Parts that caught my attention are green. If readers point out interesting passages I will color them green as well.
20th December, 2010.
E-infinity communication No. 65
An up-to-date summary of the most important aspects of Prof. Mohamed El Naschie’s work in quantum physics and E-infinity theory.
As is well known from previous communications and of course his work, Mohamed El Naschie was initially not mainstream but rather a scientific framework which opens the door for a new way to view nature [1-7]. Starting from the fundamentals of nonlinear dynamics , deterministic chaos [9-11] and fractal geometry and by combining that with quantum field theory [12-14] and the standard model of high-energy particle physics [12, 15-17] he created around 1990 the nucleus embryonic form of a novel concept for the physical interpretation of spacetime [18-21]. He replaced the spacetime continuum by a transfinite fractal set of points in an infinite dimensional space [22,23], thus in particular effectively replacing Euclidean geometry by the geometry of infinite dimensional but hierarchal Cantor set [24,25].
This is a generalization of the so called transfinite sets initially introduced by the mathematician Georg Cantor who incidentally also invented set theory which is the basis of all modern mathematics including K-theory and n-categories as well as the very recent 2009-2010 work of El Naschie . It is interesting to note that there had been earlier attempts to introduce spacetime with dimensions larger than four. The first model of that kind was formulated by the mathematicians Kaluza and Klein  with the goal to unify gravity with electromagnetism more than 80 years ago . Their spacetime, however, represents merely a continuous manifold. The work on the ten dimensional super string theory is well known and need not be discussed here . The idea to use a transfinite set of discrete points in combination with an infinite dimensional space is indeed totally new. The only somewhat similar theories are those of the Canadian physics Professor G.N. Ord  and the French cosmologist Dr. L. Nottale  but both used continuous Peano-like fractals unlike the totally disjointed Cantor sets of El Naschie . In the mathematical terminology of the theory of quasi manifolds, El Naschie’s micro spacetime is a noncommutative transfinite quotient manifold akin to that developed in the work of the leading French mathematician Alain Connes [33,34]. From around 1995 onwards El Naschie started developing his ideas to a sophisticated general theory.
One very interesting point about Professor El Naschie’s spacetime is that viewed at macroscopic resolution it appears as if it were only three plus one or four dimensional, thus mimicking the spacetime of classical mechanics and relativity and which agrees in the three plus one case with our daily experience with the macroscopic world. In other words the dimensionality of this spacetime is resolution dependent and the invariance theorem of dimensions must be relaxed. The high dimensional character of spacetime starts to become noticeable only at sub-microscopic resolution. The near infinite number of dimensions is expected to appear only at interaction energies of 10 exp 19 GeV. This is the so called Planck energy, a fantastically high energy – way out of reach of any conceivable accelerator. This is the energy at which we assume that nature becomes simplest as all fundamental forces unify to a single force, sometimes called super force and we are left with only one hypothetical particle, namely the Planckton, that is to say mini black holes with Planck mass of the order of 1019 GeV. El Naschie was able to find an exact value for the coupling constant at the point at which all the forces melt into one [35,36]. At much lower energies he furthermore could use this to determine important physical effects. By studying particles and their interactions in the framework of his theory El Naschie succeeded to determine the mass spectrum of elementary particles at realistic man made conditions showing that the multitude of elementary particles is nothing but a manifestation of the transfinite discrete fractal structure of the vacuum , i.e. the fine structure of a noncommutative geometry [33,34]. Because of the inevitable vacuum fluctuations described by quantum mechanics spacetime itself represents a fluctuation motion. In a sense the excited states of the fluctuating spacetime are what we perceive in our microscopic experiments as elementary particles. While string theory alleges that elementary particles are highly localized vibrations of tiny strings the theory of El Naschie goes even deeper by building the strings themselves from sizzling sets of Cantor points of which spacetime is made. Quantum gravity is interpreted accordingly as the effect of the passing of a fractal time. Using this line of reasoning he was able to determine the maximum coupling constant of quantum gravity to be = 1 while the largest inverse coupling constant for the unification of all fundamental forces was determined to be = 26.18 for the super symmetric case in full agreement with numerically obtained estimations using ad hoc theories. For grand unification without gravity and super symmetry he found on the other hand that = 42.36 and the Planck mass is reduced to Polyakov-‘t Hooft unification magnetic monopole with a mass equal 1016 GeV again in remarkable agreement with many other ad hoc results [24,25,38].
El Naschie does not stop here. His theory not only predicts with astonishing precision the mass spectrum of all elementary particles of the standard model, which by the way thought not to be possible without additional input and hypothesis. He also was able to determine directly a number of fundamental constants. One example definitely worthwhile mentioning is the exact relation between the dimensionless Newton constant and the coupling constants at the electro weak unification . This he achieves by using symmetry, dimensionality and other topological invariants which he translates from continuous geometry to the transfinite fractal geometry of his theory. We can now hope that following this methodology we will be able to understand the deep meaning behind certain magical numbers and values of constants of nature which was convincingly revealed by El Naschie’s papers [24,25,40]. In fact in a series of relatively recent papers El Naschie was able to give an exact analysis for quarks confinement as a spacetime phase transition [41-43]. The negative sign of the strong coupling found first by ‘t Hooft and independently later on by D. Gross and F. Wilczek as well as D. Politzer is now a proven theorem due to El Naschie’s work.
Starting from about 2005 El Naschie’s work acquired an even stronger momentum and ventured in new directions culminating in his set theoretical resolution of various fundamental as well as epistemological problems in quantum mechanics, particularly the state vector reduction or wave collapse which provided considerable discomfort to E. Schrödinger and many generations of excellent scientists after him. The research in this direction was particularly motivated by the connection found between the work on noncommutative geometry, quotient manifold and the transfinite version of E8 E8 exceptional Lie groups [44-46]. Based on the properties of the E-line of exceptional groups, namely E1, E2, E3, E4, E5, E6, E7 and E8 as well as the 17 two and three Stein spaces El Naschie was able to predict the existence of an 8 dimensional Higgs-like field which materializes in at least one or five Higgs particles and a Higgs maximal mass at 169 GeV. We could mention on passing some fascinating numerical results connecting a fuzzy Cantorian fractal Ei exceptional Lie group and Stein spaces to the inverse of the electromagnetic fine structure constant 137 . The first result is that the sum of all the fuzzy Ei groups dimensions amount to exactly 4 multiplied with the inverse electromagnetic fine structure constant 4 548 while the sum of all the fuzzy Stein spaces dimension is 5 685. Based on this result he conjectured against conventional wisdom that there may also exist an E12 exceptional Lie symmetry group with dimension equal (26.18)2 = 685 where (26.18) is the curvature of spacetime. Furthermore he gave the number of gauge gluons on the holographic boundary of his bulk to be equal to the 336 degrees of freedom of Klein’s modular curve plus 3 making a total of 339 particle-like states. In addition El Naschie modeled the quantum wave function as the empty transfinite set with a Menger-Urysohn topological dimension equal minus one (−1) and a Hausdorff dimension equal that means the square of the golden ratio [24-26]. By contrast a quantum particle was modeled by the zero set with Menger-Urysohn dimension zero and a Hausdorff dimension = in full agreement with the work on noncommutative geometry of Penrose fractal tiling as well as some experimental work conducted at the Helmholtz Centre, Germany in cooperation with the University of Oxford which found evidence of the golden ratio in quantum mechanics  as predicted by El Naschie some 20 years ago [24,25]. This identification of the wave function with the empty set and the quantum particle with the zero set is a bold move and a revolutionary discovery with far reaching consequences difficult to assess at the present time for future research. It may be important for the understanding of how the golden mean enters in the theory to know that by a well known mathematical theorem due to Mauldin and Williams, a random Cantor set posses with a probability equal to one a Hausdorff dimension equal to the golden mean. A spacetime geometry based on such random Cantor set as that of El Naschie will therefore have the golden mean at its roots [24,25]. We see here a relation to the KAM stability theorem of nonlinear dynamics which led to the El Naschie VAK or vague attractor of Kolmogorov as the stationary states of quantum mechanics as conjectured for the first time by French topologist Rene Thom [53-55].
To be able to work with totally disjointed point set geometry El Naschie had to develop a mathematical replacement for calculus [24,25]. He did this by inventing his own quantized calculus, similar to what A. Connes did for noncommutative geometry [33,49]. El Naschie replaced calculus with Wyle-Suslin scaling. That way integration became scaling up and differentiation a scaling down in the sense of H. Weyl and M. Suslin’s descriptive set theory .
There are too many other interesting aspects which El Naschie’s work has covered, in fact too many to be able to mention within this short report. However we could not possibly overlook the fact that his theory is one of the at best two or three general theories which can be used to give theoretical justification for the results of the COBE experiment of the microwave background radiation . Another result connected to cosmology is his analysis that our universe must be a compact manifold  as well as his use of the Banach-Tarski theorem .
Professor El Naschie has worked on these fundamental ideas for about 25 years. What appeared at the beginning to be a daring hypothesis has now matured into a fully fledged general theory that convincingly addresses questions which have puzzled scientists around the world for many decades with regard to fundamental aspects of quantum mechanics as well as applications regarding the prediction of new particles and their masses. In fact using his theory El Naschie was able to derive rather than put by hand the inverse electromagnetic fine structure constant 137 as well as the topological dimensionality of spacetime D =4 the average Hausdorff self similarity dimension D = 4 + = 4.23606799 and the spectral dimension of R. Loll and J. Ambjorn D = 4.01999 from first principles . One of the derivations of is based on a neat fundamental equation which is easily reproduced within this report. El Naschie’s fundamental equations say that if the bulk is given by the 496 massless gauge bosons of E8 E8 and particle physics which is 339 gluons lives on the holographic boundary, then adding the 20 principle curvatures or pure gravity of Einstein, the difference of (496) minus (339 plus 20) must be the(137) of electromagnetism [45-47]. I know of no simpler derivation than that. To the best of our knowledge this was never achieved before and is the first time ever to derive the most fundamental constant of nature = 137 from a general theory. Using similar fundamental considerations El Naschie was able to show that the exact number of first massless level of particles using his transfinite version of super strings is 8872 and the classical result of the conventional theory , namely 8064 is only an approximation which ignores the fractal fine structure of spacetime. The value 8872 could be used as an energy functional in some complex way and by applying Weyl scaling to it, many fundamental results follow [25, 52]. This is all explained in details in the corresponding papers. In my eyes El Naschie’s theory constitutes a grand design to a deeper understanding of nature.
We attach to this communication a list of a limited number of the most important publications of Professor El Naschie, chosen from nearly 600 or so papers.
 M.S. El Naschie, O.E. Rössler and I. Prigogine: Quantum Mechanics, Diffusion and Chaotic Fractals. Elsevier Science Ltd. (Pergamon in print), Oxford (1995). ISBN: 0-08-042027-3.
 Max Jammer: Concepts of Space. Dover Publications, New York (1969).
 B.G. Sidharth: The universe of fluctuations (The architecture of spacetime and the universe). Springer, Dordvecht (2005).
 M.S. El Naschie: Deterministic quantum mechanics versus classical mechanical indeterminism. In. Journal of Nonlinear Science & Numerical Simulation, 8(1),(2007), p. 5-10.
 M.S. El Naschie: Average exceptional Lie group hierarchy and high energy physics. In ‘Frontiers of fundamental and Computational physics’. American Inst. of Physics, 9th Int. Symposium Proceedings, 7-9 June (2008). AIP Conferences 101018, pp. 15-20.
 Ji-Huan He: Transfinite physics: A collection of publication on E-infinity Cantorian Spacetime Theory. China Education and Culture Publishing Co., Beijing (2005).
 Ji-Huan He, E. Goldfain, L.D. Sigalotti and A. Jejias: Beyond the 2006 Physics Nobel Price for COBE: An introduction to E-infinity theory. China Education and Culture Publishing Co., Beijing (2006).
 J. Brindly, T. Kapitaniak and M.S. El Naschie: Analytical conditions for strange chaotic and non-chaotic attractors of the quasi periodically forced van der Pol equation. Physica D, 51 (1991), p. 28-38. (North Holland-Elsevier Publications).
 M.S. El Naschie and S. Al Athel: On the connection between statical and dynamical chaos. Zeitschrift fur Naturforshung, 44a (1989), p. 645-650. Reprinted in “Chaotic Oscillators – Theory and Application”, Editor: T. Kapitaniak. World Scientific (1992).
 D.K. Campbell (Editor): Chaos. AIP, New York (1990).
 G. Cherbit (Editor): Fractals. J. Wiley, Chichester (1991).
 S. Weinberg: The quantum theory of fields, Parts I, II, III. Cambridge Press. (1998) and (2000).
 M.S. El Naschie: Quantum golden field theory – ten theorems and various conjectures. Chaos, Solitons & Fractals. 36(5), (2008), p. 1121-1125.
 M.S. El Naschie: Towards a quantum golden field theory. Int. J. of Nonlinear Sci. & Numerical Simulation, 8(4), (2007), p. 477-482.
 M.S. El Naschie: High energy physics and the standard model from the exceptional Lie groups. Chaos, Solitons & Fractals, 36, (2008), p. 1-17.
 M.S. El Naschie: P-Adic unification of the fundamental forces and the standard model. Chaos, Solitons & Fractals, 38, (2008), pp. 1011-1012.
 M.S. El Naschie: On a canonical equation for all fundamental interactions, Chaos, Solitons & Fractals, 36(5), (2008), p. 1200-1204.
 M.S. El Naschie: Statistical mechanics of multi-dimensional Cantor sets Gödel Theorem and Quantum Spacetime. J. of Franklin Institute, vol. 33, No. 1 (1993), p. 199-211.
 M.S. El Naschie: Complex Dynamics in 4D Peano-Hilbert Space. Il Nuovo Cimento, vol. 1, 107, B. N.5. (1992), p. 583-594.
 M.S. El Naschie: Peano dynamics as a model for turbulence and strange nonchaotic behavior. Acta Physica Polonica A, No. 1, Vol. 80 (1991).
 M.S. El Naschie: Quantum mechanics and the possibility of a Cantorian space-time. Chaos, Solitons & Fractals, 1 (1991), p. 485-487.
 M.S. El Naschie: Renormalization semi-groups and the dimension of Cantorian spacetime, Chaos, Solitons & Fractals, Vol. 4, No. 7 (1994), p. 1141-1145.
 M.S. El Naschie: On a class of general theories for high energy particle physics. Chaos, Solitons & Fractals, 14, (2002), p. 649-668.
 M.S. El Naschie: A review of E-infinity theory and the mass spectrum of high energy particle physics. Chaos, Solitons & Fractals, 19, (2004), p. 209-236.
 M.S. El Naschie: The theory of Cantorian spacetime and high energy particle physics (an informal review). Chaos, Solitons & Fractals, 41, (2009), p. 2635-2646.
 M.S. El Naschie: Application of chaos and fractals in fundamental physics and set theoretical resolution of the two-slit experiment and wave collapse. The 3rd Int. Symposium on Nonlinear Dynamics, Organized by Donghua University, P.R. China (2010), p. 7-8.
 M.S. El Naschie: Kaluza-Klein unification – Some possible extensions. Chaos, Solitons & Fractals, 37,(2008), p. 16-22.
 M.S. El Naschie: On dualities between Nordstrom-Kaluza-Klein Newtonian and quantum gravity. Chaos, Solitons & Fractals, 36, (2008), p. 808-810.
 M.S. El Naschie: Superstring Theory: What it cannot do but E-infinity could. Chaos, Solitons & Fractals, 29, (2006), p. 65-68.
 G.N. Ord: Fractal Spacetime. J. Phys. A: Math. Gen, 16, (1983), p. 1869.
 L. Nottale: Fractal Space-Time and Microphysics. World Scientific, Singapore (1993).
 M.S. El Naschie: Quantum mechanics, Cantorian Space-Time and the Heisenberg Uncertainty Principle. Vistas in Astronomy, vol. 37, (1993), p. 249-252.
 A. Connes: Noncommutative Geometry. Academic Press, San Diego (1994).
 M.S. El Naschie: Penrose universe and Cantorian spacetime as a model for Noncommutative quantum geometry. , Solitons & Fractals, 9(6), (1998), p. 931-933.
 M.S. El Naschie: Quantum gravity unification via transfinite arithmetic and geometrical averaging. Chaos, Solitons & Fractals. 35(2),(2008), p. 252-256.
 M.S. El Naschie: On a transfinite symmetry group with 10 to the power of 19 dimensions. Chaos, Solitons & Fractals, 36(3), (2008), p. 539-541.
 Y. Tanaka: The mass spectrum of hadrons and E-infinity theory. Chaos, Solitons & Fractals, 27, (2006), p. 851-863.
 M.S. El Naschie: Transfinite harmonization by taking the dissonance out of the quantum field symphony. Chaos, Solitons & Fractals, 36, (2008), p. 781-786.
 M.S. El Naschie: Quantum E-infinity field theoretical derivation of Newton’s gravitational constant. Int. J. Nonlinear Sci. & Numerical Simulation, 8(3), (2007), p. 469-474.
 M.S. El Naschie: From E eight to E-infinity. Chaos, Solitons & Fractals, 35 (2008), p. 285-290.
 M.S. El Naschie: Higgs mechanism, quarks confinement and black holes as a Cantorian spacetime phase transition scenario. Chaos, Solitons & Fractals, 41, (2009), p. 869-874.
 M.S. El Naschie: On phase transition to quarks confinement. Chaos, Solitons & Fractals, 38, (2008), p. 332-333.
 M.S. El Naschie: On quarks confinement and asymptotic freedom. Chaos, Solitons & Fractals, 37, (2008), p. 1289-1291.
 M.S. El Naschie: The internal dynamics of the exceptional Lie symmetry groups hierarchy and the coupling constants of unification. Chaos, Solitons & Fractals, 38, (2008), pp. 1031-1038.
 M.S. El Naschie: The exceptional Lie symmetry groups hierarchy and the expected number of Higgs bosons. Chaos, Solitons & Fractals, 35(2), (2008), p. 268-273.
 M.S. El Naschie: On D. Gross’ criticism of S. Eddington and an exact calculation of 137. Chaos, Solitons & Fractals, 32, (2007), p. 1245-1249.
 M.S. El Naschie: Rigorous derivation of the inverse electromagnetic fine structure constant = 1/137.036 using super string theory and the holographic boundary of E-infinity. Chaos, Solitons & Fractals, 32, (2007), p. 893-895.
 Ian Affleck: Golden ratio seen in a magnet. Nature, Vol. 464, 18 March (2010), p. 362-363.
 M.S. El Naschie: Von Neumann geometry and E-infinity quantum spacetime. Chaos, Solitons & Fractals, Vol. 9, No. 12, (1998), p. 2023-2030.
 M.S. El Naschie: Arguments for the compactness and multiple connectivity of our cosmic spacetime. Chaos, Solitons & Fractals, 41, (2009), p. 2787-2789.
 M.S. El Naschie: Banach-Tarski theorem and Cantorian micro spacetime. Chaos, Solitons & Fractals, 5(8), (1995), p. 1503-1508.
 M.S. El Naschie: A review of applications and results of E-infinity theory. Int. J. Nonlinear Sci. & Num. Sim., 8(1), (2007), p. 11-20.
 M.S. El Naschie: The VAK of vacuum fluctuation, spontaneous self-organisation and complexity theory interpretation of high energy particle physics and the mass spectrum. Chaos, Solitons & Fractals, 18, (2003), p. 401-420.
 M.S. El Naschie: VAK, vacuum fluctuation, and the mass spectrum of high energy particle physics. Chaos, Solitons & Fractals, 17, p. 797-807 (2003).
 Rene Thom: Structural stability and Morphogenisis. Benjamin, London (1975). (In particular see p. 27).
21st December, 2011.
E-infinity communication No. 66
Some of the most important results of M.S. El Naschie’s research in E-infinity theoretical quantum physics
1. E-Infinity theory (E (∞)) has clearly shown that random Cantor sets are the basic building blocks or atoms of quantum spacetime. The intrinsic topological dimension of these building blocks is zero, its Hausdorff fractal dimension is the golden ratio (0.618033989…) and its embedding dimension is unity.
2. Quantum spacetime is described fully by not one but three dimensions. First a topological (Menger-Urysohn) dimension equal to exactly 4. Second a Hausdorff dimension equal to 4 plus the golden ratio to the power of 3 which means 4.236067977… This is effectively a 4D cube inside a 4D cube and so on ad infinitum. Third a formal dimension equal to infinity. In other words our quantum spacetime is an infinite dimensional but hierarchal Cantor set of measure zero.
3. Because orthodox quantum mechanics is totally insensitive to fractals and does not consider the Cantorian nature of quantum spacetime, many paradoxes follow.
4. The mass spectrum of all known elementary particles of the standard model as well as some composite particles was predicted with astonishing accuracy compared to the known experimental value. Because of KAM theorem they are always a function of the golden ratio as confirmed experimentally by the quantization and the experiment at the Helmholtz Centre.
5. Many of the fundamental constants of nature were derived from first principles. This includes the coupling constant of quantum gravity as well as the electromagnetic fine structure constant and Newton’s gravity constant. In particular a fundamental equation was established relating the Bulk (E8E8) with the holographic boundary and gravity from which = 137 was derived. In addition the inverse coupling of unification of all fundamental forces were found to be = 26 + k ≃ 26.
6. An exact renormalization equation was constructed which derived quarks confinement as an exact result. Confinement is a theorem of the golden quantum field theory.
7. The relationship between Lie symmetry groups as well as two and three Stein spaces and high energy physics was outlined. In particular the role played by E8 in this respect was analyzed. The total number of elementary particles in an extended standard model was shown to be 137 particles. This includes 10 spacetime quasi dimensions. E-infinity P-Adic reasoning was also employed.
8. The theory predicted the existence of 8 dimensional Higgs field with at least one Higgs particle or five Higgs particles becoming manifest. The Higgs mass was determined to be approximately 169 Gev.
9. The Euler characteristic as well as the curvature of the quantum spacetime was shown to be equal to 26 + k 26. In fact even the real part of the dilaton is given by S® = 26. In addition arguments were given to show that our universe is likely to be compact. An alternative theory using instanton density to calculate the first massless particle-like states corrected the classical well known solution of Heterotic string theory, namely 8064 to the exact solution 8872.
10. The stationary state of quantum mechanics was shown to be that of the VAK, i.e. the vague attractor of Kolmogorov.
11. A quantum particle may be modelled as a fractal point represented by the two dimensions of a zero set dim P (0,ϕ) where . The quantum wave is the surface of a fractal point, i.e. a fractal surface and may be represented by the two dimensions of an empty set dim W . The empty set accounts for the paradoxal outcome of the two-slit experiment and the disappearance of the interference fringes.
12. It is reasoned that in a totally disjointed infinite dimensional but hierarchal Cantor-like space time manifold, calculus must be replaced by Weyl-like golden mean scaling which effectively represents a new version of quantized calculus. The set theoretical analogue of this is Suslin scaling of descriptive set theory.
13. El Naschie made several suggestions regarding a Banach-Tarski-like big bang theory based on paradoxal decomposition of spheres.
14. Mohamed El Naschie and his group proved the equivalence of his basic equations of E-infinity theory and the dimensional function of von Neumann’s continuous geometry and A. Connes’ noncommutative geometry.
15. Very recently El Naschie discovered several facts connecting his E-infinity theory to K-theory and the mathematical E-infinity theory of highly structured ring spectrum and the E4 of operad of 4 dimensional cubes.
16. It is conjectured that E-infinity building blocks represent at a minimum a O-category but could also be n-category for n → ∞.
17. The symmetry group of E-infinity is fuzzy and is a fractal symmetry given by the sum of all the dimensions of the E-line exceptional Lie groups 584 and not 496 as well as the sum of all Stein spaces with dimension equal 685. The integer value of all symmetry groups including Stein spaces of the first kind is 3689. When adding the super strings 10 dimensions then one finds (27)(137) = 3699.
18. E-infinity is shown to combine numbers and number theoretical reasoning with topology exactly as in K-theory.
19. Ervin Goldfain demonstrated that measurement of cosmic radiation confirms the correctness of E-infinity theory. El Naschie showed the consistency of the COBE measurement with E-infinity predictions. Malcolm H. MacGregor demonstrated the consistency of the experimental value of the mass spectrum with his alpha quantization of the masses (book published by World Scientific 2007). Alpha quantization is coarse graining of the golden mean quantization. The dependency on the golden mean was demonstrated for the first time experimentally in the Helmholtz-University of Oxford joint work announced earlier this year.
20. Recent work by Ali Chamseddine and Alain Connes (arXiv paper 2010) reinforces some fundamental results of E-infinity theory.
21. The recent results published by Ambjorn and R. Loll are particularly appealing confirmation of E-infinity theory. The spectral dimension of their work, namely 4.02 is a direct Weyl-Suslin scaling in exactly 16 steps from the first massless level of Heterotic strings (8872 integer value) which corresponds to the classical Green-Schwartz and Witten value (8064). The difference comes from the fact that E-infinity uses the entire 8 exceptional Lie group line which adds to 548 while the classical value (8064) considers only a subset of the exceptional Lie groups leading to only 504.
21st December, 2010.
E-infinity communication No. 67
Foreward [sic. This foreword was written by El Naschie for Chaos, Solitons and Fractals Volume 10, Issues 2-3, Pages 163-614 (February-March 1999). It's a remarkable issue, probably deserving its own article.]
In around 1930 Einstein wrote in an article published in the Journal of the Franklin Institute: “ . . . it has been pointed out that the introduction of a space-time continuum may be considered as contrary to nature in view of the molecular structure of everything which happens on a small scale. It is maintained that perhaps the success of the Heisenberg method points to a purely algebraic method of description of nature that is to the elimination of continuous functions from physics. Then however, we must also give up, by principle the space-time continuum. It is not unimaginable that human ingenuity will some day find methods which will make it possible to proceed along such a path. At present time however, such a program looks like an attempt to breath in empty space”. I must say, seeing the present collection of articles on strings and quantum field theory, and reading the preceding lines almost seventy years after Einstein first wrote them, it is becoming evident to me that we have came a long way since then. We can no longer hide that we have “almost” given up the continuum. Nevertheless we are not breathing in vacuum thanks to the pioneering work of physicists like D. Bohm, A. Wheeler, R. Feynman, D. Finkelstein, R. Penrose, E. Witten, L. Nottale and many others. These scientists have virtually revolutionized our conventional notion of spacetime geometry. The task could not have been successful in revolutionizing quantum physics if it would not have been for an equally vigorous and not less revolutionary development on the “pure” side by mathematicians like G. Cantor, K. Menger, F. Hausdorff, J. von Neumann, M. Atiyah, W. Thurston, V. Jones, S. Donaldson, M. Freedman and A, Connes.
Finally, the unbelievable advances in the power of our electronic computational and experimental capabilities have enabled us to make progress in fields where the mere mention of a real experimental verification or computer simulation would have triggered the laughter of disbelieve only three or four decades ago.
So it is not disrespectful that we are trying and have partially proven many of the giants of quantum physics to be wrong on various occasions. Schrodinger was wrong to think that we could not experiment with a single quantum particle. Heisenberg was wrong to think that quantum particles have at best a reduced reality, and at worse there are no real particles at all. We know now that elementary particles are different from classical objects, but they are very real and can be captured and tamed. Finally even Feynman and other well-known names of theoretical physics were wrong about superstrings. There are already reasonably realistic proposals for indirect experimental verification of strings which could be carried out in the foreseeable future.
In the present volume most of the contributions are basically challenging many of Einstein’s ideas about spacetime being a continuum something which was also taken for granted in classical quantum physics. This is however the heart of Einsteinian thinking because he has always had the courage and the foresight to challenge any conventional wisdom if he thought that there ought to be a better way to describe nature. Famous names never worried Einstein. Following this example I would like to gather my courage and state clearly that I believe that it is wrong to think that micro spacetime has a fixed finite dimension.
In this sense the present volume is in the best tradition of Einstein’s thinking in more than one sense and I would like to extend our thanks to all those who have contributed to this volume and give our highest esteem to Prof. Carlos Castro on behalf of the Editorial Board for his excellent effort in producing this issue. Particular words of thanks are also due to Prof. Sir R. Penrose for contributing to the present volume, even though he was travelling abroad and under enormous time pressure.
M.S. El Naschie
22nd December, 2010.
E-infinity communication No. 68
The dilaton in Mohamed El Naschie’s early papers and the very recent paper by G. ‘t Hooft incorporating the dilaton in quantum field theory
We already very briefly discussed the recent proposal of Nobel Laureate in Physics (1999) Gerard ‘t Hooft regarding the introduction of a dilaton scalar field into his version of deterministic quantum field theory. In the present short note we return to this subject and comment on it again, this time in the light of El Naschie’s early papers about the role of dilaton in E-infinity theory.
The idea of a scalar dilaton field and the hypothetical particle dilaton goes back most probably to the work of Kaluza and may be earlier on in connection with the work of the Finish theoretical physicist and colleague of Kaluza, Gunar Nordstrom. The writers remember seeing the television science program of Prof. El Naschie during which he told an engaging story about the man who invented the fifth dimension. This man is not T. Kaluza but it is G. Nordstrom. However Nordstrom space is flat which is extremely important for ‘t Hooft’s work as pointed out by El Naschie. In his first paper dedicated to the subject in 2008 El Naschie considered certain dualities pertinent to the Nordstrom-Kaluza-Klein theories which were obviously important for quantum field theory. The paper in question is “On dualities between Nordstrom-Kaluza-Klein, Newtonian and quantum gravity”, CS&F, 36 (2008), pp. 808-810. A second paper followed where the need for a coupling constant and consequently coupling “particle” between gravity and electromagnetism was discussed. This turned out to probably be a pseudo Goldstone boson. This paper was entitled “Kaluza-Klein unification – some possible extensions”, CS&F, 37 (2008), pp. 16-22. El Naschie calculated the number of elements of the matrix and found that 14 of them are due to Einstein + Maxwell. The additional pseudo Goldstone boson resulting from fusing both theories is what makes the 14 + 1 = 15. Subsequently El Naschie showed in this paper that the K-K theory can be generalized to Witten’s five Brane in eleven dimensions theory and calculates the well known 528 states.
The preceding discussion may be considered the beginning of the motivation for a dilaton field and particle. The direct predecessor was thus the Jordan, Brans and Dicke scalar. The scalar appears in the form of a dilaton in all perturbative string theories. The exception is M theory. Dilaton is important for conformal invariance. That means dilaton is important for scale invariance. The introduction of the dilaton field in quantum field theory is almost equivalent to the introduction of L. Nottale’s scale relativity into it or making quantum field theory a little more “fractal field-like”. Using the dilaton field one gains two things indirectly. First a fractal-like scale invariance and second an empty set-like vacuum with an expectation value and negative dimension. This is of course a very loose non-mathematical description of the hidden idea behind the introduction of the dilaton. The dilaton featured in El Naschie’s older publication at many locations to show the connections between his Cantorian spacetime approach and the dilaton for which there is no urgent need what so ever in E-infinity theory because Cantorian set theoretical formulation has a natural inbuilt dilaton in it so to speak. The first substantial discussion of the dilaton in E-infinity formalism was given by Prof. El Naschie ten years ago in his first outline for a general transfinite spacetime theory in a paper entitled “A general theory for the topology of transfinite Heterotic strings and quantum gravity” published in CS&F, 12 (2001), pp. 969. In section 7.3 on page 975 under the heading “Relation to 10 dimensional Newton’s constant and dilaton” El Naschie gave the exact transfinite value of g(dilaton) to be equal 0.723606797. That means 10 copies of g gives the dimension of a Milnor fractal sphere in seven fractal dimensions. Subsequently he calculates the real part of the dilaton S(D) namely S(R) and finds S(R) = 26.18033989 = 26 + k where k is a function of the golden mean to the power 3, namely k = 0.18033989. Incidentally in this paper El Naschie considers at some length the important work of J. Ambjorn of dynamical triangulation and introduces fractal fuzziness to it, something which Ambjorn et al incorporated in their excellent paper published in Scientific American in 2008 with Dr. R. Loll. In a second, mainly survey paper entitled “On a class of general theories for high energy particle physics”, CS&F, 14 (2002), pp. 649-669, El Naschie consider the dilaton again. For instance in Table 5 on page 657 he reports on the values of S® and T® estimated in a paper by F. Quevedo. These were found to be S (R) 25 and T(R) 1. These are quite near the exact transfinite values found by El Naschie, namely S(R) = 26 + k. It is tempting at this point to speculate that S(R) + T(R) = 26 + k. This is supported by the numeric (S(R) 25) + (T(R) 1) = 26 but the more important fact is that the frequent appearance of 26 in E-infinity theory must have some hidden reason. We recall the 26 + k = 26.18033989 was found to be the value for all of the following. It is the value for the bosonic dimension of Heterotic string theory, the inverse super symmetric unification of all fundamental forces, the curvature of the core of E-infinity spacetime, the Euler characteristic of the fuzzy K3-Kähler manifold of E-infinity theory as well as the instanton density of the same. Now that even the real part of the dilaton turned out to be 26, we think the reason is the topology and dynamics of the holographic boundary of E-infinity. This boundary is a compactified Klein modular curve with 336 + 16k degrees of freedom equal nearly to 336 + 3 = 339. Two of the most important orbits in this modular curve are the 42 and the 26 orbit.
In conclusion we think incorporating the dilaton in quantum field theory ‘t Hooft is essentially realizing very deeply that his deterministic quantum mechanics is a homomorphism of E-infinity indetermanistic, chaotic and fractal mechanics.
14th January, 2011.
E-infinity communication No. 69
Physics and metaphysics in transfinite set theory and E-infinity
Before anyone jumps to conclusions and misconstrues what we want to say here, we should better start by clarifying first two points. Metaphysics has nothing to do here with the supernatural. In fact at the time of G. Cantor for instance, metaphysics meant more or less fundamental science or stringent mathematical physics dealing with foundational issues. Naturally questions dealing with philosophical issues were included. Equally naturally, the role of religion and the existence of a higher being referred to as Deity were touched upon. In fact fundamental science used to be called natural philosophy and even until our time universities awarded degrees in all possible fields but they are all called doctor of philosophy and this could be in engineering rather than philosophy.
The second point is regarding Mohamed El Naschie who is deeply religious and counts himself as being a Moslem without any doubt but that is all. He is deeply distrustful of all forms of organized religion as well as mingling science, politics and religion in a demagogical way. Perhaps we should give two examples of his writing on this subject before we go any further. At the end of his paper “A note on quantum field theory and P-Branes in n dimensions published in CS&F, Vol. 10 (8), pp. 1413-1417 (1999) he wrote “It may seem that by reducing the contra intuitive physics of quantum mechanics to the contra intuitive geometry of the space of quantum mechanics, nothing is gained. However this is not the right way of looking at the problem. The physics of quantum mechanics has frequently opened the door to the wrong kind of mysticism which is wholly foreign to scientific thinking. By contrast all of the mind-baffling and contra intuitive results of modern geometry in hyperspaces are the end product of the most stringent rational thinking and leave no place for uncalled-for mystical beliefs what so ever. Thus we think a great deal is achieved by viewing quantum physics in what we think is the right geometrical way proposed here”. I think Prof. El Naschie made it clear where he stands. An additional quotation at the end of another paper by him may help to dispel any lingering doubts about a basically renaissance Egyptian who tolerates religious beliefs of the personal type. In CS&F, Vol. 10(2-3), pp. 567-580 (1999), at the end of a paper entitled ‘Nuclear spacetime theories, superstrings, monster groups and application’ El Naschie says “Philosophically inclined theoretical physicists have frequently wondered if, in creating the world, God had any choice in determining it. Assuming that our hypothesis about E-infinity geometry is correct, then it would seem to us that once God had created logic, he had as good as no option but to create the universe as we see it now.”
Having said all that, the situation is slightly different when we come to consider transfinite set theory, the continuum hypothesis, the axiom of choice and particularly the meaning of infinity. There is no doubt that there are two kinds of personalities among serious scientists. Those who do not believe in any form of creation or any form of non-materialistic existence and those who believe in non-materialistic existence as well as the possibility of creation. The first kind is more or less like Kronecker, the famous German mathematician. The second kind is more like G. Cantor. There are people who may be in between. In E-Infinity Mohamed El Naschie finds that the spiritual value and even religious beliefs can be an inspiration to solving extremely difficult problems. In general, one has to keep religion out of science. However there is one exception where we must be tolerant and that is when someone identifies the Almighty with the set of all sets. In fact El Naschie believes that the concept of a set of all sets leads to severe contradictions in set theory. He accepts the class of all classes [I think this was meant to say "He accepts the class of all sets"] but not the set of all sets.
If we search for a deep difference between Kronecker and ‘t Hooft on the one side and Cantor and El Naschie on the other side, then the line separating them scientifically is that Cantor and El Naschie believe in the infinite and yes they believe in God. This is indirectly reflected in the philosophical depth of their work where there is no more any artificial separation between physics and mathematics. In fact we recently spoke to El Naschie and he said in a recent interview that God is without doubt a mathematician and definitely not a theoretical physicist. In the same way El Naschie regards the mathematician Prof. A. Connes as the deepest physicist working among us today. In fact the history of set theory and even the mundane descriptive set theory and naturally the history of the Moscow School of Mathematics could not be understood without the role of religion. However we will talk about that in other communications.
17th January, 2011.
E-infinity communication No. 70
On ‘t Hooft’s recent work and the development of Yang-Mills instanton via El Naschie’s E-infinity theory
E-infinity theory gave considerable attention quite early on to integrating Yang-Mills theory and ‘t Hooft’s instantons into its scale invariant fractal frame work. It is probably a fact to say that Mohamed El Naschie’s relations to ‘r Hooft must have stimulated him to attempt this integration. On the other hand it must equally be a fact that the recent work of ‘t Hooft in which he introduces a dilaton field to the Lagrangian of quantum field theory was motivated primarily by the realization which he must have gained by reading El Naschie’s work on a “fractal-Cantorian” instanton [oh, this is a gem].
In the present short communication we would like to direct the reader to the relevant sources from which he can come to the above very important conclusion. Dilaton is used extensively in string theory. However it would be a misconception to think that ‘t Hooft used it in the same way. That would be underestimating the importance of ‘t Hooft’s proposal which in our view is nothing less than “fractalizing” the frame work of classical quantum field theory.
El Naschie felt relatively late that Lagrangian formulation in particle physics was wrongly elevated to the status not of merely an alternative to the differential equations but to a supreme and the only right way to do theoretical particle physics. El Naschie was originally trained in the rigorous tradition of the classical theory of elasticity and Koiter’s general theory of elastic stability where a potential energy formulation goes hand in hand with differential equations and under estimated the deep seated prejudice of the community of particle physics, particularly those working in quantum field theory and that they regard a Lagrangian as sacred. We do not know exactly when he started addressing these problems but latest in 2006 he wrote a paper “Intermediate prerequisites for E-infinity theory”, CS&F, 30 (2006), pp. 622-628 in which he attempted in section 5 to clarify the issue of a Lagrangian in E-infinity theory. It is in this paper where he introduced the concept of a fuzzy Lagrangian and a fuzzy or average instanton. In the above mentioned paper as well as somewhat earlier and later papers, El Naschie made a profound discovery liking string states with instantons. The most elementary form of this discovery was stated in earlier communications and is essentially the equality of multiplying left and right movers to obtain the first massless level of Heterotic string particle-like states, namely N(0) = (504)(16) = 8064 as well as multiplying the degree of freedom of the noncompactified holographic boundary given by Klein-modular curve (336) and the instanton density of K3 Kähler manifold (24) so that we find the same value of No namely N(0) = (336)(24) = 8064. A former student of El Naschie, Dr. A. Elokaby explained the above theory in a very lucid paper “On the deep connection between instantons and string states encoded in Klein’s modular curve”, CS&F, 42 (2009), pp. 303-305. Having come that far one can easily find the exact N(0) by remembering that the exact instanton density for a fuzzy K3 Kähler is not 24 but 24 + 2 + k, which means 26 + k = 26.18033989. Similarly the 336 should be 336 + 16k = 338.8854382. Consequently, No is No = (338.8854382)(26.18033989) = 8872.135957. This is 808.1359569 states or instantons more than the classical 8064 found first by Green, Schwarz and Witten.
We could do the same for Heterotic strings by noting that 504 should be (4)( ) = 548.328157 where = 137 + ko which is the sum of all the exceptional Lie symmetry groups in the E-line (548), plus 0.328157 transfinite or fuzziness added to them. In addition the 16 should be changed to 16 + k = 16.18033989. That way one finds N(0) = (4 )(16 + k) = 8872.135956 exactly as before. El Naschie reasoned that the same procedure could be applied to ‘t Hooft’s famous instanton result, namely I(instanton) = = 78.95683521. Since the instanton density 26 + k must remain the same, all that changed is the dimensionality of the holographic boundary. In the instanton interpretation of the Heterotic strings theory leading to No = 8064 the boundary had 336 degrees of freedom (or 338.885 ≃ 339 in the compactified fuzzy exact case). For ‘t Hooft instanton our space is only 3 + 1 dimensional as is the case in classical quantum field theory. Therefore the holographic boundary must be d = (3 + 1) ̶ 1 = 3. The exact fully Yang-Mills instanton value must therefore be <> = (3)(26 + k) = 78.54101967. This may also be expressed as ( /2) + 10 = 78.54101967. Note that the classical value of ‘t Hooft is slightly larger by 0.41581554. Notice also that if we take only the integer value, namely 26 one finds <> = (3)(26) = 78. This is exactly equal to the dimension of the exceptional E6 symmetry group of the exceptional E-line. Another note worthy observation is that using the classical instanton density 24 instead of 26 one finds <> = (3)(24) = 72. This is the roots number of E6. In other words the roots number is the lowest approximation for obtaining a value for I.
Summing up we can say all the values 72, 78 as well as are equal to the volume of 16 five dimensional Ball or four dimensional spheres. The only exact value is the fuzzy value obtained by El Naschie, namely <> = (3)(26 + k) = 78.54101967. These are the elementary and obvious but deep results of Prof. Mohamed El Naschie which prompted ‘t Hooft’s courageous and mathematically elaborate and justified attempt to patch up classical quantum field theory. We, the proponents of E-infinity, wish Prof. ‘t Hooft to succeed because it is firstly good for science and secondly because it is in the interest of E-infinity’s world wide acceptance.
In conclusion we draw the attention of the reader to various important publications on the subject:
1. The relation between Yang-Mills instanton and the exceptional groups is given by El Naschie in CS&F, 38 (2008), pp. 925-927.
2. A useful dictionary translating gauge field language to fiber bundle language may be found in CS&F, 30, (2006), pp. 656-663.
3. The Weyl scaling of E-infinity is explained in CS&F, 41 (2009), pp. 869-874.
4. The relation between ‘t Hooft’s work and that of El Naschie is discussed in CS&F, 38, (2008), pp. 980-985.
5. It should be noted that Prof. El Naschie is by no means the first to consider a connection between Yang-Mills theory and fractals. In fact already in 1992 there was a paper in Physical Review Letters (23 March, 1992, Vol. 68, No. 12, pp. 1811-1813) with the title “Evidence for a Yang-Mills fractal” in which numerical evidence for fractals in a Yang-Mills system in (2 + 1) dimension was observed. Even before that T. Kawabe and S. Ohta from Japan wrote a paper on Chaos in Yang-Mills theory “Onset of chaos in time-dependent spherically symmetric SU(2) Yang-Mills theory”, Physical Review D, Vol. 41, No. 6, 15 March, 1990, pp. 1983-1988.
18th January, 2011.
E-infinity communication No. 71
Fractal spacetime – some historical remarks to fractal spacetime deniers
We occasionally hear that fractal spacetime has nothing to do with quantum physics. These people may even go as far as saying that fractals have nothing to do with physics. Of course everyone is free to think what he likes but the unpleasant surprise comes when one finds out that most of those who make these statements are mentors of mainly those who work on fractals in physics although they are extremely inventive when it comes to giving different names to these things which we call scale relativity, Cantorian spacetime and fractal spacetime. This point could not be explained by mathematically based science. It needs the tool of social sciences and psychoanalysis. It could of course be far more straight forward than that and is mainly related to science policy and science funding coupled to the very scarce resources available to scientific research in general and theoretical and mathematical physics in particular. This subject is not particularly the source of joy to most of us and we will not dwell on it any further. However it is important to recall some important facts about the history and substance of fractal spacetime theory.
The first comprehensive paper published in an international journal with the title Fractal Spacetime was by the English-Canadian Garnet Ord who discussed this subject with Nobel Laureate Richard Feynman and was strongly influenced by Feynman’s views on the subject. Ord’s paper was published in 1983 in one of the journal of the Institute of Physics, namely J. Phys. A: Math. Gen., 16 (1983), pp. 1869-1884 and was entitled “Fractal space-time: a geometric analogue of relativistic quantum mechanics”. A little later and seemingly independently, a young but well known French astrophysicist Laurent Nottale published in 1989 a paper entitled “Fractals and quantum theory of spacetime” in Int. J. Mod. Physics !, 4, (1989), pp. 5047-5117. This was a sequel and generalization of his 1984 paper with J. Schneider entitled “Fractals and nonstandard analysis”, J. Math. Phys., 25, (1984), pp. 1296-1300. It seems however that Nottale had the same ideas as Ord at nearly the same time but for some reason he could not publish his paper except later on. His famous book “Fractal Space-Time and Microphysics” published by World Scientific in 1993 made up for this delay. Mohamed El Naschie on the other hand is at least ten years older than Nottale and Ord and came to fractals in quantum mechanics via nonlinear dynamics. El Naschie obtained his Ph.D. in engineering and started working in physics much later after having reached the position of Full Professor of structural engineering. All the same he was keenly interested in philosophy and science in general and quantum mechanics in particular. An encounter as a student of engineering in Hannover, Germany with the work and personality of Werner Heisenberg and K. von Weizacker changed his scientific interests completely. However serious work had to be postponed until he became aware of the mathematics of Cantor sets and their relation with number theory, topology and symmetry groups. El Naschie moved from applied mechanics to applied nonlinear dynamics, chaos, singularity theories and fractals to quantum mechanics and finally high energy physics. His first efforts were in understanding turbulence via fractals and Cantor sets in a way similar to the work of Kolmogorov. He was also familiar with stability theory of Poincaré, Köiter and Andronov as well as R. Thom’s catastrophe theory which helped him to move from engineering to physics. Turbulence was the first problem which Heisenberg tackled but could not solve. El Naschie used turbulence as a paradigm for vacuum fluctuation following Wheeler. The first paper by El Naschie which was relevant to fractal spacetime indirectly was the 1991 paper entitled “Multi-dimensional Cantor-like sets and ergodic behavior” in Speculations in Science & Technology, Vol. 15, No. 2, pp. 138-142. This was followed by two papers with direct relevance to quantum mechanics. First “Quantum mechanics and the possibility of a Cantorian spacetime” published in Chaos, Solitons & Fractals, Vol. 1, No. 5 (1991), pp. 485-487. This was followed by “Multi-dimensional Cantor sets in classical and quantum mechanics”, CS&F, Vol. 2, No. 2, (1992), pp. 211-220. After that an important paper in computational and applied mechanics was published entitled “Physics-like mathematics in four dimensions – implications for classical and quantum mechanics”, Computational and Applied Math II. W.F. Ames and P. van der Houwen (Editors), Elsevier (North Holland), (1992), IMACS.
Since these efforts by Ord, Nottale and El Naschie to establish a new field, namely the field of quantum-fractal physics, many papers were published intermittently. The author of these papers seems to be unaware of previous similar publications by the trio Ord-Nottale-El Naschie or for some reason or another chose not to make reference to these contributions. Whilst this was maybe understandable before the world wide web, in the meantime it is not understandable at all. In what follows we list some of the interesting work on fractal quantum and high energy physics which shows that this field is vibrant and is becoming unusually combative in a not very pleasant manner, at least occasionally.
There are many papers in quantum mechanics which mention the concept of the Hausdorff dimension as well as the Hausdorff dimension of a quantum path. These papers are relatively well known from the work of scientists like Parisi and will not be mentioned here. We may start by mentioning the paper of M. Wellner “Evidence for a Yang-Mills fractal” in Physical Review Letters, Vol. 68, No. 12 (1992), pp. 1811-1813. An earlier similar paper is “Onset of chaos in time-dependent spherically symmetric SU(2) Yang-Mills theory”, Physical Review D, Vol. 41, No. 6, (1990), pp. 1983-1988. A 2010 paper by S. Carlip entitled “The small scale structure of spacetime” (arXiv: 1009.1136VL[gr-qc]6 Sept 2010 makes the usual remark about Wheeler foam, mentions Loll and Amjborn’s work and completely overlooks the work of Ord-Nottale and their associates. Similar remarks apply to the work of Fotini Markopoulou. The paper with the almost identical title of that of Nottale and Ord “Fractal space-time and black-body radiation” published in Astrophysics and Space Science, 124 (1986), pp. 203-205 by A. Grassi, G. Sironi and G. Strini is also oblivious to the work of Ord and Nottale. Strangely Benedetti’s paper which uses the same quantum group concept of El Naschie did not mention the work of Ord, Nottale or El Naschie. The paper on fractal spacetime by O. Lauscher and M. Reuter seems to be totally unaware of the work of Nottale and Ord let alone El Naschie. Equally very strange is the absence of any reference to Nottale or Ord in the paper “Fractal geometry of quantum spacetime ar large scales” by I. Antoniadis, P. Mazier and E. Mottola although the first author works in France where L. Nottale is well known. We have not mentioned more than one percent of the large body of literature on fractals in quantum mechanics, relativity and quantum gravity. This shows that fractals are indeed relevant and may be too relevant to the extent that competition is not only fierce but slightly unfair to say the least.
We have not mentioned fractals in all other fields of physics. There are more publications on fractals in physics than on quantum mechanics when you consider that fractals were discovered in physics no more than 25 years ago while quantum mechanics is with us since more than 80 years. For all these reasons we think that the hard work our group had done and continues to do is more than justified and worthwhile.
In conclusion we should mention one of the most important recent papers by G.N. Ord “Quantum mechanics in two dimensional spacetime: What is a wave function”, published in Annals of Physics, 324 (2009), pp. 1211-1218. It may also be interesting to give Mohamed El Naschie’s answer to Ord’s question using his transfinite set theory. A wave function is an empty set with a topological Menger-Urysohn dimension equal minus one ( ̶ 1) which is the surface area or the neighborhood cobordism of the quantum particle which is the zero set. Two cultures divided by a common language, namely mathematics.
19th January, 2011.
E-infinity communication No. 72
Category theory in El Naschie’s E-infinity
Category is mentioned explicitly in connection with ribbon category in El Naschie’s important paper on quantum groups. Utilizing results connected to tensor categories he arrives at a quantum dimension which takes the exact value of 4 when q is set equal to the inverse of the golden mean. The theory is strongly connected to knot theory, Hopf algebra and of course tangle category. The close connection to noncommutative geometry and E-infinity becomes trivially obvious when setting the Eigenvalue q = ½ in El Naschie’s expectation value of the Hausdorff dimension which is identical to A. Connes’ noncommutative dimension and finding that the dimension in this case is exactly 4 which corresponds to exactly 4 for the quantum dimension provided q = ½ is replaced by q equal to the inverse golden mean. The duality between the two dimensions becomes apparent when one realizes that for q equal the golden mean the noncommutativity Hausdorff dimension becomes equal to the famous dimension of an infinitely fractal four dimensional space, namely 4.23606799. This situation is explained in detail in “Quantum groups and Hamiltonian sets on a nuclear spacetime Cantorian manifold”, CS&F, Vol. 10(7), (1999), pp. 1251-1256. In general however El Naschie does not make explicit reference to to n-categories nor its coffee shop direct by Dr. J. Baez. The fact is E-infinity results are all obtained by elementary methods but could have been obtained using categories theory. It is also a fact that Prof. El Naschie did not refer to many relevant papers in category theory which may have ignited the unprecedented anger of Dr. John Baez of Riverside University, California. In fact we in E-infinity feel that we should consider n-categories in future despite the claim of some that it is simply too abstract to be physics. This is a claim which we do not accept in principle because at the deep level of trying to understand the building blocks of nature and figure out the meaning of the quantum, any separation between physics and mathematics is artificial and in general mathematical, logical reasoning is paramount.
19th January, 2011.
E-infinity Communication No. 73
K-theory (of Grothendieck) as a theoretical framework of E-infinity and extension of the Riemann-Roch index theorem
As is well known, K-theory as well as the work of M. Atiyah are extensions of the celebrated Riemann-Roch index theorem. K-theory is largely due to the mathematician (almost cult figure) Alexander Grothendieck (born 1928, Field Medal 1966) who presently lives as a recluse and who declined receiving the Crafoord Prize or in fact any prizes since 1988 because he was disillusioned with the scientific community and society at large. He even resigned his prestigious position at the very prestigious IHES in France. Those interested in the very deep mathematical foundations of E-infinity theory should consider the role of the Riemann-Roch index theorem in the work of Mohamed El Naschie. During his Cambridge time, El Naschie published in 2000 a paper entitled “On the unification of Heterotic strings, M-theory and E-infinity theory”, CS&F, Vol. 11 (2000), pp. 2397-2408. In this paper El Naschie dealt with various mathematical aspects including the Riemann-Roch theorem which he evaluated on page 2406 for n = 4 and q = 1 + ϕ where = is the golden mean and found that the index is
Index = (2 n ̶ 1)(q ̶ 1)
Numerical evaluation taking 1200 times by Castro and Granik using gamma function to find the average Hausdorff dimension of E-infinity spacetime gave 4.32 which is very close to the above exact theoretical value. Now the exact Hausdorff dimension of E-infinity spacetime core is well established and is given by 4.236067977 which is quite close to both the Riemann-Roch index as well as the gamma function approximation using these functions as weights in analogy to the golden mean weight of the E-infinity derivation. The reason for this proximity between 4.23606799 of E-infinity and 4.3262 of the Riemann-Roch index and the 4.32 of the gamma function numerical approximation is explained as follows by Prof. El Naschie: As observed by André Weil there is a deep connection between the topological characteristic of a variety and its number theoretic aspect and this blend between numbers and topology is a main line in E-infinity theory. The same is true for K-theory as we will discuss shortly. In a second paper published a year later in 2001 El Naschie returned to Riemann-Roch, this time in connection to moduli spaces. The paper is entitled “Remarks to moduli spaces, virtual dimensions and Heterotic strings”, CS&F, 12 (2001), pp. 1607-1610. He considers on page 1607 moduli spaces of bundles and derives the dimensions of super strings in integer form. Subsequently he derives the exact transfinite values including the Euler characteristic of E-infinity, namely 26 + k = 26.18033989. We repeatedly stressed that El Naschie showed that Penrose tiling is the prototype example for both Connes’ noncommutative geometry as well as E-infinity theory.
Let us look a little closer at the K-theory connection of Penrose tiling. E-infinity researchers are familiar with the construction of a Cantor set. However only those working in deterministic chaos will be familiar with Steve Smale’s horseshoe. From this construction of a horseshoe one will realize the relation between numerical sequences and a Cantor set. The Smale construction as a mapping between a Cantor set and infinite sequences of zeros and ones. There is an important result permitting to parameterize a Penrose tiling with a set K of infinite sequence of zero and one satisfying certain conditions which will turn out to describe the Connes dimensional function or El Naschie’s bijection formula. This K is a compact and of course totally disjoint space homeomorphic to a Cantor set and we also have a relation of equivalence on R. The space x of Penrose tiling is the quotient space x = K/R. In E-infinity theory, it is easily shown that the dimension of K is given by the dimension of the real line plus the dimension of the resulting Cantor set while the dimension of R is equal to that of the real minus the dimension of the Cantor set. Assuming a random Cantor set with a Hausdorff dimension equal ϕ as per Mauldin-Williams theorem we find that
dim x = dim K/dim R
= (1 + ϕ)/(1 ̶ ϕ)
= 4 + the golden mean to power 3
This is the well known Hausdorff dimension of E-infinity and corresponds to a 4 Menger-Urysohn topological dimension. For further more technical discussion of the relation of Penrose tiling, noncommutative geometry and K-theory, the reader is referred to the nice mathematical papers of D. Bigatti for instance. El Naschie was frequently asked ‘what is the simplest most direct way to start making contact with E-infinity theory?’ Here is his answer. The deep problems are the continuum, the denumerable infinity and the not denumerable infinity which is related to the cardinality of the continuum. In E-infinity we have our cake and eat it so to speak. We have countably infinite numbers of Cantor sets which we sum over. In each Cantor set we have uncountably infinite Cantor points. Thus we are counting not over points but over equivalence classes. In his latest work to salvage quantum field theory ‘t Hooft adopted this view point and introduced a dilaton field akin to that of El Naschie’s compactified Klein modular space. This space has an inbuilt dilaton field and is homeomorphic to Penrose fractal tiling, i.e. to the x = K/R space. For a new theory connecting string states with the theory of instantons using x = K/R, the reader is referred to A. Elokaby’s paper “On the deep connection between instantons and string states encoded in Klein’s modular space”, CS&F, 42 (2009), pp. 303-305.