December 10, 2010
E-infinity communication No. 46
The road to Nottale scale relativity and some remarks on a recent paper by Nobel Laureate G. ‘t Hooft.
Various statements and new mathematical and physical ingredients in a relatively recent paper by Nobel Laureate G. ‘t Hooft (see Quantum gravity without space-time singularities – arXiv: 0909.3426V1[gr-qc], 18 Sept. 2009) are effectively a compliment to the work of Laurent Nottale. It is well known that Nottale is one of the first not only to appreciate the crucial role of scale invariance but also to give a general theory of relativity within a scale invariance program which is relevant to quantum and high energy physics. A very readable popular account of Nottale work may be found in “Ciel et Espace”, February (1994), pp. 28-32. The interview is entitled “Espace-Temps Fractal – La Nouvelle theorie de l’univers”. A formal introduction to Nottale’ theory including references to the work of Garnet Ord as well as Mohamed El Naschie may be found in his by now classical book “Fractal Space-Time and Microphysics”, World Scientific, Singapore (1993). Unitl recently Nottale considered his papers in CS&F to be the most up to date and comprehensive but in 2010 he has published at least two new very long papers in Springer journals which may be found on the internet, all apart from his excellent blog called “Scale Relativity Corner” where highly scientific and informative high level, civilized discussion is going on without the usual vulgarity and aggressiveness characteristic of the run of the mill pseudo scientific blogs nowadays. The idea behind Nottale’s work is basically the same as that of Ord and El Naschie although the mathematics is slightly different. The idea is as simple as it is ingenious. Some mathematical physicists call it the Biedenharn conjecture. It is the fact that any space which is a fractal is devoid of any natural scale. Consequently it is explicitly scale invariant. That means it is a gauge theory in the original sense of the work of H. Weyl. This is the reason by Mohamed El Naschie replaces calculus by Weyl scaling. In fact this is the reason by in E-infinity we can use the set theoretical Suslin scaling. In the particular case of E-infinity theory this leads naturally to golden mean renormalization groups used so successfully by Mitchell Feigenbaum in finding his structural universalities. The pure mathematical background for these E-infinity relations may be found in the theory of semi-groups. The research related to Lucasian numbers by Hilton as well as Fibonacci groups by Nikolova is particularly interesting. It is strange, if not quite disturbing, that J.C. Baez takes such a hostile position to E-infinity theory although he occasionally works on loop spaces, E-infinity ring spaces and spectra all apart of groupoids. However maybe this aggressive rejection has a more personal than scientific basis.
In the paper of ‘t Hooft mentioned above he states already in the abstract that he “…finds that exact invariance under scale transformation is an essential new ingredient….”. Then he continues by saying “These differences can be boiled down to conformal transformation”. He points out that Newton’s constant is not scale-invariant and neither is the Einstein-Hilbert action. Consequently he cannot use the Riemann curvature nor its Ricci components. Apart of the Weyl component nothing is left except to follow the road outlines by L. Nottale, M. Agop and E. Goldfain, namely scale relativity, fractal spacetime and E-infinity transfinite sets.