_{1}

^{*}

In this short contribution, a reciprocity relation between mass constituents of the universe was explained governed by
*Hardy*’s maximum entanglement probability of
*φ*
^{5} = 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.

Since Mermin [

the form of a power of the Golden Mean φ = 1 2 ( 5 − 1 ) giving

P = γ max = 1 2 ( 5 5 − 11 ) = 5 φ − 3 = φ 5 = 0.090169943 ⋯ , (1)

this quantum entanglement probability P was shown by the outstanding Egyptian physicist El Naschie to describe the many puzzling features of our universe very well such as dark energy, negative gravity or accelerated expansion of the universe [

In this short contribution, I want to complete a previous publication [

Besides the E-infinity theory of El Naschie [^{5} [

e m e o = 1 − β 1 + β β 2 = 1 − φ 1 + φ φ 2 = φ 5 , using β = φ (2)

Indeed, this transformation equation for the kinetic energy density in terms of the recession velocity β resembles the maximum quantum entanglement probability P of two quantum particles given by Hardy [

P ( 2 G , 2 G ) = 1 − x 1 + x x 2 [

Both mentioned theories (E respectively IR), if they can be brought into line, in the end, will have a lasting effect on our thinking and the perception of our existence.

According to given results of the set-theoretical approach of E-infinity describing the five-dimensional Kaluza-Klein spacetime [

Ω b = 1 2 φ 5 = 0.04508 … (about 4.51%). (4)

The dark matter amount can be recast in the very simple form of

Ω d = 1 100 2 φ − 5 = 0.22180 … (about 22.18%). (5)

It may not be pure fortuity but nevertheless surprising that both amounts show a reciprocity relation. This can be seen, if we write down the remaining dark energy amount as the difference to the entire mass in a more persuasive form

Ω Λ = 1 − 1 10 ( 5 φ 5 + ( 5 φ 5 ) − 1 ) (about 73.31%). (6)

Such coincidence means that both mass constituents should not be considered independent of each other. Mathematically, reciprocity is found, for instance, if one considers volume in comparison to surface, or particles in comparison to waves. Relevant topological arguments from the set theory are summarized by El Naschie [

The entire dark constituents yields

Ω d + Λ = 5 2 φ 2 = 0.954915 … (7)

By the way, the simple result of 5φ^{2} represents the five-dimensional surface of the pre-quantum wave being the cobordism of the topological volume of φ^{5} of the Kaluza-Klein five-dimensional manifold [

The estimated constituents are fairly well consistent with measurements of the Wilkinson Microwave Anisotropy Probe mission (WMAP) [

El Naschie recently pointed out that D = φ^{−5} = 11 + 0.0901699… obviously represents the fractal dimension D_{M} of Witten’s M-theory [

Because φ^{−5} = 11 + φ^{5}, its beautiful hierarchical form (or continued fraction representation) can be expressed as

φ − 5 = 11 + 1 11 + 1 11 + 1 11 + … . = 11 + 0.0 9 0 1699 … ,

in contrast to the most unique representation for the Golden mean

φ = 5 − 1 2 = 1 1 + 1 1 + 1 1 + … . = 0. 618 0 33989 …

An alternative approach for the mass constituents of the universe by applying the Archimedes’ constant π instead of φ yields almost equal amounts in comparison to results according to Equations (4) to (7) [

Ω b = π − 3 π = 0.0450703 … , (8)

Ω d = π ( π − 3 ) 100 = 0.221875 … , (9)

Ω Λ = 1 − 1 10 ( 10 ( π − 3 ) π + π 10 ( π − 3 ) ) = 0.733054 … , (10)

and finally

Ω d + Λ = 3 π = 0.9549296 … (11)

for the sum of the dark constituents (matter and energy). It may turn out in the future whether this replacement may be of any physical importance. However, some numerical approximations connect Hardy’s entanglement probability φ 5 with Archimedes’ constant π and the inverse of Sommerfeld’s fine structure constant α ¯ 0 , respectively [

π − 3 ≈ 16 137 − 24 = 0.14159292 (12)

α ¯ 0 ≈ 137 + 2 5 φ 5 = 137.03606 … (13)

If one deals with particle entanglement, superconductivity represents a physical phenomenon suspecting such property. Some years ago, the present author suggested linking the optimum hole doping σ_{o} of high-T_{c} superconductors with Hardy’s φ^{5} entanglement probability [_{1} = 8.7210972 … of the renormalized quadratic Hénon map (remember the quadrilateral layer structure of the cuprates)

σ o ≈ 2 δ 1 = 0.2293 , (14)

on the other hand with Hardy’s entanglement probability

σ o ≈ 8 π φ 5 = 0.2296 , (15)

this time connecting φ and π, the most important universal numbers of the cosmos.

The quotient of the Fermi speed to the Klitzing speed would yield

ν F / ν K ≈ 2 π φ 5 = 0.0571 , (16)

which is again proportional to φ 5 [

The fractal nature of electronic response in superconductors was documented some years ago by scanning tunneling microscopy [

This contribution points to a reciprocity relation between the mass constituents of the universe that suggests their common physical interrelation. The obtained baryonic respectively dark mass amounts are related to Hardy’s maximum particle entanglement probability, namely the fifth power of the Golden Mean. Importantly, it is recommended once more to replace the Golden Mean φ in the mass relations by the Archimedes’ constant π to utilize very simple expressions that await further interpretation. The competition between the two numbers obviously determines what the realities of our universe are. However, a Nature article just published reporting on a galaxy lacking dark matter halo is reason enough to re-evaluate the statements given here [

Otto, H.H. (2018) Reciprocity Relation between the Mass Constituents of the Universe and Hardy’s Quantum Entanglement Probability. World Journal of Condensed Matter Physics, 8, 30-35. https://doi.org/10.4236/wjcmp.2018.82003