K-theoretical foundation f E-infinity and the real meaning of counterfactuals and contexuality

6th December, 2010
E-infinity Communication No. 42
K-theoretical foundation f E-infinity and the real meaning of counterfactuals and contexuality

As the reader becomes more and more familiar with E-infinity and provided he has a strong interest in pure mathematics and set theoretical foundational work he will realize the K-theoretical origin of E-infinity theory. This should not come as a surprise to anyone who already sees E-infinity as a concrete realization of von Neumann-Connes’ noncommutative geometry. In fact there are many other unsuspected relations to E-infinity theory stemming from the relation of Yang-Mills theory and quantum gravity with the notion of connection on vector bundles. There is complete equivalence between the category of vector bundles over compact space and bundle maps as well as projective modulus of all sorts in the smooth case. They may explain the reaction of people like John Baez against E-infinity which was more elementary and easier to understand without abstract mathematics and achieved substantial and testable results in high energy physics while others, with the exception of Connes and of course Penrose were still deeply involved in abstraction. In fact much of the recent work by Nobel Laureate G. ‘t Hooft and particularly his 2006 paper “The mathematical basis for deterministic quantum mechanics”, arXiv: quant-ph/0604008V2, 26th June 2006 could be understood at least partially as translating E-infinity theory and K-theory to his language, namely the language of the very successful quantum field theory. We in E-infinity gratefully acknowledge the unique achievements of classical quantum field theory but it is time to expand this theory either piecemeal as is happening most of the time or in one great quantum jump to noncommutative geometry, E-infinity and K-theory. For instance the problem of indistinguishable large sets of quantum data forming equivalence classes is understood in E-infinity or K-theory as an algebraic K-theory analogous to topological invariant of corresponding virtual spaces and sets such as the infinitely many sets with negative Menger-Urysohn dimension and yet finite positive Hausdorff dimension. The minus one empty set for instance has a dimension equal to the golden mean in E-infinity theory as well as the K-theory of noncommutative Penrose tiling or Penrose universe as pointed out some time ago by M.S. El Naschie and Ji-Huan He. The totally empty or truly empty set at minus infinity where all kinds of geometrical objects including any Cantor set or dust disappears completely still has a non-negative Hausdorff dimension, namely exactly zero.
We note here on passing that the notion of counterfactuals was introduced to quantum mechanics only because quantum mechanics did not directly consider negative topological dimensions nor empty sets. Similarly contexuality is only the consequence of the indistinguishability between union and intersection in the basic geometrical-topological structure of a geometry which accounts not only for the zero set (0,ϕ) but also for all empty sets ( ̶ 1,ϕ2), ( ̶ 2, ϕ3), … and so on until ( ̶ ∞, 0) on a very fundamental level.
To see the connection between G. ‘t Hooft’s limit cycles given in the 2006 paper mentioned above and the work of E-infinity, the reader is advised to read in depth the paper of Mohamed El Naschie “Coupled oscillations and mode locking of quantum gravity fields, scale relativity and E-infinity space”, Chaos, Solitons & Fractals, 12 (2001), p. 179-192. For instance equation (7.4) on page 13 of ‘t Hooft’s paper is exactly identical to Dunkerley’s theorem of equation (10) and the elasto-plastic Rankin-Merchant formula given by equation (11) in El Naschie’s paper. Prof. El Naschie knew these equations from his research in engineering science and his work in buckling and vibration of elastic and elasto-plastic structures. We find it really fascinating how pure mathematics such as K-theory can be connected however indirectly with applied-applied engineering research while in between we have theories such as gauge theory and Yang-Mills theory. This cross fertilization is not unique to E-infinity but is very characteristic for the working of our E-infinity group.
In the next communication we will expand on all the aspects which we have only touched upon in this communication.
E-infinity Group.