_{1}

^{*}

We utilize two different theories to prove that cosmic
dark energy density is the complimentary Legendre transformation of ordinary
energy and vice versa as given by E(dark) = mc^{2} (21/22) and
E(ordinary) = mc^{2}/22. The first theory used is based on G ‘t Hooft’s
remarkably simple renormalization procedure in which a neat mathematical
maneuver is introduced via the dimensionality of our four dimensional
spacetime. Thus, ‘t Hooft used

‘t Hooft’s dimensional regularization as well as the role of Einstein’s field equations in connection with quantum particles high energy physics was discussed by the author as well as ‘t Hooft himself on numerous previous occasions [

The present work is exclusively devoted to clarifying the above and we will start with dimensional regularization and then proceed from there to the pure gravity Witten theory based solution [

In this section we rederive the preceding complimentary densities

Now we make the following observation:

a) for d = 3 we have

b) for d = 2 we have the unintuitive case of negative degrees of freedom

c) for relativity d = 4 spacetime we have

d) for the complete unification M-theory d = 11 we have the well known degrees of freedom [

Recalling what we established in previous numerous publications, namely that the quantum particle may be modelled by the zero set and leads to the entangled and measurable ordinary energy while the quantum wave is modelled by the empty set

while

In other words, the same results of Section 2 are reproduced using an entirely different method based upon the pure gravity of general relativity [

What could possibly be the connection between so radically different theories such as dimensional regularization of the electroweak theory of the standard model of high energy physics and the empty field equations of Einstein’s pure gravity as to give exactly the same result for a seemingly distant theory such as that of the cosmological problem of dark energy. The short answer is the geometry and topology of spacetime underlying both theories. We think normally of quantum mechanics as a non-spacetime theory. By contrast all forms of Einstein’s relativity theories are the spacetime theories par excellence. However at grass roots level both theories could be advantageously thought of in terms of a fractal spacetime geometry and topology. We think the present work is an excellent example illustrating the above. There is an almost one to one correspondence between the degrees of freedom of pure gravity for d = 3 and d = 2 as well as d = 1 and the zero set and the empty set of E-infinity Cantorian spacetime theory respectively. The final result is that the

Either way the result for the energy density

and

are in astonishing agreement with all the known cosmological measurements of COBE, WMAP, Planck as well as the supernova analysis [