Cantorian-Fractal Kinetic Energy and Potential Energy as the Ordinary and Dark Energy Density of the Cosmos Respectively

Mohamed S. El Naschie

doi:10.4236/ns.2016.812052

Natural Science > Vol.8 No.12, December 2016

Cantorian-Fractal Kinetic Energy and Potential Energy as the Ordinary and Dark Energy Density of the Cosmos Respectively

Mohamed S. El Naschie
Department of Physics, Faculty of Science, University of Alexandria, Alexandria, Egypt.
DOI: 10.4236/ns.2016.812052 PDF HTML XML 50,197 Downloads 52,175 Views Citations

Abstract

In a one-dimension Mauldin-Williams Random Cantor Set Universe, the Sigalotti topological speed of light is where . It follows then that the corresponding topological acceleration must be a golden mean downscaling of c namely . Since the maximal height in the one-dimensional universe must be where is the unit interval length and note that the topological mass (m) and topological dimension (D) where m = D = 5 are that of the largest unit sphere volume, we can conclude that the potential energy of classical mechanics translates to . Remembering that the kinetic energy is , then by the same logic we see that when m = 5 is replaced by for reasons which are explained in the main body of the present work. Adding both expressions together, we find Einstein’s maximal energy . As a general conclusion, we note that within high energy cosmology, the sharp distinction between potential energy and kinetic energy of classical mechanics is blurred on the cosmic scale. Apart of being an original contribution, the article presents an almost complete bibliography on the Cantorian-fractal spacetime theory.

Keywords

Potential Dark Energy, Kinetic Ordinary Energy, Motion as Illusion, Zenonparadoxa, E-Infinity Theory, Noncommutative Geometry, Topological Acceleration, Cantorian Universe, Accelerated Cosmic Expansion

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Naschie, M. (2016) Cantorian-Fractal Kinetic Energy and Potential Energy as the Ordinary and Dark Energy Density of the Cosmos Respectively. Natural Science, 8, 511-540. doi: 10.4236/ns.2016.812052.

1. Introduction

Space, time, matter and energy are concepts far from being trivial or obvious even within Newtonian classical mechanics [1] - [6] . This view was amply confirmed and deeply pondered in the wonderful writing of scientists such as H. Weyl and Max Jammar [1] [3] . Starting more or less from there it became the Author’s lifelong work and even magical fascination to incorporate the basic structure of quantum mechanics into the very topology and geometry of space and time [7] - [428] . To do this, the author followed a path inspired by the work of Richard Feynman and its development by the Canadian Physicist G. Ord and French Astrophysicist L. Nottale [7] [12] [13] .

The crucial turning point for E-infinity was when the Author’s basic work came in touch with the work on non-commutative geometry [14] [74] . In particular the superb analysis which the great French mathematician Alain Connes undertook on Penrose’s Fractal Tiling Universe using Von-Neumann’s pointless geometry [14] [140] is in retrospect the most important central piece in our current understanding of high energy physics and cosmology [7] - [270] . It turned out that the bijection formula which relates the Hausdorff dimension of an n-dimensional Cantor manifold to its topological dimension n [7]

(1)

where is just a compact version of Von-Neumann-Connes’ dimensional function of a Penrose tiling universe [7] [14] [165] .

(2)

In addition this dimensional function is generic and can be used to understand some of the most complex and difficult problems in Physics and Astrophysics [9] - [429] . In particular, it is easily shown using the above that the quantum particle may be described by the zero set as given by the bi-dimension [7] [27]

(3)

while the quantum wave maybe modeled by the empty set given by the bi-dimension [7] [27]

(4)

In other words, the zero set quantum particle is described by a bi-dimension, zero for the topological dimension and for the Hausdorff dimension. On the other hand, the empty set quantum wave is fixed by the bi-dimension minus one for the topological dimension and for the Hausdorff dimension [7] [9] [73] .

From this simple mental and mathematical picture, we were able to show that the volume of the quantum particle zero set in Kaluza-Klein spacetime is simply, while the corresponding volume of the quantum wave is [23] [24] . In other words, the measure of the particle is multiplicative while understandably the surface of this volume is a hyper-surface constituting the additive measure of the quantum wave. Since particle and wave in this picture, which is a ball, have a hyper spherical border and are therefore necessarily inseparable, it follows then that the total volume of the wave particle “quantum” structure is simply the sum. Inserting in Newton’s kinetic energy one finds [137] [138] [139]

(5)

where c is the speed of light. In that way, we were able to show that [48] - [400]

(6)

By contrast in the present work, we will take another route to arrive at the same result by stressing an optional separation between kinetic energy and potential energy in fractal spacetime.

2. Fractal Potential Energy and Fractal Kinetic Energy of Quantum Spacetime

The following is a “post-modern” and quite novel approach to the same fundamental problems connected to the total accepted theoretical energy density of the universe versus that which was measured and which gave rise to the new concepts of dark energy and dark matter. This problem was previously solved using a plethora of mathematical techniques. However and as we anticipated in the previous section, we are making in the present analysis a strict although optional distinction between potential energy and kinetic energy [430] .

For this reason we start from a one-dimensional Cantor set. For this set everything is zero with the exception of one fundamental thing. The bi-dimension indicated already that the topological dimension is zero. The only thing which is not zero is the Hausdorff dimension which is equal to, but what about where is embedded? That means where is the nothingness which is left from removing iteratively but randomly parts of the unit interval? This zero set “nothing” is not really nothing but rather some- thing and is embedded in the complementary empty set. Since the dimension of the unit interval is, then the dimension of this complementary empty set must be a trivial. This agrees perfectly with the bijection formula and the dimensional function for n = −1 which gives [154] [155]

(7)

or equivalently

(8)

In addition the measure i.e. the length of the complementary set is a trivial 1 − 0 = 1. In other words this empty set is a fat Cantor set [7] [154] [155] .

Now let us look at the velocity in D(0). This was established by the work of the notable Italian physicist L. Sigalotti to be which is not surprisingly the only non-zero quantity in the Cantor set. Next we like to determine the acceleration corresponding to V or say the acceleration analogous to Newtonian gravity on earth.

Now in E-Infinity we have a technique similar to non standard analysis were differentiation is equivalent to golden mean down scaling while integration is a golden mean scaling up [7] [9] [21] [28] .

In this case we have to down scale by multiplication with. Therefore the acceleration is simply

(9)

Again, not surprisingly this corresponds in elasticity to a torsional term and is numerically equal to the Hausdorff dimension of the empty set quantum wave [19] [28] [119] .

Our next step is to determine the height of the mass in the gravity field which is endowed with a positive energy i.e. a potential energy. Since the edges of the unit Cantor interval corresponding to the limit of the universe at a nominal infinity, then the maximum length of the unit interval is simply one half (1/2).

Now we can write heuristically a fractal expression for conventional potential energy for

(10)

provided we know what m is. This is easily reasoned if we get access or an insight into the real meaning of mass. This is clearly connected to energy and energy is related to entropy. On the other hand entropy maybe measured via the Hausdorff dimension which is for the empty set. Now a mass in 5-dimensional spacetime becomes a Kaluza-Klein spacetime five dimensional mass. Consequently, our empty set mass cannot be which is nothing but 1. Therefore it must be m = (1) (5). Inserting in to, we find the familiar expression of ordinary cosmic energy density

(11)

exactly as shown previously using various other methods. Let us stress this point again. We have just established the potential energy nature of dark energy and squared it with the energy of the quantum particle via a mathematical tautology. This is because at the end of the day it is completely the same thing to say

(12)

or to say

(13)

Thus could be interpreted as topological empty set mass m = 5 multiplied with the acceleration or as the volume of the 5D quantum wave empty set namely the additive volume. In other words we have a different mental picture leading to the same result [212] [426] .

Returning to the kinetic energy, this is relatively simpler because real energy of real zero set quantum particles is sensibly interpreted as a 3D mass. In this case we have then to the power of 3D which give us as a multiplicative 3D mass i.e. volume. In turn this can be interpreted as the inverse of the spacetime core Hausdorff dimension [7] [9]

(14)

Inserting in Newton’s Kinetic energy, we find the expected result

(15)

In agreement with expectations, the total energy which is the sum of the Kinetic and the potential energy is equal to Einstein’s maximal energy density [426] [427]

(16)

In concluding this part of our analysis we stress the subtlety of various interpretations of E(D) which could be potentially confusing. This is because could be interpreted

in equal measure as the quantum wave kinetic energy or

the spacetime potential energy. In both cases the result is the same but the “pictures” are different. In fact we could go as far as claiming that within quantum cosmology the difference between kinetic energy and potential energy is fuzzy and so is the difference between the state of motion or being at rest which resonates with the old philosophy of Zeno [430] .

3. Conclusion

Conflicts of Interest

The authors declare no conflicts of interest.

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[82]	El Naschie, M.S. (2006) Linderhof Room of Mirrors, Thurston Three-Manifolds and the Geometry of Our Universe. International Journal of Nonlinear Sciences and Numerical Simulation, 7, 97-100. https://doi.org/10.1515/ijnsns.2006.7.1.97
[83]	El Naschie, M.S. (2015) A Resolution of the Black Hole Information Paradox via Transfinite Set Theory. World Journal of Condensed Matter Physics, 5, 249. https://doi.org/10.4236/wjcmp.2015.54026
[84]	El Naschie, M.S. (2014) Why E Is Not Equal to mc2. Journal of Modern Physics, 5, 743-750. https://doi.org/10.4236/jmp.2014.59084
[85]	El Naschie, M.S. (2005) On a Class of Fuzzy Kähler-Like Manifolds. Chaos, Solitons & Fractals, 26, 257-261. https://doi.org/10.1016/j.chaos.2004.12.024
[86]	El Naschie, M.S. (2005) Gödel Universe, Dualities and High Energy Particles in E-Infinity. Chaos, Solitons & Fractals, 25, 759-764. https://doi.org/10.1016/j.chaos.2004.12.010
[87]	El Naschie, M.S. (1998) On the Irreducibility of Spatial Ambiguity in Quantum Physics. Chaos, Solitons & Fractals, 9, 913-919. https://doi.org/10.1016/S0960-0779(97)00165-3
[88]	El Naschie, M.S. (2013) The Quantum Entanglement behind the Missing Dark Energy. Journal of Modern Physics and Applications, 2, 88-96.
[89]	El Naschie, M.S. (2005) Deriving the Essential Features of the Standard Model from the General Theory of Relativity. Chaos, Solitons & Fractals, 24, 941-946. https://doi.org/10.1016/j.chaos.2004.10.001
[90]	El Naschie, M.S. (2014) Einstein’s General Relativity and Pure Gravity in a Cosserat and De Sitter-Witten Spacetime Setting as the Explanation of Dark Energy and Cosmic Accelerated Expansion. International Journal of Astronomy and Astrophysics, 4, 332. https://doi.org/10.4236/ijaa.2014.42027
[91]	El Naschie, M.S. (2006) The Unreasonable Effectiveness of the Electron-Volt Units System in High Energy Physics and the Role Played by a0 = 137. International Journal of Nonlinear Sciences and Numerical Simulation, 7, 119-128. https://doi.org/10.1515/IJNSNS.2006.7.2.119
[92]	El Naschie, M.S. (1998) Superstrings, Knots, and Noncommutative Geometry in E(∞) Space. International Journal of Theoretical Physics, 37, 2935-2951. https://doi.org/10.1023/A:1026679628582
[93]	El Naschie, M.S. (2013) The Missing Dark Energy of the Cosmos from Light Cone Topological Velocity and Scaling of the Planck Scale. Open Journal of Microphysics, 3, 64-70. https://doi.org/10.4236/ojm.2013.33012
[94]	El Naschie, M.S. (2008) The Fundamental Algebraic Equations of the Constants of Nature. Chaos, Solitons & Fractals, 35, 320-322. https://doi.org/10.1016/j.chaos.2007.06.110
[95]	Iovane, G. (2006) El Naschie Ε-Infinity Cantorian Spacetime and Lengths Scales in Cosmology. International Journal of Nonlinear Sciences and Numerical Simulation, 7, 155-162. https://doi.org/10.1515/IJNSNS.2006.7.2.155
[96]	El Naschie, M.S. (2014) The Meta Energy of Dark Energy. Open Journal of Philosophy, 4, 157-159. https://doi.org/10.4236/ojpp.2014.42022
[97]	El Naschie, M.S. (2007) From Symmetry to Particles. Chaos, Solitons & Fractals, 32, 427-430. https://doi.org/10.1016/j.chaos.2006.09.016
[98]	El Naschie, M.S. (2008) Kaluza-Klein Unification-Some Possible Extensions. Chaos, Solitons & Fractals, 37, 16-22. https://doi.org/10.1016/j.chaos.2007.09.079
[99]	El Naschie, M.S. (2015) On a Non-Perturbative Quantum Relativity Theory Leading to a Casimir-Dark Energy Nanotech Reactor Proposal. Open Journal of Applied Sciences, 5, 313. https://doi.org/10.4236/ojapps.2015.57032
[100]	He, J.H. (2007) Nonlinear Dynamics and the Nobel Prize in Physics. International Journal of Nonlinear Sciences and Numerical Simulation, 8, 1-4. https://doi.org/10.1515/IJNSNS.2007.8.1.1
[101]	El Naschie, M.S. (2004) Small World Network, ε (∞) Topology and the Mass Spectrum of High Energy Particles Physics. Chaos, Solitons & Fractals, 19, 689-697. https://doi.org/10.1016/S0960-0779(03)00337-0
[102]	El Naschie, M.S. (2014) From Chern-Simon, Holography and Scale Relativity to Dark Energy. Journal of Applied Mathematics and Physics, 2, 634-638. https://doi.org/10.4236/jamp.2014.27069
[103]	El Naschie, M.S. (2005) Experimental and Theoretical Arguments for the Number and the Mass of the Higgs Particles. Chaos, Solitons & Fractals, 23, 1091-1098. https://doi.org/10.1016/j.chaos.2004.08.001
[104]	He, J.H. (2006) Application of E-Infinity Theory to Biology. Chaos, Solitons & Fractals, 28, 285-289. https://doi.org/10.1016/j.chaos.2005.08.001
[105]	He, J.H. and Marek-Crnjac, L. (2013) Mohamed El Naschie’s Revision of Albert Einstein’s E = m0c2: A Definite Resolution of the Mystery of the Missing Dark Energy of the Cosmos. International Journal of Modern Nonlinear Theory and Application, 2, 55-59. https://doi.org/10.4236/ijmnta.2013.21006
[106]	El Naschie, M.S. (1998) Dimensional Symmetry Breaking, Information and Fractal Gravity in Cantorian Space. Biosystems, 46, 41-46. https://doi.org/10.1016/S0303-2647(97)00079-8
[107]	El Naschie, M.S. (2005) On Einstein’s Super Symmetric Tensor and the Number of Elementary Particles of the Standard Model. Chaos, Solitons & Fractals, 23, 1521-1525. https://doi.org/10.1016/j.chaos.2004.09.003
[108]	El Naschie, M.S. (2001) A General Theory for the Topology of Transfinite Heterotic Strings and Quantum Gravity. Chaos, Solitons & Fractals, 12, 969-988. https://doi.org/10.1016/S0960-0779(00)00263-0
[109]	El Naschie, M.S. (2006) Fuzzy Dodecahedron Topology and E-Infinity Spacetime as a Model for Quantum Physics. Chaos, Solitons & Fractals, 30, 1025-1033. https://doi.org/10.1016/j.chaos.2006.05.088
[110]	El Naschie, M.S. (2013) Determining the Missing Dark Energy Density of the Cosmos from a Light Cone Exact Relativistic Analysis. Journal of Physics, 2, 18-23.
[111]	El Naschie, M.S., Marek-Crnjac, L., Helal, M.A. and He, J.H. (2014) A Topological Magueijo-Smolin Varying Speed of Light Theory, the Accelerated Cosmic Expansion and the Dark Energy of Pure Gravity. Applied Mathematics, 5, 1780-1790. https://doi.org/10.4236/am.2014.512171
[112]	Sigalotti, L.D.G. and Mejias, A. (2006) The Golden Ratio in Special Relativity. Chaos, Solitons & Fractals, 30, 521-524. https://doi.org/10.1016/j.chaos.2006.03.005
[113]	Castro, C., El-Naschie, M.S. and Granik, A. (2000) Why We Live in 3 + 1 Dimensions. CERN Document Server. (No. hep-th/0004152).
[114]	Marek Crnjac, L. and El Naschie, M.S. (2013) Quantum Gravity and Dark Energy Using Fractal Planck Scaling. Journal of Modern Physics, 4, 31-38. https://doi.org/10.4236/jmp.2013.411A1005
[115]	El Naschie, M.S. (2016) Einstein-Rosen Bridge (ER), Einstein-Podolsky-Rosen Experiment (EPR) and Zero Measure Rindler-KAM Cantorian Spacetime Geometry (ZMG) Are Conceptually Equivalent. Journal of Quantum Information Science, 6, 1-9. https://doi.org/10.4236/jqis.2016.61001
[116]	El Naschie, M.S. (1993) On Certain Infinite Dimensional Cantor Sets and the Schrödinger Wave. Chaos, Solitons & Fractals, 3, 89-98. https://doi.org/10.1016/0960-0779(93)90042-Y
[117]	El Naschie, M.S. (1995) Statistical Geometry of a Cantor Discretum and Semiconductors. Computers & Mathematics with Applications, 29, 103-110. https://doi.org/10.1016/0898-1221(95)00062-4
[118]	El Naschie, M.S. (2003) Kleinian Groups in E(∞) and Their Connection to Particle Physics and Cosmology. Chaos, Solitons & Fractals, 16, 637-649. https://doi.org/10.1016/S0960-0779(02)00489-7
[119]	El Naschie, M.S. (2014) Electromagnetic—Pure Gravity Connection via Hardy’s Quantum Entanglement. Journal of Electromagnetic Analysis and Applications, 6, 233. https://doi.org/10.4236/jemaa.2014.69023
[120]	El Naschie, M.S. (2013) Experimentally Based Theoretical Arguments That Unruh’s Temperature, Hawking’s Vacuum Fluctuation and Rindler’s Wedge Are Physically Real. American Journal of Modern Physics, 2, 357-361. https://doi.org/10.11648/j.ajmp.20130206.23
[121]	El Naschie, M.S. (2015) Kerr Black Hole Geometry Leading to Dark Matter and Dark Energy via E-Infinity Theory and the Possibility of a Nano Spacetime Singularities Reactor. Natural Science, 7, 210. https://doi.org/10.4236/ns.2015.74024
[122]	Castro, C. (2000) Is Quantum Space-Time Infinite Dimensional. Chaos, Solitons & Fractals, 11, 1663-1670. https://doi.org/10.1016/S0960-0779(00)00018-7
[123]	El Naschie, M.S. (2014) Calculating the Exact Experimental Density of the Dark Energy in the Cosmos Assuming a Fractal Speed of Light. International Journal of Modern Nonlinear Theory and Application, 3, 1-5. https://doi.org/10.4236/ijmnta.2014.31001
[124]	El Naschie, M.S. (2004) Topological Defects in the Symplictic Vacuum, Anomalous Positron Production and the Gravitational Instanton. International Journal of Modern Physics E, 13, 835-849. https://doi.org/10.1142/S0218301304002429
[125]	El Naschie, M.S. (2000) Towards a Geometrical Theory for the Unification of All Fundamental Forces. Chaos, Solitons & Fractals, 11, 1459-1469. https://doi.org/10.1016/S0960-0779(99)00194-0
[126]	El Naschie, M.S. (2014) From Modified Newtonian Gravity to Dark Energy via Quantum Entanglement. Journal of Applied Mathematics and Physics, 2, 803. https://doi.org/10.4236/jamp.2014.28088
[127]	El Naschie, M.S. (2001) On a Heterotic String-Based Algorithm for the Determination of the Fine Structure Constant. Chaos, Solitons & Fractals, 12, 539-549. https://doi.org/10.1016/S0960-0779(00)00187-9
[128]	El Naschie, M.S. (2005) Determining the Number of Higgs Particles Starting from General Relativity and Various Other Field Theories. Chaos, Solitons & Fractals, 23, 711-726. https://doi.org/10.1016/j.chaos.2004.06.048
[129]	El Naschie, M.S. (2015) Quantum Fractals and the Casimir-Dark Energy Duality—The Road to a Clean Quantum Energy Nano Reactor. Journal of Modern Physics, 6, 1321. https://doi.org/10.4236/jmp.2015.69137
[130]	Iovane, G. and Giordano, P. (2007) Wavelets and Multiresolution Analysis: Nature of ε (∞) Cantorian Space-Time. Chaos, Solitons & Fractals, 32, 896-910. https://doi.org/10.1016/j.chaos.2005.11.097
[131]	He, J.H., Liu, Y., Xu, L. and Yu, J.Y. (2007) Micro Sphere with Nanoporosity by Electrospinning. Chaos, Solitons & Fractals, 32, 1096-1100. https://doi.org/10.1016/j.chaos.2006.07.045
[132]	Chen, W. (2006) Time-Space Fabric Underlying Anomalous Diffusion. Chaos, Solitons & Fractals, 28, 923-929. https://doi.org/10.1016/j.chaos.2005.08.199
[133]	El Naschie, M.S. (2006) Is Gravity Less Fundamental than Elementary Particles Theory? Critical Remarks on Holography and E-Infinity Theory. Chaos, Solitons & Fractals, 29, 803-807. https://doi.org/10.1016/j.chaos.2006.01.012
[134]	El Naschie, M.S. (2008) Average Exceptional Lie and Coxeter Group Hierarchies with Special Reference to the Standard Model of High Energy Particle Physics. Chaos, Solitons & Fractals, 37, 662-668. https://doi.org/10.1016/j.chaos.2008.01.018
[135]	El Naschie, M.S. (2015) Hubble Scale Dark Energy Meets Nano Scale Casimir Energy and the Rational of Their T-Duality and Mirror Symmetry Equivalence. World Journal of Nano Science and Engineering, 5, 57. https://doi.org/10.4236/wjnse.2015.53008
[136]	El Naschie, M.S. (2005) Determining the Mass of the Higgs and the Electroweak Bosons. Chaos, Solitons & Fractals, 24, 899-905. https://doi.org/10.1016/j.chaos.2004.11.003
[137]	El Naschie, M.S. (2015) From Kantian-Reinen Fernunft to the Real Dark Energy Density of the Cosmos via the Measure Concentration of Convex Geometry in Quasi Banach Spacetime. Open Journal of Philosophy, 5, 123. https://doi.org/10.4236/ojpp.2015.51014
[138]	El Naschie, M.S. (2014) Rindler Space Derivation of Dark Energy. Journal of Modern Physics and Applications, 6, 1-10.
[139]	Marek-Crnjac, L. and El Naschie, M.S. (2013) Chaotic Fractal Tiling for the Missing Dark Energy and Veneziano Model. Applied Mathematics, 4, 22. https://doi.org/10.4236/am.2013.411A2005
[140]	Nottale, L. (1999) The Scale-Relativity Program. Chaos, Solitons & Fractals, 10, 459-468. https://doi.org/10.1016/S0960-0779(98)00195-7
[141]	El Naschie, M.S. (2013) The Hydrogen Atom Fractal Spectra, the Missing Dark Energy of the Cosmos and Their Hardy Quantum Entanglement. International Journal of Modern Nonlinear Theory and Application, 2, 167. https://doi.org/10.4236/ijmnta.2013.23023
[142]	El Naschie, M.S. (2005) A New Solution for the Two-Slit Experiment. Chaos, Solitons & Fractals, 25, 935-939. https://doi.org/10.1016/j.chaos.2005.02.029
[143]	He, J.H. (2007) On the Number of Elementary Particles in a Resolution Dependent Fractal Spacetime. Chaos, Solitons & Fractals, 32, 1645-1648. https://doi.org/10.1016/j.chaos.2006.08.015
[144]	Gottlieb, I., Agop, M., Ciobanu, G. and Stroe, A. (2006) El Naschie’s ε (∞) Space-Time and New Results in Scale Relativity Theories. Chaos, Solitons & Fractals, 30, 380-398. https://doi.org/10.1016/j.chaos.2005.11.018
[145]	El Naschie, M.S. (2015) The Cantorian Monadic Plasma behind the Zero Point Vacuum Spacetime Energy. American Journal of Nano Research and Application, 3, 66-70.
[146]	Gottlieb, I., Agop, M. and Jarcau, M. (2004) El Naschie’s Cantorian Space-Time and General Relativity by Means of Barbilian’s Group: A Cantorian Fractal Axiomatic Model of Space-Time. Chaos, Solitons & Fractals, 19, 705-730. https://doi.org/10.1016/S0960-0779(03)00244-3
[147]	El Naschie, M.S. (2009) On Zero-Dimensional Points Curvature in the Dynamics of Cantorian-Fractal Spacetime Setting and High Energy Particle Physics. Chaos, Solitons & Fractals, 41, 2725-2732. https://doi.org/10.1016/j.chaos.2008.10.001
[148]	El Naschie, M.S. (2008) High Energy Physics and the Standard Model from the Exceptional Lie Groups. Chaos, Solitons & Fractals, 36, 1-17. https://doi.org/10.1016/j.chaos.2007.08.058
[149]	El Naschie, M.S. (2001) On Twistors in Cantorian E (∞) Space. Chaos, Solitons & Fractals, 12, 741-746. https://doi.org/10.1016/S0960-0779(00)00193-4
[150]	El Naschie, M.S. (2005) Non-Euclidean Spacetime Structure and the Two-Slit Experiment. Chaos, Solitons & Fractals, 26, 1-6. https://doi.org/10.1016/j.chaos.2005.02.031
[151]	El Naschie, M.S. and Rossler, O.E. (1994) Quantum Mechanics and Chaotic Fractals. Chaos, Solitons & Fractals, 4, 307-309. https://doi.org/10.1016/0960-0779(94)90049-3
[152]	Nottale, L. (1995) Scale Relativity: From Quantum Mechanics to Chaotic Dynamics. Chaos, Solitons & Fractals, 6, 399-410. https://doi.org/10.1016/0960-0779(95)80047-K
[153]	Marek-Crnjac, L. (2009) A Short History of Fractal-Cantorian Space-Time. Chaos, Solitons & Fractals, 41, 2697-2705. https://doi.org/10.1016/j.chaos.2008.10.007
[154]	Marek-Crnjac, L. (2015) On El Naschie’s Fractal-Cantorian Space-Time and Dark Energy— A Tutorial Review. Natural Science, 7, 581. https://doi.org/10.4236/ns.2015.713058
[155]	He, J.H. (2014) A Tutorial Review on Fractal Spacetime and Fractional Calculus. International Journal of Theoretical Physics, 53, 3698-3718. https://doi.org/10.1007/s10773-014-2123-8
[156]	El Naschie, M.S. (2005) Kähler-Like Manifolds, Weyl Spinor Particles and E-Infinity High Energy Physics. Chaos, Solitons & Fractals, 26, 665-670. https://doi.org/10.1016/j.chaos.2005.01.018
[157]	El Naschie, M.S. (2005) A P-Brane Vindication of the Two Higgs-Doublet Minimally Super-Symmetric Standard Model and Related Issues. Chaos, Solitons & Fractals, 23, 1511-1514. https://doi.org/10.1016/j.chaos.2004.08.008
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[159]	Giordano, P., Iovane, G. and Laserra, E. (2007) El Naschie ∈(∞) Cantorian Structures with Spatial Pseudo-Spherical Symmetry: A Possible Description of the Actual Segregated Universe. Chaos, Solitons & Fractals, 31, 1108-1117. https://doi.org/10.1016/j.chaos.2006.03.114
[160]	El Naschie, M.S. (2005) The Supersymmetric Components of the Riemann-Einstein Tensor as Nine Dimensional Spheres in Ten Dimensional Space. Chaos, Solitons & Fractals, 24, 29-32. https://doi.org/10.1016/j.chaos.2004.09.002
[161]	He, J.H. (2007) E-Infinity Theory and the Higgs Field. Chaos, Solitons & Fractals, 31, 782-786. https://doi.org/10.1016/j.chaos.2006.04.041
[162]	Iovane, G., Giordano, P. and Salerno, S. (2005) Dynamical Systems on El Naschie’s ε (∞) Cantorian Space-Time. Chaos, Solitons & Fractals, 24, 423-441. https://doi.org/10.1016/j.chaos.2004.09.068
[163]	El Naschie, M.S. (2003) On John Nash’s Crumpled Surface. Chaos, Solitons & Fractals, 18, 635-641. https://doi.org/10.1016/S0960-0779(03)00007-9
[164]	El Naschie, M.S. (2016) On a Fractal Version of Witten’s M-Theory. International Journal of Astronomy and Astrophysics, 6, 135. https://doi.org/10.4236/ijaa.2016.62011
[165]	El Naschie, M.S. (2008) The Exceptional Lie Symmetry Groups Hierarchy and the Expected Number of Higgs Bosons. Chaos, Solitons & Fractals, 35, 268-273. https://doi.org/10.1016/j.chaos.2007.07.036
[166]	El Naschie, M.S. (2015) The Casimir Topological Effect and a Proposal for a Casimir-Dark Energy Nano Reactor. World Journal of Nano Science and Engineering, 5, 26. https://doi.org/10.4236/wjnse.2015.51004
[167]	El Naschie, M.S. (2008) Exact Non-Perturbative Derivation of Gravity’s Fine Structure Constant, the Mass of the Higgs and Elementary Black Holes. Chaos, Solitons & Fractals, 37, 346-359. https://doi.org/10.1016/j.chaos.2007.10.021
[168]	El Naschie, M.S. (2015) From Fusion Algebra to Cold Fusion or from Pure Reason to Pragmatism. Open Journal of Philosophy, 5, 319. https://doi.org/10.4236/ojpp.2015.56040
[169]	El Naschie, M.S. (2015) If Quantum “Wave” of the Universe Then Quantum “Particle” of the Universe: A Resolution of the Dark Energy Question and the Black Hole Information Paradox. International Journal of Astronomy and Astrophysics, 5, 243. https://doi.org/10.4236/ijaa.2015.54027
[170]	Rössler, O.E. (1996) Relative-State Theory: Four New Aspects. Chaos, Solitons & Fractals, 7, 845-852. https://doi.org/10.1016/0960-0779(95)00117-4
[171]	Nottale, L. (1998) Scale Relativity and Schrödinger’s Equation. Chaos, Solitons & Fractals, 9, 1051-1061. https://doi.org/10.1016/S0960-0779(97)00190-2
[172]	El Naschie, M.S. (2005) On Penrose View of Transfinite Sets and Computability and the Fractal Character of E-Infinity Spacetime. Chaos, Solitons & Fractals, 25, 531-533. https://doi.org/10.1016/j.chaos.2005.01.001
[173]	Iovane, G. (2006) Cantorian Space-Time and Hilbert Space: Part II—Relevant Consequences. Chaos, Solitons & Fractals, 29, 1-22. https://doi.org/10.1016/j.chaos.2005.10.045
[174]	Czajko, J. (2000) On Conjugate Complex Time—I: Complex Time Implies Existence of Tangential Potential That Can Cause Some Equipotential Effects of Gravity. Chaos, Solitons & Fractals, 11, 1983-1992. https://doi.org/10.1016/S0960-0779(99)00091-0
[175]	El Naschie, M.S. (2005) Dead or Alive: Desperately Seeking Schrödinger’s Cat. Chaos, Solitons & Fractals, 26, 673-676. https://doi.org/10.1016/j.chaos.2005.02.030
[176]	Nottale, L. (1994) Scale Relativity, Fractal Space-Time and Quantum Mechanics. Chaos, Solitons & Fractals, 4, 361-388. https://doi.org/10.1016/0960-0779(94)90051-5
[177]	El Naschie, M.S. (2015) Application of Dvoretzky’s Theorem of Measure Concentration in Physics and Cosmology. Open Journal of Microphysics, 5, 11. https://doi.org/10.4236/ojm.2015.52002
[178]	El Naschie, M.S. (2004) Quantum Collapse of Wave Interference Pattern in the Two-Slit Experiment: A Set Theoretical Resolution. Nonlinear Science Letter A, 2, 1-9.
[179]	Iovane, G. (2006) Cantorian Spacetime and Hilbert Space: Part I—Foundations. Chaos, Solitons & Fractals, 28, 857-878. https://doi.org/10.1016/j.chaos.2005.08.074
[180]	El Naschie, M.S. (1992) On the Uncertainty of Information in Quantum Space-Time. Chaos, Solitons & Fractals, 2, 91-94. https://doi.org/10.1016/0960-0779(92)90050-W
[181]	Iovane, G., Gargiulo, G. and Zappale, E. (2006) A Cantorian Potential Theory for Describing Dynamical Systems on El Naschie’s Space-Time. Chaos, Solitons & Fractals, 27, 588-598. https://doi.org/10.1016/j.chaos.2005.05.015
[182]	El Naschie, M.S. (1998) COBE Satellite Measurement, Hyperspheres, Superstrings and the Dimension of Spacetime. Chaos, Solitons & Fractals, 9, 1445-1471. https://doi.org/10.1016/S0960-0779(98)00120-9
[183]	El Naschie, M.S. (2006) On the Vital Role Played by the Electron-Volt Units System in High Energy Physics and Mach’s Principle of “Denkökonomie”. Chaos, Solitons & Fractals, 28, 1366-1371. https://doi.org/10.1016/j.chaos.2005.11.001
[184]	El Naschie, M.S. (2015) Computing Dark Energy and Ordinary Energy of the Cosmos as a Double Eigenvalue Problem. Journal of Modern Physics, 6, 384. https://doi.org/10.4236/jmp.2015.64042
[185]	Elnaschie, M.S. (2005) The Feynman Path Integral and Ε-Infinity from the Two-Slit Gedanken Experiment. International Journal of Nonlinear Sciences and Numerical Simulation, 6, 335-342. https://doi.org/10.1515/IJNSNS.2005.6.4.335
[186]	Iovane, G., Chinnici, M. and Tortoriello, F.S. (2008) Multifractals and El Naschie E-Infinity Cantorian Space-Time. Chaos, Solitons & Fractals, 35, 645-658. https://doi.org/10.1016/j.chaos.2007.07.051
[187]	El Naschie, M.S. (2015) A Cold Fusion-Casimir Energy Nano Reactor Proposal. World Journal of Nano Science and Engineering, 5, 49. https://doi.org/10.4236/wjnse.2015.52007
[188]	El Naschie, M.S. (2014) From Highly Structured E-Infinity Rings and Transfinite Maximally Symmetric Manifolds to the Dark Energy Density of the Cosmos. Advances in Pure Mathematics, 4, 641. https://doi.org/10.4236/apm.2014.412073
[189]	Selvam, A.M. and Fadnavis, S. (1999) Superstrings, Cantorian-Fractal Spacetime and Quantum-Like Chaos in Atmospheric Flows. Chaos, Solitons & Fractals, 10, 1321-1334. https://doi.org/10.1016/S0960-0779(98)00150-7
[190]	El Naschie, M.S. (2006) Advanced Prerequisite for E-Infinity Theory. Chaos, Solitons & Fractals, 30, 636-641. https://doi.org/10.1016/j.chaos.2006.04.044
[191]	He, J.H. (2006) Application of E-Infinity Theory to Turbulence. Chaos, Solitons & Fractals, 30, 506-511. https://doi.org/10.1016/j.chaos.2005.11.033
[192]	Marek-Crnjac, L. (2013) Modification of Einstein’s E = mc2 to E = (1/22)mc2. American Journal of Modern Physics, 2, 255-263. https://doi.org/10.11648/j.ajmp.20130205.14
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[194]	El Naschie, M.S. (2016) Quantum Dark Energy from the Hyperbolic Transfinite Cantorian Geometry of the Cosmos. Natural Science, 8, 152. https://doi.org/10.4236/ns.2016.83018
[195]	Selvam, A.M. and Fadnavis, S. (1999) Cantorian Fractal Spacetime, Quantum-Like Chaos and Scale Relativity in Atmospheric Flows. Chaos, Solitons & Fractals, 10, 1577-1582. https://doi.org/10.1016/S0960-0779(98)00209-4
[196]	He, J.H. and Marek-Crnjac, L. (2013) The Quintessence of El Naschie’s Theory of Fractal Relativity and Dark Energy. Fractal Spacetime and Noncommutative Geometry in Quantum and High Energy Physics, 3, 130-137.
[197]	El Naschie, M.S. (2008) Noether’s Theorem, Exceptional Lie Groups Hierarchy and Determining 1/α ≌ 137 of Electromagnetism. Chaos, Solitons & Fractals, 35, 99-103. https://doi.org/10.1016/j.chaos.2007.05.005
[198]	El Naschie, M.S. (2007) Quantum Probability without a Phase and a Topological Resolution of the Two-Slit Experiment. International Journal of Nonlinear Sciences and Numerical Simulation, 8, 195-198. https://doi.org/10.1515/IJNSNS.2007.8.2.195
[199]	El Naschie, M.S. (2009) Higgs Mechanism, Quarks Confinement and Black Holes as a Cantorian Spacetime Phase Transition Scenario. Chaos, Solitons & Fractals, 41, 869-874. https://doi.org/10.1016/j.chaos.2008.04.013
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[205]	Babchin, A.J. and El Naschie, M.S. (2015) On the Real Einstein Beauty E = kmc2. World Journal of Condensed Matter Physics, 6, 1. https://doi.org/10.4236/wjcmp.2016.61001
[206]	He, J.H. (2008) String Theory in a Scale Dependent Discontinuous Space-Time. Chaos, Solitons & Fractals, 36, 542-545. https://doi.org/10.1016/j.chaos.2007.07.093
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[209]	El Naschie, M.S. (2015) A Casimir-Dark Energy Nano Reactor Design—Phase One. Natural Science, 7, 287. https://doi.org/10.4236/ns.2015.76032
[210]	Ord, G.N. (1997) Classical Particles and the Dirac Equation with an Electromagnetic Field. Chaos, Solitons & Fractals, 8, 727-741. https://doi.org/10.1016/S0960-0779(96)00059-8
[211]	Agop, M., Paun, V. and Harabagiu, A. (2008) El Naschie’s ε (∞) Theory and Effects of Nanoparticle Clustering on the Heat Transport in Nanofluids. Chaos, Solitons & Fractals, 37, 1269-1278. https://doi.org/10.1016/j.chaos.2008.01.006
[212]	El Naschie, M.S., Marek-Crnjac, L., He, J.H. and Helal, M.A. (2013) Computing the Missing Dark Energy of a Clopen Universe Which Is Its Own Multiverse in Addition to Being Both Flat and Curved. Fractal Spacetime and Noncommutative Geometry in Quantum and High Energy Physics, 3, 3-10.
[213]	El Naschie, M.S. (2005) A Tale of Two Kleins Unified in Strings and E-Infinity Theory. Chaos, Solitons & Fractals, 26, 247-254. https://doi.org/10.1016/j.chaos.2005.01.016
[214]	Nottale, L. (2005) On the Transition from the Classical to the Quantum Regime in Fractal Space-Time Theory. Chaos, Solitons & Fractals, 25, 797-803. https://doi.org/10.1016/j.chaos.2004.11.071
[215]	El Naschie, M.S. (2007) Rigorous Derivation of the Inverse Electromagnetic Fine Structure Constant Using Super String Theory and the Holographic Boundary of E-Infinity. Chaos, Solitons & Fractals, 32, 893-895. https://doi.org/10.1016/j.chaos.2006.09.055
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[218]	Mejias, A., Sigalotti, L.D.G., Sira, E. and De Felice, F. (2004) On El Naschie’s Complex Time, Hawking’s Imaginary Time and Special Relativity. Chaos, Solitons & Fractals, 19, 773-777. https://doi.org/10.1016/S0960-0779(03)00273-X
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[221]	El Naschie, M.S. (2003) The Cantorian Interpretation of High Energy Physics and the Mass Spectrum of Elementary Particles. Chaos, Solitons & Fractals, 17, 989-1001. https://doi.org/10.1016/S0960-0779(03)00006-7
[222]	El Naschie, M.S. (2006) Thomas Mann and Heinrich Mann, Dual Brothers and Complimentary Genius Embraced by Complex Reality. International Journal of Nonlinear Sciences and Numerical Simulation, 7, 1-6. https://doi.org/10.1515/IJNSNS.2006.7.1.1
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[224]	Selvam, A.M. (2005) A General Systems Theory for Chaos, Quantum Mechanics and Gravity for Dynamical Systems of All Space-Time Scales. arXiv Preprint Physics/0503028.
[225]	Iovane, G. and Benedetto, E. (2006) A Projective Approach to Dynamical Systems, Applications in Cosmology and Connections with El Naschie ∈(∞) Cantorian Space-Time. Chaos, Solitons & Fractals, 30, 269-277. https://doi.org/10.1016/j.chaos.2005.11.005
[226]	Goldfain, E. (2005) Local Scale Invariance, Cantorian Space-Time and Unified Field Theory. Chaos, Solitons & Fractals, 23, 701-710. https://doi.org/10.1016/j.chaos.2004.05.020
[227]	El Naschie, M.S. (2016) On a Quantum Gravity Fractal Spacetime Equation: QRG ~ HD + FG and Its Application to Dark Energy—Accelerated Cosmic Expansion. Journal of Modern Physics, 7, 729. https://doi.org/10.4236/jmp.2016.78069
[228]	El Naschie, Μ.S. (2007) Deterministic Quantum Mechanics versus Classical Mechanical Indeterminism. International Journal of Nonlinear Sciences and Numerical Simulation, 8, 5-10. https://doi.org/10.1515/IJNSNS.2007.8.1.5
[229]	El Naschie, M.S. (2009) Arguments for the Compactness and Multiple Connectivity of Our Cosmic Spacetime. Chaos, Solitons & Fractals, 41, 2787-2789. https://doi.org/10.1016/j.chaos.2008.10.011
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[231]	El Naschie, M.S. (2015) The Casimir Effect as a Pure Topological Phenomenon and the Possibility of a Casimir Nano Reactor—A Preliminary Conceptual Design. American Journal of Nano Research and Applications, 3, 33-40.
[232]	El Naschie, M.S. (2000) Scale Relativity in Cantorian E (∞) Space-Time. Chaos, Solitons & Fractals, 11, 2391-2395. https://doi.org/10.1016/S0960-0779(99)00209-X
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[235]	El Naschie, M.S. (2016) Einstein’s Dark Energy via Similarity Equivalence, ’tHooft Dimensional Regularization and Lie Symmetry Groups. International Journal of Astronomy and Astrophysics, 6, 56. https://doi.org/10.4236/ijaa.2016.61005
[236]	El Naschie, M.S. (2005) A Few Hints and Some Theorems about Witten’s M Theory and T-Duality. Chaos, Solitons & Fractals, 25, 545-548. https://doi.org/10.1016/j.chaos.2005.01.009
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[238]	El Naschie, M.S. (2004) The Higgs—Physical and Number Theoretical Arguments for the Necessity of a Triple Elementary Particle in Super Symmetric Spacetime. Chaos, Solitons & Fractals, 22, 1199-1209. https://doi.org/10.1016/j.chaos.2004.04.026
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[242]	El Naschie, M.S. (2008) Using Witten’s Five Brane Theory and the Holographic Principle to Derive the Value of the Electromagnetic Fine Structure Constant. Chaos, Solitons & Fractals, 38, 1051-1053. https://doi.org/10.1016/j.chaos.2008.06.001
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[249]	Greene, B. (2004) The Fabric of the Cosmos. Penguin Books, London.
[250]	El Naschie, M.S. (2016) From Witten’s 462 Supercharges of 5-D Branes in Eleven Dimensions to the 95.5 Percent Cosmic Dark Energy Density behind the Accelerated Expansion of the Universe. Journal of Quantum Information Science, 6, 57. https://doi.org/10.4236/jqis.2016.62007
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[254]	El Naschie, M.S. (2015) The Counterintuitive Increase of Information Due to Extra Spacetime Dimensions of a Black Hole and Dvoretzky’s Theorem. Natural Science, 7, 483. https://doi.org/10.4236/ns.2015.710049
[255]	Marek-Crnjac, L. (2003) The Mass Spectrum of High Energy Elementary Particles via El Naschie’s E (∞) Golden Mean Nested Oscillators, the Dunkerly-Southwell Eigenvalue Theorems and KAM. Chaos, Solitons & Fractals, 18, 125-133. https://doi.org/10.1016/S0960-0779(02)00587-8
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