- Email: [email protected]
Curvature, Lagrangian and holonomy of Cantorian-fractal spacetime
Chaos, Solitons and Fractals 41 (2009) 2163–2167
Contents lists available at ScienceDirect
Chaos, Solitons and Fractals journal homepage: www.elsevier.com/locate/chaos
Curvature, Lagrangian and holonomy of Cantorian-fractal spacetime M.S. El Naschie * Frankfurt Institute for the Advancement of Fundamental Research in Theoretical Physics, University of Frankfurt, Germany Department of Physics, University of Alexandria, Egypt
a r t i c l e
i n f o
a b s t r a c t The special holonomy of the Cantorian-fractal manifold of spacetime is considered in some detail. The expression for spacetime curvature, energy and the corresponding Lagrangian formulation is given. Numerous special aspects of the theory are discussed. Ó 2009 Elsevier Ltd. All rights reserved.
1. Introduction As we have stressed in various previous publications, the Cantorian-fractal E-infinity theory was largely developed from a rather topological and geometrical view point [1–3]. Energy, Lagrangian and the equations of motion were implicitly there but were seldom stressed explicitly [2]. In various recent publications we diverted our attention to energy considerations and gave an elementary discussion of the associated Lagrangian using Weyl-like gauge theory [4,5]. In the present work we explore and expand our energy considerations starting from the curvature of Cantorian spacetime and the special holonomy of E-infinity manifold. 2. The Gauss–Bonnet formula and the curvature of spacetime As is well know, the Euler characteristic is the most important information which one needs to know about a surface [6]. If v > 0 the geometry is elliptic, v = 0 means it is Euclidean and v < 0 means we have hyperbolic geometry. In general with v at our disposal, we know what global topology a surface has [6,7]. There is a remarkable formula, the Gauss–Bonnet formula which relates v to the area A and the curvature K of the surface [6,7]. For the simple case of a surface with constant curvature v, A and K are related by [6,7]
KA ¼ 2pv: Now let us just suppose that we could manipulate the situation so that A could be made to equal exactly 2p. If this is at all possible then we have a remarkable result, namely that the curvature K is equal to the Euler characteristic v [6–9]. This result will bring us a great deal forward because we know that the Euler characteristic of the K3 Kähler manifold is v = 24 and pffiffiffi 5 1 =2. At the same time when this Kähler is made fuzzy, then v changes to v = 26 + k where k = /3(1 /3) and / ¼ we know that the fuzzy K3 is a Kähler model for E-infinity spacetime which is supposed to be our real quantum spacetime [1–5]. Consequently v of our quantum spacetime is v = 26 + k. It follows then from the equality of K and v that the quantum curvature of the Cantorian-fractal quantum spacetime is also [5–9]
K ¼ 26 þ k: The preceding deduction is by no means a mathematical derivation, not least because we generalized from low dimensional surface to higher dimensional hierarchal and fuzzy manifold. It is at best a physicist mathematical elucidation and we need
* Address for correspondence: P.O. Box 272, Cobham, Surrey KT11 2FQ, UK. E-mail address: [email protected] 0960-0779/$ - see front matter Ó 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.chaos.2008.08.015
2164
M.S. El Naschie / Chaos, Solitons and Fractals 41 (2009) 2163–2167
a far more rigorous derivation. Never the less the above result will remain valid as we will see later on. The important thing gs ¼ 26 þ k. Thus now is that we know that the inverse coupling of super symmetric unification of all fundamental forces is a we could write the pseudo internal energy of E-infinity manifold as [5,8,9] 2
gs Þ2 ¼ ð26 þ kÞ2 ¼ 685:410197 W in k ða Remembering that [10] 17 X ðSteinÞ ¼ 658:410197 1
we see that the internal energy of our manifold is simply proportional to the sum over all dimensions of the 17 two and three Stein spaces well known from our previous work on compact and non-compact exceptional Lie and Stein spaces [10,11]. 3. The curvature of a Kähler manifold Before moving back to the fuzzy Kähler of Cantorian-fractal spacetime manifold, we review the general properties of the curvature of a crisp Kähler. In general a Riemann curvature tensor has two parts, the Ricci curvature and the scalar curvature [6,7]. A general tensor T abcd has 16 complex components. For a Kähler 12 of these components vanish leaving only 4. Utilizing the various symmetries of the Riemann curvature we are left with a single component Rabcd . Further consideration subsequently leads us to the realization that the Ricci curvature may be recovered from q which is a closed 2-form and that the cohomology class [q] is equal to 2pC1(M) where C1 (M) is the first Chern class of M [12]. Setting C1(M) = 24 as in the case of K3, one finds (2 p)(24). The next step is to transfinitely extend this expression to the fuzzy setting in the usual E-infinity way by setting
2p ! 2ð3 þ /3 Þ and
24 ! 26 þ k: Consequently one finds that
2pC 1 ðMÞ ! 168 þ 16k ¼ DimPs j SLð2; 7Þjc ¼
1 1 ½7ð72 1Þ þ 16/3 ð1 /3 Þ ¼ ½336 þ 2:885438 ¼ 168 þ 1:442719 2 2
¼ 169:442719: Again this is yet another remarkable result connecting the Ricci curvature of K3 in the fuzzy form with the dimension of the holographic boundary [11]. This boundary is nothing else but the compactified Klein modular curve described by the symmetry group LS (2,7). In this case our [q] is equal to the number of automorphisms which is half the number of dimensions. This is similar to the scalar curvature in E-infinity given by [8,9]
R¼
1 1 1 ðG=HÞ ¼ j F 4 j¼ ð52Þ ¼ 26 2 2 2
while here we have
½q ¼
1 1 j SLð2; 7Þjc ¼ ð336 þ 16kÞ ¼ 168 þ 8k: 2 2
Using both expressions we can deduce everything we need from our E-infinity spacetime model [1–4]. In fact 168 + 8k, when gauged in Gev is the expected mass of the Higgs boson. It is exactly half the expected constituent mass of the up and down 0 as a quasi curvature for the electromagquarks. They could be obtained from a quasi-Lagrangian using Weyl scaling using a netic section as follows:
0 Þ2 kð10Þa2 L ¼½ða dL 0 Þ2 kð10Þ ¼ 0: ¼0 ! ða da Thus we have
k ¼ ð137 þ k0 Þ2 =10 and
mH ¼ kð/Þ5 ¼ 168 þ 8k ¼ 169:44 Gev:
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M.S. El Naschie / Chaos, Solitons and Fractals 41 (2009) 2163–2167
4. The hyper Kähler manifold of E-infinity We start by defining S to be the fermat quartic. Subsequently one finds that S is a K3 by definition. From the Riemann– Roch formula we obtain v(S) = 24 while b0 = 1,b1 = 0 and b2 = 22. In addition the most important two Betti numbers are þ b2 ¼ 3 and b2 ¼ 19 [2,11]. þ
Theorem 1. Let (x, J) be a K3 surface. Then x is simply connected with Betti numbers b2 = 22, b2 ¼ 3 and b2 ¼ 19. Also x is a Kähler and the corresponding Hodge numbers are h2,0 = h0,2 = 1 and h1,1 = 20. Moving to our fuzzy K3 we have the following [2,11]: 2
þ
Theoremp 2.ffiffiffiTakea surface for which b0 ¼ 1; b1 ¼ 0; b3 ¼ 0; b4 ¼ 1; b ¼ 22; b2 ¼ 5 þ /3 and b2 ¼ 19 /6 where k = /3(1 /3) and / ¼ 5 1 =2, then this surface is the fuzzy K3 of E-infinity theory [2,11]. From Theorem 2 the fact that
v¼
X ð1Þi bi ¼ 26 þ k ¼ 26:18033989 i
follows. This was used earlier on to find the curvature k = v and then the energy. 5. The relation between vol M4, the holographic boundary of E-infinity and the quasi exceptional E12 Lie group The possibility, first suggested by R. Munroe for the existence of a quasi exceptional E12 Lie symmetry group was considered in various recent publications [11,13]. The dimensions of this group was found to be jE12j = 685. This is exactly equal to summing all the dimensions of the 17 two and three Stein spaces when transfinite corrections are included
DimE12 ¼
17 X ðSteinÞ ¼ 685: 1
At the same time the volume of the hyperbolic M4 fuzzy manifold is exactly half this value [13]
2volM 4H ¼ 685 ¼ Dim j E12 j : The important point in all of that is the fact that embedding the holographic boundary manifold in D = 4 we find the same value of vol M. Consequently we have the note worthy relation
DimE12 ¼ 2½SLð2; 7Þc þ ð4 kÞ ¼ 2½ð336 þ 16kÞ þ ð4 kÞ ¼ 2½340 þ 15k ¼ 2volM 4H ¼
17 X ðSteinÞ ffi 685: 1
We note that all the preceding remarkable coincidences between volume and dimensions is a natural consequence of the use of the Hausdorff dimension on a fundamental level. This dimension connects in a subtle way the notions volume and dimensions. In addition the use of the fuzzy geometry of fractals does the rest and goes hand in hand with the Hausdorff dimension which was invited initially for precisely the purpose of describing fractals. 6. The chern classes and ponteriagen index There is a very insightful relation between the Chern classes and the Ponteriagen index as far as the subject of the present work is concerned [12]. This is
P1 ðER Þ ¼ C 21 ðEÞ 2C 2 ðEÞ: Setting the first and second Chern-class equal to v = 24 one finds
P1 ðER Þ ¼ ð24Þ2 2ð24Þ ¼ 576 48 ¼ 528: Surprisingly this is exactly the number of states in the 5-Brane model of Duff and Witten. This is sufficient reason to warrant a more detailed analysis. First the 24 may be viewed as Dim SU(5) = jE4j of the exceptional E-line of Lie symmetry groups. However (2)(24) = 48 could be seen as the roots number of F4 exceptional group with jF4j = 52. Furthermore (24)2 = 576 may P be interpreted as 8i¼1 Ei ¼ 548 plus D4 = 28 which we discussed in earlier papers [11]. The Ponteriagen index so defined is indirectly a sum over a family of exceptional Lie group dimensions and may therefore be seen to have the meaning of a dimensionless energy expression. From the preceding discussion it is natural to investigate the situation for the fuzzy K3 with v = C1 = C2 = 26 + k. In this case one finds
0 j F 4 jc P1 ðER Þ ¼ ð26 þ kÞ2 ð2Þð26 þ kÞ ¼ 5a 0 ¼ 137 þ k0 is the inverse electromagnetic fine structure constant, jF4jc = 52 + 2k and k0 = /5(1 /5). This could also where a be written as
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M.S. El Naschie / Chaos, Solitons and Fractals 41 (2009) 2163–2167
P1 ðER Þ ¼j E12 j j F 4 jc ¼ 685:410197 52:3606799 ¼ 2 j M 4F j 2ð26 þ kÞ ¼ 633:049517: It is highly interesting to see that scaling P1 (ER) using /5 gives us the integer intrinsic dimension of E8 exceptional Lie group
½P 1 ðER Þð/5 Þ ¼ 57 þ k0 ’ 57: It is needless to stress that P1 (ER) max is attained for C1 = 26 + k and C2 = 0 and is equal to K2, i.e. the energy stored in the fuzzy K3 Kähler as discussed earlier on. It is exceedingly interesting to note the Dim E12 is found immediately from
j E12 j¼
pffiffiffiffiffi a 0 ð57 þ 1 þ 3kÞ ¼ 685:410197:
7. Discussion and conclusions The Cantorian-fractal spacetime theory as applied to high energy physics possessed a quartet base which we may represent diagrammatically as follows:
Symmetry
Number of elementary particles
Dimensions
Coupling constant
Thus we may move from one concept to another with incredible ease. Let us take as an example the inverse electromag 0 ’ 137. On the other hand a0 is a probability that an electron absorbs or emits a photon which netic fine structure constant a is as much as being a cross-section. This particular geometrical probability concept is called in this case a coupling constant. However this coupling constant may be treated exactly as the dimension of E8E8 or SL (2,7) or the number of independent 0 ¼ 137 is a dimension of a corresponding manifold as is obvious components in the Riemannian tensor. In other words a from the equation
a 0 ¼j E8 E8 j ½SLð2; 7Þc þ Rð4Þ ¼ 496 ½339 20 ¼ 137: At the same time we have shown on many previous occasions that 137 is also the maximum number of elementary particles in the standard model or the degrees of freedom of our theory where spin up and down are counted as different particles. 0 within our Cantorian fractal theory has a multitude of superficially different interpretations depending on the sitThus a uation considered. The situation with the relation between topology, geometry and dimensions is not dissimilar to the above. The Hausdorff dimension for instance has of course in the first instance the meaning of a dimension. However it is still somehow related via the procedure of covering to the meaning of an Ara. In addition the Hausdorff dimension being a measure for complexity also has the meaning of entropy and at the end of a chain of thought via thermodynamics, also a strong relation to energy. In the present paper we have unearthed some truly non-conventional connections. We found that energy could be expressed as the square of the coupling constant of super symmetric quantum gravity unification of all fundamental forces a gs because it is equal to the Euler characteristic which in turn is identical in this case to the curvature K. As if this is not enough, we found that the sum of the dimensions of all the 17 two and three Stein spaces is equal to the energy which is proportional to K2. Similar connections were recently found using knot theory [8,9] leading us to suspect that knots and exceptional Lie groups and Stein spaces are the building blocks of high energy physics like random Cantor sets are the building blocks of Cantorian-fractal spacetime [1–4]. Note that K = v only for volumeless fractals. References [1] El Naschie MS. A review of E-infinity theory and the mass spectrum of high energy particle physics. Chaos, Solitons & Fractals 2004;19:209–36. [2] El Naschie MS. Elementary prerequisites for E-infinity: recommended background reading in nonlinear dynamics, geometry and topology. Chaos Solitons & Fractals 2006;30(3):579–605. [3] Tanaka Y. The mass spectrum of hadrons and E-infinity theory. Chaos, Solitons & Fractals 2006;27:851–63. [4] El Naschie MS. From classical gauge theory back to Weyl scaling via E-infinity spacetime. Chaos, Solitons & Fractals 2008;38:980–5. [5] El Naschie MS. An energy balance Eigenvalue equation for determining super strings dimensional hierarchy and coupling constants. Chaos, Solitons & Fractals 2008;38(5):1283–5. [6] Nash C, Sen S. Topology and geometry for physicists. London: Academic Press; 1983. [7] Nakahara M. Geometry, topology and physics. Bristol: IOP; 2003. [8] El Naschie MS. Fuzzy knot theory interpretation of Yang-Mills instantons and Witten’s 5-Brane model. Chaos, Solitons & Fractals 2008;38(5):1349–54.
M.S. El Naschie / Chaos, Solitons and Fractals 41 (2009) 2163–2167
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[9] El Naschie MS. Fuzzy multi-instanton knots in the fabric of space-time and Dirac’s vacuum fluctuation. Chaos, Solitons & Fractals 2008;38(5):1260–8. [10] El Naschie MS. One and two-Stein space hierarchies in high energy physics. Chaos, Solitons & Fractals 2008;36(5):1189–90. [11] El Naschie MS. The exceptional Lie symmetry groups hierarchy and the expected number of Higgs bosons. Chaos, Solitons & Fractals 2008;35(2):268–73. [12] Gross M, Huybrechts D, Joyce D. Calabi-Yau manifolds and related geometries. Berlin: Springer; 2003. [13] El Naschie MS. Quasi exceptional E12 Lie symmetry groups with 685 dimensions, KAC-Moody algebra and E-infinity Cantorian spacetime. Chaos, Solitons & Fractals 2008;38:990–2.
Contents lists available at ScienceDirect
Chaos, Solitons and Fractals journal homepage: www.elsevier.com/locate/chaos
Curvature, Lagrangian and holonomy of Cantorian-fractal spacetime M.S. El Naschie * Frankfurt Institute for the Advancement of Fundamental Research in Theoretical Physics, University of Frankfurt, Germany Department of Physics, University of Alexandria, Egypt
a r t i c l e
i n f o
a b s t r a c t The special holonomy of the Cantorian-fractal manifold of spacetime is considered in some detail. The expression for spacetime curvature, energy and the corresponding Lagrangian formulation is given. Numerous special aspects of the theory are discussed. Ó 2009 Elsevier Ltd. All rights reserved.
1. Introduction As we have stressed in various previous publications, the Cantorian-fractal E-infinity theory was largely developed from a rather topological and geometrical view point [1–3]. Energy, Lagrangian and the equations of motion were implicitly there but were seldom stressed explicitly [2]. In various recent publications we diverted our attention to energy considerations and gave an elementary discussion of the associated Lagrangian using Weyl-like gauge theory [4,5]. In the present work we explore and expand our energy considerations starting from the curvature of Cantorian spacetime and the special holonomy of E-infinity manifold. 2. The Gauss–Bonnet formula and the curvature of spacetime As is well know, the Euler characteristic is the most important information which one needs to know about a surface [6]. If v > 0 the geometry is elliptic, v = 0 means it is Euclidean and v < 0 means we have hyperbolic geometry. In general with v at our disposal, we know what global topology a surface has [6,7]. There is a remarkable formula, the Gauss–Bonnet formula which relates v to the area A and the curvature K of the surface [6,7]. For the simple case of a surface with constant curvature v, A and K are related by [6,7]
KA ¼ 2pv: Now let us just suppose that we could manipulate the situation so that A could be made to equal exactly 2p. If this is at all possible then we have a remarkable result, namely that the curvature K is equal to the Euler characteristic v [6–9]. This result will bring us a great deal forward because we know that the Euler characteristic of the K3 Kähler manifold is v = 24 and pffiffiffi 5 1 =2. At the same time when this Kähler is made fuzzy, then v changes to v = 26 + k where k = /3(1 /3) and / ¼ we know that the fuzzy K3 is a Kähler model for E-infinity spacetime which is supposed to be our real quantum spacetime [1–5]. Consequently v of our quantum spacetime is v = 26 + k. It follows then from the equality of K and v that the quantum curvature of the Cantorian-fractal quantum spacetime is also [5–9]
K ¼ 26 þ k: The preceding deduction is by no means a mathematical derivation, not least because we generalized from low dimensional surface to higher dimensional hierarchal and fuzzy manifold. It is at best a physicist mathematical elucidation and we need
* Address for correspondence: P.O. Box 272, Cobham, Surrey KT11 2FQ, UK. E-mail address: [email protected] 0960-0779/$ - see front matter Ó 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.chaos.2008.08.015
2164
M.S. El Naschie / Chaos, Solitons and Fractals 41 (2009) 2163–2167
a far more rigorous derivation. Never the less the above result will remain valid as we will see later on. The important thing gs ¼ 26 þ k. Thus now is that we know that the inverse coupling of super symmetric unification of all fundamental forces is a we could write the pseudo internal energy of E-infinity manifold as [5,8,9] 2
gs Þ2 ¼ ð26 þ kÞ2 ¼ 685:410197 W in k ða Remembering that [10] 17 X ðSteinÞ ¼ 658:410197 1
we see that the internal energy of our manifold is simply proportional to the sum over all dimensions of the 17 two and three Stein spaces well known from our previous work on compact and non-compact exceptional Lie and Stein spaces [10,11]. 3. The curvature of a Kähler manifold Before moving back to the fuzzy Kähler of Cantorian-fractal spacetime manifold, we review the general properties of the curvature of a crisp Kähler. In general a Riemann curvature tensor has two parts, the Ricci curvature and the scalar curvature [6,7]. A general tensor T abcd has 16 complex components. For a Kähler 12 of these components vanish leaving only 4. Utilizing the various symmetries of the Riemann curvature we are left with a single component Rabcd . Further consideration subsequently leads us to the realization that the Ricci curvature may be recovered from q which is a closed 2-form and that the cohomology class [q] is equal to 2pC1(M) where C1 (M) is the first Chern class of M [12]. Setting C1(M) = 24 as in the case of K3, one finds (2 p)(24). The next step is to transfinitely extend this expression to the fuzzy setting in the usual E-infinity way by setting
2p ! 2ð3 þ /3 Þ and
24 ! 26 þ k: Consequently one finds that
2pC 1 ðMÞ ! 168 þ 16k ¼ DimPs j SLð2; 7Þjc ¼
1 1 ½7ð72 1Þ þ 16/3 ð1 /3 Þ ¼ ½336 þ 2:885438 ¼ 168 þ 1:442719 2 2
¼ 169:442719: Again this is yet another remarkable result connecting the Ricci curvature of K3 in the fuzzy form with the dimension of the holographic boundary [11]. This boundary is nothing else but the compactified Klein modular curve described by the symmetry group LS (2,7). In this case our [q] is equal to the number of automorphisms which is half the number of dimensions. This is similar to the scalar curvature in E-infinity given by [8,9]
R¼
1 1 1 ðG=HÞ ¼ j F 4 j¼ ð52Þ ¼ 26 2 2 2
while here we have
½q ¼
1 1 j SLð2; 7Þjc ¼ ð336 þ 16kÞ ¼ 168 þ 8k: 2 2
Using both expressions we can deduce everything we need from our E-infinity spacetime model [1–4]. In fact 168 + 8k, when gauged in Gev is the expected mass of the Higgs boson. It is exactly half the expected constituent mass of the up and down 0 as a quasi curvature for the electromagquarks. They could be obtained from a quasi-Lagrangian using Weyl scaling using a netic section as follows:
0 Þ2 kð10Þa2 L ¼½ða dL 0 Þ2 kð10Þ ¼ 0: ¼0 ! ða da Thus we have
k ¼ ð137 þ k0 Þ2 =10 and
mH ¼ kð/Þ5 ¼ 168 þ 8k ¼ 169:44 Gev:
2165
M.S. El Naschie / Chaos, Solitons and Fractals 41 (2009) 2163–2167
4. The hyper Kähler manifold of E-infinity We start by defining S to be the fermat quartic. Subsequently one finds that S is a K3 by definition. From the Riemann– Roch formula we obtain v(S) = 24 while b0 = 1,b1 = 0 and b2 = 22. In addition the most important two Betti numbers are þ b2 ¼ 3 and b2 ¼ 19 [2,11]. þ
Theorem 1. Let (x, J) be a K3 surface. Then x is simply connected with Betti numbers b2 = 22, b2 ¼ 3 and b2 ¼ 19. Also x is a Kähler and the corresponding Hodge numbers are h2,0 = h0,2 = 1 and h1,1 = 20. Moving to our fuzzy K3 we have the following [2,11]: 2
þ
Theoremp 2.ffiffiffiTakea surface for which b0 ¼ 1; b1 ¼ 0; b3 ¼ 0; b4 ¼ 1; b ¼ 22; b2 ¼ 5 þ /3 and b2 ¼ 19 /6 where k = /3(1 /3) and / ¼ 5 1 =2, then this surface is the fuzzy K3 of E-infinity theory [2,11]. From Theorem 2 the fact that
v¼
X ð1Þi bi ¼ 26 þ k ¼ 26:18033989 i
follows. This was used earlier on to find the curvature k = v and then the energy. 5. The relation between vol M4, the holographic boundary of E-infinity and the quasi exceptional E12 Lie group The possibility, first suggested by R. Munroe for the existence of a quasi exceptional E12 Lie symmetry group was considered in various recent publications [11,13]. The dimensions of this group was found to be jE12j = 685. This is exactly equal to summing all the dimensions of the 17 two and three Stein spaces when transfinite corrections are included
DimE12 ¼
17 X ðSteinÞ ¼ 685: 1
At the same time the volume of the hyperbolic M4 fuzzy manifold is exactly half this value [13]
2volM 4H ¼ 685 ¼ Dim j E12 j : The important point in all of that is the fact that embedding the holographic boundary manifold in D = 4 we find the same value of vol M. Consequently we have the note worthy relation
DimE12 ¼ 2½SLð2; 7Þc þ ð4 kÞ ¼ 2½ð336 þ 16kÞ þ ð4 kÞ ¼ 2½340 þ 15k ¼ 2volM 4H ¼
17 X ðSteinÞ ffi 685: 1
We note that all the preceding remarkable coincidences between volume and dimensions is a natural consequence of the use of the Hausdorff dimension on a fundamental level. This dimension connects in a subtle way the notions volume and dimensions. In addition the use of the fuzzy geometry of fractals does the rest and goes hand in hand with the Hausdorff dimension which was invited initially for precisely the purpose of describing fractals. 6. The chern classes and ponteriagen index There is a very insightful relation between the Chern classes and the Ponteriagen index as far as the subject of the present work is concerned [12]. This is
P1 ðER Þ ¼ C 21 ðEÞ 2C 2 ðEÞ: Setting the first and second Chern-class equal to v = 24 one finds
P1 ðER Þ ¼ ð24Þ2 2ð24Þ ¼ 576 48 ¼ 528: Surprisingly this is exactly the number of states in the 5-Brane model of Duff and Witten. This is sufficient reason to warrant a more detailed analysis. First the 24 may be viewed as Dim SU(5) = jE4j of the exceptional E-line of Lie symmetry groups. However (2)(24) = 48 could be seen as the roots number of F4 exceptional group with jF4j = 52. Furthermore (24)2 = 576 may P be interpreted as 8i¼1 Ei ¼ 548 plus D4 = 28 which we discussed in earlier papers [11]. The Ponteriagen index so defined is indirectly a sum over a family of exceptional Lie group dimensions and may therefore be seen to have the meaning of a dimensionless energy expression. From the preceding discussion it is natural to investigate the situation for the fuzzy K3 with v = C1 = C2 = 26 + k. In this case one finds
0 j F 4 jc P1 ðER Þ ¼ ð26 þ kÞ2 ð2Þð26 þ kÞ ¼ 5a 0 ¼ 137 þ k0 is the inverse electromagnetic fine structure constant, jF4jc = 52 + 2k and k0 = /5(1 /5). This could also where a be written as
2166
M.S. El Naschie / Chaos, Solitons and Fractals 41 (2009) 2163–2167
P1 ðER Þ ¼j E12 j j F 4 jc ¼ 685:410197 52:3606799 ¼ 2 j M 4F j 2ð26 þ kÞ ¼ 633:049517: It is highly interesting to see that scaling P1 (ER) using /5 gives us the integer intrinsic dimension of E8 exceptional Lie group
½P 1 ðER Þð/5 Þ ¼ 57 þ k0 ’ 57: It is needless to stress that P1 (ER) max is attained for C1 = 26 + k and C2 = 0 and is equal to K2, i.e. the energy stored in the fuzzy K3 Kähler as discussed earlier on. It is exceedingly interesting to note the Dim E12 is found immediately from
j E12 j¼
pffiffiffiffiffi a 0 ð57 þ 1 þ 3kÞ ¼ 685:410197:
7. Discussion and conclusions The Cantorian-fractal spacetime theory as applied to high energy physics possessed a quartet base which we may represent diagrammatically as follows:
Symmetry
Number of elementary particles
Dimensions
Coupling constant
Thus we may move from one concept to another with incredible ease. Let us take as an example the inverse electromag 0 ’ 137. On the other hand a0 is a probability that an electron absorbs or emits a photon which netic fine structure constant a is as much as being a cross-section. This particular geometrical probability concept is called in this case a coupling constant. However this coupling constant may be treated exactly as the dimension of E8E8 or SL (2,7) or the number of independent 0 ¼ 137 is a dimension of a corresponding manifold as is obvious components in the Riemannian tensor. In other words a from the equation
a 0 ¼j E8 E8 j ½SLð2; 7Þc þ Rð4Þ ¼ 496 ½339 20 ¼ 137: At the same time we have shown on many previous occasions that 137 is also the maximum number of elementary particles in the standard model or the degrees of freedom of our theory where spin up and down are counted as different particles. 0 within our Cantorian fractal theory has a multitude of superficially different interpretations depending on the sitThus a uation considered. The situation with the relation between topology, geometry and dimensions is not dissimilar to the above. The Hausdorff dimension for instance has of course in the first instance the meaning of a dimension. However it is still somehow related via the procedure of covering to the meaning of an Ara. In addition the Hausdorff dimension being a measure for complexity also has the meaning of entropy and at the end of a chain of thought via thermodynamics, also a strong relation to energy. In the present paper we have unearthed some truly non-conventional connections. We found that energy could be expressed as the square of the coupling constant of super symmetric quantum gravity unification of all fundamental forces a gs because it is equal to the Euler characteristic which in turn is identical in this case to the curvature K. As if this is not enough, we found that the sum of the dimensions of all the 17 two and three Stein spaces is equal to the energy which is proportional to K2. Similar connections were recently found using knot theory [8,9] leading us to suspect that knots and exceptional Lie groups and Stein spaces are the building blocks of high energy physics like random Cantor sets are the building blocks of Cantorian-fractal spacetime [1–4]. Note that K = v only for volumeless fractals. References [1] El Naschie MS. A review of E-infinity theory and the mass spectrum of high energy particle physics. Chaos, Solitons & Fractals 2004;19:209–36. [2] El Naschie MS. Elementary prerequisites for E-infinity: recommended background reading in nonlinear dynamics, geometry and topology. Chaos Solitons & Fractals 2006;30(3):579–605. [3] Tanaka Y. The mass spectrum of hadrons and E-infinity theory. Chaos, Solitons & Fractals 2006;27:851–63. [4] El Naschie MS. From classical gauge theory back to Weyl scaling via E-infinity spacetime. Chaos, Solitons & Fractals 2008;38:980–5. [5] El Naschie MS. An energy balance Eigenvalue equation for determining super strings dimensional hierarchy and coupling constants. Chaos, Solitons & Fractals 2008;38(5):1283–5. [6] Nash C, Sen S. Topology and geometry for physicists. London: Academic Press; 1983. [7] Nakahara M. Geometry, topology and physics. Bristol: IOP; 2003. [8] El Naschie MS. Fuzzy knot theory interpretation of Yang-Mills instantons and Witten’s 5-Brane model. Chaos, Solitons & Fractals 2008;38(5):1349–54.
M.S. El Naschie / Chaos, Solitons and Fractals 41 (2009) 2163–2167
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[9] El Naschie MS. Fuzzy multi-instanton knots in the fabric of space-time and Dirac’s vacuum fluctuation. Chaos, Solitons & Fractals 2008;38(5):1260–8. [10] El Naschie MS. One and two-Stein space hierarchies in high energy physics. Chaos, Solitons & Fractals 2008;36(5):1189–90. [11] El Naschie MS. The exceptional Lie symmetry groups hierarchy and the expected number of Higgs bosons. Chaos, Solitons & Fractals 2008;35(2):268–73. [12] Gross M, Huybrechts D, Joyce D. Calabi-Yau manifolds and related geometries. Berlin: Springer; 2003. [13] El Naschie MS. Quasi exceptional E12 Lie symmetry groups with 685 dimensions, KAC-Moody algebra and E-infinity Cantorian spacetime. Chaos, Solitons & Fractals 2008;38:990–2.