Quantum Disentanglement as the Physics behind Dark Energy

Abstract

A straightforward simple proof is given that dark energy is the natural conse-quence of a quantum disentanglement physical process. Thus while the ordinary energy density of the cosmos is equal to half that of Hardy’s quantum probability of Entanglement i.e. where , the density of cosmic dark energy is consequently one minus divided by two i.e. . This result is in full agreement with all the numerous previous theoretical predictions as well as being in remarkable agreement with the overwhelming majority of cosmic accurate measurements and observations.

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Naschie, M. (2017) Quantum Disentanglement as the Physics behind Dark Energy. Open Journal of Microphysics, 7, 1-27. doi: 10.4236/ojm.2017.71001.

1. Introduction

Although relatively short we should say from the outset that this paper covers a large part of modern cutting edge research in quantum physics and cosmology [1] - [432] . The work is essentially and mainly motivated by the desire to show more clearly than ever before the deep connection between quantum entanglement [47] [423] and the absence of almost 95.5% of the energy supposed to be contained in our cosmos [290] . We intend to give a short, simple and exact theoretical proof based on the reverse of quantum entanglement with which we mean of course Quantum Disentanglement [1] [2] . In particular we start from Hardy’s exact experimentally well-established probability of quantumly entangled of two particles [22] [26] [28] where [196] [212] and then reason that the corresponding quantum probability of disentanglement [1] [2] is. Subsequently we show that while the ordinary measureable energy density of the cosmos is given by half of Hardy’s quantum entanglement, i.e., the corresponding Dark Energy density is given by half of the quantum probability of disentanglement i.e. [185] [194] . This is all in full agreement with highly accurate cosmic measurement and observations as well as numerous previous derivations [167] [169] [177] . The strategy and details of our analysis will be given in the next two sections.

2. Background Information and Outline of the Paper

Hardy’s probability of entanglement is one of the most important exact results in quantum mechanics and was found to be exactly equal to for two quantum particles [139] [154] . It is thus an elementary almost trivial step to conclude from this result that the probability of not being quantumly entangled must be [123] [126] [135] . Subsequently it is not difficult to show that could be written as [424] . Now remembering that is the Hausdorff dimension of a Zero set modeled by a one-dimensional random Mauldin-Williams random Cantor set [7] , then could be interpreted as an entropic measure. It follows then that maybe seen as a five-dimensional entropy from which we could deduce the energy density after multiplication with a dimensional constant. In analogy to the above and knowing that is the Hausdorff dimension of an empty set modeled by the Cantor set left from the unit interval used in constructing the said Random Mauldin-Williams Cantor set [73] [80] , we see that is also a five-dimensional entropy [21] - [29] . The only difference between and is that the first is multiplicative intersection and represents an entangled state, while is an additive union which represents a disentangled state [26] [248] . In fact even the reader who is not familiar with our previous work on fractal Cantorian spacetime and Dark Energy must have guessed by now that the entanglement probability would lead to the ordinary measureable energy density of the cosmos [29] [119] [121]

(1)

while will lead us to the Dark Energy density of the cosmos which due to this very disentangled nature of cannot be measured in any direct way at least with our present technology [29] [70] [72] [78]

(2)

Finally it is also not difficult to guess that it will turn out as a surprise which on little reflection is not really a surprise that the dimensional constant needed to move from entropy to energy is given by nothing else but Einstein’s marvelous equation

(3)

so that at the end we will find from Equation (1) that [39] [62] [65]

(4)

and from Equation (3) we find that

(5)

In other words Einstein’s beauty derived long before quantum mechanics harbored all the time two quantum components namely E(O) and E(D) which when added together give the most famous formula in physics [32] [33]

(6)

3. Analysis and Proof of Ordinary Energy and Dark Energy Theorems

In the following we give in all earnest an embarrassingly short analysis leading to a proof of the following theorems:

Theorem One:

The ordinary energy density of the cosmos is half of Hardy’s probability of quantum entanglement

Theorem Two:

The Dark Energy Density of the Cosmos is half of the Hardy type Quantum Probability of disentanglement.

To prove the first Theorem we could do nothing better for the sake of brevity than repeat any of the two dozen or so previous proofs published in numerous papers over the last 4 years [23] [24] [28] However we recommend References [23] [28] and [39] as well as [32] .

On the other hand proving Theorem Two becomes trivial because which we just considered proven is the complement of which we want to prove. In other words proving that Hardy’s quantum entanglement means is automatically a proof that Hardy’s disentanglement probability means that the Dark Energy density is simply [24] [26] [28]

(7)

This is the end of the proof which has the unusual disadvantage of being too simple to believe and we have only to mention the additional obvious insight that can be measured because it is coherent while cannot be directly measured and we only infer its existence from the accelerated expansion of the universe because it is disentangled [1] [2] . This is a different view of the same good old particle-wave duality [7] . We recall our earlier conclusion that is the kinetic energy of the pre-quantum particle modeled by the Zero set while is the position or potential energy of the quantum wave modeled by the empty set [26] [28] . Now since any interference or measurement on an empty set quantum wave make the set non empty, we have to invite first quantum wave non-demolishing measuring devices before being in a position to measure dark energy directly [23] [26] [28] .

4. Zeno’s Paradox and Dark Energy

We mentioned on passing in the previous section a distinction between the kinetic energy of the particle and potential energy of the wave [432] . This seems a little odd because it is the quantum wave which is responsible in quantum mechanics for propagation. We have touched on this subject in a recent paper and here we should give a clear cut answer to his contradictory viewpoint [432] . This clear cut answer will resonate century old philosophical problems connected to Zeno’s [43] [431] and reflection on the notion that motion is illusion [43] . From the viewpoint of the entire universe motion could be considered an illusion indeed or maybe we should express this in a more conservative way and say that the distinction between kinetic energy and potential energy when it comes to regarding dark energy and the entire universe is fuzzy and fundamentally so [432] . This is easily demonstrated when we realize that in five dimensional unit universe, the largest height must be half the unit radius and that the topological acceleration [432] is the down scaling of the topological (Sigalotti) speed of light [344] which means. Now let us look at Kinetic energy [432]

(8)

where v is the Velocity and c is the speed of light. Taking m to 3D we find the topological so that the topological energy becomes

(9)

Next we look at the potential energy [432]

(10)

Setting as reasoned earlier on we find

(11)

which is the same formula as. Of course this is a fundamentally different fuzzy situation and is not the same as the conservation of Energy Theorem of classical mechanics. To stress this quantum fuzziness when it comes to regarding the entire cosmos and the possibility for a rational resolution of Zeno’s paradox [431] , let us do the same thing for the “Universe” i.e. for of the Kaluza-Klein manifold. This would lead to [432]

(12)

which is of the quantum wave as we though initially should be. However even for the potential energy, we find [432]

(13)

which is the same result [432] .

In a sense we could conclude from the above that the most important modern result in the quantum physics is that of Hardy’s quantum entanglement probability [21] [23] [28]

(14)

With that we rest our case at least for the moment.

5. Conclusion

In this paper we have stated two Theorems and proved them. The First Theorem asserts that the measurable ordinary energy density of the cosmos is half that of the Hardy Probability of quantum entanglement. The second Theorem is complimentary to the first and states that the Dark energy density of the cosmos is half the quantum probability of the Hardy disentanglement. In addition we have shown that when regarding the universe as a whole, the sharp distinction between Kinetic energy and Potential energy of classical Newtonian mechanics ceases to be true and we are faced with a fundamentally and irreducibly fuzzy situation.

Acknowledgements

Without the work on non-commutative geometry and Prof. A. Connes’ analysis of Sir R. Penrose’s Fractal Tiling, this present work could not have been possible.

Conflicts of Interest

The authors declare no conflicts of interest.

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[85] 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
[86] El Naschie, M.S. (2005) On a Class of Fuzzy Kahler-Like Manifolds. Chaos, Solitons & Fractals, 26, 257-261.
https://doi.org/10.1016/j.chaos.2004.12.024
[87] El Naschie, M.S. (2005) Godel 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
[88] 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
[89] El Naschie, M.S. (2013) The Quantum Entanglement behind the Missing Dark Energy. Journal of Modern Physics and Applications, 2, 88-96.
[90] 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
[91] 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
[92] 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
[93] 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
[94] 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
[95] 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
[96] 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
[97] 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
[98] 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
[99] 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
[100] 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
[101] 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
[102] 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
[103] 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
[104] 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
[105] 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
[106] 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
[107] 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
[108] 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
[109] 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
[110] 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
[111] 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.
[112] 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
[113] 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
[114] Castro, C., El-Naschie, M.S. and Granik, A. (2000) Why We Live in 3 + 1 Dimensions. CERN Document Server. (No. hep-th/0004152).
[115] 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
[116] 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
[117] El Naschie, M.S. (1993) On Certain Infinite Dimensional Cantor Sets and the Schrodinger Wave. Chaos, Solitons & Fractals, 3, 89-98.
https://doi.org/10.1016/0960-0779(93)90042-Y
[118] 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
[119] 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
[120] 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
[121] 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
[122] 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
[123] 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
[124] 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
[125] 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
[126] 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
[127] 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
[128] 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
[129] 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
[130] 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
[131] 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
[132] 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
[133] 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
[134] 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
[135] 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
[136] 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
[137] 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
[138] 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
[139] El Naschie, M.S. (2014) Rindler Space Derivation of Dark Energy. Journal of Modern Physics and Applications, 6, 1-10.
[140] 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
[141] Nottale, L. (1999) The Scale-Relativity Program. Chaos, Solitons & Fractals, 10, 459-468.
https://doi.org/10.1016/S0960-0779(98)00195-7
[142] 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
[143] 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
[144] 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
[145] 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
[146] 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.
[147] 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
[148] 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
[149] 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
[150] 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
[151] 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
[152] 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
[153] 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
[154] 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
[155] 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
[156] 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
[157] El Naschie, M.S. (2005) Kahler-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
[158] 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
[159] Agop, M., Griga, V., Ciobanu, B., Ciubotariu, C., Buzea, C.G., Stan, C. and Buzea, C. (1998) Gravity and Cantorian Space-Time. Chaos, Solitons & Fractals, 9, 1143-1181.
https://doi.org/10.1016/S0960-0779(98)80005-2
[160] Giordano, P., Iovane, G. and Laserra, E. (2007) El Naschie ε(∞) 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
[161] 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
[162] 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
[163] 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
[164] 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
[165] 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
[166] 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
[167] 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
[168] 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
[169] 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
[170] 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
[171] Rossler, 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
[172] Nottale, L. (1998) Scale Relativity and Schrodinger’s Equation. Chaos, Solitons & Fractals, 9, 1051-1061.
https://doi.org/10.1016/S0960-0779(97)00190-2
[173] 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
[174] 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
[175] 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
[176] El Naschie, M.S. (2005) Dead or Alive: Desperately Seeking Schrodinger’s Cat. Chaos, Solitons & Fractals, 26, 673-676.
https://doi.org/10.1016/j.chaos.2005.02.030
[177] 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
[178] 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
[179] 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.
[180] 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
[181] 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
[182] 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
[183] 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
[184] 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 “Denkokonomie”. Chaos, Solitons & Fractals, 28, 1366-1371.
https://doi.org/10.1016/j.chaos.2005.11.001
[185] 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
[186] 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
[187] 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
[188] 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
[189] 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
[190] 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
[191] 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
[192] 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
[193] 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
[194] El-Ahmady, A.E. and Rafat, H. (2006) A Calculation of Geodesics in Chaotic Flat Space and Its Folding. Chaos, Solitons & Fractals, 30, 836-844.
https://doi.org/10.1016/j.chaos.2005.05.033
[195] 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
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[206] Babchin, A.J. and El Naschie, M.S. (2015) On the Real Einstein Beauty E = Kmc2. World Journal of Condensed Matter Physics, 6, 1-6.
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[207] He, J.H. (2008) String Theory in a Scale Dependent Discontinuous Space-Time. Chaos, Solitons & Fractals, 36, 542-545.
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https://doi.org/10.1016/j.chaos.2004.04.019
[210] El Naschie, M.S. (2015) A Casimir-Dark Energy Nano Reactor Design—Phase One. Natural Science, 7, 287-298.
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[213] 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.
[214] El Naschie, M.S. (2005) A Tale of Two Kleins Unified in Strings and E-Infinity Theory. Chaos, Solitons & Fractals, 26, 247-254.
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[222] 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.
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[223] 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.
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[225] 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.
[226] Iovane, G. and Benedetto, E. (2006) A Projective Approach to Dynamical Systems, Applications in Cosmology and Connections with El Naschie ε(∞) Cantorian Space-Time. Chaos, Solitons & Fractals, 30, 269-277.
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[228] 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, 729-736.
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[232] 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.
[233] El Naschie, M.S. (2000) Scale Relativity in Cantorian ε(∞) Space-Time. Chaos, Solitons & Fractals, 11, 2391-2395.
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[236] El Naschie, M.S. (2016) Einstein’s Dark Energy via Similarity Equivalence, ‘t Hooft Dimensional Regularization and Lie Symmetry Groups. International Journal of Astronomy and Astrophysics, 6, 56-81.
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[243] 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.
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