21st December, 2011.
E-infinity communication No. 66
Some of the most important results of M.S. El Naschie’s research in E-infinity theoretical quantum physics
1. E-Infinity theory (E (∞)) has clearly shown that random Cantor sets are the basic building blocks or atoms of quantum spacetime. The intrinsic topological dimension of these building blocks is zero, its Hausdorff fractal dimension is the golden ratio (0.618033989…) and its embedding dimension is unity.
2. Quantum spacetime is described fully by not one but three dimensions. First a topological (Menger-Urysohn) dimension equal to exactly 4. Second a Hausdorff dimension equal to 4 plus the golden ratio to the power of 3 which means 4.236067977… This is effectively a 4D cube inside a 4D cube and so on ad infinitum. Third a formal dimension equal to infinity. In other words our quantum spacetime is an infinite dimensional but hierarchal Cantor set of measure zero.
3. Because orthodox quantum mechanics is totally insensitive to fractals and does not consider the Cantorian nature of quantum spacetime, many paradoxes follow.
4. The mass spectrum of all known elementary particles of the standard model as well as some composite particles was predicted with astonishing accuracy compared to the known experimental value. Because of KAM theorem they are always a function of the golden ratio as confirmed experimentally by the quantization and the experiment at the Helmholtz Centre.
5. Many of the fundamental constants of nature were derived from first principles. This includes the coupling constant of quantum gravity as well as the electromagnetic fine structure constant and Newton’s gravity constant. In particular a fundamental equation was established relating the Bulk (E8E8) with the holographic boundary and gravity from which = 137 was derived. In addition the inverse coupling of unification of all fundamental forces were found to be = 26 + k ≃ 26.
6. An exact renormalization equation was constructed which derived quarks confinement as an exact result. Confinement is a theorem of the golden quantum field theory.
7. The relationship between Lie symmetry groups as well as two and three Stein spaces and high energy physics was outlined. In particular the role played by E8 in this respect was analyzed. The total number of elementary particles in an extended standard model was shown to be 137 particles. This includes 10 spacetime quasi dimensions. E-infinity P-Adic reasoning was also employed.
8. The theory predicted the existence of 8 dimensional Higgs field with at least one Higgs particle or five Higgs particles becoming manifest. The Higgs mass was determined to be approximately 169 Gev.
9. The Euler characteristic as well as the curvature of the quantum spacetime was shown to be equal to 26 + k 26. In fact even the real part of the dilaton is given by S® = 26. In addition arguments were given to show that our universe is likely to be compact. An alternative theory using instanton density to calculate the first massless particle-like states corrected the classical well known solution of Heterotic string theory, namely 8064 to the exact solution 8872.
10. The stationary state of quantum mechanics was shown to be that of the VAK, i.e. the vague attractor of Kolmogorov.
11. A quantum particle may be modelled as a fractal point represented by the two dimensions of a zero set dim P (0,ϕ) where . The quantum wave is the surface of a fractal point, i.e. a fractal surface and may be represented by the two dimensions of an empty set dim W . The empty set accounts for the paradoxal outcome of the two-slit experiment and the disappearance of the interference fringes.
12. It is reasoned that in a totally disjointed infinite dimensional but hierarchal Cantor-like space time manifold, calculus must be replaced by Weyl-like golden mean scaling which effectively represents a new version of quantized calculus. The set theoretical analogue of this is Suslin scaling of descriptive set theory.
13. El Naschie made several suggestions regarding a Banach-Tarski-like big bang theory based on paradoxal decomposition of spheres.
14. Mohamed El Naschie and his group proved the equivalence of his basic equations of E-infinity theory and the dimensional function of von Neumann’s continuous geometry and A. Connes’ noncommutative geometry.
15. Very recently El Naschie discovered several facts connecting his E-infinity theory to K-theory and the mathematical E-infinity theory of highly structured ring spectrum and the E4 of operad of 4 dimensional cubes.
16. It is conjectured that E-infinity building blocks represent at a minimum a O-category but could also be n-category for n → ∞.
17. The symmetry group of E-infinity is fuzzy and is a fractal symmetry given by the sum of all the dimensions of the E-line exceptional Lie groups 584 and not 496 as well as the sum of all Stein spaces with dimension equal 685. The integer value of all symmetry groups including Stein spaces of the first kind is 3689. When adding the super strings 10 dimensions then one finds (27)(137) = 3699.
18. E-infinity is shown to combine numbers and number theoretical reasoning with topology exactly as in K-theory.
19. Ervin Goldfain demonstrated that measurement of cosmic radiation confirms the correctness of E-infinity theory. El Naschie showed the consistency of the COBE measurement with E-infinity predictions. Malcolm H. MacGregor demonstrated the consistency of the experimental value of the mass spectrum with his alpha quantization of the masses (book published by World Scientific 2007). Alpha quantization is coarse graining of the golden mean quantization. The dependency on the golden mean was demonstrated for the first time experimentally in the Helmholtz-University of Oxford joint work announced earlier this year.
20. Recent work by Ali Chamseddine and Alain Connes (arXiv paper 2010) reinforces some fundamental results of E-infinity theory.
21. The recent results published by Ambjorn and R. Loll are particularly appealing confirmation of E-infinity theory. The spectral dimension of their work, namely 4.02 is a direct Weyl-Suslin scaling in exactly 16 steps from the first massless level of Heterotic strings (8872 integer value) which corresponds to the classical Green-Schwartz and Witten value (8064). The difference comes from the fact that E-infinity uses the entire 8 exceptional Lie group line which adds to 548 while the classical value (8064) considers only a subset of the exceptional Lie groups leading to only 504.
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