_{1}

^{*}

We introduce an ultra high energy combined KAM-Rindler fractal spacetime quantum manifold, which increasingly resembles Einstein’s smooth relativity spacetime, with decreasing energy. That way we derive an effective quantum gravity energy-mass relation and compute a dark energy density in complete agreement with all cosmological measurements, specifically WMAP and type 1a supernova. In particular we find that ordinary measurable energy density is given by E_{1}= mc^{2} /22 while the dark energy density of the vacuum is given by E_{2} = mc^{2} (21/22). The sum of both energies is equal to Einstein’s energy E = mc^{2}. We conclude that E= mc^{2} makes no distinction between ordinary energy and dark energy. More generally we conclude that the geometry and topology of quantum entanglement create our classical spacetime and glue it together and conversely quantum entanglement is the logical consequence of KAM theorem and zero measure topology of quantum spacetime. Furthermore we show via our version of a Rindler hyperbolic spacetime that Hawking negative vacuum energy, Unruh temperature and dark energy are different sides of the same medal.

A Kolmogorov-Arnold-Moser (i.e. KAM) Cantorian spacetime manifold rather than a simple fractal is the right starting point for an exact formulation of a quantumrelativity gravity theory which resolves the challenging task of explaining the increased rather than decreased rate of cosmic expansion and calculating accurately the measured 95.5% missing energy density of the universe [1-5]. Within the KAM formulation the thin fractal, i.e. zero measure random Cantor set is modelled by the zero set which is found from K-category theory to be given by the bi-dimension and identified physically with a virtual quantum pre-particle [3-28]. On the other hand the fat fractal, i.e. the positive measure empty random Cantor set is given by and is identified with a virtual pre-quantum wave being the cobordism of the quantum pre-particle zero set and [2-28]. Continuing the K-theoretical analysis [11-22] to encompass Penrose’s noncommutative fractal tiling which constitutes a compactified Klein modular holographic boundary for our KAM spacetime bulk [8,9], we can define mass as the inverse of isomorphic diameters of the relevant spaces, for instance of Penrose tiling [11,16,19]. Simple analogies with positive and negative Van der Waals fluctuation, instantons dynamics and Hawking’s negative energy vacuum fluctuation as well as Rindler coordinates at a toy black hole horizon lead us to understand negative gravity and dark energy [29-59]. Our formal analysis starts with an inspiring idea due to D. Gross [_{p} and a corresponding topological Planck length effectively scaling the Planck scale from esoterically large Gev and equally esoterically small cm numbers to a manageably

where is the famous Hardy’s probability for quantum entanglement which amounts to almost 9 percent and [

The theory advocated in this paper is a synthesis of several fundamental ideas. The first idea is the recognition that using the KAM manifold well known and quite familiar from the theory of nonlinear dynamical systems and quantum chaos [8,80] as the geometry of our quantum spacetime is by far a mathematically more firm and solid ground to build our physics upon than ordinary continuous fractal spacetime [

In the present work we show how is related to Gross’ proposal [1,2] and discusses its various ramifications for dark energy [2,3] and the connection between general relativity’s Einstein-Rosen bridges [24, 25] and the quantum mechanics of Einstein-PodolskyRosen nonlocality [3,4,6]. This insight leads to an effective quantum gravity theory [69,70] and an explanation for the true meaning of the quantum wave function and the missing dark energy of the cosmos by fusing Einstein’s relativity and quantum mechanics entanglement together [3,4]. Our basic philosophy and technical strategy may be summed up succinctly as follows: By facing infinity and zero head on and endorsing them we eliminate the drawback of both and suddenly everything in physics looks right [44-54]. Although the present work is reasonably self contained, for a deep understanding of the transfinite nonlinear techniques used to develop our theory it is helpful to read carefully at least Refs. [1,3,5] as well as Refs. [58-72] for a bird’s eye view of the general theory [5,16]. We may also note that the theory of quantum sets was extensively developed by D. Finkelstein who also introduced the notion of quantum relativity [69-77]. We should add at this point that the KAM Cantorian spacetime theory has wider ramifications than only physics and cosmology and has found applications in brain and consciousness research by M. Persinger, C. Lavalle [

As mentioned in the introduction, the present analysis consists of several interconnected ideas and vital steps which need to be explained consecutively with a reasonable degree of logical order as follows:

Let us start by constructing a random Cantor set from the unit interval [7-10]. Proceeding in the usual way well known from the theory of fractals for the classical triadic Cantor set but adding uniform randomness to the construction procedure. That way we end up with a zero measure “thin” Cantor set of a topological dimension D_{T} = 0 and a Hausdorff dimension very close to the classical deterministic set, namely [7-10]. Notice now that the gaps left from the iterative deletion of parts of the unit interval forms a second Cantor set. By contrast to the first “thin” Cantor set this one is a “fat” Cantor set with a positive measure equal to the length of the original unit interval because. However in this case this Hausdorff dimension is not but obviously because for the original unit interval both the topological dimension and the Hausdorff dimension coincide and are equal to unity so that

as should be [7-10]. Noticing on the other hand that the thick Cantor set is made of totally empty gaps, it is clear that it is the epitome of the empty set which due to deductive dimensional and consistency reasons is assigned a topological dimension equal minus one, i.e. D_{T} (empty) = −1 [

In what follows we would like to make the KAM spacetime picture more specific. As hinted at in the previous section Penrose fractal tiling universe [5,11,24] is the prototype par excellence of a noncommutative geometry reflecting the essence of K-theory [

For a = 0 and b = 1 one finds the bi-dimensions of the zero set, i.e.

where

Similarly for the unit set we find [11-27]

Subsequent dimensions are found using the Fibonacci prescription. For instance D_{2} is found as

and so on. For the empty set D −1 the same procedure holds true but we need to mind the correct negative sign. That way one finds the Hausdorff dimension of the empty set to be [11-27]

The bijection formula on the other hand leads to the same result as mentioned at the beginning of this section however in a far simpler way because of the far more compact and economical notation. Thus from the bijection [5,11,27]

we find the zero set by simply setting n = 0

while the empty set is found by setting

Finally we mention on passing that the same result may be found using the gap labelling DIS method [

Thus

which is probably the most recognizable formula of E-infinity theory [

Consider a space given by an infinite number of unions and intersections of elementary Cantor sets resembling a Susslin operation [

so that

where and is the Hausdorff of the Cantor sets. Since, the infinite series can be summed and one finds the topological dimension to be determined by the expectation value or a centre of gravity given by [5,15-24]

Noting that the average Hausdorff dimension is [5, 15-24]

then by requiring space filling we see that we must have equal and one finds the following condition [5,15-24]

This leads to a quadratic equation in with two solutions of which the only positive one is

Inserting in one finds that [

In other words our E-infinity space is essentially the same space which we gained by considering the two Cantor sets constructed from the unit interval discussed earlier on [

Let us derive a general expression for the probability that two different points in co-exist at the same location, i.e. being geometrically entangled. To do this we consider the probability of finding a single Cantor point in an isolated elementary Cantor set. The probability for that is obviously and for n point this is therefore given by the multiplication theorem to be [3,16,27]

On the other hand the global probability and sometimes the contrafactual probability effect of finding a Cantor point in E-infinity is clearly the inverse of namely [3,16,27]

Consequently to find n point in Cantor set in E-infinity is the multiplication of the local probability with the global “contrafactual” probability. This means the total probability is given by [3,17-19]

(20)

Now we can distinguish various cases of P with definite physical meaning corresponding to different numbers of particles n. The first is for n = 2 which means the quantum entanglement of Hardy

This result was verified experimentally to a very high accuracy in various recognized laboratories [3,30-33]. The second value is for n = 3 which gives us the famous Immirzi parameter of loop quantum gravity [28,70]

The third result is that of the celebrated Unruh temperature for which we must set and consequently [30,34]

In addition for one finds the measurable ordinary energy density of the cosmos, i.e. the energy of the quantum particle [2,34,35]

Finally the microwave background radiation of the cosmos is found for to be related to geometrical self entanglement [36-38,67]

as discussed in more detail elsewhere [40-51]. Now we are actually in a position to answer Nobel Laureate G. ‘t Hooft’s deep question “What are the building blocks of Nature?” The building blocks of nature and the building blocks of spacetime are the elementary random Cantor sets [

By raising the zero set modelling the quantum particle characterized by the bi-dimension to literally the quintessence, i.e. the 5 dimensional Kaluza-Klein core of E-infinity, one easily arrives at an expression of energy density of the universe in complete agreement with cosmic measurement of COBE, WAMP and supernova burst analysis [34,35]. To show that we calculate the pseudo Hausdorff volume of Do which is a straight forward naive generalization of classical volume to [26,27]

Using Newton’s kinetic energy as a template or simplistically speaking as a “Newtonian” rather than Hamiltonian operator one finds [26,27,61]

where m is the mass and c is the speed of light. Several vital points should be stressed at this stage. First while it is useful to distinguish sometime between mass, rest mass and relativistic mass the previous formula stresses that physical real mass does not change and that such concepts are only mathematical. The only “rest” quantity is the rest energy namely itself. Second the speed of light c is quantitatively the same one we always used however its meaning here is different. The speed of light in E-infinity is variable and c is an expectation i.e. average value [

where is the Hausdorff component of the empty set bi-dimension. The kinetic energy of the quantum wave in D = 5 is thus [27,61]

The incredible fact which we should have noticed long ago but we did not is not only that is the missing energy density of the cosmos (95% of the total) but that which is the classical expression found by Einstein using not solely mathematical deduction but also a quantum “leap” of “faith”. In other words Einstein included quantum mechanical features in his famous formula although at the time of quantum mechanics was not yet invented and that later on when quantum mechanics was around Einstein did not believe in it because of the spooky action at distance of quantum entanglement. Ironically the energy expression consists of two parts, namely multiplied with where is the Hardy probability of quantum entanglement of two quantum particles so that accounts for the effect of quantum entanglement of the one particle energy expression. There is an even simpler interpretation of and consequently

when involving the mathematics and physics of Nambu-Veneziano’s old bosonic string theory of strong interaction. According to this theory the spacetime dimensions needed are not 4 but 26. Consequently if we understand as a theory for D = 4 then according to the D = 26 theory we have ignored the effect of the 264 = 22 “compactified” dimensions of spacetime [27,57]. Since E is an Eigenvalue, then by Rayleigh theorem we should reduce by division which means Weyl-Nottale scale [

Noting that the exact value is

We see that is an excellent integer approximation [27,28,34,35]. Similarly the dark energy density of the quantum wave

could be approximated to [

The preceding two results are probably the most important achievements of many research efforts of many scientists [1-51] rather than a single person and could be truly said to stand on the shoulders of giants [

To make a long story short, the surprising discovery of the accelerating rather than decelerating expansion of the universe is argued here to be due to the well known geometrical effect of anticlastic, i.e. negative curvature of the spacetime manifolds of the cosmos. In turns this anticlastic curvature accumulates at the edge of the world to a maximum as can be seen from the analogy with a long elastic thin walled cylinder squeezed at the middle as shown in

The Planck energy Gev and not dissimilarly the Planck length cm are quite esoteric scales difficult to visualize and totally outside present or near future experimental capacities if at all [

we realize the following vital topological facts which amount to a realization of D. Gross’ idea of scaling the Planck scale [

1) The topological Planck energy is nothing but Hardy’s quantum entanglement, i.e. the quantum glue which sticks the patches of spacetime together

2) The topological expectation value of the speed of light in a multi-fractal spacetime medium is

3) The topological Planck length is equal to the dimension of a fractal M-theory

since is equal to where is the topological Plank energy we see that E-infinity space is infinitely multi-connected and that all these fractal gaps and voids in their space are essentially wormholes [

A truly brief derivation of dark energy which may also highlight strong evidence that the Rindler-Unruh effects [

The topological ordinary energy density is simply half the Unruh topological mass multiplied with the square of the topological speed of light [

At the beginning there was the word topology constituting the blueprint for existence. However for the Pythagoreans it was the number which may be golden mean number system in which the topological properties of elementary Cantor sets are expressed so that we can do calculations with them and draw general conclusions [4,17,45]. The author hopes that the present work makes it clear that we can deal with infinity without processing an infinite number of information [17,54,56]. In fact written in decimal expansion 0.618033989… is infinite.

However written as we can easily work out that and that [

Finally the inspiring idea of D. Gross to scale the Planck scale [

. We note on passing that the golden mean is fundamental in KAM theorem [3,8,80]. In that way we tame infinity and extend what D. Hilbert called Cantor’s paradise to physics [17,54]. The author for one firmly believes in the preceding concept of the role of infinity in physics and the incredibly deep work of Hugh Wooden [