**5. Construction of a random Cantor set and the Cantorian space-time**

The main idea of the Cantorian space-time theory is in fact a sweeping generalization of what Einstein did in his general relativity, namely introducing a new geometry of spacetime which differs considerably from the space-time of our sensual experience. This spacetime is taken to be Euclidean. By contrast, general relativity persuaded us that the Euclidean 3+1 dimensional space-time is only an approximation and that the true geometry of the universe in the large is in reality a four-dimensional curved manifold [1, 11].

In the Cantorian space-time theory we take a similar step and allege that space-time at quantum scales is far from being the smooth, flat and passive space which we use in the classical physics. On extremely small scales, at very high observational resolution equivalent to a very high energy, space-time resembles a vacuum fluctuation and in turn modeling this fluctuation using the mathematical tools of non-linear dynamics, complexity theory and chaos. In particular, the geometry of chaotic dynamics, namely the fractal geometry is reduced to its quintessence, i.e. Cantor sets. A Cantor set has no ordinary real physical existence, because its Lebegue measure is zero and nonetheless it exists indirectly because it does have a well defined non-zero quantity, namely its Hausdorff-Besicovitch dimension. The triadic Cantor set possesses a Hausdorff-Besicovitch dimension equal to

$$D = \frac{\log 2}{\log 3} \cong 0.63.$$

92 Quantum Gravity

2 1 (2) 1 1 1.61803 *Cd*

A most remarkable 3D fractal is the Menger sponge which is shown in Fig. 3. The

ln 3 *dM*

The volume of the Menger sponge is zero. The random version of the Menger sponge has a

31 2 (3) 1 1 <sup>2</sup> *Cd*

 

One of the most far reaching and fundamental discoveries using the zero measure Cantor sets is undoubtedly that of El Naschie probability of quantum entanglement. His result for two entangled particles is a generic and universal value of the golden mean to the power of five. This is exactly equal to the famous result of Lucien Hardy [8, 9]. Quantum entanglement is thus explained as a consequence of zero measure gravity. Similarly one

The main idea of the Cantorian space-time theory is in fact a sweeping generalization of what Einstein did in his general relativity, namely introducing a new geometry of spacetime which differs considerably from the space-time of our sensual experience. This spacetime is taken to be Euclidean. By contrast, general relativity persuaded us that the Euclidean 3+1 dimensional space-time is only an approximation and that the true geometry of the

In the Cantorian space-time theory we take a similar step and allege that space-time at quantum scales is far from being the smooth, flat and passive space which we use in the classical physics. On extremely small scales, at very high observational resolution equivalent to a very high energy, space-time resembles a vacuum fluctuation and in turn modeling this fluctuation using the mathematical tools of non-linear dynamics, complexity theory and chaos. In particular, the geometry of chaotic dynamics, namely the fractal geometry is reduced to its quintessence, i.e. Cantor sets. A Cantor set has no ordinary real physical existence, because its Lebegue measure is zero and nonetheless it exists indirectly because it does have a well defined non-zero quantity, namely its Hausdorff-Besicovitch dimension. The triadic Cantor set possesses a Hausdorff-Besicovitch dimension equal to

  Using the bijection formula we can calculate any higher dimensional fractals [8, 11].

**5. Construction of a random Cantor set and the Cantorian space-time** 

universe in the large is in reality a four-dimensional curved manifold [1, 11].

Hausdorff-Besicovitch dimension of this fractal is given by [2]

could explain any velocity larger than the speed of light [8, 9].

log 2 0.63. log 3 *D*

Hausdorff- Besicovitch dimension equal to [1, 13]

 

, the random contra part of the Sierpinski triangle will have the Hausdorff-

 

ln 20 2.7268

For (0) *Cd* 

Besicovitch dimension equal to [13]

For a randomly constructed Cantor set on the other hand, the Hausdorff-Besicovitch dimension is found to take the surprising value of the inverse of the golden mean 5 1 0.61803 2 *<sup>D</sup>* by virtue of the Mauldin-Williams theorem [10].

In 1986 R. Mauldin and S. Williams proved a remarkable theorem which confirmed the main conclusion of the Hausdorff-Besicovitch dimension of the Cantorian space-time. To explain the Mauldin-Williams theorem let us construct a Cantor set of the interval *[0, 1]* via a random algorithm as follows. First we chose at random an *x* according to the uniform distribution on *[0, 1],* then between *x* and *1* we chose *y* at random according to the uniform distribution on *[x, 1].* That way we obtain two intervals *[0, X]* and *[Y, 1].* Next we repeat the same procedure on *[0, X]* and *[Y, 1]* independently and so on. Continuation of this procedure leads then to a random Cantor dust and the Hausdorff-Besicovitch dimension of

this set will be with a probability one equal to 5 1 0.61803 2 [10].

Cantorian space-time is made of an infinite number of intersections and unions of the randomly constructed Cantor sets. Let us denote the Hausdorff-Besicovitch dimension of these Cantor sets by (0) *Cd* . Next we use (0) ( )*<sup>n</sup> Cd* as a statistical weight for the topological dimension *n=1* to and determine the average dimension *n* , i.e. the expectation value of *n*. This value is easy to find following the centre of gravity theorem of probability theory to be [1, 8, 11]

$$\langle n \rangle = \frac{\sum\_{n=0}^{\infty} n^2 (d\_{\mathbb{C}}^{(0)})^n}{\sum\_{n=0}^{\infty} n (d\_{\mathbb{C}}^{(0)})^n}$$

Since

$$\sum\_{n=0}^{\infty} n (d\_{\mathbb{C}}^{(0)})^n = \frac{d\_{\mathbb{C}}^{(0)}}{(1 - d\_{\mathbb{C}}^{(0)})^2}$$

and

$$\sum\_{n=0}^{\infty} n^2 (d\_\mathbb{C}^{(0)})^n = \frac{d\_\mathbb{C}^{(0)} (1 + d\_\mathbb{C}^{(0)})}{(1 - d\_\mathbb{C}^{(0)})^3}$$

one finds that

$$\langle n \rangle = \frac{1 + d\_{\mathbb{C}}^{(0)}}{1 - d\_{\mathbb{C}}^{(0)}}.$$

Next let us calculate the average Hausdorff-Besicovitch dimension . *Cd* We sum together all the Hausdorff-Besicovitch dimensions (0) (0) ( ) *Cd* , (0) (1) ( ) *Cd* , (0) (2) ( ) *Cd* ....., following the

Quantum Gravity in Cantorian Space-Time 95

The solution of this problem comes from the fact that the generalisation of the triadic set to two dimensions is the Sierpinski gasket. The Hausdorff-Besicovitch dimension of the gasket is the inverse value of the Hausdorff-Besicovitch dimension of the triadic set log <sup>2</sup>

> (0) (0) 1 1 . *<sup>C</sup> C C*

> > (0) <sup>1</sup> . *n*

 

*C*

*d*

(4) <sup>1</sup> <sup>3</sup> 4 4.236067. *Cd* 

This is a remarkable result which means that the formally infinite dimensional but hierarchical Cantorian space-time looks from a distance as if it were four-dimensional with

The preceding derivation could be regarded as a proof for the essential four-dimensionality of our physical space-time. We perceive space-time to be four-dimensional because this is

**6. Summing over paths and summing over all dimensions in the Cantorian** 

We recall that Feynman gave an alternative formulation of quantum mechanics in which one calculates amplitudes by summing over all possible trajectories of a system weighted by

*e* where *s* is the classical action, *i* 1 and *ħ* the Planck quantum. For one particle the

 *is Z e dx*

*d d* 

2 1

1

*Cd* 

and four-dimensionality.

*Cd* to higher

log <sup>3</sup> .

Our next aim is to solve the problem of lifting the random Cantor set (0)

*Cd* .

(2)

( )

*d*

*n C*

This way we obtain the Hausdorff-Besicovitch dimension of the Cantorian space-time

3

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

the expectation value of our infinite dimensional Cantorian space-time.

*d*

The generalisation by analogy and induction can thus be written as [12]

Now let us examine the case for space filling, i.e. (0)

*Cd* for a given (0)

where <sup>2</sup> 

**space-time** 

path integral is thus [15]

, *is*

 1, 5 1. <sup>2</sup> 

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

Therefore one could write [1, 11]

formula for the infinite convergent geometric sequence, (0) (0) ( ) *Cd* , (0) (1) ( ) *Cd* , (0) (2) ( ) *Cd* ,....., where (0) 0 1 *<sup>C</sup> <sup>d</sup>* , we obtain [1, 11]

$$\sum\_{n=0}^{\infty} (d\_{\mathbb{C}}^{(0)})^n = \frac{1}{1 - d\_{\mathbb{C}}^{(0)}}.$$

The average Hausdorff-Besicovitch dimension is thus

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

If the Cantorian space-time is to be without gapes and overlapping [1, 14] then we must set *n* equal to *Cd* . Proceeding that way one finds from [1, 14] the following Peano-Hilbert space filling condition *n* = *Cd* that

$$\frac{1+d\_{\mathbb{C}}^{(0)}}{1-d\_{\mathbb{C}}^{(0)}} = \frac{1}{d\_{\mathbb{C}}^{(0)}(1-d\_{\mathbb{C}}^{(0)})}.$$

Thus we have

$$(1 + d\_{\mathbb{C}}^{(0)}) \, d\_{\mathbb{C}}^{(0)} = 1$$

or

$$(d\_{\mathbb{C}}^{(0)})^{(2)} \, \, \star d\_{\mathbb{C}}^{(0)} \, \, \cdot \, 1 = 0.$$

This is a quadratic equation with two solutions

$$d\_{\mathbb{C},1}^{(0)} = \frac{\sqrt{5} - 1}{2} = \phi$$

$$d\_{\mathbb{C},2}^{(0)} = -\frac{1}{\phi}.$$

Inserting back in *<sup>n</sup>* and *Cd* the solution (0) ,1 , *Cd* one finds that

$$\left< n \right> = \frac{1 + \rho}{1 - \rho} \quad = \frac{1}{\rho^3} = 4 + \left. \phi^3 \right>$$

and

$$\left\langle d\_{\mathbb{C}} \right\rangle = \frac{1}{\phi(1-\phi)} = \frac{1}{\phi^3} = 4 + \left\langle \phi^3 \right\rangle$$

where <sup>2</sup> 1, 5 1. <sup>2</sup> 

94 Quantum Gravity

formula for the infinite convergent geometric sequence, (0) (0) ( ) *Cd* , (0) (1) ( ) *Cd* , (0) (2) ( ) *Cd* ,.....,

(0) <sup>0</sup> <sup>1</sup> () . <sup>1</sup> *n C n C*

*d*

(0) (0) (0) ( ) <sup>1</sup> . (1 )

> <sup>1</sup> . (1 ) *C C d d*

> > *Cd* = 1

*Cd* - 1 = 0.

one finds that

= 4+ <sup>3</sup> 

= 4 + <sup>3</sup> 

5 1

*C CC*

*d dd*

(0)

(0)

*d*

*n C*

If the Cantorian space-time is to be without gapes and overlapping [1, 14] then we must set *n* equal to *Cd* . Proceeding that way one finds from [1, 14] the following Peano-Hilbert

(0) (0)

0

(0) (0)

(1+ (0) *Cd* ) (0)

(0) (2) ( ) *Cd*<sup>+</sup> (0)

(0) ,1

*n* = 1 1 

<sup>2</sup> *Cd*

(0) ,2 <sup>1</sup> . *Cd* 

1 (1 ) *Cd* <sup>3</sup>

,1 , *Cd* 

 = 3 1 

> 1

*C C d d* 

*<sup>n</sup> <sup>C</sup>*

1 1

*d*

*d*

The average Hausdorff-Besicovitch dimension is thus

where (0) 0 1 *<sup>C</sup> <sup>d</sup>* , we obtain [1, 11]

space filling condition *n* = *Cd* that

This is a quadratic equation with two solutions

Inserting back in *<sup>n</sup>* and *Cd* the solution (0)

Thus we have

or

and

Our next aim is to solve the problem of lifting the random Cantor set (0) *Cd* to higher dimensions *n* and find ( ) *<sup>n</sup> Cd* for a given (0) *Cd* .

The solution of this problem comes from the fact that the generalisation of the triadic set to two dimensions is the Sierpinski gasket. The Hausdorff-Besicovitch dimension of the gasket is the inverse value of the Hausdorff-Besicovitch dimension of the triadic set log <sup>2</sup> log <sup>3</sup> . Therefore one could write [1, 11]

$$d\_{\mathbb{C}}^{(2)} = \frac{1}{d\_{\mathbb{C}}^{(0)}} = \left(\frac{1}{d\_{\mathbb{C}}^{(0)}}\right)^{2-1}.$$

The generalisation by analogy and induction can thus be written as [12]

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

Now let us examine the case for space filling, i.e. (0) *Cd* and four-dimensionality.

This way we obtain the Hausdorff-Besicovitch dimension of the Cantorian space-time

$$d\_{\mathbb{C}}^{(4)} = \left(\frac{1}{\phi}\right)^3 = 4 + \phi^3 \equiv 4.236067.$$

This is a remarkable result which means that the formally infinite dimensional but hierarchical Cantorian space-time looks from a distance as if it were four-dimensional with the Hausdorff-Besicovitch dimension equal to <sup>3</sup> 4 4.236067. 

The preceding derivation could be regarded as a proof for the essential four-dimensionality of our physical space-time. We perceive space-time to be four-dimensional because this is the expectation value of our infinite dimensional Cantorian space-time.
