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.
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.
18th December, 2010.
E-infinity communication No. 63
Introduction to the philosophy of E-infinity
Two years ago Mohamed El Naschie started writing a paper on the philosophy of E-infinity. The paper was too long and a great many unpleasant events took place which prevented him from completing this paper which in its present form is too long and only stands as a very rough draft which cannot be easily reproduced not even in summary form as an E-infinity communication. It was no doubt the intention of those well known internet thugs and parasites to distract us from science and derail us from our road. This was the brief given to them by you know who. [Renate Loll? Just guessing. He sometimes accuses her of stealing his work to write a Scientific American article.] Never the less we will attempt to give here what can only amount to a summary of the summary of what El Naschie considers to be the philosophical background to his theory.
The real and maybe sobering truth is that there is no one single deep philosophical reason which prompted E-infinity. Scientists are not motivated by deep seated philosophical conviction. They may write in a way suggesting epistemological reasons for their theories but the real, real reality is that scientists are motivated by curiosity and a wish to succeed where other famous scientists have failed. Scientists, like all human beings, also want to understand. [This section suggests El Naschie wrote his latest Rosa column at the same time he wrote this communication.] El Naschie frequently used to quote Goethe’s Faust (to understand “was die Welt zusammenhält”, i.e. what keeps the universe in one piece). The rest is chains of accidents and the painstaking assembly of thousands of pieces of information to form a mosaic picture which can hopefully help to develop a mental picture and a feeling of understanding.
On the intellectual trip of any thinking man there are main stations and memorable events. According to his own writings, one of the main intellectual adventures of Mohamed El Naschie was reading J.P. Sartre’s monumental book “Being and Nothingness”. Arabic and German are the only two languages he has mastered to an extremely high degree although he speaks or has knowledge of some 10 languages. None the less he speaks all of them with the exception of Arabic and German with a relatively strong accent and his spelling in all languages is rivaled only by his bad hand writing which is the main reason for so many secretaries resigning! He first read Sartre in Arabic translated by Abulrahman Badawy, Egypt’s most famous existentialistic philosophers and the uncle of one of El Naschie’s best friends, Mohsen Badawi. Later on he mainly read Sartre in German but also occasionally in French. We dare to say that the second most important event was when as an engineer he came across the triadic Cantor set for the first time. He said in many of his writings in English, German and Arabic that a Cantor set was a Sartarian thing. It is not really there because it has no measure, i.e. no length and it was there because for a measure zero it has a very large non-zero dimension. The third station with respect to E-infinity was reading J.A. Wheeler’s Borel set proposal and even before that, reading the writing of Heisenberg, Weizsäcker and particularly D. Finkelstein. Deriving the dimensionality of spacetime from a primitive monadic assumption like we derive the concept of temperature in statistical mechanics became El Naschie’s program.
Meeting Otto Rössler, M. Feigenbaum, I. Prigogine, I. Procaccia and finally Binnig and ‘t Hooft brought Mohamed’s thinking to that of first Garnet Ord and then L. Nottale. El Naschie admits that without Ord and Nottale he would not have had the courage to continue his work alone. The late Prof. Werner Martienssen, Prof. Rössler, Prof. Ord, Sir H. Bondi, Sir J. Lighthill, Prof. Walter Greiner, Prof. G. ‘t Hooft as well as Prof. Ji-Huan He, L. Marek-Crnjac and E. Goldfain constantly encouraged Mohamed to go on in his quest for a spacetime theory for quantum mechanics which is similar to relativity as well as to the Feynman path integral. His relationship with his close friend Nobel Laureate G. ‘t Hooft is complex. In fact, too complex to consider here due to the tremendous difference in temperament, personality and attitude toward the science of the infinite and of course Mohamed is an extremely religious person although with no commitment to and a great distrust of organized religion, exactly like his father who was an army General [Was he in fact a general?] from a noble Egyptian family. Never the less the work of ‘t Hooft is of great importance to El Naschie and they seem to have had a great influence on each other. [On each other, LOL]
More general reasons to become a scientist and to move from applied engineering to fundamental science are probably connected to his interest in the scientific work of J.W. Goethe as well as meeting K.F. von Weizsäcker [Danger! Died in 2007!] and getting acquainted with the philosophical views and personality of W. Heisenberg [Oh, no! Died in 1976!] particularly the role of symmetry as well as determinism in nature and science. He put many of his memories about these subjects in several of his Forwards to special issues and general papers published in CS&F over the years the reader may go back to them on Elsevier’s Science Direct: www.sciencedirect.com.
In conclusion we must understand that philosophy is strengthened by the nature of reality. Therefore Mohamed El Naschie points out that the gamma distribution of the random Cantor sets which he used to model quantum spacetime (for r = 2) is effectively the same distribution of the intensity fluctuation of black body radiation but for r = 3. Thus E-infinity is physically real and consequently philosophically correct.
19th December, 2010.
E-infinity communication No. 64
Prof. Ji-Huan He – The twenty-six dimensional cube operad of a scientist and a gentleman
One of the most outstanding young generation founders of E-infinity theory is undoubtedly Prof. Ji-Huan He. The present communication will attempt to give justice to his great contribution to E-infinity theory and to shed more light on the deep mathematical though [sic] of his 26 dimensional polytope as well as his Hilbert model which is lurking behind a deceptive simplicity.
Prof. Mohamed El Naschie described Prof. He as a true Chinese scientist and a gentleman whose scientific journey to E-infinity theory took him from the romantic idealistic world of Freiherr Georg Friedrich Philipp von Hardenberg (Elias [sic. alias] Novalis) to the work of probably the greatest and most enigmatic mathematician of all time Alexander Grothendieck. In 2007 Prof. He published his paper “Twenty-six dimensional polytope and high energy spacetime physics”. This paper is only nine pages long but contains ten beautiful colored computer graphics of an n-dimensional cube and how its geometry transforms from order to deterministic chaos. The paper may be found on Elsevier’s Science Direct. It is contained in Vol. 33, Issue No. 1, pp. 5-13 published in 2007 in CS&F with extracts published in several other international journals and books including the American Inst. of Physics. Two years later an equally beautifully illustrated paper entitled “Hilbert cube model for fractal spacetime” was published in CS&F, Vol. 42 (2009), pp. 2754-2759. These two papers are closely related to a paper by O. Zmeskal, M. Weiter and M. Vala as well as a paper by El Naschie, namely “An irreducibly simple derivation of the Hausdorff dimension of spacetime”, (CS&F, 41 (2009), pp. 1902-1904). Zmeskal et al’s paper was entitled “Note to an irreducibly simple derivation by El Naschie” also in CS&F. There are various fundamental issues which lie deep at the root of these papers to which we would like to at least touch upon here.
First Prof. He’s 26 dimensional polytope is de facto an explicit generic example for a fundamental theorem in nonlinear dynamics and as far as we are aware was first stated by the outstanding mathematician and engineering scientist Prof. D. Ruelle [who is attending CHAOS 2011] from the Inst. of High Scientific Studies in Sur-Yvette, France. Prof. Ruelle’s theorem effectively says that when we put any classical mechanical system in an infinite dimensional setting, then it will become spontaneously chaotic in the sense of the theory of deterministic chaos and its fractal-Cantorian geometry. This explains the magic of the power of the expectation value of the Hausdorff dimension of E-infinity derived for the first time by El Naschie, namely 4.236067977. This value can only be obtained for an infinite dimensional system and corresponds exactly to only 4 topological dimensions in the Menger-Urysohn meaning of topological dimensions. The reduction from infinite dimensions to a core of only four dimensions is a central and recurrent theme in the work of El Naschie and the work in E-infinity. However what scientist working on the physical side of E-infinity should realize is that this reduction of infinity to 4 is far more general in pure mathematics than most people think. Let us start with a simple and obvious example from the topological quantum field theory. It is well known that El Naschie replaced current algebra by fusion algebra. However the most important fusion algebra is the four dimensional fusion algebra as documented in the work of V. Sunder and V. Kodiyalant. This work is closely connected to the work of Sir M. Atiyah as well as Field Medalist V. Jones who discovered the relation between knot theory and statistical mechanics used in the recent work on E-infinity by El Naschie. A more mathematical and abstract relation which E-infinity scientists could use more in future development is the relation to the mathematical theory of highly structured ring spectrums. This theory leads to a so called cube operad of finite dimensions. This is a fancy pure mathematical name for infinite dimensional cubes in infinite dimensional space similar to the work of Ruelle and He. On the other hand E-infinity could be replaced in this mathematical theory by E4 exactly as in El Naschie’s physical geometrical E-infinity theory. Thus E-4 is defined by replacing the operad of an infinite dimensional cube in 4 dimensional space and keeping the same basic idea of E-infinity ring spectra. The entire theory is of course related to K-theory, Grothendieck-Rieman-Roch’s theorem and n-categories. This may partially explain the unusual reaction “to say the least “ of an otherwise good mathematician like John Baez. It must have come as a mild shock that E-infinity of El Naschie and He is smack on the physical target of quantum gravity while others are still beating around the abstract mathematical bush. It is a fact that random Cantor sets are specific and measurable. More over they are O-categories and (n + 1) categories are just more complex monoidal categories. One cannot give any quantative results using the posets of F. Dowker who works with R. Loll and J. Baez. Posets are immensely important theoretically. However they do not lead directly and easily to quantative results which are needed in high energy physics. For instance in 4-dimensional fusion algebra we have the golden mean as an Eigenvalue. This makes all calculations trivial to the extent that no computer is required at all. Similarly the dimensional function of K-theory of the Penrose-‘t Hooft-El Naschie holographic boundary is a direct function of the golden mean which again makes otherwise complex calculation trivial.
The K-holographic boundary, like all fractals, has an inbuilt dilaton field in it so we need not worry about scale relativity as in the classical form of quantum field theory. All that is encoded in the beautiful work of Prof. Ji-Huan He. We should also mention that Prof. He does not work on E-infinity only. The scope of his research is staggering, spanning perturbation methods in numerical analysis to experimental methods in nano technology. Prof. He is also inclined towards literature and poetry like his great friend Prof. El Naschie. [Sample El Naschie poem here.] Readers may recall that El Naschie likened the indistinguishability condition of E-infinity which makes it impossible to say if a particle is at a point 1 or a point 2 or both points at the same time like in quantum mechanics with the novel of Robert Musil “The man without qualities”. Prof. He always finds wonderful examples from ancient Chinese literature and culture to illustrate his scientific research. Great success in the real world unfortunately brings some unpleasant things with it like envy. [An allusion to our campaign to remove He from editorial boards.] In recent times Prof. He, similar to many of us and notably El Naschie and Nottale have had more than their fair share of it. To them we can only repeat an old Nordic proverb “what does the moon care when dogs are barking down there”.
With our best wishes for a merry Christmas and a very happy New Year (2011).
El Naschie has written several articles about Novalis and his influence on Wagner and nonlinear dynamics which is hinted at here and will be discussed in detail later on.