Fractal properties of quantum spacetime in the Perimeter Inst. of Theoretical Physics and El Naschie’s quantum group dimension

9th December, 2010.
E-infinity communication No. 48

Fractal properties of quantum spacetime in the Perimeter Inst. of Theoretical Physics and El Naschie’s quantum group dimension

Dario Benedetti of the Perimeter Institute, partly founded due to the efforts of Prof. Lee Smolin, published a highly interesting paper in 2009 which is quite revealing. In this paper he seems to have carefully studied the literature used mainly by the E-infinity group, for instance what Mohamed El Naschie calls the Biedenharn conjecture. Benedetti’s paper is freely available on the net “Fractal properties of quantum spacetime”, arXiv: 0811.1396V2[hep-th], 25th March 2009. The main conclusion is that taking certain limits the dimensionality of spacetime, namely exact 4 as well as space only, namely exactly 3 may be obtained using quantum groups is obtained. To show how this follows immediately from a well known quantum group dimension when setting the monadic dimension of the atoms of the concerned space we do not need much nor even referring to old papers published by Mohamed El Naschie in many international journals. All what one needs is to go through the following steps.
Step one is to take any good book on quantum groups, for instance C. Kassel book “Quantum Groups” published by Springer 1995. Step two open page 364 and there you will find a formula for quantum dimension for a simple model given explicitly to be the said monade q to the power of n + 1 minus the same but with negative power ( ̶ n ̶ 1) then all divided by q minus the inverse of q. Setting the monade q = ϕ = the quantum dimension is found to be exactly 4. One could find the result of all dimensions, namely the space dimension 3 as well as the Hausdorff dimension 4.2367977 and R. Loll’s spectral dimension 4.01999 ≃ 4.02 in a similarly very simple way. Philosophical discussion of these dimensions were given by El Naschie in a paper written when he was in Cambridge, UK in 1998 (see Bio systems (an Elseiver journal), No. 46 (1998), pp. 4-46). This paper was entitled Dimensional symmetry breaking, information and fractal gravity in Cantorian space. A second paper is in a book published by Gordon and Breach “The quest for a unified theory of information”, Edited by Wolfgang Hofkirchner. The shortest and most condensed paper is “Quantum groups and Hamiltonian sets on nuclear spacetime Cantorian manifold” in CS&F, Vol. 10, No. 7, (1999), pp. 125-1256. In particular Table No. 1 compares all the various relevant dimensions. It is important to see how completely different theories lead to essentially similar conclusions. The only question is what is the most simple theory. We have no doubt that it is E-infinity because it is free from traditions and is obliged only to the mathematical logic with no regard to the political correctness of scientific grouping and science funding policy.
E-infinity group.