- Number 38. November 30, 2010. Nobel laureate ‘tHooft’s recent incorporation of fractals.
- Number 39. December 2, 2010. Misconception spreading about Wheeler foam being a fractal.
- Number 40. December 3, 2010. The new article by Garrett Lisi et al in Scientific American, December 2010 and the work of the E-infinity Group on E8 exceptional Lie groups.
3rd December, 2010.
E-infinity Communication No. 40.
The new article by Garrett Lisi et al in Scientific American, December 2010 and the work of the E-infinity Group on E8 exceptional Lie groups.
The first article which appeared on the arXiv by garret Lisi entitled ‘An exceptionally simple theory of everything’ dated 6th November, 2007. This article seems to have been recommended on the arXiv by Prof. Lee Smolin working at that time intensively on establishing the Parameter [sic. Perimeter] Inst. Waterloo, Canada. Subsequent to the publication and unprecedented media hype followed mainly due to an article in the Telegraph. There was an enormous publicity and controversy. A second article followed in the Telegraph. The first article in the Telegraph announced that Garrett Lisi is the new Einstein. The second article announced that he is no Einstein. Of course all that has nothing to do with science. However scientists are sometimes, and maybe wrongly so, touchy about priority. They are also touchy about people using their work and not giving them credit. It is understandable to a point. Payment and everything worldly in science is very little. The main thing which remains for scientists is reputation, a sense of being appreciated and the hope that they will be remembered for their scientific contributions. It is really not much to ask when you think about the benefits and financial gain which other people make of science. Commercial publishers alone make billions in profit from publishing scientific work. The people who invented computers died in poverty. Computer companies are multi-national billion dollar enterprises. The guy who invented the internet is anything but a millionaire. Do not ask how much Google earns a day. For this reason maybe it is not unreasonable that there was a great deal of controversy about Lisi’s paper not giving credit to anyone who used E8, his exceptional Lie symmetry group. String theoreticians used it long ago. Of course they used it differently. To super strings, E8 E8 gives an otherwise non-super symmetric theory super symmetry and that way they get rid of many contradictions and maladies which the 26 dimensional Bosonic theory possesses. Lisi however uses E8 itself. He did not need to refer to strings, quantum gravity loops or any other physical model. The only model he had was the model of E8 itself only which is a pure geometrical model. Well you could argue that any model is at the end a mathematical model. That is true but Lisi did not invent any model, physical or mathematical. He simply took over the mathematical model of E8 as established by a mathematician thinking about mathematical structures with little if any physical thoughts in the back of their mind. Some critics thought that this is the ingenuity of Lisi. Other critics thought that this is the naivety of Lisi. You can argue any way you like. It is however a fact that one of the principle members of our group, Prof. M.S. El Naschie did the same thing within E-infinity theory long ago around 2004. Other members, for instance Prof. Marek-Crnjac and Prof. He as well as Prof. Iovane also considered the exceptional Lie group in a far wider context than Lisi and this was before Lisi published his paper. Referring explicitly to E8 and publishing papers with E8 in the title was also done long before Lisi. For instance one of the early papers on E-line exceptional Lie group was published in Int. J. of Nonlinear Sci. & Num. Simulation, 8(3), p. 445-450 (2007), entitled ‘Exceptional Lie groups hierarchy and the structure of the micro universe’. This is several months before the appearance of the first paper by Lisi on the arXiv. From the end of 2006 to roughly the time when Lisi published his paper on 6th November, 2007 Prof. El Naschie and his group published more than a dozen papers on the exceptional Lie groups and its application and physics and cosmology. He also recognized many things which are now coming to play a role in the newest of Lisi’s paper in Scientific American entitled ‘A geometric theory of everything’, December 2010. Needless to say, Lisi did not acknowledge any work of Prof. El Naschie or our group not in 2007 and not now although it is absolutely clear from even a superficial reading of his newest article in Scientific American that he benefited from the publications of the E-infinity group. This is of course very regrettable but it is not the end of the world and we take it in the same spirit in which we took and endured many things before, mostly coming from the same sources. Before we turn our attention to the scientific mathematical part of our discussion, we must note with an even greater regret that Scientific American is not allowing many people to lodge comments on this site unless they are certain that there will be nothing in the comment revealing the connection to our work. In fact the Editors of Scientific American made a claim on behalf of Lisi which Dr. Lisi himself did not make, namely that Lisi pioneered what in their words they call ‘showcases striking patterns in particle physics’. This is quite unfair on behalf of whoever is responsible in Scientific American for writing that while ignoring the efforts of so many other people. Let us come now to scientific issues.
1. Lisi still does not recognize that there are 8 exceptional Lie symmetry groups and not only 5. He still thinks that the exceptional Lie symmetry groups are E8, E7, E6, F4, G2. The corresponding dimensions are 248, 133, 78, 52 and 14. This is wrong. There are 8 E-line exceptional Lie symmetry groups and when you add F4 and G2, then we have 10. However the E-line has only 8. El Naschie is not the first to find out this fact which is not well known. However El Naschie is the first to emphasize this fact and use it in physics. Lisi overlooks this because he does not appreciate the importance of what mathematicians call Dynkin diagrams. This is explained thoroughly in the following paper by Mohamed El Naschie ‘Exceptional Lie groups hierarchy and some fundamental high energy physics equations’, Vol. 35 (2008), p. 82-84, Chaos, Solitons & Fractals. This paper appeared on the net of Science Direct in 2007 before Lisi’s paper appeared. You can see clearly from this paper that there is an E5 which is identical to the unification group SO(10). In addition there is an E4 which is identical to the unification group SU(5). With unification we mean grand unification theories which only excludes gravity. The fact that these are exceptional groups is exciting and has many ramifications for E-infinity theory once they are extended transfinitely. It seems that the first person to notice that was Howard Georgi. The dimension of E5 is naturally 45 and that of E4 is 24 which matches the additional bosons of this unification theory. Later on El Naschie noticed that there is an E3, E2 and E1. Again based on the Dynkin program, E3 is nothing but SU(3) SU(2). It has a dimension 11. Then there is an E2 which is equal to SU(3) with the well known dimension 8 and finally we have E1 equal to U(1) with the dimension 1. All this may be found in many of El Naschie’s papers and there are a few nice summaries and review articles of which we give the following: Symmetry group prerequisite for E-infinity in high energy physics, CS&F, 35 (2008), p. 202-211, Average exceptional Lie and Coxeter group hierarchies with special reference to the standard model of high energy particle physics, CS&F, 37 (2008), p. 662-668, High energy physics and the standard model from the exceptional Lie groups, CS&F, 36 (2008), p. 1-17. All these papers were published on Science Direct at the beginning of 2007.
2. The most remarkable thing about the hierarchy of the E-line of exceptional Lie groups is that the sum of all its dimensions is exactly four multiplied with the inverse of the electromagnetic fine structure constant 137. This is by no means numerology. It shows that electromagnetism is an averaging over symmetries. We know from Feynman’s path integral that you can sum over paths. This could be extended to summing over topologies as used by many mathematical physicists. The same idea could be extended to summing over Lie symmetry groups. El Naschie is the first to find this explicit result. Please try it yourself and see that 248 + 133 + 78 + 45 + 24 + 11 + 8 + 1 = 548 = 4 multiplied with 137. To show that this has nothing to do with numerology is very simple. Just use the transfinite version of all these numbers. A transfinite version is always an integer plus a transfinite tail represented by an irrational number. The number of digits fixing an irrational number is infinitely large. Because of that, to fit together by a mere numerical coincidence would require a probability which is exactly equal to zero. A numerical coincidence is by virtue of this transfinite nature exclusive as an impossible event. There are far more physically accessible explanations for why this is not numerology. It is not numerology because there are theories behind it . The theory behind it also has another theory behind it. And all these theories fit together seamlessly. If this agreement is coincidence and numerology, then it would be advisable to study only the science of numerology, a proposition which we definitely refute.
3. Lisi acknowledges in his newest paper SO(10) and gives it a prominent place but he does not recognize or mention that it is equivalent to E5. The same may be said about SU(5) which he does not recognize as E4. However he follows an Italian colleague of his in appreciating SO(11) which he made only recently and is mentioned again in an article in Scientific American under the title ‘Rummaging for a final theory, September 2010. The first suggestion which makes use in this context of SO(11) came however from El Naschie. It is well documented in the literature on the subject. For instance in his paper From E-eight to E-infinity, CS&F, 35 (2008), p. 285-290 El Naschie draws attention to SO(11) with a dimension 55 which plays an important role in finding the total number of particles in the standard model as well as playing a role in Edward Witten’s 5-Bran theory with 528 particle-like states. Lisi does not make full use of these facts but he partially makes use of it.
4. Lisi never spoke about fiber bundles except in his last paper. Of course we appreciate very much fiber bundle theory because it is mathematically well developed. We dare however to say that transfinite set theory is far more mathematically developed and far more fundamental. Consequently fractals are far more developed and far more fundamental than fiber bundles. In fact fiber bundles is not simple enough nor is it complex enough. It moves from smooth manifold to somewhat hairy manifold. However it does not go to infinity like fractals. In a sense it is a middle of the way approach. In a sense it is not fish nor meat but something in between. We most definitely think that the natural geometry and the most sophisticated geometry required by high energy physics is fractals. That is to say transfinite set theory interpreted geometrically.
There are many other things which we would like to talk about but we leave that for future communications. Thank you.
2nd December, 2010.
E-infinity Communication No. 39
Misconception spreading about Wheeler foam being a fractal
This communication is prompted by a fundamental misconception among theoretical physicists not really specialized in nonlinear dynamics and fractals who think that the Wheeler foam is a fractal. Of course there is no mathematical definition for what Wheeler foam is except that it is multi-connected and not simply connected space and in addition that such structure is a consequence of the vacuum fluctuation and that happens near to the Planck length. The mistake or misconception would be harmless if it would not lead to excellent physicists thinking wrongly that fractals exist only at the Planck length and are therefore relevant only to quantum gravity. This is completely wrong. In fact highly sophisticated theoretical physicists who also know about fractals mathematically, such as the singularly clever Gerard ‘t Hooft told Prof. El Naschie on several occasions that fractals are irrelevant for quantum mechanics. When he asked why is it used by some of his collaborators, he answered this is only maybe relevant for quantum gravity at the Planck length. For this reason it is obvious that we need to clarify this point.
First we should say that most of those working outside our group and the group of Prof. Nottale and Prof. Ord mention fractals but do not generally use much of what the powerful theory of fractal sets offers. They may occasionally use a Hausdorff dimension or some kind of fractal dimension to specify an important point here or there, but they do not really use the powerful machinery of transfinite set theory and transfinite theory of dimensions. The exception here is without a trace of a doubt is Prof. Alain Connes. Mathematically speaking Alain Connes’ noncommutative geometry is the most powerful and also the most complex machinery known to our trade. Alain Connes knew that fractals exist long before one reaches the Planck length. Fractals mean zero and empty sets. Any mechanics will somehow incorporate zero and empty sets. Some physics can muddle through without these concepts being said clearly from the outset. Other physical disciplines would be a muddle without a clear conception of the set theoretical basis, namely zero set and empty set. Quantum mechanics is such a physical subject which would be hopelessly complicated to understand without clear understanding of the role of the empty and the zero set.
Let us try to go into the historical reason of how this misconception came about. When Wheeler drew a picture of metric fluctuation due to vacuum fluctuation, it looked like foam. Some people called it Wheeler foam. Somehow it entered into the book of Mandelbrot, ‘The fractal geometry of nature’. Mandelbrot calculated the fractal dimension for a cube made of 27 smaller cubes when we iteratively remove the core cube. In other words, we get a fractal dimension equal to ln26 divided by ln3 which is 2.9656. For comparison the Menger sponge has a dimension equal to ln20 divided by ln3 which equals 2.7… From now on the misconception multiplies. Since it is now called fractal foam it is a fractal and since it is the foam of Wheeler, it is connected to metric fluctuation at the Planck length. Forgive us for saying this is complete nonsense. This misunderstanding crept into the work of some of the best and
30th November, 2010.
E-infinity Communication No. 38
Nobel laureate ‘tHooft’s recent incorporation of fractals.
It is extremely interesting, in fact more than that, that Prof. ‘tHooft probably one of the most important high energy physicists of our time made several contacts with fractals recently. To appreciate the importance of these contacts for the task of incorporating nonlinear dynamics and deterministic chaos properly in high energy physics cannot be over emphasized. It is simply a fact of life that most if not all of the main stream high energy physics completely misunderstands fractals and have only a cocktail party appreciation of deterministic chaos. They simply do not think it is relevant to what they are doing. The present communication is not concerned in the first instance with the work of the group around Prof. ‘tHooft, for instance Prof. Loll and her associates. It is also not concerned with Prof. ‘tHooft’s proposal for a deterministic quantum mechanics. The new development is more concrete than all of that. To help the reader follow the discussion here we refer explicitly to a 2008 paper by Prof. ‘tHooft entitled ‘A locally finite model for gravity’. A version of this paper is freely available on ArXiv. Again to help the reader concentrate on the subtle points we refer explicitly to certain passages in this remarkable paper.
1. In the Abstract on page 1 Prof. ‘tHooft says “…. Globally however the model is not finite because solution tends to generate infinite fractals….. this model is not yet quantized”.
2. On page 2 he says “… however a confrontation with fundamental aspects of quantum mechanics appears to be inevitable….one can study systems with many particles by viewing spacetime as tissilation”.
3. On page 3 he says “… particles and strings as topological defects”.
4. On page 5 he says “… angles deficit and angles surplus…”.
5. On page 22 in the Conclusion ‘tHooft says “… strings could terminate in infinitely dense fractals of string segments where they could close the universe”.
Clearly Prof. ‘tHooft who appears from his Home Page to have appreciated the beauty and the mathematics of fractals some time ago has started to appreciate the profound relevance of fractals in physics. We would be extremely honored if this considerable change of heart on behalf of one of the most important physicists of our time was brought about by the dedicated work of the E-infinity group which was initiated by the wonderful work of Garnet Ord, Laurent Nottale and their colleagues.
The point about infinity and fractals is particularly relevant. If you want to have points and yet escape all the philosophical, mathematical unpleasant features of points including the production of infinities, then you have one thing, completely disconnected fractals which means Cantor sets. Of course you can use continuous fractals but then you are carrying unnecessary ballast. As Jim Yorke emphasized, the backbone of any nonlinear dynamics is always a Cantor set. Everything is something multiplied with a Cantor set. In his theory Mohamed El Naschie stripped everything down to Cantor sets and that way he can avoid infinity without resorting to finite stringy objects. In addition you have all the rigorous theorems connected to transfinite set theory and the transfinite theory of dimension at your disposal. Oscar Wilde used to say the best way to get rid of temptation is to give in to it. In a sense that is what fractals do. The best thing to get rid of the unpleasant infinities is work them and not to try to bar them out of the analysis. You have to accept that they are there. You have to tame them. The great Russian set theoretician tamed wild topology by understanding its transfinite set theoretical basis. It turns out that the topology of high energy physics is wild topology. It all ramifies into a dense Cantor set. ‘tHooft is rephrasing what we said here in a homomorphic language. Tissilation [sic. tessellation] stands for fractal tiling such as Penrose universe. The boundary of El Naschie’s interpretation of ‘tHooft’s holographic boundary is indeed a dense fractal. At the end it is a Cantor set. You can iterated string segments, frogs or elephants , at the end you end with a Cantor set closing the universe. In E-infinity El Naschie as well as Crnjac showed that the 336 degrees of freedom of a Klein modular curve increases to around 339 due to compactification at infinity and that is what Prof. ‘tHooft was saying in his conclusion. It remains only to mention that the golden mean binary is naturally quantized. In addition fractals are naturally self-similar or at least self-affined. In other words fractals are naturally scale invariant due to the lack of natural states. In a fractal spacetime Weyl’s ideas are valid and Einstein’s objections against them are no longer valid. We sincerely hope that Prof. ‘tHooft will continue in this direction which we are certain is feasible and promising for the future of high energy physics.
My guess is that the author of these is El Naschie himself.