Super Yang-Mills, super gravity and super strings. Spectroscopy and the impossibility of number coincidence

14th December, 2010.
E-infinity communication No. 59.

Super Yang-Mills, super gravity and super strings. Spectroscopy and the impossibility of number coincidence

If the reader is not very familiar with the theory of super strings, it would be advisable to have a text book to one side whilst reading this communication. For convenience we recommend M. Kaku’s book “Introduction to superstrings and M-theory”, Springer, New York (1999).
Let us look at the Heterotic string and consider its spectrum. For instance following pages 384 and 385 of Kaku’s book we have the following: For the left moving sector there are three distinct states. First 480 states equal in number to the kissing of two E8 and also equivalent to their roots number. In addition we have another 16 states. These make together the 496 of E8E8 dimension where each E8 has clearly 248 dimensions. Finally we have another 8 states so that the total number of the left moving section is 496 + 8 = 504. Note that this is exactly the sum of the dimensions of E8 plus E7 plus E6 plus E5 as found by M. El Naschie (248 + 133 + 78 + 45 = 504). Now we look at the right moving sector where we have only 16s states. Following the rule of Fock space of quantum field theory (where Fock space is the generalization of Hilbert space of quantum mechanics to quantum field), then the total number of all states is the multiplication of the left and right sectors. This gives us the famous number of the first level of massless particle-like states of the Heterotic string theory, namely No = (504)(16) = 8064. Now page 385 gives the number of states in a super Yang-Mills theory as 3968. These could be thought of as lifting E8E8 dimensions to super space by taking 8 copies of the 496 dimensions leading to (8)(496) = 3968. On the other hand we know from the spectrum of the theory of super gravity that we have 512 states. This could be thought of as raising the two dimensional average Hausdorff dimension of a quantum particle path to the ten dimensions of super strings. Using E-infinity bijection, this means we have 2 to the power of 10 minus 1. This is 2 to the power of 9 which is equal 512. This is only 16 more than 496 and 8 more than 504. Let us lift these 512 to super space like we did with super Yang-Mills. Proceeding in this way we find (512)(8) = 4096. The next step taken by Mohamed El Naschie is to consider No to be the sum of super Yang-Mills and a super-super gravity. That means Heterotic strings could be thought of as combing Yang-Mills and super gravity. In all events the spectrum indeed gives the correct result, namely 3968 + 4096 = 8064. In other words No = 8 (496 + 512) = 16 (248 + 256) = (16)(504).
At this point El Naschie makes the following argument: Why stop at E5 and the sum 504? It seems more natural to take the entire exceptional E-line of Lie symmetry groups. In other words we should add the dimensions of E4, E3, E2 and E1. In his extensive work on the exceptional Lie symmetry groups’ hierarchy, El Naschie shows that the sum is 548. Consequently 504 should be replaced with 548. However El Naschie also knows that the 548 is 4 times 137. This is the integer value of the inverse electromagnetic fine structure constant. The exact E-infinity value is however = 137.082039325. In addition El Naschie also knows that due to the main scaling sequence of this ( /2) we must have 16 + k rather than simply 16. This means 16 should be ‘corrected transfinitely’ to 16.18033989. The exact No value is thus No = (16.18033989)(548.3281573) = 8872.135956. This last value may be expressed differently as No = (16 + k) (4) ( ) = (64 + 4k)(137 + ko) where k = 0.18033989 and ko = 0.082039325 are all higher order values of the golden mean. Recalling that 64 is the number of different spin directions and 137 was shown to be the number of elementary particles in an extended standard model, then one could easily conjecture that a return to the classical No = 8064 of Green, Schwarz and Witten may be obtained by replacing 137 with the number of the conventional standard model, namely 126 and 64 + 4k by simply 64. Proceeding this way one finds No = (64)(126) = 8064. This is a delightful result showing the logical consistency of the entire theory.
Anyone suspecting any form of numerology in the intended bad sense of the word should remember that = 137.0820393 and all other transfinitely corrected values harmonize with each and every equation. If one makes a single error in the calculation it is impossible not to notice it immediately after two steps and never, ever more than six steps. It is the well known butterfly effect of chaos manifested in the most irrational number of all irrational numbers, namely the golden mean. The charge of numerology is the cheapest shot possible and comes mainly from those with not so well hidden agendas. These people are well known to all the E-infinity researchers and we need not name them here. They will also be eternalized as infamous in the history books for their non-scientific, disgraceful attacks on innocent people in their literally obscene blogs.
E-infinity Group.