Abstract:
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].

Abstract:
The paper presents an exact analysis leading to an accurate theoretical prediction of the amount of the mysteriously missing hypothetical dark energy density in the cosmos. The value found, namely 95.4915028% is in full agreement with earlier analysis, the WMAP and the supernova cosmic measurements. The work follows first the strategy of finding a critical point which separates a semi-classical regime from a fully relativistic domain given by topological unit interval velocity parameter then proceeds to wider aspects of a topological quantum field of fractal unit interval. This idea of a critical velocity parameter was first advanced by Sigalotti and Mejias in 2006 who proposed a critical value equal . A second interesting proposal made in 2012 by Hendi and Sharifzadeh set the critical point at 0.8256645. The present analysis is based upon a light cone velocity quantized coordinate. This leads to the same quantum relativity energy mass relation found in earlier publications by rescaling that of Einstein’s special relativity. Two effective quantum gravity formulae are obtained. The first is for the ordinary measurable energy of the quantum particle while the second is for dark energy density of the quantum wave which we cannot measure directly and we can only infer its existence from the measured accelerated expansion of the universe E(D)=where . The critical velocity parameter in this case arises naturally to be . The results so obtained are validated using a heuristic Lorentzian transformation. Finally the entire methodology is put into the wider perspective of a fundamental scaling theory for the Planck scale proposed by G. Gross.

We reason that in quantum cosmology there are two kinds of energy. The first is the ordinary energy of the quantum particle which we can measure. The second is the dark energy of the quantum wave by quantum duality. Because measurement collapses the Hawking-Hartle quantum wave of the cosmos, dark energy cannot be detected or measured in any conventional manner. The quantitative results are confirmed using some exact solutions for the hydrogen atom. In particular the ordinary energy of the quantum particle is given by E(0) = (/2)(mc^{2}) where is Hardy’s probability of quantum entanglement,^{ }=( - 1)/2 is the Hausdorff dimension of the zero measure thin Cantor set modeling the quantum particle, while the dark energy of the quantum wave is given by E(D) = (5/2)(mc^{2}) where is the Hausdorff dimension of the positive measure thick empty Cantor set modeling the quantum wave and the factor five (5) is the Kaluza-Klein spacetime dimension to which the measure zero thin Cantor set D(0) = (0,) and the thick empty set D(-1) = (1,) must be lifted to give the five dimensional analogue sets namely

The paper concludes that the energy given by Einstein’s famous formula E=mc^{2} consists of two parts. The first part is the positive energy of the quantum particle modeled by the topology of the zero set. The second part is the absolute value of the negative energy of the quantum Schr?dinger wave modeled by the topology of the empty set. We reason that the latter is nothing else but the so called missing dark energy of the universe which accounts for 94.45% of the total energy, in full agreement with the WMAP and Supernova cosmic measurement which was awarded the 2011 Nobel Prize in Physics. The dark energy of the quantum wave cannot be detected in the normal way because measurement collapses the quantum wave.

Dark
energy is shown to be the absolute value of the negative kinetic energy of the
halo-like quantum wave modeled mathematically by the empty set in a five
dimensional Kaluza-Klein (K-K) spacetime. Ordinary or position energy of the
particle on the other hand is the dual of dark energy and is contained in the
dynamic of the quantum particle modeled
by the zero set in the same five dimensional K-K spacetime. The sum of both
dark energy of the wave and the ordinary
energy of the particle is exactly equal to the energy given by the well known
formula of Einstein E=mc^{2 }which is set in a four
dimensional spacetime. Various interpretations of the results are presented and
discussed based on the three fundamental energy density equations developed. In
particular where E is the energy, m is the
mass and c is the speed of light, is Hardy’s quantum entanglement and gives results in
complete agreement with the cosmological measurements of WMAP and Supernova. On
the other hand gives an intuitive explanation of
negative gravity and the observed increased rate of cosmic expansion. Adding E (ordinary) to E (dark

The supposedly missing dark energy of the cosmos is found quantitatively in a direct analysis without involving ordinary energy. The analysis relies on five dimensional Kaluza-Klein spacetime and a Lagrangian constrained by an auxiliary condition. Employing the Lagrangian multiplier method, it is found that this multiplier is equal to the dark energy of the cosmos and is given by where E is energy, m is mass, c is the speed of light, and λ is the Lagrangian multiplier. The result is in full agreement with cosmic measurements which were awarded the 2011 Nobel Prize in Physics as well as with the interpretation that dark energy is the energy of the quantum wave while ordinary energy is the energy of the quantum particle. Consequently dark energy could not be found directly using our current measurement methods because measurement leads to wave collapse leaving only the quantum particle and its ordinary energy intact.

Abstract:
The 95.5 percent of discrepancy between theoretical prediction based on Einstein’s theory of relativity and the accurate cosmological measurement of WMAP and various supernova analyses is resolved classically using Newtonian mechanics in conjunction with a fractal Menger sponge space proposal. The new energy equation is thus based on the familiar kinetic energy of Newtonian mechanics scaled classically by a ratio relating our familiar three dimensional space homology to that of a Menger sponge. The remarkable final result is an energy equation identical to that of Einstein’s E=mc^{2}but divided by 22 so that our new equation reads as . Consequently the energy Lorentz-like reduction factor of percent is in astonishing agreement with cosmological measurements which put the hypothetical dark energy including dark matter at percent of the total theoretical value. In other words our analysis confirms the cosmological data putting the total value of measured ordinary matter and ordinary energy of the entire universe at 4.5 percent. Thus ordinary positive energy which can be measured using conventional methods is the energy of the quantum particle modeled by the Zero set in five dimensions. Dark energy on the other hand is the absolute value of the negative energy of the quantum Schrodinger wave modeled by the empty set also in five dimensions.

A new quantum gravity formula accurately predicting the actually measured cosmic energy content of the universe is presented. Thus by fusing Hardy’s quantum entanglement and Einstein’s energy formula we have de facto unified relativity and quantum mechanics in a single equation applicable to predicting the energy of the entire universe. In addition the equation could be seen as a simple scaling of Einstein’s celebrated equation when multiplied by a scaling parameter where is Hardy’s quantum entanglement and. Furthermore could be approximated to and thus may be interpreted as the inverse of the compactified bosonic strings dimension .

We introduce an ultra high
energy combined KAM-Rindler fractal spacetime quantum manifold, which increasingly resembles
Einstein’s smooth relativity spacetime, with decreasing energy. That way we derive
an effective quantum gravity energy-mass relation and
compute a dark energy density in complete agreement with all cosmological measurements,
specifically WMAP and type 1a supernova. In particular we find that ordinary
measurable energy density is given by E_{1}= mc^{2} /22 while the dark
energy density of the vacuum is given by E_{2} = mc^{2} (21/22). The sum of both energies is equal to Einstein’s
energy E = mc

In this letter,I outline the intimate connection
between the fractal spectra of the exact solution of the hydrogen atom and the
issue of the missing dark energy of the cosmos. A proposal for a dark energy
reactor harnessing the dark energy of the Schrodinger wave via a quantum wave
nondemolition measurement is also presented.