_{1}

^{*}

't Hooft-Veltman Wilson dimensional regularization depends crucially upon Borel summability which entails strong links to the modern mathematical theory of transfinite sets and consequently to the fractal-Cantorian spacetime proposal of Ord-Nottale-El Naschie. Starting from the above, we interpret the main step of the mathematical analysis in terms of elementary particles interaction. Thus 't Hooft-Veltman “perturbation” parameter which measures the deviation of the regulated space from the four dimensionality of spacetime is interpreted as an elementary particle with a topological mass charge equal to 0.18033989, i.e. double the magnitude of Hardy’s quantum entanglement. In turn, Hardy’s quantum entanglement which may be interpreted geometrically as a consequence of the zero set embedded in an empty set could also be interpreted as an exchange of pseudo elementary particles with a topological mass charge equal to Hardy’s entanglement where is the Hausdorff dimension of the zero set of the corresponding 't Hooft-Veltman spacetime.

‘t Hooft-Veltman-Wilson dimensional regularization is a highly successful method in its analytical form as well as in the computerized counterpart extensively used in solid state physics [

First To avoid troublesome singularities and to be able to extract a finite answer from an otherwise diverging series, dimensional regularization resorts to some ingenious mathematics due to E. Borel [

This could be considered to approach D = 5 from above with

where

subsequently reasoned geometrically to be the density of the dark energy of our cosmos provided the K-K fractal theory is an accurate topological description of our universe. In this respect our hope was greatly fulfilled and we were rewarded by a result in full agreement with cosmic measurements and observations as well as all previous derivations, namely [

As mentioned a moment ago the situation with

where

to be the ratio of the ‘regulated’ dimension

which is exactly our previous result apart from being what has been measured by WMAP and PLANCK.

The rational question is now to ask what kind of magical number system is involved in the preceding calculation and how come that this numeric fits seamlessly to physics and everything else? This we explain in the next section and we hasten to say that at the end we should find out that the magic is nothing else but the revival of Kantian pure mathematical reasons upon which our very existence is based.

Let us reconsider our last result for

Noting that we started with [

we see that we can let

and

However if we had worked from the very beginning with

Again, how is this possible? The short answer is that we used one of the seven pillars of wisdom which is traditionally ignored in physics, namely the number system employed by nature to construct a logical universe [

The preceding assertion needs considerable elaboration to be fully or minimally understood. In such a case we could not do better than use a generic example which happens to be the very case we are dealing with here. In essence and in a nutshell, without going into the maize of abstract mathematical arguments characteristic for transfinite se theory, measure theory and the continuum hypothesis the answer is that we will be replacing Borel resummation by what is for physicists, more familiar Weyl scaling [

As known from E-infinity, differentiation and integration are replaced by down scaling and up scaling respectively [

For a Cooper pair, we have

Notice we always have an integer plus or minus a multiple of

Define The disintegration of the simplictic vacuum was studied in several earlier publications in connection with paradoxical decomposition and fractal Cantorian spacetime as a source of exotic particles [

M(K) = 0.18033989 Mev (12)

as reported by the present Author in Ref. [

The fuzzy logic related notion of fractal counting of quantum particles which is based on fractal logic [

where ^{2} turn out to be equal 3 − k_{o} as an elementary computation easily reveal which is a fractal spatial dimension found from some fundamental equations.

With the benefit of hindsight we see that the present result, i.e. the existence of an exotic quasi particle M(k) = 2ϕ^{5} where ϕ^{5} is Hardy’s generic value for the entanglement of two quantum particles [

In view of all the aforementioned, we cannot hesitate to express our strong view that quantum spacetime is a Cantorian fractal manifold and that without this fact, dimensional regularization could not be applied in the way it is applied and would not have given the right answer to the problem at hand as it did and in full agreement with measurements and observations.