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The paper presents a very simple and straight forward yet pure mathematical derivation of the structure of actual spacetime from quantum set theory. This is achieved by utilizing elements of the topological theory of cobordism and the Menger-Urysohn dimensional theory in conjunction with von Neumann-Connes dimensional function of Klein-Penrose modular holographic boundary of the E8E8 exceptional Lie group bulk of our universe. The final result is a lucid sharp mental picture, namely that the quantum wave is an empty set representing the surface, i. e. boundary of the zero set quantum particle and in turn quantum spacetime is simply the boundary or the surface of the quantum wave empty set. The essential difference of the quantum wave and quantum spacetime is that the wave is a simple empty set while spacetime is a multi-fractal type of infinitely many empty set s with increasing degrees of emptiness.

Renown Austrian writer Ferdinand Kürnberger [

In the present short paper the author goes even further than Wittgenstein by taking the three words to literally mean three steps leading to the emergence of spacetime from the quantum [

The following derivation consists of exactly three steps as alluded to in our short introduction:

Step 1

We start with the quantum pre-particle as modelled by the zero set [

where the zero is the Menger-Urysohn topological dimension and

Step 2

From step one it follows naturally that the surface of D(O) is given by the empty set [

A short explanation for the negative minus one topological dimension of the empty set is given in Appendix 2.

Step 3

Now it may come as a slight surprise that continuing in the same manner as above, the surface of the quantum wave turns out to be nothing else but our quantum spacetime [

In other words, quantum spacetime is an emptier set than the empty set. Not only that but the average empty set from

The preceding three steps give the quintessence of our theory and explain both the quantum wave and quantum spacetime in one stroke in terms of each other [

Let us consider the dimensions corresponding to the unfolding of our three basic steps or basic sets.

Set 1

Unfolding the zero set with which we mean moving from the negative topological dimensions domain to the positive one by inversion [

This we interpret as a one dimensional classical string plus an irrational tail (f). In other words

Set 2

In analogy to the preceding zero set, our empty set leads to [

This may be interpreted as a classical world sheet plus an irrational tail. Again this may be seen as a fractal world sheet [

as well as

Consequently we conclude from the above that there is an intrinsic indistinguishability latent in our Cantorian manifold modelling quantum spacetime with regard to the operations of union and intersection, which explains the superficially paradoxical outcome of the two-slit experiment with quantum particles [

and for the Hausdorff counterpart

respectively. The time dimension is consequently the difference between the two:

The preceding remarkable result could be used to elucidate the strong link between number theory and physics. We could for instance argue that our classical 3D space is simply an “integer” approximation of the basic two dimensions

and

leading to

On the other hand Einstein’s 4D could be seen as a rational approximation

and

leading to

An even more striking feature of the deep relation between number theory as well as transfinite set theory and physics as seen through the mathematics of our present analysis is the following result which follows from the inversion of the zero set and empty set at the averaging level [

and

This leads clearly to

and

In other words the empty set quantum wave gives us directly the topological dimension of spacetime while the zero set particle gives us the topological dimension of the spacetime world sheet [

There are no other two integers which could stimulate the basic interaction of our two irrational numbers

“God made the bulk ‘but’ the surface was invented by the devil”. This is a well known quotation ascribed to Wolfgang Pauli which may be viewed as the theme for the present work. On the other hand the present work showed how the word bulk could be replaced by the quantum particle and then concluded that the quantum wave is simply the “surface” of this particle. Going one step further it was shown here in unheard of simplicity that spacetime is the multilayer (multi-fractal) surface of the quantum wave. This demonstrates how in three simple steps spacetime emerges from the quantum. Seen that way the surface as well as ‘tHooft-Susskind holography is definitely not an invention by the devil but a great idea of deep, subtle beauty worthy of the great pure mathematician who created existence.

Mohamed S. El Naschie, (2016) The Emergence of Spacetime from the Quantum in Three Steps. Advances in Pure Mathematics,06,446-454. doi: 10.4236/apm.2016.66032

The present analysis and derivation depends fundamentally upon the von Neumann-Connes recursive dimensional function [

This function was used to describe superficially pathological x spaces such as that of Penrose tiling in noncommutative geometry [

where

The following illustrates and derives the basic results of cobordism as applied to our theory in an elementary fashion. We start from a three dimensional cube. The surfaces of the cube are evidently square, i.e. two dimensional. This means we have an equation stating that [

where n = 3 for a cube and it follows then that its “borders” or surface have

For a line the “borders” are the end points so that our elementary equation still holds

The next step is on the other hand not trivial. We ask our self what is the border” or the surface of a point? The point is a zero dimensional object, which in theory is the best model of a pre-quantum particle and now we are de facto asking what is the dimension of the neighbourhood of a zero point? Our equation says then that it is

This is exactly how K. Menger and P. Urysohn defined the empty set [