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The present paper utilizes the similarity between the non-perturbative Julian Schwinger - Efimov-Fredkin approach and that of E-infinity Cantorian spacetime theory to give an exact solution to the problem of cosmic dark energy via a golden mean scaling-super quantization of the electromagnetic field.

As is well known, there are many situations in which summing over all Feynman graphs is not only a daunting task but actually an impossibility [

The most important E-infinity equation in an analogous context to a QED based vacuum energy of the Schwinger-Zymanzik-Fredkin formulation is the renormalization equation of α ¯ o [

α ¯ o = ( α ¯ 1 ) ( 1 / ϕ ) + ( α ¯ 2 = α ¯ 1 / 2 ) + ( α ¯ 3 = 8 + 1 ) + ( α ¯ 4 = 1 ) (1)

Here ( 1 / ϕ ) is the transfinite Clebsh factor, α ¯ 1 = 60 is the ideal inverse electromagnetic coupling, α ¯ 2 = 30 is the inverse weak force coupling and α ¯ i = 1 is the maximal quantum gravity-Planck coupling. Noting that 1 / ϕ is equal to the expectation average value corresponding to the D = 2 quantum path Hausdorff dimension, then we could investigate the various limits of physical meaning for α ¯ o in the light of the remarkable P-Adic expansion [

‖ α ¯ o ‖ 2 = ‖ 2 8 − 1 + 4 4 − 1 + 2 1 − 1 ‖ = ‖ 2 7 + 2 3 + 2 0 ‖ = ‖ 128 + 8 + 1 ‖ = ‖ 137 ‖ 2 = 1 (2)

This result is effectively a T-duality in the sence used by E. Witten in his M-theory [

Let us look at the limit 1 / ϕ → 1 . In this case we find α ¯ o = 100 and that represents the normed degree of freedom or total core dimension of the E-infinity Penrose universe [

| α ¯ o | = ∑ i = 1 4 α ¯ i = 100 = 26 + 74 = 4 + 22 + 74 = γ ( O ) + γ ( D M ) + γ ( P D ) = γ ( total ) (3)

where γ ( O ) is the ordinary energy density, γ ( D M ) is the dark matter energy density and γ ( P D ) is the pure dark energy density given in percentage of the normed 100 core dimension. There is one more important limit of α ¯ o which reconciles α ¯ 4 = 1 of the Planck energy scale with ‖ α ¯ o ‖ = 1 , namely when α ¯ 1 = α ¯ 2 = α ¯ 3 = 0 so that one finds the limit

Lim ( α ¯ o ) = α ¯ 4 = 1 = ‖ α ¯ o ‖ 2 (4)

As we said earlier on, this is basically a T-duality-like statement [

γ ( P D ) = 100 − ( 4.508497187 + 22 + k ) = 100 − 26.68883708 = 73.1116292 % (5)

In fact the exact transfinite values of the three different energy densities can be found when taking on board an important ordinary energy density-pure dark matter energy duality noticed for the first time by Prof. Herman Otto [

γ ( O ) = 100 22 + k = 4.508497187 (6)

as per Otto’s duality. That way pure dark energy density can be trivially deduced as

γ ( P D ) = 100 − ( 4.508497187 + 22 + k ) = 100 − 26.68883708 = 73.31116292 % (7)

exactly as should be [

The next main important step is now to show how γ ( O ) and γ ( D ) may be found directly from the golden mean super quantized QED via α ¯ o = 137 + k o where ϕ 5 = ϕ 5 ( 1 − ϕ 5 ) , and ϕ 5 is the Hardy probability of quantum entanglement as proclaimed at the very beginning of this paper [

α ¯ o = 10 + 127 + k o = ( 6 + k ) + ( 4 − k ) + 127 + k o = ( 6 + k ) + 130.9016994 (8)

In the above form it is clear that the ordinary energy density must be the ratio of the 6 + k to the 137 + k o super quantized QED value of α ¯ o . This means [

γ ( O ) = ( 6 + k ) / ( 137 + k o ) = 1 / ( 22 + k ) = ϕ 5 / 2 = 0.04508497 = 4.508497 % (9)

Note here that 6 + k is the transfinite value found from the famous superstring equation 4 n + 2 when setting n = 1 and adding the transfinite correction k due to the interplay of ‘tHooft’s renormalon [

γ ( D ) = ( 137 + k o ) − ( 6 + k ) 137 + k o = 21 + k 22 + k = 0.95491502 = 5 ϕ 2 / 2 = 95.49150281 % (10)

in full agreement with our previous calculations based on entirely different models and theories [

Having reached this stage in our mathematical-physical analysis, it is natural to ask why the present E-infinity theory is so miraculously effective [

The first example is that taken from the theory of phyllotaxis of plants. That way we find that the observed golden mean angles are a consequence of the stationary minimization condition of the corresponding action [

The second example is the sphere packing density in four dimensions [

On the other hand our result regarding ordinary cosmic energy density γ = ϕ 5 / 2 and dark cosmic energy γ = 5 ϕ 2 / 2 could be considered as a proof of our conjecture because it is quite close to measurement and observation [

Finally we must state the following important point related to Coxeter’s geometry, Buckyballs nano particles and the platonic bodies which connected the role of f in number system to the role of f in Lie symmetry groups [

Drawing on various ideas going back to J. Schwinger and E. Fredkin and utilizing a variety of transfinite methods and techniques such as P-Adic quantum mechanics, Saller operational quantum theory and E-infinity super quantization, we showed how the various cosmic energy densities could be found rigorously from the electromagnetic field represented by α ¯ o = 137 + k o while considering the dimensionality equation of string theory 4 n + 2 for n = 1. The results are all in complete agreement with all the previously obtained ones, both the theoretical and measuremental.

El Naschie, M.S. (2018) QED Cosmic Dark Energy Density Using Schwinger-Fredkin and E-Infinity Theory. Journal of Applied Mathematics and Physics, 6, 621-627. https://doi.org/10.4236/jamp.2018.64054