Chaos, Solitons and Fractals 35 (2008) 252–256

www.elsevier.com/locate/chaos

Quantum gravity unification via transfinitearithmetic and geometrical averaging

M.S. El Naschie *

Department of Physics, University of Alexandria, Egypt

Donghua University, Shanghai, PR China

Department of Astrophysics, University of Cairo, Egypt

Dedicated to the memory of Prof. Dr. Thomas Barta

Abstract

In E-Infinity theory, we have not only infinitely many dimensions but also infinitely many fundamental forces. How-ever, due to the hierarchical structure of eð1Þ spacetime we have a finite expectation number for its dimensionality andlikewise a finite expectation number for the corresponding interactions. Starting from the preceding fundamental prin-ciples and using the experimental findings as well as the theoretical value of the coupling constants of the electroweakand the strong forces we present an extremely simple averaging procedure for determining the quantum gravity unifi-cation coupling constant with and without super symmetry. The work draws heavily on previous results, in particular apaper, by the Slovian Prof. Marek-Crnjac [Marek-Crnjac L. On the unification of all fundamental forces in a funda-mentally fuzzy Cantorian eð1Þ manifold and high energy physics. Chaos, Solitons & Fractals 2004;4:657–68].� 2007 Elsevier Ltd. All rights reserved.

1. Introduction

In [1], Marek-Crnjac gave a very lucid as well as elementary discussion of Grand as well as Quantum Gravity uni-fication of all fundamental forces. In particular, Crnjac used unheard of simple numerical procedures to arrive at sur-prisingly accurate predictions [1]. In addition, Crnjac gave great effort to give sophisticated and non-classical theoreticaljustifications of the numerical results.

In the present work we will follow Marek-Crnjac’s spirit of simplicity [1] while aiming at an exact determination ofthe coupling constant rather than mere acceptable approximations.

2. The expectation number of fundamental interactions

It is well known that eð1Þ is formally infinite dimensional although it is uniquely determined by two finite expectationvalues for the topological and Hausdorff dimensions, namely [2–4]

0960-0779/$ - see front matter � 2007 Elsevier Ltd. All rights reserved.doi:10.1016/j.chaos.2007.07.019

* Address for correspondence: P.O. Box 272, Cobham, Surrey KT11 2FQ, United Kingdom.E-mail address: Chaossf@aol.com

M.S. El Naschie / Chaos, Solitons and Fractals 35 (2008) 252–256 253

DT ¼ 4

and

hdci ¼� hni ¼ 4þ /3

where / ¼ ðffiffiffi5p� 1Þ=2.

Less well known is the fact that in E-Infinity the number of fundamental interactions is also infinite. Neverthelessand similar to dimensions we have the following finite expectation numbers:

Excluding gravity the expectation number is [2–4]

n1 ¼ ð4þ /3Þ � 1 ¼ 3þ /3

Including gravity and considering the electric force and the magnetic force to be two different forces, the expectationnumber is [2–4]

n2 ¼ ð4þ /3Þ þ 1 ¼ 5þ /3

Finally, excluding gravity from n2, the expectation number is

n3 ¼ n2 � 1 ¼ n1 þ 1 ¼ 4þ /3

3. The electroweak and the strong interactions

3.1. The experimental value of the coupling constant

Following, for instance, Peskin [5] modern experimental data for the electroweak and strong coupling constants aregiven by

a1 ¼ 1=98:4;

a2 ¼ 1=29:6

and

a3 ¼ 1=8:5

Using different notations for the inverse coupling and noting that the Clebsh coefficient is C = 5/3 one could write that

�a1 ¼ 59:04

�a2 ¼ 29:6

�a3 ¼ 8:5

Let us reconstruct the inverse electromagnetic fine structure constant from the above. This is easily found to be [2–4]

�ao ¼ ðCÞð�a1Þ þ �a2 þ �a3 ¼ ð5=3Þð59:04Þ þ 29:6þ 8:5 ¼ 98:4þ 29:6þ 8:5 ¼ 128þ 8:5 ¼ 136:5

This is just 0.5 short of the well-known integer value of �ao namely 137. We recall that the eð1Þ theoretical value�ao ¼ 137þ ko ¼ 137:0820393 could easily be found using well-defined rules which we shall discuss in the following section.

3.2. Theoretical consideration regarding the reconstruction of the electromagnetic fine structure constant of the electroweak

and strong coupling charges

Following E-Infinity standard analysis �ao may be found by noting that C ¼ 5=3 should be replaced by1=/ ¼ 1:618033989 as follows [2–4]:

�ao ¼ ð1=/Þð�a1Þ þ �a2 þ �aS þ �aQG ¼ ð1=/Þð�a1sÞ þ �a2 þ �a3

where

�a1 ¼ 60

�a2 ¼ ð1=2Þð�a1Þ ¼ 30

�aS ¼ 9

�aQG ¼ 1

254 M.S. El Naschie / Chaos, Solitons and Fractals 35 (2008) 252–256

and

�a3 ¼ �aS þ �aQG ¼ 10

We note that �aQG ¼ 1 is the theoretical maximal coupling of the hypothetical Planck mass.Inserting in �ao one finds

�ao ¼ ð1=/Þð60Þ þ 30þ 9þ 1 ¼ ð97þ koÞ þ 30þ 10 ¼ 137þ ko ¼ 137:082039325

where ko ¼ /5ð1� /5Þ ¼ 0:082039325.The exact experimental value of �ao is found by simple E-Infinity projection as follows:

�aoðexperimentalÞ ¼ ð�ao � koÞ=ðcosðp=�aoÞÞ ¼ 137:0359852

It is clear from above that the experimental value for �ai is fairly close to the E-Infinity theoretical one. Needless to say,�a4 ¼ 1 has never been experimentally determined in any direct way.

We could also draw several more fundamental conclusions from the reconstruction of �ao. For instance, excluding thestrong force, one finds the electroweak fine structure constant:

�aew ¼ �ao � �a3 ¼ ð137þ koÞ � 9 ¼ 128þ ko ’ 128

This is in rather good agreement with the experimental evidence. In addition, one notices that without the theoreticalvalue of the Planck mass coupling �a4 ¼ 1 we would have �aew ’ 127 while �ao ¼ 137þ ko would be reduced to�ao ¼ 136þ ko.

In other words, the coupling to the Planck mass field plays, at least theoretically, an important role in understandingthe experimental value of �ao ’ 137. In fact, this point may become more pronounced when contemplating the structureof the P-Adic expansion of �ao ’ 137, namely [2–4]

kj37k2 ¼ 28�1

þ 24�1

þ 21�1

¼ 27 þ 23 þ 20 ¼ 128þ 8þ 1 ¼ 136þ 1 ¼ 137:

4. Exact calculation of the inverse quantum gravity coupling by transfinite arithmetic mean averaging

Naively speaking, averaging between a set of unequal elements is finding a pseudo element which could be describedas belonging to all elements. Applying this simplistic notion to the different coupling constants of the various funda-mental interactions is obviously equivalent to finding the common unification coupling charge.

Let us start by determining the simplest case. This is paradoxically the exact super symmetric quantum gravity con-stant. Since in this case the total charge must be equal, all what we need for all the five fundamental constants is todivide �ao by the expectation number of all the infinitely many fundamental interactions which are not simply 5 but5þ /3 as explained in earlier sections. Consequently, one finds

�ags ¼�ae þ �am þ �a2 þ �as þ �aQG

n2

¼ ð1=/Þð�a1Þ þ �a2 þ �a3 þ �a4

5þ /3

¼ 137þ ko

5þ /3

¼ 26þ k

which is the well-known exact result of E-Infinity theory and in fair agreement with almost all approximate resultsfound in the literature [2–4] which range between 24 and 27. The preceding result could be found in a different wayby assuming that gravity is excluded from being counted as a fundamental force and setting C ¼ 1. The expectationnumber of the interaction is in this case equivalent to the E-Infinity spacetime dimension Dð4Þ ¼ 4� k rather thanð5þ /3Þ; ð4þ /3Þ or 4. Proceeding this way one finds

�ags ¼�a1 þ �a2 þ �a3 þ �a4

ðD4 ¼ 4� kÞ ¼ 60þ 30þ 9þ 1

4� k¼ 100=ð4� kÞ ¼ 26þ k

where k ¼ /3ð1� /3Þ ¼ 0:18033989.Next we look at the case of non-super symmetric coupling to find �ag. We proceed as in the first analysis but consider

the electric force and the magnetic force to be only one force and exclude gravity from our calculations. Then dividingagain the total inverse charge by 3þ /3 and not by three forces one finds

�ag ¼ ð137þ koÞ=ð3þ /3Þ ¼ 42þ 2k:

In summary, �ag and �ags could be found from simple transfinite averaging to be

M.S. El Naschie / Chaos, Solitons and Fractals 35 (2008) 252–256 255

�ags ¼X4

i¼1

�ai=ðDð4Þ � kÞ ¼ 26þ k

and

�ag ¼X4

i¼1

�ai=ð2þ 2kÞ ¼ 42þ k:

Alternatively, we can work with �ao as the inverse total charge and find that

�ags ¼ �ao=ð5þ /3Þ ¼ 26þ k

and

�ag ¼ �ao=ð3þ /3Þ ¼ 42þ 2k:

5. Efficient approximate determination of the quantum gravity inverse coupling

Although there is no need to find �ags approximately because we have found the exact values, one can gain deeperinsight into the meaning of it all by considering the following simple procedure based on geometrical mean averaging.

Let us start by �ags. Taking the geometrical mean averaging of our �a1; �a2 and ��a3 ¼ �a3 þ �a4 ¼ 10, one finds

�ags ’ffiffiffiffiffiffiffiffiffiffiffiffiffi�a1�a2 ��a3

p¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffið30Þð60Þð10Þ

pffi 26:2

This is an excellent approximation to �ags ¼ 26þ k ¼ 26:1803.Next we look at �ag of the non-super symmetric QG coupling. To do that we must use the following simple formula

which ignores �a3 and �a4

�agi ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffið�a1Þð�a2Þ

p¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffið60Þð30Þ

p¼ 42:426 ’ 42

This is quite close to �ag ¼ 42þ 2k. On the other hand this procedure could be applied to all the three intersections:

�ag2 ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffið�a2Þð�a3Þ

p¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffið30Þð10Þ

pffi 17:3

and

�ag3 ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffið�a1Þð�a3Þ

p¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffið60Þð10Þ

pffi 24:49

Subsequently, one finds once again an accurate estimation of the super symmetric inverse coupling

�ags ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffið�ag1Þð�ag2Þð�ag3Þ3

q¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffið42:4Þð17:3Þð24:49Þ

pffi 26:18

in remarkable agreement with the exact value �ags ¼ 26:18033989.

6. Discussion and conclusion

Unification seems to be closely connected to the electromagnetic fine structure constant �ao ’ 137 in a variety ofways. Consider, for instance, superstring theory which is probably the most popular theory of unification at the presenttime. Specifically, let us consider the anomaly cancellation condition [3,4]

nH þ 29nt � nv ¼ 273

where nH is the hyper multiplet, nt the tensor multiplet and nv the vector multiplet. This representation is valid for boththe E8E8 and the SO(32) heterotic super strings. Setting nH ¼ 625; nt ¼ 1 and nv ¼ 381 one finds [3,4]

625þ 29� 381 ¼ 273

Let us group nH and the 29 together and add one to the left and right-hand sides of the equality. That way one finds thatð625þ 29Þ � ð381þ 1Þ ¼ 273þ 1.

Dividing by 2 one finds

327� 190 ¼ 137:

256 M.S. El Naschie / Chaos, Solitons and Fractals 35 (2008) 252–256

Adding 12 to the positive and to the negative terms of the left-hand side one finds

339� 202 ¼ 137:

The above equation means the compactified jSLð2; 7Þcjminus the compactified or fractal Riemaniann tensor componentin seven dimensions Rð7Þc is equal to the electromagnetic fine structure constant.

This is the boundary manifold statement of the Holographic principle. To find the more familiar bulk statement, we addthe number of compactified automorphism of Klein’s modular curve Cð7Þc namely 169 instead of the 168 and find that

ð327þ 169Þ � ð190þ 169Þ ¼ 137:

That means

496� 359 ¼ 137

or more explicitly

496� ð339þ 20Þ ¼ 137:

In other words, we have our familiar application of the holographic principles for determining the fine structure con-stant, namely [3,4]

jE8E8j � ½jSLð2; 7Þc þ Rð4Þj� ¼ �ao

The present work presents yet another example for the power of the geometrical topological method when coupled tononlinear dynamics, chaos and fractals. In this theory which is known as E-Infinity or Cantorian spacetime theory theconcepts state like particles, probability, isometries and dimensions are in various subtle ways exchangeable.

Utilizing this new found freedom and flexibility we were able to determine the exact value of the inverse quantumgravity coupling constant with unheard of simplicity. For instance, �ags is found exactly from

�ags ¼X4

i¼1

�ai=ðDð4Þ � kÞ ¼ 26þ k

or equivalently

�ags ¼ ð�aoÞ=ð5þ /3Þ ¼ 26þ k

A third noteworthy formula which is almost exact is the following

�ags ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffið�ag1Þð�ag2Þð�ag3Þ3

q

where

�ag1 ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffið�a1Þð�a2Þ2

p

�ag2 ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffið�a2Þð��a3Þ2

q

and

�ag3 ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffið�a1Þð��a3Þ2

q

with �a1 ¼ 60; �a2 ¼ 30 and ��a3 ¼ 9þ 1 ¼ 10 one finds that

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffið42:4Þð17:3Þð24:49Þ3

pffi 26:18

This is an excellent approximation of the exact value �ags ¼ 26þ k ¼ 26:18033989.

References

[1] Marek-Crnjac L. On the unification of all fundamental forces in a fundamentally fuzzy cantorian manifold and high energy physics.Chaos, Solitons & Fractals 2007;20(4):669–82. May.

[2] Ji-Huan He, Goldfain E, Sigalotti L, Mejias A. Beyond the 2006 physics Nobel Prize for COBE. Shanghai, PR China; 2006, ISBN988-97681-9-4/0.4.

[3] El Naschie MS. Elementary prerequisites of E-Infinity. Chaos, Solitons & Fractals 2006;30(3):570–605.[4] El Naschie MS. A review of applications and results of E-Infinity theory. Int. J. Nonlinear Sci Numer Simulat 2007;8(1):1–20.[5] Peskin ME. Supersymmetry: the next spectroscopy. In: Buschhorn G, Wess J, editors. Fundamental physics – Heisenberg and

beyond. Berlin: Springer; 2004. p. 99–133.