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We compute the dark energy and ordinary energy density of the cosmos as a double Eigenvalue problem. In addition, we validate the result using two different theories. The first theory is based on Witten’s 11 dimensional spacetime and the second is based on ‘tHooft’s fractal renormalization spacetime. In all cases, the robust result is E(O) = mc
^{2}/22 for ordinary energy and E(D) = mc
^{2}(21/22) for dark energy. Adding E(O) to E(D) we obtain Einstein’s famous equation which confirms special relativity, although it adds a quantum twist to its interpretation. This new interpretation is vital because it brings relativity theory in line with modern cosmological measurements and observations. In particular, we replace calculus by Weyl scaling in all computation which is essentially transfinite discrete.

The present work looks at the major cosmological problems associated with the energy density measurement of the entire universe [

In what follows and for later use, we will put SU(2) and E8 Lie symmetry groups as well as two and three Stein spaces under a transfinite microscope to reveal their inner fine structure as E-infinity Cantorian fractal form [

For E8 on the other hand we have 248 generators, which means 496 generators for E8E8. Both groups have extremely important applications outside of pure mathematics, in particular SU(2) is a subgroup of the standard model of high energy quantum particles dealing with the weak force while E8 underpins the prominent theory of superstrings [

where E is the degrees of freedom of the electromagnetic field as represented by the integer value of the inverse fine structure constant

exactly as should be. However we know that the different quantities of the right hand side could not be Weyl scaled, i.e. “differentiated” or “integrated” by being converted into each other unless they are all the same quantities but “measured” at different energies which means at different scales. It was the systematic application of E-infinity theory and its golden mean counting number system which revealed that adding or subtracting the exact transfinite values arsing from the various quantum entanglement probabilities such as that of Hardy

That way however we see that

The first number on the right hand side is what gave |SL(2,7)| its fine structure adjustment, the second number is the |E8E8| transfinite correction while the third number is what gives

where ^{2} and k_{o} are probably entitled to be called quasi particle as _{5} = k_{o} and k^{2} provided we count in the fractal-fuzzy or transfinite logical way [_{5} = k_{o} = ^{2} [

Next we would like to reason why 16k could also be interpreted as the additional fine structure of SL(2,7). This comes from the fact that |SL(2,7)| = 336 is actually eight copies of the Klein orbit 42 so that 336 = (8)(42). On the other hand 42 is really the truncation of ten copies of the Hausdorff dimension = 4.23606799 so that ten copies of

exactly as anticipated. In fact we could count the 8 copies also using the fractal fuzzy logical way as

Now for 8 +

Clearly two copies of dim M gives us the sum of the dimensions of the 17 Stein spaces

in complete agreement with what we said earlier on [

Now is the time for us to look in some detail at the Weyl scaling calculus form of E-infinity [

Theorem 1: Any positive integer can be written uniquely as the sum of nonconservative Fibonacci numbers of the “F” series (i.e. the Fibonacci series).

Theorem 2: Any positive real number can be represented uniquely as a sum of nonconservative numbers from the infinite series between

From the above and particularly theorem No. 2 we see that the

Therefore we can write

In fact something similar to the above recursive formula turned out to be an excellent example of noncommutative space dimensional function [

The exact E-infinity formula corresponding to the above has been used extensively because it is a more compact and superior notation, termed the bijection formula

Applying

empty set

and

so that we may write

while

where D_{MU} is the Menger Urysohn topological dimension and D_{H} is the Hausdorff dimension [

The next most important tool in our mathematical tool kit is Weyl scaling [

Let us start by scaling the inverse fine structure constant

" " n = 2 26 + k Super symmetric quantum gravity coupling or the 26 bosonic string dimensions

" " 3 16 + k The extra 16 boson dimensions of Heterotic strings [

" " 4 10 The dimensions of super string spacetime [

" " 5 6 + k Compactified dimensions of super strings.

" " 6 4 ? k ‘tHooft dimensional regularization fractal spacetime [

" " 7 2 + 2k Fractal string world sheet

" " 8 2 ? 3k Complement of the string workd sheet [

Note that 26 + k, 16 + k, 10, 6 + k and

To see that this is the correct value we demonstrate first that

However the exact transfinite instanton density is 26 + k rather than 24 and the 336 is really 336 + 16k as explained earlier on. Consequently

exactly as anticipated [_{o}.

We will give here only the most important results, namely for n = 1, 2, 3, 4, 5 and 6 as well as the remarkable result for n = 16. Thus we have [

(i.e. R. Loll et al spectral dimension). Now the first five results when divided by superstrings dimensionality D = 10 give us the sum of all the E-line 8 exceptional groups, namely (4)

We start with an epistemological reappraisal of Einstein’s formula from a Lagrangian approach viewpoint which incidentally Einstein never attempted. This Lagrangian is clearly dependant on a single generalized coordinate, namely the ordinary photon

That way one finds the Eigenvalue to be the famous equation

However we know in the meantime extremely well that nature harbours far more than our classical photon and is really functioning based on 12 rather than a single messenger particle. In other words our submitted Einstein Lagrangian is missing eleven more generalized coordinates

To be precise the supposed Einstein Langrangian should have included 3 more SU(2) particles of the electroweak, i.e. the experimentally verified_{o} as well as the equally experimentally confirmed 8 gluons of SU(3) which together with the U(1) conventional photon form the basis of our current SU(3) SU(2) U(1) standard model of high energy physics. Not only that but from the physical theory of E-in- finity which is based on the mathematical theory of the highly structured E-infinity golden mean rings we know that the exact fractal number weight of the 12 bosons of the standard model is exactly ^{2} could be obtained by scaling down the eigenvalue using the number of the missing generalized coordinates, i.e. messenger particles of the standard model [

This must be taken in conjunction with either super symmetry or simply Newton’s kinetic energy so that we find at the end that

That means the ordinary energy-mass relationship should be [

E(O) = mc^{2}/22, (28)

rather than simply E = mc^{2} found for a single photon ^{2}(21/22)? We could speculate scientifically that there is no rest of energy and that is all what we have. However we could let ourselves be guided by cosmological measurements and observations that the rest is the mysterious dark energy which is suspected to be behind the accelerated rather than decelerated expansion of the cosmos. That could indeed be the case for two reasons. First, it is because 21/22 mc^{2} is indeed the indirectly measured missing dark energy. Second, and that is even more general, we know that the particle-wave duality is a fundamental real aspect of quantum physics and that the wave, although devoid of anything we could call ordinary matter, momentum or energy, has a real physical effect which goes as far as telling the particle where to go. Thus we could make a second educated guess and wonder if we are facing what in mathematics is called a double Eigenvalue problem rather than a single Eigenvalue problem. That would mean that our nonconstructively found 12 equilibrium equations defined via a 12 by 12 solvability determinant

possesses two Eigenvalues and not merely one Eigenvalue. The first Eigenvalue is that found earlier on E(O) = mc^{2}/22 while the second is inferred from the physical-mathematical scenario to be

Rearranging one finds

That means

Assuming that c in a fractal spacetime must be itself an expectation fractal value of a speed that varies between zero and infinity, we see that we can make the following identification ‘transformation’, namely [

or the dual core correspondence

It is an elementary task to insert these values in the double Eigenvalue formula and find that the result agrees completely with what we obtain from the corresponding Magueijo-Smolin famous extension of Einstein’s E = mc^{2} [^{2} and ^{2} of the Cantorian point in spacetime is ^{2} becomes now

Consequently the average is

which leads to Einstein’s original equation

E = mc^{2}. (37)

The two other equations are consequently

for ordinary energy and

for dark energy. The sum of both energies gives Einstein’s equation

We stress again that E(O) and E(D) can be considered experimentally confirmed with high accuracy because they totally agree with the cosmological measurements of COBE, WMAP and Planck as well as the supernova observations [

Noting that Einstein’s equation lives in d = 4 and Witten’s equation in d = 11 one finds the following ratio for the Lorentzian factor of E = mc^{2}, namely

and

Let us first recall the relation between Einstein spacetime and that of ‘tHooft fractal renormalization spacetime. The ratio of the respective dimensionality, i.e. D(‘tHooft) to D(Einstein) defines the dark energy coupling. Thus from [

where

where

and therefore our coupling between the two spaces is given by

To show that the preceding in the meantime well known result for the dark energy density of the cosmos is far from being an ad hoc one, we show in what follows how it can be logically deduced from the general framework of superstrings and Witten’s fractal M-theory.

We recall that |E8| = Dim E8 = 248 and that the transfinitely corrected version is given by

Some readers may still feel awkward about why we had to account for the k coupling in this way. To alleviate any doubt let us compute

Now that we did not take any super symmetric intersection, we must take the square root because the above expression represents

Consequently

That way ‘tHooft k is not needed and the same result is of course found, namely that

exactly as before.

Einstein’s equation E = mc^{2}, without Einstein or in fact anyone else realizing until recently, consists of two quantum components. The first is the energy of the quantum particle E(O) = mc^{2}/22, which can be readily measured. The second is the energy of what is actually energyless quantum waves E(D) = mc^{2}(21/22). Obviously we say energyless because this is not ordinary energy and it is not a coincidence that it agrees exactly with the energy density of the presumed dark energy of the cosmos. Adding both energies together we are back to the classical relativistic equation of Einstein. These fundamental results were obtained in the last three years or so using various methods.

In the present paper the problem was also nonconstructively solved as a double Eigenvalue problem using the comparison theorems of Southwell and Dunkerley. In addition, we validated the results via Witten’s eleven dimensional theory as well as ‘tHooft’s fractal renormalization spacetime theory. The main computational tool introduced here was the golden mean Weyl scaling which could be viewed as a substitute for calculus [

Mohamed S. ElNaschie， (2015) Computing Dark Energy and Ordinary Energy of the Cosmos as a Double Eigenvalue Problem。 Journal of Modern Physics，06，384-395. doi: 10.4236/jmp.2015.64042