E-infinity communication No. 9
Wyle scaling and deriving the spectral dimension 4.02 of Loll, Ambjorn and Jurkiewicz using E-infinity
There is no reason why we should not continue with our discussion and give further examples of the use of golden Wyle scaling in E-infinity. We have to take the opportunity first to draw the attention of the readers to the literature where details are given. The most important three papers which can be consulted on this subject are the following: From classical gauge theory back to Weyl scaling via E-infinity spacetime by M. S. El Naschie, Chaos, Solitons & Fractals (CS&F), 38, 2008, p. 980, A Feynman path integral-like method for deriving the four dimensionality of spacetime from first principles by L. Marek-Crnjac, CS&F, 41, 2009, p. 2471 and Density manifolds, geometric measures and high-energy physics in transfinite dimensions by S.I. Nada, CS&F, 42, 2009, p. 1539.
To explain the main idea in the most excessively simplistic terms you could say the following. There is a fundamental difference between changing direction in space and stretching a line in space. In the first case nothing really physical happened while in the second case something almost physical happened. That was the crux of Einstein's objection against Wyle's gauge theory. Lacking a natural scale, in a fractal setting a stretching can be set on the same footing as changing an angle. You read the rest please in the relevant literature. By the way many people know this trick and that is why fractal spacetime is becoming fashionable and will become even more fashionable as time goes by, so you can play it again Sam.
Let me now return to examples of Wyle scaling and we can do two things for the price of one. We give another example and derive the spectral dimension given for instance in the paper of Ambjorn, Jurkiewicz and Loll in Scientific American or the improved version they published in a book Edited by Daniele Oriti entitled Approaches to Quantum Gravity, published by Cambridge, 2009 a few years after the first derivation by El Naschie using Bose Einstein statistics. The excellent paper by the three authors who are world renowned for simplictic triangulation, in other words, tiling the space with simplexes which again means Regge calculus or finite element of John Argyris who was one of Mohamed El Naschie's teachers in Stuttgart, Germany and Imperial College, London is entitled Quantum Gravity: the art of building spacetime on page 341-359. The important formula is on page 352.
Let's begin at the beginning. In Heterotic string theory anomaly cancellation requires either O(32) or E8 E8. In this theory there are left and right moving sectors. The left is purely bosonic. We start with 26 dimensional strings where 16 have been compactified on a lattice, for instance E8 E8 lattice or spin 32 divided by Z2 lattice. The right movers on the other hand are super symmetric. Thus the right movers are explicitly spacetime symmetric. Recall that spacetime symmetry of super space is found until this moment by trial and error. Now in the right moving sector at the lowest level we have 8 plus 8 equals 16 states. In the left moving sector we have three distinct objects leading to 8 states plus 16 states plus 480 states. This makes altogether 504 states. To obtain the entire spectrum we follow Fock space rules of quantum field theory and multiply left movers with right movers. That way we obtain from 504 times 16 the well known 8064 states. This is a very well known result. You can find this result in popular literature like Scientific American or text books on string theory like Kaku's book. Knowing that there are truly childish people hanging around on internet blogs with an enormous chips on their shoulders, we would like to give you the literature to check everything yourself so that these children do not write that we are telling you fibs. OK, here they are. Scientific American, Superstrings by M.B. Green, p. 183-203, in the caption of Figure 11.14. Reprinted in a book called Particles and Forces, Editor Richard A. Carrigan, published by W. Freeman, New York (1990). M. Kaku's book Introduction to Superstrings and M-theory, published by Springer, 1999, see page 384. Now enter Mohamed El Naschie. He realized that the same result is easily obtained in an entirely different manner and in some respects a far simpler geometrical way by assigning the right amount of instantons to each site of Klein's modular curve as a holographic boundary of string theory. Without going into details the instanton number in this case is 24. The number of triangles as you know from earlier communications is 336. Multiplying the two numbers you get the total number of instantons, namely 8064. Mohamed was under the false impression that this must be a well known method. He published it and mentioned the whole thing just as a marginal thing. He published it a few times without paying too much attention to any novelty. It was then after heated discussion that Nobel laureate Gerard 'tHooft convinced him that it is an entirely new theory. I forgot to mention some of the extensions and the insight which El Naschie gave to this point also without realizing that it is an entirely new theory. First look at 504. This was realized by Mohamed as the summing of an exceptional Lie group. Here is a sum about which some people who are ignorant about the exceptional Lie group hierarchy rejected off hand and of course wrongly so. Add the dimensions of E8, E7, E6 and E5 together then you will have 248 plus 133 plus 78 plus 45 equals exactly 504. What most people did not know is the fact that E5 is nothing else but SO(10) of grand unification which is competing with SU(5). SO(10) is really an E5 based on its Dykin diagram. This fact was known to El Naschie as well as many other careful people which do not include of course John Baez who had too much to do on his shows on his diverse blogs for entertaining people. It was of course a mistake to stop at E5. In mathematics it would mean you are not using a complete set. To use a complete set you have to take all the exceptional groups. The sum of that as everyone knows in the meantime was found by El Naschie to be 4 times 137 equals 548. Similar reasoning would show that classical Heterotic strings are only approximation and the real numbers of massless states is not 8064 but 8872 when we ignore everything after the dot. Now El Naschie did not calculate it in this way. He reasoned differently using the holographic boundary. Taking compactification into account you do not have 336 triangles but 338.885438 weighted number of triangles. In addition he derived the exact instanton density and found that it is not 24 but exactly 26.18033989. Multiplying both exact numbers you find 8872.135951. Wonderful. This is now our numerical Lagrangian or potential or whatever you want to call it. Sixteen times differentiation is an E-infinity equal to sixteen times scaling with the golden mean. Multiplying our Lagrangian with the golden mean to the power of sixteen, you get 4.01999999 on a pocket calculator. This is Loll, Ambjorn and co's result. For all practical considerations it is 4.02. Please do this calculation yourself.
It is interesting to ask why the pioneer of the holographic boundary did not find this result first. I do not know but we cannot all work on everything simultaneously. A reasonable explanation may be the following. The expert on the holographic boundary did not care about the 336 because they know once they compactify we get infinity and physicists do not like infinity and do not work with it. Mohamed El Naschie on the other hand took a gamma distribution weighted infinitely compactified Klein modular curve which added approximately 3 to the 336 to get approximately 339 which is a finite result and meaningful. He was also able to deal with a fuzzy K3 manifold and find the instanton number to change from the classical 24 to 26.18033. This is incidentally equal to the dimension of transfinite Heterotic strings as well as the corresponding Euler constant as well as the curvature which we discussed in an earlier communication. 26.18033 turned out to be an extremely important number. People interested in number theory knew that much earlier but did not know about any relevance in physics. Nobel laureate David Gross must be credited with super natural intuition to have invented Heterotic string theory. Witten and his friends were probably the first to introduce K3 to physics. So you can see string theory is not in any way as useless as some of its opponents want us to believe. Nothing is useless except underestimating people. Nothing is as harmful as belittling the achievements of other people. Nothing is as revolting as the yellow jokes of envious souls who ceaselessly inundate us with their superfluous comments, polluting the blogosphere with their inferiority complexes. You can find explicit calculations of the spectral dimension in several papers recently published in Chaos, Solitons & Fractals. For instance From Menger-Urysohn to Hausdorff dimensions in high energy physics by G. Iovane, 42, 2009, p. 2338 and On the Menger-Urysohn theory of Cantorian manifolds and transfinite dimensions in physics by Guo-Cheng Wu and Ji-Huan He, 42, 2009, p. 781. Also A Feynman path integral-like method for deriving the four dimensionality of spacetime from first principles by L. Marek-Crnjac, 41, 2008, p. 2471 and Density manifolds, geometric measures and high-energy physics in transfinite dimensions by S.I. Nada, 42, 2009, p. 1539.
I think by now anyone who has attentively read the last nine communications must be able to derive on his own the basic and fundamental dimensions of fractal spacetime, namely the topological dimension 4, the Hausdorff dimension 4 plus the golden mean to the power of 3, the scaling dimension 4 minus k equal to the golden mean square to the power of 10 and finally the spectral dimension which is 4.02. The work of Loll and her crew is remarkable in that they got the exact result without having the complete theory. They were right to think that only a computer can get this result right. I say only a computer or E-infinity theory and nothing in between. They may have wrongly led people to think that 4.02 is a Hausdorff dimension or a topological dimension for quantum spacetime. It is not. It is an exact dimension for the spectral dimension of quantum spacetime. In a sense this is an even more important dimension because it measures how spacetime unfolds. It looks at it as a dynamical process, not as a statical process. It looks at it like ink spreading in water. They should be commended for their handiwork. What a pity that they were not aware of all what we have done in the last 15 years using E-infinity theory and fractal spacetime and what a pity that people can sometimes be so individualistic that they cannot see the achievements of others. Cooperation is always much better than meaningless struggle against others. At the end of the day, at least in E-infinity, we have benefitted from all the sound ideas of well developed careful theories like string theory, loop quantum mechanics, holographic principles, Penrose tiling, noncommutative geometry and so on and so forth. I sincerely hope you agree with us, at least on this.
