Author and name of theory

A. Connes: Non commutative geometry applied to Penrose fractal universe tiling

M. Duff and others: Einstein filed equation for pure gravity

El Naschie et al. E-Infinity Cantorian-fractal spacetime

The Fundamental “Master” equation.

D = Hausdorff dimension, a and b are integers.

, Topological dimension, degrees of freedom for Einstein filed equation in the absence of matter filed, i.e. pure gravity.

,. This is the bijection formula relating the topological Manger-Uhryson dimension n to the Hausdorff dimension

Thus,.

The Zero Set bi-dimension:

It models the quantum particle in 5D-Kaluza Klein spacetime.

a = 0, b = 1 gives

where

d =3 gives Zero locale degrees of freedom is a quantum like behavior or a topological Witten-like quantum field theory.

i.e..

The Empty Set bi-dimensions:

the empty set model the quantum wave in 5D Klein-Kaluza spacetime.

Setting:

a = 1 and b = −1 gives

.

Two dimensional pure gravity gives us. A negative degree of freedom is a quasi empty set i.e. a pre quantum wave.

i.e.

where.

References and Literature

A. Connes. Non Commutative Geometry Academic Press San Diego, USA (1994) see in particular Page 4 92 and 93.

M. Duff: The World in Eleven Dimensions. Inst. of Phys. Publications, Bristol (1999).

M.S. El Naschie: A review of E-infinity and the mass spectrum of high energy particle physics. Chaos, Solitons & Fractals, 19(1), (2004), p. 209-236.