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

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

, *is e* where *s* is the classical action, *i* 1 and *ħ* the Planck quantum. For one particle the path integral is thus [15]

$$Z = \int e^{\frac{\stackrel{\text{\twoheadrightarrow}}{\hbar}}{\hbar}} \left[ dx \right]$$

(

bijection formula

gives for *N* = *n* =

*<sup>G</sup>* which is defined as

Planck mass.

That means

solving for *n* one finds that

*ew* 128 the following

*d*

Quantum Gravity in Cantorian Space-Time 97

*ew* 128 is the inverse coupling constant measured at the electroweak scale) then the

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

 

*C*

*d*

(128) 127 38

<sup>1</sup> 2 (1.70141)(10) *<sup>C</sup>*

where (128) <sup>2</sup>*<sup>N</sup> C dC* . The value <sup>38</sup> (1.70141)(10) is the non-dimensional gravity constant

<sup>2</sup> (1.7)(10) *<sup>G</sup> p c Gm*

It is of interest to mention that a similar result was found empirically by F. Parker Rhodes

Here *ħ* is the Planck quantum, *c* the speed of light, *G* the Newton's gravity constant, *mp* the

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

<sup>127</sup> <sup>38</sup> 2 137 (1.7)(10)

( )

*d*

(0)

*C*

which was the subject of extensive discussions by Noyes [18]

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

the logarithm of both sides of the equation

*d*

*n C*

128 1

 

*<sup>G</sup>*

( )

*d*

( )

*d n*

*C*

*n*

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

( )

*n C*

*d*

ln 1.

(0)

*C*

*d* 

<sup>1</sup> ln

*n C*

1

38

1

(0)

*C*

*d* 

*n*

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

*C*

*d* 

where [dx] means that we are summing over all possible paths of the concerned particle. What is important here is to realize that from all of infinitely many paths which a quantum particle can take some are more probable than others. The probability of the actual path, that is to say the amplitude of an event is the sum over the amplitude corresponding to all paths. Thus we have a weight assigned to each path in the Feynman formulation of quantum mechanics.

In the Cantorian space-time theory we proceed in an analogous way. However, instead of summing over all paths, we sum over all dimensions of infinite dimensional hierarchical Cantorian space-time. El Naschie has recently demonstrated that E-Infinity is a Suslin operation and the so-called Suslin A operation [9]. In this theory Suslin scaling replace the classical Lagrangian and the classical calculus using descriptive set theory [16, 17].
