_{1}

^{*}

Realizing the physical reality of ‘tHooft’s self similar and dimensionaly regularized fractal-like spacetime as well as being inspired by a note worthy anecdote involving the great mathematician of Alexandria, Pythagoras and the larger than life man of theoretical physics Einstein, we utilize some deep mathematical connections between equivalence classes of equivalence relations and E-infinity theory quotient space. We started from the basic principles of self similarity which came to prominence in science with the advent of the modern theory of nonlinear dynamical systems, deterministic chaos and fractals. This fundamental logico-mathematical thread related to partially ordered sets is then applied to show how the classical Newton’s kinetic energy
E = 1/2
mv
^{2} leads to Einstein’s celebrated maximal energy equation
E =
mc
^{2} and how in turn this can be dissected into the ordinary energy density
E(
O) =
mc
^{2}/22 and the dark energy density
E(
D) =
mc
^{2}(21/22) of the cosmos where
m is the mass;
v is the velocity and
c is the speed of light. The important role of the exceptional Lie symmetry groups and ‘tHooft-Veltman-Wilson dimensional regularization in fractal spacetime played in the above is also highlighted. The author hopes that the unusual character of the analysis and presentation of the present work may be taken in a positive vein as seriously attempting to propose a different and new way of doing theoretical physics by treating number theory, set theory, group theory, experimental physics as well as conventional theoretical physics on the same footing and letting all these diverse tools lead us to the answer of fundamental questions without fear of being labelled in one way or another.

The present paper was initially motivated by the wider implications of a small discovery attributed to a young Albert Einstein and the imagined wider implications for his later work on his famous formula E = mc^{2} which we allege could have taken place. However the main thrust of the paper is to demonstrate in a concrete way a general methodology or maybe a new theory in theoretical physics which we could have labelled “Simplixity method of quantum complexity via topological E-infinity theory”. To explain as briefly as possible what we mean by that, it is best to start with a concise resume of basic theories connecting self similarity, fractals and physics [

In mathematics, it is well known that not only Cantorian fractals probabilistic equivalence relation (see Appendixs 1-4) can be based on similarities and we will show here how this can lead to surprizing connections to physics via Penrose fractal tiling based self similar E-infinity quotient spacetime manifold [

where

tion derived from and in this spacetime will also be self similar or at a minimum self affine [

similar in one way or another [^{2} where m is the mass and v is the velocity and probably the most famous formula in physics, with which we mean Einstein’s mass-energy equation E = mc^{2} where c is the speed of light (see

The title of this section is not in praise of being sloppy in physics but simply a somewhat provocative invitation to take fuzzy logic [

this naïve question and the puzzling fractal logical reality, we hasten to say that E8E8 super string theory starts with 496 quasi massless gauge bosons and not 12 and one has normally to show how these 496 isometries are reduced via symmetry breaking to our observed 12 gauge bosons of the standard model [

as in the E8E8 where E8 is the largest exceptional Lie symmetry group [

The standard model contains 126 particle-like states when considering super symmetry and disregarding fractal logic counting [

Taking our previous discussion into account as well as realizing that all energies must ultimately be a scaling of Einstein’s maximal energy E = mc^{2}, we are inclined to conclude that a good estimate of the ordinary, real measurable energy density of the universe must be E = mc^{2} scaled by the ratio of real space D = 3 to the space of the standard model plus gravity which is in the afore mentioned fuzzy meaning D = 64 so that one finds approximately [

of the total Einstein energy in reasonable agreement with the cosmic measurements [

graviton making them 14 particles corresponding to the dimension of

Yet, again in fair agreement with the limit which cosmic measurements set on the dark matter energy density [

Following the pictorial logic of the one dimensionally embedded iterative triadic Cantor set of

situation remains the same for a random Cantor set (see

It follows then from the geometry and topology of the above that each black point is essentially a pre-quan- tum particle with a positive topological attracting pressure equal to the Hausdorff dimension of the zero set which is

where D(T) = 4 is the topological dimension and

quantum spacetime. We conclude that the above is an accurate one dimensional Cantorian spacetime which could now be expanded to the fully fledged 5 dimensional Kaluza-Klein fractal spacetime of our reality [^{2}) which leads to a dark energy density equal to [

This is self evidently the exact dark energy density found via accurate measurement and cosmological observation. Our second most import conclusion is that the Casimir energy is the local form of dark energy which by the well known theorem of Dvoretzky must be concentrated at the boundary of the holographic boundary of the universe [

It is an illusion to think that a Cantor set is intuitively simple. A Cantor set is in fact both the simplest and the most mind boggling thing that there is on the foundational level of pure mathematics and mathematical physics.

To start with a Cantor set construction produces at the end two contradictory Cantor sets. Let us look at the classical iterative procedure of Figures 1-3 which is rather familiar to those working in any of the many branches of nonlinear dynamics, chaos and fractals [

[

which is not much smaller than the 0.63 of the classical Cantor triadic set [^{2} is actually the sum of two quantum components, namely E(O) = mc^{2}/22 for ordinary energy [^{2}(21/22) for dark energy so that at the end we have the energy of a quantum particle E(O) added to the energy of a quantum wave E(D) giving us the maximal E of Einstein (see

In this sense E(O) could be regarded as the position or potential energy and E(D) as the kinetic energy while E = mc^{2} is the maximal total energy with the possible interpretation of converting mass to energy, only in theory or also in actual real world, but this is not the point we want to discuss in the present context. To enhance understanding of the role of fractals in the present work and for a deeper understanding of the role of dimensions, the reader is referred to Appendixes 1-3. We should also stress at this point the fundamental role played by Hardy’s quantum entanglement [

The logarithmic spiral with its self similar geometrical structure creating golden mean rectangles step by step as shown in

and its exact E-infinity counterpart [

This subject is discussed in considerable details in Refs. [

this case corresponds to a ‘tHooft-Polyakovmonopole

which is very close to the expected accurate value [

Again it is an excellent result compared to the exact one, namely 26 + k = 26.180339 so far for the logarithmic scaling of quantum field theory. For the E-infinity golden mean scaling on the other hand we find [

and [

exactly as expected [

tion. This becomes clear when we take in the above calculation ^{16} GeV and replace

using the golden mean scaling. On the other hand things are quite straight forward leading to [

just as in the grand unification case [

Clearly there can be no energy without spacetime volume to contain it. This statement remains true even if volume and energy are one and the same thing as in general relativity curvature of spacetime and gravity are one and the same. Now we have two volumes. First we have the volume of the five dimensional pre-quantum par-

ticle [

spacetime [

Consequently E = mc^{2} could be rewritten as [

as claimed by us at the very beginning of this section [

There are a few things that we need to know before noticing at once that the Dvoretzky theorem [^{2} is the maximal energy density possible in the universe when m is the mass and c is the speed of light [^{2} when 1/2 ® 1 and v ® c. Second we need to reason that a spherical multi-dimensional ball is the most likely shape of our universe. Thus our universe must be at least a five dimensional sphere which may be regarded as a sufficiently high dimensionality to make Dvoretzky’s theorem work. Now we see that the theorem employs a volume measure concentration at the surface of the sphere amounting to 96% while the rest of the universe contains only 4% of the volume. Since energy must be proportional to the volume of spacetime, it follows directly that E is 96% at the end of the universe, i.e. at the boundary of the holographic boundary [

It does not seem farfetched at all that our real spacetime must be at least five dimensional. The 4 dimensions of Einstein are self evidently 3 space and one time dimension which were merely fused together in special and general relativity. On the other hand the phenomenal success of Kaluza-Klein’s five dimensional theories, let alone 10 dimensional super strings and eleven dimensional super gravity and Witten’s M-theory [

where

gravity coupling all measured at the electroweak scale and results in the exact theoretical

This could be interpreted as the normed total dimensions of the universe and could be sub-divided into the following [

D_{1} = 4% for spacetime of ordinary energy;

D_{2} = 26% − 4% = 22% for compactified bosonic dimensions of dark matter energy;

D_{3} = 100% − 26% = 74% for the pure dark energy.

Again this agrees with a very high accuracy with the COBE, WMAP and Type 1a supernova measurements and observations [

A cow is definitely not a sphere but for all topological purposes it may be approximated to a sphere and glossing over biologically indispensible holes as well as the four stretched legs. This may be a drastic way to start arguing for an overwhelming self similarity of the cosmos and may also be the mathematical equations describing the cosmos. From this somewhat too general view point to be of any practical value, the manifest similarity between Newton’s kinetic energy E = 1/2mv^{2} on the one side and celebrated Einstein’s maximal energy equation E = mc^{2} and El Naschie’s two components equations E(O) = mc^{2}/22 and E(D) = mc^{2}(21/22) which sum up to that of Einstein E(O) + E(D) = mc^{2} [

The first step in this direction could have been showing that four dimensions seems to be the expectation number of weighted integers from zero to infinity in the sense of E-infinity theory. This is really the reason behind the E-infinity Hausdorff dimension expectation [

where

In such a space with infinite hierarchal topological dimensions, self similarity is basic and most fundamental so that it should not come as a surprise to observe that all formulas related to something as fundamental as energy (E) should be subject to a minimum of self similarity or self affinity. To stress the point we recall that E-infinity spacetime is made of an infinite number of unions and intersection of random Cantor sets with a Hausdorff dimension equal to ^{2} (see ^{2} except the appearance and that E = mc^{2} can be meaningful only in the context of Lorentzian invariance. On this count however we beg to differ. Admittedly biased by our own result E = mc^{2}/22 + mc^{2}(21/22) [

Let us start by quoting what is written on page 4 of the excellent book of Schroeder [

“The ‘resemblance’ of equation 3 to Einstein’s later discovery, his famous E = mc^{2}, is of course entirely fortuitous. The equivalence of mass m and energy E which is at the basis of nuclear power in all its guises is a consequence of Lorentz invariance. This invariance which underlies special relativity was predicted by Einstein in 1905 after it seems, several false starts……”.

Now analysing the above in a liberal way consistent with the big fuzzy picture approach of E-infinity, we can take the resemblance to mean self similarity or self affinity while the Lorentz invariance could be understood as a gauge which means scaling and renormalization and these notions are in turn just another more or less sophisticated form of self similarity which Einstein used to devise an alternative proof for the Pythagoras theorem [^{2} as used in another excellent popular book on the subject by Brian Cox and Jeff Forshaw “Why Does E = mc^{2}” [^{2} of Newton compared to E = mc^{2} of Einstein we do not need much persuasion to see that it is only a matter of scale where the 1/2 ® 1 because we are integrating in the case of v = c over a constant velocity, namely that of light and v ® c is also a matter of scale. For this reason maximal energy equation E = mc^{2} and El Naschie’s two components equations E(O) = (1/22)mc^{2} and E(D) = (21/22)mc^{2} could be seen as scaling of E = mc^{2} when due to certain conditions the Lorentzian factor g = 1 of maximal “Einstein” energy takes two other values ^{2}/22) + mc^{2}(21/22) = mc^{2} long ago. After all it is sometimes good to think superficially. This dialectic statement could be illustrated by a well known observation of Einstein himself who explained time as that quantity that we measure with a clock. A second not so good example of the above is that when v is constant then it goes out of the integration belonging to the definition of energy and 1/2 becomes unity and v may be replaced by a new notation, namely the constant c so that at the end E = 1/2mv^{2} is replaced by E = mc^{2}. From a formal view point that is really all and our conclusion is that self similarity and generalized equivalence relations [

To end this section we should add that according to Manfred Schroeder, the story of Einstein’s proof of Pythagoras’ theorem was recounted to him by SchneiovLifson of the Weizmann Institute in Tel Aviv who had it from Einstein’s assistant Ernst Strauss who was told the story by Albert Einstein himself [

It is remarkable that our self similarity principle which we used to justify moving from E = 1/2mv^{2} to E = (mc^{2}/22) + mc^{2}(21/22) = mc^{2} can be extended to the realm of Newton gravity and quantum gravity [

which is very close to most of the results found in the literature in the non-super symmetric case. Similarly the number of isometries of the manifold combining classical gravity with all other fundamental forces is a staggering number, namely 10^{19} as we will show later on. We also know that at the point of total unification of all fundamental forces we are simply dealing with the Planck energy scale and that the coupling constant in this case is a maximum equal 1. This is the only degree of freedom. Thus the equation corresponding to [

must now be [

That means

In other words Newton’s degrees of freedom is

It follows then that

That means

in a rather excellent agreement with the inverse value of the dimensionless Newtonian gravity. As for the spacetime manifold with 10^{19} isometries which we mentioned earlier on, this was investigated about eight years ago [

looking at how many Planck length mini black holes ^{19} GeV which means nearly

Let us attempt to extend our present concepts and analysis to become truly cosmic. The idea or rather our hypothesis is that similar to the quantum mechanical Dirac’s vacuum or “sea” which led to the prediction and subsequent discovery of the positron and thus to anti-particles, one could propose that the cosmos is abound with regions of positive energy and regions of negative energy. Since almost 95.5% of the energy density is due to the quantum wave, i.e. quantum energy of propagation and since the quantum wave is the cobordism of the quantum particle, then it follows that most of the energy will be located at the boundary of the holographic boundary of the universe that lies at the hyperbolic infinity [

Lie symmetry group and its associated fuzzy manifold with

ticle. Second we must recall the formula for the degrees of freedom of pure gravity, i.e. the formula of the Vierbien description of graviton, namely [

It is then a trivial matter to reason that for tangible energy and matter d must be d = 4 which leads to D = 2 while the full space of Witten’s M-theory may be found from d = 11 to be

Consequently the density Lorentz factor must be

for ordinary matter and energy and

for dark matter and energy as reasoned in previous sections. Here however we want to expand our argument and start with something more insightful than E = mc^{2}, namely the formula from which Einstein deduced E = mc^{2}, namely [

where A = p^{2}c^{2} and P = mv is the momentum [^{2} and finding that

That way we can relate the previous discussion and the finding ^{2} then we must note that while our universe is fully described by E8E8 we should subtract from it either D = 4 of Einstein spacetime in an exact analysis or alternatively we take out the 12 messenger particles of the standard model in an integer approximate analysis. Similarly we note for the same purpose that for the dark section we have to subtract the 44 degrees of

freedom of pure gravity plus the 11 dimension of M-theory, i.e. we will be left with

grees of freedom or isometries. Combining what we have just calculated with E^{2} one finds the squared dark energy density of the cosmos as

It is then rather gratifying to find our previous result strongly confirmed by

exactly as expected but with an additional negative sign besides the well known positive sign. On the other hand Einstein’s celebrated formula follows as a trivial result when we ignore the true vacuum, i.e. pure gravity as well

as the

So far for the integer theory but the really very nice aspect of our theory comes to the fore when we considered the exact transfinite analysis and discovered the role of dimensional regularization of that at the time young Dutch Ph.D. student and presently senior Nobel laureate Gerardus ‘tHooft [

In the preceding analysis we used for dim E8E8 the integer value 496 and invoked it in finding E^{2}. However the exact transfinite value is not 496 but

where

Now it comes as no surprise to note that we could have cut the preceding analysis to exactly two lines of trivial calculation by simply pausing a minute to reason that the energy density must be simply the ratio of a completely solid spacetime, i.e. Einstein’s spacetime dimension D = 4 to ‘tHooft-Veltman-Wilson fractal spacetime

The drawback of this shortcut is that the plus minus double sign of taking the square root of E is not as obvious as in the previous analysis.

It is useful to reformulate our preceding analysis in terms of combinatorics. This way we can write [

which is the total quantum-like state of Witten’s 5-Brane in eleven dimensional theory which is also given by [

killing vector fields. Furthermore we have

which is equal to the dimension of E8E8 as well as SO(32) where [

Adding the two results together one finds that

which is nothing but a membrane in eleven dimensions which gives us the remarkable E-infinity result

For n = 11 and

That way we find a simple combinatoric Brane formula for dark energy via

This means

exactly as it should be. The corresponding exact transfinite expression is

and since

where

ue of the entangleon corresponding to k of ‘tHooft’s renormalon hypothetical particle of dimensional regularization. Finally, motivated by the facts that

and that our string bosonic spacetime is D = 26 similar to one of the two directions of Heterotic superstrings, we will use a two Stein space representing a non-standard Riemannian manifold with a Lie symmetry group dimension given by [

where |E6| = 133 and |F4| = 52 correspond to the well known exceptional Lie groups family. Noting further more that for the non-compactified Klein-Kaluza section of D = 5 and the Einstein section D = 4 we have first the two Stein spaces, namely first:

and second the simply connected symmetric harmonic k = 1 Stein space, i.e. Einstein space [

then we can write the ordinary and the dark energy Lorentz factor

and

where ^{2}. Some readers may find the following relation interesting in their own right within the context of dark energy research

, (58)

Thus

and finally

In addition we draw the readers’ attention to identical results obtained by the author several years ago using the holonomy and co-holonomy of Kähler manifolds where the ratio of the relevant Betti number for smooth

Einstein manifold is one and for a rugged Kähler is 22 leading to

Although the great man spoke German with a characteristic Danish accent, the author could not translate the eloquence of Niels Bohr’s German sentence as recounted in the memories of the young Werner Heisenberg [

“Nur die FüllefürtzurKlarheit”.

In English this could mean something like “only abundance leads to clarity”. The somewhat contradictory statement reflects of course Niels Bohr’s admiration for F. Hegel’s dialectic philosophy [

Starting from nonlinear dynamics, chaos and fractals and linking that with the pure mathematical results of the theory of equivalence relations [

The author is deeply indebted to the work of Prof. G. ‘tHooft on dimensional regularization which is far more than an ingeniously simple mathematical manoeuvre concealing a great deal of physical reality about the fine structure of quantum spacetime. He is also grateful to all the discoverers of the completely beautiful theory of strong interaction that has clear marks on the present paper. Last but not least, the work of A. Connes was indispensible for the present author.

Mohamed S. El Naschie, (2016) Einstein’s Dark Energy via Similarity Equivalence, ‘tHooft Dimensional Regularization and Lie Symmetry Groups. International Journal of Astronomy and Astrophysics,06,56-81. doi: 10.4236/ijaa.2016.61005

Illustration of the first five iterations of the construction of a deterministic Cantor set

Referring to

In other words nothing remains of the initial black line of unit length except an uncountable infinite number of “Cantorian” black points (or bars) with a total length equal zero. Thus we could imagine the black line to have been replaced by a white line of the length one minus zero which is the original line of the length of the black line. Therefore we could say that we have two “Cantor” sets. The first is an infinite number of black “points” with length, i.e. measure equal zero and a Hausdorff dimension equal 0.63 apart of a topological dimension equal zero because it consists only of points at the limiting infinity.

The second set by contrast is a white line of a unit length of a Hausdorff dimension equal

Generalizing the classical Cantor set to two dimensions

The two dimensional counterpart of the one dimensional triadic Cantor set is neither the two dimensional Swiss flag fractal (see

the two dimensional random Cantor set which is also found from the inverse of

Generalizing the classical Cantor set to three dimensions

Referring to

The generic generalization of Cantor’s triadic set to 3 dimensions is neither that shown in

theory

Equivalence relations in mathematics and physics

The interest of the present work in the pure mathematics and mathematical logic of equivalence relations stem from numerous expected and less expected connections to theoretical and mathematical physics [^{2}, as well as the Planck radiation expression E = hf and possibly Newton’s second law F = ma as well as the similarity of Newton’s kinetic energy E = 1/2mv^{2} and our dissected Einstein formula E = (mc^{2}/22) + mc^{2}(21/22) = mc^{2} [

Geodolian role in the triality of logic

This short appendix is intended to show how fuzziness enters into physics via logic. Let us start with classical “ordinary logic”. Here we have zero and one which may be taken to correspond to yes or nor or equivalently, right and wrong. On the other hand Gödel’s theorem establishes at a minimum the possibility of a three valued logic corresponding to the above, i.e. yes and not plus a third possibility of undecidability which we may call in plain English “do not know”. The E-infinity set theoretical realization of this triality is aptly set theory, namely the zero set and the empty set. It is remarkable that this seemingly unconventional logic leads to a disarmingly simple and intuitive realization, namely yes for sets and do not know for the zero set. The first and the second identifications are obvious and self evident. By contrast the third possibility may need some further elaboration. This is because undecidability has an element of fuzziness while a zero set is crisp, i.e. sharp. Our response to this point is that a zero set is not zero but a set containing zeros as “elements”. When the zeros disappear completely we are still left with a set, albeit an empty set. Consequently a zero set is not as sharp as a zero but represents the logical location of the zero being the border separating positiveness and negativeness on a number world line. Seen from this viewpoint it is thus the dynamics created by set, empty set and zero sets that represent a true fuzzy dynamics and incidentally it also corresponds to G. Ord’s anti-Bernulli processes, i.e. 1, 0, −1 which gives us quantum mechanics while the Bernulli processes gives us a mere diffusion.

Important remarks on Schrödinger’s equation and special relativity

Fusing Schrödinger’s equation, i.e. quantum mechanics with E = mc^{2}, i.e. special relativity does not result in a quantum gravity theory but rather in a Dirac equation, i.e. a Schrödinger equation without its nonlocal character. Similarly a quantum gravity theory is not a complete unification of all fundamental forces because for instance it leaves out electromagnetism. For this reason one has to be quite critical and on guard from undue generalization to avoid later misconceptions arising from reading too much into partial unification of the five different fundamental interactions.

Remarks on the dependence of the empty Cantor set on the dimensionality

Let us consider a one dimensional random Cantor set. The topological dimension is clearly zero and its Hausdorff dimension is

or the bijection equation of E-infinity theory [

where

An attempt towards a simple intuitive elucidation and elementary derivation of E = mc^{2}/22 as an Eigenvalue problem

We have here a remarkable Eigenvalue problem. It is E = mc^{2} which almost everybody knows because it is arguably the most famous formula in the history of physics. Energy in this way is comparable to the buckling

load in an Eigenvalue stability problem. As we know it is

strut [

All that would have remained unimportant or at a maximum of minor academic value if it were not for the following.

When they tested the equation of Einstein in the laboratory within a room not larger than a sitting room, they found experimentally that it represents the energy density expected. Yes Einstein equation of converting mass into pure energy although mostly theoretical, never the less represented a practically correct maximal energy density. The shock came some 15 years or so ago when in repeated experiments using the entire universe as a laboratory cosmologists attempted to calculate the energy density of the entire universe using super nova events and as revealed by the WMAP experiment and confirmed by Planck and other cosmic measurements. They found approximately 4.5% only of the expected 100% density [

If the analogy between the buckling load and Einstein’s formula as an Eigenvalue holds, then the discrepancy could be explained as restricting the system and forcing the energy density of Einstein to be much higher than the real energy density is. The reason for the small amount of energy of 4.5% they found was suspected by the author immediately when he heard and understood the problem that it might be related to the 11 extra degrees of freedom which Einstein did not take into account. Einstein was limited by history and his training. He did not know at the time of the experimental facts and quantum mechanics was not invented yet and CERN did not yet exist. Moreover, when he got to know quantum mechanics he was not ready to accept it and the fight between him on the one side and Bohr, Heisenberg and Schrodinger on the other is well known in the history of science [

Now the author solved this problem using quite sophisticated ad complex mathematical physics as shown for example in the present paper. However the author believes that a simpler and more intuitive explanation could lead us to the correct result. The author had such a simple idea and he even presented it conferences but normally he published the sophisticated mathematics.

If there is a simple watertight solution based on similarity and number of degrees of freedom then this must be

based on the similarity between ^{2} could be thought of as a scaling

agrees not only with cosmic measurements but also with the results obtained using quite sophisticated mathematics similar to what was done in the main body of the present paper. In short we need to make the present simple scaling argument mathematically watertight because the author feels it is not yet and wrote this short appendix mainly in the hope that some of the readers with more mathematical skills could find a mathematical way to argue the case.