4th December, 2010.
E-infinity communication No. 41
Comments on Lisi’s new 2010 paper in Scientific American ‘The vital importance of the concept of average symmetry and summing over Lie symmetry groups
One could of course use a superficially mathematical language and terminology to suggest that Dr. G. Lisi’s work on E8 is pure numerology. For instance why E8 and why did he never mention E7. Maybe because E8 is the largest with 248 dimensions. If this is the reason then he may be mistaken because in the Max Planck Inst. for Gravitation near Berlin, Germany as well as elsewhere people have found E10 and E11. In fact there are many papers by leading mathematical physicists suggesting a relation between E11 and the eleven dimensional M-theory of E. Witten, so why not E11? This is by no means the end either. There is a well documented 12 dimensional F-theory and both American physicist Ray Munroe and Mohamed El Naschie conjectured that there may be out there a 685 dimensional E12 symmetry group. Worse still if we grant that E8 is the main relevant exceptional group, then Lisi’s work could be termed numerology because he merely fits a certain number of real or hypothetical elementary particles to certain numbers fixing the geometry and topology of E8, for instance the dimension 248 or the number of roots 240. Of course we do not view Lisi’s work in this way. We were only playing the devil’s advocate because of all the trivial and cheap charges of numerology which emanate from certain quarters for reasons best known to themselves. However it is important to emphasize the importance of super positioning many exceptional compact and non-compact Lie symmetry groups to obtain fuzzy or fractal-like larger symmetry groups which represent a symmetry on average or a real world average symmetry like the well known average symmetry of chaotic strange attractors (see for instance Symmetry in Chaos by M. Field and M. Golubitsky, Oxford (1992)). The chapter on page 27 is entitled ‘Symmetry on Average’. It is worth summarizing what the author says on pages 27-30:
“To make 100 iterations we require 5000 arithmetic computations and if we work by hand without a computer we need 3 days and this does not include plotting and colouring of the strange attractor and yet we will not observe any symmetry. The symmetry hidden in these strange attractors needs at least 1500 iterations and to see it clearly we need 60,000 iterations which is a hopeless task without a computer and that is why the concept of symmetry in chaos or average symmetry was not discovered earlier.”
One of the earliest papers on average symmetry in physics and E-infinity nonlinear dynamics was El Naschie’s 1994 paper entitled ‘Average symmetry, stability and ergodicity of multi-dimensional Cantor sets’, Il Nuovo Cimento, Vol./ 109, N.2, February (1994), p. 149-157. In this paper both the golden mean and random Cantor sets were involved on a very fundamental level. Later on El Naschie considered summing over two and three Stein spaces as well as exceptional Lie groups. This led to the discovery that the sum of the dimension of these Stein spaces is 5 multiplied with the inverse fine structure constant 137. This means 685 which is the dimension of the conjectured E12 exceptional Lie group. The second discovery was that the sum of all of the E-line exceptional Lie groups is equal to 4 multiplied by 137 which is 548. Having spoken so strongly against numerology and number coincidence we fear we could be giving a wrong impression about the role of number theory and number coincidence in physics. It was not only the well known Cambridge Professor Sir Arthur Stanley Edington who was deeply engaged in what many suspected of being numerology in his fundamental theory. In fact Nobel Laureate of Physics of the caliber of Paul Dirac we well as more recently Steven Weinberg were considering Eddington’s work and certain large number coincidence. Even more surprisingly Richard Feynman had great interest in the work of Eddington but he was talked out of it, probably by his friend Herbert C. Corben around 1942 as recounted in the autobiography of Feynman “The Beat of a Different Drum” by J. Mihra (Oxford 1994).
There are many excellent papers written by members of the E-infinity group explaining what is numerology and what is not. For instance L. Marek-Crnjac “On the vital difference between number theory and numerology in physics”, CS&F, 38 (2008), p. 1239-1242 as well as the paper by Ting Zhong “From the numerics of dynamics to the dynamics of numerics and visa versa in high energy particle physics”, CS&F, 42 (2009), p. 1780-1783. Next we would like to briefly mention the paper of Prof. A. Mukhamedov of Kazan University, Russia which connects E-infinity with the fiber bundle mentioned by Lisi. This paper is entitled “E-infinity as a fiber bundle and its thermodynamics”, CS&F, 33 (2007), p. 717-724.
Finally we have to again consider the embedding of E8 in the context of E-infinity which was strangely not taken up by Lisi although the view point was communicated to him in many comments years ago. The point is that once we embed E8 with its 248 dimensions in the 26 bosonic dimensions of string’s spacetime, we find that 248 + 26 = 274. Taking a shadow world like E8 E8 sper symmetry theory then we find (2)(274) = 548 which is exactly 4 times 137 found earlier on in summing over all exceptional Lie symmetry groups. There is an even more striking embedding this time with the kissing number of all the E-line exceptional groups. The sum of these kissing numbers is 522 and embedding in 26 leads to 522 + 26 = 548 exactly as above. This kissing number which may be interpreted physically as the contact point of hyper-sphere packing could be thought of as a tangible geometrical picture for the massless gauge boson of higher order theory of particle physics. Thank you.