Einstein’s energy mass formula is shown to consist of two basically quantum components E(O) = mc2/22 and E(D) = mc2(21/22). We give various arguments and derivations to expose the quantum entanglement physics residing inside a deceptively simple expression E = mc2. The true surprising aspect of the present work is however the realization that all the involved “physics” in deriving the new quantum dissection of Einstein’s famous formula of special relativity is actually a pure mathematical necessity anchored in the phenomena of volume concentration of convex manifold in high dimensional quasi Banach spaces. Only an endophysical experiment encompassing the entire universe such as COBE, WMAP, Planck and supernova analysis could have discovered dark energy and our present dissection of Einstein’s marvelous formula.

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From E = mc 2 to E = mc 2 /22—A Short Account of the Most Famous Equation in Physics and Its Hidden Quantum Entanglement Origin *

Starting from Witten’s eleven dimensional M-theory, the present work develops in an analogous way a corresponding dimensional fractal version where . Subsequently, the new fractal formalism is utilized to determine the measured ordinary energy density of the cosmos which turns out to be intimately linked to the new theory’s fractal dimension via non-integer irrational Lorentzian-like factor: where is Hardy’s probability of quantum entanglement. Consequently, the energy density is found from a limiting classical kinetic energy to be Here, is ‘tHooft’s renormalon of dimensional regularization. The immediate logical, mathematical and physical implication of this result is that the dark energy density of the cosmos must be in astounding agreement with cosmic measurements and observations.

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On a Fractal Version of Witten’s M-Theory

Based on the Banach spaces motivated theory of convex geometry in high dimensionality we give a new confirmation of the derivation of the 95.5 per cent so called dark energy density of the cosmos. The result derives directly from the purely geometric-topological phenomenon of measure-mass concentration and gives an unqualified complete confirmation of our previous results including the hidden quantum nature of Einstein’s celebrated equation E = mc2. Thus the quantum dissection into ordinary energy of the quantum particle E(O) = mc2/22 and dark energy of the quantum wave E(D) = mc2(21/22) were validated by first a purely mathematical theorem of convex geometry and second by conservation of energy leading to E(O) + E(D) = mc2 as in Einstein’s special theory of relativity.

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The Measure Concentration of Convex Geometry in a Quasi Banach Spacetime behind the Supposedly Missing Dark Energy of the Cosmos

We start from Banach spaces motivated theory of convex geometry in high dimensionality and give a new additional confirmation of previous derivations of the 96 per cent dark energy density of the cosmos. The result derives directly from the purely geometric-topological phenomenon of measure-mass concentration and gives an unqualified complete confirmation of our previous analysis including the hidden quantum nature of Einstein’s celebrated equation E = mc2. The straight forward relation to holographic dark energy and its endophysical nature is also outlined.

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Banach Spacetime-Like Dvoretzky Volume Concentration as Cosmic Holographic Dark Energy

Realizing the physical reality of ‘tHooft’s self similar and dimensionaly regularized fractal-like spacetime as well as being inspired by a note worthy anecdote involving the great mathematician of Alexandria, Pythagoras and the larger than life man of theoretical physics Einstein, we utilize some deep mathematical connections between equivalence classes of equivalence relations and E-infinity theory quotient space. We started from the basic principles of self similarity which came to prominence in science with the advent of the modern theory of nonlinear dynamical systems, deterministic chaos and fractals. This fundamental logico-mathematical thread related to partially ordered sets is then applied to show how the classical Newton’s kinetic energy E = 1/2mv2 leads to Einstein’s celebrated maximal energy equation E = mc2 and how in turn this can be dissected into the ordinary energy density E(O) = mc2/22 and the dark energy density E(D) = mc2(21/22) of the cosmos where m is the mass; v is the velocity and c is the speed of light. The important role of the exceptional Lie symmetry groups and ‘tHooft-Veltman-Wilson dimensional regularization in fractal spacetime played in the above is also highlighted. The author hopes that the unusual character of the analysis and presentation of the present work may be taken in a positive vein as seriously attempting to propose a different and new way of doing theoretical physics by treating number theory, set theory, group theory, experimental physics as well as conventional theoretical physics on the same footing and letting all these diverse tools lead us to the answer of fundamental questions without fear of being labelled in one way or another.

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Einstein’s Dark Energy via Similarity Equivalence, ‘tHooft Dimensional Regularization and Lie Symmetry Groups

't Hooft-Veltman Wilson dimensional regularization depends crucially upon Borel summability which entails strong links to the modern mathematical theory of transfinite sets and consequently to the fractal-Cantorian spacetime proposal of Ord-Nottale-El Naschie. Starting from the above, we interpret the main step of the mathematical analysis in terms of elementary particles interaction. Thus 't Hooft-Veltman “perturbation” parameter which measures the deviation of the regulated space from the four dimensionality of spacetime is interpreted as an elementary particle with a topological mass charge equal to 0.18033989, i.e. double the magnitude of Hardy’s quantum entanglement. In turn, Hardy’s quantum entanglement which may be interpreted geometrically as a consequence of the zero set embedded in an empty set could also be interpreted as an exchange of pseudo elementary particles with a topological mass charge equal to Hardy’s entanglement where is the Hausdorff dimension of the zero set of the corresponding 't Hooft-Veltman spacetime.

https://file.scirp.org/pdf/WJNST_2014101611420739.pdf

Mohamed S. El Naschie

Papers

On a New Elementary Particle from the Disintegration of the Symplectic &#39t Hooft-Veltman-Wilson Fractal Spacetime