Dear Bob, A. Kasim, and Sonja,
I have been reading some of Prof. El Naschie's work on E Infinity, and I think I have a better understanding of the similarities between us. When we use the symplictic transformation of a square proportioned according to the dimensional hierarchy of heterotic string theory, we get 10, 16+k, 26+k and 42+2k string dimensions (Chaos, Solitons & Fractals 30 (2006) 579-605, pg. 594, Fig. 15). If we truncate these numbers, then 16 x 42 gives us the 672 roots of E12 (except that E12 has condensed from the 16 dimensional 16 x 42 down to a 12 dimensional 12 x 56 that might be more compatible with the SO(8) 28-plets of Hyperflavor). By construction, (42+2k)/(16+k) = phi^(-2) = phi + 2, the inverse golden mean squared, 2.618. If we keep our decimal places, then (16+k) x (42+2k) = 685.41 ~ 5 alpha bar. Of Course, Bob noted the similarities with 5 alpha bar, and A. Kasim noted the similarities with (26+k) x (26+k) = 685.41, and Sonja noted the similarities between the transfinitely exact 685.41 and the integer part (2)(342) = 684 of the total elements in E12. E12 might be the closest representation to E Infinity in an integer number of dimensions.
Is it a problem that our apparently 16-dimensional 16 x 42 has condensed into the apparently 12-dimensional 12 x 56 roots of E12? In my book "New Approaches Towards A Grand Unified Theory", I expected our 26 dimensional string to be composed of 4-dimensional Spacetime plus a dominant 3-brane (that decomposes into gravity and a 2-brane Weakbrane with sequestered Higgs and Hyperflavor bosons) plus a less-dominant 3-brane (our WIMP-gravitons and Grand bosons are sequestered on this Gravity-brane) plus three hierarchal 2-branes plus two very weak 5-branes (that may also decompose into 3-branes and 2-branes). Effectively, we are modeling the three hierarchal 2-branes (dimensions 11 through 16) as one 2-brane (dimensions 11 and 12), and collapsing 16 dimensions down into 12.
In my book, I expected the 12-dimensional E12 to condense into two 6-dimensional E6-Primes (yes, I used another exceptional group that doesn't properly exist). The "surface area" of a unit radius hypersphere is maximized for 7 dimensions, whereas the "volume" of such a hypersphere is maximized for 5 dimensions (see the same El Naschie article above). Six dimensions are the ideal compromise between maximum area and maximum volume.
Dear Bob,
I concede that I overlooked a possibility with gravitinos, although I don't think that this option applies to Higgsinos (look up "Minimal Supersymmetric Standard Model" on Wikipedia), and that is Majorana spinors. If our gravitino is a Majorana spinor, then we have 260 degrees of freedom (dgf). If our gravitino is a massless Dirac spinor, then we have 264 dgf's. If our gravitino is a massive Dirac spinor with spin 3/2 and spin 1/2 projections, then we have 272 dgf's. Which is the minimal choice? 260, Of Course! Which is the most likely choice? 272 for three reasons: 1) Considering the fact that neutrinos have mass, there are no clear examples of Majorana spinors in Nature, 2) Most, if not all, Sparticles are expected to be massive, and 3) It works better with E8 and 8-dimensional singlets than 260 does - the fact that 260 is not divisible by the rank 8 of E8 implies that something is missing.
Now 272 divided by two is 136, which is very close to alpha bar in the low-energy limit, 137.036. I'm not sure how to make up the difference. I have noticed that the non-integer part of alpha bar, 0.035999679, is close to 1/28 (to within 1% difference), and Hyperflavor theory is full of 28-plets. Do you have any ideas?
If I don't hear anything from you or Garrett, I will assume you are busy publishing your ideas...
Sincerely, Ray Munroe