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At its most basic level physics starts with space-time topology and geometry. On the other hand topology’s and geometry’s simplest and most basic elements are random Cantor sets. It follows then that nonlinear dynamics i.e. deterministic chaos and fractal geometry is the best mathematical theory to apply to the problems of high energy particle physics and cosmology. In the present work we give a short survey of some recent achievements of applying nonlinear dynamics to notoriously difficult subjects such as quantum entanglement as well as the origin and true nature of dark energy, negative absolute temperature and the fractal meaning of the constancy of the speed of light.

The closing report of a notable Nobel conference [

Besides M-theory, super strings and loop quantum gravity, A. Connes’ non-commutative geometry is generally accepted as a most promising theory for high energy particle physics and quantum gravity [

where and. Starting with the seed

and then proceeding in a Fibonacci manner one finds [

and so on.

Proceeding inductively one finds

where a, b represents the classical Fibonacci numbers and consequently moving in the direction of negative Menger-Urysohn dimension one finds [9-11]

Thus

is just another way of writing the bijection formula of Einfinity Cantorian space-time [9-11]

where is the Hausdorff dimension of a randomly constructed elementary triadic Cantor set, i.e. a chaotic version of the classical triadic set [10,11]. We note the proximity of the magnitude of to the golden mean [10,11]. Two important results can immediately be established, namely that [7,9-11] the Hausdorff dimension of the classical empty set [2,10,11] while is the Hausdorff dimension of the truly empty set [2,10,11].

Consequently the distance between and

measures the degree of emptiness of an empty Cantor set (dust) [9-11].

One of the most fundamental connections between noncommutative geometry and E-infinity theory is undoubtedly the famous Penrose quasi-periodic tiling universe with its golden mean proportionality [

where and is an arbitrary radius. In other words, latest after moving a distance equal the space reproduces itself exactly. This is clearly a universe which is its own multi-verse. We see here how the self-similarity and self-affinity of fractals can solve seemingly and superficial contradictory notions such as whether we live in a multi-verse or a universe.

From the above we see that in a fractal universe of the Penrose non-commutative type the problem resolves itself.

The totally incomprehensive riddle of spatial separation in quantum mechanics may easily be resolved using the property of zero Lebesque measure of all totally disjointed Cantor sets [12,13]. There is irrefutable theoretical and experimental proof for this E-infinity based proposal [12,13]. The story goes as follows: Using an ingenious Gedanken experiment L. Hardy [

where is the golden mean. The E-infinity interpretation stems from the general E-infinity formula for the probability of quantum entanglement [

where n is the number of quantum particles and is the inverse of the Hausdorff dimension of the E-infinity fractal space-time core [10,11]

For two particles this means

Seen that way quantum entanglement can be understood as a natural consequence of the zero length (i.e. zero measure) of a Cantor set and the problem of spatial separation in quantum mechanics is swept away. In a zero measure space-time manifold there is simply no meaning for spatial separation [

At present the problem, of dark energy or the missing energy in the universe, constitutes the most challenging problem in physics and cosmology alike [3,5,6]. Accurate measurement has shown that only 4.5% of the total energy thought to be contained in the universe is detectable [3,5]. The simple conclusion for these results which were awarded the 2011 Nobel Prize in Physics is that either Einstein’s equation contains some error or 95.5% of the energy in the universe is due to mysterious dark matter and dark energy which cannot be detected with any known methods [3,5,6]. The nonlinear-dynamical fractal resolution of this problem however is unbelievably simple, more than one could imagine [3,5]. The rationale behind this is as follows: If space-time itself is a real Cantorian fractal then it resembles an unimaginably large cotton candy [5,7,11,13]. The majority of this cosmic cotton candy is naturally voids containing nothing, not even space or time. Consequently Einstein’s famous equation [

must be modified to take all these fractal voids into consideration. This can be done in various equivalent ways. The simplest is to take bosonic strings compactified “dark” dimensions into account in the form of a Weyl-Nottale scaling factor. Since bosonic string space has 26 dimensions and Einstein’s relativity is only 4 dimensional then the “dark” dimensions are 26 – 4 = 22 and our scaling factor must be [

Consequently the revised E is

Noting that, we see that the new E_{QR} accurately accounts for the cosmological measurements [

Another way to come to the same conclusion is to reason that from high energy particle physics point of view ^{ }is based on the existence of one messenger particle, namely the photon However this was in 1905 when Einstein conceived his theory. In the meantime we know that we have 12 messenger photon-like particles given by the Lie symmetry groups of the standard model of particle physics [10,11]

Consequently by inserting in Newton’s kinetic energy and letting one finds

exactly as in the first derivation, namely Equation (14).

The previous derivations of the revised Einstein equation

were only very accurate approximations. However an exact derivation can be obtained when taking the exact fractal nature of quantum entanglement in deriving E_{QR}. Again this could be done in several equivalent ways. Here we give two methods only. The first is based upon formal analogy between the E-formula of the theory of varying speeds of light (see

and the Cantor set unit interval physics of Ultimate L and F and Taiji-El Naschie theory [18,19] (see

In other words the exact of quantum relativity is equal Einstein multiplied with half of Hardy’s [12-15]. That means

Seen that way the reduction of E from the 100% of Einstein’s theory to the 4.5% of the exact quantum relativity theory is due to quantum entanglement at the Hubble cosmic distances (see

where is a boost which needs not be specified at this point. Setting in Newton’s kinetic energy one finds [

Taking to be Sigalotti’s critical value [20, 21] one finds the same previous result

For the last thirty years or so nonlinear dynamics became an indispensible tool for countless branches of engineering and applied sciences as well as mathematics [_{QR}; are merely self-similar scaling of each other in the sense of modern nonlinear dynamical theories [

energy formula for dark energy is given directly by

This obviously is the complementary energy of the ordinary energy

(24) Adding both expressions we find that (see

We draw here attention to the T-duality (see

In turn this mathematics is nothing more than taming all singularities using fractal self-similarity [1,2].

This conclusion has momentous ramifications going as far as showing the existence of negative gravity (see

Nonlinear dynamics, chaos and fractals have enriched science and gave theoreticians meantime various new indispensable mathematical tools such as self-similarity, average symmetry and fuzzy group theories [2,11]. High energy physics was relatively late in utilizing these new methods, but things have flourished in the last five years thanks to the dedicated work of various schools which applied nonlinear dynamics to particle physics and cosmology [1,8,9,12]. We were able to reason that dark energy is related to compactified and fractal extra-dimensions zero and empty sets which employ fractals and Cantorian sets [5,22]. That way we were in a position to derive a fundamental quantum relativity equation

and a second complementary equation (see

The sum of both equations is exactly equal to Einstein’s famous equation which doesn’t distinguish between ordinary energy and dark energy i.e. It is important to notice that the ratio of E(Dark) to E(Ordinary) is exactly equal to

.

In other words this is the 26 + k of transfinite bosonic string theory minus the five dimensions of Kaluza-Klein theory. Seen that way dark energy is related to the compactified section of space-time of bosonic strings which represents negative curvature and thus negative energy and negative gravity (see

the deep problems of the true meaning of time, gravity and the constancy of the speed of light which we outlined here in Figures 3, 4, 6 and 7. In addition we tackled negative absolute temperature and it’s dual relation to negative dimensions and relevance to negative gravity and dark energy [4,23] (see