Nobel laureate ‘tHooft’s recent incorporation of fractals

30th November, 2010.
E-infinity Communication No. 38
Nobel laureate ‘tHooft’s recent incorporation of fractals.
It is extremely interesting, in fact more than that, that Prof. ‘tHooft probably one of the most important high energy physicists of our time made several contacts with fractals recently. To appreciate the importance of these contacts for the task of incorporating nonlinear dynamics and deterministic chaos properly in high energy physics cannot be over emphasized. It is simply a fact of life that most if not all of the main stream high energy physics completely misunderstands fractals and have only a cocktail party appreciation of deterministic chaos. They simply do not think it is relevant to what they are doing. The present communication is not concerned in the first instance with the work of the group around Prof. ‘tHooft, for instance Prof. Loll and her associates. It is also not concerned with Prof. ‘tHooft’s proposal for a deterministic quantum mechanics. The new development is more concrete than all of that. To help the reader follow the discussion here we refer explicitly to a 2008 paper by Prof. ‘tHooft entitled ‘A locally finite model for gravity’. A version of this paper is freely available on ArXiv. Again to help the reader concentrate on the subtle points we refer explicitly to certain passages in this remarkable paper.
1. In the Abstract on page 1 Prof. ‘tHooft says “…. Globally however the model is not finite because solution tends to generate infinite fractals….. this model is not yet quantized”.
2. On page 2 he says “… however a confrontation with fundamental aspects of quantum mechanics appears to be inevitable….one can study systems with many particles by viewing spacetime as tissilation”.
3. On page 3 he says “… particles and strings as topological defects”.
4. On page 5 he says “… angles deficit and angles surplus…”.
5. On page 22 in the Conclusion ‘tHooft says “… strings could terminate in infinitely dense fractals of string segments where they could close the universe”.
Clearly Prof. ‘tHooft who appears from his Home Page to have appreciated the beauty and the mathematics of fractals some time ago has started to appreciate the profound relevance of fractals in physics. We would be extremely honored if this considerable change of heart on behalf of one of the most important physicists of our time was brought about by the dedicated work of the E-infinity group which was initiated by the wonderful work of Garnet Ord, Laurent Nottale and their colleagues.
The point about infinity and fractals is particularly relevant. If you want to have points and yet escape all the philosophical, mathematical unpleasant features of points including the production of infinities, then you have one thing, completely disconnected fractals which means Cantor sets. Of course you can use continuous fractals but then you are carrying unnecessary ballast. As Jim Yorke emphasized, the backbone of any nonlinear dynamics is always a Cantor set. Everything is something multiplied with a Cantor set. In his theory Mohamed El Naschie stripped everything down to Cantor sets and that way he can avoid infinity without resorting to finite stringy objects. In addition you have all the rigorous theorems connected to transfinite set theory and the transfinite theory of dimension at your disposal. Oscar Wilde used to say the best way to get rid of temptation is to give in to it. In a sense that is what fractals do. The best thing to get rid of the unpleasant infinities is work them and not to try to bar them out of the analysis. You have to accept that they are there. You have to tame them. The great Russian set theoretician tamed wild topology by understanding its transfinite set theoretical basis. It turns out that the topology of high energy physics is wild topology. It all ramifies into a dense Cantor set. ‘tHooft is rephrasing what we said here in a homomorphic language. Tissilation stands for fractal tiling such as Penrose universe. The boundary of El Naschie’s interpretation of ‘tHooft’s holographic boundary is indeed a dense fractal. At the end it is a Cantor set. You can iterated string segments, frogs or elephants , at the end you end with a Cantor set closing the universe. In E-infinity El Naschie as well as Crnjac showed that the 336 degrees of freedom of a Klein modular curve increases to around 339 due to compactification at infinity and that is what Prof. ‘tHooft was saying in his conclusion. It remains only to mention that the golden mean binary is naturally quantized. In addition fractals are naturally self-similar or at least self-affined. In other words fractals are naturally scale invariant due to the lack of natural states. In a fractal spacetime Weyl’s ideas are valid and Einstein’s objections against them are no longer valid. We sincerely hope that Prof. ‘tHooft will continue in this direction which we are certain is feasible and promising for the future of high energy physics.
E-infinity Group