Electromagnetic—Pure Gravity Connection via Hardy’s Quantum Entanglement


Pure gravity in eight dimensional super space possesses twenty degrees of freedom. Gauging these generalized coordinates in terms of a single self entangled Hardy type quantum particle we retrieve one hundred and thirty seven isometries corresponding to the well known value of the inverse electromagnetic fine structure constant. The deep implications of this thought provoking result are discussed in a general physical context.

Share and Cite:

Naschie, M. (2014) Electromagnetic—Pure Gravity Connection via Hardy’s Quantum Entanglement. Journal of Electromagnetic Analysis and Applications, 6, 233-237. doi: 10.4236/jemaa.2014.69023.

“Cosmology’s dark apparitions raise the question of whether Einstein’s theory is the right one”.

Jacob Bekenstein

1. Introduction

The extraordinary importance of Hardy’s quantum entanglement in clarifying a host of fundamental problems in theoretical physics and cosmology was researched extensively in the last three years [1] - [6] . Hardy’s result or rather theorem which states that the maximal probability of entanglement for two quantum particles is given by where was confirmed by numerous theoretical analysis and experimental tests with very high accuracy [7] [8] . In addition this result was generalized within the framework of a fractal-Cantorian spacetime theory to encompass n particles leading to a simple general expression for quantum entanglement [8] [9] . The key point in obtaining and understanding these results is undoubtedly the surreal nature of the “Cantorian” points representing the space in question which is a physical realization of the mathematical E-infinity theory [10] [11] . A point of this KAM-like Cantorian space where zero measure thin fractals are interwoven with positive measure fat fractals [12] - [14] is impossible to locate. It is something of a geometrical topological “Scarlet Pimpernel” [15] for being here and there and everywhere simultaneously so to speak [8] [13] [16] . The physical mechanics behind this almost esoterically unreal quasi manifold is the fundamental fact of quantum entanglement [8] .

In what follows we will see how gauging pure gravity in terms of Hardy’s quantum entanglement leads directly to the exact theoretical value of the inverse electromagnetic fine structure constant introduced to fundamental physics by the teacher of both Wolfgang Pauli and Werner Heisenberg, namely Arnold Sommerfeld [3] [17] - [22] . The physical implication of this new result in the context of the unification program of the fundamental interactions as well as ordinary and dark energy is discussed in some details.

2. Hardy’s Self Entanglement of a Single Quantum Particle

The general formula for Hardy’s entanglement is given by the multiplication of two parts, the local and the global where n is the number of particles [8] . Thus from


we find that for n = 2, Hardy’s celebrated result follows [8]


where. On the other hand for a single particle we obviously have [8] [9] [23]


We note that this particular result is essentially a statement about the magnitude and nature of the dimensionless topological Unruh thermal temperature which a Rindler observer experiences [9] [13] . Once we have established the magnitude of the degrees of freedom of pure gravity in D = 8 super space of unification [22] [23] we will be gauging it in terms of the Hardy-Unruh as mentioned in the introduction [8] [9] [23] . We also note that gives that exact value for the ordinary energy density of the cosmos while its Legendre transformation, i.e. gives the measured cosmic dark energy density [1] - [6] .

3. Pure Gravity in Unification Super Space

The degrees of freedom of the state of pure gravity, i.e. gravity in the absence of matter field are given dependently of the dimension d of spacetime by [24] [25]


For d = 8 one finds

. (5)

This happens to be numerically identical to the number of independent components of the Riemann tensor in four dimensional space. In the next section we will see how gauging D(8) in terms of P(Hardy) of oneself entangled quantum particle [8] [9] [12] will lead to the fundamental Sommerfeld “constant” [16] [17] .

4. Electromagnetism from Pure Gravity via Quantum Entanglement

The electromagnetic fine structure constant is one of the most fundamental “constants” in nature and has a direct bearing on the existence and appearance of life and obviously on our entire range of modern technological civilization [22] . There are in the meantime numerous ways to derive the exact E-infinity value of the inverse fine structure constant where is a small irrational tail given by k = 0.082033925 which is directly related to Hardy’s quantum entanglement as we will see momentarily [2] - [9] . In fact could easily be reconstructed from the theoretical value of the electroweak couplings using the exact renormalization equation of E-infinity theory as follows [16] :


where, , , and. Inserting in one finds


We note that and are very close indeed to the corresponding electroweak value. However of the strong interaction is on the other hand slightly off the experimental value due to the nature of quarks confinement. On the other hand corresponding to quantum gravity Planck coupling does not enter into the conventional approximate renormalization equation of. In our present exact equation however is essential for the consistency of our theory. This can be seen from three subtle points. First the P-Adic expansion of gives us the following T-duality relation between weak range and Planck range as is obvious from [26]


where, and. In other words [16] [26]

. (9)

where T stands for Witten’s T-duality [23] . Second we know from previous fundamental results that the ultra high Planck energy Epl = 1019 Gev corresponds to a topological value which is nothing else but Hardy’s probability of quantum entanglement [13] [14] [23] . Our point became manifestly clear when we notice that the irrational tail in is given exactly by


Thus is the intersection of quantum entanglement as well as quantum disentanglement showing the deep involvement of with Hardy’s entanglement as well as the theory of prime numbers because of the remarkable exact value of, namely [16] [26]


However this is by no means the only surprising aspect between the geometrical fusion and overcoming spacial separation by quantum entanglement and the attractive force of opposite charges of electromagnetic interaction. To show this we recall our earlier results regarding pure gravity in unification d = 8 super symmetric space and gauge the corresponding degrees of freedom in terms of a single quantum particle self entanglement, namely. That way we find


This is a remarkable result confirming a new aspect of the undeniable correctness of the unification program of high energy physics of quantum gravity [23] [25] and the dark energy mystery of cosmology [1] - [28] .

5. Conclusion

Unification is not only a simple and powerful idea which follows directly from Mach’s principle of Denk Economie taken to its ultimate but it is also a visible reality which can be inferred directly from the experimental facts of the standard model [9] [22] [25] . In the present short work we showed yet again the fundamental role of Hardy’s magnificent discovery of the exact value of probability of entanglement of two quantum particles [7] [8] [23] . Seen via the mental picture afforded by Hardy’s work, quantum entanglement [7] is a consequence of the transfinite fractal-Cantorian nature of quantum spacetime [1] [6] which glues the various fundamental interactions to a monolithic unity where electromagnetism and pure gravity are but two different sides of the same coin as amply demonstrated by the present new derivation of [3] [4] . In this sense we could extend the quotation of Prof. J. Bekenstein adorning the present work that Einstein’s pure gravity implicitly contains quantum mechanics and electromagnetism in a way which would have surprised even Einstein himself [27] [28] .

Conflicts of Interest

The authors declare no conflicts of interest.


[1] Marek-Crnjac, L. and He, J.-H. (2013) An Invitation to El Naschie’s Theory of Cantorian Space-Time and Dark Energy. International Journal of Astronomy and Astrophysics, 3, 464-471.
[2] 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 & Application, 2, 167-169.
[3] El Naschie, M.S., He, J.-H. and Marek-Crnjac, L. (2012) The Fine Structure Alpha Electromagnetism as the Most Fundamental Constant of Theoretical Physics. Fractal Spacetime and Noncommutative Geometry and High Energy Physics, 2, 143-153.
[4] El Naschie, M.S. and Nada, S. (2012) A Note on a Topological Geometrical Interpretation of Bell’s Inequality and Hardy’ Quantum Entanglement. Fractal Spacetime and Noncommutative Geometry and High Energy Physics, 2, 50-55.
[5] El Naschie, M.S. (2011) Electroweak Connection and Universality of Hardy’s Quantum Entanglement. Fractal Spacetime and Noncommutative Geometry and High Energy Physics, 1, 25-30.
[6] Helal, M.A., Marek-Crnjac, L. and He, J.-H. (2013) The Three Page Guide to the Most Important Results of M.S. El Naschie’s Research in E-Infinity Quantum Physics and Cosmology. Open Journal of Microphysics, 3, 141-145.
[7] Hardy, L. (1993) Nonlocality of Two Particles without Inequalities for Almost All Entangled States. Physical Review Letters, 71, 1665-1668.
[8] El Naschie, M.S. (2011) Quantum Entanglement as a Consequence of a Cantorian Micro Spacetime Geometry. Journal of Quantum Information Science, 1, 50-53.
[9] El Naschie, M.S. (2013) A Unified Newtonian-Relativistic Quantum Resolution of the Supposedly Missing Dark Energy of the Cosmos and the Constancy of the Speed of Light. International Journal of Modern Nonlinear Theory and Application, 2, 43-54.
[10] May, J.P. (2009) What Precisely Are E-Infinity Ring Spaces and E-Infinity Ring Spectra?
arXiv preprint: arXiv.0903.2813V1[math.A7]
[11] Fresse, B. (2010) The Bar Complex of an E-Infinity Algebra. Advances in Mathematics, 223, 2049-2096.
[12] El Naschie, M.S. (2013) A Rindler-KAM Spacetime Geometry and Scaling the Planck Scale Solves Quantum Relativity and Explains Dark Energy. International Journal of Astronomy and Astrophysics, 3, 483-493.
[13] El Naschie, M.S. (2014) Rindler Space Derivation of Dark Energy. Journal of Modern Physics & Applications, 6, 1-10.
[14] El Naschie, M.S. (2014) Entanglement of E8E8 Exceptional Lie Symmetry Group Dark Energy, Einstein’s Maximal Total Energy and the Hartle-Hawking No Boundary Proposal as the Explanation for Dark Energy. World Journal of Condensed Matter Physics, 4, 74-77.
[15] Orczy, B.E. (1905) The Scarlet Pimpernel. Hutchinson, UK.
[16] El Naschie, M.S. (2004) A Review of E-Infinity and the Mass Spectrum of High Energy Particle Physics. Chaos, Solitons & Fractals, 19, 209-236.
[17] El Naschie, M.S. (2007) Rigorous Derivation of the Inverse Electromagnetic Fine Structure Constant α = 1/137.036 Using Superstring Theory and the Holographic Boundary of E-Infinity. Chaos, Solitons & Fractals, 32, 893-895.
[18] El Naschie, M.S. (2004) Quantum Gravity, Clifford Algebras, Fuzzy Set Theory and the Fundamental Constants of Nature. Chaos, Solitons & Fractals, 20, 437-450.
[19] 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.
[20] El Naschie, M.S. (2008) On a Canonical Equation for All Fundamental Interactions. Chaos, Solitons & Fractals, 36, 1200-1204.
[21] El Naschie, M.S. (2007) On D. Gross’ Criticism of S. Eddington and an Exact Calculation of . Chaos, Solitons & Fractals, 32, 1245-1249.
[22] Penrose, R. (2004) The Road to Reality. Jonathan Cape, London.
[23] 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.
[24] El Naschie, M.S. (2014) Cosmic Dark Energy from ‘t Hooft’s Dimensional Regularization and Witten’s Topological Quantum Field Pure Gravity. Journal of Quantum Information Science, 4, 83-91.
[25] 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, 332-339.
[26] El Naschie, M.S. (2008) P-Adic Analysis and the Transfinite E8 Exceptional Lie Symmetry Group Unification. Chaos, Solitons & Fractals, 38, 612-614.
[27] Bekenstein, J. (2013) Trouble with Physics: Time to Discard Relativity? New Scientist, 217, 42.
[28] El Naschie, M.S. (2014) Why E Is Not Equal to mc2. Journal of Modern Physics, 5, 743-750.

Copyright © 2024 by authors and Scientific Research Publishing Inc.

Creative Commons License

This work and the related PDF file are licensed under a Creative Commons Attribution 4.0 International License.