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SU(5) grand unification in a transfinite form
Chaos, Solitons and Fractals 32 (2007) 370–374 www.elsevier.com/locate/chaos
SU(5) grand unification in a transfinite form M.S. El Naschie
1
King Abdul Aziz City of Science & Technology, Riyadh, Saudi Arabia
Abstract The SU(5) grand unification with its jSU(5)j = 24 gauge Bosons is partially reformulated in a transfinite setting. By means of transfinite continuation it is shown that a new version of the theory yields an expectation value hjSU(5)jci = 26 + k instead of the classical 24. By systematically exploring the non-super symmetric SU(5) scheme and transforming many of its fundamental aspects, it becomes plausible that it is a fundamental theory which could be integrated in various other fundamental theories including the transfinite forms of super strings and M theory. 2006 Elsevier Ltd. All rights reserved.
1. Introduction Despite failure to predict the right time scale for protons decay, the non-super symmetric SU(5) grand unification is a highly elegant theory and the seamless fitting of the standard model SU(3) SU(2) U(1) into its fundamental symmetry group SU(5) makes it extremely difficult to believe that the theory could be entirely wrong [1–3]. In fact, and as we will be arguing in the present short work, the theory is substantially correct, at least in its basic structure. Using the technique of transfinite continuation we will show that SU(5) unification could be given a twist along the lines of E-infinity theory and link it to a family of other similar theories formulated in a fuzzy setting [3–23]. 2. The SU(5) grand unification The standard model of high energy physics is based on a combination of three symmetry groups, the order of which gives us 12 gauge Bosons [3,10,11] jSUð3Þ SUð2Þ Uð1Þj ¼ 12 By contrast the SU(5) model has double as much, namely [1,2] jSUð5Þj ¼ 24 where besides the photon (c), the W±, the Z0 and the 8 Gluons we have an additional 12 exotic gauge Bosons. Thus while we have 12 flavour and colour forces in the standard model [1–3] 1
P.O. Box 272, Cobham, Surrey KT11 2FQ, United Kingdom.
0960-0779/$ - see front matter 2006 Elsevier Ltd. All rights reserved. doi:10.1016/j.chaos.2006.09.018
M.S. El Naschie / Chaos, Solitons and Fractals 32 (2007) 370–374
371
Gal ; W l ; Z l ; Al an additional 12 super weak and lepton-quark forces e il ; Y e il Xil ; Yil ; X are assumed to exist in SU(5). The anomaly free combination of SU(5) representation is given by { 5} + {10} and the relevant multiplets are shown in Fig. 8 of Ref. [3]. The {24} of SU(5) under SU(3)c · SU(2)L · U(1) decomposes as follows [1–3] f24g ¼ f8; 1g þ f1; 3g þ f3; 2g þ f3 ; 2g þ f1; 1g To find jSU(5)j from the above we just need to proceed formally as below: jSUð5Þj ¼ ð8Þð1Þ þ ð1Þð3Þ þ ð3Þð2Þ þ ð3 Þð2Þ þ ð1Þð1Þ ¼ 8 þ 3 þ 6 þ 6 þ 1 ¼ 24: We will refer to this as the multiplication order which is normally given by N2 1 = 25 1 = 24. For later use we mention that a second formal way to find our 24 is to write jSUð5Þj ¼ ½8 þ 1 þ 1 þ 3 þ 3 þ 2 þ 3 þ 2 þ 1 þ 1 1 ¼ ½25 1 ¼ 24: This is nothing but a trivial way of rewriting jSU(5)j = N2 1 as N2 = jSU(5)j + 1 where [25] = N2. The value N2 will be labelled the additive order. By choosing SU(5), the couplings of the electroweak unifications are fixed [1,2] rffiffiffi 3 gSUð5Þ g1 ¼ 5 and because gSUð5Þ ¼ g and tan hw g1 =g one finds that tan hw ¼
rffiffiffi 3 5
Therefore the Weinberg mixing angle is given in this case by sin2 hw ¼ 3=8 A further prediction of SU(5) grand unification is that a1 ðM x Þ ¼ a2 ðM x Þ ¼ a3 ðM x Þ ¼ au where Mx ’ 1015 Gev is the unification energy and au ’ 1/50 is the unification coupling. It should be noted however that extrapolation of measured values at LEP and SLC shows a lack of unification of gauge coupling for the standard model, but that averaging based on the centre of gravity of the triangle formed by the three lines of the running coupling gives au ’ 1/42, rather than au ’ 1/50. Interestingly, L. Marek-Crnjac in Slovenia also found that au ’ 1/42 while the exact value predicted by E-infinity theory is ag =1/au = 42.36067977. 3. Transfinite continuation of SU(5) At a minimum, fuzzy sets, fuzzy logic and fuzzy geometry have been for decades infiltrating numerous branches of engineering and technology [5–7] and in a relatively lesser measure, also physics [4,8,9]. The concepts of fuzziness come under various names and a particularly important category increasingly used in high energy physics is that connected to fractal statistics and fractal geometry [23]. In particular E-infinity theory used a form of transfinite fuzziness which was utilized to solve numerous problems in quantum mechanics [14] and high energy physics. Thus the notion of a non-integer fractal dimension has been generalized to the extent of talking in precise terms about 0.23606799 particles or 6.1803398977 elementary particles. Translated
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to our classical world, these statements are simply statistical or probabilistic expectation values and as such, not as strange as they sound. On the other hand it comes initially as a little surprise that many well known mathematical formulas envisaged exclusively for integer values, could be extended with a minimal effort to the irrational transfinite or fractal domain to yield highly surprising results with new physical insights. A trivial example of this kind of transfinite continuation with far reaching consequences may be the following one connected to the SO(N) symmetry groups. Taking 10 copies of SO(N = 4) one finds 10 dim SOðNÞj4 ¼ ð10Þ½N ðN 1Þ=2 ¼ 10½ð4Þð3Þ=2 ¼ 60: However for the transfinite expectation value of the Hausdorff dimension of E-infinity, namely 4 + /3 one finds ð10Þ dim SOð4 þ /3 Þ ¼ ð10ð4 þ /3 Þð3 þ /3 Þ=2 ¼ 68:54101965 ¼ a0 =2 where a0 ’137 is the inverse electromagnetic fine structure constant [14]. This and numerous other transfinite continuations form an important part of results upon which E-infinity is leaning [14]. A second example is given by the simple linear group SL(2, n). For n = 5 we have jSLð2; 5Þj ¼ nðn2 1Þ ¼ 5ð25 1Þ ¼ 120: However for n = 5 + /3, we obtain the exact value of the electromagnetic fine structure constant a0 ¼ 137:08203939 jSLð2; 5 þ /3 Þj ¼ ð5 þ /3 Þ½ð5 þ /3 Þ2 ð1 þ /3 Þ ¼ Now let us apply our transfinite procedure in an ad hoc manner to the {24} decomposition of SU(5) which we gave earlier on. This decomposition pffiffiffi will be denoted by a subscript c, namely {n}c. Adding our transfinite corrections k = /3 (1 /3) where / ¼ ð 5 1Þ=2 in the usual way, one could write fngc ¼ f8; 1g þ f1 þ /3 ; 3 þ /3 g þ f1; 1 þ /3 g þ f3 þ /3 ; 2g þ f3 þ /3 ; 2g: Now we calculate the multiplicative order of {n}c and find that jfngc j ¼ ð8Þð1Þ þ ð1 þ /3 Þð3 þ /3 Þ þ ð1Þð1 þ /3 Þ þ ð3 þ /3 Þð2Þ þ ð3 þ /3 Þð2Þ ¼ 26 þ k ¼ 26:18033989: Thus j{n}cj is not equal 24 as in the non-transfinite crisp case. It is however equal to the transfinite version of the string dimension D(26) = 26 and is obtained by adding k to D(26). Before discussing this remarkable result, let us calculate the additive order of {n}c. This is clearly equal to jfngc j ¼ 8 þ 1 þ 1 þ /3 þ 3 þ /3 þ 1 þ 1 þ /3 þ 3 þ /3 þ 2 þ 3 þ /3 þ 2 ¼ 26 þ k: That may be another surprise because both values, the multiplicative and the additive are exactly equal. Not only that, but also because 26 + k could be interpreted in a variety of ways. First it is the expectation number of gauge Bosons of a transfinite grand unification theory based on SU(5). Second it is equal to the Euler class of the fuzzy K3 Ka¨hler manifold of E-infinity as well as the number of instanton density. Third it is equal to the instanton density of fuzzy K3 as well as being the Bosonic dimension of the transfinite Heterotic string theory. This flexibility in interpretation is taken to be a sign of the universality class of E-infinity theory which is all encompassing as far as this category of theories is concerned [8,14]. Now we are in a position to transfinitely continue jSUð5Þj ¼ ð5Þ2 1 to jSUc ð5Þj ¼ ð5 þ /3 Þ2 ð1 þ /3 Þ ¼ 26 þ k so that 5 ! 5 þ /3 and 1 ! 1 þ /3 : 3 We recall from E-infinity theory that 5 + /3 may be interpreted as the bþ 2 ¼ 5 þ / Betti number of fuzzy K3 [14] or as the fermionic expectation value of E-infinity [14,23]
hnif ¼ ð4 þ /3 Þð1 þ /3 Þ ¼ ð4 þ /3 Þ þ 1 ¼ 5 þ /3 : Now let us transfinitely continue the other representation of SU(5) for instance that given by (n) (n 1)/2 which leads to the 10 dimensional representation in the crisp non-fuzzy case of n = 5. However for n = 5 + /3, one finds a most important result, namely
M.S. El Naschie / Chaos, Solitons and Fractals 32 (2007) 370–374
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ð5 þ /3 Þð4 þ /3 Þ=2 ¼ 11 þ k=2 Setting k ffi 0 compared to unity we obtain an 11 dimensional representation related to the 11 dimensional M theory which 11 + k/2 is the dimensionality of Ji-Huan He’s fractal M theory [20–22]. Next we look at the (n) (n + 1)/2 representation in the crisp case which leads to {15} multiplets. In the case of n = 5 + /3 we have to modify the formula to ð5 þ /3 Þð5 þ /3 þ ð1 /6 ÞÞ=2 ¼ ð5 þ /3 Þð6 þ kÞ=2 ¼ 16 þ k ¼ 16:18033989 which are the extra 16 + k Bosons of the Heterotic string theory and may be obtained from 26 + k by scaling as (26 + k) / = 16 + k as shown in E-infinity theory. We stress again that one can introduce fuzziness or transfinite correction generally in a multitude of equivalent ways. That is after all the deep meaning of being fuzzy [5]. 4. Maxing angle and the number of particles in the standard model pffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffi 3=5 ¼ 0:6 , it is not difficult to guess that the transfinite value must be [1–3] pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffi tan hw ¼ 0:6 þ k=10 ¼ / pffiffiffiffi where k = /3 (1 /3) as before. Consequently we have a triangle with a = 1, b ¼ / and qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffi ffi pffiffiffiffiffiffiffiffiffiffiffiffi c ¼ ð1Þ2 þ ð /Þ2 ¼ 1 þ /: From hw ¼
Therefore 1 sin hw ¼ pffiffiffiffiffiffiffiffiffiffiffiffi 1þ/ and thus sin2 hw ¼
1 ¼ /2 ¼ 0:381966011: 1þ/
These values have to be reduced by projection to obtain that found from experimental measurement. By contrast the coupling constant of unification is obtained directly from 26 + k by scaling using the well known exponents of E-infinity, namely (1//)n. Proceeding in this way one finds ð26 þ kÞð1=/Þ ¼ 42 þ 2k ¼ ag ¼ 1=ag : Proceeding in the same way ao /2 is obtained and directly interpreted as a number of elementary particles: ð26 þ kÞð1=/Þ2 ¼ ð42 þ 2kÞð1=/Þ ¼ 68:54101967 ¼
137:0820393 ¼ ao =2: 2
On the other hand scaling ag ¼ 26 þ k down, one finds ð26 þ kÞð/Þ ¼ 16 þ k ’ 16 ð26 þ kÞð/Þ2 ¼ 10 ¼ 10 ð26 þ kÞð/3 Þ ¼ 6 þ k ’ 6 ð26 þ kÞð/4 Þ ¼ 4 k ’ 4 which are the well known Heterotic string hierarchy. Now may be it is of interest to ask about the number of elementary particles which one may expect to find in the low energy standard model based on the SU(5) unification. Taking the {45} and the {24} representation to mean standard model Fermions and Bosons respectively, then we should expect the standard model to contain 45 þ 24 ¼ 69 particles. This seems to agree squarely with the 69 particles predicted by E-infinity theory. However the division into 45 Fermions and 24 Bosons does not seem right because we are short of the 48 Fermions of the standard model by three elementary particles. Of course we could argue that at this low energy level, 3 Bosons would have changed to 3 Fermions but maybe we could argue the case in a different way by involving the {50} representation of SU(5) and excluding the 12 exotic Bosons. That way we end up with
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50 þ 12 ¼ 72 particles as argued by L. Marek-Crnjac in a recent paper [9]. However what are these two extra Fermions? If they are taken very seriously they may help deepen our understanding of many things, including the discredited Axion.
5. Conclusion The non-super symmetric SU(5) is far too beautiful a theory to be disregarded even when a very important experimental contradiction, such as the lifetime of the proton is currently unresolved. To show that this is quite a reasonable attitude to take, we have converted the essential features of SU(5) to a transfinite version of SU(5) and found that it is in essence homomorphic to E-infinity Cantorian spacetime theory. In a work by L. Marek-Crnjac it was found that including super symmetry in grand unification gives the correct inverse ags ffi 26. Interestingly the result remains the same whether or not gravity is included. This seems to validate the expectation that super symmetry is physically relevant and that including super symmetry means implicitly including gravity in an indirect manner.
References [1] ‘t Hooft G. In search of the ultimate building blocks. Cambridge;1997. [2] Zee A. Quantum field theory. Princeton: Princeton University Press; 2003. [3] El Naschie MS. Determining the number of Higgs particles starting from general relativity and various other field theories. Chaos, Solitons & Fractals 2005;23:711–26. [4] El Naschie MS. Quantum gravity, Clifford algebras, fuzzy set theory and the four dimensional constraints of nature. Chaos, Solitons & Fractals 2004;20:437–50. [5] Kosko B. Fuzzy thinking. (Flamingo), London: Harper Collins Publishers; 1994. [6] Zadeh LA. Fuzzy sets. Inform Contrl 1965;8:338–53. [7] Klir G, Folger T. Fuzzy sets, uncertainty and information. New Jersey: Prentice Hall; 1988. [8] El Naschie MS. On a fuzzy Ka¨hler-like manifold which is consistent with the two slit experiment. Int J Nonlinear Sci Numer Simul 2005;6(2):95–8. [9] Marek-Crnjac L. Different Higgs models and the number of Higgs particles. Chaos, Solitons & Fractals 2006;27: 575–9. [10] El Naschie MS. Spinorial content of the standard model, a different look at super symmetry and fuzzy E-infinity hyper Ka¨hler. Chaos, Solitons & Fractals 2005;26:303–11. [11] El Naschie MS. The Higgs-physical and number theoretical arguments for the necessity of a triple elementary particle in super symmetric spacetime. Chaos, Solitons & Fractals 2004;22:1199–209. [12] Tanaka Y. The mass spectrum of heavier hadrons and E-infinity theory. Chaos, Solitons & Fractals, in press, doi:10.1016/ j.chaos.2006.09.010. [13] Marek-Crnjac L. The maximum number of elementary particles in a super symmetric extension of the standard model. Chaos, Solitons & Fractals, in press, doi:10.1016/j.chaos.2006.09.020. [14] Sigalotti L, Mejias A. On El Naschie’s conjugate complex time, fractal e(1) spacetime and faster-than-light particles. Int J Nonlinear Sci Numer Simul 2006;7(4):467–72. [15] El Naschie MS. Holographic dimensional reduction: Centre manifold theorem and E-infinity. Chaos, Solitons & Fractals 2006;29:816–22. [16] El Naschie MS. On gauge invariance, dissipative quantum mechanics and self adjoint set. Chaos, Solitons & Fractals 2007;32(2):271–3. [17] El Naschie M.S. Determining the number of Fermions and the number of Bosons separately in an extended standard model. Chaos, Solitons & Fractals, in press, doi:10.1016/j.chaos.2006.09.069. [18] Tanaka Y, et al. The mass spectrum of heavier hadrons and E-infinity theory. Chaos, Solitons & Fractals. In press, doi:10.1016/ j.chaos.2006.09.010. [19] El Naschie M.S. On the universality class of all universality classes and E- infinity spacetime physics. Chaos, Solitons & Fractals. In press, doi:10.1016/j.chaos.2006.08.017. [20] He Ji-Huan. The number of elementary particles in a fractal M-theory of 11.236067977 dimensions. Chaos, Solitons & Fractals 2007;32:346–51. [21] El Naschie MS. On an eleven dimensional E-infinity fractal spacetime theory. Int J Nonlinear Sci Numer Simul 2006;7(4):407–9. [22] El Naschie MS. The ‘‘discrete’’ charm of certain eleven dimensional spacetime theory. Int J Nonlinear Sci Numer Simul 2006;7(4):477–81. [23] El Naschie MS. Elementary prerequisites for E-infinity. Chaos, Solitons & Fractals 2006;30:579–605.
SU(5) grand unification in a transfinite form M.S. El Naschie
1
King Abdul Aziz City of Science & Technology, Riyadh, Saudi Arabia
Abstract The SU(5) grand unification with its jSU(5)j = 24 gauge Bosons is partially reformulated in a transfinite setting. By means of transfinite continuation it is shown that a new version of the theory yields an expectation value hjSU(5)jci = 26 + k instead of the classical 24. By systematically exploring the non-super symmetric SU(5) scheme and transforming many of its fundamental aspects, it becomes plausible that it is a fundamental theory which could be integrated in various other fundamental theories including the transfinite forms of super strings and M theory. 2006 Elsevier Ltd. All rights reserved.
1. Introduction Despite failure to predict the right time scale for protons decay, the non-super symmetric SU(5) grand unification is a highly elegant theory and the seamless fitting of the standard model SU(3) SU(2) U(1) into its fundamental symmetry group SU(5) makes it extremely difficult to believe that the theory could be entirely wrong [1–3]. In fact, and as we will be arguing in the present short work, the theory is substantially correct, at least in its basic structure. Using the technique of transfinite continuation we will show that SU(5) unification could be given a twist along the lines of E-infinity theory and link it to a family of other similar theories formulated in a fuzzy setting [3–23]. 2. The SU(5) grand unification The standard model of high energy physics is based on a combination of three symmetry groups, the order of which gives us 12 gauge Bosons [3,10,11] jSUð3Þ SUð2Þ Uð1Þj ¼ 12 By contrast the SU(5) model has double as much, namely [1,2] jSUð5Þj ¼ 24 where besides the photon (c), the W±, the Z0 and the 8 Gluons we have an additional 12 exotic gauge Bosons. Thus while we have 12 flavour and colour forces in the standard model [1–3] 1
P.O. Box 272, Cobham, Surrey KT11 2FQ, United Kingdom.
0960-0779/$ - see front matter 2006 Elsevier Ltd. All rights reserved. doi:10.1016/j.chaos.2006.09.018
M.S. El Naschie / Chaos, Solitons and Fractals 32 (2007) 370–374
371
Gal ; W l ; Z l ; Al an additional 12 super weak and lepton-quark forces e il ; Y e il Xil ; Yil ; X are assumed to exist in SU(5). The anomaly free combination of SU(5) representation is given by { 5} + {10} and the relevant multiplets are shown in Fig. 8 of Ref. [3]. The {24} of SU(5) under SU(3)c · SU(2)L · U(1) decomposes as follows [1–3] f24g ¼ f8; 1g þ f1; 3g þ f3; 2g þ f3 ; 2g þ f1; 1g To find jSU(5)j from the above we just need to proceed formally as below: jSUð5Þj ¼ ð8Þð1Þ þ ð1Þð3Þ þ ð3Þð2Þ þ ð3 Þð2Þ þ ð1Þð1Þ ¼ 8 þ 3 þ 6 þ 6 þ 1 ¼ 24: We will refer to this as the multiplication order which is normally given by N2 1 = 25 1 = 24. For later use we mention that a second formal way to find our 24 is to write jSUð5Þj ¼ ½8 þ 1 þ 1 þ 3 þ 3 þ 2 þ 3 þ 2 þ 1 þ 1 1 ¼ ½25 1 ¼ 24: This is nothing but a trivial way of rewriting jSU(5)j = N2 1 as N2 = jSU(5)j + 1 where [25] = N2. The value N2 will be labelled the additive order. By choosing SU(5), the couplings of the electroweak unifications are fixed [1,2] rffiffiffi 3 gSUð5Þ g1 ¼ 5 and because gSUð5Þ ¼ g and tan hw g1 =g one finds that tan hw ¼
rffiffiffi 3 5
Therefore the Weinberg mixing angle is given in this case by sin2 hw ¼ 3=8 A further prediction of SU(5) grand unification is that a1 ðM x Þ ¼ a2 ðM x Þ ¼ a3 ðM x Þ ¼ au where Mx ’ 1015 Gev is the unification energy and au ’ 1/50 is the unification coupling. It should be noted however that extrapolation of measured values at LEP and SLC shows a lack of unification of gauge coupling for the standard model, but that averaging based on the centre of gravity of the triangle formed by the three lines of the running coupling gives au ’ 1/42, rather than au ’ 1/50. Interestingly, L. Marek-Crnjac in Slovenia also found that au ’ 1/42 while the exact value predicted by E-infinity theory is ag =1/au = 42.36067977. 3. Transfinite continuation of SU(5) At a minimum, fuzzy sets, fuzzy logic and fuzzy geometry have been for decades infiltrating numerous branches of engineering and technology [5–7] and in a relatively lesser measure, also physics [4,8,9]. The concepts of fuzziness come under various names and a particularly important category increasingly used in high energy physics is that connected to fractal statistics and fractal geometry [23]. In particular E-infinity theory used a form of transfinite fuzziness which was utilized to solve numerous problems in quantum mechanics [14] and high energy physics. Thus the notion of a non-integer fractal dimension has been generalized to the extent of talking in precise terms about 0.23606799 particles or 6.1803398977 elementary particles. Translated
372
M.S. El Naschie / Chaos, Solitons and Fractals 32 (2007) 370–374
to our classical world, these statements are simply statistical or probabilistic expectation values and as such, not as strange as they sound. On the other hand it comes initially as a little surprise that many well known mathematical formulas envisaged exclusively for integer values, could be extended with a minimal effort to the irrational transfinite or fractal domain to yield highly surprising results with new physical insights. A trivial example of this kind of transfinite continuation with far reaching consequences may be the following one connected to the SO(N) symmetry groups. Taking 10 copies of SO(N = 4) one finds 10 dim SOðNÞj4 ¼ ð10Þ½N ðN 1Þ=2 ¼ 10½ð4Þð3Þ=2 ¼ 60: However for the transfinite expectation value of the Hausdorff dimension of E-infinity, namely 4 + /3 one finds ð10Þ dim SOð4 þ /3 Þ ¼ ð10ð4 þ /3 Þð3 þ /3 Þ=2 ¼ 68:54101965 ¼ a0 =2 where a0 ’137 is the inverse electromagnetic fine structure constant [14]. This and numerous other transfinite continuations form an important part of results upon which E-infinity is leaning [14]. A second example is given by the simple linear group SL(2, n). For n = 5 we have jSLð2; 5Þj ¼ nðn2 1Þ ¼ 5ð25 1Þ ¼ 120: However for n = 5 + /3, we obtain the exact value of the electromagnetic fine structure constant a0 ¼ 137:08203939 jSLð2; 5 þ /3 Þj ¼ ð5 þ /3 Þ½ð5 þ /3 Þ2 ð1 þ /3 Þ ¼ Now let us apply our transfinite procedure in an ad hoc manner to the {24} decomposition of SU(5) which we gave earlier on. This decomposition pffiffiffi will be denoted by a subscript c, namely {n}c. Adding our transfinite corrections k = /3 (1 /3) where / ¼ ð 5 1Þ=2 in the usual way, one could write fngc ¼ f8; 1g þ f1 þ /3 ; 3 þ /3 g þ f1; 1 þ /3 g þ f3 þ /3 ; 2g þ f3 þ /3 ; 2g: Now we calculate the multiplicative order of {n}c and find that jfngc j ¼ ð8Þð1Þ þ ð1 þ /3 Þð3 þ /3 Þ þ ð1Þð1 þ /3 Þ þ ð3 þ /3 Þð2Þ þ ð3 þ /3 Þð2Þ ¼ 26 þ k ¼ 26:18033989: Thus j{n}cj is not equal 24 as in the non-transfinite crisp case. It is however equal to the transfinite version of the string dimension D(26) = 26 and is obtained by adding k to D(26). Before discussing this remarkable result, let us calculate the additive order of {n}c. This is clearly equal to jfngc j ¼ 8 þ 1 þ 1 þ /3 þ 3 þ /3 þ 1 þ 1 þ /3 þ 3 þ /3 þ 2 þ 3 þ /3 þ 2 ¼ 26 þ k: That may be another surprise because both values, the multiplicative and the additive are exactly equal. Not only that, but also because 26 + k could be interpreted in a variety of ways. First it is the expectation number of gauge Bosons of a transfinite grand unification theory based on SU(5). Second it is equal to the Euler class of the fuzzy K3 Ka¨hler manifold of E-infinity as well as the number of instanton density. Third it is equal to the instanton density of fuzzy K3 as well as being the Bosonic dimension of the transfinite Heterotic string theory. This flexibility in interpretation is taken to be a sign of the universality class of E-infinity theory which is all encompassing as far as this category of theories is concerned [8,14]. Now we are in a position to transfinitely continue jSUð5Þj ¼ ð5Þ2 1 to jSUc ð5Þj ¼ ð5 þ /3 Þ2 ð1 þ /3 Þ ¼ 26 þ k so that 5 ! 5 þ /3 and 1 ! 1 þ /3 : 3 We recall from E-infinity theory that 5 + /3 may be interpreted as the bþ 2 ¼ 5 þ / Betti number of fuzzy K3 [14] or as the fermionic expectation value of E-infinity [14,23]
hnif ¼ ð4 þ /3 Þð1 þ /3 Þ ¼ ð4 þ /3 Þ þ 1 ¼ 5 þ /3 : Now let us transfinitely continue the other representation of SU(5) for instance that given by (n) (n 1)/2 which leads to the 10 dimensional representation in the crisp non-fuzzy case of n = 5. However for n = 5 + /3, one finds a most important result, namely
M.S. El Naschie / Chaos, Solitons and Fractals 32 (2007) 370–374
373
ð5 þ /3 Þð4 þ /3 Þ=2 ¼ 11 þ k=2 Setting k ffi 0 compared to unity we obtain an 11 dimensional representation related to the 11 dimensional M theory which 11 + k/2 is the dimensionality of Ji-Huan He’s fractal M theory [20–22]. Next we look at the (n) (n + 1)/2 representation in the crisp case which leads to {15} multiplets. In the case of n = 5 + /3 we have to modify the formula to ð5 þ /3 Þð5 þ /3 þ ð1 /6 ÞÞ=2 ¼ ð5 þ /3 Þð6 þ kÞ=2 ¼ 16 þ k ¼ 16:18033989 which are the extra 16 + k Bosons of the Heterotic string theory and may be obtained from 26 + k by scaling as (26 + k) / = 16 + k as shown in E-infinity theory. We stress again that one can introduce fuzziness or transfinite correction generally in a multitude of equivalent ways. That is after all the deep meaning of being fuzzy [5]. 4. Maxing angle and the number of particles in the standard model pffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffi 3=5 ¼ 0:6 , it is not difficult to guess that the transfinite value must be [1–3] pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffi tan hw ¼ 0:6 þ k=10 ¼ / pffiffiffiffi where k = /3 (1 /3) as before. Consequently we have a triangle with a = 1, b ¼ / and qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffi ffi pffiffiffiffiffiffiffiffiffiffiffiffi c ¼ ð1Þ2 þ ð /Þ2 ¼ 1 þ /: From hw ¼
Therefore 1 sin hw ¼ pffiffiffiffiffiffiffiffiffiffiffiffi 1þ/ and thus sin2 hw ¼
1 ¼ /2 ¼ 0:381966011: 1þ/
These values have to be reduced by projection to obtain that found from experimental measurement. By contrast the coupling constant of unification is obtained directly from 26 + k by scaling using the well known exponents of E-infinity, namely (1//)n. Proceeding in this way one finds ð26 þ kÞð1=/Þ ¼ 42 þ 2k ¼ ag ¼ 1=ag : Proceeding in the same way ao /2 is obtained and directly interpreted as a number of elementary particles: ð26 þ kÞð1=/Þ2 ¼ ð42 þ 2kÞð1=/Þ ¼ 68:54101967 ¼
137:0820393 ¼ ao =2: 2
On the other hand scaling ag ¼ 26 þ k down, one finds ð26 þ kÞð/Þ ¼ 16 þ k ’ 16 ð26 þ kÞð/Þ2 ¼ 10 ¼ 10 ð26 þ kÞð/3 Þ ¼ 6 þ k ’ 6 ð26 þ kÞð/4 Þ ¼ 4 k ’ 4 which are the well known Heterotic string hierarchy. Now may be it is of interest to ask about the number of elementary particles which one may expect to find in the low energy standard model based on the SU(5) unification. Taking the {45} and the {24} representation to mean standard model Fermions and Bosons respectively, then we should expect the standard model to contain 45 þ 24 ¼ 69 particles. This seems to agree squarely with the 69 particles predicted by E-infinity theory. However the division into 45 Fermions and 24 Bosons does not seem right because we are short of the 48 Fermions of the standard model by three elementary particles. Of course we could argue that at this low energy level, 3 Bosons would have changed to 3 Fermions but maybe we could argue the case in a different way by involving the {50} representation of SU(5) and excluding the 12 exotic Bosons. That way we end up with
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50 þ 12 ¼ 72 particles as argued by L. Marek-Crnjac in a recent paper [9]. However what are these two extra Fermions? If they are taken very seriously they may help deepen our understanding of many things, including the discredited Axion.
5. Conclusion The non-super symmetric SU(5) is far too beautiful a theory to be disregarded even when a very important experimental contradiction, such as the lifetime of the proton is currently unresolved. To show that this is quite a reasonable attitude to take, we have converted the essential features of SU(5) to a transfinite version of SU(5) and found that it is in essence homomorphic to E-infinity Cantorian spacetime theory. In a work by L. Marek-Crnjac it was found that including super symmetry in grand unification gives the correct inverse ags ffi 26. Interestingly the result remains the same whether or not gravity is included. This seems to validate the expectation that super symmetry is physically relevant and that including super symmetry means implicitly including gravity in an indirect manner.
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