On ‘t Hooft’s recent work and the development of Yang-Mills instanton via El Naschie’s E-infinity theory

17th January, 2011.
E-infinity communication No. 70
On ‘t Hooft’s recent work and the development of Yang-Mills instanton via El Naschie’s E-infinity theory
E-infinity theory gave considerable attention quite early on to integrating Yang-Mills theory and ‘t Hooft’s instantons into its scale invariant fractal frame work. It is probably a fact to say that Mohamed El Naschie’s relations to ‘r Hooft must have stimulated him to attempt this integration. On the other hand it must equally be a fact that the recent work of ‘t Hooft in which he introduces a dilaton field to the Lagrangian of quantum field theory was motivated primarily by the realization which he must have gained by reading El Naschie’s work on a “fractal-Cantorian” instanton.
In the present short communication we would like to direct the reader to the relevant sources from which he can come to the above very important conclusion. Dilaton is used extensively in string theory. However it would be a misconception to think that ‘t Hooft used it in the same way. That would be underestimating the importance of ‘t Hooft’s proposal which in our view is nothing less than “fractalizing” the frame work of classical quantum field theory.
El Naschie felt relatively late that Lagrangian formulation in particle physics was wrongly elevated to the status not of merely an alternative to the differential equations but to a supreme and the only right way to do theoretical particle physics. El Naschie was originally trained in the rigorous tradition of the classical theory of elasticity and Koiter’s general theory of elastic stability where a potential energy formulation goes hand in hand with differential equations and under estimated the deep seated prejudice of the community of particle physics, particularly those working in quantum field theory and that they regard a Lagrangian as sacred. We do not know exactly when he started addressing these problems but latest in 2006 he wrote a paper “Intermediate prerequisites for E-infinity theory”, CS&F, 30 (2006), pp. 622-628 in which he attempted in section 5 to clarify the issue of a Lagrangian in E-infinity theory. It is in this paper where he introduced the concept of a fuzzy Lagrangian and a fuzzy or average instanton. In the above mentioned paper as well as somewhat earlier and later papers, El Naschie made a profound discovery liking string states with instantons. The most elementary form of this discovery was stated in earlier communications and is essentially the equality of multiplying left and right movers to obtain the first massless level of Heterotic string particle-like states, namely N(0) = (504)(16) = 8064 as well as multiplying the degree of freedom of the noncompactified holographic boundary given by Klein-modular curve (336) and the instanton density of K3 Kähler manifold (24) so that we find the same value of No namely N(0) = (336)(24) = 8064. A former student of El Naschie, Dr. A. Elokaby explained the above theory in a very lucid paper “On the deep connection between instantons and string states encoded in Klein’s modular curve”, CS&F, 42 (2009), pp. 303-305. Having come that far one can easily find the exact N(0) by remembering that the exact instanton density for a fuzzy K3 Kähler is not 24 but 24 + 2 + k, which means 26 + k = 26.18033989. Similarly the 336 should be 336 + 16k = 338.8854382. Consequently, No is No = (338.8854382)(26.18033989) = 8872.135957. This is 808.1359569 states or instantons more than the classical 8064 found first by Green, Schwarz and Witten.
We could do the same for Heterotic strings by noting that 504 should be (4)( ) = 548.328157 where = 137 + ko which is the sum of all the exceptional Lie symmetry groups in the E-line (548), plus 0.328157 transfinite or fuzziness added to them. In addition the 16 should be changed to 16 + k = 16.18033989. That way one finds N(0) = (4 )(16 + k) = 8872.135956 exactly as before. El Naschie reasoned that the same procedure could be applied to ‘t Hooft’s famous instanton result, namely I(instanton) = = 78.95683521. Since the instanton density 26 + k must remain the same, all that changed is the dimensionality of the holographic boundary. In the instanton interpretation of the Heterotic strings theory leading to No = 8064 the boundary had 336 degrees of freedom (or 338.885 ≃ 339 in the compactified fuzzy exact case). For ‘t Hooft instanton our space is only 3 + 1 dimensional as is the case in classical quantum field theory. Therefore the holographic boundary must be d = (3 + 1) ̶ 1 = 3. The exact fully Yang-Mills instanton value must therefore be <> = (3)(26 + k) = 78.54101967. This may also be expressed as ( /2) + 10 = 78.54101967. Note that the classical value of ‘t Hooft is slightly larger by 0.41581554. Notice also that if we take only the integer value, namely 26 one finds <> = (3)(26) = 78. This is exactly equal to the dimension of the exceptional E6 symmetry group of the exceptional E-line. Another note worthy observation is that using the classical instanton density 24 instead of 26 one finds <> = (3)(24) = 72. This is the roots number of E6. In other words the roots number is the lowest approximation for obtaining a value for I.
Summing up we can say all the values 72, 78 as well as are equal to the volume of 16 five dimensional Ball or four dimensional spheres. The only exact value is the fuzzy value obtained by El Naschie, namely <> = (3)(26 + k) = 78.54101967. These are the elementary and obvious but deep results of Prof. Mohamed El Naschie which prompted ‘t Hooft’s courageous and mathematically elaborate and justified attempt to patch up classical quantum field theory. We, the proponents of E-infinity, wish Prof. ‘t Hooft to succeed because it is firstly good for science and secondly because it is in the interest of E-infinity’s world wide acceptance.
In conclusion we draw the attention of the reader to various important publications on the subject:
1. The relation between Yang-Mills instanton and the exceptional groups is given by El Naschie in CS&F, 38 (2008), pp. 925-927.
2. A useful dictionary translating gauge field language to fiber bundle language may be found in CS&F, 30, (2006), pp. 656-663.
3. The Weyl scaling of E-infinity is explained in CS&F, 41 (2009), pp. 869-874.
4. The relation between ‘t Hooft’s work and that of El Naschie is discussed in CS&F, 38, (2008), pp. 980-985.
5. It should be noted that Prof. El Naschie is by no means the first to consider a connection between Yang-Mills theory and fractals. In fact already in 1992 there was a paper in Physical Review Letters (23 March, 1992, Vol. 68, No. 12, pp. 1811-1813) with the title “Evidence for a Yang-Mills fractal” in which numerical evidence for fractals in a Yang-Mills system in (2 + 1) dimension was observed. Even before that T. Kawabe and S. Ohta from Japan wrote a paper on Chaos in Yang-Mills theory “Onset of chaos in time-dependent spherically symmetric SU(2) Yang-Mills theory”, Physical Review D, Vol. 41, No. 6, 15 March, 1990, pp. 1983-1988.