Quantum Gravity and Dark Energy Using Fractal Planck Scaling

Following an inspiring idea due to D. Gross, we arrive at a topological Planck energy Ep and a corresponding topological Planck length p  33 10 effectively scaling the Planck scale from esoterically large and equally esoterically small numbers to a manageably 19 10 GeV p E  cm p     5 5 1 11 1 p p E P H         where P(H) is the famous Hardy’s probability for quantum entanglement which amounts to almost 9 percent and   2 5 1  .   Based on these results, we conclude the equivalence of Einstein-Rosen “wormhole” bridges and Einstein’s Podolsky-Rosen’s spooky action at a distance. In turn these results are shown to be consistent with distinguishing two energy components which results in , namely the quantum zero set particle component 2 c E m    2 22 E O mc  which we can measure and the quantum empty set wave component which we cannot measure     2 11 22 E D mc  , i.e. the missing dark energy. Together the two components add to       2 Einstein E E O E D E mc     where E is the total energy, m is the mass and c is the speed of light. In other words, the present new derivation of the world’s most celebrated formula explains in one stroke the two most puzzling problems of quantum physics and relativistic cosmology, namely the physicomathematical meaning of the wave function and the nature of dark energy. In essence they are one and the same when looked upon from the view point of quantum-fractal geometry.

     

Introduction
In a remarkable essay [1] the founding father of Heterotic superstring theory and Nobel Laureate in physics D. Gross gave without any mathematical elaboration in a novel general direction for the unification of all fundamental forces and arriving at quantum gravity via an imaginative, unconventional and in fact revolutionary idea of scaling the almost esoteric Planck scale [1,2].In the present work which tackles one of the currently most researched and hotly debated problems in quantum physics and cosmology , we take this idea literally and seriously to mean a mathematical topological formulation of physics and non-smooth transfinite discrete geometry and topology [2][3][4][5].Proceeding in this way, we arrive at a quintessential, i.e. five dimensional Kaluza-Klein space-time [26,30] and a general formula for quantum entanglement which reduces in the case of two particles n = dy's famous probability [16][17][18][19][20][21]  .This result w ork we show how peatedly verified experimentally with a very high degree of accuracy [3,4].
In the present w   5 P H   is rela ramifications for dark energy [2,3] and the connection ted to Gross' proposal [1,2] and disc arious usses its v starting from a unit interval spacetime with non-classical between general relativity's Einstein-Rosen bridges [24,25] and the quantum mechanics of Einstein-Podolsky-Rosen non-locality [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][45][46][47][48][49][50][51][52][53][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] and [5] as well as Ref. [58][59][60][61][62][63][64][65][66][67][68][69][70][71][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].

Analysis
The present an vital steps whic alysis consists of several interconnected h need to be explained consecutively with rval he ell a reasonable degree of logical order as follows:

Space-Time Topology in the Unit Inte
Let us start by constructing a random Cantor set from t unit interval [7][8][9][10].Proceeding in the usual way w known from the theory of fractals for the classical triadic   [7][8][9][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 L = 1 -0 = 1.However, in this case this Hausdorff dimension is not  but obviously because for the original unit interval both the topological dimension and the Hausdo f dimension cod are equal to unity so that as should be [7][8][9][10].Noticing on the other hand that the thick Cantor set is made of totally empty gaps, it is cle

nd the Empty Set Quantum Wave from a Generic f
As h g univ excellence of a ar 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 [22].All the preceding result follows smoothly from Grothendieck Ktheoretical (i.e. the general theory of classes) conception of Penrose's fractal tiling universe [11][12][13][14][15][16][17][18][19][20][21][22][23][24][25] and Connesvon Neumann dimensional function [11][12][13][14] as well as E-infinity Cantorian space-time bijection formula developed by the present author [5,16].We showed previously how quantum physics could be formulated using a quantum set theory based on the preceding thin and thick Cantor sets.These particles and waves arise naturally from a KAM-like fractal-Cantorian space-time which is the ultimate source of matter and fields [18,22,61].In this picture the origin of the quantum particle-like state is modelled by the zero set while the quantum wave-like state is modelled by the empty set [10,14] as will be made more precise in the next subsections.

Non-Commutative Space and the Speed o Light as an Expectation Value
inted at in the previous section Penrose fractal tilin erse [5,11,24] is the prototype par noncommutative geometry reflecting the essence of Ktheory as well as E-infinity theory and its bijection formula which gives the Hausdorff dimension as a function of the zero set topological dimension  and the topological dimension n [5,11].The corresponding dimension function is given by A. Connes in [11] For a = 0 and b = 1 one finds the bi-dimensi zero set, i.e. where Similarly for the unit set we find [11][12][13][14][15][16][17][18][19][20][21][22][23][24][25][26][27] Subsequent dimensions are found using the prescription.For instance D 2 is found as and so on.For the empty set D −1 the same holds true but we need to mind the correct negat procedure ive sign.That way one finds the Hausdorff dimension of the empty set to be [11][12][13][14][15][16][17][18][19][20][21][22][23][24][25][26][27] The bijection formula on the other hand l same result as mentioned at the beginning of t ho eads to the his section wever in a far simpler way because of the far more compact and economical notation.Thus from the bijec-tion [5,11,27] we find the zero set by simply setting n = 0 while the empty set is found by setting n = −1 Finally we mention on passing that may be found using the gap labelling DIS m is the same result ethod [12].It also obvious from the above that the empty set is the cobordism i.e. the surface of the zero set.In addition both the union and the intersection of the inverse dimensions of the zero and the empty set give rise to Hilbert-He's 4D fractal cube which is the core of our E-infinity space [5,17].That means [16] which is probably the most recognizable formula of E-infinity theory [5].Another immensely important reng but th finity Space and Suslin Operation ions and sets resembling a (12) so that sult of the von Neumann-Connes K-formula [11,18] and E-infinity bijection is that for n   we have . That means we have an infinity of empty sets with an increasing degree o tiness.This onsequences regarding the speed of light.For   0 c d   the speed of light must be infinite.Consequently the observed constancy of the speed of light is nothi e average speed of light in a fractal spacetime harbouring infinitely many empty sets [57].Some experimental evidence for the reality of Cantorian spacetime is discussed by E. Goldfain in [62,67].

The Topological Invariants of E-In f emp has physical c
Consider a space given by an infinite number of un intersections of elementary Cantor Suslin operation [29] of the form where 0,1, 2, n   sets S (n) .Since ed and one fi and is the Hausdorff of the Cantor , the infinite series can be summ nd ologica determined by the expectation value or a center of gravn by [5,15-2 This lead Inserting in n one finds that [5]       ntially the same space which we gained by considering t Cantor sets constructed from the unit interval earlier on [27].We mention here without derivation that de In other words our E-infinity space is esse he two discussed tailed analysis of E-infinity established a remarkable result, namely the equality of the curvature of the E-infinity manifold and its Euler characteristic which amounts to 26 + k = 26.18033989[55].

The Space E-Infinity (E ∞ ) as a Probability
Space and Quantum Entanglement-The Immirzi Parameter and Unruh Let us derive a general expression for the probability th two different points in E  co-exist at the same locati being geometrically entangled.To do this we conthe probability of finding a single Cantor point in on, i.e. sider an isolated elementary Cantor set.The probability for that is obviously P   an n d for n points this is therefore given by the mu theorem to be [3,16,27] ltiplicatio On the other hand the global probability and sometimes the contraf al probability effect of finding a Cantor point in E-infinity is clearly the inverse of actu n namely [3,16,27] 3 7977 .
Consequently to find n points in a Cantor set in E-infinity is the multiplication of the local probability .This means the total probability is given by [3, No we can distinguish various cases of P with defiysical meaning corresponding to different numbers of particles n.The first is for n = 2 which means the quantum ich gives us the famous Immirzi parameter of loop quantum gr (21) This result was verified experimentally to a very high accuracy in various recognized laboratories [3,[30][31][32][33].The second value is for n = 3 wh avity [28,70]   3 3 6   Immirzi .
The third result is that of the celebrated Unruh temperature for which we must set n = 1 and consequently [30,34]   1 3 4   Unruh .
In addition for 5 2 P   one finds the measurable ordinary energy density of the cosmos, i.e. the energy of the quantum particle [2,34,35] 2 Einstein E Finally the microwave background radiation of the cosmos is found for n = 0 to be related to geometrical self entanglement [36][37][38]67]   3 microwave P   as of nature and the building blocks of spacetime are the elemen ed by the bi-dimension (25) discussed in more detail elsewhere [40][41][42][43][44][45][46][47][48][49][50][51].Now we are actually in a position to answer Nobel Laureate G. 'tHooft's deep question "What are the building blocks of Nature?" Te building blocks tary random Cantor sets [64].core of E-infinity, one easily arrives at an expression gy density of t with cosmic measurement of COBE, WAMP and supernova burst analysis [34,35].To show tha calculate the pseudo Hausdorff volume of D o which is a straight forward naive generalization of classical volume to [26,27]     5 5 5 .

The Dark Energy of
Using Newton's kinetic energy as a template or simplistically speaking as a "Newtonian" rather than Hamiltonian operator one finds [26,27,61] between mass, rest mass and relativistic mass the previous formula that physical real mass does not change and t 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 stresses hat such concepts are only mathematical.The only "rest" quantity is the rest energy namely E(O) 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 [10] as much as is the expectation for the formal topological dimension f d   of E-infinity [5].In this sense E(O) represents a potential energy of the momentarily at resting measured quantum particle and we are entitled to ask now where is the kinetic energy?The answer of this question is as simple as it is surprisingly because it is the energy of the Einstein-Bohm "ghost" quantum wave responsible for the propagation of the quantum particle [30,61].To find the magnitude of this energy we calculate again a corresponding volume which turned out to be the dual value for 5   namely an additive volume of the empty set The in fact which we should have noticed long ago but we did not is no n D) is the missing energy density of the cosmos (95% of the total) but that E(O) + E(D) = mc 2 which is the classical expression athematical deducn but also a quantum "leap" of "faith".In other words nstein included quantum mechanical features in his famous formula although at the time of E = mc 2 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 a distance of quantum entanglement.Ironically the energy expression E(O) consists of two parts, namely mc 2 multiplied with 5 2  where 5   is the Hardy probability of quantum entanglement of two quantum particles so that 5 2  accounts for the effect of quantum entanglement of the one particle energy expression.There is an even simpler in- when involving the mathematics and physics of Na Venezi 's old bosonic string theory of strong interaction.According to this theory the space-time di sions needed are not 4 but 26.Consequently if we understand E = mc 2 as a theory for D = 4 then according to the D = 26 theory we have ignored the effect of the 26 mbuano men "compactified" dimensions of space-time [27,57].Since E is an eigenvalue, then by Rayleigh theorem we should reduce by division E = mc 2 which means Weyl-Nottale scale [14] it using a scaling exponent 1 22   so that we find Noting that the exact value is We see that   2

E O mc
 is an excellent integer ilarly the dark energy approximation [27,28,34,35].Sim density of the quantum wave The preceding two results are probably the mo portant achievements of many research efforts o scientists  rather than a single person and could be ders of giants [53].From this privileged position we can now deep question by G. Ord [63], namely " fu ng expansion of the universe is argued here to be due to the well known e of th antic ates at the edge of the world The Planck energy and not dissimilarly ic scale esent or near Never the l and lt arising from Hardy's st imf many truly said to stand on the shoul answer a second What is the wave nction?"From the above it is clear that it is the empty set or the surface of the zero set quantum particle.In addition the wave function is the source of dark energy [61,63].In other words the surface of the zero set quantum particle is the source of dark energy.

Accelerated Cosmic Expansion, Anticlastic Curvature and Hawking's Negative Vacuum Energy Fluctuation
To make a long story short, the surprising discovery of the accelerating rather than decelerati geometrical effect of anticlastic, i.e. negative curvatur e spacetime manifolds of the cosmos.In turns this lastic curvature accumul to a maximum as can be seen from the analogy with a long elastic thin walled cylinder squeezed at the middle as shown in Figure 14 of Ref [2].The situation is thus analogous to that of the negative energy of vacuum fluctuation found by S. Hawking to exist at the horizon of a black hole.In other words when we look at the two dimensional projection of a ramified i.e. compactified Klein modular curve or a fractal Penrose universe [65], then at the edges being in the hyperbolic plane at infinity surrounded by a Cantorian circle we have anticlastic curvature, i.e. negative curvature producing negative dark energy causing negative gravity pushing the universe apart rather than pulling it together.The preceding geometrical picture could be translated to a topological picture by reasoning that in simplistic terms, while the visible 4 dimensions of our space produce positive curvature and ordinary gravity, the 26 − 4 = 22 compactified bosonic string dimensions are responsible for dark energy, i.e. negative curvature and consequently negative gravity which is the force behind the measured increased acceleration of cosmic expansion at the edge of the holographic Klein-Penrose universe [65].The situation can be explained quantitatively by taking the ratio between Heterotic string theory 504 states and the 528 states of maximally symmetric spaces such as Witten's M-theory [2][3][4]10,27,28].

Scaling the
Open Access JMP  [30].It is ironic as well as surreal to see that the Einstein-Podolsky-Rosen spoo alway ein-R t work makes deal with infinity without processing of information [17,54,56].In fact ultimate resolution of the ky action at a distance existed s in the Einst osen bridges and consequently in quantum entanglement which Einstein as well as Schrödinger rejected so vehemently [5].

Conclusion
At the beginning there was topology, or was it the 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 authors hope that the presen it clear that we can an infinite number  written in decimal expansion 0.618033989… is infinite.
However written as   tly the movi tic" empty set [2,61].Together E(O) + E(D) = mc 2 give Einstein's iconic formula for the maximal total energy of a classical particle [4,26,61].It is important to note the E(O) density is only 4.5% of the total expected energy density of the universe in agreement t sophisticated accurate cosmological measurements [34,35,61].On the other hand present measurement technologies do not permit measurement of E(D) which constitutes 95.5% of the total expected energy because of the lapse at measurement [10,20,33].For this reason E(D) is dubbed dark energy and baring new developments in wave collapse free non-demolition measurement instruments [2,4,10,20], dark it will remain [30,34,35].Negative dark energy is linked here to the anticlastic curvature at the edge of the universe and is therefore a cousin of Hawking's negative energy fluctuation of the vacuum at the horizon of a black hole which constitutes a circular edge of the hyperbolic compactified edge lying at infinity of a Klein-Penrose holographic universe [2,61,65].It is negative gravity force which is responsible for the unexpected measured and accelerated expansion of the cosmos [10,34,35].Technically speaking we could not have obtained accurate quantitative results for the preceding effect if it would not have been for L. Hardy's magnificent quantum entanglement exact solutions for two particles [31][32][33] as well as the work of Sigalotti [10] leading to the topological speed of light c ng "quin-  .In addition, the general theory of noncommutative space of A. Connes and the realization that Penrose universe is a K-theoretical explicit realization of a generic noncommutative space which played a pivotal role in reaching our conclusion.Finally the inspiring idea of D. Gross to scale the Planck scale [1] gave an overall intuitive justification for to realize that E-infinity fractal geometry is the geometry of quantum entanglem inking Einstein-Rosen bridges with quantum entanglement, zero measure geometry and Magueijo-Smolin theory of varying speed of light [44].The entire analysis and result obtained has testified once more to Einstein's dictum "Raffiniert is der Herr Gott aber boshaft ist er nicht".Einstein added later that "Die Natur verbirgt ihr Geheimnis dur rhawesen, aber nicht durch List" [52].In fact nature is simple to a degree that at least initially defies human expectation, in order not to say human comprehension.However, as H.R.H. the Prince of Wales, T. Juniper and I. Shelly emphasized in their book [53], harmony is the way to comprehension and nothing could be more harmonious than a golden ratio based number system which we used in E-infinity theory [2,17,53].In E-infinity the most irrational number

10 
effectively scaling the Planck scale from esoterically large and equally esoteri-H) is the famous Hardy's probability for quantum entanglement which amounts to almost 9 percent and conclude the equivalence of Einstein-Rosen "wormhole" bridges and Einstein's Podolsky-Rosen's spooky action at a distance.In turn these results are shown to be consistent with distinguishing two energy components which results in , namely the quantum zero set particle component 2 e. the missing dark energy.Together the two components add to randomness to the construction procedure.That way we end up with Cantor set of a topological dimension D T = 0 and a Hausdorff dimension very close to the classical deterministic set, namely a zero measure "thin" as requiring space filling we see that we must have c d equal n and one finds the following co [5 Quantu Kaluza-Klein Space-TimeBy raising the zero set modelling the quantum particle characteriz energy of the quantum wave in D = 5 is thus [27that E( found by Einstein using not solely m tio Ei

1 
we realize the following vital topological facts which amount to a realization of D. Gross' idea of scaling the Planck scale[10]:a) The topological Planck energy is noth Hardy'topological Planck length is equal to the dimension of a fractal Mwhere is the E p we see that E-infinity s that all these fractal ga pace is infinitely multi-connected and ps es and voids in their space are sentially wormholes with length equal to which constitutes Einstein-Rosen bridges . In any event we showed here beyond any reasonable doubt that E = mc 2 consists of two parts, E(O) = mc 2 /22 which is the position or potential energy of a quantum particle or equivalently a five dimensional "quintic" zero set e quantum wave or equivalen with the most recen well documented but ill understood quantum wave col-of th we tame infinity and extend what D. Hilbert called Cantor's paradise to physics[17,54].The authors firmly believe in the preceding concept of the role of infinity in physics and the work of Hugh Wo