22nd January, 2011
E-infinity Communication No. 75
Fake R(4) and exotic Milnor seven Spheres S(7) in the fuzzy or average knot Yang-Mills instantons of E-infinity
Donaldson fake R(4) was considered in the work of El Naschie in E-infinity quite early on. A little later he considered the exotic Milnor seven spheres. In a paper published in CS&F6, 19 (2004), pp. 17-25 influenced by the work of El Naschie entitled “On Milnor seven dimensional sphere, El Naschie E-infinity theory and energy of a Bianchi universe” by Gamal Nashed of Ainshams University in Cairo, Egypt the particular relation between exotic geometry and E-infinity was discussed and an interesting summary was given in a very nice illustrative form in Fig. 1 on page 23. Also following El Naschie, Nashed made important use of the maximum sphere surface area and maximum sphere volume given in his figures 2 and 3 on page 24. El Naschie remarked that Nash formula gives a seven sphere for an Euclidean embedding of a one dimensional object because D = (0.5)(n)(3n ̶ 11) = 14/2 = 7. In addition he introduced the fractal seven dimensional sphere with the dimension 7 plus phi to the power 3, i.e. 7.23606799 which played a role in his fractal black hole theory. We recommend reading the paper entitled “Fractal black holes and information” by M.S .El Naschie, CS&F 29, (2006), pp. 23-35 and consider the explanation of Fig. 1 on page 25 and Fig. 3 on page 27. The most important conclusion of all these attempts for E-infinity research was the deep realization that the idea of moving from the factorial function to the gamma function should be generalized as done in moving from a topological dimension to a Hausdorff dimension. In fact doing this systematically one moves from classical quantum field theory to K-theory which is the mathematical realization of E-infinity theory. El Naschie proclaimed that Nottale’s idea of giving up classical differentiability and replacing it with Robinson’s non-standard analysis should be considered much deeper. El Naschie was familiar with non-standard analysis from his work on the canard of catastrophe theory. Therefore he was convinced that moving to Nottale’s frame work is a first step. The second step was to move to exotic ‘differentiability’. However this was not sufficient in his view and that is when he moved to point set geometry with cardinality equal to that of the continuum and that is how he arrived at Cantor sets and Cantorian spacetime of E-infinity theory.
It follows then that Yang-Mills theory must be modified to account for the true transfinite nature of high energy particle physics. This modification is what most probably inspired ‘t Hooft recently to include a dilaton field in his quantum field theory while hoping to refine classical calculus which is in principle of course possible when accepting some difficulties as the price. On the other hand random Cantor sets with their golden mean Hausdorff dimension offers natural quantization coupled with incredible computational ease due to the inbuilt golden mean number system which we explained in many previous communications. The usual mathematical way of thinking about fiber bundle theory is that we start with point set then move to a topological manifold, then smooth manifold, then geometric manifold, then bundle. We may start before point set and end beyond bundles. E-infinity is both the prior point set and the beyond bundle. Let us argue the case for an E-infinity action which is far more physical than ‘t Hooft’s S = 82 and at the same time much easier to hand, all apart from the unexpected fact that using E-infinity, hidden connection which would have passed totally unnoticed become obvious and trivially visible.
We reconsider again 82. This is obviously exactly 16 four dimensional sphere volumes. The volume of a four dimensional sphere with unit radius is as is well known, vol S(4) = 2/2. Consequently (16)( 2/2) = 8p2 = S, the action of ‘t Hooft’s Yang-Mills instanton. In E-infinity however we make a much richer relation when we take average everything. This average is a transfinite average. You could call it fuzzy values if you want. First we replace the volume of the spheres with the fuzzy hyperbolic volume of knot. We take K(82). For this knot the hyperbolic fuzzy volume is 5 ̶ ϕ4 where ϕ is the golden mean 0.618033989. Instead of taking 16 spheres we take an average of 16 + k = 16.18033989 knots of the 82 type. That way the total volume is exactly SF = ( /2) + 10 = 78.5419966. This is the value corresponding to 8p2 = 78.95683521 of ‘t Hooft. However we see here relations which we cannot see when using the classical analysis of ‘t Hooft. In particular we see that SF ̶ 10 when multiplied with 2 gives the exact theoretical inverse electromagnetic fine structure constant namely = 137 + ko = 137.082039325. From that a plethora of other relations follow, for instance (3 + ϕ)( ) = E8 E8 and remembering that E6 = 78 is already an integer approximation to S = (3)(26) = 78 we see that the net of interrelations with the exceptional Lie symmetry groups and not only SO(3) where we noted in a previous communication that volume SO(3) = 8p2. In general we can say the El Naschie fuzzy K3 is a K-theory K3 and that the transfinite Feynman diagrams of E-infinity are the equivalent of Feynman motives which was developed recently. Thus E-infinity could be called K-infinity and El Naschie’s fuzzy golden field theory is nothing else but a Grothendieck motives applied to the theory behind the standard model of high energy physics. In this sense Mohamed El Naschie was deadly right that ‘t Hooft’s dimensional regularization implies E-infinity spacetime which means noncommutative physical spacetime. The same conclusion was recently made by A. Connes. It is interesting to note that ‘t Hooft did not agree initially but he may reconsider the situation in view of the compelling E-infinity results.