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By viewing spacetime as a transfinite Turing computer, the present work is aimed at a generalization and geometrical-topological reinterpretation of a relatively old conjecture that the wormholes of general relativity are behind the physics and mathematics of quantum entanglement theory. To do this we base ourselves on the comprehensive set theoretical and topological machinery of the Cantorian-fractal E-infinity spacetime theory. Going all the way in this direction we even go beyond a quantum gravity theory to a precise set theoretical understanding of what a quantum particle, a quantum wave and quantum spacetime are. As a consequence of all these results and insights we can reason that the local Casimir pressure is the difference between the zero set quantum particle topological pressure and the empty set quantum wave topological pressure which acts as a wormhole “connecting” two different quantum particles with varying degrees of entanglement corresponding to varying degrees of emptiness of the empty set (wormhole). Our final result generalizes the recent conceptual equation of Susskind and Maldacena ER = EPR to become ZMG = ER = EPR where ZMG stands for zero measure Rindler-KAM geometry (of spacetime). These results were only possible because of the ultimate simplicity of our exact model based on Mauldin-Williams random Cantor sets and the corresponding exact Hardy’s quantum entanglement probability P(H) = where is the Hausdorff dimension of the topologically zero dimensional random Cantor thin set, i.e. a zero measure set and . On the other hand the positive measure spatial separation between the zero sets is a fat Cantor empty set possessing a Hausdorff dimension equal while its Menger-Urysohn topological dimension is a negative value equal minus one. This is the mathematical quintessence of a wormhole paralleling multiple connectivity in classical topology. It is both physically there because of the positive measure and not there because of the negative topological dimension.

The present paper had its roots around six years ago in our work on the geometrical-topological interpretation of Hardy’s famous quantum entanglement and transfinite golden mean Turing computers [

One idea led to another and the logic of the situation set the course at attempting not only to validate the remarkable suggestions of Susskind and Maldacena but also to reduce their results to a natural consequence of a Cantorian Rindler-KAM zero measure and empty set geometry. Actually this is a grossly simplified label because the required E-infinity-Rindler-KAM spacetime is a multi fractal containing sets with positive and zero measure, u thin and fat fractals as well as positive and negative topological Menger-Urysohn dimensions as explained in the extensive literature on E-infinity theory and its application [

Our general theory of quantum spacetime follows in a rigorous way from the von Neumann-Connes by now famous dimensional group function [

where a, b,

where d_{c} = D_{H} is the Hausdorff dimension corresponding to the Menger-Urysohn topological dimension n. For instance if we take n = O which is a zero set modelling the quantum (pre)particle, we see that [

On the other hand, for

Two further particular dimensions are of relevance to the present work, namely

and

Adding to the above the realization gained from the elementary theory of co-bordism [

empty set is the emptier set

must be spacetime itself. To see this subtle point we look at the inversion of

In other words it gives us the average spacetime dimension [

That way we can construct a very simple and beautiful mental picture of our universe [

surrounding the zero set is the empty set (pre)quantum wave

wave is a multi fractal with average bi dimension

It is almost a trivial conclusion that a wormhole [

The Casimir topological pressure

boundary of the holographic boundary and is given exactly by the dark energy density

while the rest is inside the “universe” and is given by the ordinary energy density

As mentioned before, there are two basic sets in E-infinity theory, the zero set

or

This is the remarkable indistinguishability condition of E-infinity spacetime [

EP = EPR = ZMG (11)

is just another manifestation of this geometrical-topological fractal-Cantorian fuzziness.

We have known since a long time that orthodox quantum mechanics has no place for our human intuitive need for a concept related to spatial separation. However, even general relativity evades this concept and leaves the possibility open by contemplating a multiply connected topology and consequently the possible existence of wormholes [

Without the work of Professors Susskind, Maldacena, ‘tHooft and the excellent book of Prof. Holland this paper could not have seen the light. To them and all the authors referred to in this paper I am truly indebted.

Mohamed S. ElNaschie, (2016) Einstein-Rosen Bridge (ER), Einstein-Podolsky-Rosen Experiment (EPR) and Zero Measure Rindler-KAM Cantorian Spacetime Geometry (ZMG) Are Conceptually Equivalent. Journal of Quantum Information Science,06,1-9. doi: 10.4236/jqis.2016.61001