**9. Fundamental constants of Cantorian space-time**

The fine-structure constant usually denoted with *α*, is a fundamental physical constant, namely the coupling constant characterizing the strength of the electromagnetic interaction. It is a dimensionless quantity and is defined as

$$\alpha = \frac{e^2}{(4\pi\varepsilon\_0)\hbar c} = \frac{1}{137.035999074}$$

or as the inverse fine-structure constant

$$\overline{\alpha} = \frac{1}{\alpha} = 137.03599907$$

where *e* is the unit electron, *ħ* = *h*/2π is the Planck constant, *c* is the speed of light, *ε*<sup>0</sup> permittivity of free space.

In the Cantorian space-time theory the inverse fine-structure constant <sup>0</sup> can be written in a remarkable short form based upon the multiplication and addition theorems of probability theory [1]. This is done by interpreting (0) *Cd* as a topological probability of a Cantor set formed by the ratio of the Hausdorff- Besicovitch dimension (0) *Cd* and the embedding topological dimension (1) 1. *Cd* 

That way one finds

$$\overline{a\_0} = (2)(10)(\frac{1}{d\_C^{(0)}})^4$$

or

$$
\overline{\alpha\_0} = (2)(10)(\frac{1}{\phi})^4 = 137.082039... 
$$

The value 137.082039 is in excellent agreement with the measured experimental value.

98 Quantum Gravity

*Cd* =Z, where Z is the partition function, one finds

ln 1. ln 2 *Z n*

1

4

*<sup>G</sup>* as being the expectation value of the

as a topological probability of a Cantor set

*Cd*  <sup>0</sup> can be written in a

and the embedding

The above formula is very well-known in the combinatorial topology [14, 19, 20] and is called

ln 1 128

This is the inverse of the Sommerfeld electromagnetic fine structure constant measured at

The fine-structure constant usually denoted with *α*, is a fundamental physical constant, namely the coupling constant characterizing the strength of the electromagnetic interaction.

(4 ) 137.035999074

137.03599907

where *e* is the unit electron, *ħ* = *h*/2π is the Planck constant, *c* is the speed of light, *ε*<sup>0</sup>

remarkable short form based upon the multiplication and addition theorems of probability

0 (0) <sup>1</sup> (2)(10)( ) *Cd*

4

 (2)(10)( ) 137.082039. 

1

The value 137.082039 is in excellent agreement with the measured experimental value.

*Cd* 

*<sup>G</sup> D ew*

partition function of the observable universe then the connectivity dimension would be

ln 2

2

*e*

0

1 

*c*

Setting (0)

<sup>1</sup> <sup>2</sup> *Cd* 

the electroweak scale [1, 13].

and ( ) *<sup>n</sup>*

the connectivity dimension. Now if we conceive of

**9. Fundamental constants of Cantorian space-time** 

In the Cantorian space-time theory the inverse fine-structure constant

formed by the ratio of the Hausdorff- Besicovitch dimension (0)

0

It is a dimensionless quantity and is defined as

or as the inverse fine-structure constant

theory [1]. This is done by interpreting (0)

permittivity of free space.

topological dimension (1) 1. *Cd* 

That way one finds

or

From the inverse fine-structure constant 0 we can derive the inverse coupling constant of the non-super symmetric unification of all forces *<sup>g</sup>* and the inverse coupling constant of the super symmetric unification of all forces *gs* using the scaling arguments in the Cantorian space-time [1] . The scaling factor in the Cantorian space-time is . To derive the inverse coupling constant of the non-super symmetric unification of all forces*<sup>g</sup>* we start with the

Cooper pair. That means we multiply <sup>0</sup> 2 with and obtain the following result [20]

$$\overline{\frac{\alpha\_0}{2}}\phi = 42.360679 = \overline{\alpha\_{\mathcal{S}}}\ . $$

Proceeding in this way one finds the inverse coupling constant of the super symmetric unification of all forces *gs* . We multiply <sup>0</sup> 2 with <sup>2</sup> and obtain

$$
\frac{\overline{\alpha\_0}}{2} \phi^2 = 26.18033989 = \overline{\alpha\_{\mathfrak{g}^s}}.
$$

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