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Notes on exceptional lie symmetry groups hierarchy and possible implications for E-Infinity high energy physics
Chaos, Solitons and Fractals 35 (2008) 67–70 www.elsevier.com/locate/chaos
Notes on exceptional lie symmetry groups hierarchy and possible implications for E-Infinity high energy physics M.S. El Naschie
1
Department of Physics, University of Alexandria, Alexandria, Egypt Donghua University, Shanghai, PR China
Abstract The classical exceptional lie group hierarchy E8 ; E6 ; E7 ; F 2 and G2 is reconsidered in connection with SU(5) and SO(10) unification. Subsequently, various interpretations are presented which are relevant to the standard model and E-Infinity physics. Ó 2007 Elsevier Ltd. All rights reserved.
1. Introduction In various recent publications the exceptional lie symmetry groups [1–3] were considered in connection with the standard model and E-Infinity physics [4,5]. In the present note we continue this discussion and highlight various highly interesting features of some extended exceptional lie groups hierarchy [1–3]. In particular, we start by a fundamental topological–geometrical picture of cantorian spacetime fractal manifold which possesses infinitely many hierarchical dimensions with four as the expectation number. To each of these four dimensions, there are a0 ¼ 137 þ k 0 extra dimensions joined to them. The total dimensions of this space is therefore equal to the expectation topological a0 which gives 4 a0 ¼ 548 þ 4k 0 pffiffiffi dimension of E-Infinity namely 4 multiplied by where k0 = /5(1 /5) and / ¼ ð 5 1Þ=2. This leads to a total number of quasi-dimensions or what is equivalently total number of isometries equal to [4–6] NðtotalÞ ¼ 4a0 ¼ 548:3281572 ’ 548 Interestingly the grand total is exactly 20 isometries more than the total number of states given by the p = 5 Brane of Witten’s M theory 11 11 11 ð11Þ N ðBraneÞ ¼ þ þ ¼ 528 1 2 5 and 32 isometries more than the number of mass-less gauge bosons of the Green, Schwartz superstring theory namely N = 496. Since this value is nothing else but the dimension of the largest of the exceptional lie groups E8E8 where [1–6]
1
E-mail address: [email protected] Address for correspondence: P.O. Box 272, Cobham, Surrey KT11 2FQ, UK.
0960-0779/$ - see front matter Ó 2007 Elsevier Ltd. All rights reserved. doi:10.1016/j.chaos.2007.05.013
68
M.S. El Naschie / Chaos, Solitons and Fractals 35 (2008) 67–70
¼ Dim E8 E8 ¼ 2ð248Þ ¼ 496 then it is quite reasonable to ask if more general relationships do exist between the exceptional lie groups hierarchy E8 ; E6 ; E7 ; F2 and G2 and the total number of states 548. This way of thinking is clearly motivated by the essentially hierarchical structure of E-Infinity theory which we suspect to be the reason behind the hierarchical structure of the exceptional lie symmetry groups and the success of E8 E8 in ridding the old string theory from all its maladies.
2. The first exceptional lie group hierarchy As in previous works, we consider the first hierarchy [1–3] jE8 j ¼ 248;
jE7 j ¼ 133;
jE6 j ¼ 78;
jE5 j ¼ jSOð10Þj ¼ 45 and jE4 j ¼ jSUð5Þj ¼ 24
Adding all dimensions (see Fig. 1) one finds, i¼8 X
jEi j ¼ 528
i¼4
We note that this is exactly equal to N(11), (Brane) as well as to the isometries of a maximally symmetric manifold [5] Next adding the standard model, one finds 528 þ SUð3ÞSUð2ÞUð1Þ ¼ 528 þ 12 ¼ 540 This is eight isometries short of 548 and one could speculate that this may be related to a Higgs field with eight degrees of freedom which will at the end contribute only one Higgs boson to the standard model because [4–6] N ðSMÞ ¼ 540=8 ¼ 67:5 While N ðSMÞ ¼ 548=8 ¼ 68:5 which is almost equal to the exact result of E-Infinity theory namely, N ðSMÞ ¼ a0 =2 ¼ 68:54101965 ’ 69 All of the above is summarized in Fig. 1.
3. The second exceptional lie group hierarchy The second possibility which we will consider here is shown in Fig. 2. Thus our hierarchy is given by [1–6] jE8 j;
jE7 j;
jE6 j;
jF 4 j;
and jE4 j
248 133 78
Exceptional lie groups hierarchy
45
E8
24
E7
E6
12
E5 E 4
8 SU(3)SU(2)U(1)
548= (4) (137)
Fig. 1.
M.S. El Naschie / Chaos, Solitons and Fractals 35 (2008) 67–70
69
248 133 78
Exceptional lie groups hierarchy
52
E8
24 12
E7
1
E6
SU(3)SU(2)U(1)
F4
E4
548= (4) (137)
Fig. 2.
Adding all dimensions, one finds N¼
i¼8 X
jEi j þ jF 4 j þ jE4 j ¼ 459 þ 52 þ 24 ¼ 535
i¼4
Adding the standard model, one finds N ¼ 535 þ SUð3ÞSUð2ÞUð1Þ ¼ 535 þ 12 ¼ 547 In other words only a single isometry is missing to complete the picture and reach 548. This ‘‘particle’’ contributes only 1/8 real particle to the standard model.
4. The third exceptional lie group hierarchy As displayed in Fig. 3, the third exceptional lie hierarchy to be considered is given by [1–6] 11 jE8 j; jE7 j; jE6 j; ¼ 55; jG2 j ¼ 14 2
Exceptional lie groups hierarchy
E8
248 133 78
E7
55 14
E6 2
12
G2
8 SU(3)SU(2)U(1)
548= (4) (137)
Fig. 3.
70
M.S. El Naschie / Chaos, Solitons and Fractals 35 (2008) 67–70
11 ¼ 55 is not an exceptional lie group which is the only exception in the present computation. However, 2 the 55 are clearly motivated by P = 2 Brane theory in D = 11. Consequently, one finds 11 þ jG2 j ¼ 528 N ¼ jE8 j þ jE7 j þ jE6 j þ 2 Of course
Adding the standard model and the 8 extra states, one finds our earlier results of the first hierarchy.
5. The Fuzzy Ka¨hler corresponding to the 548 isometries Remembering the two fundamental fuzzy ka¨hler manifolds of E-Infinity theory [5,7] we can conjecture the existence of a third fuzzy manifold corresponding to jE1j = 548. This manifold has jKE1j = 548 + 4k0 and a dual instanton den0 Þð16þkÞ sity n = 16 + k. Consequently N ðSMÞ ¼ ð548þ4k ¼ ao =2 exactly as should be. ð128þ8kÞ
6. The transfinitly exact lie groups hierarchy To obtain the transfinitly exact exceptional lie group hierarchy we note the following transfinite corrections: jE8 j ¼ 248 ! 248 K 2 jE7 j ¼ 133 ! 133 8K /12 jE6 j ¼ 78 ! ð137 þ K 0 Þ=2 þ 10 jE5 j ¼ 52 ! 52 þ 2K jE4 j ¼ 24 ! 26 þ K SUð3ÞSUð2ÞUð1Þ ¼ 12 ! 11:70820393 a0 =2. Added together gives 4a0 ¼ 548:3281572. Dividing by 8 one finds the exact expectation NðSMÞ ¼
7. Conclusion The exceptional lie symmetry groups form a nested Russian doll-like hierarchy involving the standard model as well as the two most prominent grand unification proposals, the SU(5) and the SO(10). The fact that this hierarchy led to physically meaningful results in high energy physics may indicate that this hierarchy is a consequence of the E-Infinity spacetime hierarchy. Finally we have conjectured the existence of a new large fuzzy manifold with a dual instanton number n = 16 + k and a total dimension 548 + 4k0.
References [1] Elnaschie MS. Symmetry group prerequisite for E-Infinity in high energy physics. Chaos, Solitons & Fractals, in press. doi:10.1016/ j.chaos.2007.05.006. [2] Georgi H. Lie algebras in particle physics. Boulder: Westview press; 1999. [3] Elnaschie MS. Noether’s theorem, exceptional lie group hierarchy and determining 1/a ffi 137 of electromagnetism. Chaos, Solitons & Fractals, in press. doi:10.1016/j.chaos.2007.05.005. [4] Elnaschie MS. The exceptional lie symmetry groups hierarchy and the expected number of Higgs bosons. Chaos, Solitons & Fractals [in press]. [5] He Ji-Huan. Transfinite physics. Shanghai, PR China. ISBN988-98846-5-8; 2005. [6] He Ji-Huan, Sigalotti LD, Mejias M. Beyond the 2006 physics Nobel prize for Cobe. Shanghai, PR China. ISBN 988-97681-9-4/04; 2006. [7] Elnaschie MS. On D. Gross’ criticism of S. Eddington and the exact calculation of a0 ¼ 137. Chaos, Solitons & Fractals 2007;32:1245–9.
Notes on exceptional lie symmetry groups hierarchy and possible implications for E-Infinity high energy physics M.S. El Naschie
1
Department of Physics, University of Alexandria, Alexandria, Egypt Donghua University, Shanghai, PR China
Abstract The classical exceptional lie group hierarchy E8 ; E6 ; E7 ; F 2 and G2 is reconsidered in connection with SU(5) and SO(10) unification. Subsequently, various interpretations are presented which are relevant to the standard model and E-Infinity physics. Ó 2007 Elsevier Ltd. All rights reserved.
1. Introduction In various recent publications the exceptional lie symmetry groups [1–3] were considered in connection with the standard model and E-Infinity physics [4,5]. In the present note we continue this discussion and highlight various highly interesting features of some extended exceptional lie groups hierarchy [1–3]. In particular, we start by a fundamental topological–geometrical picture of cantorian spacetime fractal manifold which possesses infinitely many hierarchical dimensions with four as the expectation number. To each of these four dimensions, there are a0 ¼ 137 þ k 0 extra dimensions joined to them. The total dimensions of this space is therefore equal to the expectation topological a0 which gives 4 a0 ¼ 548 þ 4k 0 pffiffiffi dimension of E-Infinity namely 4 multiplied by where k0 = /5(1 /5) and / ¼ ð 5 1Þ=2. This leads to a total number of quasi-dimensions or what is equivalently total number of isometries equal to [4–6] NðtotalÞ ¼ 4a0 ¼ 548:3281572 ’ 548 Interestingly the grand total is exactly 20 isometries more than the total number of states given by the p = 5 Brane of Witten’s M theory 11 11 11 ð11Þ N ðBraneÞ ¼ þ þ ¼ 528 1 2 5 and 32 isometries more than the number of mass-less gauge bosons of the Green, Schwartz superstring theory namely N = 496. Since this value is nothing else but the dimension of the largest of the exceptional lie groups E8E8 where [1–6]
1
E-mail address: [email protected] Address for correspondence: P.O. Box 272, Cobham, Surrey KT11 2FQ, UK.
0960-0779/$ - see front matter Ó 2007 Elsevier Ltd. All rights reserved. doi:10.1016/j.chaos.2007.05.013
68
M.S. El Naschie / Chaos, Solitons and Fractals 35 (2008) 67–70
¼ Dim E8 E8 ¼ 2ð248Þ ¼ 496 then it is quite reasonable to ask if more general relationships do exist between the exceptional lie groups hierarchy E8 ; E6 ; E7 ; F2 and G2 and the total number of states 548. This way of thinking is clearly motivated by the essentially hierarchical structure of E-Infinity theory which we suspect to be the reason behind the hierarchical structure of the exceptional lie symmetry groups and the success of E8 E8 in ridding the old string theory from all its maladies.
2. The first exceptional lie group hierarchy As in previous works, we consider the first hierarchy [1–3] jE8 j ¼ 248;
jE7 j ¼ 133;
jE6 j ¼ 78;
jE5 j ¼ jSOð10Þj ¼ 45 and jE4 j ¼ jSUð5Þj ¼ 24
Adding all dimensions (see Fig. 1) one finds, i¼8 X
jEi j ¼ 528
i¼4
We note that this is exactly equal to N(11), (Brane) as well as to the isometries of a maximally symmetric manifold [5] Next adding the standard model, one finds 528 þ SUð3ÞSUð2ÞUð1Þ ¼ 528 þ 12 ¼ 540 This is eight isometries short of 548 and one could speculate that this may be related to a Higgs field with eight degrees of freedom which will at the end contribute only one Higgs boson to the standard model because [4–6] N ðSMÞ ¼ 540=8 ¼ 67:5 While N ðSMÞ ¼ 548=8 ¼ 68:5 which is almost equal to the exact result of E-Infinity theory namely, N ðSMÞ ¼ a0 =2 ¼ 68:54101965 ’ 69 All of the above is summarized in Fig. 1.
3. The second exceptional lie group hierarchy The second possibility which we will consider here is shown in Fig. 2. Thus our hierarchy is given by [1–6] jE8 j;
jE7 j;
jE6 j;
jF 4 j;
and jE4 j
248 133 78
Exceptional lie groups hierarchy
45
E8
24
E7
E6
12
E5 E 4
8 SU(3)SU(2)U(1)
548= (4) (137)
Fig. 1.
M.S. El Naschie / Chaos, Solitons and Fractals 35 (2008) 67–70
69
248 133 78
Exceptional lie groups hierarchy
52
E8
24 12
E7
1
E6
SU(3)SU(2)U(1)
F4
E4
548= (4) (137)
Fig. 2.
Adding all dimensions, one finds N¼
i¼8 X
jEi j þ jF 4 j þ jE4 j ¼ 459 þ 52 þ 24 ¼ 535
i¼4
Adding the standard model, one finds N ¼ 535 þ SUð3ÞSUð2ÞUð1Þ ¼ 535 þ 12 ¼ 547 In other words only a single isometry is missing to complete the picture and reach 548. This ‘‘particle’’ contributes only 1/8 real particle to the standard model.
4. The third exceptional lie group hierarchy As displayed in Fig. 3, the third exceptional lie hierarchy to be considered is given by [1–6] 11 jE8 j; jE7 j; jE6 j; ¼ 55; jG2 j ¼ 14 2
Exceptional lie groups hierarchy
E8
248 133 78
E7
55 14
E6 2
12
G2
8 SU(3)SU(2)U(1)
548= (4) (137)
Fig. 3.
70
M.S. El Naschie / Chaos, Solitons and Fractals 35 (2008) 67–70
11 ¼ 55 is not an exceptional lie group which is the only exception in the present computation. However, 2 the 55 are clearly motivated by P = 2 Brane theory in D = 11. Consequently, one finds 11 þ jG2 j ¼ 528 N ¼ jE8 j þ jE7 j þ jE6 j þ 2 Of course
Adding the standard model and the 8 extra states, one finds our earlier results of the first hierarchy.
5. The Fuzzy Ka¨hler corresponding to the 548 isometries Remembering the two fundamental fuzzy ka¨hler manifolds of E-Infinity theory [5,7] we can conjecture the existence of a third fuzzy manifold corresponding to jE1j = 548. This manifold has jKE1j = 548 + 4k0 and a dual instanton den0 Þð16þkÞ sity n = 16 + k. Consequently N ðSMÞ ¼ ð548þ4k ¼ ao =2 exactly as should be. ð128þ8kÞ
6. The transfinitly exact lie groups hierarchy To obtain the transfinitly exact exceptional lie group hierarchy we note the following transfinite corrections: jE8 j ¼ 248 ! 248 K 2 jE7 j ¼ 133 ! 133 8K /12 jE6 j ¼ 78 ! ð137 þ K 0 Þ=2 þ 10 jE5 j ¼ 52 ! 52 þ 2K jE4 j ¼ 24 ! 26 þ K SUð3ÞSUð2ÞUð1Þ ¼ 12 ! 11:70820393 a0 =2. Added together gives 4a0 ¼ 548:3281572. Dividing by 8 one finds the exact expectation NðSMÞ ¼
7. Conclusion The exceptional lie symmetry groups form a nested Russian doll-like hierarchy involving the standard model as well as the two most prominent grand unification proposals, the SU(5) and the SO(10). The fact that this hierarchy led to physically meaningful results in high energy physics may indicate that this hierarchy is a consequence of the E-Infinity spacetime hierarchy. Finally we have conjectured the existence of a new large fuzzy manifold with a dual instanton number n = 16 + k and a total dimension 548 + 4k0.
References [1] Elnaschie MS. Symmetry group prerequisite for E-Infinity in high energy physics. Chaos, Solitons & Fractals, in press. doi:10.1016/ j.chaos.2007.05.006. [2] Georgi H. Lie algebras in particle physics. Boulder: Westview press; 1999. [3] Elnaschie MS. Noether’s theorem, exceptional lie group hierarchy and determining 1/a ffi 137 of electromagnetism. Chaos, Solitons & Fractals, in press. doi:10.1016/j.chaos.2007.05.005. [4] Elnaschie MS. The exceptional lie symmetry groups hierarchy and the expected number of Higgs bosons. Chaos, Solitons & Fractals [in press]. [5] He Ji-Huan. Transfinite physics. Shanghai, PR China. ISBN988-98846-5-8; 2005. [6] He Ji-Huan, Sigalotti LD, Mejias M. Beyond the 2006 physics Nobel prize for Cobe. Shanghai, PR China. ISBN 988-97681-9-4/04; 2006. [7] Elnaschie MS. On D. Gross’ criticism of S. Eddington and the exact calculation of a0 ¼ 137. Chaos, Solitons & Fractals 2007;32:1245–9.