25th January, 2011.
E-infinity communication No. 76
‘t Hooft-Veltman dimensional regularization implies E-infinity Cantorian spacetime (we told you so!).
At least for the last ten years or to be exact since his paper “’t Hooft dimensional regularization implies transfinite Heterotic string theory and dimensional transmutation”, Mohamed El Naschie maintained that the ‘t Hooft-Veltman method is not a mere mathematical trick but something physically more profound, namely a strong indication that our real physical quantum spacetime has a Cantorian-fractal geometry and topology. The paper which is included in the Proceedings of a conference also attended by ‘t Hooft is also published in various refined and modified versions in CS&F and other international journals. For ten years ‘t Hooft seemed unconvinced and as more often than not, skeptical which is a healthy scientific attitude in general although in this particular case not so clear why because the situation is rather straight forward and clear at least to those familiar with nonlinear dynamics and fractals. It was about three to four years ago that the great French mathematician A. Connes came to the same realization of El Naschie but of course in a far more mathematical and stringent way, namely that dimensional regularization is a clear indication of noncommutative geometry. In less abstract mathematical language, this means it is an indication of Cantorian-fractal spacetime geometry and topology.
In what follows we give a very short outline of El Naschie’s papers and direct the reader to the relevant literature where one can also find a copy of a letter from Richard Feynman talking about fractal curves in quantum mechanics to Prof. Garnet Ord. Rather than assuming like ‘t Hooft did that spacetime dimensionality is slightly less than 4 and approached the exact topological value from below, El Naschie de facto assumed it to be slightly larger than 4 and approached it from above. Interestingly he recovered two dimensions, namely 4 for the Menger-Urysohn topological dimension and a corresponding effective Hausdorff dimension slightly larger than 4, namely the famous number 4.23606799… He starts by assuming an unknown theory M with a spacetime dimension D(M) which may be used to replace the pole terms so that 1/(D ̶ 4) goes to → D(M). Replacing D by D(M) everywhere we find a quadratic equation (D(M) ̶ 4)D(M) = 1 with two solutions D(M) = 4 + ϕ3 = 4.236067977 and D(M) = ̶ ϕ3 = ̶ 0.236067977. Added together one finds the topological 4 of which 4.236067977 is the Hausdorff dimension of E-infinity spacetime. For details and discussions the reader is referred to the work of M.S. El Naschie and Alain Connes. Here we give first the classical literature on this extremely important evidence for the correctness of E-infinity and noncommutative geometry as being truly physical theories.
1. M.S. El Naschie: ‘t Hooft Dimensional Regularization implies transfinite heterotic string theory and dimensional transmutation. Frontiers of Fundamental Physics, 4, p. 81-86. Edited by B.G. Sidharth and M.V. Altaisky. Kluwer Academic Publishers, New York (2001).
2. M.S. El Naschie: On 't Hooft dimensional regularization in E-infinity space. Chaos, Solitons & Fractals, Vol. 12, Issue 5, 4 January (2001), P. 851-858.
3. M.S. El Naschie: Dimensional Regularization implies transfinite heterotic string theory. Chaos, Solitons & Fractals, Vol. 12, (2001), P. 1299-1303.
4. A. Connes and M. Marcolli: Renormalization and motivic Galois theory. Int. Math Research Notices, (2004), No. 76, P. 4073-4091.
Now we come to the highly interesting new development regarding where Nobel Laureate Gerard ‘t Hootf currently stands vis-à-vis this fundamental and crucial connection between the method which he invented together with his thesis supervisor (Nobel Laureate M. Veltman and for which they shared the Nobel Prize) and fractal noncommutative spacetime geometry..
In 2009 in a book entitled “Approaches to Quantum Gravity” published by Cambridge Press and edited by D. Oriti, ‘t Hooft was asked by L. Crane the following question: “Do you think of dimensional regularization as a particularly effective trick or do you believe that it is a hint as to the fine structure of spacetime? In particular have you thought about the possibility of quantum spacetime having a non-integral Hausdorff dimension distinct from its topological dimension?” ;t Hooft answered as follows: “We thought of such a possibility. As far as the real world is concerned, dimensional regularization is nothing but a trick…. Veltman once thought there might be real physics in non-integer dimension, but he never got anywhere with that”.
Of course we, following El Naschie, beg to differ. Rather than quoting El Naschie to counter balance ‘t Hooft’s statement, we will quote A. Connes verbatim on this subject. Connes said: “We show that a careful investigation of dimensional regularization leads us to an interpretation that it is not just a formal procedure, but is an actual geometry….”. Somehow we hope that ‘t Hooft will validate the insight of El Naschie and Connes to the benefit of the development of unified theory of quantum gravity which we think is substantially complete in E-infinity, noncommutative geometry. Knowing ‘t Hooft’s genius, modesty and his ability to concede an error, we are very hopeful indeed.
26th January, 2011.
E-infinity communication No. 77
A moonshine conjecture from E-infinity (number theoretical motivation)
One of Alexander Grothendieck’s greatest insights was to follow Andre Weil’s hint at the deep connection between topological characteristic of a variety and its number theoretical aspect, i.e. its diplomatic aspects. Topologizing physics within a number theoretical framework seems to be an obvious characteristic of El Naschie’s E-infinity theory.
In the present communication we discuss a surprising relation between the totality of all Stein spaces, the compact and non-compact Lie symmetry groups on the one side and super string theory, path integral and the summing over dimensions procedure of E-infinity theory as well as the inverse fine structure constant = 137. The relation seems at first sight so bizarre and unreal that it is justifiably called the moonshine conjecture. In fact it has some similarity with the original moonshine conjecture and it is best to start by introducing the relation between the monster symmetry group and the coefficient of the j-function. The story starts when it was noticed that the minimal dimension for the monster is only one less than the first coefficient in the j-function. Thus we have D(min monistor) = b ̶ 1 = 196884 = 196883. The relation was clarified and the conjecture proven by Borcherds, a student of Conway (see El Naschie’s paper on the subject, CS&F, 32, (2007), pp. 383-387 as well as his paper “Symmetry groups prerequisite for E-infinity”, CS&F, 35, (2008), pp. 202-211 as well as “On the sporadic 196884-dimensional group, strings and E-infinity spacetime”, CS&F, 10(6), (1999), pp. 1103-1109.
We start by observing that the sum of the dimensions of the 17 two and three Stein spaces is exactly 686. This is equal 5 times 137 plus one. On the other hand the sum of the dimensions of the 12 compact and non-compact Lie symmetry groups is 1151. This is one short of 1152 which is 9 times 128, the electroweak inverse coupling of electromagnetics. This value (9)(128) = 1152 plays an important role in calculating the quantum states spectrum of the Heterotic string theory as can be seen in the excellent book of M. Kaku. Adding 686 to 1151 one finds 1837. Next we consider the total number of dimensions of the 12 non-compact Lie groups which comes to 1325. On the other hand the total number of the 8 non-compact 2 and 3 Stein spaces is given by 527. This is one short of Witten’s 528 states of a 5-Bran theory in 11 dimensions. Adding 527 and 1325 one finds 1852. The grand total is thus 1837 + 1852 = 3689. Now we embed 3689 in the ten dimensions of super strings and find that 3689 + 10 = 3699. Here comes the first incredible surprise because 3699 = (27)(137) = 3699 where = 137.
The second surprise in when we consider the “energy” stored in the “isometries” of the symmetry groups. Starting with the curvature of E-infinity spacetime = 26 + k we see that ( )( ) = (26 + k)(26 + k) which comes to 685.5. This is almost equal to 686 of the sum over all two and three Stein spaces. This is one of the best and simplest justifications ever for the theory of summing over symmetry group dimensions. Next we consider the intrinsic dimension of E7. This is dim E8(intrinsic) = 57. The transfinitely corrected compactified value is 57 + 1 + 3k 58.5. The energy is thus given by (58.54101966)2. This gives us 3427.050983. Here comes our next and final surprise for this communication. Dividing the energy by 25 one finds = 137.082039. The numerics indicate that there is indeed a deep connection between energy, symmetry and the electromagnetic fine structure constant. Members of the E-infinity group may like to think about a water tight proof for the above as well as pointing to more intricate relations.