‘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 [1] -[5] , all a part of the alternatives furnished by different theories employing an inbuilt scale invariance such as fractals and Cantor sets [5] -[7] . Following recent work on clarification of the origin and nature of measurable ordinary energy density accounting for only 4.5% of the total “theoretical” energy density of the universe and the “missing” 95.5% density which are at a minimum we cannot measure directly [9] -[11] , the author realized somewhat belatedly that ‘t Hooft’s renormalization or perhaps more accurately ‘t Hooft-Veltman dimensional regularization procedure offered a neat elegant shortcut derivation of the density of this “missing” dark energy [9] -[11] . It was also at almost the same time or a very short time after that to the author’s own surprise, he realized that the whole subject could be made intimately connected to Witten’s topological field theory and M-theory via an elegantly simple equation connecting the degree of freedom of the graviton in Einstein four and the eleven dimensions of M-theory.

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 [12] -[14] .

In this section we rederive the preceding complimentary densities and by piecing together ideas due to E. Witten and his school [12] -[14] regarding the field equations of Einstein in the complete absence of a matter field with ideas from noncommutative geometry and fractal-Cantorian spacetime theories [18] -[25] . Naively, it would seem that Einstein’s equation in the absence of a matter field is absolutely trivial and surely without any physical interest. However this seemingly reasonable initial opinion was questioned by many deep thinkers from the earliest of times and led among other things to the work on co-ordinates dependent singularities [10] -[15] . Here the most important aspect of research findings that concerns us is that connected to the topological quantum field theory [12] -[14] . In short like massless gravitons pure gravity has degrees of freedom dependent on the dimension d and given by a simple equation which is far from being simple to derive, namely [12] -[14]

Now we make the following observation:

a) for d = 3 we have which implies a quantum-like behaviour or a topological “field” theory [12] -[14] .

b) for d = 2 we have the unintuitive case of negative degrees of freedom [12] -[14] . Remarkably for d = 1 we have the same result, i.e..

c) for relativity d = 4 spacetime we have [12] -[14] . That means that Einstein’s vacuum is a two dimensional string sheet.

d) for the complete unification M-theory d = 11 we have the well known degrees of freedom [12] -[14] of a massless graviton [12] .

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 and leads to the dark energy density which we cannot measure, then we could relax our stringent distinction between dimension of degree of freedom and treat them in this case on equal footing so that represents a quantum particle zero set and as well as represents a quantum wave empty set. Consequently the ratios and may be seen as an equivalent substitute to the ratio of and 4 or the earlier analysis of Section 2. Consequently is in this case found first as

while is its Legendre transformation

In other words, the same results of Section 2 are reproduced using an entirely different method based upon the pure gravity of general relativity [12] -[14] . A comparison between the different theories used in the present work is summarized in Table 1 and Flow Chart 1.

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 factor measuring the relative fractality of spacetime could be expressed in terms of the ratio of the two dimensions, namely 4 − k and 4 leading to give or in terms of the degrees of freedom for d = 4 and d = 11, i.e. and:

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 [8] -[11] [26] [27] . Finally, in view of the above, the author finds it to be justified to call the Hausdorff dimension of ‘t Hooft-Veltman fractal spacetime.