Prof. Ji-Huan He – The twenty-six dimensional cube operad of a scientist and a gentleman

19th December, 2010.
E-infinity communication No. 64

Prof. Ji-Huan He – The twenty-six dimensional cube operad of a scientist and a gentleman

One of the most outstanding young generation founders of E-infinity theory is undoubtedly Prof. Ji-Huan He. The present communication will attempt to give justice to his great contribution to E-infinity theory and to shed more light on the deep mathematical though of his 26 dimensional polytope as well as his Hilbert model which is lurking behind a deceptive simplicity.
Prof. Mohamed El Naschie described Prof. He as a true Chinese scientist and a gentleman whose scientific journey to E-infinity theory took him from the romantic idealistic world of Freiherr Georg Friedrich Philipp von Hardenberg (Elias Novalis) to the work of probably the greatest and most enigmatic mathematician of all time Alexander Grothendieck. In 2007 Prof. He published his paper “Twenty-six dimensional polytope and high energy spacetime physics”. This paper is only nine pages long but contains ten beautiful colored computer graphics of an n-dimensional cube and how its geometry transforms from order to deterministic chaos. The paper may be found on Elsevier’s Science Direct. It is contained in Vol. 33, Issue No. 1, pp. 5-13 published in 2007 in CS&F with extracts published in several other international journals and books including the American Inst. of Physics. Two years later an equally beautifully illustrated paper entitled “Hilbert cube model for fractal spacetime” was published in CS&F, Vol. 42 (2009), pp. 2754-2759. These two papers are closely related to a paper by O. Zmeskal, M. Weiter and M. Vala as well as a paper by El Naschie, namely “An irreducibly simple derivation of the Hausdorff dimension of spacetime”, (CS&F, 41 (2009), pp. 1902-1904). Zmeskal et al’s paper was entitled “Note to an irreducibly simple derivation by El Naschie” also in CS&F. There are various fundamental issues which lie deep at the root of these papers to which we would like to at least touch upon here.
First Prof. He’s 26 dimensional polytope is de facto an explicit generic example for a fundamental theorem in nonlinear dynamics and as far as we are aware was first stated by the outstanding mathematician and engineering scientist Prof. D. Ruelle from the Inst. of High Scientific Studies in Sur-Yvette, France. Prof. Ruelle’s theorem effectively says that when we put any classical mechanical system in an infinite dimensional setting, then it will become spontaneously chaotic in the sense of the theory of deterministic chaos and its fractal-Cantorian geometry. This explains the magic of the power of the expectation value of the Hausdorff dimension of E-infinity derived for the first time by El Naschie, namely 4.236067977. This value can only be obtained for an infinite dimensional system and corresponds exactly to only 4 topological dimensions in the Menger-Urysohn meaning of topological dimensions. The reduction from infinite dimensions to a core of only four dimensions is a central and recurrent theme in the work of El Naschie and the work in E-infinity. However what scientist working on the physical side of E-infinity should realize is that this reduction of infinity to 4 is far more general in pure mathematics than most people think. Let us start with a simple and obvious example from the topological quantum field theory. It is well known that El Naschie replaced current algebra by fusion algebra. However the most important fusion algebra is the four dimensional fusion algebra as documented in the work of V. Sunder and V. Kodiyalant. This work is closely connected to the work of Sir M. Atiyah as well as Field Medalist V. Jones who discovered the relation between knot theory and statistical mechanics used in the recent work on E-infinity by El Naschie. A more mathematical and abstract relation which E-infinity scientists could use more in future development is the relation to the mathematical theory of highly structured ring spectrums. This theory leads to a so called cube operad of finite dimensions. This is a fancy pure mathematical name for infinite dimensional cubes in infinite dimensional space similar to the work of Ruelle and He. On the other hand E-infinity could be replaced in this mathematical theory by E4 exactly as in El Naschie’s physical geometrical E-infinity theory. Thus E-4 is defined by replacing the operad of an infinite dimensional cube in 4 dimensional space and keeping the same basic idea of E-infinity ring spectra. The entire theory is of course related to K-theory, Grothendieck-Rieman-Roch’s theorem and n-categories. This may partially explain the unusual reaction “to say the least “ of an otherwise good mathematician like John Baez. It must have come as a mild shock that E-infinity of El Naschie and He is smack on the physical target of quantum gravity while others are still beating around the abstract mathematical bush. It is a fact that random Cantor sets are specific and measurable. More over they are O-categories and (n + 1) categories are just more complex monoidal categories. One cannot give any quantative results using the posets of F. Dowker who works with R. Loll and J. Baez. Posets are immensely important theoretically. However they do not lead directly and easily to quantative results which are needed in high energy physics. For instance in 4-dimensional fusion algebra we have the golden mean as an Eigenvalue. This makes all calculations trivial to the extent that no computer is required at all. Similarly the dimensional function of K-theory of the Penrose-‘t Hooft-El Naschie holographic boundary is a direct function of the golden mean which again makes otherwise complex calculation trivial.
The K-holographic boundary, like all fractals, has an inbuilt dilaton field in it so we need not worry about scale relativity as in the classical form of quantum field theory. All that is encoded in the beautiful work of Prof. Ji-Huan He. We should also mention that Prof. He does not work on E-infinity only. The scope of his research is staggering, spanning perturbation methods in numerical analysis to experimental methods in nano technology. Prof. He is also inclined towards literature and poetry like his great friend Prof. El Naschie. Readers may recall that El Naschie likened the indistinguishability condition of E-infinity which makes it impossible to say if a particle is at a point 1 or a point 2 or both points at the same time like in quantum mechanics with the novel of Robert Musil “The man without qualities”. Prof. He always finds wonderful examples from ancient Chinese literature and culture to illustrate his scientific research. Great success in the real world unfortunately brings some unpleasant things with it like envy. In recent times Prof. He, similar to many of us and notably El Naschie and Nottale have had more than their fair share of it. To them we can only repeat an old Nordic proverb “what does the moon care when dogs are barking down there”.
With our best wishes for a merry Christmas and a very happy New Year (2011).
E-infinity Group
P.S.
El Naschie has written several articles about Novalis and his influence on Wagner and nonlinear dynamics which is hinted at here and will be discussed in detail later on.