Wednesday, August 19, 2015

Could one define dynamical homotopy groups in WCW?

I learned that Agostino Prastaro has done highly interesting work with partial differential equations, also those assignable to geometric variational principles such as Kähler action in TGD. I do not understand the mathematical details but the key idea is a simple and elegant generalization of Thom's cobordism theory, and it is difficult to avoid the idea that the application of Prastaro's idea might provide insights about the preferred extremals, whose identification is now on rather firm basis.

One could also consider a definition of what one might call dynamical homotopy groups as a genuine characteristics of WCW topology. The first prediction is that the values of conserved classical Noether charges correspond to disjoint components of WCW. Could the natural topology in the parameter space of Noether charges zero modes of WCW metric) be p-adic and realize adelic physics at the level of WCW? An analogous conjecture was made on basis of spin glass analogy long time ago. Second surprise is that the only the 6 lowest dynamical homotopy/homology groups of WCW would be non-trivial. The Kähler structure of WCW suggets that only Π0, Π2, and Π4 are non-trivial.

The interpretation of the analog of Π1 as deformations of generalized Feynman diagrams with elementary cobordism snipping away a loop as a move leaving scattering amplitude invariant conforms with the number theoretic vision about scattering amplitude as a representation for a sequence of algebraic operation can be always reduced to a tree diagram. TGD would be indeed topological QFT: only the dynamical topology would matter.

For details see the chapter Recent View about K\"ahler Geometry and Spin Structure of "World of Classical Worlds" of "Quantum physics as infinite-dimensional geometry" or the article Could One Define Dynamical Homotopy Groups in WCW?.

For a summary of earlier postings see Links to the latest progress in TGD.


Anonymous said...

I meant to post that comment on the other blog post to this one.. oops, here it is. I read some of the paper.. this is incredible...

this seems related to 3.8#SpectralFunctions of @Book{bochner1955harmonic,
Title = {Harmonic Analysis and the Theory of Probability},
Author = {Bochner, Solomon},
Publisher = {University of California Press},
Year = {1955},
Series = {California Monographs in mathematical sciences},

or am I getting that confused with something else? I remember thinking it was the appearance of the polynomials and the anti-symmetrization involing the binomial processes and theorem, etc.

$\lim_{n \rightarrow \infty} \sum_{j = 1}^n \sum_{k = 1}^n \frac{f_{\gamma}\left( \frac{j}{n} \right) f_{\gamma}^{\ast} \left( \frac{k}{n} \right) Q\left( \frac{j - k}{n} \right)}{n^2} = \lim_{n \rightarrow \infty} E \left|\sum_{j = 1}^n \frac{f_{\gamma} \left( \frac{j}{n} \right) g \left(\frac{j}{n} \right)}{n} \right|^2$

see a (much more readable image) at ;-)


Anonymous said...

oops, wrong book reference... here is book with the formula I transliterated