An up-to-date summary of the most important aspects of Prof. Mohamed El Naschie’s work in quantum physics and E-infinity theory

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 [8], 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 [26]. 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 [27] with the goal to unify gravity with electromagnetism more than 80 years ago [28]. 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 [29]. 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 [30] and the French cosmologist Dr. L. Nottale [31] but both used continuous Peano-like fractals unlike the totally disjointed Cantor sets of El Naschie [32]. 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 [37], 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 [39]. 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 [47]. 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 [48] 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 [25].
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 [7]. Another result connected to cosmology is his analysis that our universe must be a compact manifold [50] as well as his use of the Banach-Tarski theorem [51].
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 [25]. 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 [25], 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.

References:

[1] 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.
[2] Max Jammer: Concepts of Space. Dover Publications, New York (1969).
[3] B.G. Sidharth: The universe of fluctuations (The architecture of spacetime and the universe). Springer, Dordvecht (2005).
[4] M.S. El Naschie: Deterministic quantum mechanics versus classical mechanical indeterminism. In. Journal of Nonlinear Science & Numerical Simulation, 8(1),(2007), p. 5-10.
[5] 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.
[6] Ji-Huan He: Transfinite physics: A collection of publication on E-infinity Cantorian Spacetime Theory. China Education and Culture Publishing Co., Beijing (2005).
[7] 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).
[8] 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).
[9] 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).
[10] D.K. Campbell (Editor): Chaos. AIP, New York (1990).
[11] G. Cherbit (Editor): Fractals. J. Wiley, Chichester (1991).
[12] S. Weinberg: The quantum theory of fields, Parts I, II, III. Cambridge Press. (1998) and (2000).
[13] M.S. El Naschie: Quantum golden field theory – ten theorems and various conjectures. Chaos, Solitons & Fractals. 36(5), (2008), p. 1121-1125.
[14] M.S. El Naschie: Towards a quantum golden field theory. Int. J. of Nonlinear Sci. & Numerical Simulation, 8(4), (2007), p. 477-482.
[15] M.S. El Naschie: High energy physics and the standard model from the exceptional Lie groups. Chaos, Solitons & Fractals, 36, (2008), p. 1-17.
[16] M.S. El Naschie: P-Adic unification of the fundamental forces and the standard model. Chaos, Solitons & Fractals, 38, (2008), pp. 1011-1012.
[17] M.S. El Naschie: On a canonical equation for all fundamental interactions, Chaos, Solitons & Fractals, 36(5), (2008), p. 1200-1204.
[18] 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.
[19] M.S. El Naschie: Complex Dynamics in 4D Peano-Hilbert Space. Il Nuovo Cimento, vol. 1, 107, B. N.5. (1992), p. 583-594.
[20] M.S. El Naschie: Peano dynamics as a model for turbulence and strange nonchaotic behavior. Acta Physica Polonica A, No. 1, Vol. 80 (1991).
[21] M.S. El Naschie: Quantum mechanics and the possibility of a Cantorian space-time. Chaos, Solitons & Fractals, 1 (1991), p. 485-487.
[22] M.S. El Naschie: Renormalization semi-groups and the dimension of Cantorian spacetime, Chaos, Solitons & Fractals, Vol. 4, No. 7 (1994), p. 1141-1145.
[23] M.S. El Naschie: On a class of general theories for high energy particle physics. Chaos, Solitons & Fractals, 14, (2002), p. 649-668.
[24] 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.
[25] 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.
[26] 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.
[27] M.S. El Naschie: Kaluza-Klein unification – Some possible extensions. Chaos, Solitons & Fractals, 37,(2008), p. 16-22.
[28] M.S. El Naschie: On dualities between Nordstrom-Kaluza-Klein Newtonian and quantum gravity. Chaos, Solitons & Fractals, 36, (2008), p. 808-810.
[29] M.S. El Naschie: Superstring Theory: What it cannot do but E-infinity could. Chaos, Solitons & Fractals, 29, (2006), p. 65-68.
[30] G.N. Ord: Fractal Spacetime. J. Phys. A: Math. Gen, 16, (1983), p. 1869.
[31] L. Nottale: Fractal Space-Time and Microphysics. World Scientific, Singapore (1993).
[32] M.S. El Naschie: Quantum mechanics, Cantorian Space-Time and the Heisenberg Uncertainty Principle. Vistas in Astronomy, vol. 37, (1993), p. 249-252.
[33] A. Connes: Noncommutative Geometry. Academic Press, San Diego (1994).
[34] M.S. El Naschie: Penrose universe and Cantorian spacetime as a model for Noncommutative quantum geometry. , Solitons & Fractals, 9(6), (1998), p. 931-933.
[35] M.S. El Naschie: Quantum gravity unification via transfinite arithmetic and geometrical averaging. Chaos, Solitons & Fractals. 35(2),(2008), p. 252-256.
[36] 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.
[37] Y. Tanaka: The mass spectrum of hadrons and E-infinity theory. Chaos, Solitons & Fractals, 27, (2006), p. 851-863.
[38] M.S. El Naschie: Transfinite harmonization by taking the dissonance out of the quantum field symphony. Chaos, Solitons & Fractals, 36, (2008), p. 781-786.
[39] 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.
[40] M.S. El Naschie: From E eight to E-infinity. Chaos, Solitons & Fractals, 35 (2008), p. 285-290.
[41] 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.
[42] M.S. El Naschie: On phase transition to quarks confinement. Chaos, Solitons & Fractals, 38, (2008), p. 332-333.
[43] M.S. El Naschie: On quarks confinement and asymptotic freedom. Chaos, Solitons & Fractals, 37, (2008), p. 1289-1291.
[44] 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.
[45] 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.
[46] 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.
[47] 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.
[48] Ian Affleck: Golden ratio seen in a magnet. Nature, Vol. 464, 18 March (2010), p. 362-363.
[49] M.S. El Naschie: Von Neumann geometry and E-infinity quantum spacetime. Chaos, Solitons & Fractals, Vol. 9, No. 12, (1998), p. 2023-2030.
[50] M.S. El Naschie: Arguments for the compactness and multiple connectivity of our cosmic spacetime. Chaos, Solitons & Fractals, 41, (2009), p. 2787-2789.
[51] M.S. El Naschie: Banach-Tarski theorem and Cantorian micro spacetime. Chaos, Solitons & Fractals, 5(8), (1995), p. 1503-1508.
[52] 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.
[53] 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.
[54] M.S. El Naschie: VAK, vacuum fluctuation, and the mass spectrum of high energy particle physics. Chaos, Solitons & Fractals, 17, p. 797-807 (2003).
[55] Rene Thom: Structural stability and Morphogenisis. Benjamin, London (1975). (In particular see p. 27).