It is usual for E-infinity to spam the screeds here and there about the Internet, for example on A Geometric Theory of Everything, much to the chagrin of Garrett Lisi; but at the moment of this writing communications 44 and 45 appear only on EIHEC.
December 8, 2010
E-infinity communication No. 44
Schrödinger’s cat meets Arnold’s cat in El Naschie’s infinity garden.
No one knows exactly why E. Schrödinger the Austrian Nobel Laureate in physics and founder of wave mechanics chose a cat to subject to a most cruel type of quantum measurement. The reasons for the Russian V. Arnold, probably one of the greatest mathematicians of all time to deform a cat’s picture beyond recognition using his well known map are even less clear. In all events both cats are alive, well and jumping in E-infinity’s golden garden. The two eigenvalues of Arnold’s cat map happened to be the inverse of the golden mean (1/ϕ) = 1 + ϕ and the inverse of the golden mean squared (1/ϕ)2 = 2 + ϕ. Now adding both values or multiplying them gives exactly the very same result, namely (1 + ϕ) + (2 + ϕ) = 4.23606799 and (1 + ϕ) (2 + ϕ) = 4.23606799. In other words union and intersection of the two transfinite sets gives the same result. One can easily see that the first set have the topological dimension of an area just like a world sheet in string theory. This area has a fractal dimension equal 1 + ϕ = 1.61833989. The second set on the other hand is topologically three dimensional, however its Haudorff dimension is 2 + ϕ = 2.168033989. Topologically the union of the two sets is 2 + 3 = 5 dimensional. However seen from the Hausdorff dimension view point, the two sets give a dimension equal 4 + ϕ3 = 4.23606799 which corresponds to exactly 4 and only 4 topological dimensions. In other words one topological dimension is hidden. The 5 dimensions are used only for embedding the 4.23606799 fractal dimension. On the other hand we know very well that 2 + ϕ = 2.61803389 corresponds exactly to 3 topological dimensions of the tangible world. The only thing left for interpretation is that 1 + ϕ which is topologically 2 dimensional is that these two dimensions stand for time and the spin ½ fermionic dimension or alternatively for the fifth Kaluza-Klien compactified or the cyclic dimension of electromagnetism.
The reader is referred to the following paper for simple geometrical visualization of the Russian doll-like E-infinity space with 4.23606799 Hausdorff dimension corresponding to exactly only 4 topological Menger-Urysohn dimensions. See for instance “An irreducibly simple derivation of the Hausdorff dimension of spacetime”, Chaos, Solitons & Fractals, 41, (2009), pp. 1902-1904, particularly Table 1 on page 1903. See also “The theory of Cantorian spacetime and high energy particle physics (an informal review)”, Chaos, Solitons & Fractals, 41 (2009), pp. 2635-2646, in particular Fig. 1 and Fig. 2 on pages 2636 and 2639 respectively.
It remains to say that it is the fine structure of fractals contributing 0.23606799 = ϕ3 where ϕ is the golden mean = 0.618033989 which causes the equality of union and intersection of the Cantor sets spanning E-infinity spacetime and leading to the infinite but hierarchal dimensionality of this fractal Cantorian spacetime manifold which is the cause of the persistent illusion that a quantum object on a fractal particle can be said to be in two different spacetime fractal ‘points’ at the very same fractal ‘time’. Looked upon it from a distance, this intricate non-smooth and chaotic Cantorian-fractal spacetime appears smooth with only four spacetime topological dimensions. To show this in the most clear quantative way we just need to approximate the irrational number to the simplest rational number. That means 2 + ϕ = 2.618033 will be 2.5 = 5/2 and 1 + ϕ = 1.618033 will be 1.5 = 3/2. Adding together we find 2.5 + 1.5 = 4. On the other hand multiplication leads to a different dimension known from the theory of dimensional regularization, namely (5/2)(3/2) = 3.75 = 4 ̶ 0.25. This corresponds in E-infinity to 4 ̶ k where k = 0.18033989 of the transfinite Heterotic super string theory. It is vital to recall the importance of Arnold’s cat map in the study of quantum chaos as well as the discussion of the role of irrational numbers in combined harmonic oscillators by Prof. G. ‘t Hooft in his paper mentioned in an earlier communication “The mathematical basis for deterministic quantum mechanics”. In fact on page 7 of the arXiv paper he writes “….If the frequencies have an irrational ratio (which in the terminology of El Naschie and nonlinear dynamics means irrational winding number), the period of the classical system is infinite and so a continuous spectrum would be expected.” Sooner or later we are confident that those working in quantum field theory and have come as far as this will realize that K-theory, noncommutative geometry and E-infinity is the right way to come to what Nobel Laureate ‘t Hooft hopes to find, namely some kind of deterministic quantum mechanics.
December 8, 2010
E-infinity communication No. 45
Dial E or K for dilation and an apology to fiber bundle
The present communication, not in any way related to a Hitchcock film, is a very short bird’s eye view of the connection between E-infinity and K theory as well as the relation to ‘t Hooft’s very recent dilation proposal to improve quantum field theory. We start with a short sincere apology over a hasty statement made with regard to the theory of fiber bundles in a communication related to the new Lisi paper in Scientific American, December 2010. In the communication we wrote that fiber bundles are neither fish nor meat. This was an unnecessarily harsh statement and what is worse, it is incorrect and worse still, E-infinity is a fiber bundle theory. Fiber bundles and Yang-Mill theories are extremely fundamental and this statement has to be withdrawn. It is true we live in a different world and electronic computation has revolutionized science and the methods of proof. We now have subjects like experimental mathematics as well as computer assisted proof. In the light of all of that fractals may appear to be the only way to model complex behavior in high energy physics. However fundamental thinking from which transfinite set theory and subsequently fractals were created is needed now more than ever.
Let us start by recalling some informal explanation for what K-theory is. It is a devise by which one examines mathematical structures such as rigs or topological spaces via a parameterized vector space. In particular A. Grothendick initiated the subject by extending a theorem well known in the theory of 4-manifold namely the Riemann-Roch theorem. We may recall that M.S. El Naschie as well as C. Castro considered this theorem in connection with their research on E-infinity theory. Very loosely speaking much of the work on K-theory is a generalization of Riemann-Roch theorem to a theory of indexes as shown in the work of Sir M. Atiyah and E. Hirzebruch. In fact Mohamed El Naschie’s interest in K-theory stems from a meeting with Prof. Sir. M. Atiyah in the year 2000. In a Cairo conference organized by the Egyptian Mathematical Society Sir Atiyah gave the opening lecture followed by Prof. El Naschie’s second invited lecture which was on Cantorian-fractal spacetime. The discussion between Sir Atiyah and Prof. El Naschie continued the next day in the evening at the home of the Minister of Industry, Prof. Dr. Ibrahim Fawzi. The lecture of Sir Atiyah is contained in the Proceedings published by World Scientific (2001), Edited by A. Ashour and A. Obada, the two Chairmen of the conference and is entitled “Mathematics and the 21st century”.
Coming back to ‘t Hooft’s paper “The conformal constraint in canonical quantum gravity” it seems that to include the fractal-like dilaton field he had to consider a flat Kaluza-Klein space. This is the idea of the Finish G. Nordström who according to a paper by Prof. El Naschie published in CS&F, proposed the fifth dimension before T. Kaluza and of course before O. Klein. In addition a vanishing beta function became necessary. Again the role of beta function in E-infinity was discussed in one of El Naschie’s rare publications on quantum field theory. In doing all that ‘t Hooft was hoping that the landscape of his quantum field became denumerable. In the nonlinear science terminology of E-infinity this means make it countable infinity instead of uncountably infinite. Once more we must recall that the landscape of E-infinity or the infinitely many ground states can all be summed exactly as shown in two papers by El Naschie published in CS&F, one of them entitled “On the universality class of all universality classes and E-infinity spacetime physics”.
Finally we may point out that the dimensional function of von Neumann and Connes used by El Naschie in the form of his bijection formula and the golden mean average theorem may be regarded very loosely as an index theorem of K-theory. The same may be said also very loosely and informally about El Naschie’s Cantorian spacetime and Penrose tessellation for being based on K-theory.
P.S.: Relevant literature:-
1. M.S. El Naschie: Penrose universe and Cantorian spacetime as a model for noncommutative quantum geometry Chaos, Solitons & Fractals, Vol. 9(6), (1998), pp. 931-933.
2. M.S. El Naschie: Quantum Groups and Hamiltonian Sets on a Nuclear Spacetime Cantorian Manifold Chaos, Solitons, & Fractals, Vol. 10(7), (1999), pp. 1251-1256.
3. M.S. El Naschie: A Note on Quantum Field Theory and P-brans in Dimensions Chaos, Solitons, & Fractals, Vol. 10(8), (1999), pp. 1413-1417.
4. A. Mukhamedov: E-infinity as a fiber bundle and its thermodynamics, Chaos, Solitons & Fractals, 33, (2007), pp. 717-724.
5. M.S. El Naschie: On the universality class of all universality classes and E-infinity spacetime physics, Chaos, Solitons & Fractals, 32, (2007), pp. 927-936.
6. M.S. El Naschie: On the topological ground state of E-infinity spacetime and the super string connection Chaos, Solitons & Fractals, Vol. 32(2), (2007), pp. 468-470.