On the Need for Fractal Logic in High Energy Quantum Physics
Affiliation(s)
Department of Physics, Faculty of Science, University of Alexandria, Alexandria, Egypt. College of Central Florida, Ocala, USA. Nantong Textile Institute, National Engineering Laboratory for Modern Silk, College of Textile and Clothing, Soochow University, Suzhou, China. Mathematics Department, Qatar University, Doha, Qatar. Technical School Center of Maribor, Maribor, Slovenia. Department of Mathematics, Faculty of Science, Cairo University, Giza, Egypt.
Department of Physics, Faculty of Science, University of Alexandria, Alexandria, Egypt. College of Central Florida, Ocala, USA. Nantong Textile Institute, National Engineering Laboratory for Modern Silk, College of Textile and Clothing, Soochow University, Suzhou, China. Mathematics Department, Qatar University, Doha, Qatar. Technical School Center of Maribor, Maribor, Slovenia. Department of Mathematics, Faculty of Science, Cairo University, Giza, Egypt.
ABSTRACT
Modern advances in pure mathematics and particularly in transfinite set theory have introduced into the fundamentals of theoretical physics many novel concepts and devices such as fractal quasi manifolds with non-integer (Hausdorff) dimension for its geometry as well as infinite dimensional wild topology and non classical fuzzy logic. In the present work transfinite fractal sets and fuzzy logic are combined to enable the introduction of a new theory termed fractal logic to the foundation of high energy particle physics. This leads naturally to a new look at quantum gravity. In particular we will show that to understand and develop quantum gravity we have to bring various fields together, particularly fractals and nonlinear dynamics as well as sphere packing, fuzzy set theory, number theory and quantum entanglement and irrationally q-deformed algebra.
Modern advances in pure mathematics and particularly in transfinite set theory have introduced into the fundamentals of theoretical physics many novel concepts and devices such as fractal quasi manifolds with non-integer (Hausdorff) dimension for its geometry as well as infinite dimensional wild topology and non classical fuzzy logic. In the present work transfinite fractal sets and fuzzy logic are combined to enable the introduction of a new theory termed fractal logic to the foundation of high energy particle physics. This leads naturally to a new look at quantum gravity. In particular we will show that to understand and develop quantum gravity we have to bring various fields together, particularly fractals and nonlinear dynamics as well as sphere packing, fuzzy set theory, number theory and quantum entanglement and irrationally q-deformed algebra.
Cite this paper
M. Naschie, S. Olsen, J. He, S. Nada, L. Marek-Crnjac and A. Helal, "On the Need for Fractal Logic in High Energy Quantum Physics," International Journal of Modern Nonlinear Theory and Application, Vol. 1 No. 3, 2012, pp. 84-92. doi: 10.4236/ijmnta.2012.13012.
M. Naschie, S. Olsen, J. He, S. Nada, L. Marek-Crnjac and A. Helal, "On the Need for Fractal Logic in High Energy Quantum Physics," International Journal of Modern Nonlinear Theory and Application, Vol. 1 No. 3, 2012, pp. 84-92. doi: 10.4236/ijmnta.2012.13012.
References
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Ord, “Fractal Space-Time a Geometric Analog of Relativistic Quantum Mechanics,” Journal of Physics A, Vol. 16, No. 9, 1983, pp. 1869-1884. doi:10.1088/0305-4470/16/9/012 [8] L. Nottale, “Fractal Space-Time and Micro Physics,” World Scientific, Singapore City, 1993. [9] A. Stakhov, “The Mathematics of Harmony”, World Scientific, Singapore, 2009. doi:10.1142/6635 [10] M. S. El Naschie, “A Review of E-Infinity Theory and the Mass Spectrum of High Energy Particle Physics,” Chaos, Solitons & Fractals, Vol. 19, No. 1, 2004, pp. 209-236. doi:10.1016/S0960-0779(03)00278-9 [11] R. L. Devaney, “An Introduction to Chaotic Dynamical Systems,” Addison-Wesley, Redwood City, 1989. [12] M. S. El Naschie, O. E. Rossler and I. Prigogine, “Quantum Mechanics, Diffusion and Chaotic Fractals,” Pergamon Press—Elsevier Publishing, Oxford, 1995. [13] L. B. Crowell, “Quantum Fluctutations of Space Time,” World Scientific, Singapore City, 2005. doi:10.1142/5952 [14] L. Glass and M. 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El Naschie, “Quantum Entanglement as a Consequence of a Cantorian Micro Space-time Geometry,” Journal of Quantum Information Science, Vol. 1, No. 2, 2011, pp. 50-53. doi:10.4236/jqis.2011.12007 [31] G. Ord, M. S. El Naschie and J. H. He, Fractal Space-Time and Non Commutative Geometry in High Energy Physics, Asian Academic Publishing Ltd., Hong Kong, Vol. 1, No. 1, 2011, pp. 1-46. [32] G. Ord, M. S. El Naschie and J. H. He, Fractal Space-Time and Non Commutative Geometry in High Energy Physics, Asian Academic Publishing Ltd., Hong Kong, Vol. 2, No. 1, 2012, pp. 1-79. [33] L. Zadeh, “Fuzzy Logic and Approximate Reasoning,” Synthese, Vol. 30, No. 3-4, 1975, pp. 407-428. doi:10.1007/BF00485052 [34] L. Zadeh, “Fuzzy Sets,” Information and Control, Vol. 8, 1965, pp. 338-353. doi:10.1016/S0019-9958(65)90241-X [35] K. Kosko, “Fuzzy Thinking,” The New Science of Fuzzy Logic, Hyperion, New York, 1993. [36] M. S. El Naschie, “Fuzzy Knot interpretation of Yang-Mills Instantons and Witten’s 5 Brane Model,” Chaos, Solitons & Fractals, Vol. 38, No. 5, 2008, pp. 1349-1354. doi:10.1016/j.chaos.2008.07.002 [37] M. S. El Naschie, “From Experimental Quantum Optics to Quantum Gravity via a Fuzzy Kahler Manifold,” Chaos, Solitons & Fractals, Vol. 25, No. 5, 2005, pp. 969-977. doi:10.1016/j.chaos.2005.02.028 [38] M. S. El Naschie, “Fuzzy Dedochaedron Topology and E-Infinity Space-Time as a Model for Quantum Physics,” Chaos, Solitons & Fractals, Vol. 30, No. 5, 2006, pp. 1025-1033. doi:10.1016/j.chaos.2006.05.088 [39] N. M. Ahmed, “George Cantor: The Father of Set Theory,” The Post Graduate Magazine, 2007, pp. 4-14. [40] M. S. El Naschie, “The Brain and E-Infinity”, International Journal of Nonlinear Sciences and Numerical Simulation, Vol. 7, No. 2, 2006, pp. 128-131. [41] M. S. El Naschie, “From Symmetry to Particle,” Chaos, Solitons & Fractals, Vol. 32, No. 2, 2007, pp. 427-430. doi:10.1016/j.chaos.2006.09.016 [42] M. S. El Naschie, “Kac-Moody Exceptional E12 from Simplectic Tiling,” Chaos, Solitons & Fractals, Vol. 41, No. 4, 2009, pp. 1569-1571. doi:10.1016/j.chaos.2008.06.020 [43] J. H. He, “Transfinite Physics,” China Culture and Science Publishing, Shanghai, 2005. [44] M. S. El Naschie, “Knots and Exceptional Lie Groups as Building Blocks of High Energy Particle Physics,” Chaos, Solitons & Fractals, Vol. 41, No. 4, 2009, pp. 1799-1803. doi:10.1016/j.chaos.2008.07.025 [45] M. S. El Naschie, “Symmetry Group Prerequisite for E-Infinity in High Energy Physics,” Chaos, Solitons & Fractals, Vol. 35, No. 1, 2008, pp. 202-211. doi:10.1016/j.chaos.2007.05.006 [46] M. S. El Naschie, “Quantum Groups and Hamiltonian Sets on Nuclear Space-Time Cantorian Manifold,” Chaos, Solitons & Fractals, Vol. 10, No. 7, 1999, pp. 1251-1256. doi:10.1016/S0960-0779(99)00009-0 [47] M. S. El Naschie, “On a Class of Fuzzy Kahler-Like Manifold”, Chaos, Solitons & Fractals, Vol. 26, No. 2, 2005, pp. 257-261. doi:10.1016/j.chaos.2004.12.024 [48] M. S. El Naschie, “On a Class of General Theories for High Energy Particle Physics,” Chaos, Solitons & Fractals, Vol. 14, No. 4, 2002, pp. 649-668. doi:10.1016/S0960-0779(02)00033-4 [49] M. S. El Naschie, “On an Eleven Dimensional E-Infinity Fractal Space-Time,” International Journal of Nonlinear Sciences and Numerical Simulation, Vol. 7, No. 4, 2006, pp. 407-409. doi:10.1515/IJNSNS.2006.7.4.407 [50] M. S. El Naschie, “The Discrete Charm of Certain Eleven Dimensional Space-Time Theory,” International Journal of Nonlinear Sciences and Numerical Simulation, Vol. 7, No. 4, 2006, pp. 477-481. [51] M. S. El Naschie, “On Fuzzy Kahler-Like Manifold Which Is Consistent with the Two-Slit Experiment,” International Journal of Nonlinear Sciences and Numerical Simulation, Vol. 6, No. 2, 2005, pp. 8895-8898. [52] M. S. El Naschie, “The Symplectic Vacuum, Exotic Quasi Particles and Gravitational Instanton,” Chaos, Solitons & Fractals, Vol. 22, No. 1, 2004, pp. 1-11. doi:10.1016/j.chaos.2004.01.015 [53] J. H. He, “Non Linear Dynamics and the Nobel Prize in Physics,” International Journal of Nonlinear Sciences and Numerical Simulation, Vol. 8, No. 1, 2007, pp. 1-4. doi:10.1515/IJNSNS.2007.8.1.1 [54] L. Sigalotti and A. Mejias, “On El Naschie’s Conjugate Complex Time, Fractal E-Infinity Space-Time and Faster than Light Particles,” International Journal of Nonlinear Sciences and Numerical Simulation, Vol. 7, No. 4, 2006, pp. 467-472. doi:10.1515/IJNSNS.2006.7.4.467 [55] M. Kaku, “Introduction to Superstrings and M-Theory,” Springer, New York, 1999. [56] S. Weinberg, “The Quantum Theory of Fields,” Cambridge, Vol. II, 1996. [57] S. Weinberg, “The Quantum Theory of Fields,” Cambridge, Vol. III, 2000. [58] M. S. El Naschie, “Quantum Golden Field Theory—Ten Theorems and Various CONJECTURES,” Chaos, Solitons & Fractals, Vol. 36, No. 5, 2008, pp. 1121-1125. doi:10.1016/j.chaos.2007.09.023 [59] M. S. El Naschie, “An Outline for a Quantum Golden Field Theory,” Chaos, Solitons & Fractals, Vol. 37, No. 2, 2008, pp. 317-323. doi:10.1016/j.chaos.2007.09.092 [60] M. S. El Naschie, “Asymptotic Freedom and Unification in a Golden Field Theory,” Chaos, Solitons & Fractals, Vol. 36, No. 3, 2008, pp. 521-525. doi:10.1016/j.chaos.2007.09.004 [61] M. S. El Naschie, “A Guide to the Mathematics of E-Infinity Cantorian Space-Time Theory,” Chaos, Solitons & Fractals, Vol. 25, No. 5, 2005, pp. 995-964. doi:10.1016/j.chaos.2004.12.033 [62] M. S. El Naschie, “Elementary Prerequisites for E-Infinity,” Chaos, Solitons & Fractals, Vol. 30, No. 3, 2006, pp. 579-605. doi:10.1016/j.chaos.2006.03.030 [63] M. S. El Naschie, “The Theory of Cantorian Space-Time and High Energy Particle Physics (An Informal Review),” Chaos, Solitons & Fractals, Vol. 41, No. 5, 2009, pp. 2635-2646. doi:10.1016/j.chaos.2008.09.059 [64] K. Svozil, “Quantum Field Theory on Fractal Space-Time,” Journal of Physics A, Vol. 20, No. 12, 1987, pp. 3861-3875. doi:10.1088/0305-4470/20/12/033 [65] M. S. El Naschie, “Transfinite Harmonization by Taking the Dissonance Out of the Quantum Field Symphony,” Chaos, Solitons & Fractals, Vol. 36, No. 4, 2008, pp. 781-786. doi:10.1016/j.chaos.2007.09.018 [66] M. S. El Naschie, “Extended Renormalization Group Analysis for Quantum Gravity and Newton’s Gravitational Constant,” Chaos, Solitons & Fractals, Vol. 35, No. 3, 2008, pp. 425-431. doi:10.1016/j.chaos.2007.07.059 [67] M. S. El Naschie, “Exact Non-Perturbative-Derivation of Gravity’S G4 Fine Structure Constant, the Mass of the Higgs and Elementary Black Holes,” Chaos, Solitons & Fractals, Vol. 37, No. 2, 2008, pp. 346-359. doi:10.1016/j.chaos.2007.10.021 [68] M. S. El Naschie, “Quantum E-Infinity Field Theoretical Gravitational Constant,” International Journal of Nonlinear Sciences and Numerical Simulation, Vol. 8, No. 3, 2007, pp. 496-474. [69] M. S. El Naschie, “Towards a Quantum Golden Field theory,” International Journal of Nonlinear Sciences and Numerical Simulation, Vol. 8, No. 4, 2007, pp. 477-482. doi:10.1515/IJNSNS.2007.8.4.477
[1] L. Marek-Crnjac, “The Hausdorff Dimension of the Penrose Universe,” Physics Research International, Vol. 2011, 2011, Article ID: 874302. [2] L. Marek-Crnjac, “A Short History of Fractal Cantorian Space-Time,” Chaos, Solitons & Fractals, Vol. 41, 2009, pp. 2697-2705. doi:10.1016/j.chaos.2008.10.007 [3] J. Hocking and G. Young, “Topology,” Dover Publishing, New York, 1961. [4] S. L. Lipscomb, “Fractals and Universal Spaces in Dimension Theory,” Springer, New York, 2009. doi:10.1007/978-0-387-85494-6 [5] M. S. El Naschie, “Complexity Theory Interpretation of High Energy Physics and Elementary Particle Mass Spectrum,” In: B. G. Sidharth, Ed., Frontiers of Fundamental Physics, Vol. 3, Universities Press, Hyderbad, 2007, pp. 1-32. [6] M. Heller and W. H. Woodin, “Infinity: New Research Frontiers,” Cambridge University Press, Cambridge, 2011. [7] G. N. Ord, “Fractal Space-Time a Geometric Analog of Relativistic Quantum Mechanics,” Journal of Physics A, Vol. 16, No. 9, 1983, pp. 1869-1884. doi:10.1088/0305-4470/16/9/012 [8] L. Nottale, “Fractal Space-Time and Micro Physics,” World Scientific, Singapore City, 1993. [9] A. Stakhov, “The Mathematics of Harmony”, World Scientific, Singapore, 2009. doi:10.1142/6635 [10] M. S. El Naschie, “A Review of E-Infinity Theory and the Mass Spectrum of High Energy Particle Physics,” Chaos, Solitons & Fractals, Vol. 19, No. 1, 2004, pp. 209-236. doi:10.1016/S0960-0779(03)00278-9 [11] R. L. Devaney, “An Introduction to Chaotic Dynamical Systems,” Addison-Wesley, Redwood City, 1989. [12] M. S. El Naschie, O. E. Rossler and I. Prigogine, “Quantum Mechanics, Diffusion and Chaotic Fractals,” Pergamon Press—Elsevier Publishing, Oxford, 1995. [13] L. B. Crowell, “Quantum Fluctutations of Space Time,” World Scientific, Singapore City, 2005. doi:10.1142/5952 [14] L. Glass and M. Mackey, “The Rhythms of Life,” Princeton University Press, Princeton, 1988. [15] B. B. Mandelbrot, “The Fractal Geometry of Nature,” Freeman, New York, 1983. [16] J. H. He, E. Goldfain, L. D. G. Sigalotti and A. Mejias, “Beyond the 2006 Physics Nobel Prize for COBE,” China Culture and Science Publishing, Shanghai, 2006. [17] C. Beck, “Spation-Temporal Chaos and Vacuum Fluctuations of Quantized Fields,” World Scientific, Singapore City, 2002. doi:10.1142/4853 [18] Y. Baryshev and P. Terrikorpi, “Discovery of Cosmic Fractals,” World Scientific, Singapore City, 2002. doi:10.1142/4896 [19] J. Nicolis, G. Nicolis and C. Nicolis, “Non Linear Dynamics and the Two Slit Delayed Experiment,” Chaos, Solitons & Fractals, Vol. 12, 2001, pp. 407-416. doi:10.1016/S0960-0779(00)00190-9 [20] M. S. El Naschie, “Quantum Collapse of Wave Interference Pattern in the Two-Slit Experiment: A Set of Theoretical Resolution,” Nonlinear Science Letter A, Vol. 2, No. 1, 2011, pp. 1-9. [21] R. Elwes, “Ultimate Logic,” New Scientist, Vol. 211, No. 2183, 2011, pp. 30-33. doi:10.1016/S0262-4079(11)61838-1 [22] J. Ambjorn, J. Jurkiewicz and R. Loll, “The Self-Organizing Universe,” Scientific American, 2008, pp. 24-31. [23] S. Kranz and H. Park, “Geometric Integration Theory,” Birkhauser, Boston, 2008. doi:10.1007/978-0-8176-4679-0 [24] T. Jech, “Set Theory,” Springer, Berlin, 2003. [25] A. Kanamori, “The Higher Infinite,” Springer, Berlin, 2003. [26] A. Kechris, “Classical Descriptive Set Theory,” Springer, New York, 1995. doi:10.1007/978-1-4612-4190-4 [27] L. Graham and J. Kantor, “Naming Infinity,” Harvard University Press, Cambridge, 2009. [28] L. M. Wapner, “The Pea and the Sun,” A. K. Peters Ltd., Natick, 2005. [29] F. Morgan, “Geometric Measure Theory,” Elsevier, Amsterdam, 2009. [30] M. S. El Naschie, “Quantum Entanglement as a Consequence of a Cantorian Micro Space-time Geometry,” Journal of Quantum Information Science, Vol. 1, No. 2, 2011, pp. 50-53. doi:10.4236/jqis.2011.12007 [31] G. Ord, M. S. El Naschie and J. H. He, Fractal Space-Time and Non Commutative Geometry in High Energy Physics, Asian Academic Publishing Ltd., Hong Kong, Vol. 1, No. 1, 2011, pp. 1-46. [32] G. Ord, M. S. El Naschie and J. H. He, Fractal Space-Time and Non Commutative Geometry in High Energy Physics, Asian Academic Publishing Ltd., Hong Kong, Vol. 2, No. 1, 2012, pp. 1-79. [33] L. Zadeh, “Fuzzy Logic and Approximate Reasoning,” Synthese, Vol. 30, No. 3-4, 1975, pp. 407-428. doi:10.1007/BF00485052 [34] L. Zadeh, “Fuzzy Sets,” Information and Control, Vol. 8, 1965, pp. 338-353. doi:10.1016/S0019-9958(65)90241-X [35] K. Kosko, “Fuzzy Thinking,” The New Science of Fuzzy Logic, Hyperion, New York, 1993. [36] M. S. El Naschie, “Fuzzy Knot interpretation of Yang-Mills Instantons and Witten’s 5 Brane Model,” Chaos, Solitons & Fractals, Vol. 38, No. 5, 2008, pp. 1349-1354. doi:10.1016/j.chaos.2008.07.002 [37] M. S. El Naschie, “From Experimental Quantum Optics to Quantum Gravity via a Fuzzy Kahler Manifold,” Chaos, Solitons & Fractals, Vol. 25, No. 5, 2005, pp. 969-977. doi:10.1016/j.chaos.2005.02.028 [38] M. S. El Naschie, “Fuzzy Dedochaedron Topology and E-Infinity Space-Time as a Model for Quantum Physics,” Chaos, Solitons & Fractals, Vol. 30, No. 5, 2006, pp. 1025-1033. doi:10.1016/j.chaos.2006.05.088 [39] N. M. Ahmed, “George Cantor: The Father of Set Theory,” The Post Graduate Magazine, 2007, pp. 4-14. [40] M. S. El Naschie, “The Brain and E-Infinity”, International Journal of Nonlinear Sciences and Numerical Simulation, Vol. 7, No. 2, 2006, pp. 128-131. [41] M. S. El Naschie, “From Symmetry to Particle,” Chaos, Solitons & Fractals, Vol. 32, No. 2, 2007, pp. 427-430. doi:10.1016/j.chaos.2006.09.016 [42] M. S. El Naschie, “Kac-Moody Exceptional E12 from Simplectic Tiling,” Chaos, Solitons & Fractals, Vol. 41, No. 4, 2009, pp. 1569-1571. doi:10.1016/j.chaos.2008.06.020 [43] J. H. He, “Transfinite Physics,” China Culture and Science Publishing, Shanghai, 2005. [44] M. S. El Naschie, “Knots and Exceptional Lie Groups as Building Blocks of High Energy Particle Physics,” Chaos, Solitons & Fractals, Vol. 41, No. 4, 2009, pp. 1799-1803. doi:10.1016/j.chaos.2008.07.025 [45] M. S. El Naschie, “Symmetry Group Prerequisite for E-Infinity in High Energy Physics,” Chaos, Solitons & Fractals, Vol. 35, No. 1, 2008, pp. 202-211. doi:10.1016/j.chaos.2007.05.006 [46] M. S. El Naschie, “Quantum Groups and Hamiltonian Sets on Nuclear Space-Time Cantorian Manifold,” Chaos, Solitons & Fractals, Vol. 10, No. 7, 1999, pp. 1251-1256. doi:10.1016/S0960-0779(99)00009-0 [47] M. S. El Naschie, “On a Class of Fuzzy Kahler-Like Manifold”, Chaos, Solitons & Fractals, Vol. 26, No. 2, 2005, pp. 257-261. doi:10.1016/j.chaos.2004.12.024 [48] M. S. El Naschie, “On a Class of General Theories for High Energy Particle Physics,” Chaos, Solitons & Fractals, Vol. 14, No. 4, 2002, pp. 649-668. doi:10.1016/S0960-0779(02)00033-4 [49] M. S. El Naschie, “On an Eleven Dimensional E-Infinity Fractal Space-Time,” International Journal of Nonlinear Sciences and Numerical Simulation, Vol. 7, No. 4, 2006, pp. 407-409. doi:10.1515/IJNSNS.2006.7.4.407 [50] M. S. El Naschie, “The Discrete Charm of Certain Eleven Dimensional Space-Time Theory,” International Journal of Nonlinear Sciences and Numerical Simulation, Vol. 7, No. 4, 2006, pp. 477-481. [51] M. S. El Naschie, “On Fuzzy Kahler-Like Manifold Which Is Consistent with the Two-Slit Experiment,” International Journal of Nonlinear Sciences and Numerical Simulation, Vol. 6, No. 2, 2005, pp. 8895-8898. [52] M. S. El Naschie, “The Symplectic Vacuum, Exotic Quasi Particles and Gravitational Instanton,” Chaos, Solitons & Fractals, Vol. 22, No. 1, 2004, pp. 1-11. doi:10.1016/j.chaos.2004.01.015 [53] J. H. He, “Non Linear Dynamics and the Nobel Prize in Physics,” International Journal of Nonlinear Sciences and Numerical Simulation, Vol. 8, No. 1, 2007, pp. 1-4. doi:10.1515/IJNSNS.2007.8.1.1 [54] L. Sigalotti and A. Mejias, “On El Naschie’s Conjugate Complex Time, Fractal E-Infinity Space-Time and Faster than Light Particles,” International Journal of Nonlinear Sciences and Numerical Simulation, Vol. 7, No. 4, 2006, pp. 467-472. doi:10.1515/IJNSNS.2006.7.4.467 [55] M. Kaku, “Introduction to Superstrings and M-Theory,” Springer, New York, 1999. [56] S. Weinberg, “The Quantum Theory of Fields,” Cambridge, Vol. II, 1996. [57] S. Weinberg, “The Quantum Theory of Fields,” Cambridge, Vol. III, 2000. [58] M. S. El Naschie, “Quantum Golden Field Theory—Ten Theorems and Various CONJECTURES,” Chaos, Solitons & Fractals, Vol. 36, No. 5, 2008, pp. 1121-1125. doi:10.1016/j.chaos.2007.09.023 [59] M. S. El Naschie, “An Outline for a Quantum Golden Field Theory,” Chaos, Solitons & Fractals, Vol. 37, No. 2, 2008, pp. 317-323. doi:10.1016/j.chaos.2007.09.092 [60] M. S. El Naschie, “Asymptotic Freedom and Unification in a Golden Field Theory,” Chaos, Solitons & Fractals, Vol. 36, No. 3, 2008, pp. 521-525. doi:10.1016/j.chaos.2007.09.004 [61] M. S. El Naschie, “A Guide to the Mathematics of E-Infinity Cantorian Space-Time Theory,” Chaos, Solitons & Fractals, Vol. 25, No. 5, 2005, pp. 995-964. doi:10.1016/j.chaos.2004.12.033 [62] M. S. El Naschie, “Elementary Prerequisites for E-Infinity,” Chaos, Solitons & Fractals, Vol. 30, No. 3, 2006, pp. 579-605. doi:10.1016/j.chaos.2006.03.030 [63] M. S. El Naschie, “The Theory of Cantorian Space-Time and High Energy Particle Physics (An Informal Review),” Chaos, Solitons & Fractals, Vol. 41, No. 5, 2009, pp. 2635-2646. doi:10.1016/j.chaos.2008.09.059 [64] K. Svozil, “Quantum Field Theory on Fractal Space-Time,” Journal of Physics A, Vol. 20, No. 12, 1987, pp. 3861-3875. doi:10.1088/0305-4470/20/12/033 [65] M. S. El Naschie, “Transfinite Harmonization by Taking the Dissonance Out of the Quantum Field Symphony,” Chaos, Solitons & Fractals, Vol. 36, No. 4, 2008, pp. 781-786. doi:10.1016/j.chaos.2007.09.018 [66] M. S. El Naschie, “Extended Renormalization Group Analysis for Quantum Gravity and Newton’s Gravitational Constant,” Chaos, Solitons & Fractals, Vol. 35, No. 3, 2008, pp. 425-431. doi:10.1016/j.chaos.2007.07.059 [67] M. S. El Naschie, “Exact Non-Perturbative-Derivation of Gravity’S G4 Fine Structure Constant, the Mass of the Higgs and Elementary Black Holes,” Chaos, Solitons & Fractals, Vol. 37, No. 2, 2008, pp. 346-359. doi:10.1016/j.chaos.2007.10.021 [68] M. S. El Naschie, “Quantum E-Infinity Field Theoretical Gravitational Constant,” International Journal of Nonlinear Sciences and Numerical Simulation, Vol. 8, No. 3, 2007, pp. 496-474. [69] M. S. El Naschie, “Towards a Quantum Golden Field theory,” International Journal of Nonlinear Sciences and Numerical Simulation, Vol. 8, No. 4, 2007, pp. 477-482. doi:10.1515/IJNSNS.2007.8.4.477