**8. Cantorian space-time and the connectivity dimension**

Next we show the logarithmic scaling which will connect the non-dimensional gravity constant to the most fundamental equation namely the bijection formula. We start by taking the logarithm of both sides of the equation

$$\ln d\_{\mathbb{C}}^{(n)} = \ln \left( \frac{1}{d\_{\mathbb{C}}^{(0)}} \right)^{n-1}.$$

That means

$$\ln d\_{\mathbb{C}}^{(n)} = (n-1)\ln\left(\frac{1}{d\_{\mathbb{C}}^{(0)}}\right)$$

solving for *n* one finds that

$$n = \frac{\ln d\_{\mathbb{C}}^{(n)}}{\ln \left(\frac{1}{d\_{\mathbb{C}}^{(0)}}\right)} + 1.$$

Quantum Gravity in Cantorian Space-Time 99

<sup>0</sup> 42.360679 . 2 *<sup>g</sup>*

> with <sup>2</sup>

<sup>0</sup> <sup>2</sup> 26.18033989 . 2 *gs*

Both inverse coupling constants are in full agreement with the experimental values [1, 13, 20].

In the present review article we gave a short overview of ideas leading to the fractal spacetime and the Cantorian space-time theory. The triadic set, the Sierpinski gasket, the Menger sponge and their random analogous are introduced. The Cantorian space-time is determined by three dimensions, the formal *nf* , the topological *nT* = 4 and the

Feynman introduced a procedure which consists of summing over all possible paths of the concerned particle. In the Cantorian space-time theory the procedure is analogous, but instead of summing over all paths we sum over all dimensions of the infinite dimensional

We establish a conceptual and quantitative connection between classical gravity and the electro-weak field using the Cantorian space-time theory and the descriptive set theory. This

With the use of the golden mean scaling operator we derive an expectation value of the

The author would like to thank Prof. Mohamed El Naschie for the discussion and

led El Naschie to a fundamental discovery for quantum entanglement [8, 9, 21, 22].

*gs* . and obtain

<sup>0</sup> , the inverse coupling constant of the non-

*<sup>g</sup>* and the inverse coupling constant of the super

Proceeding in this way one finds the inverse coupling constant of the super symmetric

2 

0 we can derive the inverse coupling constant of the

*gs* using the scaling arguments in the Cantorian

and obtain the following result [20]

*<sup>g</sup>* and the inverse coupling constant of the

. To derive the inverse

*<sup>g</sup>* we start with the

space-time [1] . The scaling factor in the Cantorian space-time is

coupling constant of the non-super symmetric unification of all forces

2 with

 

*gs* . We multiply <sup>0</sup>

Hausdorff-Besicovitch dimension equal to <sup>3</sup> 4 4.236067.

but hierarchical Cantorian space-time.

inverse electromagnetic fine structure constant

super symmetric unification of all forces

symmetric unification of all forces

**11. Acknowledgement** 

permission to use his figures.

 

From the inverse fine-structure constant

super symmetric unification of all forces

Cooper pair. That means we multiply <sup>0</sup>

unification of all forces

**10. Conclusion** 

non-super symmetric unification of all forces

Setting (0) <sup>1</sup> <sup>2</sup> *Cd* and ( ) *<sup>n</sup> Cd* =Z, where Z is the partition function, one finds

$$n = \frac{\ln Z}{\ln 2} + 1.$$

The above formula is very well-known in the combinatorial topology [14, 19, 20] and is called the connectivity dimension. Now if we conceive of *<sup>G</sup>* as being the expectation value of the partition function of the observable universe then the connectivity dimension would be

$$D = \frac{\ln \overline{a\_{\gets}}}{\ln 2} + 1 \equiv 128 = \overline{a\_{ev}}$$

This is the inverse of the Sommerfeld electromagnetic fine structure constant measured at the electroweak scale [1, 13].
