**3. A short historical overview of ideas leading to fractal space-time**

The idea of a hierarchy and fractal-like self-similarity in science started presumably first in cosmology before moving to the realm of quantum and particle physics [1]. It is possible that the English clergyman T. Right was the first to entertain such ideas (Fig. 1). Later on the idea reappeared in the work of the Swedish scientist Emanuel Swedenborg (1688-1772) and then much later and in a more mathematical fashion in the work of another Swedish astrophysicist C. Charlier (1862-1934) (Fig. 2).

In 1983, the English-Canadian physicist Garnet Ord wrote a seminal paper [3] and coined the phrase Fractal Space-time. Ord set on to take the mystery out of analytical continuation. We should recall that analytical continuation is what converts an ordinary diffusion equation into a Schrödinger equation and a telegraph equation into a Dirac equation. Analytical continuation is thus a short cut quantization. However what really happened is totally inexplicable. Ord showed using his own (invented) quantum calculus, that analytical continuation which consist of replacing ordinary time *t* by imaginary time *it* where *i* 1 is not needed if we work in a fractal-like setting, i.e. a fractal space-time. Although rather belated Ord's work has gained wider acceptance in the mean time and was published for instance, in Physics Review [4]. Therefore one is hopeful that his message has found wider understanding. It is the transfinite geometry and not quantization which produces the equations of quantum mechanics. Quantization is just a very convenient way to reach the same result fast, but understanding suffers in the process of a formal analytical continuation.

88 Quantum Gravity

dimensional point moving in the two or three-dimensional space. Classical geometry similar to classical mechanics has made various tacit simplifications and ignored several subtle

If a line is one-dimensional and if it is made of infinite number of points then the sum of infinitely many zeros should be equal to one. That is of course not true. On the other hand we know that there is a curve called the Peano-Hilbert curve which is area filling and twodimensional [1, 7-20]. By contrast, we can construct a three dimensional cube known as the Menger sponge which has a fractal dimension more than two and less than three,

*D* as explained for instance in the classical book of Mandelbrot [2].

The existence of all these non-conventional forms described in modern parlance, following Mandelbrot, as fractals, may be traced back to the archetypal transfinite set known as Cantor

A Cantor set is a set of disjoint points which possesses the same cardinality as the continuum. It may be this coincidence that makes it an ideal compromise between the discrete and the continuum. It is transfinite discrete. Our Cantorian space-time which we will use to "topologize" physics is based on these transfinite sets. The main idea behind the Cantorian space-time approach is to replace the formal analysis of quantum mechanics and the Riemannian space-time geometry of general relativity by a transfinite fractal Cantorian

The idea of a hierarchy and fractal-like self-similarity in science started presumably first in cosmology before moving to the realm of quantum and particle physics [1]. It is possible that the English clergyman T. Right was the first to entertain such ideas (Fig. 1). Later on the idea reappeared in the work of the Swedish scientist Emanuel Swedenborg (1688-1772) and then much later and in a more mathematical fashion in the work of another Swedish

In 1983, the English-Canadian physicist Garnet Ord wrote a seminal paper [3] and coined the phrase Fractal Space-time. Ord set on to take the mystery out of analytical continuation. We should recall that analytical continuation is what converts an ordinary diffusion equation into a Schrödinger equation and a telegraph equation into a Dirac equation. Analytical continuation is thus a short cut quantization. However what really happened is totally inexplicable. Ord showed using his own (invented) quantum calculus, that analytical continuation which consist of replacing ordinary time *t* by imaginary time *it* where *i* 1 is not needed if we work in a fractal-like setting, i.e. a fractal space-time. Although rather belated Ord's work has gained wider acceptance in the mean time and was published for instance, in Physics Review [4]. Therefore one is hopeful that his message has found wider understanding. It is the transfinite geometry and not quantization which produces the equations of quantum mechanics. Quantization is just a very convenient way to reach the same result fast, but understanding suffers in the process of a formal analytical

**3. A short historical overview of ideas leading to fractal space-time** 

topological facts [2].

namely log <sup>20</sup>

triadic set [6].

continuation.

log 3

space-time manifold [1, 8, 11, 13].

astrophysicist C. Charlier (1862-1934) (Fig. 2).

Fig. 1. A vision of T. Right's cosmos as a form of sphere packing, on all scales [1].

Fig. 2. A vision of a fractal-like universe, with clusters of clusters ad infinitum as envisaged by the Swedish astronomer C. Charlier who lived between 1862 and 1934. This work was clearly influenced by the work of the Swedish astrophysicist A. Swedenborg (1688–1772) [1].

Quantum Gravity in Cantorian Space-Time 91

It is important to note that the Sierpinski triangle is a curve and its dimension lies between

**Hausdorff dimension**

*dS=ln3/ln2*

*ln20/ln3 2.7268*

*Cd* = *4.236068*

Fig. 3. In this figure we draw analogy between smooth spaces as a line, a square, a cube, a higher-dimensional cube and the Cantor set, the Sierpinski triangle, the Menger sponge and the Cantorian space-time which is difficult to draw. The calculation of the Hausdorff-Besicovitch dimension of classical fractals and their random version is presented [1, 13].

It was shown in the Cantorian space-time theory [12] that the generalization of the formula connecting the triadic Cantor set with the Sierpinski triangle is possible for *n* dimension and

( )

*d*

*n C*

1

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

 

*C*

*d*

4

ln 2 *<sup>S</sup> <sup>d</sup>*

**Menger– Urysohn dimension**

the classical line and the classical area.

*0.630929* <sup>0</sup>*Cantor* 

*1.5849625* <sup>2</sup>*Sierpinski* 

<sup>3</sup>*Menger* 

is given by the so-called bijection formula

**Geometrical shape Type of** 

**fractal** 

*Set*

*gasket*

*sponge*

*The 4 dimension random Cantor set*

ln 3 1.5849

**Corresponding random Hausdorff dimension** 

0.618033 <sup>1</sup>*dC=ln2/ln3*

<sup>1</sup> <sup>1</sup>

2

*dM=* 

1.618033

2.618033 

<sup>3</sup> 4 4.236067 (4)

**Corresponding Euclidean shape** 

*Line* 

*Square* 

*Cube* 

*Hyper cube* 

**Embedding dimension**

2

3

5

Similar work, but not identical, was carried out by the French cosmologist Laurent Nottale, fifteen years ago. Nottale connected scaling and Einstein's relativity to what is now called scale relativity theory [5]. Around 1990, M. S. El Naschie began to work on his Cantorian version of fractal space-time [6]. In M. S. El Naschie's work on high energy physics and electromagnetic weak interactions the golden mean plays a very important role.

In the year 1995 Nobel laureate Prof. Ilya Prigogine, Otto Rössler and M. S. El Naschie edited an important book [7] in which the basic principles of fractal space-time were spelled out. Sometime later El Naschie using the work of Prigogine on irreversibility showed that the arrow of time may be explained in a fractal space-time. Recently El Naschie gave for the first time a geometrical explanation of quantum entanglement and calculated a probability of the golden mean to the power of five <sup>5</sup> for the entanglement of two quantum particles [8, 9].
