Fractal spacetime – some historical remarks to fractal spacetime deniers

18th January, 2011.
E-infinity communication No. 71
Fractal spacetime – some historical remarks to fractal spacetime deniers
We occasionally hear that fractal spacetime has nothing to do with quantum physics. These people may even go as far as saying that fractals have nothing to do with physics. Of course everyone is free to think what he likes but the unpleasant surprise comes when one finds out that most of those who make these statements are mentors of mainly those who work on fractals in physics although they are extremely inventive when it comes to giving different names to these things which we call scale relativity, Cantorian spacetime and fractal spacetime. This point could not be explained by mathematically based science. It needs the tool of social sciences and psychoanalysis. It could of course be far more straight forward than that and is mainly related to science policy and science funding coupled to the very scarce resources available to scientific research in general and theoretical and mathematical physics in particular. This subject is not particularly the source of joy to most of us and we will not dwell on it any further. However it is important to recall some important facts about the history and substance of fractal spacetime theory.
The first comprehensive paper published in an international journal with the title Fractal Spacetime was by the English-Canadian Garnet Ord who discussed this subject with Nobel Laureate Richard Feynman and was strongly influenced by Feynman’s views on the subject. Ord’s paper was published in 1983 in one of the journal of the Institute of Physics, namely J. Phys. A: Math. Gen., 16 (1983), pp. 1869-1884 and was entitled “Fractal space-time: a geometric analogue of relativistic quantum mechanics”. A little later and seemingly independently, a young but well known French astrophysicist Laurent Nottale published in 1989 a paper entitled “Fractals and quantum theory of spacetime” in Int. J. Mod. Physics !, 4, (1989), pp. 5047-5117. This was a sequel and generalization of his 1984 paper with J. Schneider entitled “Fractals and nonstandard analysis”, J. Math. Phys., 25, (1984), pp. 1296-1300. It seems however that Nottale had the same ideas as Ord at nearly the same time but for some reason he could not publish his paper except later on. His famous book “Fractal Space-Time and Microphysics” published by World Scientific in 1993 made up for this delay. Mohamed El Naschie on the other hand is at least ten years older than Nottale and Ord and came to fractals in quantum mechanics via nonlinear dynamics. El Naschie obtained his Ph.D. in engineering and started working in physics much later after having reached the position of Full Professor of structural engineering. All the same he was keenly interested in philosophy and science in general and quantum mechanics in particular. An encounter as a student of engineering in Hannover, Germany with the work and personality of Werner Heisenberg and K. von Weizacker changed his scientific interests completely. However serious work had to be postponed until he became aware of the mathematics of Cantor sets and their relation with number theory, topology and symmetry groups. El Naschie moved from applied mechanics to applied nonlinear dynamics, chaos, singularity theories and fractals to quantum mechanics and finally high energy physics. His first efforts were in understanding turbulence via fractals and Cantor sets in a way similar to the work of Kolmogorov. He was also familiar with stability theory of Poincaré, Köiter and Andronov as well as R. Thom’s catastrophe theory which helped him to move from engineering to physics. Turbulence was the first problem which Heisenberg tackled but could not solve. El Naschie used turbulence as a paradigm for vacuum fluctuation following Wheeler. The first paper by El Naschie which was relevant to fractal spacetime indirectly was the 1991 paper entitled “Multi-dimensional Cantor-like sets and ergodic behavior” in Speculations in Science & Technology, Vol. 15, No. 2, pp. 138-142. This was followed by two papers with direct relevance to quantum mechanics. First “Quantum mechanics and the possibility of a Cantorian spacetime” published in Chaos, Solitons & Fractals, Vol. 1, No. 5 (1991), pp. 485-487. This was followed by “Multi-dimensional Cantor sets in classical and quantum mechanics”, CS&F, Vol. 2, No. 2, (1992), pp. 211-220. After that an important paper in computational and applied mechanics was published entitled “Physics-like mathematics in four dimensions – implications for classical and quantum mechanics”, Computational and Applied Math II. W.F. Ames and P. van der Houwen (Editors), Elsevier (North Holland), (1992), IMACS.
Since these efforts by Ord, Nottale and El Naschie to establish a new field, namely the field of quantum-fractal physics, many papers were published intermittently. The author of these papers seems to be unaware of previous similar publications by the trio Ord-Nottale-El Naschie or for some reason or another chose not to make reference to these contributions. Whilst this was maybe understandable before the world wide web, in the meantime it is not understandable at all. In what follows we list some of the interesting work on fractal quantum and high energy physics which shows that this field is vibrant and is becoming unusually combative in a not very pleasant manner, at least occasionally.
There are many papers in quantum mechanics which mention the concept of the Hausdorff dimension as well as the Hausdorff dimension of a quantum path. These papers are relatively well known from the work of scientists like Parisi and will not be mentioned here. We may start by mentioning the paper of M. Wellner “Evidence for a Yang-Mills fractal” in Physical Review Letters, Vol. 68, No. 12 (1992), pp. 1811-1813. An earlier similar paper is “Onset of chaos in time-dependent spherically symmetric SU(2) Yang-Mills theory”, Physical Review D, Vol. 41, No. 6, (1990), pp. 1983-1988. A 2010 paper by S. Carlip entitled “The small scale structure of spacetime” (arXiv: 1009.1136VL[gr-qc]6 Sept 2010 makes the usual remark about Wheeler foam, mentions Loll and Amjborn’s work and completely overlooks the work of Ord-Nottale and their associates. Similar remarks apply to the work of Fotini Markopoulou. The paper with the almost identical title of that of Nottale and Ord “Fractal space-time and black-body radiation” published in Astrophysics and Space Science, 124 (1986), pp. 203-205 by A. Grassi, G. Sironi and G. Strini is also oblivious to the work of Ord and Nottale. Strangely Benedetti’s paper which uses the same quantum group concept of El Naschie did not mention the work of Ord, Nottale or El Naschie. The paper on fractal spacetime by O. Lauscher and M. Reuter seems to be totally unaware of the work of Nottale and Ord let alone El Naschie. Equally very strange is the absence of any reference to Nottale or Ord in the paper “Fractal geometry of quantum spacetime ar large scales” by I. Antoniadis, P. Mazier and E. Mottola although the first author works in France where L. Nottale is well known. We have not mentioned more than one percent of the large body of literature on fractals in quantum mechanics, relativity and quantum gravity. This shows that fractals are indeed relevant and may be too relevant to the extent that competition is not only fierce but slightly unfair to say the least.
We have not mentioned fractals in all other fields of physics. There are more publications on fractals in physics than on quantum mechanics when you consider that fractals were discovered in physics no more than 25 years ago while quantum mechanics is with us since more than 80 years. For all these reasons we think that the hard work our group had done and continues to do is more than justified and worthwhile.
In conclusion we should mention one of the most important recent papers by G.N. Ord “Quantum mechanics in two dimensional spacetime: What is a wave function”, published in Annals of Physics, 324 (2009), pp. 1211-1218. It may also be interesting to give Mohamed El Naschie’s answer to Ord’s question using his transfinite set theory. A wave function is an empty set with a topological Menger-Urysohn dimension equal minus one ( ̶ 1) which is the surface area or the neighborhood cobordism of the quantum particle which is the zero set. Two cultures divided by a common language, namely mathematics.
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