By viewing spacetime as a transfinite Turing computer, the present work is aimed at a generalization and geometrical-topological reinterpretation of a relatively old conjecture that the wormholes of general relativity are behind the physics and mathematics of quantum entanglement theory. To do this we base ourselves on the comprehensive set theoretical and topological machinery of the Cantorian-fractal E-infinity spacetime theory. Going all the way in this direction we even go beyond a quantum gravity theory to a precise set theoretical understanding of what a quantum particle, a quantum wave and quantum spacetime are. As a consequence of all these results and insights we can reason that the local Casimir pressure is the difference between the zero set quantum particle topological pressure and the empty set quantum wave topological pressure which acts as a wormhole “connecting” two different quantum particles with varying degrees of entanglement corresponding to varying degrees of emptiness of the empty set (wormhole). Our final result generalizes the recent conceptual equation of Susskind and Maldacena ER = EPR to become
ZMG = ER = EPR
where ZMG stands for zero measure Rindler-KAM geometry (of spacetime). These results were only possible because of the ultimate simplicity of our exact model based on Mauldin-Williams random Cantor sets and the corresponding exact Hardy’s quantum entanglement probability P(H) = where is the Hausdorff dimension of the topologically zero dimensional random Cantor thin set, i.e. a zero measure set and . On the other hand the positive measure spatial separation between the zero sets is a fat Cantor empty set possessing a Hausdorff dimension equal while its Menger-Urysohn topological dimension is a negative value equal minus one. This is the mathematical quintessence of a wormhole paralleling multiple connectivity in classical topology. It is both physically there because of the positive measure and not there because of the negative topological dimension.
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Mohamed S. El Naschie: The Measure Concentration of Convex Geometry in a Quasi Banach Spacetime behind the Supposedly Missing Dark Energy of the Cosmos. American Journal of Astronomy & Astrophysics, 2(6), 2014, pp. 72-77.
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Mohamed S. El Naschie: From E = mc2 to E = mc2/22—A Short Account of the Most Famous Equation in Physics and Its Hidden Quantum Entanglement Origin. Journal of Quantum Information Science, 4, 2014, pp. 284-291.
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Jean-Paul Auffray: On an Intriguing Invention Albert Einstein Made Which Has Gone Unnoticed Hitherto. Journal of Modern Physics, 6(11), 2015, pp. 1478-1491.
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M. S. El Naschie: The Cantorian Monadic Plasma behind the Zero Point Vacuum Spacetime Energy. American Journal of Nano Research & Application, 3, 2015, pp. 66-70.
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A. J. Babchin and M. S. El Naschie: On the Real Einstein Beauty E = kmc2. World Journal of Condensed Matter Physics, 6(1), 2016.
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Mohamed S. El Naschie: Three Quantum Particles Hardy Entanglement from the Topology of Cantorian-Fractal Spacetime and the Casimir Effect as Dark Energy—A Great Opportunity for Nanotechnology. American Journal of Nano Research and Applications, 3(1), 2015, pp. 1-5.
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Mohamed S. El Naschie: Deriving E = mc2/22 of Einstein’s Ordinary Quantum Relativity Energy Density from the Lie Symmetry Group SO(10) of Grand Unification of All Fundamental Forces and without Quantum Mechanics. American Journal of Mechanics & Applications, 2(2), 2014, pp. 6-9.
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Mohamed S. El Naschie: Cosserat-Cartan Modification of Einstein-Riemann Relativity and Cosmic Dark Energy Density. American Journal of Modern Physics, 3(2), 2014, pp. 82-87.
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Mohamed S. El Naschie: 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(6), 2013, pp. 357-361.
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M. S. El Naschie: The Quantum Gravity Immirzi Parameter—A General Physical and Topological Interpretation. Gravitation and Cosmology, 19(3), 2013, pp. 151-155.
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M. S. El Naschie: To Dark Energy Theory from a Cosserat-Like Model of Spacetime. Problems of Nonlinear Analysis in Engineering Systems, 1(41), Vol. 20, 2014, pp. 79-98.
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M. S. El Naschie, S. Olsen, J. H. He, S. Nada, L. Marek-Crnjac, A. Helal: On the Need for Fractal Logic in High Energy Quantum Physics. International Journal of Modern Nonlinear Theory and Application, 1(3), 2012, pp. 84-92.
![\"\"](\"Edit_1176e24c-c475-4071-abac-8d7cc2cc5b76.bmp\")
where ![\"\"](\"Edit_8eafc795-6855-43b8-9afa-76e53f7ad7ae.bmp\")
is the Hausdorff dimension of the topologically zero dimensional random Cantor thin set, i.e. a zero measure set and ![\"\"](\"Edit_95e8bb96-0c6c-4e60-9540-73d09e7eefbd.jpg\")
. On the other hand the positive measure spatial separation between the zero sets is a fat Cantor empty set possessing a Hausdorff dimension equal ![\"\"](\"Edit_ddef6026-6df9-4cf1-9c45-873a843ab669.bmp\")
while its Menger-Urysohn topological dimension is a negative value equal minus one. This is the mathematical quintessence of a wormhole paralleling multiple connectivity in classical topology. It is both physically there because of the positive measure and not there because of the negative topological dimension.
