Page 1 of 6
European Journal of Applied Sciences – Vol. 10, No. 1
Publication Date: February 25, 2022
DOI:10.14738/aivp.101.11579. Naschie, M. S. E. (2022). Superstrings and Loop Quantum Gravity are Two Sides of the Same Golden Coin. European Journal of
Applied Sciences, 10(1). 387-392.
Services for Science and Education – United Kingdom
Superstrings and Loop Quantum Gravity are Two Sides of the
Same Golden Coin
M. S. El Naschie
Distinguished Professor of Physics and Engineering
Department of Physics, Faculty of Science
University of Alexandria, Alexandria, Egypt
ABSTRACT
The general framework of the paper is to rekindle the profound mathematical and
philosophical revelations of the Ancient Greek-Egyptian school of Alexandria where
the Egyptian, Persian, Indian, Chinese, Greek, Roman, Islamic and Arabic
civilizations melted together to form a shrine of science, art and wisdom
culminating in the Platonic-Pythagorean golden mean number system utilized in E- Infinity Cantorian spacetime of present modern science. Based on the above, we
introduce a new universal mean which seems not only to have numerous
applications in statistics, but also goes as far as showing that not only superstrings
and loop quantum gravity are simply two sides of a certain harmonic golden coin.
In fact, the work goes far beyond that by encompassing science, art and
consciousness into a magnificent single entity.
Keywords: Golden mean number system. Golden harmonic mean, Unification of
superstrings and loop quantum gravity, Fractal space time, The Immirzi parameter.
1. INTRODUCTION
The present work is a synthesis of the work of a great number of pioneers and leading
researchers [1-28] spanning those of antiquity like Pythagoras, Aristotle and Plato [3-5] to
mention only a few who are known to us, up to present day scientific leaders [1],[6-14] such as
the great Sir R. Penrose [7,13], Nobel Laureates David Gross [6] and Gerardus 'tHooft [6,11] as
well as celebrated scientists like Carlo Rovelli [16], the late Michael Feigenbaum [18] and Fields
Medalist Alain Connes [9].
In view of the above and if the Author mistakenly says in the text "my theory" then it is an
unintended and horrible exaggeration. This is because the Author genuinely believes that he
has done nothing more than put together the work of hundreds of serious great scientists. Most
of these scientists not even known to him but somehow their wisdom and findings came to him
from the fortunate scientific atmosphere in which he lived and worked [19, 20, 21]. To make a
very long story short, the crux of the matter came from a controversy between the Author and
his friend Prof. Scott Olsen [3,4] who was a student of the legendary David Bohm [11] when
despicable racial and political discrimination forced him to be exiled to Birbeck University in
London, UK [19, 20]. The result was, as the great Albert Einstein used to say, that one can only
quarrel with friends because the others are too foreign. The Author admits he was wrong to
dismiss Scott's proposition and hence the Author came to re-establish the well known harmonic
mean on the basis of the golden mean number system [3-5]. This in turn became a primary tool
Page 2 of 6
388
European Journal of Applied Sciences (EJAS) Vol. 10, Issue 1, February-2022
Services for Science and Education – United Kingdom
to connect to the fundamental work of D. Gross and E. Witten to that of Lee Smolin [12] and
Rovelli [16,21]. That way the pioneers of superstring theories and loop quantum gravity are
shown to be talking about the same idea using different mental pictures and thus leading to
what the Author considers to be a great synthesis [1-28] as he hopes to show in the next few
sections.
2. Preliminary analysis and conceptual substantiation of a fundamental conjecture
Let us consider the possibility that except for scale invariance Albert Einstein's spacetime is
basically substantially correct, i.e. the four dimensionality is more than a very reasonable
approximation to spacetime dimensional reality. One way of arriving at the exact scale
invariance of D = 4 is to assume that time is a fractal [3-28] which implies the following
continued fraction [23-28]
(1)
Adding to that the other 3 space dimensions, we find the by now well- known dimensions [23-
28]
(2)
exactly as derived previously using other methods of E-Infinity Cantorian spacetime [23-28].
Next, let us assume that all of Einstein's four dimensions are fractal similar to the previous
analysis. Furthermore, let us look at the spacetime spanned by the union of these four fractal
dimensions which leads to the following simple result [14,15][23-26]
4(1+ ) = 4 + 4
= 4 + 0.944271902 (3)
The remarkable fact is that the above equation (3) amounts to:
D(union) = 5 - 0.55728092 (4)
= 5 -
where 5 is the dimensionality of Kaluza-Klein spacetime and is the exact transfinite value of
the Barbero-Immirzi parameter [23-28] connecting superstring theory with loop quantum
gravity as shown in various previous publications by the Author and others [21],[23-28].
From the last well-established result we can conclude that by assuming either super-symmetry
or the validity of the transactional interpretation of quantum mechanics [7], it is almost an
obvious validation for a correct fundamental conjecture that both D = 10 superstings theory
and loop quantum gravity are indeed two different sides of a harmonic golden mean implied by
and the golden mean number system utilized extensively in E-Infinity Cantorian spacetime
theory [1-5].
3
1 1 1 4
4 ...
1
+
+
+
= + f
3
3
D 31
4
= ++ f
= + f
3 f 3 f
6 f
6 f
6 f
Page 3 of 6
389
Naschie, M. S. E. (2022). Superstrings and Loop Quantum Gravity are Two Sides of the Same Golden Coin. European Journal of Applied Sciences,
10(1). 387-392.
URL: http://dx.doi.org/10.14738/aivp.101.11579
Next we look at the intersectional outcome analogous to the above analysis namely:
Considering five copies of the above equation (5), one finds:
D = (5)(1.888543819)
= 9.442719095 (6)
In terms of D =10 of superstring and [27] this amounts to a "miraculous" number theoretical
result namely:
10(1 - ) = 10 - 10
= 9.442719095 (7)
Again, we stress that the same results will obviously be obtained, whether we involve super- symmetry or the transactional interpretation which goes back to J. Wheeler and R. Feynman
[2],[7],[19-20],[23-26] and yields the same relation between D=10 of superstrings on the one
hand and the Barbero-Immirzi [27, 29] golden mean based loop quantum gravity parameter on
the other [23-26].
3. Mathematical derivation of the relationship between the arithmetic mean, the
geometric mean and the harmonic mean of two numbers
Let us denote the three well-known means [29] as following:
AM for the Arithmetic mean.
GM for the Geometric mean
and HM for the Harmonic mean.
Now noting the well-known relation [29]:
(8)
as defined above and converting to the golden mean number system [3-5] where
and are the minor and the major forms of the golden ratio
respectively, we find the following rather beautiful, subtle and meaningful results:
(9)
( ) ( )
( )
( )
4 1 3
3 3
6 39
D intersectional 1
1
13 3
1.888543821 5
-
= + f
= + f
= + f + f + f
=
6 f
6 f 6 f
2 AM HM (GM) Ä =
f = ( 5 1) / 2 - 1 5
2
+ F =
AM ( ) / 2
1 ( )/2
= f + F
= f +
f
= 0.5
Page 4 of 6
390
European Journal of Applied Sciences (EJAS) Vol. 10, Issue 1, February-2022
Services for Science and Education – United Kingdom
which corresponds to the average of all possible cantor sets of the Mauldin-Williams types [14,15],
[18-21],[23-28].
Finally taking the relative average connecting 0.5 with the standard Hausdorff dimension M-W cantor
set i.e. one finds:
(10)
where is the E-infinity value connecting loop quantum gravity with superstrings. Furthermore we
have:
=
= 1 (11)
Consequently,
=
=
= 1.79442719
= (10) (12)
In other words, HM is equal to ten times the inverse exact transfinite Barbero-Immirzi
parameter [27, 28] which has now a definite physical meaning reinforcing what we explained
at the beginning of this section.
4. Conclusion
Superstings and loop quantum mechanics are indeed two faces of the same golden truth as
confirmed and seen via the golden eyes of the Harmonic mean derived within the number
theoretical framework of the Alexandrian golden mean number system discovered a long time
ago by the mathematical-philosophical giants of the international Egyptian-Greek school in
Alexandria. This great heritage which is the result of a peaceful dedication of the entire human
race was unfortunately lost only to be rediscovered again also through the dedicated work of
modern scholars and scientists, artists and philosophers all over the planet earth testifying to
what the human race can achieve when they all work together peacefully and abandoning wars,
a destruction which is good for absolutely nothing.
f
AM 0.5 / 2 ( )
0.5590169945
= f+
=
( )( ) 6 = 10 f
6 f
GM ( )( ) = f F
( )(1/ ) f f
2 (GM) HM
AM =
2
6
(1)
(10)( ) f
6 (1 )
10
+ f
6 ( ) f
Page 5 of 6
391
Naschie, M. S. E. (2022). Superstrings and Loop Quantum Gravity are Two Sides of the Same Golden Coin. European Journal of Applied Sciences,
10(1). 387-392.
URL: http://dx.doi.org/10.14738/aivp.101.11579
5. References
[1] Edward Witten. Editor. Einstein in Alexandria. The Scientific Symposium (2005). Publisher: Bibliotheca
Alexandrina. Alexandria-Egypt [2006]. 179 pages.
[2] Mohamed Saladin El Naschie. Quantum two-slit Gedanken Experiment Revisited. In the proceedings of
“Einstein in Alexandria” symposium. Edited by Edward Witten [2006] pp. 141-153. (See Ref. 1).
[3] Scott, Olsen, L. Marek-Crnjac, Je-Huan He and M.S. El Naschie. A Grand Unification of Science, Arts and
Consciousness: Rediscovering the Pythagorean Plato’s golden mean number system. First published 2021 by
Scott Olsen. P.O. Box 3612. Ocala, Florida 34478. Kindle Edition (sold online by Amazon.com Service LLC). 195
pages. (Kindle Press. 19.95 USD).
[4] Scott Olsen, L. Marek-Crnjac, Je-Huan He and M. S. El Naschie. A Grand Unification of the Sciences, Arts and
Consciousness: Rediscovering the Pythagorean Plato’s Golden mean number system. Journal of Progressive
Research in Mathematics. ISSN: 2395-0218. Vol. 16, issue 2. Published online. May 13, 2029, pp. 2888-2931.
(Note that Ref. 3 is the extended version of Ref. 4).
[5] M. S. El Naschie. L. Da Vinci’s Turbulence, Multi-dimensional golden spinors, Prandtl’s Boundary layer and
quantum mechanical spin are all implied by the golden mean number system. European Journal of Applied
Science. Vol. No 6. December 25 [2021] pp. 134-136.
[6] G. ‘tHooft. Under the Spell of Gauge Principle. World Scientific . Singapore. [1984].
[7] Roger Penrose. The Road to Reality. Jonathan-Cape - London [2004].
[8] Joseph Polchinski. String Theory. Volume 1. Cambridge University Press. Cambridge [1998].
[9] Alain Connes. Non-Commutative Geometry. Academic Press. San Diego [1994].
[10] K. Becker, M. Becker and J. H. Schwarz. String Theory and M-Theory. Cambridge University Press -
Cambridge [2007].
[11] Gerardus ‘tHooft. In Search of the Ultimate Building Blocks. Cambridge University Press, Cambridge [1987].
[12] Lee Smolin. Three Roads to Quantum Gravity. Weidenfeld and Nicholson, London [2009].
[13] S. Hawking and R. Penrose. The Nature of Space and Time. Princeton University Press, Princeton and Oxford
[1996].
[14] M. S. El Naschie, O. E. Rossler and I. Prigogine. Pergamo. Quantum Mechanics and Chaotic Fractals. Elsevier
Science Ltd. Oxford - New York [1995].
[15] M. S. El Naschie. A review of E-Infinity theory and the mass spectrum of high energy particle physics. C, S &
F. Vol. 19. Issue 1. January [2004] pp. 209-236.
[16] David Gross and Carlo Rovelli. String Theory or Loop Gravity? Video on YouTube.
https://YouTube.com/watch? V=AUYYIRSRPZW. December 19, 2021. (See also Mohamed El Naschie facebook.
https=//www.Facebook.com/profile. php?id=100007609216640.
[17] Thomas Nowotny and Manfred Raquardt. Pre-geometric concept on graphs and cellular networks as
possible models of space-time at the Planck scale. C, S & F 19 (1999) 469.
[18] M. S. El Naschie. Feigenbaum scenario for Turbulence and Cantorian E-Infinity theory of high energy particle
physics. C, S & F [2007] Vol 32. Issue 3 pp. 911-915.
[19] G. Ord. Transfinite physics - Trading the path of Cantor and Einstein. In celebration of Mohamed Salah El
Naschie’s 69th birthday). C, S & F. Volume 25. Issue 4. August 2005.
[20] Peter Weibel, Garnet Ord, Otto E. Rossler. Spacetime physics and fractality. Festschrift in honor of Mohamed
El Naschie on the occasion of his 60th birthday. Springer Verlag. Vienna and New York [2005].
[21] M. S. El Naschie. Proving superstrings theory using loop quantum mechanics (gravity). C, S & F. Volume 26
[2005] pp. 43-45.
[22] F. David Peat. (1997) Infinite Potential: The Life and Times of David Bohm. Basic Books, New York, 133.
Illustrated edition. (November 1997).
Page 6 of 6
392
European Journal of Applied Sciences (EJAS) Vol. 10, Issue 1, February-2022
Services for Science and Education – United Kingdom
[23] Mohamed S. El Naschie. Elements of a new set theory based on quantum mechanics with applications in high
energy quantum physics and cosmology. International Journal of High Energy Physics. Volume 4 issue 6
December 2019. pp. 65-74.
[24] Mohamed S. El Naschie and Leila Marek-Crnjac. Set Theoretical foundation of quantum mechanics, NASA’s
EM Drive Technology and Minimal surface interpretation of the state vector reduction of the quantum wave
collapse. Chaos and Complexity Letters. Volume 12. No. 2 [2018] pp. 85-100.
[25] M. S. El Naschie. From Nikolay Umov via Albert Einstein’s to Dark Energy density of
the cosmos . World Journal of Mechanics [2018]. Volume 8 No. 4. April 2018 pp. 73-81.
[26] Mohamed S. El Naschie. Symmetria Massima of the fractal M-Theory via the Golden mean number system - A
new language for a deep dialogue between Man and Nature. International Journal of Artificial Intelligence and
Mechatronics. Volume 7. Issue 3, ISSN 2329-5121. [2018] pp. 11-14.
[27] M.S. El Naschie. The Quantum Gravity Immirzi parameter - A general physical and topological interpretation.
Gravitation and Cosmology. Vol. 19. Issue 3 [2013] pp 151-155.
Note: it is more accurate to refer to this parameter as Barbero-Immirzi parameter.
[28] Mohamed S. El Naschie. On a fractal version of Witten’s M-Theory. International Journal of Astronomy and
Astrophysics. Vol. 6 No. 2 June 2016, pp 135-144.
[29] L. N. Bronstein and K. A. Semendjaew. Handbuch der Mathematik (pocket book of mathematics) Harri
Deutsch Publishing. Zurich und Frankfurt [1965] (in German).
2 E Kmc = 2 E mc = g
2 E (21/ 22)mc =