itself give us ideal and pure numbers, especially in the case of singularities or a complex of them as often a competing concept that requires tweaking a foundational fundamental physics?
Must we forever separate the concept of gauge and knots and the like that may weave through them unto some point of assumed orientation and breaking? Surely the bare charge, a bare non necessarily quasi singularity can only be adjusted to what shields them as a constant beyond the simplicity of binary continuum powers and primes of the groups in ways that can be intelligibly explained as well as of it we observe a standard but dimensionless measure.
From this view, we certainly can see that in the quantum world that quaternions may be applied, and where we extend that world to octonions it is essential to do so if we are to understand why nature has given particles at least three fundamental generations. This of course suggests that we can use the group numbers themselves as things that we can treat without our ordinary laws of arithmetic and extend to infinity such group laws without restriction, rather quasi restrictions should we understand the why of them in generalizations beyond the octonions.
I chose the name for this trial idea as Ottonians, not so much to name something after myself as to be humble in the misspelling of octonians... And perhaps it is not an idea worthy of my name.
All of this from a number theoretical viewpoint is here concerned with factorials where if we had infinite eyes the primes would be totally described, we imagine. But less than that we need to understand such factorials from a finite growth in relation to ideas of changes within the growing dimensions. It seems to me that should we look beyond four space into eight space and more the count of such seemingly all too obvious and trivial calculations is most intelligible.
When we apply my interpretation of the Conway matrix of colored cube matching, that matrix of 36 things which I intuit breaks the usual observed particles in its own sort of possible symmetry breaking between what is the 24 or 48 of the orthogonal groups we find a good beginning to the description for our new particle zoo. We also find some things which may have been a candidate for particles, like Penrose's zig-zags, there in spirit but not true particles when they are reduced to the idea of twistors.
In fact what I propose here is the labeling of the cells or faces of the hypercube so as to match a normal labeled three space cube with overall colors and with the permutations of six sets of 24 colors on the faces. Thus what occurs in the hyperspace of its six directions, three as translations and three as rotations where it is a shadow down into three space representation, has influence on the rigid spinning of the three space cube.
We find that the perpendicular invariants of all such limited three space descriptions are in the z direction of imagined translation motions, in a rather discrete manner or quasi discrete of things moving through the hypercube.
It is not clear to me this enumerates all the permutations or what it would suggest for any doubling of things, or what series integrated, to make the totality symmetrical in operations.
But we know these cubes, at least as rigid cubes in three space, will in the hyper-rotation in the hypercube exhibit a sort of classical understanding of spin. Or in the hyper-translation exhibit a zig zag or wave like motion that covers only three of its four toric faces thus rocking back and forth, that is the observed oscillation as if a foundation for discrete waves, does not exceed one half pi, a right angle in one direction or the other. Here at the heart of simple motion we observe the essential distinction between 3+1 models and 2+2 models in 4 space.
The reading of the order of which face in the hyper cube is a quasi-concept which also means that we can perhaps benefit from a different ordering that may be useful to apply to a linear connection of things instead of the powerful and clear sequence over a two space of many compressed dimensions. But having thought I saw this once cannot recall how to do it or if it did not suffer from the usual defaulting to loops which no longer showed the themes of the whole.
So, in the 6 x 6 grid of 30 cubes we observe that these can be rigidly rotated into 24 positions, and there are 30 permutations and this equals 6!. We note in this quaternion space that this = 12 x 60, and it = 3 x 240 for 6 x 5!.
If we continue, to the edges of the 8x8 grid we find 21 instead of 15 cube like things in which case we have 21 x 240 with four space for 5040 or 7! and while we are considering such arithmetic let us note that 80 x 1172 is 7! Let us not forget as we have here 3 axes of the 60 that 24 x 20 = 480 which you may recognize in some of the physics formulas.
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Full of Time and Stars
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I received a long reply from Pitkanen as I asked so please visit the comment section there for his kind reply:
to which my return reply I post here (otherwise we have a long forum) which reminds me if our blogs are not visited at least once a year before the end of june on the blog buzz they will vanish. Does this mean an older blog like Arcadian Functor or mine before I exceeded the photo limit and google started a new one?
Thank you for a most interesting reply.
I am reminds also of those science magazine articles a few years back that tried to see the whole of our visible universe as inside a black hole of so many dimensions. Five I think.
This holography idea for me is the old question where does the information go? Lincoln says on the sciencechatforum that once a particle is adsorbed it is no longer connected to its pair partner. Some have said that there is no inside to the BH.
This challenging paper did catch your eye as I thought it might. But considering I can play two dimensional chess in three space, 4^3 rather than 8^2 I do not put much stock in holography as a fundamental principle. That game btw is harder to play than 4D chess with the restrictions that seem to happen to the moves.
You grasp the issues and I am impressed- and yes it is time to think again about such interiors of which there is a certain fractal aspect of which maybe what is relevant is only some sort of dynamics visible that is not curved for such analogs to relativity, quite besides consideration of what we mean by acceleration and so on.
In any case the Casmir effect can be linked on a very small scale to the very large scale whose description btw could equally be viewed as if an ensemble of point like particles.
More added here: Of course some of the names you referred to I will have to look up. I assume some are about spinning black-holes. But I agree with Penrose as the values need not be Minkowski like but can all be say positive. A matter of view. I also, in my own posts considered that we can have a translation of discrete values across some subspace of say a hypercube... Is all this not just a question of when we can view something as a particle or fluid? And I am considering that space not quite to be considered inside or outside of anything, and relatively so. Somewhere perhaps between a vague singularity and our cherished dynamics of continuity.
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