Sunday, September 5, 2010

Alternative Use of Color Dimensions In Math and Physics

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Alternative Use of Color Dimensions In Math and Physics

There is not much on my mind this morning but a small thought or two on how some abstract or ideal geometric structures may be colored as an added dimension to show more clearly the symmetries involved (here I do not want to forbid or suggest my use of color is that similar to the abstract idea of QCD).

I vaguely suggest that there is absolutely idea just as if an absolute vs a relative vacuum. May we take color literally to represent charge conjugation? May we say that some structure such as a nucleon can have parton shells (the earlier model) of which there would be so many possibilities as to which are absolute or relative shells of vacua?

In my post to Lubos I vaguely mentioned that what is surface can be the depth and that this may not be as strictly determined and may even be a four way deal- that is the holographic idea needs this further generalization. Also, the added "color" dimension (scolor?) may indeed touch on some of our ideas of supersymmetry vaguely.

Here in the simple example of a tetrahedron with colored edges we see that the inversion of the ideal shape is not necessarily that of the color and its chirality where applied- nor what is the bkd gin scheme of things (also now in four directions with respect to a cube or the 30 cubes or 45 squares and so on) that what is the surface here represented as a triangle, can convert into a point with three rays from a local perspective. Does a neutrino for example know what such state it is in?

I realize that my last comment to Lubos may seem a little less intelligible (nevertheless it tends to support his position on the perils of an anthrocentric position for science and global warming- yet this too is hard to see and pin down.)
Anyway I hope some of you check out the debate and insights in his comments on the reference frame blog.

Also Ulla, I like what comments you posted there (if you are reading this). I also wanted to point out the idea of the codons bases as stacked (and it certainly is an inspiring idea even if as entanglement only finally applied to the DNA and coherence of the helix- still we are a long way from deep resolution of the molecule and I feel it is deeper still for stuff within that base as also a flattened geometry. But no doubt as Rowlands hints and your posted article it does apply to the reading of the code- and in some ways that I have shown. But materially or not, that is as if a particle thing and not a vague force thing- certainly there is an analog here to the Casmir forces as if the bases were close plates of metal and the paths or pressure outside them were preferred or excluded quantum frequencies. Again, from my view we are dealing here with string theory, black holes, and dark matter-energy that applies to organic things a little deeper than our ideas of quantum and relativistic theories.

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I also made a comment on

I found it most fruitful to use higher dimensional matrix notations and my friend who studied string theory in 95 saw them as useful but not applying to anything he was working on... however other methods on finding resonate states of particles so rapidly he found very interesting and wondered what in the hell matrix I had as it beat the computers.

I think this is the right track... but it is only the beginning and a crack in the door into a more general and yes topological physics of which speculations here look rather conservative to me.

But we can take the linearity of it all too far from the foundations as there can be other forms of expression by nature of determinants. It all depends on how far we are willing to stray from our traditions that are way to restricted.

Some mathematical idea or proof say of Poinclaire or Riemann's zeta at 1/2 may only describe our limited world view in a multiverse.

How far can we use such a spatial notation or matrix say to describe nucleons and so on? Can we say that such matrices cannot exceed octonians? What sort of determinants moreover would be needed to describe curves in planes if multilnear in paths- similar to recent ideas of "topological insulation" in the spin of electrons as room temperature?

The PeSla

* * * Next Morning: an informative and gracious reply from someone who knows the terain and what he is talking about too. Such minds I feel are capable of understanding what needs to be done and I fancy those areas of the issue of which I have made independent contemplations for objective criticism. But my post of this morning forthcoming will be of a decidedly philosphical nature as a metaphysical interlude- and why not the universe to inspire us this way? In any case something is afoot here on the foundation and fundementals of matrix algebra- but as far as simple counting here and an intelligible method of enquiry, I know it is but a beginning but is a suggestion as a mechanism for physics and perhaps more. I have thought more on the span and depth of things... in a sense we really gain little by matrices beyond the two dimensions as these are also hyperdimensional in depth- other than perhaps a picture of higher space, and the steps to comprehend the significance of what we mean by the reduction of say 8x8 to 4x4 or even such spaces for our prefered Dirac algebraic notations. It is a matter of taste really, if we claim a vanishing is a reduction to zero or it is a potential for wide varieties of intelligible geometric structures and processes in the universe.

At 6:15 PM, Matti Pitkanen said...
The proposed application of the notion of generalized hyper-matrices and multilinearity (in generic case non-linearity) appears only for deformations of the preferred extremals , which are extremely non-linear objects as such. Multilinearity in this case serves as a signature for absence of infinities and for exact solvability too. Vanishing of hyper-determinant would be a a signature for criticality against n:th order phase transition.

The number of tensor factors is infinite in the generic case. The number of field variables is 8 (reducing to 4 by general coordinate invariance) and being very naive one could say that something analogous to infinite tensor product of 8x8 or 4x4 matrices over space-time points is in question.

The infinite-D hyper-determinants would be rather intricate objects as compared to their finite-D variants. Generalizations of Gaussian determinants of quantum field theories. One should be able to define them in practical manner.

Second application of hyper-determinants is to the description of quantum entanglement. For pure states the matrix describing entanglement between two systems has minimum rank for pure state and thus vanishing determinant. Hyper-matrix and hyper-determinant come into play when one speaks about entanglement between n quantum systems. If I have understood correctly, the vanishing of hyper-determinant means that the state is not maximally non-pure.

For the called hyper-finite factor defined by second quantized induced spinor fields one has very formally infinite tensor product of 8-D H-spinor space. The hyper-determinant would characterize entanglement of fermionic modes. Could quantum classical correspondence mean that the vanishing of n-particle hyper-determinant for fermionic entanglement has as a space-time correlate n:th order criticality. Probably not.

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