IJAA  Vol.6 No.2 , June 2016
On a Fractal Version of Witten’s M-Theory
Abstract: Starting from Witten’s eleven dimensional M-theory, the present work develops in an analogous way a corresponding dimensional fractal version where . Subsequently, the new fractal formalism is utilized to determine the measured ordinary energy density of the cosmos which turns out to be intimately linked to the new theory’s fractal dimension via non-integer irrational Lorentzian-like factor: where is Hardy’s probability of quantum entanglement. Consequently, the energy density is found from a limiting classical kinetic energy to be Here, is ‘tHooft’s renormalon of dimensional regularization. The immediate logical, mathematical and physical implication of this result is that the dark energy density of the cosmos must be in astounding agreement with cosmic measurements and observations.
Cite this paper: El Naschie, M. (2016) On a Fractal Version of Witten’s M-Theory. International Journal of Astronomy and Astrophysics, 6, 135-144. doi: 10.4236/ijaa.2016.62011.

[1]   W. Rindler: Relativity (Special, General and Cosmological). Second Edition, Oxford University Press, Oxford, 2006.

[2]   A. Zee: Einstein’s Universe. Oxford University Press, Oxford, UK, 1989.

[3]   P.S. Wesson: Five Dimensional Physics, World Scientific, Singapore, 2006.

[4]   S. Hawking and R. Penrose: The Nature of Space and Time. Princeton University Press, Princeton, USA, 1996.

[5]   S. Weinberg: Cosmology. Oxford University Press, Oxford, UK. 2008.

[6]   E.F. Tayler and J.A. Wheeler: Spacetime Physics. W.H. Freeman Company, New York, USA, 1966.

[7]   J. Moffat: A Physicist Goes Beyond Einstein. Smithsonian Books, New York, USA, 2008.

[8]   M. Carmeli: Cosmological Relativity. World Scientific, Singapore, 2006.

[9]   Ø. Grøn and S. Hervik: Einstein’s General Theory of Relativity—With ModernApplications in Cosmology. Springer, New York, USA, 2007.

[10]   Y. Baryshev and P. Teerikorpe: Discovery of Cosmic Fractals. World Scientific, New Jersey, USA, 2002.

[11]   G. Ellis and P. Williams: Flat and curved space-time. Second Edition. Oxford University Press, Oxford, UK, 2000.

[12]   M.S. El Naschie: Einstein’s dream and fractal gravity. Chaos, Solitons & Fractals, 24(1), 2005, pp. 1-5.

[13]   M.S. El Naschie: A resolution of cosmic dark energy via a quantum entanglement relativity theory. Journal of Quantum Information Science, 3(1), 2013, pp. 23-26.

[14]   M.S. El Naschie: A review of E-infinity and the mass spectrum of high energy particle physics. Chaos, Solitons & Fractals, 19, 2004, pp. 209-236.

[15]   M.S. El Naschie: The theory of Cantorian spacetime and high energy particle physics (an informal review). Chaos, Solitons & Fractals, 41, 2009, pp. 2635-2646.

[16]   L. Marek-Crnjac, et al.: Chaotic fractals at the root of relativistic quantum physics and cosmology. International Journal of Modern Nonlinear Theory & Application, 2(1A), 2013, pp. 78-88.

[17]   M.S. El Naschie: The concepts of E-infinity: An elementary introduction to the Cantorian-fractal theory of quantum physics. Chaos, Solitons & Fractals, 22, 2004, pp. 495-511.

[18]   M.S. El Naschie: A unified newtonian-relativistic quantum resolution of supposedly missing dark energy of the cosmos and the constancy of the speed of light. International Journal of Modern Nonlinear Theory & Application, 2013, 2, pp. 43-54.

[19]   M.S. El Naschie: Nash embedding of Witten’s M-theory and the Hawking-Hartle quantum wave of dark energy. Journal of Modern Physics, 4(10), 2013, pp. 1417-1428.

[20]   M.S. El Naschie: Cosserat-Cartan and de Sitter-Witten spacetime setting for dark energy. Quantum Matter, 5(1), 2016, pp. 1-4.

[21]   K. Becker, M. Becker and J.H. Schwarz: String theory and M-theory. Cambridge University Press, Cambridge, 2007.

[22]   P. West: Introduction to strings and branes. Cambridge University Press, Cambridge, UK, 2012.

[23]   M. Kaku: Strings, Conformal fields and M-theory. Springer, New York, USA, 2000.

[24]   A. Connes: Noncommutative geometry. Academic Press, San Diego, 1994.

[25]   C. Rovelli: Quantum gravity. Cambridge Press, Cambridge, 2004.

[26]   D. Freedman and A. van Proeyen: Super gravity. Cambridge University Press, Cambridge, 2004.

[27]   C.V. Johnson: D-Branes. Cambridge University Press, Cambridge, UK, 2003.

[28]   M.S. El Naschie: Penrose universe and Cantorian spacetime as a model for noncommutative quantum geometry. Chaos, Solitons & Fractals, 9(6), 1998, pp. 931-933.

[29]   M.S. El Naschie: Superstrings, knots and noncommutative geometry in E-infinity space. International Journal of Theoretical Physics, 37(12), 1998, pp. 2935-2951.

[30]   L. Marek-Crnjac, J.-H. He: An invitation to El Naschie’s theory of Cantorian space-time and dark energy. International Journal of Astronomy and Astrophysics, 3(4), 2013, pp. 464-471.

[31]   J.-H. He, et al.: The importance of the empty set and noncommutative geometry in underpinning the foundations of quantum physics. Nonlinear Science B, 1(1), 2011, pp. 14-23.

[32]   M. Duff (Editor): The world in eleven dimensions. IOP Publishing, Bristol, UK, 1999.

[33]   M.S. El Naschie: A Rindler-KAM spacetime geometry and scaling the planck scale solves quantum relativity and explains dark energy. International Journal of Astronomy and Astrophysics, 3(4), 2013, pp. 483-493.

[34]   M. Gardner: Penrose tiles to trapdoor ciphers. W.H. Freeman, New York, USA, 1989. (See in particular Chapter 1, pp. 10).

[35]   E-Infinity Group: The exact renormalization equation for E-infinity unification of fundamental forces, quantum mechanics and the golden mean in theoretical physics. E-infinity-High Energy Communication No. 62, 15 December 2010.

[36]   M.S. El Naschie: Elementary number theory in superstring loop quantum mechanics, twistors and E-infinity high energy physics. Chaos, Solitons & Fractals, 27(2), 2006, pp. 297-330.

[37]   M.S. El Naschie: String theory, exceptional Lie groups hierarchy and the structural constant of the universe. Chaos, Solitons & Fractals, 35(1), 2008, pp. 7-12.

[38]   L. Marek-Crnjac: Stein spaces in connection with El Naschie’s exceptional Lie groups hierarchies in high energy physics. Chaos, Solitons & Fractals, 38(2), 2008, pp. 309-315.

[39]   T. Zhong: From the numeric of dynamics to the dynamics of numeric and visa versa in high energy particle physics. Chaos, Solitons & Fractals, 42, 2009, pp. 1780-1783.

[40]   M.S. El Naschie: On a new elementary particle from the disintegration of the Symplectic ‘t Hooft-Veltman-Wilson fractal spacetime. World Journal of Nuclear Science and Technology, 4(4), 2014, pp. 216-221.

[41]   M.S. El Naschie: Mathematical models and methods in dark energy theory: Dvoretzky’s theorem, Casimir effect, Mobius geometry. Problems of Nonlinear Analysis in Engineering Systems, 2(44), 2015, pp. 1-16. University of Kazan Press, Russia. (Published in Russian and English language versions).

[42]   M.S. El Naschie: Dark energy and its cosmic density from Einstein’s relativity and gauge fields renormalization leading to the possibility of a new ‘tHooft quasi particle. The Open Journal of Astronomy, 8, 2015, pp. 1-17.

[43]   M.S. El Naschie: Compactified dimensions as produced by quantum entanglement, the four dimensionality of Einstein’s smooth spacetime and ‘tHooft’s 4-ε fractal spacetime. American Journal of Astronomy & Astrophysics, 2(3), 2014, pp. 34-37.

[44]   M.S. El Naschie: The quantum gravity immirzi parameter—A general physical and topological interpretation. Gravity and Cosmology, 19(3), 2013, 151-153.

[45]   J.-P. Auffray: E-infinity, the zero set, absolute space and the photon spin. Journal of Modern Physics, 6(5), 2015, pp. 536-545.

[46]   M.-W. Ho, M.S. El Naschie and M.W. Giuseppe Vitello: Is spacetime fractal and quantum coherent in the golden mean? Global Journal of Science Frontier Research, 15(1), 2015, pp. 61-80.

[47]   J. Ambjorn, J. Jurkiewicz and R. Loll: Quantum gravity: The act of building spacetime. In “Quantum Gravity”, Edited by D. Oriti. Cambridge University Press, Cambridge, UK, 2009, pp. 342-359.

[48]   L. Amendola and S. Tsujikawa: Dark energy: Theory and observation. Cambridge University Press, Cambridge, UK, 2010.

[49]   M.S. El Naschie: On the unification of heterotic strings, M theory and E (∞) theory. Chaos, Solitons & Fractals, 11(14), 2000, pp. 2397-2408.