- Email: [email protected]
Number of elementary particles using exceptional Lie symmetry groups hierarchy
Chaos, Solitons and Fractals 39 (2009) 2119–2124 www.elsevier.com/locate/chaos
Number of elementary particles using exceptional Lie symmetry groups hierarchy Ji-Huan He *, Lan Xu Modern Textile Institute, Donghua University, Shanghai 200051, China Accepted 21 June 2007
Abstract This paper suggests several approaches to predict the number of elementary particles via a remarkable finite exceptional Lie symmetry groups hierarchy. This result confirms the earlier finding namely that nine elementary particles are still missing from the standard model. Ó 2007 Elsevier Ltd. All rights reserved.
1. Introduction Mathematicians have just mapped the inner structure of one of the most complicated structures ever studied: an object known as the exceptional Lie group E8 [1–6]. This development is significant both as an advance in basic mathematical knowledge and because of the many connections between E8 and other areas, including string theory and geometry. E8 is an extraordinary complicated group: it is the symmetries of a particular 57-dimensional object, while E8 itself is 248-dimensional. Furthermore, E8 is completely abstract and no one knows how it looks geometrically. But now things have completely changed and we link it here to the hierarchy of E-infinity theory, which is one of the most promising candidates for a unified description of all fundamental forces including gravity [7–12].
2. Predicting the number of elementary particles A concerted analysis was undertaken in the last few years within the theoretical framework of E-infinity theory [13–17] to determine the most likely number of yet to be discovered elementary particles of a moderately extended standard model. In particular many of the results obtained using E-infinity crucially depend upon earlier results obtained in quantum field theory and super strings [18,19]. In what follows, we review briefly this subject before proceeding to explain how to apply the exceptional Lie groups hierarchy to this problem.
*
Corresponding author. E-mail address: [email protected] (J.-H. He).
0960-0779/$ - see front matter Ó 2007 Elsevier Ltd. All rights reserved. doi:10.1016/j.chaos.2007.06.088
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2.1. Three steps symmetry breaking [20–30] The role of jE8E8j = 496 is well known and understood in heterotic superstring theory. Considering a maximally symmetric four manifold in a super space, we see that the dimensionality m must be m = (4) (8) = 32. Consequently, we have jSOð32Þj ¼ Dim E8 E8 ¼ mðm 1Þ=2 ¼ ð32Þð31Þ=2 ¼ 496 Since super string theory is accredited with unifying all fundamental forces, then it may be useful to investigate the particles content of this theory. In order to include fermions in string theory, there must be a special kind of symmetry called supersymmetry, which means that for every boson there is a corresponding fermion. Thus, supersymmetry relates the particles that transmit forces to the particles that make up matter. String theory starts with 496 massless states. This number is simply the dimension of the exceptional Lie group E8E8. It is clear that we cannot observe these 496 massless particles because they exist at fantastically high unification energy. In fact, exactly half of these particles, namely 248 are totally unobservable because they belong to a shadow universe. Secondly, we have not observed any super symmetric partners and since super string is a super symmetric theory, half of the 248 particles cannot be observed at the energy scale of the present standard model. We say super symmetry is broken and that leaves us only half of 248 particles which means 124 particles. Finally, in our counting of particles, we made no statement about helicity, thus we break this symmetry as well. Consequently, a three-step symmetry breaking of this exceptional Lie symmetry group leads to an estimation of the number of elementary particles [20–30]: N ðSMÞ ¼ 496=8 ¼ 62 It is well-known that there are 60 experimentally verified particles of the standard model namely [31–34]: 6 Leptons + 6 anti-leptons + 3 (color) (6 quarks) + 3 (color) (6 anti-quarks) + (12 messenger particles) = 60 elementary particles all together. The additional two particles could be interpreted as one graviton, and one Higgs particle [35–42]. In the forthcoming sections, we should give exact results using E-infinity theory that reveals E-infinity theory is superior to any other [43–51]. 2.2. Translations and boosts [52] Considering the 32 dimensional space to be quasi-Mikowskian, the number of Killing Vector Field is [52] 1 ð32Þð32 þ 1Þ ¼ 528: 2 We could break down the 528 vector fields as follows [52]: m ¼ 32 are translations m 1 ¼ 31 are boosts and ðm 1Þðm 2Þ=2 ¼ 465 are space rotations: We focus our attention on the boosts and translation part of the above, which amounts to [52] b þ t ¼ 31 þ 32 ¼ 63 We could interpret this number as that found for the elementary particle content of the standard model based on the well-known results of heterotic string theory namely 8064/128 = 63[1,2,18,19]. 2.3. Factoring the standard model out of the exceptional Lie hierarchy [52] Witten [18,19] revealed some time ago that embedding SU(3)SU(2)U(1) of the standard model needs seven dimensions which when added to our 3 + 1 spacetime results naturally in the 11 dimensions of his M-theory [47,48]. This is just one dimension more than the 10 of superstring theory. Consequently, if we start by Dm = 11 and proceed with all the exceptional Lie groups upwards until we reach E8, then this hierarchy adds to the following maximal total number of states [52] Dm þ jSUð3ÞSUð2ÞUð1Þj þ jG2 j þ jF 4 j þ jE6 j þ jE7 j þ jE8 j ¼ 11 þ 12 þ 14 þ 52 þ 78 þ 133 þ 248 ¼ 548 Consequently, a three-step symmetry breaking gives [8,9]
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NðSMÞ ¼ 549=8 ¼ 68:5 69; exactly as should be. 2.4. The 57-dimensional object of E8 Considering the 12 gauge bosons of the SU(3) SU(2) U(1) and adding that to the 57-dimensional object of E8, we have the following amazing result: NðSMÞ ¼ jSUð3ÞSUð2ÞU ð1Þj þ Dim Eð8Þ ¼ 12 þ 57 ¼ 69 exactly as should be[8]. 2.5. P-Branes Following Witten’s theory of p = 5-Branes in 11 dimensions, it is possible to replace 57 by 11 N ðSMÞ ¼ jSUð3ÞSUð2ÞU ð1Þj þ Z ð2Þ ¼ 12 þ ¼ 12 þ 55 ¼ 67 2
11 , and obtain [52] 2
particles in an M-like theory in 11 dimensions [47,48].
3. Geometrical visualization of E8 and the inverse electromagnetic fine structure constant [46]
3.1. The 6D cube of jE8E8j Fig. 1 shows a 6D cube, which is relevant to the campactified space of string theory. In this theory, it is given by a six-dimensional Calabi–Yau manifold or an orbifold [46]. It is vital to note that the number of vertices plus edges plus
Fig. 1. The 6D cube. (1) The number of elements is 496 which is the sum of the number of vortices, edges and faces. This is exactly the same number as that of the massless Gauge bosons of the Hetoretic superstring theory. (2) The number of routes for E8 is 240. This is exactly the same number as the sphere kissing number in eight dimensions. So in E8 E8 we could imagine that we have de facto 480 routes or kissing points. (3) Kissing points could be understood as a kind of bosons, so the routes could also be understood as a kind of bosons. Thus in breaking the symmetry we find 60 particles because 480 divided by 8 equals 60. This is exactly the number of particles discovered in the standard model. (4) The other two particles come from the additional 16 isometries which are not kissing points so we have here a distinction between the Higgs and the graviton and the rest of the elementary particles.
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faces is exactly equal to 496. This is quite remarkable. The vertices are zero brane, the edges are one-dimensional brane or a string, the faces are two-dimensional brane or membrane and the whole thing is a six-dimensional cube. It is well known that within the six dimensions of the string, massive particles are created via symmetry breaking. 3.2. El Naschie–t’ Hooft Holographic equation [53–55] 0 ’ 137, the inverse electromagUsing the basic equation of E-infinity one could give a very elegant derivation of a netic fine structure constant. The argument is very simple and goes back to the work of t’ Hooft and El Naschie [10]. Since the bulk contains all fundamental interactions, while the holographic boundary have only particle physics then one can translate this situation to the following global equation: All interactions ½particle physics þ gravity ¼ electromagnetism The equivalent symmetry groups dimensional equation is thus [10] Dim E8 E8 ½Dim Cc ð7Þ þ Rð4Þ ¼ a0 where 1 Dim E8 E8 ¼ ð32Þð32 1Þ ¼ 496 2 Dim Cð7Þ ¼ 82 ð82 1Þ=12 ¼ 336 Dim Cc ð7Þ ¼ 336 þ 16k ¼ 339 and 2 Rð4Þ ¼ ð4Þ ð42 1Þ=12 ¼ 20
That means a0 ¼ 496 ½339 þ 20 ¼ 137 The preceding derivation clearly shows that E-infinity’s conception of ao as a dimension related to the symmetry of some higher dimensional spacetime is fully justified and agrees with both super strings theory and the holographic principle.
4. Conclusions An extremely simple and elementary but rigorous derivation of the number of elementary particles is given using the exceptional Lie symmetry groups hierarchy and E-infinity. The result clearly confirms that nine particles which are not included in Standard Model are still missing. These may include one graviton and at least one Higgs particle.
Acknowledgement The project is supported by the Program for New Century Excellent Talents in University.
References [1] [2] [3] [4]
Nakahara M. Geometry, topology and physics. I.O.P. Bristol 2004. Kaku M. Introduction to superstrings and M-Theory. New York: Springer; 1999. Kaku M. Highlights of supersymmetry, superstrings, and M-theory. Foundat Phys 2003;33(5):689–706. El Naschie MS. The Fibonacci code behind super strings and P-Branes. An answer to M. Kaku’s fundamental question. Chaos, Solitons & Fractals 2007;31(3):537–47. [5] El Naschie MS. SO(10) grand unification in a fuzzy setting. Chaos, Solitons & Fractals 2007;32(3):958–61. [6] El Naschie MS. The unreasonable effectiveness of the electron-volt units system in high energy physics and the role played by a0 ¼ 137. Int J Nonlinear Sci Numer Simulat 2006;7(2):119–28. [7] El Naschie MS. On the vital role played by the electron-volt units system in high energy physics and Mach’s principle of Denkokonomie. Chaos, Solitons & Fractals 2006;28(5):1366–71.
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[8] He JH et al. Transfinite physics. China Education & Culture Publisher; 2005. [9] He JH et al. Beyond the 2006 Physics Nobel Prize for COBE: an introduction to E-infinity spacetime theory. China Education & Culture Publisher; 2006. [10] El Naschie MS. A review of applications and results of E-infinity theory. Int J Nonlinear Sci Numer Simulat 2007;8(1):11–20. [11] El Naschie MS. A review of E-infinity theory and the mass spectrum of high energy particle physics. Chaos, Solitons & Fractals 2004;19(1):209–36. [12] Yang CD, Han SY, Hsiao FB. Nonlinear dynamics governing quantum transition behavior. Int J Nonlinear Sci Numer Simulat 2007;8(3):397–412. [13] Iovane G. E1 Naschie E-infinity Cantorian spacetime and lengths scales in cosmology. Int J Nonlinear Sci Numer Simulat 2006;7(2):155–62. [14] El Naschie MS. Thomas Mann and Heinrich Mann, dual brothers and complimentary genius embraced by complex reality. Int J Nonlinear Sci Numer Simulat 2006;7(1):1–6. [15] El Naschie MS. Linderhof room of mirrors, Thurston three-manifolds and the geometry of our universe. Int J Nonlinear Sci Numer Simulat 2006;7(1):97–100. [16] Iovane G, Benedetto E. El Naschie E-infinity cantorian space-time, Fantappie’s group and applications in cosmology. Int J Nonlinear Sci Numer Simulat 2005;6(4):357–70. [17] El Naschie MS. Einstein in a complex time – some very personal thoughts about E-infinity theory and modern physics. Int J Nonlinear Sci Numer Simulat 2005;6(3):331–3. [18] Horava P, Witten E. Heterotic and Type I string dynamics from eleven dimensions. Nucl Phys B 1996;460(3):506–24. [19] Witten E. String theory dynamics in various dimensions. Nucl Phys B 1995;443(1-2):85–126. [20] El Naschie MS. Probability set particles. Int J Nonlinear Sci Numer Simulat 2007;8(1):117–9. [21] Marek-Crnjac L. The maximum number of elementary particles in a super symmetric extension of the standard model. Chaos, Solitons & Fractals 2007;32(5):1631–6. [22] El Naschie MS. Determining the number of Fermions and the number of Boson separately in an extended standard model. Chaos, Solitons & Fractals 2007;32(3):1241–3. [23] El Naschie MS. On an eleven dimensional E-infinity fractal spacetime theory. Int J Nonlinear Sci Numer Simulat 2006;7(4):407–9. [24] El Naschie MS. The elementary particles content of quantum spacetime via Feynman graphs and higher dimensional polytops. Chaos, Solitons & Fractals 2007;31(1):1–4. [25] El Naschie MS. Gauge anomalies, SU(N) irreducible representation and the number of elementary particles of a minimally extended standard model. Chaos, Solitons & Fractals 2007;31(1):14–6. [26] Marek-Crnjac L. Pentaquarks and the mass spectrum of the elementary particles of the standard model. Chaos, Solitons & Fractals 2006;30(2):332–41. [27] El Naschie MS. Superstrings, entropy and the elementary particles content of the standard model. Chaos, Solitons & Fractals 2006;29(1):48–54. [28] El Naschie MS. Deriving the essential features of the standard model from the general theory of relativity. Chaos, Solitons & Fractals 2005;24(4):941–6. [29] El Naschie MS. On 336 kissing spheres in 10 dimensions, 528 P-Brane states in 11 dimensions and the 60 elementary particles of the standard model. Chaos, Solitons & Fractals 2005;24(2):447–57. [30] El Naschie MS. Deterministic quantum mechanics versus classical mechanical indeterminism. Int J Nonlinear Sci Numer Simulat 2007;8(1):5–10. [31] Giordano P. Numerical analysis of hypersingular integral equations in the E-infinite Cantorian spacetime. Int J Nonlinear Sci Numer Simulat 2006;7(4):451–60. [32] Sigalotti LDG, Mejias A. On El Naschie’s conjugate complex time, fractal E-(infinity) space-time and faster-than-light particles. Int J Nonlinear Sci Numer Simulat 2006;7(4):467–72. [33] Naschie MS. The ’discreet’ charm of certain eleven dimensional spacetime theories. Int J Nonlinear Sci Numer Simulat 2006;7(4):477–81. [34] El Naschie MS. The brain and E-infinity. Int J Nonlinear Sci Numer Simulat 2006;7(2):129–32. [35] He JH. E-infinity theory and the Higgs field. Chaos, Solitons & Fractals 2007;31(4):782–6. [36] El Naschie MS. The missing particles of the standard model via a unified particle-field framework. Int J Nonlinear Sci Numer Simulat 2006;7(1):101–4. [37] El Naschie MS. The Feynman path integral and e-infinity from the two-slit Gedanken experiment. Int J Nonlinear Sci Numer Simulat 2005;6(4):335–42. [38] El Naschie MS. Higgs number from anomaly cancellation and super Riemann tensor. Chaos, Solitons & Fractals 2005;24(2):653–7. [39] El Naschie MS. The Higgs and the expectation value of the number of elementary particles in a supersymmetric extensions of the standard model of high energy physics. Chaos, Solitons & Fractals 2005;23(2):363–71. [40] Marek-Crnjac L. Different Higgs models and the number of Higgs particles. Chaos, Solitons & Fractals 2006;27(3):575–9. [41] El Naschie MS. Tadpoles, anomaly cancellation and the expectation value of the number of the Higgs particles in the standard model. Chaos, Solitons & Fractals 2005;24(3):659–63. [42] Goldfain E. Higgs-free derivation of gauge boson masses using complex dynamics of Levy flows. Int J Nonlinear Sci Numer Simulat 2005;6(4):351–6.
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He JH. In search of 9 hidden particles. Int J Nonlinear Sci Numer Simulat 2005;6(2):93–4. He JH. Space, time and beyond. Int J Nonlinear Sci Numer Simulat 2005;6(4):343–6. He JH. Nonlinear dynamics and the Nobel Prize in Physics. Int J Nonlinear Sci Numer Simulat 2007;8(1):1–4. He JH, Xu L, Zhang LN, et al. Twenty-six dimensional polytope and high energy spacetime physics. Chaos, Solitons & Fractals 2007;33(1):5–13. He JH. On the number of elementary particles in a resolution dependent fractal spacetime. Chaos, Solitons & Fractals 2007;32(5):1645–8. He JH. The number of elementary particles in a fractal M-theory of 11. 2360667977 dimensions. Chaos, Solitons & Fractals 2007;32(2):346–51. He JH. Application of E-infinity theory to turbulence. Chaos, Solitons & Fractals 2006;30(2):506–11. He JH. Application of E-infinity theory to biology. Chaos, Solitons & Fractals 2006;28(2):285–9. Saladin El Naschie M. Intermediate prerequisites for E-infinity theory (Further recommended reading in nonlinear dynamics and mathematical physics). Chaos, Solitons & Fractals 2006;30(3):622–8. El Naschie MS. Exceptional Lie groups hierarchy and the structure of the micro universe. Int J Nonlinear Sci Numer Simulat 2007;8(3):445–50. El Naschie MS. On a fuzzy Kahler-like manifold which is consistent with the two slit experiment. Int J Nonlinear Sci Numer Simulat 2005;6(2):95–8. El Naschie MS. Quantum probability without a phase and a topological resolution of the two-slit experiment. Int J Nonlinear Sci Numer Simulat 2007;8(2):,195–198. El Naschie MS. The Cosmic da Vinci Code for the Big Bang – a Mathematical Toy model. Int J Nonlinear Sci Numer Simulat 2007;8(2):191–4.
Number of elementary particles using exceptional Lie symmetry groups hierarchy Ji-Huan He *, Lan Xu Modern Textile Institute, Donghua University, Shanghai 200051, China Accepted 21 June 2007
Abstract This paper suggests several approaches to predict the number of elementary particles via a remarkable finite exceptional Lie symmetry groups hierarchy. This result confirms the earlier finding namely that nine elementary particles are still missing from the standard model. Ó 2007 Elsevier Ltd. All rights reserved.
1. Introduction Mathematicians have just mapped the inner structure of one of the most complicated structures ever studied: an object known as the exceptional Lie group E8 [1–6]. This development is significant both as an advance in basic mathematical knowledge and because of the many connections between E8 and other areas, including string theory and geometry. E8 is an extraordinary complicated group: it is the symmetries of a particular 57-dimensional object, while E8 itself is 248-dimensional. Furthermore, E8 is completely abstract and no one knows how it looks geometrically. But now things have completely changed and we link it here to the hierarchy of E-infinity theory, which is one of the most promising candidates for a unified description of all fundamental forces including gravity [7–12].
2. Predicting the number of elementary particles A concerted analysis was undertaken in the last few years within the theoretical framework of E-infinity theory [13–17] to determine the most likely number of yet to be discovered elementary particles of a moderately extended standard model. In particular many of the results obtained using E-infinity crucially depend upon earlier results obtained in quantum field theory and super strings [18,19]. In what follows, we review briefly this subject before proceeding to explain how to apply the exceptional Lie groups hierarchy to this problem.
*
Corresponding author. E-mail address: [email protected] (J.-H. He).
0960-0779/$ - see front matter Ó 2007 Elsevier Ltd. All rights reserved. doi:10.1016/j.chaos.2007.06.088
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2.1. Three steps symmetry breaking [20–30] The role of jE8E8j = 496 is well known and understood in heterotic superstring theory. Considering a maximally symmetric four manifold in a super space, we see that the dimensionality m must be m = (4) (8) = 32. Consequently, we have jSOð32Þj ¼ Dim E8 E8 ¼ mðm 1Þ=2 ¼ ð32Þð31Þ=2 ¼ 496 Since super string theory is accredited with unifying all fundamental forces, then it may be useful to investigate the particles content of this theory. In order to include fermions in string theory, there must be a special kind of symmetry called supersymmetry, which means that for every boson there is a corresponding fermion. Thus, supersymmetry relates the particles that transmit forces to the particles that make up matter. String theory starts with 496 massless states. This number is simply the dimension of the exceptional Lie group E8E8. It is clear that we cannot observe these 496 massless particles because they exist at fantastically high unification energy. In fact, exactly half of these particles, namely 248 are totally unobservable because they belong to a shadow universe. Secondly, we have not observed any super symmetric partners and since super string is a super symmetric theory, half of the 248 particles cannot be observed at the energy scale of the present standard model. We say super symmetry is broken and that leaves us only half of 248 particles which means 124 particles. Finally, in our counting of particles, we made no statement about helicity, thus we break this symmetry as well. Consequently, a three-step symmetry breaking of this exceptional Lie symmetry group leads to an estimation of the number of elementary particles [20–30]: N ðSMÞ ¼ 496=8 ¼ 62 It is well-known that there are 60 experimentally verified particles of the standard model namely [31–34]: 6 Leptons + 6 anti-leptons + 3 (color) (6 quarks) + 3 (color) (6 anti-quarks) + (12 messenger particles) = 60 elementary particles all together. The additional two particles could be interpreted as one graviton, and one Higgs particle [35–42]. In the forthcoming sections, we should give exact results using E-infinity theory that reveals E-infinity theory is superior to any other [43–51]. 2.2. Translations and boosts [52] Considering the 32 dimensional space to be quasi-Mikowskian, the number of Killing Vector Field is [52] 1 ð32Þð32 þ 1Þ ¼ 528: 2 We could break down the 528 vector fields as follows [52]: m ¼ 32 are translations m 1 ¼ 31 are boosts and ðm 1Þðm 2Þ=2 ¼ 465 are space rotations: We focus our attention on the boosts and translation part of the above, which amounts to [52] b þ t ¼ 31 þ 32 ¼ 63 We could interpret this number as that found for the elementary particle content of the standard model based on the well-known results of heterotic string theory namely 8064/128 = 63[1,2,18,19]. 2.3. Factoring the standard model out of the exceptional Lie hierarchy [52] Witten [18,19] revealed some time ago that embedding SU(3)SU(2)U(1) of the standard model needs seven dimensions which when added to our 3 + 1 spacetime results naturally in the 11 dimensions of his M-theory [47,48]. This is just one dimension more than the 10 of superstring theory. Consequently, if we start by Dm = 11 and proceed with all the exceptional Lie groups upwards until we reach E8, then this hierarchy adds to the following maximal total number of states [52] Dm þ jSUð3ÞSUð2ÞUð1Þj þ jG2 j þ jF 4 j þ jE6 j þ jE7 j þ jE8 j ¼ 11 þ 12 þ 14 þ 52 þ 78 þ 133 þ 248 ¼ 548 Consequently, a three-step symmetry breaking gives [8,9]
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NðSMÞ ¼ 549=8 ¼ 68:5 69; exactly as should be. 2.4. The 57-dimensional object of E8 Considering the 12 gauge bosons of the SU(3) SU(2) U(1) and adding that to the 57-dimensional object of E8, we have the following amazing result: NðSMÞ ¼ jSUð3ÞSUð2ÞU ð1Þj þ Dim Eð8Þ ¼ 12 þ 57 ¼ 69 exactly as should be[8]. 2.5. P-Branes Following Witten’s theory of p = 5-Branes in 11 dimensions, it is possible to replace 57 by 11 N ðSMÞ ¼ jSUð3ÞSUð2ÞU ð1Þj þ Z ð2Þ ¼ 12 þ ¼ 12 þ 55 ¼ 67 2
11 , and obtain [52] 2
particles in an M-like theory in 11 dimensions [47,48].
3. Geometrical visualization of E8 and the inverse electromagnetic fine structure constant [46]
3.1. The 6D cube of jE8E8j Fig. 1 shows a 6D cube, which is relevant to the campactified space of string theory. In this theory, it is given by a six-dimensional Calabi–Yau manifold or an orbifold [46]. It is vital to note that the number of vertices plus edges plus
Fig. 1. The 6D cube. (1) The number of elements is 496 which is the sum of the number of vortices, edges and faces. This is exactly the same number as that of the massless Gauge bosons of the Hetoretic superstring theory. (2) The number of routes for E8 is 240. This is exactly the same number as the sphere kissing number in eight dimensions. So in E8 E8 we could imagine that we have de facto 480 routes or kissing points. (3) Kissing points could be understood as a kind of bosons, so the routes could also be understood as a kind of bosons. Thus in breaking the symmetry we find 60 particles because 480 divided by 8 equals 60. This is exactly the number of particles discovered in the standard model. (4) The other two particles come from the additional 16 isometries which are not kissing points so we have here a distinction between the Higgs and the graviton and the rest of the elementary particles.
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faces is exactly equal to 496. This is quite remarkable. The vertices are zero brane, the edges are one-dimensional brane or a string, the faces are two-dimensional brane or membrane and the whole thing is a six-dimensional cube. It is well known that within the six dimensions of the string, massive particles are created via symmetry breaking. 3.2. El Naschie–t’ Hooft Holographic equation [53–55] 0 ’ 137, the inverse electromagUsing the basic equation of E-infinity one could give a very elegant derivation of a netic fine structure constant. The argument is very simple and goes back to the work of t’ Hooft and El Naschie [10]. Since the bulk contains all fundamental interactions, while the holographic boundary have only particle physics then one can translate this situation to the following global equation: All interactions ½particle physics þ gravity ¼ electromagnetism The equivalent symmetry groups dimensional equation is thus [10] Dim E8 E8 ½Dim Cc ð7Þ þ Rð4Þ ¼ a0 where 1 Dim E8 E8 ¼ ð32Þð32 1Þ ¼ 496 2 Dim Cð7Þ ¼ 82 ð82 1Þ=12 ¼ 336 Dim Cc ð7Þ ¼ 336 þ 16k ¼ 339 and 2 Rð4Þ ¼ ð4Þ ð42 1Þ=12 ¼ 20
That means a0 ¼ 496 ½339 þ 20 ¼ 137 The preceding derivation clearly shows that E-infinity’s conception of ao as a dimension related to the symmetry of some higher dimensional spacetime is fully justified and agrees with both super strings theory and the holographic principle.
4. Conclusions An extremely simple and elementary but rigorous derivation of the number of elementary particles is given using the exceptional Lie symmetry groups hierarchy and E-infinity. The result clearly confirms that nine particles which are not included in Standard Model are still missing. These may include one graviton and at least one Higgs particle.
Acknowledgement The project is supported by the Program for New Century Excellent Talents in University.
References [1] [2] [3] [4]
Nakahara M. Geometry, topology and physics. I.O.P. Bristol 2004. Kaku M. Introduction to superstrings and M-Theory. New York: Springer; 1999. Kaku M. Highlights of supersymmetry, superstrings, and M-theory. Foundat Phys 2003;33(5):689–706. El Naschie MS. The Fibonacci code behind super strings and P-Branes. An answer to M. Kaku’s fundamental question. Chaos, Solitons & Fractals 2007;31(3):537–47. [5] El Naschie MS. SO(10) grand unification in a fuzzy setting. Chaos, Solitons & Fractals 2007;32(3):958–61. [6] El Naschie MS. The unreasonable effectiveness of the electron-volt units system in high energy physics and the role played by a0 ¼ 137. Int J Nonlinear Sci Numer Simulat 2006;7(2):119–28. [7] El Naschie MS. On the vital role played by the electron-volt units system in high energy physics and Mach’s principle of Denkokonomie. Chaos, Solitons & Fractals 2006;28(5):1366–71.
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[8] He JH et al. Transfinite physics. China Education & Culture Publisher; 2005. [9] He JH et al. Beyond the 2006 Physics Nobel Prize for COBE: an introduction to E-infinity spacetime theory. China Education & Culture Publisher; 2006. [10] El Naschie MS. A review of applications and results of E-infinity theory. Int J Nonlinear Sci Numer Simulat 2007;8(1):11–20. [11] El Naschie MS. A review of E-infinity theory and the mass spectrum of high energy particle physics. Chaos, Solitons & Fractals 2004;19(1):209–36. [12] Yang CD, Han SY, Hsiao FB. Nonlinear dynamics governing quantum transition behavior. Int J Nonlinear Sci Numer Simulat 2007;8(3):397–412. [13] Iovane G. E1 Naschie E-infinity Cantorian spacetime and lengths scales in cosmology. Int J Nonlinear Sci Numer Simulat 2006;7(2):155–62. [14] El Naschie MS. Thomas Mann and Heinrich Mann, dual brothers and complimentary genius embraced by complex reality. Int J Nonlinear Sci Numer Simulat 2006;7(1):1–6. [15] El Naschie MS. Linderhof room of mirrors, Thurston three-manifolds and the geometry of our universe. Int J Nonlinear Sci Numer Simulat 2006;7(1):97–100. [16] Iovane G, Benedetto E. El Naschie E-infinity cantorian space-time, Fantappie’s group and applications in cosmology. Int J Nonlinear Sci Numer Simulat 2005;6(4):357–70. [17] El Naschie MS. Einstein in a complex time – some very personal thoughts about E-infinity theory and modern physics. Int J Nonlinear Sci Numer Simulat 2005;6(3):331–3. [18] Horava P, Witten E. Heterotic and Type I string dynamics from eleven dimensions. Nucl Phys B 1996;460(3):506–24. [19] Witten E. String theory dynamics in various dimensions. Nucl Phys B 1995;443(1-2):85–126. [20] El Naschie MS. Probability set particles. Int J Nonlinear Sci Numer Simulat 2007;8(1):117–9. [21] Marek-Crnjac L. The maximum number of elementary particles in a super symmetric extension of the standard model. Chaos, Solitons & Fractals 2007;32(5):1631–6. [22] El Naschie MS. 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