Building upon the pioneering work of J. Bell [1] and an incredible result due to L. Hardy [2] it was shown that the probability of quantum entanglement of two particles is a maximum of 9.0169945 percent [2]. This happens to be exactly the golden mean to the power of five (?5) [3-7]. Although it has gone largely unnoticed for a long time, this result was essentially established independently in a much wider context by the present author almost two decades ago [3-6]. The present work gives two fundamentally different derivations of Hardy’s beautiful result leading to precisely the same general conclusion, namely that by virtue of the zero measure of the underlying Cantorian-fractal spacetime geometry the notion of spatial separability in quantum physics is devoid of any meaning [7]. The first derivation is purely logical and uses a probability theory which combines the discrete with the continuum. The second derivation is purely geometrical and topological using the fundamental equations of a theory developed by the author and his collaborators frequently referred to as E-infinity or Cantorian spacetime theory [3-7]
Time dilation, space contraction and relativistic mass are combined in a novel fashion using Newtonian dynamics. In this way we can surprisingly retrieve an effective quantum gravity energy-mass equation which gives the accurate experimental value of vacuum density. Furthermore Einstein's equation of special relativity E = mc 2 , where m is the mass and c is the velocity of light developed assuming smooth 4D space time is transferred to a rugged Calabi-Yau and K3 fuzzy Kähler manifolds and revised to become, where the division factor 22 maybe interpreted as the compactified bosonic dimensions of Veneziano-Nambu strings. The result is again an accurate effective quantum gravity energy-mass relation akin to the one found using Newtonian dynamics which correctly predicts that 95.4915028% of the energy in the cosmos is the hypothetical missing dark energy. The agreement with WMAP and supernova measurements is in that respect astounding. In addition different theories are used to check the calculations and all lead to the same quantitative result. Thus the theories of varying speed of light, scale relativity, E-infinity theory, M-theory, Heterotic super strings, quantum field in curved space time, Veneziano's dual resonance model, Nash Euclidean embedding and super gravity all reinforce, without any reservation, the above mentioned theoretical result which in turn is in total agreement with the most sophisticated cosmological measurements which was deservingly awarded the 2011 Nobel Prize in Physics. Finally and more importantly from certain viewpoints, we reason that the speed of light is constant because it is a definite probabilistic expectation value of a variable velocity in a hierarchical fractal clopen, i.e. closed and open micro space time.
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