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American Journal of Nano Research and Applications

Three Quantum Particles Hardy Entanglement from the Topology of Cantorian-Fractal Spacetime and the Casimir Effect as Dark Energy – A Great Opportunity for Nanotechnology

Mohamed Saladin El Naschie

Published Feb 6, 2015• 5 citations
Publisher

Abstract

The present work brings together three different fields which depend crucially upon nano hardware under the umbrella of E-infinity theoretical framework. We start by following E-infinity topological methodology by dividing Hardy’s entanglement into two parts, a global ‘counterfactual’ part given by Φ3 where Φ = 2/(1+ √5) and a ‘local’ part Φn where n is the number of quantum particles. For Hardy’s celebrated gedankenexperiment with two quantum particles, which was moreover experimentally confirmed with high accuracy, the quantum probability is found for n = 2 to be P(2) (Hardy) = Φ3+2= Φ5 exactly as calculated by Hardy using orthodox quantum mechanics. Applying the same topological E-infinity entanglement theory to three quantum particles give a maximal Φ6 as well as a three partite much smaller value equal Φ3(1− Φ3)/ = 0.018033989. We conclude by outlining the relevant and extremely timely ideas and remarks on the possible connection, via a state of the art nanotechnology, to the Casimir effect as a conjectured origin of dark energy.

DOI
10.11648/j.nano.20150301.11

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The present work brings together three different fields which depend crucially upon nano hardware under the umbrella of E-infinity theoretical framework. We start by following E-infinity topological methodology by dividing Hardy’s entanglement into two parts, a global ‘counterfactual’ part given by Φ3 where Φ = 2/(1+ √5) and a ‘local’ part Φn where n is the number of quantum particles. For Hardy’s celebrated gedankenexperiment with two quantum particles, which was moreover experimentally confirmed with high accuracy, the quantum probability is found for n = 2 to be P(2) (Hardy) = Φ3+2= Φ5 exactly as calculated by Hardy using orthodox quantum mechanics. Applying the same topological E-infinity entanglement theory to three quantum particles give a maximal Φ6 as well as a three partite much smaller value equal Φ3(1− Φ3)/ = 0.018033989. We conclude by outlining the relevant and extremely timely ideas and remarks on the possible connection, via a state of the art nanotechnology, to the Casimir effect as a conjectured origin of dark energy.
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].
In this short contribution, a reciprocity relation between mass constituents of the universe was explained governed by Hardy’s maximum entanglement probability of φ5 = 0.09017. While well explainable through a set-theoretical argumentation, the relation may also be a consequence of a coupling factor attributed to the normed dimensions of the universe. Also, very simple expressions for the mass amounts were obtained, when replacing the Golden Mean φ by the Archimedes’ constant π. A brief statement was devoted to the similarity between the E-Infinity Theory of El Naschie and the Information Relativity Theory of Suleiman. In addition, superconductivity was also linked with Hardy’s entanglement probability.
Time, space and entanglement are the main characters in this work. Their nature is still a great mystery in physics and we study here the possibility that these three phenomena are closely connected, showing how entanglement can be at the basis of the emergence of time and space within closed quantum systems. We revisit and extend the Page and Wootters theory that was originally introduced in order to describe the emergence of time through entanglement between subsystems in a globally static, quantum Universe. In the book, after providing a complete review of the salient aspects of the theory, we establish a connection with recent research on the foundations of statistical mechanics and we propose a new understanding of the thermalization process. Furthermore, we generalize the framework in order describe the spatial degree of freedom and we provide a model of 3+1 dimensional, quantum spacetime emerging from entanglement among different subsystems in a globally "timeless" and "positionless" Universe. Finally, via the Page and Wootters theory, the evolution of quantum clocks within a gravitational field is treated and a time dilation effect is obtained in agreement with the Schwarzschild solution.
We demonstrate that quantum mechanics emerges naturally from the five-dimensional hydrodynamic spacetime framework. The central insight is that quantum probability arises not from fundamental indeterminacy but from observers' confinement to three spatial dimensions. An observer measuring a particle's position integrates over all possible values of the fourth spatial coordinate w (the time density dimension), creating an apparent probability distribution even for particles with definite five-dimensional positions. Starting from the classical geodesic equations in five-dimensional spacetime, we derive the Schrödinger equation through dimensional projection and the assumption of w-confinement at the scale Δw ~ ℏ/(mc). The Planck constant ℏ emerges as a geometric quantity setting the width of the observational window in the w-direction. We show that Heisenberg's uncertainty principle, wave-particle duality, quantum tunneling, and entanglement all follow from the five-dimensional geometric structure without additional postulates. The theory makes a striking prediction: particles should possess excited states in the w-direction with masses m_n = (2n+1)m₀, where m₀ is the ground state mass and n = 0,1,2,... For electrons, this predicts states at 1.53 MeV/c², 2.56 MeV/c², and higher energies. Searches for these resonances in high-energy collisions provide direct experimental tests of the geometric origin of quantum mechanics. This work resolves the measurement problem by showing that wavefunction collapse is the process of dimensional projection from five to three dimensions, constrained by the observer's limited w-resolution. Quantum mechanics is thus revealed as the statistical mechanics of observers confined to lower-dimensional hypersurfaces within a deterministic five-dimensional spacetime. This is the second paper of a trilogy. The foundational framework is developed in "Five-Dimensional Hydrodynamic Spacetime: Time Density and Dark Energy Complementarity." Dark matter implications are addressed in "Dark Matter as Temporal Displacement in Five-Dimensional Hydrodynamic Spacetime."