_{1}

^{*}

Using von Neumann’s continuous geometry in conjunction with A. Connes’ noncommutative geometry an exact mathematical-topological picture of quantum spacetime is developed ab initio. The final result coincides with the general conclusion of E-infinity theory and previous results obtained in the realm of high energy physics. In particular it is concluded that the quantum particle and the quantum wave spans quantum spacetime and conversely quantum particles and waves mutates from quantum spacetime.

Modern philosophy has always had a privileged place for the picture theory as exemplified for instance by the work of Ludwig Wittgenstein [

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 [

it is surprisingly straight forward and simple to picture the topology of quantum spacetime (see

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 [

In the above the bi-dimension consists of two numbers [

also on passing that while the classical deterministic triadic Cantor set has

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 [

cumbersome and does not make the value of a and b which gives us for instance the zero set

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 [

where

exactly as should be. For the quantum pre-wave on the other hand, we have to realize that the empty set must be

exactly as expected. Now without the need to work through a tedious recursive calculation, we can work out

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

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

our

That way we have developed in projection a remarkable picture made of three concentric circles as can be seen in

ticle

so that the inversion is obviously

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

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

same time, nothing but the pre-spacetime itself on average as manifestly clear from

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

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

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 [

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 [

sion in this case is equal to the sum of all the dimension functions [

In this paper we made little attempt to cover the vast literature but some additional important references are given for the sake of completeness [

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