12th December, 2010.
E-infinity communication No. 53
The first E8 proposal for complete unification of all fundamental forces
It is abundantly clear from a quick study of the large body of literature published by Mohamed El Naschie, Ji-Huan He, L. Marek-Crnjac and many of those working on E-infinity theory that the first implicit proposal for unification using E8 exceptional Lie group was published around 2005 or even earlier. The exact date needs time to nail down more accurately because of the large volume of papers by the prolific author. In a paper entitled “Determining the number of Hiss particles starting from general relativity and various other field theories” which was published in CS&F, 23, (2005), pp. 711-725 El Naschie set outs to explain the idea of unification in paragraph 11 which is labeled Discussion and Conclusion on pages 724 and 725.
In what follows we give a summary of what was said on this subject. At that time El Naschie chose to call his idea, conservation of the dimensional symmetry. He wrote that in what was a rudimentary form simply equating Dim E8E8 to N(R(8)) plus alpha bar naught of electromagnetism, that is to say 137 plus N(R(4)). With R(8) he means the number of independent components of a Riemannian tensor in eight dimensions. These are 336 before compactification and 338.885438, nearly equal 339 after compactification. They are the net difference between the compactified dimensions of E8E8, namely 496 ̶ k2 = 495.9674775 and the compactified dimensions of E6E6 which are equal to 156 + 6k = 157.0820393 when symmetry is broken and E8E8 goes to E6E6 of the exceptional E-line of those important Lie symmetry groups. It is historically interesting to note that at this time El Naschie did not use ‘t Hooft’s holographic boundary where all the 339 particles of the 496 bulk lives. Thus El Naschie used a kind of super gravity argument at that time. He then adds the usual 20 representing the independent components of the Einstein gravity. There are exactly 20 for the case of four dimensions. Thus he has in essence equated the 496 dimensions of the E8E8 bulk with N(R(8)) plus alpha bar naught of electromagnetism plus N(R(4)) = 339 + 137 + 20. This is exactly equal 496 when we consider the integer approximation. In later publications the said 20 were interpreted as the number of isometries of pure gravity in 8 dimensions and the 339 as the number of elementary particles living on the holographic boundary. This is a real unification attempt using the E8E8 group and is many years before any similar attempt by anyone and definitely years before G. Lisi. It is also interesting to note that calculation without compactification leading to 336 instead of 339 particles was interpreted in this particular article as suggesting the existence of 3 Higgs particles H(+), H( ̶ ) and H(0) to make up the deficit of 339 ̶ 336 = 3. El Naschie for some reason never repeated this interesting interpretation again as far as we are aware.
12th December, 2010.
E-infinity communication No. 54
Suslin set theoretical foundation of E-infinity theory
Descriptive set theory which forms an important basis and deep theoretical underpinning of E-infinity theory was the joint discovery of Nikolai Luzin, the real founder of the famous Moscow School of Mathematics and a young brilliant student of his, Mikhail Suslin in the summer of 1916. In fact there is a birth certificate for this theory with both Luzin and Suslin as parents and no one less than the famous Polish mathematician Waclaw Sierpinski (the father of the Sierpinski fractal triangle or gasket) was a witness. The certificate is dated the afternoon of October 1916 but no day is given. There is a serious reason for this joke. It seems that P.S. Alexandrov, the great Russian topologist whose work on wild topology combining topology with Cantor sets (which is crucial to El Naschie’s E-infinity) wanted to claim Luzin and Suslin’s discovery for himself. Alexandrov was a great mathematician and was actually a very close friend of P. Urysohn whose dimensional theory is about the most important thing in E-infinity theory. Never the less it seems Alexandrov had a character deficiency. Dishonesty among the greatest of scientists seems to be as old as the history of science and it is still with us today. There are far worse plagiarists than this and far more famous scientific disputes engulfing as famous people as Newton versus Leipnitz, Einstein versus Poincaré as well as Lorenz and even D. Hilbert and so on and so on. This is extremely sad but it is fact. Great minds do not always necessarily imply great characters. Scientists are only human and sometimes very human.
A good place to start studying descriptive set theory is to start with trees and trees on products. After that we have to concentrate on polish spaces. The most important examples of polish spaces for E-infinity are the n-dimensional cube studied by M. El Naschie and the Hilbert cube studied by Ji-Huan He. After that comes the Cantor space used in several papers by El Naschie on E-infinity and descriptive set theory, all published in CS&F. Finally we should mention the Bair space. In E-infinity ordinary differentiation and integration are replaced by Weyl scaling while Suslin scaling is used as a fundamental operation on sets. These things sound more difficult than they are and in praxis the situation is much simpler when we are dealing with a concrete problem. Finally we should mention that E-infinity may be regarded as a space made of a Borel set. El Naschie repeatedly mentioned that this idea was given without any mathematics at all by the very great theoretical physicist J.A. Wheeler. El Naschie did not do more than adding the mathematics using random Cantor sets with golden mean Hausdorff dimension.
13th December, 2010.
E-infinity communication No. 55
Crystallographic groups and Heterotic strings in E-infinity.
The following is a notice on a very short paper published in CS&F, 42 (2008), pp. 2282-2284. Although very short and at first glance may appear unassuming, it is extremely important to grasp the idea behind it. It is well known that there are 17 different ‘tiling’ groups in two dimensions corresponding to 17 two and three Stein spaces with total dimensions of 685. In three dimensions there are 230 groups and 219 from that correspond exactly to the 17 groups in two dimensions. The question now is how many dimensions belong to these 219? In this paper El Naschie answers this question. The answer is 8872 which is exactly the number of states of the first massless level of particle-like states of the Heterotic string theory. It is extremely interesting to see that these seemingly abstract 8872 states have such a strong and deep connection with something as real as 3D crystals.
13th December, 2010.
E-infinity communication No. 56
Lee Smolin’s trouble with physics and fractal spacetime.
As things stand at this point in time, Prof. Lee Smolin is probably the most influential scientist in theoretical physics. Smolin has a talent for both serious fundamental science as well a popular science writing in a beyond measure manner. A handful of people could be compared to Smolin’s dual role in science such as R. Penrose but then he is (wrongly) considered by physicists to be only a mathematician. His latest book which is yet again a best seller is not only worth reading for those working in E-infinity, it is a must. Another reason for discussing his book is the involvement of Prof. Lee Smolin with fractals in high energy physics in general as well as the involvement of many of his associate in the Perimeter Inst. and elsewhere with the subject of fractal spacetime.
There are countless points in Lee Smolin’s book which we would like to comment upon and therefore we have to be very choosy and concentrate on what is useful and important to E-infinity. First and foremost there is no mention what so ever of fractal spacetime in the book of Smolin. This is surprising given his documented interest in this subject. In fact there is no mention at all of the word fractal in the entire book. This could be understandable in a mathematical book where pure mathematicians exchange the word fractal with other words like foliation for instance. However Smolin’s book is a popular book and fractal is the most popular word for foliation and continuous geometry or noncommutative geometry. Even the book of Field Medalist A. Connes who is held by Smolin and our group in high esteem occasionally speaks of fractals. We very much hope that no misconception or misunderstanding has prejudiced Prof. Lee Smolin against fractal spacetime because our group needs and hopes for his scientific weight and support.
On page 313 Smolin considers the category of researcher referred to by him as seers and considers their opinion about background-independent approach. E-infinity was developed using the quantum sets methodology of D. Finkelstein. On page 322 Smolin says that D. Finkelstein is a deep thinker (a seer) who does physics differently from anyone else. He talks of the problems that people like Finkelstein would face today if they tried to get funding or a professorship like Finkelstein did. He says it would be impossible today. We in E-infinity do not care about funding or professorships. However we do care that Lee Smolin extends to our work the same tolerance he extended to Finkelstein so that we can at least see that one person, Smolin, is still conducting science ethics and science policy in the same way as in the golden age of science and physics.
On pages 245-247 Smolin gives an apt description of A. Connes and his noncommutative geometry with which we agree completely. There is only one point here. Alain Connes works is a de facto fractal universe. His work is exactly like that of Mohamed El Naschie anchored in von Neumann’s continuous geometry, K-theory and E-infinity rings and groups. On page 313 Smolin mentions the work of Fay Dowker and R. Sorkin. Well both work with partially ordered sets. In E-infinity we work with random Cantor sets. In our case we do not even need a computer to reach the same excellent results which R. Loll and J. Ambjorn reached. The work of both of these two excellent scientists is considered by L. Smolin on pages 242 and 243. Both Loll and Ambjorn worked indirectly in fractal spacetime and published many papers on that including one in Scientific American which we hoped and wished they had acknowledged our work but they did not. This is sad but everyone could overlook important papers in the heat of publication fever. It is not tragic. However and this is a most polite and respectful hint to Prof. Smolin personally……. Is it not more often the case that those who made it big often forget and do not care about those who did not yet make it that big? We remind Prof. L. Smolin of the wonderful story of A. Connes on page 275 of his wonderful book. In all fairness, with this we rest our case, at least here.
Smolin’s book is published by Penguin, England (2006). Price £25.- with £5 discount. [By using pounds, El Naschie intentionally is acknowledging his authorship.]
13th December, 2010.
E-infinity communication No. 57
Implications of Sarkovskii and El Naschie’s number theoretical theorem for physics
In 1975 the American J. York (one of the members of the Honorary Editorial Board of Chaos, Solitons & Fractals) and his student Li wrote a paper entitled “Period Three Implies Chaos” which was one of the most important and influential papers which helped establish chaos as a field of research in engineering and physics. However the same theorem of J. York who gave the science of chaos its name was already discovered in 1964 in number theory by the Russian mathematician Sarkovskii as explained for instance in the excellent book of H.G. Schuster “Deterministic Chaos”, published by VCH, Weinheim, Germany (1989). This shows the importance of number theory in physics which should be understood as the foundation of mathematics just like set theory and therefore the foundation of physics. It should not be understood as playing with the golden mean or number acrobatics (there is nothing called number acrobatics at all) or numerology as some (truly silly) people sometimes say. There is another very important example of a theorem in number theory given by El Naschie in 1998 with deep physical connections to much of the work on spacetime physics. The paper in question is “Four as the expectation value of the set of all positive integers and the geometry of four manifolds”, published in CS&F, Vol 9(9), (1998), pp. 1625-1629. The paper may be found in the E-infinity free of charge published papers (open access) or Elsevier’s Science Direct.