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  • Post #21

Ray,

This problem is related in some ways to Perleman's proof of the homotopy equivlanece of homology spheres. I have been doing some background reading on this problem and am in many ways suspect this has some very deep mathematics to it. If you take a toy balloon and twist it up into various shapes it will smapp back into a spherical shape. I assume the surface is perfectly frictionless so that tying it up does not clamp it into shape. This Hamilton flow is such that it is always guaranteed to bring the space back to its minimal configuration. If you look at Perelman's paper you will see how he assigns entropy functionals to this! So a three sphere is guaranteed to evolved by the Ricc flow

dg_{ab}/dt = -2R_{ab} ...

This idea is carried over to higher dimensional spaces, and this evolution is conidered according to Turing machine logic. Look at the Novikov theorem quoted in Abhijnan's paper. So the logic is whether one can compute the minimal configuration of a space by Turing machine logic and assertain whether it is the same space as a known space in its minimal configuration. This is important for understanding all these strange Calabi-Yau spaces and whether one can compute them distinctly from each other.

The one thing which needs to be done is to carry this from homological theory to K-theory. K-theory is a much more powerful approach to topology, and it has connections to noncommutative geometry. So this is a vast domain which has many areas left unexplored. I am still patiently trying learn the foundations which Abhijnan quotes in this summary paper.

It is too bad this is not receiving a higher score than what it has garnered so far. This is one of the better papers in the whole lot submitted here.

Cheers LC

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  • Post #22

Dear Lawrence,

Abhijnan's paper does contain more physics than most of the papers in this competition. In my opinion, too many of these papers contain more philosophy than physics. Philosophy is certainly important to physics ("natural philosophy"), but we are past the days of Aristotle - we do have real knowledge and data to work with as well. I am tired of reading modern "hand-waving" arguments about how TOE's do or don't exist that rely exclusively on previously existent theorems such Godel's Incompleteness Theorem or the Turing Machine. I prefer the Nike slogan "Just Do It!".

Although Abhijnan's paper does contain a lot of physics, his goal of computing the string ground state is absolutely impossible with modern computers in the absence of a real breakthrough in our understanding of strings or TOE's. He indicates the possibility that patterns might simplify the computation. Perhaps my lattices ARE the patterns he needs! My alliance with Mohamed El Naschie last year led me to think that the difference between the finite K12' minimal roots and the nearly infinite Universe can be represented with a fractal approximation. My papers emphasized the minimal roots of K12', which are our nearest-neighbor lattice points, but this lattice could also have next-nearest-neighbors (similar to the long roots of K12, and also similar to the hyperflavor leptons and quarks in my book), next-next-nearest-neighbors, and so on to infinity. This is the physical reason why a fractal approximation may be appropriate.

I have been rereading your Jordan paper. You combine some interesting mathematical structures in your paper. Your 27 dimensional Jordan transformation is the natural extension to Dray and Manogue's 10 dimensional transformation. I think your physical interpretation is different from mine, but I'm not certain of myself either - I'm still considering the problem, and how it might tie into Supersymmetry or Feynman diagrams.

Have Fun!

Ray Munroe

13 days later

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  • Post #23

I would agree there needs to be some additional physics which selects for an outcome on the so called landscape. That is why I think this is in part determined by a quantum phase transition. The part in my current paper with G_2 basis elements determined by a Dirac field are a part of the physics for this quantum critical point.

So far the outcome on the landscape is similar to finding a microstate on a microcanonical distribution. Some additional physics is required to give a distribution which has low entropy determined by a phase. With standard phase transitions as temperature decreases the decrease in entropy is associated with an increased ordering in the lattice configuration of molecules or atoms in a solid state. Similarly a quantum phase transition should indicate how the "needle" was able to land on a unique low entropy configuration for the early universe.

Cheers LC

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