The exact renormalization equation for E-infinity unification of fundamental forces

15th December, 2010.
E-infinity communication No. 62
The exact renormalization equation for E-infinity unification of fundamental forces

E-infinity blends topology and number theory in a way not that familiar in theoretical or even mathematical physics except for mathematicians of the mold of A. Connes. Even exceptionally mathematically superior physicists like E. Witten rarely use extensive number theoretical arguments nor of course computer experiments nor computer aided proof as used for instance by B. Cherikov or M. El Naschie and more generally researchers in nonlinear dynamics, chaos and fractals. For example take the theory of M.H. Freedman dealing with the topology of 4-manifold which is used extensively in E-infinity theory (see M. Freedman and F. Quinn: Topology of 4-manifolds, Princeton University Press, New Jersey (1990). There you will see how his new method of capped gropes is related to both wild (non-tame) topology as well as number theory. Speed, complexity and size of height are controlled by Fibonacci numbers, the golden mean limit and a maximum words length at most 5. The authors of the said book go as far as saying on page 41 that “… their occurrence (I.e. the Fibonacci number) provides amusing evidence for the organic nature of gropes”. El Naschie remembers that the type setter of one of his papers dealing with this subject thought he had to correct “gropes” to “grapes” to suit the word organic!!! Perhaps we should mention that gropes were introduced in topology by Stan’ko to use in taming wild embedding of Cantorian topology such as that of noncommutative geometry and E-infinity in higher dimensions. Cantor set always appears somewhere, for instance in the actions of free groups, such as Kleinian groups and Schottky groups studied by El Naschie because of the important limit set constituting Cantorian structure. We recall that the end point, or boundary of E-infinity compactification is functorial and the limit set is a Cantor set. We recommend three papers by El Naschie about these subjects:
1. Wild topology, hyperbolic geometry and fusion algebra of high energy particle physics Chaos, Solitons & Fractals, Vol. 13, Issue 9, July (2002), P. 1935-1945.
2. Quantum loops, wild topology and fat Cantor sets in transfinite high-energy physics Chaos, Solitons & Fractals, Vol. 13, Issue 5, April (2002), P. 1167-1174.
3. Complex vacuum fluctuation as a chaotic “limit” set of any Kleinian group transformation and the mass spectrum of high energy particle physics via spontaneous self-organization Chaos, Solitons & Fractals, Vol. 17, Issue 4, August (2003), P. 631-638.
Now we can proceed with some confidence to introduce E-infinity’s so called number theoretical “Ocam Razor”. To do that we should realize the direct Weyl-Suslin scaling connection between the time independent inverse coupling constant of electromagnetism, the electroweak and the Lie symmetry groups involved. First the inverse fine structural constant alpha bar naught can be derived from a fundamental equation and found to be exactly equal to 20 multiplied by the inverse golden mean to the power of 4. This is equal 137.08203939. Second the inverse of alpha one is exactly 60. Second the inverse alpha two is exactly half of 60, namely 30. The inverse alpha 3 of the strong interaction is exactly 9 and the quantum gravity coupling is 1. The experimental values are all well known and are very close to the preceding value, namely 137.036, 59.4 and 29.8. The quantum gravity unity is of course only a theoretical value which cannot be obtained in any of the present day experiments. Now we reconstruct the 137.082039325 from the rest. To do that we note the E-infinity value of the Clebsch coefficient c2 = 5/3 is to be changed to 1 + ϕ = 1.618033989 where ϕ = 0.618033989 is the golden mean. The reconstruction of 137.082039325 follows the classical one known in any text book on quantum field theory only with our number theoretical make up. It is equal 60 multiplied with 1 + ϕ and added to 30 plus 9 plus 1. This comes to exactly 97.0820393 + 40 which means 137.0820393. To see that this cannot have anything to do with number coincidence we stress that all values are very near to the experimental ones and that the fuzzy E8E8 = 496 namely E8E8(fuzzy) = 496 ̶ k2 = 495.9674775 where k = 0.18033989 = ϕ3 (1 ̶ ϕ3) may be found from 3 + ϕ Weyl scaling of 137 + ko. A little pocket calculator will confirm that (137 + ko)(3 + ϕ)= 495.9674775. Because of the irrational number involved we have a second guarantee that this is not number coincidence but basic fuzzy Cantorian set theory as shown in detail in E-infinity theory.
Next we want to derive the exact equation of renormalization for unification. First we should mention that from our E-infinity view point Heisenberg’s matrix quantum mechanics was a giant quantum leap forward. By contrast Schrödinger’s wave mechanics was the work of absolute genius which was misunderstood and caused a jump backwards. We will not discuss this here but will do it later on in another communication.
As the blog does not take mathematical equations we ask the reader to make the effort and follow the verbal explanation of the equation besides the little bit of math we can print. We call all coupling constants alpha with some subscripts but we mainly use in our calculation the inverse values which we call alpha bar, that is an with a bar on it. Now we should have the following mental picture for the exact renormalization equation of E-infinity theory which is the same as the standard equation only much tidier and easy to see through its structure. On the left hand side we have alpha bar of unification. This is equal to right hand side which consists of two expressions. The first is alpha bar 3 of the strong coupling which has the theoretical exact value 9 added to alpha bar 4 of quantum gravity which is equal to one. The two terms are thus equal 10. The second expression consists of which is either 1 for non-super symmetric theory or 1/2 for the minimal super symmetric theory. Thus is multiplied with the familiar logarithmic term of the ratio between two masses. The first mass is the unification mass divided by the second mass which is the reference scale. For grand unification for instance the unification energy is 10 to the power of 16 Gev while the reference scale is the mass of the of the electroweak namely 91 Gev. Thus the logarithmic term is approximately equal 32. We take it to be exactly 32 + 2k which is the theoretical value of the inverse of the electromagnetic fine structure constant at the infrared energy scale namely 137.082033989 multiplied with the golden mean to the power 3. Taking = 1 one finds that the unification inverse coupling is 10 + 32 = 42 or accurately 10 + 32.36067977 = 42.36067977 which is exactly ten copies of the well known Hausdorff dimension of E-infinity spacetime core. For super symmetry we take = 1/2 and consequently we have 10 + (0.5)(32) = 26 in full agreement with the most accurate value found in the literature and obtained using numerical methods and extrapolations. To find the coupling of unification for quantum gravity we take the Planck energy as that of unification. This is given by the Planck mass 10 to the power of 19 Gev. The reference energy we take to be that of the mass of a Cooper pair, i.e. two electrons with a mass equal to 0.001 Gev. The logarithmic term is thus almost equal 50 which when analyzed is found to be (22)(ln10) and since the transfinite value of ln10 is 2.236067977 while 22 is 22.18033989 we find that the exact value of the logarithmic terms plus one for alpha 4 is the dimension of F4 exceptional Lie symmetry group which when corrected transfinitely gives 52.36067977. With = 0.5 we find the product to be 26 + k = 26.18033989. Now this is the value of the inverse coupling of super symmetric unification. Consequently we have 26.18033 on the left hand side equal bar plus bar plus 26.18033. Therefore bar plus bar must be equal zero. Since bar is unity, the bar of the strong coupling must be negative which means the strong coupling is negative as predicted by the standard theory of confinement. Alternatively if the unification coupling is unity then bar must be zero which means the strong coupling must be infinitely large which means confinement as expected and as it should be.
E-infinity Group.

P.S. Some very relevant papers are the following as strongly recommended reading.
1. Transfinite harmonization by taking the dissonance out of the quantum field symphony, Chaos, Solitons & Fractals, vol. 36(4), (2008), p. 781-786.
2. Quantum golden field theory - ten theorems and various conjectures, Chaos, Solitons & Fractals, 36(5), (2008), p. 1121-1125.
3. Extended renormalizations group analysis for quantum gravity and Newton’s gravitational constant Chaos, Solitons & Fractals, Vol. 35(3), (2008), p. 425-431.
4. Exact non-perturbative derivation of gravity's G4 fine structure constant, the mass of the Higgs and elementary black holes, Chaos, Solitons & Fractals, Vol. 37(2), (2008), p. 346-359.
5. Elementary number theory in superstrings, loop quantum mechanics, twistors and E-infinity high energy physics Chaos, Solitons & Fractals, Vol. 27(2), (2006), p. 297-330.