- Email: [email protected]
Confinement and asymptotic freedom seen with a golden eye
Chaos, Solitons and Fractals 41 (2009) 2592–2594
Contents lists available at ScienceDirect
Chaos, Solitons and Fractals journal homepage: www.elsevier.com/locate/chaos
Confinement and asymptotic freedom seen with a golden eye A. Elokaby Department of Physics, University of Alexandria, Alexandria, Egypt
a r t i c l e
i n f o
a b s t r a c t
Article history: Accepted 22 September 2008
The present short note is an attempt to reconcile the current conventional understanding of quarks confinement and asymptotic freedom with the results found by El Naschie using the exact renormalization equation of his quantum golden field theory. Ó 2009 Published by Elsevier Ltd.
1. Introduction In the last few years we have witnessed a new trend towards non-perturbative solutions in high energy physics based on golden differential geometry [1–3] and a corresponding quantum golden field theory [3,6,7]. This relatively new field theory is a natural consequence of a new spacetime theory based almost entirely on the geometry and topology of random Cantor sets with a golden mean Hausdorff dimension [1–3]. It is this theory, as applied to the basic problem of confinement and asymptotic freedom which is the subject of the present paper. Our aim is to reconcile the current conventional understanding of these two concepts with the exact non-perturbative result obtained by El Naschie using the afore mentioned quantum golden field theory [6,7]. 2. The exact renormalization group equation for super symmetric and non-super symmetric unification Following some of El Naschie’s recent work on confinement and asymptotic freedom [1,2,6,7], the exact relevant renormalization equation for unification of all fundamental interactions may be given in the following form [7]
a u ¼ ða 3 þ a 4 Þ þ q‘n
Mu : M— Z
u is the inverse unification coupling constant which is a u1 ¼ a g for non-super symmetric theory and a u2 ¼ a gs for the Here a 4 ¼ 1 is the Planck coupling. Furthermore Mu is the 3 is the electroweak inverse strong coupling and a super symmetric case, a unification energy (mass) and MZ is the energy at which we are taking our measurement which in this case is MZ of the electroweak Zo boson. We note that q is given by q = 1 for the non-super symmetric theory and q = 1/2 for doubling the number 3 ¼ 9 for MZ = 91 Gev and taking Mu to be the mass of the unification of particles in the super symmetric case [1,6,7]. Since a GUT monopole, one finds
a u1 ¼ ð9 þ 1Þ þ ‘n
1016 91
!
’ 10 þ 31:33 ’ 42:3
which is very close to the exact value predicted by El Naschie’s E-infinity theory, namely [1,6,7]
a u1 ¼ ða o =2Þð/Þ ¼< n > ð10Þ ¼ 42:36067977 ’ 42 o ¼ 137:08203932 is the inverse electromagnetic fine structure constant. To obtain the exact value we have to recall where a u is therefore [6,7] that E-infinity is based on golden mean rather than logarithmic scaling [4–7]. The exact value of ‘n M M— Z E-mail address: [email protected] 0960-0779/$ - see front matter Ó 2009 Published by Elsevier Ltd. doi:10.1016/j.chaos.2008.09.035
A. Elokaby / Chaos, Solitons and Fractals 41 (2009) 2592–2594
‘n
2593
Mu o Þð/3 Þ ¼ ð137 þ ko Þð/3 Þ ¼ 32 þ 2k ¼ 32:36067977 ¼ ða M— Z
pffiffiffi where k = /3(1 /3), ko = /5(1 /5) and / ¼ ð 5 1Þ=2. There is a simple way of reaching this same exact result of El Naschie by noting the following: starting from [1,6,7]
‘n
Mu 1016 ¼ ‘n M— 91 Z
! ¼ ‘n½ð1:0989Þð10Þ14 ¼ 14‘n10 þ ‘nð1:0989Þ ¼ 14‘n10 þ 0:0943 ’ 14‘n‘n10:
Now we may add to El Naschie’s dictionary of transfinite continuation the following important new transformation, namely
‘n10 ! ð/3 Þð10Þ ¼ 2:3606797: Noting that
o =10 ¼ 13:70820393: 14 ! a Then the exact transfinite expression is obviously
o =10Þ ¼ ða o Þð/3 Þ ¼ 32 þ 2k 14‘n10 ) ð/3 Þð10Þða u1 is exactly as given by El Naschie. Consequently the exact a
a u1 ¼ a g ¼ 10 þ ð32 þ 2kÞ ¼ 42 þ 2k ¼ 42:36067977 precisely as should be. Next we consider the super symmetric case. Setting q ¼ 12 one finds [6,7]
1 2
a u2 ¼ a gs ¼ 10 þ ð32 þ 2kÞ ¼ 10 þ ð16 þ kÞ ¼ 26 þ k: Again this is the exact E-infinity value. Now we are ready to look into the question of confinement and asymptotic freedom [4–7]. 3. An attempt to reconcile the two views on confinement and asymptotic freedom Let us follow Mohamed El Naschie in his reasoning of confinement. In this case he replaces MZ by the Planck mass Mp. Consequently the logarithmic term vanishes because ‘n (Mp/Mp) = ‘n1 = 0. In addition it is clear that taking measurements u ¼ a QG ¼ 1. Noting that a 4 is also equal a QG which is unity, u must be set equal to the Planck coupling a at Mp means that a our renormalization equations
a u ¼ a 3 þ a 4 þ ‘n
Mp Mp
becomes
3 þ 1 þ 0: 1¼a This means [2]
a 3 ¼ 0 and consequently
1
a3 ¼ ¼ 1: a3 In other words we have infinitely large strong coupling and therefore complete confinement [4–7]. However El Naschie did not stress the second possibility when we replace MZ by a Cooper pair made of two electron masses 2me. In one of his papers u ¼ a gs ¼ 26 and since he just briefly mentions that taking measurements at the me scale, the unification constant must be a Mp ’ 26 this leads to a negative strong coupling exactly as discovered for the first time by the then graduate student Ger‘n 2m e ardus ‘tHooft [4]. This point is very important for reconciling both interpretations, the conventional one and the E-infinity interpretation. Therefore we will analyse it in some detail. We start by evaluating the new logarithmic term transfinitely. This is
‘n
Mp ð10Þ19 Gev ¼ ‘n ¼ ‘nð10Þ22 ¼ 22‘n10: 2me ð10Þ3 Gev
The obvious transfinite continuation is
‘n10 ! ð/3 Þð10Þ
2594
A. Elokaby / Chaos, Solitons and Fractals 41 (2009) 2592–2594
and
22 ! 22 þ k: This gives us
‘n
Mp ! ð/3 Þð10Þð22 þ kÞ ¼ ð/3 Þð221:8033989Þ ¼ 52 þ 2k: 2me
gs ¼ 26 þ k one finds Since q ¼ 12 and a
1 3 þ 1 þ ð52 þ 2kÞ ¼ a 3 þ 1 þ 26 þ k: 26 þ k ¼ a 2 Consequently we have [2]
a 3 þ 1 ¼ 0 or
a 3 ¼ 1 which means the strong coupling a3 ¼ a13 is negative. 4. Conclusion Introducing an elementary transfinite transformation for the vital logarithmic term in El Naschie’s exact quantum golden field renormalization equation (‘n10 ? (/)3 (10)) leads to conclusions which reconcile the results of E-infinity theory with those of current conventional interpretations of confinement and asymptotic freedom. References [1] El Naschie MS. Higgs mechanism, quarks confinement and black holes as a Cantorian spacetime phase transition scenario. Chaos, Solitons & Fractals 2009;41(2):869–74. [2] El Naschie MS. Quarks confinement. Chaos, Solitons & Fractals 2008;37:6–8. [3] Crasmareanu M, Hretcanu C. Golden differential geometry. Chaos, Solitons & Fractals 2008;38:1229–38. [4] ‘tHooft G. When was asymptotic freedom discovered? Or the rehabilitation of quantum field theory. Nucl Phys B 1999;74:413–25 [Proceedings Supplement]. [5] Gross D, Wilczek F. Phys Rev Lett 1973;30:1343. [6] El Naschie MS. Towards a quantum golden field theory. Int J Nonlinear Sci Numer Simul 2007;8(4):477–82. [7] El Naschie MS. Asymptotic freedom and unification in a golden quantum field theory. Chaos, Solitons & Fractals 2008;30:521–5.
Contents lists available at ScienceDirect
Chaos, Solitons and Fractals journal homepage: www.elsevier.com/locate/chaos
Confinement and asymptotic freedom seen with a golden eye A. Elokaby Department of Physics, University of Alexandria, Alexandria, Egypt
a r t i c l e
i n f o
a b s t r a c t
Article history: Accepted 22 September 2008
The present short note is an attempt to reconcile the current conventional understanding of quarks confinement and asymptotic freedom with the results found by El Naschie using the exact renormalization equation of his quantum golden field theory. Ó 2009 Published by Elsevier Ltd.
1. Introduction In the last few years we have witnessed a new trend towards non-perturbative solutions in high energy physics based on golden differential geometry [1–3] and a corresponding quantum golden field theory [3,6,7]. This relatively new field theory is a natural consequence of a new spacetime theory based almost entirely on the geometry and topology of random Cantor sets with a golden mean Hausdorff dimension [1–3]. It is this theory, as applied to the basic problem of confinement and asymptotic freedom which is the subject of the present paper. Our aim is to reconcile the current conventional understanding of these two concepts with the exact non-perturbative result obtained by El Naschie using the afore mentioned quantum golden field theory [6,7]. 2. The exact renormalization group equation for super symmetric and non-super symmetric unification Following some of El Naschie’s recent work on confinement and asymptotic freedom [1,2,6,7], the exact relevant renormalization equation for unification of all fundamental interactions may be given in the following form [7]
a u ¼ ða 3 þ a 4 Þ þ q‘n
Mu : M— Z
u is the inverse unification coupling constant which is a u1 ¼ a g for non-super symmetric theory and a u2 ¼ a gs for the Here a 4 ¼ 1 is the Planck coupling. Furthermore Mu is the 3 is the electroweak inverse strong coupling and a super symmetric case, a unification energy (mass) and MZ is the energy at which we are taking our measurement which in this case is MZ of the electroweak Zo boson. We note that q is given by q = 1 for the non-super symmetric theory and q = 1/2 for doubling the number 3 ¼ 9 for MZ = 91 Gev and taking Mu to be the mass of the unification of particles in the super symmetric case [1,6,7]. Since a GUT monopole, one finds
a u1 ¼ ð9 þ 1Þ þ ‘n
1016 91
!
’ 10 þ 31:33 ’ 42:3
which is very close to the exact value predicted by El Naschie’s E-infinity theory, namely [1,6,7]
a u1 ¼ ða o =2Þð/Þ ¼< n > ð10Þ ¼ 42:36067977 ’ 42 o ¼ 137:08203932 is the inverse electromagnetic fine structure constant. To obtain the exact value we have to recall where a u is therefore [6,7] that E-infinity is based on golden mean rather than logarithmic scaling [4–7]. The exact value of ‘n M M— Z E-mail address: [email protected] 0960-0779/$ - see front matter Ó 2009 Published by Elsevier Ltd. doi:10.1016/j.chaos.2008.09.035
A. Elokaby / Chaos, Solitons and Fractals 41 (2009) 2592–2594
‘n
2593
Mu o Þð/3 Þ ¼ ð137 þ ko Þð/3 Þ ¼ 32 þ 2k ¼ 32:36067977 ¼ ða M— Z
pffiffiffi where k = /3(1 /3), ko = /5(1 /5) and / ¼ ð 5 1Þ=2. There is a simple way of reaching this same exact result of El Naschie by noting the following: starting from [1,6,7]
‘n
Mu 1016 ¼ ‘n M— 91 Z
! ¼ ‘n½ð1:0989Þð10Þ14 ¼ 14‘n10 þ ‘nð1:0989Þ ¼ 14‘n10 þ 0:0943 ’ 14‘n‘n10:
Now we may add to El Naschie’s dictionary of transfinite continuation the following important new transformation, namely
‘n10 ! ð/3 Þð10Þ ¼ 2:3606797: Noting that
o =10 ¼ 13:70820393: 14 ! a Then the exact transfinite expression is obviously
o =10Þ ¼ ða o Þð/3 Þ ¼ 32 þ 2k 14‘n10 ) ð/3 Þð10Þða u1 is exactly as given by El Naschie. Consequently the exact a
a u1 ¼ a g ¼ 10 þ ð32 þ 2kÞ ¼ 42 þ 2k ¼ 42:36067977 precisely as should be. Next we consider the super symmetric case. Setting q ¼ 12 one finds [6,7]
1 2
a u2 ¼ a gs ¼ 10 þ ð32 þ 2kÞ ¼ 10 þ ð16 þ kÞ ¼ 26 þ k: Again this is the exact E-infinity value. Now we are ready to look into the question of confinement and asymptotic freedom [4–7]. 3. An attempt to reconcile the two views on confinement and asymptotic freedom Let us follow Mohamed El Naschie in his reasoning of confinement. In this case he replaces MZ by the Planck mass Mp. Consequently the logarithmic term vanishes because ‘n (Mp/Mp) = ‘n1 = 0. In addition it is clear that taking measurements u ¼ a QG ¼ 1. Noting that a 4 is also equal a QG which is unity, u must be set equal to the Planck coupling a at Mp means that a our renormalization equations
a u ¼ a 3 þ a 4 þ ‘n
Mp Mp
becomes
3 þ 1 þ 0: 1¼a This means [2]
a 3 ¼ 0 and consequently
1
a3 ¼ ¼ 1: a3 In other words we have infinitely large strong coupling and therefore complete confinement [4–7]. However El Naschie did not stress the second possibility when we replace MZ by a Cooper pair made of two electron masses 2me. In one of his papers u ¼ a gs ¼ 26 and since he just briefly mentions that taking measurements at the me scale, the unification constant must be a Mp ’ 26 this leads to a negative strong coupling exactly as discovered for the first time by the then graduate student Ger‘n 2m e ardus ‘tHooft [4]. This point is very important for reconciling both interpretations, the conventional one and the E-infinity interpretation. Therefore we will analyse it in some detail. We start by evaluating the new logarithmic term transfinitely. This is
‘n
Mp ð10Þ19 Gev ¼ ‘n ¼ ‘nð10Þ22 ¼ 22‘n10: 2me ð10Þ3 Gev
The obvious transfinite continuation is
‘n10 ! ð/3 Þð10Þ
2594
A. Elokaby / Chaos, Solitons and Fractals 41 (2009) 2592–2594
and
22 ! 22 þ k: This gives us
‘n
Mp ! ð/3 Þð10Þð22 þ kÞ ¼ ð/3 Þð221:8033989Þ ¼ 52 þ 2k: 2me
gs ¼ 26 þ k one finds Since q ¼ 12 and a
1 3 þ 1 þ ð52 þ 2kÞ ¼ a 3 þ 1 þ 26 þ k: 26 þ k ¼ a 2 Consequently we have [2]
a 3 þ 1 ¼ 0 or
a 3 ¼ 1 which means the strong coupling a3 ¼ a13 is negative. 4. Conclusion Introducing an elementary transfinite transformation for the vital logarithmic term in El Naschie’s exact quantum golden field renormalization equation (‘n10 ? (/)3 (10)) leads to conclusions which reconcile the results of E-infinity theory with those of current conventional interpretations of confinement and asymptotic freedom. References [1] El Naschie MS. Higgs mechanism, quarks confinement and black holes as a Cantorian spacetime phase transition scenario. Chaos, Solitons & Fractals 2009;41(2):869–74. [2] El Naschie MS. Quarks confinement. Chaos, Solitons & Fractals 2008;37:6–8. [3] Crasmareanu M, Hretcanu C. Golden differential geometry. Chaos, Solitons & Fractals 2008;38:1229–38. [4] ‘tHooft G. When was asymptotic freedom discovered? Or the rehabilitation of quantum field theory. Nucl Phys B 1999;74:413–25 [Proceedings Supplement]. [5] Gross D, Wilczek F. Phys Rev Lett 1973;30:1343. [6] El Naschie MS. Towards a quantum golden field theory. Int J Nonlinear Sci Numer Simul 2007;8(4):477–82. [7] El Naschie MS. Asymptotic freedom and unification in a golden quantum field theory. Chaos, Solitons & Fractals 2008;30:521–5.