- Email: [email protected]
Asymptotic freedom and unification in a golden quantum field theory
Available online at www.sciencedirect.com
Chaos, Solitons and Fractals 36 (2008) 521–525 www.elsevier.com/locate/chaos
Asymptotic freedom and unification in a golden quantum field theory M.S. El Naschie
*
Department of Physics, University of Alexandria, United Kingdom Department of Astrophysics, University of Cairo, Egypt
Abstract By harmonizing the perturbation equations of renormalization, we find a set of exact and simple equations for determining the coupling constant of super symmetric unification. Using this exact equation, we demonstrate exact quarks confinement as energy tends toward the Planck energy. We conclude that confinement can be demonstrated without involving asymptotic freedom. Ó 2007 Elsevier Ltd. All rights reserved.
1. Introduction In virtually one stroke, G. ‘tHooft, D. Politzer, D. Gross and F. Wilczek were able to make quark confinement plausible and justify Feynman parton model as well as current algebra [1,2]. The key to all of that was the derivation of higher order corrections to the lower order equation: aS ðQ2 Þ ¼
ð33 2nf Þ log Q2 =k2 : 12p
Thus taking second order terms, one finds ( ) 36p 1 153 19nf log log Q2 =k2 2 aS ðQ Þ ¼ : ð33 2nf Þ log Q2 =k2 3 ð33 2nf Þ2 12 log Q2 =k2 We see that for Q2 ! 1 the coupling vanishes. This behavior is what is termed asymptotic freedom for which Gross, Wilczek and Politzer shared the Nobel prize for physics [3]. In the present work we give a different picture for confinement and prove rigorously without perturbation and without invoking infinite momentum for which aS is supposed to vanish [1]. Instead strong coupling in our derivation is replaced by 3 inverse couplings aS1 ¼ 8, aS2 ¼ 1 and aS3 ¼ a4 ¼ 1. This seemingly harmless modification is crucial for our exact analysis. Equally crucial is our harmonization procedure which replaces the various experimental values of the coupling constant at the electroweak scale by the idealized
*
Address for correspondence: PO Box 272, Cobham, Surrey KT11 2FQ, UK. E-mail address: [email protected]
0960-0779/$ - see front matter Ó 2007 Elsevier Ltd. All rights reserved. doi:10.1016/j.chaos.2007.09.004
522
M.S. El Naschie / Chaos, Solitons and Fractals 36 (2008) 521–525
and theoretically exact value namely a1 ¼ 60 of electromagnetism, a2 ¼ ð1=2Þð a1 Þ ¼ 30 for the weak force, a3 ¼ 9 for the strong interaction where a3 ¼ 8 þ 1 and a4 ¼ 1 for Planck mass coupling [4–7]. At the electroweak all three could be absorbed in a ¼ a3 þ a4 ¼ ð8 þ 1Þ þ 1 ¼ 10 so that we can readily reconstruct the low energy electromagnetic inverse fine structure constant [4–7] a0 ¼ ð1=/Þa1 þ a2 þ a3 þ a4 ¼ ð1:618033989Þð60Þ þ 30 þ 9 þ 1 ¼ ð127 þ k 0 Þ þ 10 ¼ 137 þ k 0 ¼ 137:082039325; pffiffiffi where 1=/ ¼ ð 5 þ 1Þ=2 has replaced the Clebsch coefficient 5/3 and k0 = /5(1 /5). We stress once more that a4 ¼ aQG ¼ 1 is an important coupling and this is reinforced by the 2-Adic expansion of a0 ¼ 137 k137k2 ¼ k128 þ 8 þ 1k2 1 as well as the result of solving the Kaluza–Klein equation and finding that a(E) = 1 for the compactified radius R = ‘ Planck. Finally we will be making use of the realization that ‘n(Mu/Ml) = 32 + 2k for grand unification. The physical nonidealized explanation is that for Mu = (10)16 Gev and Mu = 91 Gev we obtain approximately the abovementioned value. 2. The exact unification equations Let us start by establishing an accurate estimation of the quantum gravity coupling constants using very elementary mathematical tools requiring hardly any mathematical skills. To search for a common coupling geometrical averaging is in essence a unification. This is self evident when one ffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi thinks about the meaning of averaging deeply. The excellent closeness of 3 a1 a3 þ a4 Þ to exact the value a2 ð ags ¼ 26 þ k ¼ 26:18033989 should abolish any left doubt about this fact. Thus we have [8] pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ags ’ 3 ð60Þð30Þð10Þ ’ 26:20741394: In fact not only ags but an entire hierarchy of dimensions and couplings could be generated from a most elementary, though again quite deep realization that the ratio between the total number of massless states of heterotic string theory and the 45 massless unification gauge bosons of the unification Lie group SO(10) is a fundamental coupling. Thus writing N 0 Heterotic 8064 ¼ ¼ 179:2; jSOð10Þj 45
pffiffiffi and downscaling the result using (/)n as the quantum spacetime scaling exponent where / ¼ ð 5 1Þ=2 one finds the following hierarchy ð179:2Þð/Þ ¼ 110:7516908; ð. . .Þð/2 Þ ¼ 68:4483 927 ffi a0 =2; ð. . .Þð/3 Þ ¼ 42:3 ffi ag ; ð. . .Þð/4 Þ ¼ 26:14492769 ffi ag ; ð. . .Þð/5 Þ ¼ 16:1584 ’ Dð16Þ ; ð. . .Þð/6 Þ ¼ 9:9864 ’ Dð10Þ ; ð. . .Þð/7 Þ ¼ 6:1719 ’ Dð6Þ ; ð. . .Þð/8 Þ ¼ 3:81449 ’ Dð4Þ : We see that we have generated approximate values for the entire dimensional hierarchy of heterotic superstring theory in addition to a0 =2 and the non-super symmetric coupling of quantum gravity ag ’ 42. Thus we have established that ags and ag must be in the region of 26 and 42 respectively. Now before giving our exact golden quantum field theory result, we reproduce first the result for ag and ags obtained using conventional quantum field theory. The first order renormalization equation gives au as [4–8] au ¼ au ðZÞ þ
bi M u ‘n : 2p M z
M.S. El Naschie / Chaos, Solitons and Fractals 36 (2008) 521–525
523
For a1 ðZÞ ¼ 59; a2 ðZÞ ¼ 29:6 and as ðZÞ ¼ 8:5, we have bi = 33/5, b2 = 1 and bs = 3 respectively. All bi are for a super-symmetric theory. Inserting in au one finds for mu = (10)16 Gev and mz = 91 Gev that ‘n
Mu ¼ 32:33050198; Mz
and the following three approximations for au 33 au ¼ a1 ð32:33050198Þ ¼ 59 33:96069074 ¼ 25:03930926; 10p 32:33050198 ao ¼ a2 ¼ 29:6 5:1455620 ’ 24:454438; 2p and au ¼ as þ
ð3Þð32:33050198Þ ¼ 8:5 þ 15:43667761 ¼ 23:93667761: 2p
Thus far from being a perfect unification theory all three results are different. Now nothing could speak for the simplicity and exactness of our theory than to contrast the above computation with ours using the golden quantum field theory. The general formula in this case is Mu ¼ ai bi ð32 þ 2kÞ; Mz pffiffiffi where k = /3(1 /3) and / ¼ ð 5 1Þ=2 is the famous golden mean of KAM theorem. Next we use our idealized exact theoretical values au ¼ ai bi ‘n
Dð26Þ 3 23 þ k ; ¼ Dð26Þ 4 22 þ k 1 a2 ¼ 30; d2 ¼ bi ¼ ; 8 þ 2/3 a3 þ a4 ¼ 9 þ 1 ¼ 10; d3 ¼ b3 ¼ 1=2:
a1 ¼ 60;
d1 ¼ bi ¼
Inserting in au one finds ags ¼ 26 þ k ¼ 26:1803398; and that for all three cases 2 3 2 3 a2 d2 6 7 6 7 ags ¼ 4 a3 5 þ 4 d3 5ða0 Þð/Þ3 : a1 d1 For ag on the other hand, one finds ag ¼ ða3 þ a4 Þ þ ðd ¼ 1Þða0 Þð/Þ3 ¼ 10 þ ð32 þ 2kÞ ¼ 42 þ 2k; which is the exact value. It is both instructive and economical to combine all the ai and super symmetry in one equation for au Proceeding that way one finds 1=2 4 X M planck Mu ai ‘n a0 ¼ þ ‘n : M Mz i¼1 Here we set the scale of Mplanck to be 10 Gev which gives a dimensionless reduced Planck energy scale while setting Mu = 1016 Gev, namely the scale of a unification magnetic monopole and mz = 91 Gev as usual. Thus we have 4 X
ai ¼ 60 þ 30 þ 10 ¼ 100;
i
‘n
ð10Þ19 ¼ ‘nð10Þ18 ¼ 41:446 ’ 42; 10
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M.S. El Naschie / Chaos, Solitons and Fractals 36 (2008) 521–525
and sffiffiffiffiffiffiffiffiffiffiffiffi ð10Þ16 ‘n ¼ 16:1652 ’ 16: 91 Consequently, au ’ 100 ½42 þ 16 ¼ 100 58 ¼ 42 ¼ ag : In the case of super symmetry on the other hand, we have M au ¼ 100 42 þ ‘n u ¼ 100 ½42 þ 32 ¼ 10 ½74 ¼ 26 ¼ ags : Mz
3. Discussion and conclusion A very simple and exact set of equations for the inverse unification coupling has been obtained by transfinitely harmonizing the basic renormalization groups of quantum field theory in a way similar to Feigenbaum golden mean renormalization groups ags ¼ a1 þ d1 ða0 Þ/3 ; ags ¼ a2 þ d2 ða0 Þ/3 ; ags ¼ ða3 þ a4 Þ þ d3 ða0 Þ/3 ; and ag ¼ ða3 þ a4 Þ þ d4 ða0 Þ/3 ; where 23 þ k ¼ 1:045084972; 22 þ k 1 d2 ¼ ¼ 8:472135954; 8 þ 2/3 d3 ¼ 1=2; d1 ¼
and d4 ¼ 1: The so-obtained solution vector is exact and completely consistent with all our previously obtained results [4–8] 2 3 26 þ k 6 26 þ k 7 6 7 a0 ¼ 6 7: 4 26 þ k 5 42 þ k Using any of these equations it is easy to show that when our energy scale goes to the Planck scale [4–8] M u ¼ M z ¼ M ) M planck then as = a3 ) 1 which means it is impossible to force quarks out of their confinement no matter how hard one hits them in a super-collider in full agreement with the confinement hypothesis. In fact at this point there are no quarks because spacetime suffers a phase transition to a Planck Aether consisting merely of Planck black holes elementary particles. Thus although totally consistent with the experimental observation and agrees in outcome with the asymptotic freedom proposed by other theories, it does indeed differ in philosophy and the details of the underlying physical picture. Maybe it is possible to elucidate the confinement mechanism using super gravity as following: We know that the symmetry group of super gravity is OSP(1/4) which is an ortho-symplictic Lie group. The dimension of this group is given by jOSPð1=4Þj ¼ 14:
M.S. El Naschie / Chaos, Solitons and Fractals 36 (2008) 521–525
525
Let us examine the ratio of the order of this group to that of the SL(2, 7) of the non-compactified holographic boundary of the E8E8 manifold. This is given by jSL(2, 7)j = 336 and consequently SLð2; 7Þ 336 OSPð1=4Þ ¼ 14 ¼ 24: Noting that (24)2 = 576, which is 548 + 28 and representing the almost maximal sum of any consistent Exceptional Lie symmetry groups hierarchy, [7] we must conjecture that 24 is the number of instantons of the E8E8 manifold corresponding to Ka¨hler K3 of heterotic string theory compactification although we know that 26 + k is the exact solution and corresponds to ð26 þ kÞ2 ¼ ð5Þða0 Þ ¼ 685:4101965. Consequently, a transfinite correction analogous to quantum field theory radiation corrections must show that 24 ! 26 + k. If this is correct and it is correct, then 14 may be guessed to transform as follows: jOSPð1=4Þj ! ða0 Þ=10; which is the signature of the fuzzy counterpart of K3 Ka¨hler. The two preceding transformations imply then that 336 ! ða0 Þð26 þ kÞ=10. This means 336 ! 336 þ 16k þ 20: It is not surprising that we did not obtain only the compactified 336 which is 336 + 16k where k = 0.18033989, but we obtained also Einstein’ s gravity R(4) = 20. Our exact transfinite coupling is thus nothing else but ags ¼ 26 þ k given by jSLð2; 7Þjc þ 20 336 þ 16k þ 0 ¼ 26:18033989: ¼ jOSPð1=4Þjc 13:7082039325 The crucial point is now the following: The jSL(2, 7)jc have theoretically infinite elements but the expectation value is finite one given by 336 + 16k ’ 339. Geometrically, it is a compactified Klein modular curve resembling one of an Escher’s famous pictures. By contrast, the ortho-normal group has the same dimension as G2 of the exceptional Lie groups jSOPð1=4Þj ¼ jG2 j ¼ 14: In principle we could let jG2j go to the largest exceptional group. We could also exceed E8 and obtain an infinite dimensional Lie algebra. Thus in principle jSLð2; 7Þjc þ Rð4Þ ¼ ags ¼ 26:18033989 jSOPð1=4Þjc could be made to approach zero. If ags ¼ 0, then ags = 1 and since we have a point of unification, then everything is completely frozen in its place, including quarks. In fact as we mentioned many times before we are on the doorstep of a new universe, the Planck universe. Hitting the quarks extremely hard, transmutate them to the Planck masses rather than make them free. Leaving the quarks in peace, they become asymptotically free. It is for non-Abelian theories like in social sciences: violence breeds contra violence. References [1] Yndura´in FJ. The theory of quark and gluon interaction. Berlin: Springer; 1992. [2] Davies P, editor, The new physics. Cambridge; 1989. [3] ‘tHooft G. When was asymptotic freedom discovered? Or the rehabilitation of quantum field theory. Nucl Phys B (Proc Suppl) 1999;74:418–25. [4] El Naschie MS. High energy physics, the standard model from the exceptional Lie groups and quarks confinement. Chaos, Solitons & Fractals 2008;36:1–17. [5] El Naschie MS. Quantum E-Infinity field theoretical derivation of Newton’s gravitational constant. Int J Nonlinear Sci Numer Simul 2007;8(3):469–74. [6] El Naschie MS. Elementary prerequisites for E-Infinity. Chaos, Solitons & Fractals 2006;30(3):579–605. [7] El Naschie MS. Exceptional Lie groups hierarchy and some fundamental high energy physics equations. Chaos, Solitons & Fractals 2008;35:82–4. [8] El Naschie MS. Quantum gravity unification via transfinite arithmetic and geometrical averaging. Chaos, Solitons & Fractals 2008;35:252–6.
Chaos, Solitons and Fractals 36 (2008) 521–525 www.elsevier.com/locate/chaos
Asymptotic freedom and unification in a golden quantum field theory M.S. El Naschie
*
Department of Physics, University of Alexandria, United Kingdom Department of Astrophysics, University of Cairo, Egypt
Abstract By harmonizing the perturbation equations of renormalization, we find a set of exact and simple equations for determining the coupling constant of super symmetric unification. Using this exact equation, we demonstrate exact quarks confinement as energy tends toward the Planck energy. We conclude that confinement can be demonstrated without involving asymptotic freedom. Ó 2007 Elsevier Ltd. All rights reserved.
1. Introduction In virtually one stroke, G. ‘tHooft, D. Politzer, D. Gross and F. Wilczek were able to make quark confinement plausible and justify Feynman parton model as well as current algebra [1,2]. The key to all of that was the derivation of higher order corrections to the lower order equation: aS ðQ2 Þ ¼
ð33 2nf Þ log Q2 =k2 : 12p
Thus taking second order terms, one finds ( ) 36p 1 153 19nf log log Q2 =k2 2 aS ðQ Þ ¼ : ð33 2nf Þ log Q2 =k2 3 ð33 2nf Þ2 12 log Q2 =k2 We see that for Q2 ! 1 the coupling vanishes. This behavior is what is termed asymptotic freedom for which Gross, Wilczek and Politzer shared the Nobel prize for physics [3]. In the present work we give a different picture for confinement and prove rigorously without perturbation and without invoking infinite momentum for which aS is supposed to vanish [1]. Instead strong coupling in our derivation is replaced by 3 inverse couplings aS1 ¼ 8, aS2 ¼ 1 and aS3 ¼ a4 ¼ 1. This seemingly harmless modification is crucial for our exact analysis. Equally crucial is our harmonization procedure which replaces the various experimental values of the coupling constant at the electroweak scale by the idealized
*
Address for correspondence: PO Box 272, Cobham, Surrey KT11 2FQ, UK. E-mail address: [email protected]
0960-0779/$ - see front matter Ó 2007 Elsevier Ltd. All rights reserved. doi:10.1016/j.chaos.2007.09.004
522
M.S. El Naschie / Chaos, Solitons and Fractals 36 (2008) 521–525
and theoretically exact value namely a1 ¼ 60 of electromagnetism, a2 ¼ ð1=2Þð a1 Þ ¼ 30 for the weak force, a3 ¼ 9 for the strong interaction where a3 ¼ 8 þ 1 and a4 ¼ 1 for Planck mass coupling [4–7]. At the electroweak all three could be absorbed in a ¼ a3 þ a4 ¼ ð8 þ 1Þ þ 1 ¼ 10 so that we can readily reconstruct the low energy electromagnetic inverse fine structure constant [4–7] a0 ¼ ð1=/Þa1 þ a2 þ a3 þ a4 ¼ ð1:618033989Þð60Þ þ 30 þ 9 þ 1 ¼ ð127 þ k 0 Þ þ 10 ¼ 137 þ k 0 ¼ 137:082039325; pffiffiffi where 1=/ ¼ ð 5 þ 1Þ=2 has replaced the Clebsch coefficient 5/3 and k0 = /5(1 /5). We stress once more that a4 ¼ aQG ¼ 1 is an important coupling and this is reinforced by the 2-Adic expansion of a0 ¼ 137 k137k2 ¼ k128 þ 8 þ 1k2 1 as well as the result of solving the Kaluza–Klein equation and finding that a(E) = 1 for the compactified radius R = ‘ Planck. Finally we will be making use of the realization that ‘n(Mu/Ml) = 32 + 2k for grand unification. The physical nonidealized explanation is that for Mu = (10)16 Gev and Mu = 91 Gev we obtain approximately the abovementioned value. 2. The exact unification equations Let us start by establishing an accurate estimation of the quantum gravity coupling constants using very elementary mathematical tools requiring hardly any mathematical skills. To search for a common coupling geometrical averaging is in essence a unification. This is self evident when one ffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi thinks about the meaning of averaging deeply. The excellent closeness of 3 a1 a3 þ a4 Þ to exact the value a2 ð ags ¼ 26 þ k ¼ 26:18033989 should abolish any left doubt about this fact. Thus we have [8] pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ags ’ 3 ð60Þð30Þð10Þ ’ 26:20741394: In fact not only ags but an entire hierarchy of dimensions and couplings could be generated from a most elementary, though again quite deep realization that the ratio between the total number of massless states of heterotic string theory and the 45 massless unification gauge bosons of the unification Lie group SO(10) is a fundamental coupling. Thus writing N 0 Heterotic 8064 ¼ ¼ 179:2; jSOð10Þj 45
pffiffiffi and downscaling the result using (/)n as the quantum spacetime scaling exponent where / ¼ ð 5 1Þ=2 one finds the following hierarchy ð179:2Þð/Þ ¼ 110:7516908; ð. . .Þð/2 Þ ¼ 68:4483 927 ffi a0 =2; ð. . .Þð/3 Þ ¼ 42:3 ffi ag ; ð. . .Þð/4 Þ ¼ 26:14492769 ffi ag ; ð. . .Þð/5 Þ ¼ 16:1584 ’ Dð16Þ ; ð. . .Þð/6 Þ ¼ 9:9864 ’ Dð10Þ ; ð. . .Þð/7 Þ ¼ 6:1719 ’ Dð6Þ ; ð. . .Þð/8 Þ ¼ 3:81449 ’ Dð4Þ : We see that we have generated approximate values for the entire dimensional hierarchy of heterotic superstring theory in addition to a0 =2 and the non-super symmetric coupling of quantum gravity ag ’ 42. Thus we have established that ags and ag must be in the region of 26 and 42 respectively. Now before giving our exact golden quantum field theory result, we reproduce first the result for ag and ags obtained using conventional quantum field theory. The first order renormalization equation gives au as [4–8] au ¼ au ðZÞ þ
bi M u ‘n : 2p M z
M.S. El Naschie / Chaos, Solitons and Fractals 36 (2008) 521–525
523
For a1 ðZÞ ¼ 59; a2 ðZÞ ¼ 29:6 and as ðZÞ ¼ 8:5, we have bi = 33/5, b2 = 1 and bs = 3 respectively. All bi are for a super-symmetric theory. Inserting in au one finds for mu = (10)16 Gev and mz = 91 Gev that ‘n
Mu ¼ 32:33050198; Mz
and the following three approximations for au 33 au ¼ a1 ð32:33050198Þ ¼ 59 33:96069074 ¼ 25:03930926; 10p 32:33050198 ao ¼ a2 ¼ 29:6 5:1455620 ’ 24:454438; 2p and au ¼ as þ
ð3Þð32:33050198Þ ¼ 8:5 þ 15:43667761 ¼ 23:93667761: 2p
Thus far from being a perfect unification theory all three results are different. Now nothing could speak for the simplicity and exactness of our theory than to contrast the above computation with ours using the golden quantum field theory. The general formula in this case is Mu ¼ ai bi ð32 þ 2kÞ; Mz pffiffiffi where k = /3(1 /3) and / ¼ ð 5 1Þ=2 is the famous golden mean of KAM theorem. Next we use our idealized exact theoretical values au ¼ ai bi ‘n
Dð26Þ 3 23 þ k ; ¼ Dð26Þ 4 22 þ k 1 a2 ¼ 30; d2 ¼ bi ¼ ; 8 þ 2/3 a3 þ a4 ¼ 9 þ 1 ¼ 10; d3 ¼ b3 ¼ 1=2:
a1 ¼ 60;
d1 ¼ bi ¼
Inserting in au one finds ags ¼ 26 þ k ¼ 26:1803398; and that for all three cases 2 3 2 3 a2 d2 6 7 6 7 ags ¼ 4 a3 5 þ 4 d3 5ða0 Þð/Þ3 : a1 d1 For ag on the other hand, one finds ag ¼ ða3 þ a4 Þ þ ðd ¼ 1Þða0 Þð/Þ3 ¼ 10 þ ð32 þ 2kÞ ¼ 42 þ 2k; which is the exact value. It is both instructive and economical to combine all the ai and super symmetry in one equation for au Proceeding that way one finds 1=2 4 X M planck Mu ai ‘n a0 ¼ þ ‘n : M Mz i¼1 Here we set the scale of Mplanck to be 10 Gev which gives a dimensionless reduced Planck energy scale while setting Mu = 1016 Gev, namely the scale of a unification magnetic monopole and mz = 91 Gev as usual. Thus we have 4 X
ai ¼ 60 þ 30 þ 10 ¼ 100;
i
‘n
ð10Þ19 ¼ ‘nð10Þ18 ¼ 41:446 ’ 42; 10
524
M.S. El Naschie / Chaos, Solitons and Fractals 36 (2008) 521–525
and sffiffiffiffiffiffiffiffiffiffiffiffi ð10Þ16 ‘n ¼ 16:1652 ’ 16: 91 Consequently, au ’ 100 ½42 þ 16 ¼ 100 58 ¼ 42 ¼ ag : In the case of super symmetry on the other hand, we have M au ¼ 100 42 þ ‘n u ¼ 100 ½42 þ 32 ¼ 10 ½74 ¼ 26 ¼ ags : Mz
3. Discussion and conclusion A very simple and exact set of equations for the inverse unification coupling has been obtained by transfinitely harmonizing the basic renormalization groups of quantum field theory in a way similar to Feigenbaum golden mean renormalization groups ags ¼ a1 þ d1 ða0 Þ/3 ; ags ¼ a2 þ d2 ða0 Þ/3 ; ags ¼ ða3 þ a4 Þ þ d3 ða0 Þ/3 ; and ag ¼ ða3 þ a4 Þ þ d4 ða0 Þ/3 ; where 23 þ k ¼ 1:045084972; 22 þ k 1 d2 ¼ ¼ 8:472135954; 8 þ 2/3 d3 ¼ 1=2; d1 ¼
and d4 ¼ 1: The so-obtained solution vector is exact and completely consistent with all our previously obtained results [4–8] 2 3 26 þ k 6 26 þ k 7 6 7 a0 ¼ 6 7: 4 26 þ k 5 42 þ k Using any of these equations it is easy to show that when our energy scale goes to the Planck scale [4–8] M u ¼ M z ¼ M ) M planck then as = a3 ) 1 which means it is impossible to force quarks out of their confinement no matter how hard one hits them in a super-collider in full agreement with the confinement hypothesis. In fact at this point there are no quarks because spacetime suffers a phase transition to a Planck Aether consisting merely of Planck black holes elementary particles. Thus although totally consistent with the experimental observation and agrees in outcome with the asymptotic freedom proposed by other theories, it does indeed differ in philosophy and the details of the underlying physical picture. Maybe it is possible to elucidate the confinement mechanism using super gravity as following: We know that the symmetry group of super gravity is OSP(1/4) which is an ortho-symplictic Lie group. The dimension of this group is given by jOSPð1=4Þj ¼ 14:
M.S. El Naschie / Chaos, Solitons and Fractals 36 (2008) 521–525
525
Let us examine the ratio of the order of this group to that of the SL(2, 7) of the non-compactified holographic boundary of the E8E8 manifold. This is given by jSL(2, 7)j = 336 and consequently SLð2; 7Þ 336 OSPð1=4Þ ¼ 14 ¼ 24: Noting that (24)2 = 576, which is 548 + 28 and representing the almost maximal sum of any consistent Exceptional Lie symmetry groups hierarchy, [7] we must conjecture that 24 is the number of instantons of the E8E8 manifold corresponding to Ka¨hler K3 of heterotic string theory compactification although we know that 26 + k is the exact solution and corresponds to ð26 þ kÞ2 ¼ ð5Þða0 Þ ¼ 685:4101965. Consequently, a transfinite correction analogous to quantum field theory radiation corrections must show that 24 ! 26 + k. If this is correct and it is correct, then 14 may be guessed to transform as follows: jOSPð1=4Þj ! ða0 Þ=10; which is the signature of the fuzzy counterpart of K3 Ka¨hler. The two preceding transformations imply then that 336 ! ða0 Þð26 þ kÞ=10. This means 336 ! 336 þ 16k þ 20: It is not surprising that we did not obtain only the compactified 336 which is 336 + 16k where k = 0.18033989, but we obtained also Einstein’ s gravity R(4) = 20. Our exact transfinite coupling is thus nothing else but ags ¼ 26 þ k given by jSLð2; 7Þjc þ 20 336 þ 16k þ 0 ¼ 26:18033989: ¼ jOSPð1=4Þjc 13:7082039325 The crucial point is now the following: The jSL(2, 7)jc have theoretically infinite elements but the expectation value is finite one given by 336 + 16k ’ 339. Geometrically, it is a compactified Klein modular curve resembling one of an Escher’s famous pictures. By contrast, the ortho-normal group has the same dimension as G2 of the exceptional Lie groups jSOPð1=4Þj ¼ jG2 j ¼ 14: In principle we could let jG2j go to the largest exceptional group. We could also exceed E8 and obtain an infinite dimensional Lie algebra. Thus in principle jSLð2; 7Þjc þ Rð4Þ ¼ ags ¼ 26:18033989 jSOPð1=4Þjc could be made to approach zero. If ags ¼ 0, then ags = 1 and since we have a point of unification, then everything is completely frozen in its place, including quarks. In fact as we mentioned many times before we are on the doorstep of a new universe, the Planck universe. Hitting the quarks extremely hard, transmutate them to the Planck masses rather than make them free. Leaving the quarks in peace, they become asymptotically free. It is for non-Abelian theories like in social sciences: violence breeds contra violence. References [1] Yndura´in FJ. The theory of quark and gluon interaction. Berlin: Springer; 1992. [2] Davies P, editor, The new physics. Cambridge; 1989. [3] ‘tHooft G. When was asymptotic freedom discovered? Or the rehabilitation of quantum field theory. Nucl Phys B (Proc Suppl) 1999;74:418–25. [4] El Naschie MS. High energy physics, the standard model from the exceptional Lie groups and quarks confinement. Chaos, Solitons & Fractals 2008;36:1–17. [5] El Naschie MS. Quantum E-Infinity field theoretical derivation of Newton’s gravitational constant. Int J Nonlinear Sci Numer Simul 2007;8(3):469–74. [6] El Naschie MS. Elementary prerequisites for E-Infinity. Chaos, Solitons & Fractals 2006;30(3):579–605. [7] El Naschie MS. Exceptional Lie groups hierarchy and some fundamental high energy physics equations. Chaos, Solitons & Fractals 2008;35:82–4. [8] El Naschie MS. Quantum gravity unification via transfinite arithmetic and geometrical averaging. Chaos, Solitons & Fractals 2008;35:252–6.