International Journal of Astronomy and Astrophysics, 2016, 6, 56-81
Published Online March 2016 in SciRes.
http://www.scirp.org/journal/ijaa
http://dx.doi.org/10.4236/ijaa.2016.61005
How to cite this paper: El Naschie, M.S. (2016) Einstein’s Dark Energy via Similarity Equivalence, ‘tHooft Dimensional Regu-
larization and Lie Symmetry Groups. International Journal of Astronomy and Astrophysics, 6, 56-81.
http://dx.doi.org/10.4236/ijaa.2016.61005
Einstein’s Dark Energy via Similarity
Equivalence, ‘tHooft Dimensional
Regularization and Lie Symmetry Groups
Mohamed S. El Naschie
Department of Physics, Faculty of Science, University of Alexandria, Egypt
Received 25 February 2016; accepted 19 March 2016; published 22 March 2016
Copyright © 2016 by author and Scientific Research Publishing Inc.
This work is licensed under the Creative Commons Attribution International License (CC BY).
http://creativecommons.org/licenses/by/4.0/
Abstract
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 = 1/2mv
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 frac-
tal 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 ques-
tions without fear of being labelled in one way or another.
Keywords
Equivalence Relation, Scaling, Intermediate Asymptotic, Golden Mean Scaling, Einstein
Self Similarity, Fractal Scaling, E-Infinity, Special Relativity, Random Cantor Sets, ‘tHooft
Regularization, Fractal Quantum Field, Quantum Gravity, Exceptional Lie Symmetry Groups
M. S. El Naschie
57
1. The Simplexity of Quantum Complexity—An Informal Introduction
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 gen-
eral methodology or maybe a new theory in theoretical physics which we could have labelled “Simplixity me-
thod 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
[1]-[40].
In mathematics, it is well known that not only Cantorian fractals probabilistic equivalence relation (see
Ap-
pendixs 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
[1]-[20]. Self similar-
ity may be seen as trivially obvious as the number ten, being ten copies of unity and/or as quite complex as the
Julia and Mandelbrot set of fractal dynamics
[1]-[6]. Self similarity (see Appendices 1-3) is in the meantime a
well known fundamental principle of modern nonlinear dynamics, theory of fractals (see
Figures 1-3) as well as
deterministic and quantum chaos in physics and cosmology
[7]-[18]. More recently fractal-Cantorian spacetime
and E-infinity theory demonstrated how self similarity is intimately and closely related to the renormalization
procedures
[19]-[25] of quantum field theory and ‘tHooft’s dimensional regularization [26]. In fact the said
E-infinity Cantorian spacetime theory
[26]-[34] is built almost entirely on the explicit self similarity of the ex-
pectation value of its Hausdorff dimension
[10] [28]
3
1
44
1
4
4
D
φ
=+=+
+
+ ⋯
(1)
where
( )
512
φ
= −
[7]-[17]. It should therefore be expected as a matter of logical consistency that the equa-
tion derived from and in this spacetime will also be self similar or at a minimum self affine [23]-[45]. This is
actually the main theme of the present paper
[1]-[106] and at the end of a rather detailed discussion and analysis
we will rediscover a fundamental fact, namely that most of the experimentally verified theories were always self
Figure 1. A one dimensional Cantor universe (see Refs. [1] [2] and [6]).
Figure 2. Construction of a fat fractal. It has infinitely many holes but the remainders keep a to-
tal length that is greater than zero (see Ref. [6]).
M. S. El Naschie
58
(a)
(b)
Figure 3. (a) and (b)—Two random versions of the Cantor set. In (a) each interval is di-
vided into three equal parts from which some are selected at random. In (b) each interval
is replaced by two subintervals of random lengths.
similar in one way or another
[106]. Looking for instance at fields as diverse as classical gravity [17] [24] [30],
electric fields as well as magnetic fields we find that the basic inverse square law reigns supreme
[46]-[48]. In
fact and as will be analysed in substantial detail in the present work, Newton’s kinetic energy E = 1/2mv
2
where
m is the mass and v is the velocity and probably the most famous formula in physics, with which we mean Eins-
tein’s mass-energy equation E = mc
2
where c is the speed of light (see
Figure 3) differ only formally in the scale
and notation used and nothing much more than that (see
Figure 3 and Figure 4). Remembering that energy, en-
tropy and information
[46] [49]-[57] are without a doubt some of the most fundamental and interrelated notions
in physics, then it is natural that we utilize these afore mentioned self similarity to shed light on the major prob-
lem of the measurable ordinary energy density
[58]-[63] and the dark energy density [60]-[63] of the cosmos
which we cannot measure at present in any direct way
[57]-[63]. As mentioned earlier on, this and various re-
lated aspects
[60]-[74] is the subject of the present paper which although relatively short, has the character of a
survey paper due to the large number of subjects and problems we address as well as the numerous theoretical
and mathematical physics references included
[1]-[132]. On the other hand we give limited space to the pure
mathematical literature of equivalence relations which is never the less vital to an even deeper understanding of
our method and results. This short coming could be helped by consulting Refs.
[105]-[111] (see also Appendix
4
). The author would like to apologise for several quite unavoidable gaps in the presentation which he tried hard
to minimize in the included 8 appendices. In this respect we stress that reading Refs. 121 to 135 is strongly
recommended to gain a deep insight into the subject.
2. Fuzziness as a Method
The title of this section is not in praise of being sloppy in physics but simply a somewhat provocative invitation
to take fuzzy logic
[11], fuzzy sets [76]-[85] and consequently Cantor sets, fractals and transfinite set theory
very seriously in quantum physics
[70]-[106]. Nature is overwhelmingly fuzzy in an irreducible and fundamen-
tal way as reflected in the mathematics of even the most orthodox quantum mechanics theories, Hardy’s quan-
tum entanglement included
[54]. Consequently to force nature to be described mathematically in a sharp, crisp
way is really forcing it to submit to an inaccurate mathematical formulation
[11]. As an example of what we
have in mind, let us count quantum particles. At the beginning there was SU(3) SU(2) and U(1) of the standard
model
[13] [28] [46]. The number of the generators of this combined Lie symmetry group corresponds to a di-
mension equal to 8 + 3 + 1 = 12 and this number corresponds, as is well known, to eight gluons, three massive
photons and one massless ordinary photon
[47]. So are quantum particle isometries of the same Lie symmetry
spaces corresponding to Lie symmetry groups—this may be a naïve question not worthy of a question mark. In
fact nature is far stranger than fiction because fractal logic
[83] teaches us that these 12 messenger particles are
in reality 14 particles and have the fractal weight number of 11.708239325
[83]. Now regardless how we answer
M. S. El Naschie
59
Figure 4. A light hearted, not historical picture depicting the moment when the truly great
Albert Einstein was about to discover E-infinity dark energy formula E(D) = mc
2
(21/22) as
well as the ordinary energy formula of the same theory E(O) = mc
2
/22. It seems possible
that after giving an alternative proof for the theorem of Pythagoras based on the principle of
self similarity that he realized that this theory and Newton’s kinetic energy E = 1/2mv
2
are
self similar. That way he could have discovered the chaotic nonlinear dynamics beneath
quantum mechanics and found El Naschie’s dissection of E = mc
2
into its quantum compo-
nents E = (mc
2
/22) + mc
2
(21/22) (see Refs. [36] and [50]).
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
[28] [47]. In addition we
know from bosonic string theory that spacetime is 26 dimensional and when we imagine 16 dimensions running
in an opposite direction to these 26, then we find 26 − 16 = 10 which corresponds to one type of a so called he-
terotic super string theory
[28] [47] of which one may be based on E8E8 or on SO(32) which leads to [101]-
[105] [124]-[126]
( )
( )( )
32 32 32 1 2 496SO = −=
(2)
as in the E8E8 where E8 is the largest exceptional Lie symmetry group
[28] [55] [56]. Thus we could really
think of internal space dimensions as particles resulting from symmetry breaking related to Noether’s conserva-
tion laws and symmetry theorem and if we venture into even more “fuzzy” spacetime dimensions
[41] such as D
= 11 of Witten, D = 5 of Kaluza-Klein then even the D = 4 of Einstein could be seen as a kind of pre-particles
[47]. But what good could this wild mixing of basically different notions and concepts bring about except a little
more chaos added to a mess of other problems? We could answer this point by working out something specific,
namely determining the density of ordinary energy, dark energy, dark matter energy and pure dark energy den-
sity of the cosmos. This is what we will do next
[63]-[77].
The standard model contains 126 particle-like states when considering super symmetry and disregarding frac-
M. S. El Naschie
60
tal logic counting [28] [46]. In the case of super gravity this contain 2 more, namely 128 which corresponds to
half of the 256 Einstein-Riemann independent curvature tensor components as given by
( )
4
4 256=
[48].
Without super symmetric partners this corresponds to 64 particles in 4D, which we can detect only if we per-
form experiments in 4D. However, real experiments can be done only in 3D with time as a parameter
[47].
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 ap-
proximately
[36]-[45]
( )
( )
22
3
0.046875 4.7%
64
E O mc mc
≃≃ ≃
(3)
of the total Einstein energy in reasonable agreement with the cosmic measurements
[36] [47]. To account for
dark energy we have the self explanatory scaling
( )
64 3 64 95.312%
λ
≅− =
, again in fair agreement with
measurements
[37]-[45]. Finally to find the scaling associated with dark matter, we see that it is reasonable to
consider the ratio of the total number of the 12 messenger particles of the standard model plus the Higgs and the
graviton making them 14 particles corresponding to the dimension of
2
14G =
exceptional Lie groups or dim
OSP (1/4) = 14 of orthosimplectic group
[28], for which one finds
( )
( )
2
14
dark matter 21.875%
64
D mc
≃≃
(4)
Yet, again in fair agreement with the limit which cosmic measurements set on the dark matter energy density
[36]-[47] [60]. In Appendix 8 a preliminary, simple and intuitive derivation of the above is given using a radi-
cally different conception.
3. A Thin and a Fat Cantor Universe Made of One Cantor Set and Its Embedding
Following the pictorial logic of the one dimensionally embedded iterative triadic Cantor set of Figure 1 we see
that it leads us to consider two sets (see also
Appendix 1). The first is the classical triadic set consisting of an
uncountable infinite number of black Cantor points with a Hausdorff dimension equal
( )
2 3 0.63DH n n= ℓ ℓ≃
and a topological dimension equal zero and in addition the length of this set is zero, i.e. it is a zero measure
geometrical-topological structure. This is basically the zero set which models in E-infinity theory the pre-quan-
tum particle. The second set is the white gaps between the black Cantorian points of the zero set
[36]-[45]. The
Hausdorff dimension of this set is simply one minus the Hausdorff dimension of the zero (black) Cantor set, i.e.
1 0.63 0.37− ≃
. The measure of this set on the other hand is equal to one minus zero which is then equal one in-
dicating a fat Cantor set as shown in
Figure 2 [72]. Since this set represents the gaps in the “black” set, it is es-
sentially an empty set with a negative topological empty set dimension equal minus one. It follows then that this
“gaps” Cantor set is the empty set which models the pre-quantum wave in E-infinity theory
[10] [13] [28]. The
situation remains the same for a random Cantor set (see
Figure 3) except that D(H) goes to
( )
2
512
φ
= −
and thus
( )
1 DH−
goes to
2
0.381966
φ
=
instead of 0.37 [36]-[63].
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
φ
while the white gap between them is the empty set modelling the pre-quantum wave with an op-
posing topological pressure equal to the Hausdorff of the empty set, namely
2
φ
[62]-[76]. Consequently we
have a resulting net pressure equal
23
φφ φ
−=
which we can view as the topological local Casimir force. In
turn
3
φ
is equal to the Hausdorff dimension of the cobordism [35] [42], i.e. the surface of the quantum wave as
well as being the expectation value of the multi-fractal surrounding of the quantum wave giving rise to the core
of quantum spacetime as is obvious from the relation
[63]-[76]
( )
( ) ( )
41 3
4
33
1 114
c
d
φ φφ φ
−
= = = = +
(5)
where D(T) = 4 is the topological dimension and
( )
3
4DH
φ
= +
is the expectation Hausdorff dimension of