Modern philosophy has always had a privileged place for the picture theory as exemplified for instance by the work of Ludwig Wittgenstein [1] . In the present work we seek to give an exact mathematical-topological picture for quantum spacetime [2] - [38] . Ours tools to achieve such a picture are primarily the dimensional function of von Neumann’s continuous geometry as developed further by A. Connes in his noncommutative geometry to describe Penrose fractal tiling universe [8] [9] . In addition we will be using various results from the mathematical and the physical E-infinity theory as well as a variety of other powerful mathematical techniques and theories including the Menger-Urysohn dimensional theory, fuzzy sets, deterministic chaos, fusion algebra and chaotic fractals [4] [10] [22] . In addition we stress that we adopt in the present work a view point which is relatively nearer to the source theory of Schwinger [35] than the popular Feynman diagrams method [10] [13] . One important objective and implicit message of the present paper is that pure mathematics is essentially pure “physics” and is indispensible for the advancement of applied science and engineering. Thus as on previous occasions [34] , we felt it important to address the readers of advanced pure mathematical research using advanced mathematical language in the hope of reaching this enormously important community.

The following presupposes a modest degree of familiarity with the Cantorian and fractal spacetime theories of Ord, Nottale and El Naschie as well as the fundamental work of May and Connes [1] - [4] . Starting from von Neumann-Connes’ dimensional function [8]

it is surprisingly straight forward and simple to picture the topology of quantum spacetime (see Figure 1 and Figure 2). Following the above function, which is the pillar of John von Neumann and Alain Connes’ continuous noncommutative geometry, it was rather surprising how accurately and simply we can draw a definite picture based on a pre-quantum particle given by the bi-dimension [4] [8] [10] [11]

and a pre-quantum wave

enveloping the pre-quantum particle and floating in a quantum spacetime resembling the well known golden mean proportioned Penrose fractal tiling universe termed by Connes X quotient space given by the expectation (average) bi-dimension [11] [12] [18]

.

In the above the bi-dimension consists of two numbers [9] [18] . The first is the topological inductive Menger- Urysohn dimension, i.e. a consistent refinement of the ordinary topological dimension while the second number is the Hausdorff dimension corresponding to the said Menger-Urysohn topological dimension [1] - [5] [8] . It is remarkable that is identical to the Mauldin-Williams Hausdorff dimension of a one dimensional randomly but uniformly constructed version of the well known deterministically ordered triadic Cantor set. We note

also on passing that while the classical deterministic triadic Cantor set has our E-infinity uniformly random Cantor set has a slightly smaller value [4] . Never the less

this slight difference, by pure chance or providence, happens to impact calculations and theory positively be- yond the wildest expectations. On reflection however this is in agreement with the number theoretical expectation and befitting the highly structured E-infinity golden mean ring involved in the von Neumann-Connes dimensional function [8] - [11] . However it is fair to say that the notation of the dimensional function is slightly

cumbersome and does not make the value of a and b which gives us for instance the zero set pre-quan- tum particle or of the pre-quantum wave immediately obvious. It is basically masking the involved

Fibonacci growth law. This is mainly the reason for why we opt here for the far more convenient bijection notation of E-infinity theory. Thus the dimensional function could far more conveniently be rewritten as [4] [5] [11]

where is the Hausdorff dimension corresponding to an n Menger-Urysohn dimension. Consequently to obtain we simply set n = 0 in the above formula and find that [4] [5] [11]

exactly as should be. For the quantum pre-wave on the other hand, we have to realize that the empty set must be and inserting in the same bijection formula one finds [4] [5]

exactly as expected. Now without the need to work through a tedious recursive calculation, we can work out directly. Consequently it is a trivial result to find that

,

and

which is the famous expectation value of the Hausdorff dimension f our E-infinity Cantorian spacetime core. Next we can move towards drawing an exact topological picture of our spacetime.

Let us start with the simplest thing which we could in principle hold in our hand, namely the quantum pre- particle. We represent this quantum particle with a small circle. The inverse, namely which would correspond to a two dimensional object because must be originally and by virtue of our bijection formula nothing but [2] [5]

.

This corresponds to the world sheet of superstring theory, albeit a fractal one. We must not confuse ourselves by not remembering that this circle represents a point of topological dimension zero and Hausdorff dimension and that the two dimensional representation corresponds only to the inversion of to become. Now the neighbourhood of a zero point is clearly the empty set with and consequently the surface of

our fractal world sheet is the empty set characterized by the Hausdorff dimension. The inversion leads then to. In other words, the pre-quantum wave is basically a three dimensional object due to the fact that

That way we have developed in projection a remarkable picture made of three concentric circles as can be seen in Figure 1 and Figure 2. The first part of this picture is that of a circle representing the quantum pre-par-

ticle. The next circle is representative for the cobordism of the particle which is nothing but the pre- quantum wave and finally the third circle is the cobordism of the quantum wave which is clearly given by. This is easily seen from the fact

so that the inversion is obviously. In other words this is an object living in and leads therefore to

exactly as expected. Now we can go on drawing circles around circles and each circle is essentially the surface, i.e. cobordism of the circle inside it and could continue this process indefinitely so that in the limit we would have at minus infinity

as mentioned much earlier on. The inverse is clearly an infinite dimensional space but in the two dimensional project this is a zero cobordism, i.e. a surface beyond which there is absolutely “not only nothing” but not even nothing.

Now comes a most probably unexpected observation that has far reaching implications. We saw from our analysis that the surface of the zero set quantum pre-particle is the empty set pre-quantum wave and that the surface of the pre-quantum wave is an emptier still set. On the other hand the expectation value of the core of our spacetime was found to be. That means a bi-dimension which in projection looks like. However since it is an expectation value we should emphasize this point by a more accurate notation which we do here by writing. Now we are almost ready to draw our mathematical topological picture because it is manifestly clear that is neither a particle nor a

wave but simply and somewhat unexpectedly it is our quantum, micro, fractal or what ever other names one likes to give it, spacetime. This picture is extremely simple, in fact far simpler than we expected and herein may lay the difficulties of arriving at a simple but profound picture as the one at hand. Our universe consists of a

quantum pre-particle, the surface of which is a quantum pre-wave floating in a quantum pre-spacetime given by the expectation value. The surface of is which is at the

same time, nothing but the pre-spacetime itself on average as manifestly clear from Figure 1 and the accompanied artist impression of Figure 2. Thus the definite logical conclusion must be that the quantum particle and the quantum wave spans spacetime and in turn spacetime gives birth to the quantum particle and the quantum wave. On the count of the conclusion above it becomes natural that the gravity is the curvature of spacetime in Einstein’s relativity and that vacuum fluctuation of spacetime is the source of near infinite zero point energy [11] [12] .

The preceding analysis and conclusion are truly profound so that we have to study their further consequences and implications in more detail. We start by noting that while the inversion of the quantum particle Hausdorff dimension leads to and the inversion of the quantum wave Hausdorff dimension leads to, it is remarkable that union and intersection of the two leads to the same Hausdorff dimension. Thus we have

In both cases one finds the same result, namely the expectation value of the Hausdorff dimension of spacetime

This fact is fundamental to understanding the paradoxical outcome of the two-slit experiment with quantum particles as discussed on numerous previous occasions. By contrast in the present work we focus the light on another related aspect that reflects another interesting relation to superstrings and related theories. It is obvious that could be seen as a one dimensional string with a irrational fine structure tail. In a sense a fractal string representing the particle pretty much as in the string picture. On the other hand the wave could be seen as a two dimensional world sheet with a irrational fine structure. In other words the E-infinity strings are more than one dimensional and less than two dimensional while the E-infinity world sheet is more than two dimensional and less than three dimensional. Clearly the lower bound sum is 1 + 2 = 3 resembling our three dimensional world while the upper bound sum is 2 + 3 = 5 which resembles the Kaluza-Klein spacetime. The average of the two bounds is not miraculously (3 + 5)/2 = 4, i.e. the good old 4D Einstein spacetime.

Pondering E-infinity fuzziness and contrasting this to its exactness we arrive at the following dialectic conclusion:

On the one hand irrationality gives E-infinity dynamics stability which Hamiltonian systems lack but without having friction dissipative “energy” losses. On the other hand having the golden mean as the organizing center gives us two advantages. First a number which is the worse possible to approximate using rational numbers and in this sense it is maximally irrational, yet the arithmetic of the golden mean is maximally simple because the golden mean ring is maximally highly structured, as is well known from the mathematical E-infinity theory. Furthermore, it is well known that the complexity measure of maximal disorder is equal to that of maximal order so that having golden mean irrationality as the basis of our theory represents a coincidentia-oppositorium uniting the usually un-unitable similar to Fittegel dialectic philosophy, getting the best of all opposed worlds so that at the end the golden mean irrationality fuzziness becomes the hallmark of exactness as in fuzzy logic of Lotfi Zada which introduces fuzzy techniques to the quality control of Japan’s car industry, making it more reliable than any comparable car production in the USA at that time [36] . We note on passing the deep relation between fuzzy logic and our own theory of fractal logic [37] .

Fusion was observed almost exclusively on the sun or in a hydrogen bomb test explosion. That is definitely the most important single reason for people to suppose that only hot fusion reactors are a viable possibility towards this highly sought after possibility for a major almost infinite source of relatively clean energy. However an almost trivial point seems to have been overlooked in this otherwise convincing argument, namely how did the sun and not only the sun but all these stars and galaxies come into being? Well the traditional answer is a small or big bang scenario or the creational scenarios described in the Bible and Koran in impressive metaphoric language. However this is then not as convincing an argument for a scientific philosopher nor a religious thinker as we thought and a big bang could actually be conceived without the need for the heat of billions of suns as large as our sun. In fact we have advanced sometime ago a cold big bang scenario based on Banach-Tarski theorem of paradoxical decomposition [22] which is even more in harmony with logic and religion. It is thus our thesis here that our present set theoretical analysis of spacetime and its relation to the quantum particle and the quantum wave gives us the possibility of constructing a nano universe ab initio from which the latent Casimir energy could be extracted from its outer surface where by Dvoretzky’s theorem 96% of the energy is concentrated [16] . Let us elaborate this vital point a little bit more. It is almost self evident that the Casimir energy and ordinary energy is one and the same thing. It is all simply a matter of boundary conditions. On the nano scales the boundary condition is given by the two Casimir plates. However ordinary energy, and I must hasten to say and dark energy, are produced by the cosmic boundary condition of the universe. This cosmic boundary condition is by its definition a one-sided boundary like a Möbiusstrip [38] . It is the boundary which has an inside but no outside. Consequently the Casimir energy and Casimir force of the nano scale accumulates and ramifies at the end of the universe. Ninety six percent of this energy is then concentrated at the surface of the universe, that is to say at the boundary of the universe. The rest of the energy is what is inside the universe and we call it ordinary energy because we can measure it while the dark energy part of the total energy, being concentrated at the edge of the universe which for all practical purposes is at infinity, cannot be measured directly and we perceive its effects only as a negative pressure forcing the universe to accelerate its expansion. In other words Casimir energy, dark energy and ordinary energy are basically one and the same thing, manifested however in different ways depending on scale and topology of spacetime. We could do a back of an envelope calculation to come to this momentous result. Let us start with Einstein’s space of general relativity. It is 4 dimensional, D = 4. However deeper investigation shows that this D = 4 smooth spacetime is a deceptively simple picture because the core of this spacetime is basically a fractal with a dimensionality Thus there is an excess of which happens to be equal to the latent Casimir topological pressure due to the empty set quantum wave interacting with the zero set quantum particle and forming quantum spacetime. This point could be made far more precise via a five dimensional de Sitter spacetime. The exact fractal dimen-

sion in this case is equal to the sum of all the dimension functions [23] which means which is equal to 4.5 percent. For dark energy or Casimir large scale structure dark energy density one finds which comes to 95.5% in full agreement with observations.

In this paper we made little attempt to cover the vast literature but some additional important references are given for the sake of completeness [39] - [102] . Starting from the zero set a hierarchy of empty sets with increasing degrees of emptiness were found using von Neumann-Connes dimensional function which accurately describes Penrose fractal tiling universe as well as Klein’s compactified modular cures alike. Going this road to its ultimate logic, an exact mathematical-topological picture of quantum-fractal spacetime is developed from which it becomes evident that spacetime is not only the stage upon which physical phenomena involving matter and radiation takes place but it is the very stuff of which matter is made. This is the very reason why we can build a basically nano universe to function as a reactor from which free energy could be extracted [11] [12] [38] [97] [101] [102] . One final word, designing and developing a Casimir-dark energy reactor which would be the greatest achievement of human civilization to give humanity a practically infinite source of energy [89] cannot be more expensive than the needless wars which Nobel Laureate J. Stiglatz described in his book [87] and which cost the USA tax payer alone three trillion dollars.

Mohamed S. ElNaschie, (2015) An Exact Mathematical Picture of Quantum Spacetime. Advances in Pure Mathematics,05,560-570. doi: 10.4236/apm.2015.59052