Dial E or K for dilation and an apology to fiber bundle

7th December, 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 Fiber bundles and Yang-Mill theories are extremely fundamental and this statement has to be withdrawn.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.
E-infinity Group.

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.