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 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 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 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. 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. 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).
E-infinity Group
P.S.
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.
Quantum Mechanics and the Golden Mean in Theoretical Physics
Dec 20, 2010
Introduction to the philosophy of E-infinity
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. 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. 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 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.
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 and getting acquainted with the philosophical views and personality of W. Heisenberg 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.
E-infinity Group
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. 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. 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 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.
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 and getting acquainted with the philosophical views and personality of W. Heisenberg 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.
E-infinity Group
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.
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.
Dec 15, 2010
Important: E-infinity communication No. 61 – Latest News
15th December, 2010.
Important: E-infinity communication No. 61 – Latest News
Prof. Alain Connes’ new 2010 paper with Prof. Ali Chamseddine comes to the same fundamental conclusions and predictions of E-infinity, namely 9 new particles and a Higgs mass equal 170 Gev
Connes and Chamseddine published a brand new paper on 3rd April, 2010. The paper is entitled “Noncommutative geometry as a frame work for unification of all fundamental interactions including gravity. Part I is on arXiv: 1004.0464V1[hep-th], 3rd April, 2010. Researchers working in E-infinity will remember that the 2005 paper of Prof. Ji-Huan He entitled “In search of 9 hidden particles”, published in Int. J. of Nonlinear Sci. & Numerical Simulation, 6(2), (2005), pp. 93-94. Using Prof. El Naschie’s theory 9 new particles were predicted to exist in an extended and completed standard model. Prof. Alain Connes is the inventor of noncommutative geometry and one of the greatest living mathematicians of our time who was awarded the Field Medal in 1982. Prof. Ali Chamseddine is an outstanding mathematical physicist who previously worked intensively with Nobel Laureate Abdulsalam and who met Prof. El Naschie, the last time in Alexandria where they had a hearty conversation. For these reasons alone this work of the two outstanding achievers of the highest caliber should be taken very seriously by our group. On the other hand the paper is detailed and very easy to read and understand. In a nutshell they use a non-simple Lie symmetry group SU(2) SU(2)SU(4) which gives 3 + 3 + 15 = 21 massless gauge bosons. A symmetry breaking leading to the standard model with the well known standard model 12 gauge bosons shows that there are 21 – 12 = 9 particles missing. This is the start of the paper. The important result and conclusion could not be anything better than predicting the mass of the Higgs which they do. The result is identical to that found years ago by El Naschie, namely 170 Gev. This result of M.S. El Naschie can be found for instance in his paper “Experimental and theoretical arguments for the number and the mass of the Higgs particles”, CS&F, 23 (2005), pp. 1091-1098.
For additional reading we recommend the book of A. Connes as well as the following papers.
1. An elementary proof for the nine missing particles of the standard model Chaos, Solitons & Fractals, Vol. 28(5), (2006), p. 1136-1138.
2. L. Marek-Crnjac: Different Higgs models and the number of Higgs particles. CS&F, 27, (2006), pp. 575-579.
It should be remembered that both E-infinity theory and noncommutative geometry are K-theory based and both have Penrose fractal universe (Penrose tiling) as a prototype and generic example where the golden mean plays a crucial role in the dimension function.
Please send your scientific comments and evaluation. They are urgently needed.
E-infinity Group.
Important: E-infinity communication No. 61 – Latest News
Prof. Alain Connes’ new 2010 paper with Prof. Ali Chamseddine comes to the same fundamental conclusions and predictions of E-infinity, namely 9 new particles and a Higgs mass equal 170 Gev
Connes and Chamseddine published a brand new paper on 3rd April, 2010. The paper is entitled “Noncommutative geometry as a frame work for unification of all fundamental interactions including gravity. Part I is on arXiv: 1004.0464V1[hep-th], 3rd April, 2010. Researchers working in E-infinity will remember that the 2005 paper of Prof. Ji-Huan He entitled “In search of 9 hidden particles”, published in Int. J. of Nonlinear Sci. & Numerical Simulation, 6(2), (2005), pp. 93-94. Using Prof. El Naschie’s theory 9 new particles were predicted to exist in an extended and completed standard model. Prof. Alain Connes is the inventor of noncommutative geometry and one of the greatest living mathematicians of our time who was awarded the Field Medal in 1982. Prof. Ali Chamseddine is an outstanding mathematical physicist who previously worked intensively with Nobel Laureate Abdulsalam and who met Prof. El Naschie, the last time in Alexandria where they had a hearty conversation. For these reasons alone this work of the two outstanding achievers of the highest caliber should be taken very seriously by our group. On the other hand the paper is detailed and very easy to read and understand. In a nutshell they use a non-simple Lie symmetry group SU(2) SU(2)SU(4) which gives 3 + 3 + 15 = 21 massless gauge bosons. A symmetry breaking leading to the standard model with the well known standard model 12 gauge bosons shows that there are 21 – 12 = 9 particles missing. This is the start of the paper. The important result and conclusion could not be anything better than predicting the mass of the Higgs which they do. The result is identical to that found years ago by El Naschie, namely 170 Gev. This result of M.S. El Naschie can be found for instance in his paper “Experimental and theoretical arguments for the number and the mass of the Higgs particles”, CS&F, 23 (2005), pp. 1091-1098.
For additional reading we recommend the book of A. Connes as well as the following papers.
1. An elementary proof for the nine missing particles of the standard model Chaos, Solitons & Fractals, Vol. 28(5), (2006), p. 1136-1138.
2. L. Marek-Crnjac: Different Higgs models and the number of Higgs particles. CS&F, 27, (2006), pp. 575-579.
It should be remembered that both E-infinity theory and noncommutative geometry are K-theory based and both have Penrose fractal universe (Penrose tiling) as a prototype and generic example where the golden mean plays a crucial role in the dimension function.
Please send your scientific comments and evaluation. They are urgently needed.
E-infinity Group.
Highly structured ring spectrum and the mathematical foundations of E-infinity theory
14th December, 2010.
E-infinity communication No. 60
Highly structured ring spectrum and the mathematical foundations of E-infinity theory
Mohamed El Naschie did not develop his theory starting from abstract mathematics. He could not even if he had wanted to because, as stressed by him on countless occasions, he is not a mathematician and does not hold a degree in mathematics nor in fact in physics. This is not usual but it is not that unusual either. For instance Prof. E. Witten who is a Field Medalist has studied neither mathematics nor physics as an under graduate. The great inventor of green functions was not even an academic and as everyone knows, the best study is self study. All the same a mathematical foundation for E-infinity was given by El Naschie many years ago after the initial geometrical picture of a Cantorian space was established. After realizing that E-infinity theory is a particularly simple and neat example of K-theory and Rene Thom (bordism) spectra El Naschie opted for the name E-infinity suggesting that all these subjects, including infinity-categories are united in a mathematical theory called “Highly structured ring spectrum” which deals with multiplicative processes similar to E-infinity and is usually designated in the mathematical literature by E-infinity rings and E-infinity loop algebra. Prof. El Naschie says that he is particularly indebted to Prof. Sir R. Penrose because his tiling was an eye opener for him. He is equally indebted to the work of Prof. A. Connes because it made him realize the large arsenal of mathematics behind noncommutative geometry and K-theory of which he actually never dreamed when he started working on his simple model of Cantorian-fractal spacetime.
A fairly mathematical paper dealing with the foundations of E-infinity as related to the Coxeter and reflection groups is the following note: ‘Mathematical foundations of E-infinity via Coxeter and reflection groups’, M.S. El Naschie, CS&F, Vol. 37 (2008), pp. 1267-1268. After becoming familiar with the dimensional theory of F. Hausdorff as distinct from that of K. Menger and P. Urysohn, El Naschie started his own mathematical program to introduce transfiniteness everywhere. In doing so he was far ahead of almost all mathematicians because he was of course using intuitive arguments without the essential mathematical rigor which mathematicians must subject themselves to. El Naschie followed two generalizations from integer to non-integer and even irrational values, namely the Hausdorff dimension and the factorial function. The Hausdorff dimension generalizes the usual topological integer dimension to a non-integer ‘fractal’ dimension. Similarly the gamma function generalizes the factorial function to non-integer function. El Naschie did not stop at that and generalized the integer dimension of a symmetry group to a non-integer dimension of a fractal symmetry semi-group. If you think about physics very deeply, like a mathematician of the caliber of A. Connes, then you will find that except in physics integer values are mostly empty mathematical concepts not applicable to the rest of the real world. The quantum Hall effect was discovered by Nobel Laureate K. von Klitzing and generalized to a fractional Hall effect by Nobel Laureate H. Stoermer. El Naschie’s theory predicts a fractal Hall effect as the final generalization of the previous two and he has written a few papers about that and all that remains is the experimental confirmation El Naschie’s fractal Hall effect.
Another mathematical generalization which was achieved within E-infinity theory is the curvature of a fractal-Cantorian point set. This is discussed in connection with the Gauss-Bonnet theorem in one of the most important papers of E-infinity: “On zero-dimensional points curvature in the dynamics of Cantorial-fractal spacetime setting and high energy particle physics”, M.S. El Naschie, CS&F, Vol. 41 (2009), pp. 2725-2732. The detailed discussion of this paper will be given in a forth coming communication.
E—infinity Group.
E-infinity communication No. 60
Highly structured ring spectrum and the mathematical foundations of E-infinity theory
Mohamed El Naschie did not develop his theory starting from abstract mathematics. He could not even if he had wanted to because, as stressed by him on countless occasions, he is not a mathematician and does not hold a degree in mathematics nor in fact in physics. This is not usual but it is not that unusual either. For instance Prof. E. Witten who is a Field Medalist has studied neither mathematics nor physics as an under graduate. The great inventor of green functions was not even an academic and as everyone knows, the best study is self study. All the same a mathematical foundation for E-infinity was given by El Naschie many years ago after the initial geometrical picture of a Cantorian space was established. After realizing that E-infinity theory is a particularly simple and neat example of K-theory and Rene Thom (bordism) spectra El Naschie opted for the name E-infinity suggesting that all these subjects, including infinity-categories are united in a mathematical theory called “Highly structured ring spectrum” which deals with multiplicative processes similar to E-infinity and is usually designated in the mathematical literature by E-infinity rings and E-infinity loop algebra. Prof. El Naschie says that he is particularly indebted to Prof. Sir R. Penrose because his tiling was an eye opener for him. He is equally indebted to the work of Prof. A. Connes because it made him realize the large arsenal of mathematics behind noncommutative geometry and K-theory of which he actually never dreamed when he started working on his simple model of Cantorian-fractal spacetime.
A fairly mathematical paper dealing with the foundations of E-infinity as related to the Coxeter and reflection groups is the following note: ‘Mathematical foundations of E-infinity via Coxeter and reflection groups’, M.S. El Naschie, CS&F, Vol. 37 (2008), pp. 1267-1268. After becoming familiar with the dimensional theory of F. Hausdorff as distinct from that of K. Menger and P. Urysohn, El Naschie started his own mathematical program to introduce transfiniteness everywhere. In doing so he was far ahead of almost all mathematicians because he was of course using intuitive arguments without the essential mathematical rigor which mathematicians must subject themselves to. El Naschie followed two generalizations from integer to non-integer and even irrational values, namely the Hausdorff dimension and the factorial function. The Hausdorff dimension generalizes the usual topological integer dimension to a non-integer ‘fractal’ dimension. Similarly the gamma function generalizes the factorial function to non-integer function. El Naschie did not stop at that and generalized the integer dimension of a symmetry group to a non-integer dimension of a fractal symmetry semi-group. If you think about physics very deeply, like a mathematician of the caliber of A. Connes, then you will find that except in physics integer values are mostly empty mathematical concepts not applicable to the rest of the real world. The quantum Hall effect was discovered by Nobel Laureate K. von Klitzing and generalized to a fractional Hall effect by Nobel Laureate H. Stoermer. El Naschie’s theory predicts a fractal Hall effect as the final generalization of the previous two and he has written a few papers about that and all that remains is the experimental confirmation El Naschie’s fractal Hall effect.
Another mathematical generalization which was achieved within E-infinity theory is the curvature of a fractal-Cantorian point set. This is discussed in connection with the Gauss-Bonnet theorem in one of the most important papers of E-infinity: “On zero-dimensional points curvature in the dynamics of Cantorial-fractal spacetime setting and high energy particle physics”, M.S. El Naschie, CS&F, Vol. 41 (2009), pp. 2725-2732. The detailed discussion of this paper will be given in a forth coming communication.
E—infinity Group.
Super Yang-Mills, super gravity and super strings. Spectroscopy and the impossibility of number coincidence
14th December, 2010.
E-infinity communication No. 59.
Super Yang-Mills, super gravity and super strings. Spectroscopy and the impossibility of number coincidence
If the reader is not very familiar with the theory of super strings, it would be advisable to have a text book to one side whilst reading this communication. For convenience we recommend M. Kaku’s book “Introduction to superstrings and M-theory”, Springer, New York (1999).
Let us look at the Heterotic string and consider its spectrum. For instance following pages 384 and 385 of Kaku’s book we have the following: For the left moving sector there are three distinct states. First 480 states equal in number to the kissing of two E8 and also equivalent to their roots number. In addition we have another 16 states. These make together the 496 of E8E8 dimension where each E8 has clearly 248 dimensions. Finally we have another 8 states so that the total number of the left moving section is 496 + 8 = 504. Note that this is exactly the sum of the dimensions of E8 plus E7 plus E6 plus E5 as found by M. El Naschie (248 + 133 + 78 + 45 = 504). Now we look at the right moving sector where we have only 16s states. Following the rule of Fock space of quantum field theory (where Fock space is the generalization of Hilbert space of quantum mechanics to quantum field), then the total number of all states is the multiplication of the left and right sectors. This gives us the famous number of the first level of massless particle-like states of the Heterotic string theory, namely No = (504)(16) = 8064. Now page 385 gives the number of states in a super Yang-Mills theory as 3968. These could be thought of as lifting E8E8 dimensions to super space by taking 8 copies of the 496 dimensions leading to (8)(496) = 3968. On the other hand we know from the spectrum of the theory of super gravity that we have 512 states. This could be thought of as raising the two dimensional average Hausdorff dimension of a quantum particle path to the ten dimensions of super strings. Using E-infinity bijection, this means we have 2 to the power of 10 minus 1. This is 2 to the power of 9 which is equal 512. This is only 16 more than 496 and 8 more than 504. Let us lift these 512 to super space like we did with super Yang-Mills. Proceeding in this way we find (512)(8) = 4096. The next step taken by Mohamed El Naschie is to consider No to be the sum of super Yang-Mills and a super-super gravity. That means Heterotic strings could be thought of as combing Yang-Mills and super gravity. In all events the spectrum indeed gives the correct result, namely 3968 + 4096 = 8064. In other words No = 8 (496 + 512) = 16 (248 + 256) = (16)(504).
At this point El Naschie makes the following argument: Why stop at E5 and the sum 504? It seems more natural to take the entire exceptional E-line of Lie symmetry groups. In other words we should add the dimensions of E4, E3, E2 and E1. In his extensive work on the exceptional Lie symmetry groups’ hierarchy, El Naschie shows that the sum is 548. Consequently 504 should be replaced with 548. However El Naschie also knows that the 548 is 4 times 137. This is the integer value of the inverse electromagnetic fine structure constant. The exact E-infinity value is however = 137.082039325. In addition El Naschie also knows that due to the main scaling sequence of this ( /2) we must have 16 + k rather than simply 16. This means 16 should be ‘corrected transfinitely’ to 16.18033989. The exact No value is thus No = (16.18033989)(548.3281573) = 8872.135956. This last value may be expressed differently as No = (16 + k) (4) ( ) = (64 + 4k)(137 + ko) where k = 0.18033989 and ko = 0.082039325 are all higher order values of the golden mean. Recalling that 64 is the number of different spin directions and 137 was shown to be the number of elementary particles in an extended standard model, then one could easily conjecture that a return to the classical No = 8064 of Green, Schwarz and Witten may be obtained by replacing 137 with the number of the conventional standard model, namely 126 and 64 + 4k by simply 64. Proceeding this way one finds No = (64)(126) = 8064. This is a delightful result showing the logical consistency of the entire theory.
Anyone suspecting any form of numerology in the intended bad sense of the word should remember that = 137.0820393 and all other transfinitely corrected values harmonize with each and every equation. If one makes a single error in the calculation it is impossible not to notice it immediately after two steps and never, ever more than six steps. It is the well known butterfly effect of chaos manifested in the most irrational number of all irrational numbers, namely the golden mean. The charge of numerology is the cheapest shot possible and comes mainly from those with not so well hidden agendas. These people are well known to all the E-infinity researchers and we need not name them here. They will also be eternalized as infamous in the history books for their non-scientific, disgraceful attacks on innocent people in their literally obscene blogs.
E-infinity Group.
E-infinity communication No. 59.
Super Yang-Mills, super gravity and super strings. Spectroscopy and the impossibility of number coincidence
If the reader is not very familiar with the theory of super strings, it would be advisable to have a text book to one side whilst reading this communication. For convenience we recommend M. Kaku’s book “Introduction to superstrings and M-theory”, Springer, New York (1999).
Let us look at the Heterotic string and consider its spectrum. For instance following pages 384 and 385 of Kaku’s book we have the following: For the left moving sector there are three distinct states. First 480 states equal in number to the kissing of two E8 and also equivalent to their roots number. In addition we have another 16 states. These make together the 496 of E8E8 dimension where each E8 has clearly 248 dimensions. Finally we have another 8 states so that the total number of the left moving section is 496 + 8 = 504. Note that this is exactly the sum of the dimensions of E8 plus E7 plus E6 plus E5 as found by M. El Naschie (248 + 133 + 78 + 45 = 504). Now we look at the right moving sector where we have only 16s states. Following the rule of Fock space of quantum field theory (where Fock space is the generalization of Hilbert space of quantum mechanics to quantum field), then the total number of all states is the multiplication of the left and right sectors. This gives us the famous number of the first level of massless particle-like states of the Heterotic string theory, namely No = (504)(16) = 8064. Now page 385 gives the number of states in a super Yang-Mills theory as 3968. These could be thought of as lifting E8E8 dimensions to super space by taking 8 copies of the 496 dimensions leading to (8)(496) = 3968. On the other hand we know from the spectrum of the theory of super gravity that we have 512 states. This could be thought of as raising the two dimensional average Hausdorff dimension of a quantum particle path to the ten dimensions of super strings. Using E-infinity bijection, this means we have 2 to the power of 10 minus 1. This is 2 to the power of 9 which is equal 512. This is only 16 more than 496 and 8 more than 504. Let us lift these 512 to super space like we did with super Yang-Mills. Proceeding in this way we find (512)(8) = 4096. The next step taken by Mohamed El Naschie is to consider No to be the sum of super Yang-Mills and a super-super gravity. That means Heterotic strings could be thought of as combing Yang-Mills and super gravity. In all events the spectrum indeed gives the correct result, namely 3968 + 4096 = 8064. In other words No = 8 (496 + 512) = 16 (248 + 256) = (16)(504).
At this point El Naschie makes the following argument: Why stop at E5 and the sum 504? It seems more natural to take the entire exceptional E-line of Lie symmetry groups. In other words we should add the dimensions of E4, E3, E2 and E1. In his extensive work on the exceptional Lie symmetry groups’ hierarchy, El Naschie shows that the sum is 548. Consequently 504 should be replaced with 548. However El Naschie also knows that the 548 is 4 times 137. This is the integer value of the inverse electromagnetic fine structure constant. The exact E-infinity value is however = 137.082039325. In addition El Naschie also knows that due to the main scaling sequence of this ( /2) we must have 16 + k rather than simply 16. This means 16 should be ‘corrected transfinitely’ to 16.18033989. The exact No value is thus No = (16.18033989)(548.3281573) = 8872.135956. This last value may be expressed differently as No = (16 + k) (4) ( ) = (64 + 4k)(137 + ko) where k = 0.18033989 and ko = 0.082039325 are all higher order values of the golden mean. Recalling that 64 is the number of different spin directions and 137 was shown to be the number of elementary particles in an extended standard model, then one could easily conjecture that a return to the classical No = 8064 of Green, Schwarz and Witten may be obtained by replacing 137 with the number of the conventional standard model, namely 126 and 64 + 4k by simply 64. Proceeding this way one finds No = (64)(126) = 8064. This is a delightful result showing the logical consistency of the entire theory.
Anyone suspecting any form of numerology in the intended bad sense of the word should remember that = 137.0820393 and all other transfinitely corrected values harmonize with each and every equation. If one makes a single error in the calculation it is impossible not to notice it immediately after two steps and never, ever more than six steps. It is the well known butterfly effect of chaos manifested in the most irrational number of all irrational numbers, namely the golden mean. The charge of numerology is the cheapest shot possible and comes mainly from those with not so well hidden agendas. These people are well known to all the E-infinity researchers and we need not name them here. They will also be eternalized as infamous in the history books for their non-scientific, disgraceful attacks on innocent people in their literally obscene blogs.
E-infinity Group.
You call it numerology, I call you mathematical illiteratology
14th December, 2010.
E-infinity communication No. 58
You call it numerology, I call you mathematical illiteratology
Let us take one really obvious example for what we would like to illustrate here, namely how intricate the role played by number theory in physics could be and how hasty judgment could put one in an embarrassing situation. Academics in general are judgmental but mathematical physicists should not be no matter how senior they are because at a certain very deep level mostly connected to quantum mechanics rather than general relativity there is no real difference between ‘pure mathematics’ and ‘pure physics’. At this deep level as Prof. El Naschie stressed in one of his lectures, one should not relay on ‘common sense’ which he calls ‘the logic of savages’ following Lord B. Russler. The only thing one can rely on is the most stringent form of logic as in transfinite set theory and K-theory or inspiration from God or such higher instance. Such inspiration is hard to come by and cannot be depended upon in a regular way. The example we will give here is taken from Table No. 6, page 658 of the paper “On a class of general theories for high energy particle physics”, published in CS&F, (2006), pp. 649-668 by M.S. El Naschie. The table is labeled “The number 42 in mathematics and physics – A short survey”.
The importance of 42 is that it is 10 times the expectation Hausdorff dimension of E-infinity spacetime when we take the integer part only and is also equal to the inverse coupling constant of non-super symmetric grand unification of all fundamental forces excluding gravity. The first number of the list is the third Bernoulli number. I remember that one notable scientist protested against such unmotivated connection and obvious numerology. Others went as far as calling on a French scientist very close to Mohamed El Naschie and becoming impolite about this playing with numbers. One in particular who is as self confident as he is ignorant called it ‘alchemy’ although modern physics has long shown that the dream of alchemists of changing copper into gold is in principle correct because the basic building blocks are the same, namely elementary particles. Deep mathematical contemplation coupled with a great deal of knowledge of mathematical and physical facts shows that a relation between all Bernoulli numbers and quantum mechanics is by no means outlandish nor numerology or number coincidence. For instance the function guessed initially by Max Planck for his black body radiation which marked the beginning of quantum theory was a complex function involving many hidden things such as the Bernoulli numbers. That is how Planck arrived at a constant which he later called h and which we know as the Planck constant in order to obtain dimensionless numbers from physical quantities with dimension. Even more general than that, we have to recall that the Bernoulli numbers originally come from the function of integers and are thus related to K-theory topics such as the Riemann-Roch index used by C. Castro and M.S. El Naschie in E-infinity theory as well as the Atiyeh-Singer index used in topological quantum field theory of Witten. Finally for this communication, they are related to the Gauss-Bonnet theorem which El Naschie was able to generalize for a fractal Cantorian geometry and find the curvature of spacetime as a topological invariant exactly equal to the square root of the sum of all two and three Stein spaces which is in turn exactly equal to (5) multiplied with the inverse of the electromagnetic fine structure constant 137. Taking the square root of 685.4101966 we find the said curvature, namely 26 + k = 26.18033989 which is the inverse coupling constant of the super symmetric unification of all fundamental forces including gravity. Consequently 685.4101966 may be seen as a numerical potential or a Lagrangian and using Weyl-Suslin scaling as a substitute for calculus, a great deal of information of physical relevance may be obtained with an unheard of simplicity and elegance.
So in the end, the joke is on those who think they are very witty and spend most of their time making silly jokes on other people instead of spending their time on something useful. For this reason, and although he can be very witty when he wants to be, El Naschie as well as most of his colleagues and friends keep a great distance from making fun of other people’s work or personality. Trying to be witty most of the time is a clear sign of poverty of intellect and personality.
E-infinity Group
Relevant literature:
1. Curvature, Lagrangian and holonomy of Cantorian-fractal spacetime, Chaos, Solitons & Fractals, Vol. 41(4), (2009), p. 2163-2167.
2. Deriving the largest expected number of elementary particles in the standard model from the maximal compact subgroup H of the exceptional Lie group E7(-5), Chaos, Solitons & Fractals, Vol. 38(4), (2008), p. 956-961.
3. Derivation of the Euler characteristic and the curvature of Cantorian-fractal spacetime using Nash Euclidean embedding and the universal Menger sponge, Chaos, Solitons & Fractals, Vol. 41(5), (2009), p. 2394-2398.
E-infinity communication No. 58
You call it numerology, I call you mathematical illiteratology
Let us take one really obvious example for what we would like to illustrate here, namely how intricate the role played by number theory in physics could be and how hasty judgment could put one in an embarrassing situation. Academics in general are judgmental but mathematical physicists should not be no matter how senior they are because at a certain very deep level mostly connected to quantum mechanics rather than general relativity there is no real difference between ‘pure mathematics’ and ‘pure physics’. At this deep level as Prof. El Naschie stressed in one of his lectures, one should not relay on ‘common sense’ which he calls ‘the logic of savages’ following Lord B. Russler. The only thing one can rely on is the most stringent form of logic as in transfinite set theory and K-theory or inspiration from God or such higher instance. Such inspiration is hard to come by and cannot be depended upon in a regular way. The example we will give here is taken from Table No. 6, page 658 of the paper “On a class of general theories for high energy particle physics”, published in CS&F, (2006), pp. 649-668 by M.S. El Naschie. The table is labeled “The number 42 in mathematics and physics – A short survey”.
The importance of 42 is that it is 10 times the expectation Hausdorff dimension of E-infinity spacetime when we take the integer part only and is also equal to the inverse coupling constant of non-super symmetric grand unification of all fundamental forces excluding gravity. The first number of the list is the third Bernoulli number. I remember that one notable scientist protested against such unmotivated connection and obvious numerology. Others went as far as calling on a French scientist very close to Mohamed El Naschie and becoming impolite about this playing with numbers. One in particular who is as self confident as he is ignorant called it ‘alchemy’ although modern physics has long shown that the dream of alchemists of changing copper into gold is in principle correct because the basic building blocks are the same, namely elementary particles. Deep mathematical contemplation coupled with a great deal of knowledge of mathematical and physical facts shows that a relation between all Bernoulli numbers and quantum mechanics is by no means outlandish nor numerology or number coincidence. For instance the function guessed initially by Max Planck for his black body radiation which marked the beginning of quantum theory was a complex function involving many hidden things such as the Bernoulli numbers. That is how Planck arrived at a constant which he later called h and which we know as the Planck constant in order to obtain dimensionless numbers from physical quantities with dimension. Even more general than that, we have to recall that the Bernoulli numbers originally come from the function of integers and are thus related to K-theory topics such as the Riemann-Roch index used by C. Castro and M.S. El Naschie in E-infinity theory as well as the Atiyeh-Singer index used in topological quantum field theory of Witten. Finally for this communication, they are related to the Gauss-Bonnet theorem which El Naschie was able to generalize for a fractal Cantorian geometry and find the curvature of spacetime as a topological invariant exactly equal to the square root of the sum of all two and three Stein spaces which is in turn exactly equal to (5) multiplied with the inverse of the electromagnetic fine structure constant 137. Taking the square root of 685.4101966 we find the said curvature, namely 26 + k = 26.18033989 which is the inverse coupling constant of the super symmetric unification of all fundamental forces including gravity. Consequently 685.4101966 may be seen as a numerical potential or a Lagrangian and using Weyl-Suslin scaling as a substitute for calculus, a great deal of information of physical relevance may be obtained with an unheard of simplicity and elegance.
So in the end, the joke is on those who think they are very witty and spend most of their time making silly jokes on other people instead of spending their time on something useful. For this reason, and although he can be very witty when he wants to be, El Naschie as well as most of his colleagues and friends keep a great distance from making fun of other people’s work or personality. Trying to be witty most of the time is a clear sign of poverty of intellect and personality.
E-infinity Group
Relevant literature:
1. Curvature, Lagrangian and holonomy of Cantorian-fractal spacetime, Chaos, Solitons & Fractals, Vol. 41(4), (2009), p. 2163-2167.
2. Deriving the largest expected number of elementary particles in the standard model from the maximal compact subgroup H of the exceptional Lie group E7(-5), Chaos, Solitons & Fractals, Vol. 38(4), (2008), p. 956-961.
3. Derivation of the Euler characteristic and the curvature of Cantorian-fractal spacetime using Nash Euclidean embedding and the universal Menger sponge, Chaos, Solitons & Fractals, Vol. 41(5), (2009), p. 2394-2398.
Dec 13, 2010
Implications of Sarkovskii and El Naschie’s number theoretical theorem for physics
13th December, 2010.
E-infinity communication No. 57
Implications of Sarkovskii and El Naschie’s number theoretical theorem for physics
In 1975 the American J. York (one of the members of the Honorary Editorial Board of Chaos, Solitons & Fractals) and his student Li wrote a paper entitled “Period Three Implies Chaos” which was one of the most important and influential papers which helped establish chaos as a field of research in engineering and physics. However the same theorem of J. York who gave the science of chaos its name was already discovered in 1964 in number theory by the Russian mathematician Sarkovskii as explained for instance in the excellent book of H.G. Schuster “Deterministic Chaos”, published by VCH, Weinheim, Germany (1989). This shows the importance of number theory in physics which should be understood as the foundation of mathematics just like set theory and therefore the foundation of physics. It should not be understood as playing with the golden mean or number acrobatics (there is nothing called number acrobatics at all) or numerology as some (truly silly) people sometimes say. There is another very important example of a theorem in number theory given by El Naschie in 1998 with deep physical connections to much of the work on spacetime physics. The paper in question is “Four as the expectation value of the set of all positive integers and the geometry of four manifolds”, published in CS&F, Vol 9(9), (1998), pp. 1625-1629. The paper may be found in the E-infinity free of charge published papers (open access) or Elsevier’s Science Direct.
E-infinity Group.
E-infinity communication No. 57
Implications of Sarkovskii and El Naschie’s number theoretical theorem for physics
In 1975 the American J. York (one of the members of the Honorary Editorial Board of Chaos, Solitons & Fractals) and his student Li wrote a paper entitled “Period Three Implies Chaos” which was one of the most important and influential papers which helped establish chaos as a field of research in engineering and physics. However the same theorem of J. York who gave the science of chaos its name was already discovered in 1964 in number theory by the Russian mathematician Sarkovskii as explained for instance in the excellent book of H.G. Schuster “Deterministic Chaos”, published by VCH, Weinheim, Germany (1989). This shows the importance of number theory in physics which should be understood as the foundation of mathematics just like set theory and therefore the foundation of physics. It should not be understood as playing with the golden mean or number acrobatics (there is nothing called number acrobatics at all) or numerology as some (truly silly) people sometimes say. There is another very important example of a theorem in number theory given by El Naschie in 1998 with deep physical connections to much of the work on spacetime physics. The paper in question is “Four as the expectation value of the set of all positive integers and the geometry of four manifolds”, published in CS&F, Vol 9(9), (1998), pp. 1625-1629. The paper may be found in the E-infinity free of charge published papers (open access) or Elsevier’s Science Direct.
E-infinity Group.
Lee Smolin’s trouble with physics and fractal spacetime.
13th December, 2010.
E-infinity communication No. 56
Lee Smolin’s trouble with physics and fractal spacetime.
As things stand at this point in time, Prof. Lee Smolin is probably the most influential scientist in theoretical physics. Smolin has a talent for both serious fundamental science as well a popular science writing in a beyond measure manner. A handful of people could be compared to Smolin’s dual role in science such as R. Penrose but then he is (wrongly) considered by physicists to be only a mathematician. His latest book which is yet again a best seller is not only worth reading for those working in E-infinity, it is a must. Another reason for discussing his book is the involvement of Prof. Lee Smolin with fractals in high energy physics in general as well as the involvement of many of his associate in the Perimeter Inst. and elsewhere with the subject of fractal spacetime.
There are countless points in Lee Smolin’s book which we would like to comment upon and therefore we have to be very choosy and concentrate on what is useful and important to E-infinity. First and foremost there is no mention what so ever of fractal spacetime in the book of Smolin. This is surprising given his documented interest in this subject. In fact there is no mention at all of the word fractal in the entire book. This could be understandable in a mathematical book where pure mathematicians exchange the word fractal with other words like foliation for instance. However Smolin’s book is a popular book and fractal is the most popular word for foliation and continuous geometry or noncommutative geometry. Even the book of Field Medalist A. Connes who is held by Smolin and our group in high esteem occasionally speaks of fractals. We very much hope that no misconception or misunderstanding has prejudiced Prof. Lee Smolin against fractal spacetime because our group needs and hopes for his scientific weight and support.
On page 313 Smolin considers the category of researcher referred to by him as seers and considers their opinion about background-independent approach. E-infinity was developed using the quantum sets methodology of D. Finkelstein. On page 322 Smolin says that D. Finkelstein is a deep thinker (a seer) who does physics differently from anyone else. He talks of the problems that people like Finkelstein would face today if they tried to get funding or a professorship like Finkelstein did. He says it would be impossible today. We in E-infinity do not care about funding or professorships. However we do care that Lee Smolin extends to our work the same tolerance he extended to Finkelstein so that we can at least see that one person, Smolin, is still conducting science ethics and science policy in the same way as in the golden age of science and physics.
On pages 245-247 Smolin gives an apt description of A. Connes and his noncommutative geometry with which we agree completely. There is only one point here. Alain Connes works is a de facto fractal universe. His work is exactly like that of Mohamed El Naschie anchored in von Neumann’s continuous geometry, K-theory and E-infinity rings and groups. On page 313 Smolin mentions the work of Fay Dowker and R. Sorkin. Well both work with partially ordered sets. In E-infinity we work with random Cantor sets. In our case we do not even need a computer to reach the same excellent results which R. Loll and J. Ambjorn reached. The work of both of these two excellent scientists is considered by L. Smolin on pages 242 and 243. Both Loll and Ambjorn worked indirectly in fractal spacetime and published many papers on that including one in Scientific American which we hoped and wished they had acknowledged our work but they did not. This is sad but everyone could overlook important papers in the heat of publication fever. It is not tragic. However and this is a most polite and respectful hint to Prof. Smolin personally……. Is it not more often the case that those who made it big often forget and do not care about those who did not yet make it that big? We remind Prof. L. Smolin of the wonderful story of A. Connes on page 275 of his wonderful book. In all fairness, with this we rest our case, at least here.
E-infinity Group.
P.S.
Smolin’s book is published by Penguin, England (2006). Price £25.- with £5 discount.
E-infinity communication No. 56
Lee Smolin’s trouble with physics and fractal spacetime.
As things stand at this point in time, Prof. Lee Smolin is probably the most influential scientist in theoretical physics. Smolin has a talent for both serious fundamental science as well a popular science writing in a beyond measure manner. A handful of people could be compared to Smolin’s dual role in science such as R. Penrose but then he is (wrongly) considered by physicists to be only a mathematician. His latest book which is yet again a best seller is not only worth reading for those working in E-infinity, it is a must. Another reason for discussing his book is the involvement of Prof. Lee Smolin with fractals in high energy physics in general as well as the involvement of many of his associate in the Perimeter Inst. and elsewhere with the subject of fractal spacetime.
There are countless points in Lee Smolin’s book which we would like to comment upon and therefore we have to be very choosy and concentrate on what is useful and important to E-infinity. First and foremost there is no mention what so ever of fractal spacetime in the book of Smolin. This is surprising given his documented interest in this subject. In fact there is no mention at all of the word fractal in the entire book. This could be understandable in a mathematical book where pure mathematicians exchange the word fractal with other words like foliation for instance. However Smolin’s book is a popular book and fractal is the most popular word for foliation and continuous geometry or noncommutative geometry. Even the book of Field Medalist A. Connes who is held by Smolin and our group in high esteem occasionally speaks of fractals. We very much hope that no misconception or misunderstanding has prejudiced Prof. Lee Smolin against fractal spacetime because our group needs and hopes for his scientific weight and support.
On page 313 Smolin considers the category of researcher referred to by him as seers and considers their opinion about background-independent approach. E-infinity was developed using the quantum sets methodology of D. Finkelstein. On page 322 Smolin says that D. Finkelstein is a deep thinker (a seer) who does physics differently from anyone else. He talks of the problems that people like Finkelstein would face today if they tried to get funding or a professorship like Finkelstein did. He says it would be impossible today. We in E-infinity do not care about funding or professorships. However we do care that Lee Smolin extends to our work the same tolerance he extended to Finkelstein so that we can at least see that one person, Smolin, is still conducting science ethics and science policy in the same way as in the golden age of science and physics.
On pages 245-247 Smolin gives an apt description of A. Connes and his noncommutative geometry with which we agree completely. There is only one point here. Alain Connes works is a de facto fractal universe. His work is exactly like that of Mohamed El Naschie anchored in von Neumann’s continuous geometry, K-theory and E-infinity rings and groups. On page 313 Smolin mentions the work of Fay Dowker and R. Sorkin. Well both work with partially ordered sets. In E-infinity we work with random Cantor sets. In our case we do not even need a computer to reach the same excellent results which R. Loll and J. Ambjorn reached. The work of both of these two excellent scientists is considered by L. Smolin on pages 242 and 243. Both Loll and Ambjorn worked indirectly in fractal spacetime and published many papers on that including one in Scientific American which we hoped and wished they had acknowledged our work but they did not. This is sad but everyone could overlook important papers in the heat of publication fever. It is not tragic. However and this is a most polite and respectful hint to Prof. Smolin personally……. Is it not more often the case that those who made it big often forget and do not care about those who did not yet make it that big? We remind Prof. L. Smolin of the wonderful story of A. Connes on page 275 of his wonderful book. In all fairness, with this we rest our case, at least here.
E-infinity Group.
P.S.
Smolin’s book is published by Penguin, England (2006). Price £25.- with £5 discount.
Crystallographic groups and Heterotic strings in E-infinity.
13th December, 2010.
E-infinity communication No. 55
Crystallographic groups and Heterotic strings in E-infinity.
The following is a notice on a very short paper published in CS&F, 42 (2008), pp. 2282-2284. Although very short and at first glance may appear unassuming, it is extremely important to grasp the idea behind it. It is well known that there are 17 different ‘tiling’ groups in two dimensions corresponding to 17 two and three Stein spaces with total dimensions of 685. In three dimensions there are 230 groups and 219 from that correspond exactly to the 17 groups in two dimensions. The question now is how many dimensions belong to these 219? In this paper El Naschie answers this question. The answer is 8872 which is exactly the number of states of the first massless level of particle-like states of the Heterotic string theory. It is extremely interesting to see that these seemingly abstract 8872 states have such a strong and deep connection with something as real as 3D crystals.
E-infinity group.
E-infinity communication No. 55
Crystallographic groups and Heterotic strings in E-infinity.
The following is a notice on a very short paper published in CS&F, 42 (2008), pp. 2282-2284. Although very short and at first glance may appear unassuming, it is extremely important to grasp the idea behind it. It is well known that there are 17 different ‘tiling’ groups in two dimensions corresponding to 17 two and three Stein spaces with total dimensions of 685. In three dimensions there are 230 groups and 219 from that correspond exactly to the 17 groups in two dimensions. The question now is how many dimensions belong to these 219? In this paper El Naschie answers this question. The answer is 8872 which is exactly the number of states of the first massless level of particle-like states of the Heterotic string theory. It is extremely interesting to see that these seemingly abstract 8872 states have such a strong and deep connection with something as real as 3D crystals.
E-infinity group.
Suslin set theoretical foundation of E-infinity theory
12th December, 2010.
E-infinity communication No. 54
Suslin set theoretical foundation of E-infinity theory
Descriptive set theory which forms an important basis and deep theoretical underpinning of E-infinity theory was the joint discovery of Nikolai Luzin, the real founder of the famous Moscow School of Mathematics and a young brilliant student of his, Mikhail Suslin in the summer of 1916. In fact there is a birth certificate for this theory with both Luzin and Suslin as parents and no one less than the famous Polish mathematician Waclaw Sierpinski (the father of the Sierpinski fractal triangle or gasket) was a witness. The certificate is dated the afternoon of October 1916 but no day is given. There is a serious reason for this joke. It seems that P.S. Alexandrov, the great Russian topologist whose work on wild topology combining topology with Cantor sets (which is crucial to El Naschie’s E-infinity) wanted to claim Luzin and Suslin’s discovery for himself. Alexandrov was a great mathematician and was actually a very close friend of P. Urysohn whose dimensional theory is about the most important thing in E-infinity theory. Never the less it seems Alexandrov had a character deficiency. Dishonesty among the greatest of scientists seems to be as old as the history of science and it is still with us today. There are far worse plagiarists than this and far more famous scientific disputes engulfing as famous people as Newton versus Leipnitz, Einstein versus Poincaré as well as Lorenz and even D. Hilbert and so on and so on. This is extremely sad but it is fact. Great minds do not always necessarily imply great characters. Scientists are only human and sometimes very human.
A good place to start studying descriptive set theory is to start with trees and trees on products. After that we have to concentrate on polish spaces. The most important examples of polish spaces for E-infinity are the n-dimensional cube studied by M. El Naschie and the Hilbert cube studied by Ji-Huan He. After that comes the Cantor space used in several papers by El Naschie on E-infinity and descriptive set theory, all published in CS&F. Finally we should mention the Bair space. In E-infinity ordinary differentiation and integration are replaced by Weyl scaling while Suslin scaling is used as a fundamental operation on sets. These things sound more difficult than they are and in praxis the situation is much simpler when we are dealing with a concrete problem. Finally we should mention that E-infinity may be regarded as a space made of a Borel set. El Naschie repeatedly mentioned that this idea was given without any mathematics at all by the very great theoretical physicist J.A. Wheeler. El Naschie did not do more than adding the mathematics using random Cantor sets with golden mean Hausdorff dimension.
E-infinity group.
E-infinity communication No. 54
Suslin set theoretical foundation of E-infinity theory
Descriptive set theory which forms an important basis and deep theoretical underpinning of E-infinity theory was the joint discovery of Nikolai Luzin, the real founder of the famous Moscow School of Mathematics and a young brilliant student of his, Mikhail Suslin in the summer of 1916. In fact there is a birth certificate for this theory with both Luzin and Suslin as parents and no one less than the famous Polish mathematician Waclaw Sierpinski (the father of the Sierpinski fractal triangle or gasket) was a witness. The certificate is dated the afternoon of October 1916 but no day is given. There is a serious reason for this joke. It seems that P.S. Alexandrov, the great Russian topologist whose work on wild topology combining topology with Cantor sets (which is crucial to El Naschie’s E-infinity) wanted to claim Luzin and Suslin’s discovery for himself. Alexandrov was a great mathematician and was actually a very close friend of P. Urysohn whose dimensional theory is about the most important thing in E-infinity theory. Never the less it seems Alexandrov had a character deficiency. Dishonesty among the greatest of scientists seems to be as old as the history of science and it is still with us today. There are far worse plagiarists than this and far more famous scientific disputes engulfing as famous people as Newton versus Leipnitz, Einstein versus Poincaré as well as Lorenz and even D. Hilbert and so on and so on. This is extremely sad but it is fact. Great minds do not always necessarily imply great characters. Scientists are only human and sometimes very human.
A good place to start studying descriptive set theory is to start with trees and trees on products. After that we have to concentrate on polish spaces. The most important examples of polish spaces for E-infinity are the n-dimensional cube studied by M. El Naschie and the Hilbert cube studied by Ji-Huan He. After that comes the Cantor space used in several papers by El Naschie on E-infinity and descriptive set theory, all published in CS&F. Finally we should mention the Bair space. In E-infinity ordinary differentiation and integration are replaced by Weyl scaling while Suslin scaling is used as a fundamental operation on sets. These things sound more difficult than they are and in praxis the situation is much simpler when we are dealing with a concrete problem. Finally we should mention that E-infinity may be regarded as a space made of a Borel set. El Naschie repeatedly mentioned that this idea was given without any mathematics at all by the very great theoretical physicist J.A. Wheeler. El Naschie did not do more than adding the mathematics using random Cantor sets with golden mean Hausdorff dimension.
E-infinity group.
The first E8 proposal for complete unification of all fundamental forces
12th December, 2010.
E-infinity communication No. 53
The first E8 proposal for complete unification of all fundamental forces
It is abundantly clear from a quick study of the large body of literature published by Mohamed El Naschie, Ji-Huan He, L. Marek-Crnjac and many of those working on E-infinity theory that the first implicit proposal for unification using E8 exceptional Lie group was published around 2005 or even earlier. The exact date needs time to nail down more accurately because of the large volume of papers by the prolific author. In a paper entitled “Determining the number of Hiss particles starting from general relativity and various other field theories” which was published in CS&F, 23, (2005), pp. 711-725 El Naschie set outs to explain the idea of unification in paragraph 11 which is labeled Discussion and Conclusion on pages 724 and 725.
In what follows we give a summary of what was said on this subject. At that time El Naschie chose to call his idea, conservation of the dimensional symmetry. He wrote that in what was a rudimentary form simply equating Dim E8E8 to N(R(8)) plus alpha bar naught of electromagnetism, that is to say 137 plus N(R(4)). With R(8) he means the number of independent components of a Riemannian tensor in eight dimensions. These are 336 before compactification and 338.885438, nearly equal 339 after compactification. They are the net difference between the compactified dimensions of E8E8, namely 496 ̶ k2 = 495.9674775 and the compactified dimensions of E6E6 which are equal to 156 + 6k = 157.0820393 when symmetry is broken and E8E8 goes to E6E6 of the exceptional E-line of those important Lie symmetry groups. It is historically interesting to note that at this time El Naschie did not use ‘t Hooft’s holographic boundary where all the 339 particles of the 496 bulk lives. Thus El Naschie used a kind of super gravity argument at that time. He then adds the usual 20 representing the independent components of the Einstein gravity. There are exactly 20 for the case of four dimensions. Thus he has in essence equated the 496 dimensions of the E8E8 bulk with N(R(8)) plus alpha bar naught of electromagnetism plus N(R(4)) = 339 + 137 + 20. This is exactly equal 496 when we consider the integer approximation. In later publications the said 20 were interpreted as the number of isometries of pure gravity in 8 dimensions and the 339 as the number of elementary particles living on the holographic boundary. This is a real unification attempt using the E8E8 group and is many years before any similar attempt by anyone and definitely years before G. Lisi. It is also interesting to note that calculation without compactification leading to 336 instead of 339 particles was interpreted in this particular article as suggesting the existence of 3 Higgs particles H(+), H( ̶ ) and H(0) to make up the deficit of 339 ̶ 336 = 3. El Naschie for some reason never repeated this interesting interpretation again as far as we are aware.
E-infinity communication No. 53
The first E8 proposal for complete unification of all fundamental forces
It is abundantly clear from a quick study of the large body of literature published by Mohamed El Naschie, Ji-Huan He, L. Marek-Crnjac and many of those working on E-infinity theory that the first implicit proposal for unification using E8 exceptional Lie group was published around 2005 or even earlier. The exact date needs time to nail down more accurately because of the large volume of papers by the prolific author. In a paper entitled “Determining the number of Hiss particles starting from general relativity and various other field theories” which was published in CS&F, 23, (2005), pp. 711-725 El Naschie set outs to explain the idea of unification in paragraph 11 which is labeled Discussion and Conclusion on pages 724 and 725.
In what follows we give a summary of what was said on this subject. At that time El Naschie chose to call his idea, conservation of the dimensional symmetry. He wrote that in what was a rudimentary form simply equating Dim E8E8 to N(R(8)) plus alpha bar naught of electromagnetism, that is to say 137 plus N(R(4)). With R(8) he means the number of independent components of a Riemannian tensor in eight dimensions. These are 336 before compactification and 338.885438, nearly equal 339 after compactification. They are the net difference between the compactified dimensions of E8E8, namely 496 ̶ k2 = 495.9674775 and the compactified dimensions of E6E6 which are equal to 156 + 6k = 157.0820393 when symmetry is broken and E8E8 goes to E6E6 of the exceptional E-line of those important Lie symmetry groups. It is historically interesting to note that at this time El Naschie did not use ‘t Hooft’s holographic boundary where all the 339 particles of the 496 bulk lives. Thus El Naschie used a kind of super gravity argument at that time. He then adds the usual 20 representing the independent components of the Einstein gravity. There are exactly 20 for the case of four dimensions. Thus he has in essence equated the 496 dimensions of the E8E8 bulk with N(R(8)) plus alpha bar naught of electromagnetism plus N(R(4)) = 339 + 137 + 20. This is exactly equal 496 when we consider the integer approximation. In later publications the said 20 were interpreted as the number of isometries of pure gravity in 8 dimensions and the 339 as the number of elementary particles living on the holographic boundary. This is a real unification attempt using the E8E8 group and is many years before any similar attempt by anyone and definitely years before G. Lisi. It is also interesting to note that calculation without compactification leading to 336 instead of 339 particles was interpreted in this particular article as suggesting the existence of 3 Higgs particles H(+), H( ̶ ) and H(0) to make up the deficit of 339 ̶ 336 = 3. El Naschie for some reason never repeated this interesting interpretation again as far as we are aware.
Peter Woit “Not Even Wrong” is certainly right and wrong: An interview with Mohamed El Naschie
12th December, 2010.
E-infinity communication No. 52
Peter Woit “Not Even Wrong” is certainly right and wrong: An interview with Mohamed El Naschie
The following is a summary of an interview with Prof. Mohamed El Naschie conducted by Shayma, a journalist based in Cairo which will be published in full length in Arabic.
Dr. Woit’s book (published by Jonathan Cape, London (2006)) is an excellent popular account voicing strong dissent against the dominance of string theory. Never the less we think super strings are a very useful theory which has made and it still making important contributions. Dr. Woit made several important remarks and suggestions towards improving the standard model of high energy physics and reaching the number one goal of theoretical physics at present which is quantum gravity, i.e. unifying quantum mechanics and general relativity in one coherent and consistent theory. Other important aspects of modern quantum physics were omitted all together such as fractal spacetime and fractal gravity or addressed briefly and superficially like noncommutative geometry. Let us commence with the positive aspects of this important book which we believe should be read by every scientist working on modern physics.
The crucial point which Woit makes is related to the role of symmetry in quantum mechanics. He spells it out already in the introduction at the end of page 7 where he writes “The failure of the superstring theory programme can be traced to its lack of any fundamental new symmetry principle”. He then continues on page 8 by writing “…. Advances are only likely to come about if theoretist turn their attention away from this failed programme and (direct) it towards the difficult task of better understanding the symmetries of the natural world.” It is important to carefully analyze these statements of Woit summarizing all the wisdom of his book. To be sure, superstrings are making use of a very important symmetry principle which is new in physics although well known for a long time in mathematics, namely the exceptional Lie symmetry group E8. In fact superstrings take E8 E8 with dimension equal (2)(dim E8) = (20)(248) – 496. Thus this is not what Woit means. What we think he means is a completely new symmetry better suited to quantum gravity. Maybe he also means a larger group which is richer than E8. However E8 is the largest except for the monster group which is hardly understood as physics in general. If that is what Woit means then he should become a great friend of E-infinity theory and of Prof. El Naschie, that is if he took the time to read our work carefully and without prejudice. In E-infinity El Naschie, He, Crnjac and Iovane introduced much larger Lie symmetry groups than E8 by summing over all Lie symmetry groups. For instance the sum over the E-line gives a ‘fuzzy’ group, i.e. a ‘fractal’ group with 548 instead of only 496 as integer part of the dimension. During the 80’s and 90’s of the last century in what is now called the European Journal of Physics, El Naschie wrote several papers on ‘average’ symmetries. This symmetry is the symmetry of chaos and avoids the anomalies stemming from the clashes between external and internal symmetries. All that is discussed in many papers by El Naschie, particularly those related to knot theory, wild topology, von Neumann’s continuous geometry, K-theory and especially A. Connes’s noncommutative geometry and even the monster group. Woit repeats his important point regarding symmetry in the conclusion on page 266 where he write “I see as an important lesson that each generation of physicist since the advent of quantum mechanics seems to need to learn anew. This lesson is the importance of symmetry principles expressed in mathematical language of group representation theory…… The underlying source of the problems of superstring theory is that the theory is not built on a fundamental symmetry principle or expressed within the language of representation theory.” We in E-infinity theory fully endorse this point of view and believe that our reformulation of Feynman’s summing over paths theory to summing over dimensions and summing over Lie groups as well as two and three Stein spaces is the correct strategy and as such is in full harmony with the views expressed in Woit’s book.
Now we come to the point upon which we do not agree at all with Woit. E-infinity is a K-theory for spacetime. In less formal mathematical language we mainly use semi groups (being summed over groups). In popular language of computer science, we use fractal automaton. Fractal sets and K-theory are what noncommutative geometry is all about. Noncommutative geometry is not as popular as string theory. However this is not the mistake of noncommutative geometry as Woit very well knows. For this reason it is not adequate to give only a few lines in his book to noncommutative geometry. On page 256 Woit write “One other speculative research programme that deserves mention goes under the name of non-commutative geometry….”. We in E-infinity profoundly disagree. If you call noncommutative geometry speculative, what do you call superstrings with 10 spacetime dimensions of which we have only seen 3 and God knows what the fourth or the fifth could be. A. Connes is the most important mathematician alive working in physics besides Sir. R. Penrose. We are astonished about this evaluation of Woit for noncommutative geometry for another reason. In his Acknowledgement Dr. Woit thanks Sr. R. Penrose for critical help. But one of the most important and most famous contributions of Penrose is his well known Penrose tiling. This tiling is the best known example of a noncommutative space based on K-theory and it is the prototype of E-infinity space. On piece of interesting information mentioned in Woit’s book which most of us did not know is that E. Witten has no degree, not even a Bachelors in physics. He was a journalist but his father worked in relativity and was a professor. This speaks of course for Witten. However some silly people hold it against El Naschie that he is a structural engineer and has no degree in physics.
We sincerely hope there will be a new edition of Woit’s book and that noncommutative geometry, fractal spacetime and Penrose tiling will be given more attention. There are many papers published on Elsevier’s Science Direct in CS&F regarding Penrose tiling as an example for E-infinity and noncommutative geometry of which we recommend the following:
1. M.S. El Naschie: von Neumann geometry and E-infinity quantum spacetime. CS&F, Vol. 9(12), (1998), pp. 2023-2030.
2. M.S. El Naschie: Penrose universe and Cantorian spacetime as a model for noncommutative quantum geometry. CS&F, Vol. 9(6), (1998), pp. 931-933.
3. M.S. El Naschie: Penrose tiling, semi conduction and Cantorian 1/fa spectra in four and five dimensions. CS&F, Vol. 3(4), (1993), pp. 498-491.
In addition an important paper on average symmetry is
4. M.S. El Naschie: Average exceptional Lie and Coxeter group hierarchies with special reference to the standard model of high energy particle physics. CS&F, 37, (2008), pp. 662-668.
E-infinity communication No. 52
Peter Woit “Not Even Wrong” is certainly right and wrong: An interview with Mohamed El Naschie
The following is a summary of an interview with Prof. Mohamed El Naschie conducted by Shayma, a journalist based in Cairo which will be published in full length in Arabic.
Dr. Woit’s book (published by Jonathan Cape, London (2006)) is an excellent popular account voicing strong dissent against the dominance of string theory. Never the less we think super strings are a very useful theory which has made and it still making important contributions. Dr. Woit made several important remarks and suggestions towards improving the standard model of high energy physics and reaching the number one goal of theoretical physics at present which is quantum gravity, i.e. unifying quantum mechanics and general relativity in one coherent and consistent theory. Other important aspects of modern quantum physics were omitted all together such as fractal spacetime and fractal gravity or addressed briefly and superficially like noncommutative geometry. Let us commence with the positive aspects of this important book which we believe should be read by every scientist working on modern physics.
The crucial point which Woit makes is related to the role of symmetry in quantum mechanics. He spells it out already in the introduction at the end of page 7 where he writes “The failure of the superstring theory programme can be traced to its lack of any fundamental new symmetry principle”. He then continues on page 8 by writing “…. Advances are only likely to come about if theoretist turn their attention away from this failed programme and (direct) it towards the difficult task of better understanding the symmetries of the natural world.” It is important to carefully analyze these statements of Woit summarizing all the wisdom of his book. To be sure, superstrings are making use of a very important symmetry principle which is new in physics although well known for a long time in mathematics, namely the exceptional Lie symmetry group E8. In fact superstrings take E8 E8 with dimension equal (2)(dim E8) = (20)(248) – 496. Thus this is not what Woit means. What we think he means is a completely new symmetry better suited to quantum gravity. Maybe he also means a larger group which is richer than E8. However E8 is the largest except for the monster group which is hardly understood as physics in general. If that is what Woit means then he should become a great friend of E-infinity theory and of Prof. El Naschie, that is if he took the time to read our work carefully and without prejudice. In E-infinity El Naschie, He, Crnjac and Iovane introduced much larger Lie symmetry groups than E8 by summing over all Lie symmetry groups. For instance the sum over the E-line gives a ‘fuzzy’ group, i.e. a ‘fractal’ group with 548 instead of only 496 as integer part of the dimension. During the 80’s and 90’s of the last century in what is now called the European Journal of Physics, El Naschie wrote several papers on ‘average’ symmetries. This symmetry is the symmetry of chaos and avoids the anomalies stemming from the clashes between external and internal symmetries. All that is discussed in many papers by El Naschie, particularly those related to knot theory, wild topology, von Neumann’s continuous geometry, K-theory and especially A. Connes’s noncommutative geometry and even the monster group. Woit repeats his important point regarding symmetry in the conclusion on page 266 where he write “I see as an important lesson that each generation of physicist since the advent of quantum mechanics seems to need to learn anew. This lesson is the importance of symmetry principles expressed in mathematical language of group representation theory…… The underlying source of the problems of superstring theory is that the theory is not built on a fundamental symmetry principle or expressed within the language of representation theory.” We in E-infinity theory fully endorse this point of view and believe that our reformulation of Feynman’s summing over paths theory to summing over dimensions and summing over Lie groups as well as two and three Stein spaces is the correct strategy and as such is in full harmony with the views expressed in Woit’s book.
Now we come to the point upon which we do not agree at all with Woit. E-infinity is a K-theory for spacetime. In less formal mathematical language we mainly use semi groups (being summed over groups). In popular language of computer science, we use fractal automaton. Fractal sets and K-theory are what noncommutative geometry is all about. Noncommutative geometry is not as popular as string theory. However this is not the mistake of noncommutative geometry as Woit very well knows. For this reason it is not adequate to give only a few lines in his book to noncommutative geometry. On page 256 Woit write “One other speculative research programme that deserves mention goes under the name of non-commutative geometry….”. We in E-infinity profoundly disagree. If you call noncommutative geometry speculative, what do you call superstrings with 10 spacetime dimensions of which we have only seen 3 and God knows what the fourth or the fifth could be. A. Connes is the most important mathematician alive working in physics besides Sir. R. Penrose. We are astonished about this evaluation of Woit for noncommutative geometry for another reason. In his Acknowledgement Dr. Woit thanks Sr. R. Penrose for critical help. But one of the most important and most famous contributions of Penrose is his well known Penrose tiling. This tiling is the best known example of a noncommutative space based on K-theory and it is the prototype of E-infinity space. On piece of interesting information mentioned in Woit’s book which most of us did not know is that E. Witten has no degree, not even a Bachelors in physics. He was a journalist but his father worked in relativity and was a professor. This speaks of course for Witten. However some silly people hold it against El Naschie that he is a structural engineer and has no degree in physics.
We sincerely hope there will be a new edition of Woit’s book and that noncommutative geometry, fractal spacetime and Penrose tiling will be given more attention. There are many papers published on Elsevier’s Science Direct in CS&F regarding Penrose tiling as an example for E-infinity and noncommutative geometry of which we recommend the following:
1. M.S. El Naschie: von Neumann geometry and E-infinity quantum spacetime. CS&F, Vol. 9(12), (1998), pp. 2023-2030.
2. M.S. El Naschie: Penrose universe and Cantorian spacetime as a model for noncommutative quantum geometry. CS&F, Vol. 9(6), (1998), pp. 931-933.
3. M.S. El Naschie: Penrose tiling, semi conduction and Cantorian 1/fa spectra in four and five dimensions. CS&F, Vol. 3(4), (1993), pp. 498-491.
In addition an important paper on average symmetry is
4. M.S. El Naschie: Average exceptional Lie and Coxeter group hierarchies with special reference to the standard model of high energy particle physics. CS&F, 37, (2008), pp. 662-668.
Dec 10, 2010
The quantum sets of D. Finkelstein, E-infinity sets and the paper of D. Benedetti
10th December, 2010.
E-infinity communication No. 51
The quantum sets of D. Finkelstein, E-infinity sets and the paper of D. Benedetti
Prof. David Ritz Finkelstein is one of the earliest pioneers of quantum set and the construction of spacetime in a way similar to deriving thermodynamics from the motion of atoms. This is the original idea of El Naschie’s work which he acknowledges to have taken over from Finkelstein and Wizecker. Finkelstein is a deep thinker who was described in one of the acclaimed books of Lee Smolin as one of the seers in science today. The monad of Finkelstein space is the null set. The null here is not the dimension and we can consider his null set to be our null set as well as our empty set. We recall that E-infinity follows Suslin set theory and constructs everything from a Suslin tree starting with the empty set, the null set and the unity set. In E-infinity these are ( ̶ 1, ϕ2), (0, ϕ) and (1, 1). We note that in E-infinity we have Mandelbrot-El Naschie notion of the degrees of emptiness of an empty set leading to the totally empty set ( ̶ ∞, 0). The monad of E-infinity is random Cantor sets in general. Monad in the terminology of ‘t Hooft are called the building blocks of spacetime. ‘t Hooft does not work with sets. However a scientist somewhat close to him and very close to Prof. R. Loll, namely Prof. Fay Dowker works with partially ordered sets as explained in earlier communications. The work by D. Finkelstein was developed considerably by Heinrich Saller (see Quantum space-time-gravity by J. Baugh, D. Finkelstein, M. Shiri-Garakani and H. Saller). A particularly good summary of D. Finkelstein’s pioneering work is “Quantum sets and Clifford algebras”, Int. J. of Theoretical Physics, Vol. 21, No. 6/7 (1982). In the abstract of this paper Finkelstein says “… Quantum set theory may be applied to a quantum time space and quantum automaton.” In the introduction of his lecture in the presence of Feynman and Wheeler he said “several of us here including Feynman, Fredkin, Kantor, Moussouris, Perti, Wheeler and Zuse suggest that the universe may be discrete rather than continuous and more like a digital than analog computer. C.C. von Weizaecker (the student of Heisenberg and the elder brother of the past President of the Federal Republic of Germany) worked this path since the early 1950’s and we have recently benefitted from the relevant work of J. Ford (Ford was an expert in deterministic chaos and together with two of El Naschie’s friends, J. Casati and B. Cherekov pioneered the science of quantum chaos which is sometimes confused with the theory of Cantorian-fractal spacetime).
In the rest of this communication we address the technical aspect of E-infinity and show how to derive all the results of the work of D. Dario Benedetti from first principles in a far simpler and transparent manner using E-infinity. It is difficult to do this without writing equations but we will try our best.
First we consider our Cantorian E-infinity spacetime to be the union of infinitely many elementary monads Cantor sets. A single random Cantor set has, by the theorem of Mauldin-Williams, a Hausdorff dimension equal to the golden mean. The higher order Cantor sets will have a Hausdorff dimension equal ϕ to the power of n where n is 0, 1, 2… Adding all these dimensions together one finds the finite dimension of the large Cantorian space. Thus from summing from zero to infinity of ϕ to the power n, one finds 1 + ϕ + ϕ2 + ϕ3 + … and so on to infinity. The sum is exactly equal 2 + ϕ. This corresponds in the Connes-El Naschie dimensional function or bijection to a Menger-Urysohn dimension of exactly 3. However this has not been gauged in terms of the original monad ϕ. Therefore we have to divide 2 + ϕ by ϕ and this gives us the famous dimension 4.23606799 which corresponds exactly to the Menger-Urysohn topological dimension equal to 4. To show that 2 + ϕ corresponds to the topological dimension of 3 we just exclude the non-fractal dimension 1 from our summing. That means we start from n = 1 to n = ∞. That way the total dimensions become 2 + ϕ ̶ 1 = 1 + ϕ. To obtain the gauged dimension we divide by ϕ and find 2 + ϕ again which means the topological dimension is 3. Thus we have obtained from first principles the 4 and 3 dimensions of spacetime and space only with a single assumption which is that the union of all the elementary monads represents our spacetime. The monads themselves are formed by intersections of two and more monads. Unlike other theories our monads are explicit and well defined. They are random Cantor sets. The zero set for instance is given by two dimensions. First the topologically invariant Menger-Urysohn dimension is zero and second the Hausdorff-fractal dimension which is in this case ϕ. Thus the zero set is fixed by (0, ϕ). Using a simple elementary cobordism argument for (0, ϕ), the neighborhood or border is a wave given by the empty set ( ̶ 1, ϕ2). We could go on that way indefinitely. That means we have ( ̶ 2, ϕ3), ( ̶ 3, ϕ4) and finally (0, ̶ ∞) as mentioned earlier on. In fact we can determine the world sheet of fractal string theory. We did that when we summed from n = 1 to n = ∞ and found the total Hausdorff dimension to be 2 + ϕ ̶ 1 = 1 + ϕ. This corresponds in the Connes-El Naschie dimensional function bijection to a two dimensional object. The fractal world sheet is topologically two dimensions but Hausdorffly more than one dimensional and less than two dimensions. It is a fractal area-like with a Hausdorff fractal dimension equal 1 + ϕ = 1.618033989.
In fundamental scientific research there are two directions. We either keep improving and even patching and fixing our older theories or we start afresh from different or slightly different basic assumptions and principles. E-infinity is the second possibility. Dario Benedetti is the usual first possibility. Einstein’s relativity and quantum mechanics as well as string theory and loop quantum gravity started as the second possibility like E-infinity. In fact ‘t Hooft’s quantum field theory started exactly as E-infinity theory. Now string theory and quantum field theory are the establishment and the improvements and patching are characteristic for these important developments. In real life we need both methods and both philosophies. When we reach the same conclusions, this is then the promised land.
E-infinity group.
E-infinity communication No. 51
The quantum sets of D. Finkelstein, E-infinity sets and the paper of D. Benedetti
Prof. David Ritz Finkelstein is one of the earliest pioneers of quantum set and the construction of spacetime in a way similar to deriving thermodynamics from the motion of atoms. This is the original idea of El Naschie’s work which he acknowledges to have taken over from Finkelstein and Wizecker. Finkelstein is a deep thinker who was described in one of the acclaimed books of Lee Smolin as one of the seers in science today. The monad of Finkelstein space is the null set. The null here is not the dimension and we can consider his null set to be our null set as well as our empty set. We recall that E-infinity follows Suslin set theory and constructs everything from a Suslin tree starting with the empty set, the null set and the unity set. In E-infinity these are ( ̶ 1, ϕ2), (0, ϕ) and (1, 1). We note that in E-infinity we have Mandelbrot-El Naschie notion of the degrees of emptiness of an empty set leading to the totally empty set ( ̶ ∞, 0). The monad of E-infinity is random Cantor sets in general. Monad in the terminology of ‘t Hooft are called the building blocks of spacetime. ‘t Hooft does not work with sets. However a scientist somewhat close to him and very close to Prof. R. Loll, namely Prof. Fay Dowker works with partially ordered sets as explained in earlier communications. The work by D. Finkelstein was developed considerably by Heinrich Saller (see Quantum space-time-gravity by J. Baugh, D. Finkelstein, M. Shiri-Garakani and H. Saller). A particularly good summary of D. Finkelstein’s pioneering work is “Quantum sets and Clifford algebras”, Int. J. of Theoretical Physics, Vol. 21, No. 6/7 (1982). In the abstract of this paper Finkelstein says “… Quantum set theory may be applied to a quantum time space and quantum automaton.” In the introduction of his lecture in the presence of Feynman and Wheeler he said “several of us here including Feynman, Fredkin, Kantor, Moussouris, Perti, Wheeler and Zuse suggest that the universe may be discrete rather than continuous and more like a digital than analog computer. C.C. von Weizaecker (the student of Heisenberg and the elder brother of the past President of the Federal Republic of Germany) worked this path since the early 1950’s and we have recently benefitted from the relevant work of J. Ford (Ford was an expert in deterministic chaos and together with two of El Naschie’s friends, J. Casati and B. Cherekov pioneered the science of quantum chaos which is sometimes confused with the theory of Cantorian-fractal spacetime).
In the rest of this communication we address the technical aspect of E-infinity and show how to derive all the results of the work of D. Dario Benedetti from first principles in a far simpler and transparent manner using E-infinity. It is difficult to do this without writing equations but we will try our best.
First we consider our Cantorian E-infinity spacetime to be the union of infinitely many elementary monads Cantor sets. A single random Cantor set has, by the theorem of Mauldin-Williams, a Hausdorff dimension equal to the golden mean. The higher order Cantor sets will have a Hausdorff dimension equal ϕ to the power of n where n is 0, 1, 2… Adding all these dimensions together one finds the finite dimension of the large Cantorian space. Thus from summing from zero to infinity of ϕ to the power n, one finds 1 + ϕ + ϕ2 + ϕ3 + … and so on to infinity. The sum is exactly equal 2 + ϕ. This corresponds in the Connes-El Naschie dimensional function or bijection to a Menger-Urysohn dimension of exactly 3. However this has not been gauged in terms of the original monad ϕ. Therefore we have to divide 2 + ϕ by ϕ and this gives us the famous dimension 4.23606799 which corresponds exactly to the Menger-Urysohn topological dimension equal to 4. To show that 2 + ϕ corresponds to the topological dimension of 3 we just exclude the non-fractal dimension 1 from our summing. That means we start from n = 1 to n = ∞. That way the total dimensions become 2 + ϕ ̶ 1 = 1 + ϕ. To obtain the gauged dimension we divide by ϕ and find 2 + ϕ again which means the topological dimension is 3. Thus we have obtained from first principles the 4 and 3 dimensions of spacetime and space only with a single assumption which is that the union of all the elementary monads represents our spacetime. The monads themselves are formed by intersections of two and more monads. Unlike other theories our monads are explicit and well defined. They are random Cantor sets. The zero set for instance is given by two dimensions. First the topologically invariant Menger-Urysohn dimension is zero and second the Hausdorff-fractal dimension which is in this case ϕ. Thus the zero set is fixed by (0, ϕ). Using a simple elementary cobordism argument for (0, ϕ), the neighborhood or border is a wave given by the empty set ( ̶ 1, ϕ2). We could go on that way indefinitely. That means we have ( ̶ 2, ϕ3), ( ̶ 3, ϕ4) and finally (0, ̶ ∞) as mentioned earlier on. In fact we can determine the world sheet of fractal string theory. We did that when we summed from n = 1 to n = ∞ and found the total Hausdorff dimension to be 2 + ϕ ̶ 1 = 1 + ϕ. This corresponds in the Connes-El Naschie dimensional function bijection to a two dimensional object. The fractal world sheet is topologically two dimensions but Hausdorffly more than one dimensional and less than two dimensions. It is a fractal area-like with a Hausdorff fractal dimension equal 1 + ϕ = 1.618033989.
In fundamental scientific research there are two directions. We either keep improving and even patching and fixing our older theories or we start afresh from different or slightly different basic assumptions and principles. E-infinity is the second possibility. Dario Benedetti is the usual first possibility. Einstein’s relativity and quantum mechanics as well as string theory and loop quantum gravity started as the second possibility like E-infinity. In fact ‘t Hooft’s quantum field theory started exactly as E-infinity theory. Now string theory and quantum field theory are the establishment and the improvements and patching are characteristic for these important developments. In real life we need both methods and both philosophies. When we reach the same conclusions, this is then the promised land.
E-infinity group.
The placebo effect in mainstream physics
10th December, 2010.
E-infinity Communication No. 50
The placebo effect in mainstream physics
The violent effect of the establishment against new ideas is well documented and historically well researched. E-infinity has experience just such a violent effect only on the internet which gives perverted maniacs the same voice as anyone and may even be hired by sinister forces to silence competition. This is definitely the same with certain blogs hired in the north of Germany as well as a small neighbor country to scandalize as far as India and China. This possibility did not exist at the time of Einstein’s relativity and his Nazi opponents, some of whom were Nobel Laureates. Seen in this way the hired, as distinct from the real scientific opposition against E-infinity is harmless compared to that against Einstein’s relativity (see for instance the article in Physics World, April 2003).
The main reason in modern times for the rejection of a theory and unfair competition is funding money. String theoreticians may be excused for thinking that the book of Peter Woit “Not even wrong” in which he attacks string theory on all levels may have been written on the instigation of the leading scientist working in loop quantum gravity. We venture no opinion on this particular case but we think that as far as the young, innocent, unsuspecting scientists go, the problem may be related to a form of placebo effect. You expect that a theory or a scientist is bad because your supervisor told you so, then you find it bad. On the contrary, if you are told by your supervisor, or thesis advisor or Editor-in-Chief of a journal that a theory or a scientist is pure genius, then you will find it really pure genius. This is exactly the placebo effect in scientific research. Please think about it.
E-infinity Group.
E-infinity Communication No. 50
The placebo effect in mainstream physics
The violent effect of the establishment against new ideas is well documented and historically well researched. E-infinity has experience just such a violent effect only on the internet which gives perverted maniacs the same voice as anyone and may even be hired by sinister forces to silence competition. This is definitely the same with certain blogs hired in the north of Germany as well as a small neighbor country to scandalize as far as India and China. This possibility did not exist at the time of Einstein’s relativity and his Nazi opponents, some of whom were Nobel Laureates. Seen in this way the hired, as distinct from the real scientific opposition against E-infinity is harmless compared to that against Einstein’s relativity (see for instance the article in Physics World, April 2003).
The main reason in modern times for the rejection of a theory and unfair competition is funding money. String theoreticians may be excused for thinking that the book of Peter Woit “Not even wrong” in which he attacks string theory on all levels may have been written on the instigation of the leading scientist working in loop quantum gravity. We venture no opinion on this particular case but we think that as far as the young, innocent, unsuspecting scientists go, the problem may be related to a form of placebo effect. You expect that a theory or a scientist is bad because your supervisor told you so, then you find it bad. On the contrary, if you are told by your supervisor, or thesis advisor or Editor-in-Chief of a journal that a theory or a scientist is pure genius, then you will find it really pure genius. This is exactly the placebo effect in scientific research. Please think about it.
E-infinity Group.
The stationary states of quantum mechanics and the golden mean in E-infinity
10th December, 2010.
E-infinity communication No. 49
The stationary states of quantum mechanics and the golden mean in E-infinity
The stationary point in classical mechanics is well known. A complete classification was given long ago in the work of Andrenov which was frequently used by M.S. El Naschie and J.M.T. Thompson in their work on stability theory, theory of bifurcation and René Thom’s catastrophe theory. For details see El Naschie’s book “Stress, stability and chaos in structural engineering – An energy approach”, McGraw Hill, London (1990) as well as Thompson’s “Instabilities and castrophies in science and engineering”, John Wiley, Chichester (1982). However something was overlooked for which Sir James Lighthill (who had a very high regard for M.S. El Naschie and who helped to establish Chaos, Solitons & Fractals) had to apologize publicly to the public at large in an article published in the Proceedings of the Royal Society. Sir James said that by overlooking the generic chaotic nature of classical mechanics and letting so many people wrongly believe for such a long time that classical mechanics is generically deterministic, the community of theoretical and applied mechanics, of which he was the Head at the time (IUTAM President), misled the educated society at large. Five major discoveries were made through the revolution of deterministic chaos.
First the work of E. Lorenz established the one additional missing attractor of classical mechanics, namely the so called strange attractor. The great German chaos scientist Otto Rössler simplified the attractor of Lorenz. Rössler’s attractor is besides the Lorenz attractor, one of the main paradigms of chaos. Second the discovery of universalities in chaos by Mitchell Feigenbaum which extended original earlier work by S. Grossmann. Third the mathematical theory of turbulence and strange attractors of D. Ruelle and F. Takens. Fourth period 3 implies chaos by J. York who found that a single Cantor set is the backbone of all complex strange attractors and chaotic behavior and finally B. Mandelbrot who gave the correct geometry of deterministic chaos. This geometry is fractals. Again fractals is what the great G. Cantor saw with his inner eyes as the geometry of his Cantor set and the related gallery of monsters as H. Poincaré described them. All these strange attractors belong to dissipative dynamics. However what about Hamiltonian systems? These systems are more relevant to quantum mechanics. Seeing a connection between chaos in Hamiltonian systems and quantum mechanics was the achievement of René Thom and applying it to high energy physics and the mass spectrum of elementary particles was the achievement of Mohamed El Naschie and E-infinity theory. Making a great deal of bad jokes and running obscene blogs about non-science is the anti-achievement of those who do not tire from wasting their lives on defamatory allegations against E-infinity and those working in our group. At a minimum this shows the low self esteem of these internet characters who are so thick skinned that they force themselves on society just because the internet tolerates almost anything and any ignoramus.
In what follows we give briefly the most important points about the VAK in quantum mechanics and E-infinity theory:
1. The KAM nested concentric tori complex picture which Poincaré was reluctant to draw in 1899 is exactly that which R Thom proposed in 1975 as the Hamiltonian analogy of the strange attractor in differential dynamics. It was later on named the VAK. Like all fractals the VAK is almost self-similar. If one small VAK inside a large VAK is enlarged, we find another VAK in it and so on like a solenoid. In infinite dimensions as in E-infinity theory the VAK vague stability is due to the irrationality of the KAM orbits and may be used according to R. Thom as a model for the stable states of quantum mechanics.
2. Influenced by the thinking of René Thom, Mohamed El Naschie used to say in his numerous lectures which he gave in the last twenty years in Germany, Italy, Spain and particularly Egypt and Saudi Arabia “Descartes explained everything using his vortices and hooked atoms but could calculate nothing; Newton calculated everything using his inverse square law but could explain nothing. Only a geometrical theory like Einstein’s theory, string theory or E-infinity theory could explain and compute almost everything.
3. The vague attractor was studied by Kolmogorov, Moser and Arnold and that is why it is called KAM or VAK. The golden mean is easily shown to be to the most irrational number because in a continued fraction we have only the smallest non-zero integer, namely one. Consequently the golden mean is the worst irrational number which could be approximated by a rational number and that explains the stability of any periodic orbit with the golden mean winding number against perturbation.
4. The infinite complexity of geometrical form is reminiscent of the paradoxical notion of quantum field theory where the energy density of the vacuum is infinite.
5. The onset of turbulence is characterized by the replacement of a vague attractor of a Hamiltonian dynamics with finite-dimensional pseudo group of symmetries by a large ergodic set similar to a Cantor set. Thus we must distinguish two types of catastrophe point, the ordinary catastrophe and the second type of open, chaotic set with the complicated topology of a Cantor set. This second type is what is the case for noncommutative geometry and E-infinity theory.
6. We emphasize one more the wide spread misconception that Cantor sets and fractals appear only at the Planck length and Planck energy. The work of people as well established as ‘t Hooft and Lee Smolin as well as young otherwise excellent researchers such as Dario Benedetti all suffer from this misconception. Fractals have very little to do with Wheeler and Hawking’s foam and are not bound to the Planck energy. They appear generically even in classical systems at low energy.
7. As the level of the VAK there is no essential difference between quantum phase space, a Hilbert space or a fractal Cantorian spacetime. The excellent work of T.N. Palmer needlessly suffers from this misconception which is based on a widespread prejudice related to wrongly connecting fractals to the so called Wheeler-Hawking spacetime foam.
E-infinity group.
E-infinity communication No. 49
The stationary states of quantum mechanics and the golden mean in E-infinity
The stationary point in classical mechanics is well known. A complete classification was given long ago in the work of Andrenov which was frequently used by M.S. El Naschie and J.M.T. Thompson in their work on stability theory, theory of bifurcation and René Thom’s catastrophe theory. For details see El Naschie’s book “Stress, stability and chaos in structural engineering – An energy approach”, McGraw Hill, London (1990) as well as Thompson’s “Instabilities and castrophies in science and engineering”, John Wiley, Chichester (1982). However something was overlooked for which Sir James Lighthill (who had a very high regard for M.S. El Naschie and who helped to establish Chaos, Solitons & Fractals) had to apologize publicly to the public at large in an article published in the Proceedings of the Royal Society. Sir James said that by overlooking the generic chaotic nature of classical mechanics and letting so many people wrongly believe for such a long time that classical mechanics is generically deterministic, the community of theoretical and applied mechanics, of which he was the Head at the time (IUTAM President), misled the educated society at large. Five major discoveries were made through the revolution of deterministic chaos.
First the work of E. Lorenz established the one additional missing attractor of classical mechanics, namely the so called strange attractor. The great German chaos scientist Otto Rössler simplified the attractor of Lorenz. Rössler’s attractor is besides the Lorenz attractor, one of the main paradigms of chaos. Second the discovery of universalities in chaos by Mitchell Feigenbaum which extended original earlier work by S. Grossmann. Third the mathematical theory of turbulence and strange attractors of D. Ruelle and F. Takens. Fourth period 3 implies chaos by J. York who found that a single Cantor set is the backbone of all complex strange attractors and chaotic behavior and finally B. Mandelbrot who gave the correct geometry of deterministic chaos. This geometry is fractals. Again fractals is what the great G. Cantor saw with his inner eyes as the geometry of his Cantor set and the related gallery of monsters as H. Poincaré described them. All these strange attractors belong to dissipative dynamics. However what about Hamiltonian systems? These systems are more relevant to quantum mechanics. Seeing a connection between chaos in Hamiltonian systems and quantum mechanics was the achievement of René Thom and applying it to high energy physics and the mass spectrum of elementary particles was the achievement of Mohamed El Naschie and E-infinity theory. Making a great deal of bad jokes and running obscene blogs about non-science is the anti-achievement of those who do not tire from wasting their lives on defamatory allegations against E-infinity and those working in our group. At a minimum this shows the low self esteem of these internet characters who are so thick skinned that they force themselves on society just because the internet tolerates almost anything and any ignoramus.
In what follows we give briefly the most important points about the VAK in quantum mechanics and E-infinity theory:
1. The KAM nested concentric tori complex picture which Poincaré was reluctant to draw in 1899 is exactly that which R Thom proposed in 1975 as the Hamiltonian analogy of the strange attractor in differential dynamics. It was later on named the VAK. Like all fractals the VAK is almost self-similar. If one small VAK inside a large VAK is enlarged, we find another VAK in it and so on like a solenoid. In infinite dimensions as in E-infinity theory the VAK vague stability is due to the irrationality of the KAM orbits and may be used according to R. Thom as a model for the stable states of quantum mechanics.
2. Influenced by the thinking of René Thom, Mohamed El Naschie used to say in his numerous lectures which he gave in the last twenty years in Germany, Italy, Spain and particularly Egypt and Saudi Arabia “Descartes explained everything using his vortices and hooked atoms but could calculate nothing; Newton calculated everything using his inverse square law but could explain nothing. Only a geometrical theory like Einstein’s theory, string theory or E-infinity theory could explain and compute almost everything.
3. The vague attractor was studied by Kolmogorov, Moser and Arnold and that is why it is called KAM or VAK. The golden mean is easily shown to be to the most irrational number because in a continued fraction we have only the smallest non-zero integer, namely one. Consequently the golden mean is the worst irrational number which could be approximated by a rational number and that explains the stability of any periodic orbit with the golden mean winding number against perturbation.
4. The infinite complexity of geometrical form is reminiscent of the paradoxical notion of quantum field theory where the energy density of the vacuum is infinite.
5. The onset of turbulence is characterized by the replacement of a vague attractor of a Hamiltonian dynamics with finite-dimensional pseudo group of symmetries by a large ergodic set similar to a Cantor set. Thus we must distinguish two types of catastrophe point, the ordinary catastrophe and the second type of open, chaotic set with the complicated topology of a Cantor set. This second type is what is the case for noncommutative geometry and E-infinity theory.
6. We emphasize one more the wide spread misconception that Cantor sets and fractals appear only at the Planck length and Planck energy. The work of people as well established as ‘t Hooft and Lee Smolin as well as young otherwise excellent researchers such as Dario Benedetti all suffer from this misconception. Fractals have very little to do with Wheeler and Hawking’s foam and are not bound to the Planck energy. They appear generically even in classical systems at low energy.
7. As the level of the VAK there is no essential difference between quantum phase space, a Hilbert space or a fractal Cantorian spacetime. The excellent work of T.N. Palmer needlessly suffers from this misconception which is based on a widespread prejudice related to wrongly connecting fractals to the so called Wheeler-Hawking spacetime foam.
E-infinity group.
Fractal properties of quantum spacetime in the Perimeter Inst. of Theoretical Physics and El Naschie’s quantum group dimension
9th December, 2010.
E-infinity communication No. 48
Fractal properties of quantum spacetime in the Perimeter Inst. of Theoretical Physics and El Naschie’s quantum group dimension
Dario Benedetti of the Perimeter Institute, partly founded due to the efforts of Prof. Lee Smolin, published a highly interesting paper in 2009 which is quite revealing. In this paper he seems to have carefully studied the literature used mainly by the E-infinity group, for instance what Mohamed El Naschie calls the Biedenharn conjecture. Benedetti’s paper is freely available on the net “Fractal properties of quantum spacetime”, arXiv: 0811.1396V2[hep-th], 25th March 2009. The main conclusion is that taking certain limits the dimensionality of spacetime, namely exact 4 as well as space only, namely exactly 3 may be obtained using quantum groups is obtained. To show how this follows immediately from a well known quantum group dimension when setting the monadic dimension of the atoms of the concerned space we do not need much nor even referring to old papers published by Mohamed El Naschie in many international journals. All what one needs is to go through the following steps.
Step one is to take any good book on quantum groups, for instance C. Kassel book “Quantum Groups” published by Springer 1995. Step two open page 364 and there you will find a formula for quantum dimension for a simple model given explicitly to be the said monade q to the power of n + 1 minus the same but with negative power ( ̶ n ̶ 1) then all divided by q minus the inverse of q. Setting the monade q = ϕ = the quantum dimension is found to be exactly 4. One could find the result of all dimensions, namely the space dimension 3 as well as the Hausdorff dimension 4.2367977 and R. Loll’s spectral dimension 4.01999 ≃ 4.02 in a similarly very simple way. Philosophical discussion of these dimensions were given by El Naschie in a paper written when he was in Cambridge, UK in 1998 (see Bio systems (an Elseiver journal), No. 46 (1998), pp. 4-46). This paper was entitled Dimensional symmetry breaking, information and fractal gravity in Cantorian space. A second paper is in a book published by Gordon and Breach “The quest for a unified theory of information”, Edited by Wolfgang Hofkirchner. The shortest and most condensed paper is “Quantum groups and Hamiltonian sets on nuclear spacetime Cantorian manifold” in CS&F, Vol. 10, No. 7, (1999), pp. 125-1256. In particular Table No. 1 compares all the various relevant dimensions. It is important to see how completely different theories lead to essentially similar conclusions. The only question is what is the most simple theory. We have no doubt that it is E-infinity because it is free from traditions and is obliged only to the mathematical logic with no regard to the political correctness of scientific grouping and science funding policy.
E-infinity group.
E-infinity communication No. 48
Fractal properties of quantum spacetime in the Perimeter Inst. of Theoretical Physics and El Naschie’s quantum group dimension
Dario Benedetti of the Perimeter Institute, partly founded due to the efforts of Prof. Lee Smolin, published a highly interesting paper in 2009 which is quite revealing. In this paper he seems to have carefully studied the literature used mainly by the E-infinity group, for instance what Mohamed El Naschie calls the Biedenharn conjecture. Benedetti’s paper is freely available on the net “Fractal properties of quantum spacetime”, arXiv: 0811.1396V2[hep-th], 25th March 2009. The main conclusion is that taking certain limits the dimensionality of spacetime, namely exact 4 as well as space only, namely exactly 3 may be obtained using quantum groups is obtained. To show how this follows immediately from a well known quantum group dimension when setting the monadic dimension of the atoms of the concerned space we do not need much nor even referring to old papers published by Mohamed El Naschie in many international journals. All what one needs is to go through the following steps.
Step one is to take any good book on quantum groups, for instance C. Kassel book “Quantum Groups” published by Springer 1995. Step two open page 364 and there you will find a formula for quantum dimension for a simple model given explicitly to be the said monade q to the power of n + 1 minus the same but with negative power ( ̶ n ̶ 1) then all divided by q minus the inverse of q. Setting the monade q = ϕ = the quantum dimension is found to be exactly 4. One could find the result of all dimensions, namely the space dimension 3 as well as the Hausdorff dimension 4.2367977 and R. Loll’s spectral dimension 4.01999 ≃ 4.02 in a similarly very simple way. Philosophical discussion of these dimensions were given by El Naschie in a paper written when he was in Cambridge, UK in 1998 (see Bio systems (an Elseiver journal), No. 46 (1998), pp. 4-46). This paper was entitled Dimensional symmetry breaking, information and fractal gravity in Cantorian space. A second paper is in a book published by Gordon and Breach “The quest for a unified theory of information”, Edited by Wolfgang Hofkirchner. The shortest and most condensed paper is “Quantum groups and Hamiltonian sets on nuclear spacetime Cantorian manifold” in CS&F, Vol. 10, No. 7, (1999), pp. 125-1256. In particular Table No. 1 compares all the various relevant dimensions. It is important to see how completely different theories lead to essentially similar conclusions. The only question is what is the most simple theory. We have no doubt that it is E-infinity because it is free from traditions and is obliged only to the mathematical logic with no regard to the political correctness of scientific grouping and science funding policy.
E-infinity group.
The role of dissipation in the ‘t Hooft-El Naschie-Ord quantum systems
9th December, 2010.
E-infinity communication No. 47
The role of dissipation in the ‘t Hooft-El Naschie-Ord quantum systems
Almost all realistic engineering systems have friction losses or any other kind of dissipation. As a structural engineer and applied mechanics scientist El Naschie dealt as long ago as 1976 with nonconservative, dissipative mechanical systems. In his paper in ‘Solid Mechanics Archives, Vol. 4, August 1979 (published by Stijthoff & Noordhoff Int. Publishers, Holland) he developed a finite element-like method (finite element is the engineer’s version of Regge calculus of general relativity). The method he employed, invented by Belgian engineer van den Dungen (Bull. Acad. Ray Belg. Sci Ser 1945 (31), pp. 659-668) consists of joining two dissipative systems, one with energy losses and another with energy gain (a so called flutter set), balancing each other and thus forming a conservative Hamiltonian system. In a paper published in 1995 in CS&F entitled “A note on quantum mechanics, diffusional interference and information” he extended his two dissipative systems forming one conservative system idea to the Schrödinger equation by two conjugate complex Schrödinger equations, one going forwards and the other going backwards in time (see CS&F, 5(5), pp. 881-884 (1995)). The importance of this paper was immediately recognized by Prof. G. Ord whose model is essentially very similar although it may not seem to be that way without careful examination. It all boiled down to the need for an additional negative sign which will become apparent later on. It could be that the Nobel Laureate became quite interested in nonlinear dynamics which always includes dissipation and a dissipative ‘strange attractor’ after meeting El Naschie on several occasions around the year 2000 including a conference in Riyadh and another in Cairo. It is clear that ‘t Hooft must have been thinking about these things for some time before that because he realized that the notion of time in relativity is fundamentally different from that in quantum mechanics and because he had a controversy with Steven Hawkings about the information paradox of black holes. Ultimately ‘t Hooft wrote several papers connecting the loss of information with dissipation and applied that to a new quantum mechanics which he called deterministic quantum mechanics. Similar to E-infinity ‘t Hooft used fluid turbulence as a paradigm for his theory. However at that time ‘t Hooft new nothing about quasi attractors in Hamiltonian systems because the VAK (the vague attractor of Kolmogorov) was not yet recognized by Mohamed El Naschie and thus not yet incorporated into his work. The VAK first conjectured as the stationary states of quantum mechanics by French topologist and the inventor of catastrophe theory, Rene Thom was considered much later (see for instance the paper “Strange non-dissipative and non-chaotic attractors and Palmer’s deterministic quantum mechanics” by G. Iovane and S. Nada published in CS&F, 42, pp 641-642 (2009).
Part II of communication No. 47
It is important to realize that although the VAK has no physical friction to give it stability, it has a mathematical substitute for the lack of friction, namely the irrationality of the winding number. Noting that the golden mean is the most irrational number, it follows that a golden mean winding number is the most stable orbit for a dynamic system and that is the reason why the mass of the elementary particles which could realistically be observed experimentally is always a function of the golden mean and its power. This also follows from von Neumann-Connes’ dimensional function. Dimensions are related to the coupling constant and these are in turn related to energy and thus the mass of elementary particles. In 2007 M. El Naschie gave a Lagrangian formulation to his basic 1979 idea in a paper entitled “On gauge invariance, dissipative quantum mechanics and self-adjoint sets. The paper was dedicated to his by that time very close friend Gerard ‘t Hooft in celebration of his 60th birthday (see CS&F, 32 (2007), pp. 271-273). In the meantime Ord refined his work by introducing the ani-Bernulli diffusion mimicking quantum mechanics (see for instance Annals of Physics, 324 (2009), pp. 1211-1218) where he refers as usual to the very same 1995 paper of El Naschie. The fact however is that all these different formulations are basically different faces of the same multi-dimensional coin. We can obtain the needed ‘negative’ sign by considering an adjoint flutter set with energy gain or a dissipative set with energy losses. We can also change a Bernulli random walk using zero and plus one to an anti-Bernulli random walk with 0, +1 and ̶ 1. We can introduce anti-commuting Grassmanian coordinates a proposed by ‘t Hooft and used in super string theory or we can go back to fundamentals and consider all the empty sets with their negative Menger-Urysohn dimensions as done by Mohamed El Naschie
E-infinity Group
E-infinity communication No. 47
The role of dissipation in the ‘t Hooft-El Naschie-Ord quantum systems
Almost all realistic engineering systems have friction losses or any other kind of dissipation. As a structural engineer and applied mechanics scientist El Naschie dealt as long ago as 1976 with nonconservative, dissipative mechanical systems. In his paper in ‘Solid Mechanics Archives, Vol. 4, August 1979 (published by Stijthoff & Noordhoff Int. Publishers, Holland) he developed a finite element-like method (finite element is the engineer’s version of Regge calculus of general relativity). The method he employed, invented by Belgian engineer van den Dungen (Bull. Acad. Ray Belg. Sci Ser 1945 (31), pp. 659-668) consists of joining two dissipative systems, one with energy losses and another with energy gain (a so called flutter set), balancing each other and thus forming a conservative Hamiltonian system. In a paper published in 1995 in CS&F entitled “A note on quantum mechanics, diffusional interference and information” he extended his two dissipative systems forming one conservative system idea to the Schrödinger equation by two conjugate complex Schrödinger equations, one going forwards and the other going backwards in time (see CS&F, 5(5), pp. 881-884 (1995)). The importance of this paper was immediately recognized by Prof. G. Ord whose model is essentially very similar although it may not seem to be that way without careful examination. It all boiled down to the need for an additional negative sign which will become apparent later on. It could be that the Nobel Laureate became quite interested in nonlinear dynamics which always includes dissipation and a dissipative ‘strange attractor’ after meeting El Naschie on several occasions around the year 2000 including a conference in Riyadh and another in Cairo. It is clear that ‘t Hooft must have been thinking about these things for some time before that because he realized that the notion of time in relativity is fundamentally different from that in quantum mechanics and because he had a controversy with Steven Hawkings about the information paradox of black holes. Ultimately ‘t Hooft wrote several papers connecting the loss of information with dissipation and applied that to a new quantum mechanics which he called deterministic quantum mechanics. Similar to E-infinity ‘t Hooft used fluid turbulence as a paradigm for his theory. However at that time ‘t Hooft new nothing about quasi attractors in Hamiltonian systems because the VAK (the vague attractor of Kolmogorov) was not yet recognized by Mohamed El Naschie and thus not yet incorporated into his work. The VAK first conjectured as the stationary states of quantum mechanics by French topologist and the inventor of catastrophe theory, Rene Thom was considered much later (see for instance the paper “Strange non-dissipative and non-chaotic attractors and Palmer’s deterministic quantum mechanics” by G. Iovane and S. Nada published in CS&F, 42, pp 641-642 (2009).
Part II of communication No. 47
It is important to realize that although the VAK has no physical friction to give it stability, it has a mathematical substitute for the lack of friction, namely the irrationality of the winding number. Noting that the golden mean is the most irrational number, it follows that a golden mean winding number is the most stable orbit for a dynamic system and that is the reason why the mass of the elementary particles which could realistically be observed experimentally is always a function of the golden mean and its power. This also follows from von Neumann-Connes’ dimensional function. Dimensions are related to the coupling constant and these are in turn related to energy and thus the mass of elementary particles. In 2007 M. El Naschie gave a Lagrangian formulation to his basic 1979 idea in a paper entitled “On gauge invariance, dissipative quantum mechanics and self-adjoint sets. The paper was dedicated to his by that time very close friend Gerard ‘t Hooft in celebration of his 60th birthday (see CS&F, 32 (2007), pp. 271-273). In the meantime Ord refined his work by introducing the ani-Bernulli diffusion mimicking quantum mechanics (see for instance Annals of Physics, 324 (2009), pp. 1211-1218) where he refers as usual to the very same 1995 paper of El Naschie. The fact however is that all these different formulations are basically different faces of the same multi-dimensional coin. We can obtain the needed ‘negative’ sign by considering an adjoint flutter set with energy gain or a dissipative set with energy losses. We can also change a Bernulli random walk using zero and plus one to an anti-Bernulli random walk with 0, +1 and ̶ 1. We can introduce anti-commuting Grassmanian coordinates a proposed by ‘t Hooft and used in super string theory or we can go back to fundamentals and consider all the empty sets with their negative Menger-Urysohn dimensions as done by Mohamed El Naschie
E-infinity Group
The road to Nottale scale relativity and some remarks on a recent paper by Nobel Laureate G. ‘t Hooft.
8th December, 2010.
E-infinity communication No. 46
The road to Nottale scale relativity and some remarks on a recent paper by Nobel Laureate G. ‘t Hooft.
Various statements and new mathematical and physical ingredients in a relatively recent paper by Nobel Laureate G. ‘t Hooft (see Quantum gravity without space-time singularities – arXiv: 0909.3426V1[gr-qc], 18 Sept. 2009) are effectively a compliment to the work of Laurent Nottale. It is well known that Nottale is one of the first not only to appreciate the crucial role of scale invariance but also to give a general theory of relativity within a scale invariance program which is relevant to quantum and high energy physics. A very readable popular account of Nottale work may be found in “Ciel et Espace”, February (1994), pp. 28-32. The interview is entitled “Espace-Temps Fractal – La Nouvelle theorie de l’univers”. A formal introduction to Nottale’ theory including references to the work of Garnet Ord as well as Mohamed El Naschie may be found in his by now classical book “Fractal Space-Time and Microphysics”, World Scientific, Singapore (1993). Unitl recently Nottale considered his papers in CS&F to be the most up to date and comprehensive but in 2010 he has published at least two new very long papers in Springer journals which may be found on the internet, all apart from his excellent blog called “Scale Relativity Corner” where highly scientific and informative high level, civilized discussion is going on without the usual vulgarity and aggressiveness characteristic of the run of the mill pseudo scientific blogs nowadays. The idea behind Nottale’s work is basically the same as that of Ord and El Naschie although the mathematics is slightly different. The idea is as simple as it is ingenious. Some mathematical physicists call it the Biedenharn conjecture. It is the fact that any space which is a fractal is devoid of any natural scale. Consequently it is explicitly scale invariant. That means it is a gauge theory in the original sense of the work of H. Weyl. This is the reason by Mohamed El Naschie replaces calculus by Weyl scaling. In fact this is the reason by in E-infinity we can use the set theoretical Suslin scaling. In the particular case of E-infinity theory this leads naturally to golden mean renormalization groups used so successfully by Mitchell Feigenbaum in finding his structural universalities. The pure mathematical background for these E-infinity relations may be found in the theory of semi-groups. The research related to Lucasian numbers by Hilton as well as Fibonacci groups by Nikolova is particularly interesting. It is strange, if not quite disturbing, that J.C. Baez takes such a hostile position to E-infinity theory although he occasionally works on loop spaces, E-infinity ring spaces and spectra all apart of groupoids. However maybe this aggressive rejection has a more personal than scientific basis.
In the paper of ‘t Hooft mentioned above he states already in the abstract that he “…finds that exact invariance under scale transformation is an essential new ingredient….”. Then he continues by saying “These differences can be boiled down to conformal transformation”. He points out that Newton’s constant is not scale-invariant and neither is the Einstein-Hilbert action. Consequently he cannot use the Riemann curvature nor its Ricci components. Apart of the Weyl component nothing is left except to follow the road outlines by L. Nottale, M. Agop and E. Goldfain, namely scale relativity, fractal spacetime and E-infinity transfinite sets.
E-infinity communication No. 46
The road to Nottale scale relativity and some remarks on a recent paper by Nobel Laureate G. ‘t Hooft.
Various statements and new mathematical and physical ingredients in a relatively recent paper by Nobel Laureate G. ‘t Hooft (see Quantum gravity without space-time singularities – arXiv: 0909.3426V1[gr-qc], 18 Sept. 2009) are effectively a compliment to the work of Laurent Nottale. It is well known that Nottale is one of the first not only to appreciate the crucial role of scale invariance but also to give a general theory of relativity within a scale invariance program which is relevant to quantum and high energy physics. A very readable popular account of Nottale work may be found in “Ciel et Espace”, February (1994), pp. 28-32. The interview is entitled “Espace-Temps Fractal – La Nouvelle theorie de l’univers”. A formal introduction to Nottale’ theory including references to the work of Garnet Ord as well as Mohamed El Naschie may be found in his by now classical book “Fractal Space-Time and Microphysics”, World Scientific, Singapore (1993). Unitl recently Nottale considered his papers in CS&F to be the most up to date and comprehensive but in 2010 he has published at least two new very long papers in Springer journals which may be found on the internet, all apart from his excellent blog called “Scale Relativity Corner” where highly scientific and informative high level, civilized discussion is going on without the usual vulgarity and aggressiveness characteristic of the run of the mill pseudo scientific blogs nowadays. The idea behind Nottale’s work is basically the same as that of Ord and El Naschie although the mathematics is slightly different. The idea is as simple as it is ingenious. Some mathematical physicists call it the Biedenharn conjecture. It is the fact that any space which is a fractal is devoid of any natural scale. Consequently it is explicitly scale invariant. That means it is a gauge theory in the original sense of the work of H. Weyl. This is the reason by Mohamed El Naschie replaces calculus by Weyl scaling. In fact this is the reason by in E-infinity we can use the set theoretical Suslin scaling. In the particular case of E-infinity theory this leads naturally to golden mean renormalization groups used so successfully by Mitchell Feigenbaum in finding his structural universalities. The pure mathematical background for these E-infinity relations may be found in the theory of semi-groups. The research related to Lucasian numbers by Hilton as well as Fibonacci groups by Nikolova is particularly interesting. It is strange, if not quite disturbing, that J.C. Baez takes such a hostile position to E-infinity theory although he occasionally works on loop spaces, E-infinity ring spaces and spectra all apart of groupoids. However maybe this aggressive rejection has a more personal than scientific basis.
In the paper of ‘t Hooft mentioned above he states already in the abstract that he “…finds that exact invariance under scale transformation is an essential new ingredient….”. Then he continues by saying “These differences can be boiled down to conformal transformation”. He points out that Newton’s constant is not scale-invariant and neither is the Einstein-Hilbert action. Consequently he cannot use the Riemann curvature nor its Ricci components. Apart of the Weyl component nothing is left except to follow the road outlines by L. Nottale, M. Agop and E. Goldfain, namely scale relativity, fractal spacetime and E-infinity transfinite sets.
Dec 8, 2010
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.
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.
Schrödinger’s cat meets Arnold’s cat in El Naschie’s infinity garden
E-infinity communication No. 44
Schrödinger’s cat meets Arnold’s cat in El Naschie’s infinity garden.
No one knows exactly why E. Schrödinger the Austrian Nobel Laureate in physics and founder of wave mechanics chose a cat to subject to a most cruel type of quantum measurement. The reasons for the Russian V. Arnold, probably one of the greatest mathematicians of all time to deform a cat’s picture beyond recognition using his well known map are even less clear. In all events both cats are alive, well and jumping in E-infinity’s golden garden. The two eigenvalues of Arnold’s cat map happened to be the inverse of the golden mean (1/ϕ) = 1 + ϕ and the inverse of the golden mean squared (1/ϕ)2 = 2 + ϕ. Now adding both values or multiplying them gives exactly the very same result, namely (1 + ϕ) + (2 + ϕ) = 4.23606799 and (1 + ϕ) (2 + ϕ) = 4.23606799. In other words union and intersection of the two transfinite sets gives the same result. One can easily see that the first set have the topological dimension of an area just like a world sheet in string theory. This area has a fractal dimension equal 1 + ϕ = 1.61833989. The second set on the other hand is topologically three dimensional, however its Haudorff dimension is 2 + ϕ = 2.168033989. Topologically the union of the two sets is 2 + 3 = 5 dimensional. However seen from the Hausdorff dimension view point, the two sets give a dimension equal 4 + ϕ3 = 4.23606799 which corresponds to exactly 4 and only 4 topological dimensions. In other words one topological dimension is hidden. The 5 dimensions are used only for embedding the 4.23606799 fractal dimension. On the other hand we know very well that 2 + ϕ = 2.61803389 corresponds exactly to 3 topological dimensions of the tangible world. The only thing left for interpretation is that 1 + ϕ which is topologically 2 dimensional is that these two dimensions stand for time and the spin ½ fermionic dimension or alternatively for the fifth Kaluza-Klien compactified or the cyclic dimension of electromagnetism.
The reader is referred to the following paper for simple geometrical visualization of the Russian doll-like E-infinity space with 4.23606799 Hausdorff dimension corresponding to exactly only 4 topological Menger-Urysohn dimensions. See for instance “An irreducibly simple derivation of the Hausdorff dimension of spacetime”, Chaos, Solitons & Fractals, 41, (2009), pp. 1902-1904, particularly Table 1 on page 1903. See also “The theory of Cantorian spacetime and high energy particle physics (an informal review)”, Chaos, Solitons & Fractals, 41 (2009), pp. 2635-2646, in particular Fig. 1 and Fig. 2 on pages 2636 and 2639 respectively.
It remains to say that it is the fine structure of fractals contributing 0.23606799 = ϕ3 where ϕ is the golden mean = 0.618033989 which causes the equality of union and intersection of the Cantor sets spanning E-infinity spacetime and leading to the infinite but hierarchal dimensionality of this fractal Cantorian spacetime manifold which is the cause of the persistent illusion that a quantum object on a fractal particle can be said to be in two different spacetime fractal ‘points’ at the very same fractal ‘time’. Looked upon it from a distance, this intricate non-smooth and chaotic Cantorian-fractal spacetime appears smooth with only four spacetime topological dimensions. To show this in the most clear quantative way we just need to approximate the irrational number to the simplest rational number. That means 2 + ϕ = 2.618033 will be 2.5 = 5/2 and 1 + ϕ = 1.618033 will be 1.5 = 3/2. Adding together we find 2.5 + 1.5 = 4. On the other hand multiplication leads to a different dimension known from the theory of dimensional regularization, namely (5/2)(3/2) = 3.75 = 4 ̶ 0.25. This corresponds in E-infinity to 4 ̶ k where k = 0.18033989 of the transfinite Heterotic super string theory. It is vital to recall the importance of Arnold’s cat map in the study of quantum chaos as well as the discussion of the role of irrational numbers in combined harmonic oscillators by Prof. G. ‘t Hooft in his paper mentioned in an earlier communication “The mathematical basis for deterministic quantum mechanics”. In fact on page 7 of the arXiv paper he writes “….If the frequencies have an irrational ratio (which in the terminology of El Naschie and nonlinear dynamics means irrational winding number), the period of the classical system is infinite and so a continuous spectrum would be expected.” Sooner or later we are confident that those working in quantum field theory and have come as far as this will realize that K-theory, noncommutative geometry and E-infinity is the right way to come to what Nobel Laureate ‘t Hooft hopes to find, namely some kind of deterministic quantum mechanics.
E-infinity Group.
Schrödinger’s cat meets Arnold’s cat in El Naschie’s infinity garden.
No one knows exactly why E. Schrödinger the Austrian Nobel Laureate in physics and founder of wave mechanics chose a cat to subject to a most cruel type of quantum measurement. The reasons for the Russian V. Arnold, probably one of the greatest mathematicians of all time to deform a cat’s picture beyond recognition using his well known map are even less clear. In all events both cats are alive, well and jumping in E-infinity’s golden garden. The two eigenvalues of Arnold’s cat map happened to be the inverse of the golden mean (1/ϕ) = 1 + ϕ and the inverse of the golden mean squared (1/ϕ)2 = 2 + ϕ. Now adding both values or multiplying them gives exactly the very same result, namely (1 + ϕ) + (2 + ϕ) = 4.23606799 and (1 + ϕ) (2 + ϕ) = 4.23606799. In other words union and intersection of the two transfinite sets gives the same result. One can easily see that the first set have the topological dimension of an area just like a world sheet in string theory. This area has a fractal dimension equal 1 + ϕ = 1.61833989. The second set on the other hand is topologically three dimensional, however its Haudorff dimension is 2 + ϕ = 2.168033989. Topologically the union of the two sets is 2 + 3 = 5 dimensional. However seen from the Hausdorff dimension view point, the two sets give a dimension equal 4 + ϕ3 = 4.23606799 which corresponds to exactly 4 and only 4 topological dimensions. In other words one topological dimension is hidden. The 5 dimensions are used only for embedding the 4.23606799 fractal dimension. On the other hand we know very well that 2 + ϕ = 2.61803389 corresponds exactly to 3 topological dimensions of the tangible world. The only thing left for interpretation is that 1 + ϕ which is topologically 2 dimensional is that these two dimensions stand for time and the spin ½ fermionic dimension or alternatively for the fifth Kaluza-Klien compactified or the cyclic dimension of electromagnetism.
The reader is referred to the following paper for simple geometrical visualization of the Russian doll-like E-infinity space with 4.23606799 Hausdorff dimension corresponding to exactly only 4 topological Menger-Urysohn dimensions. See for instance “An irreducibly simple derivation of the Hausdorff dimension of spacetime”, Chaos, Solitons & Fractals, 41, (2009), pp. 1902-1904, particularly Table 1 on page 1903. See also “The theory of Cantorian spacetime and high energy particle physics (an informal review)”, Chaos, Solitons & Fractals, 41 (2009), pp. 2635-2646, in particular Fig. 1 and Fig. 2 on pages 2636 and 2639 respectively.
It remains to say that it is the fine structure of fractals contributing 0.23606799 = ϕ3 where ϕ is the golden mean = 0.618033989 which causes the equality of union and intersection of the Cantor sets spanning E-infinity spacetime and leading to the infinite but hierarchal dimensionality of this fractal Cantorian spacetime manifold which is the cause of the persistent illusion that a quantum object on a fractal particle can be said to be in two different spacetime fractal ‘points’ at the very same fractal ‘time’. Looked upon it from a distance, this intricate non-smooth and chaotic Cantorian-fractal spacetime appears smooth with only four spacetime topological dimensions. To show this in the most clear quantative way we just need to approximate the irrational number to the simplest rational number. That means 2 + ϕ = 2.618033 will be 2.5 = 5/2 and 1 + ϕ = 1.618033 will be 1.5 = 3/2. Adding together we find 2.5 + 1.5 = 4. On the other hand multiplication leads to a different dimension known from the theory of dimensional regularization, namely (5/2)(3/2) = 3.75 = 4 ̶ 0.25. This corresponds in E-infinity to 4 ̶ k where k = 0.18033989 of the transfinite Heterotic super string theory. It is vital to recall the importance of Arnold’s cat map in the study of quantum chaos as well as the discussion of the role of irrational numbers in combined harmonic oscillators by Prof. G. ‘t Hooft in his paper mentioned in an earlier communication “The mathematical basis for deterministic quantum mechanics”. In fact on page 7 of the arXiv paper he writes “….If the frequencies have an irrational ratio (which in the terminology of El Naschie and nonlinear dynamics means irrational winding number), the period of the classical system is infinite and so a continuous spectrum would be expected.” Sooner or later we are confident that those working in quantum field theory and have come as far as this will realize that K-theory, noncommutative geometry and E-infinity is the right way to come to what Nobel Laureate ‘t Hooft hopes to find, namely some kind of deterministic quantum mechanics.
E-infinity Group.
Dec 7, 2010
‘t Hooft dilaton proposal and the inbuilt natural conformal invariance of E-infinity and Nottale scale relativity
6th December, 2010
E-infinity Communication No. 43
‘t Hooft dilaton proposal and the inbuilt natural conformal invariance of E-infinity and Nottale scale relativity
Nobel Laureate Professor Gerard ‘t Hooft is one of a handful of leading mainstream physicists who have an enduring interest in reforming quantum field theory and therefore by necessity amending the basic structure of quantum mechanics itself. This interest continued despite considerable criticism from many other leading scientists, for instance Nobel Laureate D. Gross who recently said in an article entitled “The major unknown in particle physics and cosmology” with respect to ‘t Hooft’s deterministic quantum mechanics that if we were to go back to deterministic classical theory, things would only get worse. In our view this is a total misunderstanding of what ‘t Hooft is trying to do as well as a total misunderstanding of classical mechanics. Classical mechanics is generically deterministically chaotic. In fact this is a well known result of the major work of Poincaré on the three body problems. The four body problem is even worse and at the time of writing there is no computer in the universe which is large enough to even start numerically integrating the problem of how five planets move under mutual influence. It was in fact Nobel Laureate Max Born, the teacher of Werner Heisenberg who disputed that classical mechanics is deterministic. A very readable and relatively short paper discussing the work of Max Born and contrasting ‘t Hooft’s work on deterministic quantum mechanics with Mohamed El Naschie’s work on E-infinity theory is “Deterministic quantum mechanics versus classical mechanical indeterminism”, published in Int. J. of Nonlinear Sci. & Num. Simulation, 8(1), 5-10 (2007). The article of Max Born is “Ist die Klassische Mechanik tatsächlich deterministisch?”, published in Physikalishe Blätter II, pp. 49 (1955). A second paper by El Naschie is in the same volume where Prof. ‘t Hooft’s paper is published entitled “What is quantum mechanics?”, pp. 84-102. The volume is a publication of the American Inst. of Physics, Vol. 905 (2007). El Naschie’s paper is entitled “Deterministic quantum mechanics versus classical mechanical indeterminism and nonlinear dynamics”, pp. 56-66. In our opinion the most important new developments in the work of ‘t Hooft is that he seems to have discovered fractals and consequently El Naschie’s zero and empty transfinite-fractal sets for quantum field theory. In his 2008 paper “Locally finite model for gravity”, published on arXiv: 0804.0328U1[gr-qc]2, April, 2008 he clearly says in the abstract that the model is not finite because the solutions tend to generate infinite fractals. Furthermore in Fig. 2 on page 5 his deficit and surplus angles are the reason for introducing fractal simplicticity a la Mohamed El Naschie and thus Cantor dust and fractals. In fact it is the dilaton, which is known from all superstring theories which bring the scale invariance of the fractal geometry of Nottale and El Naschie through a backdoor into ‘t Hooft’s quantum field theoretical model. He admits trouble due to the negative sign of the new field but that is a little price to pay to enter into the paradise of zero and empty transfinite-fractal sets which are the basic building blocks of spacetime and consequently of E-infinity. We hope ‘t Hooft remains in Cantor paradise with D. Hilbert.
E-infinity Communication No. 43
‘t Hooft dilaton proposal and the inbuilt natural conformal invariance of E-infinity and Nottale scale relativity
Nobel Laureate Professor Gerard ‘t Hooft is one of a handful of leading mainstream physicists who have an enduring interest in reforming quantum field theory and therefore by necessity amending the basic structure of quantum mechanics itself. This interest continued despite considerable criticism from many other leading scientists, for instance Nobel Laureate D. Gross who recently said in an article entitled “The major unknown in particle physics and cosmology” with respect to ‘t Hooft’s deterministic quantum mechanics that if we were to go back to deterministic classical theory, things would only get worse. In our view this is a total misunderstanding of what ‘t Hooft is trying to do as well as a total misunderstanding of classical mechanics. Classical mechanics is generically deterministically chaotic. In fact this is a well known result of the major work of Poincaré on the three body problems. The four body problem is even worse and at the time of writing there is no computer in the universe which is large enough to even start numerically integrating the problem of how five planets move under mutual influence. It was in fact Nobel Laureate Max Born, the teacher of Werner Heisenberg who disputed that classical mechanics is deterministic. A very readable and relatively short paper discussing the work of Max Born and contrasting ‘t Hooft’s work on deterministic quantum mechanics with Mohamed El Naschie’s work on E-infinity theory is “Deterministic quantum mechanics versus classical mechanical indeterminism”, published in Int. J. of Nonlinear Sci. & Num. Simulation, 8(1), 5-10 (2007). The article of Max Born is “Ist die Klassische Mechanik tatsächlich deterministisch?”, published in Physikalishe Blätter II, pp. 49 (1955). A second paper by El Naschie is in the same volume where Prof. ‘t Hooft’s paper is published entitled “What is quantum mechanics?”, pp. 84-102. The volume is a publication of the American Inst. of Physics, Vol. 905 (2007). El Naschie’s paper is entitled “Deterministic quantum mechanics versus classical mechanical indeterminism and nonlinear dynamics”, pp. 56-66. In our opinion the most important new developments in the work of ‘t Hooft is that he seems to have discovered fractals and consequently El Naschie’s zero and empty transfinite-fractal sets for quantum field theory. In his 2008 paper “Locally finite model for gravity”, published on arXiv: 0804.0328U1[gr-qc]2, April, 2008 he clearly says in the abstract that the model is not finite because the solutions tend to generate infinite fractals. Furthermore in Fig. 2 on page 5 his deficit and surplus angles are the reason for introducing fractal simplicticity a la Mohamed El Naschie and thus Cantor dust and fractals. In fact it is the dilaton, which is known from all superstring theories which bring the scale invariance of the fractal geometry of Nottale and El Naschie through a backdoor into ‘t Hooft’s quantum field theoretical model. He admits trouble due to the negative sign of the new field but that is a little price to pay to enter into the paradise of zero and empty transfinite-fractal sets which are the basic building blocks of spacetime and consequently of E-infinity. We hope ‘t Hooft remains in Cantor paradise with D. Hilbert.
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