The dilaton in Mohamed El Naschie’s early papers and the very recent paper by G. ‘t Hooft incorporating the dilaton in quantum field theory

22nd December, 2010.
E-infinity communication No. 68
The dilaton in Mohamed El Naschie’s early papers and the very recent paper by G. ‘t Hooft incorporating the dilaton in quantum field theory

We already very briefly discussed the recent proposal of Nobel Laureate in Physics (1999) Gerard ‘t Hooft regarding the introduction of a dilaton scalar field into his version of deterministic quantum field theory. In the present short note we return to this subject and comment on it again, this time in the light of El Naschie’s early papers about the role of dilaton in E-infinity theory.
The idea of a scalar dilaton field and the hypothetical particle dilaton goes back most probably to the work of Kaluza and may be earlier on in connection with the work of the Finish theoretical physicist and colleague of Kaluza, Gunar Nordstrom. The writers remember seeing the television science program of Prof. El Naschie during which he told an engaging story about the man who invented the fifth dimension. This man is not T. Kaluza but it is G. Nordstrom. However Nordstrom space is flat which is extremely important for ‘t Hooft’s work as pointed out by El Naschie. In his first paper dedicated to the subject in 2008 El Naschie considered certain dualities pertinent to the Nordstrom-Kaluza-Klein theories which were obviously important for quantum field theory. The paper in question is “On dualities between Nordstrom-Kaluza-Klein, Newtonian and quantum gravity”, CS&F, 36 (2008), pp. 808-810. A second paper followed where the need for a coupling constant and consequently coupling “particle” between gravity and electromagnetism was discussed. This turned out to probably be a pseudo Goldstone boson. This paper was entitled “Kaluza-Klein unification – some possible extensions”, CS&F, 37 (2008), pp. 16-22. El Naschie calculated the number of elements of the matrix and found that 14 of them are due to Einstein + Maxwell. The additional pseudo Goldstone boson resulting from fusing both theories is what makes the 14 + 1 = 15. Subsequently El Naschie showed in this paper that the K-K theory can be generalized to Witten’s five Brane in eleven dimensions theory and calculates the well known 528 states.
The preceding discussion may be considered the beginning of the motivation for a dilaton field and particle. The direct predecessor was thus the Jordan, Brans and Dicke scalar. The scalar appears in the form of a dilaton in all perturbative string theories. The exception is M theory. Dilaton is important for conformal invariance. That means dilaton is important for scale invariance. The introduction of the dilaton field in quantum field theory is almost equivalent to the introduction of L. Nottale’s scale relativity into it or making quantum field theory a little more “fractal field-like”. Using the dilaton field one gains two things indirectly. First a fractal-like scale invariance and second an empty set-like vacuum with an expectation value and negative dimension. This is of course a very loose non-mathematical description of the hidden idea behind the introduction of the dilaton. The dilaton featured in El Naschie’s older publication at many locations to show the connections between his Cantorian spacetime approach and the dilaton for which there is no urgent need what so ever in E-infinity theory because Cantorian set theoretical formulation has a natural inbuilt dilaton in it so to speak. The first substantial discussion of the dilaton in E-infinity formalism was given by Prof. El Naschie ten years ago in his first outline for a general transfinite spacetime theory in a paper entitled “A general theory for the topology of transfinite Heterotic strings and quantum gravity” published in CS&F, 12 (2001), pp. 969. In section 7.3 on page 975 under the heading “Relation to 10 dimensional Newton’s constant and dilaton” El Naschie gave the exact transfinite value of g(dilaton) to be equal 0.723606797. That means 10 copies of g gives the dimension of a Milnor fractal sphere in seven fractal dimensions. Subsequently he calculates the real part of the dilaton S(D) namely S(R) and finds S(R) = 26.18033989 = 26 + k where k is a function of the golden mean to the power 3, namely k = 0.18033989. Incidentally in this paper El Naschie considers at some length the important work of J. Ambjorn of dynamical triangulation and introduces fractal fuzziness to it, something which Ambjorn et al incorporated in their excellent paper published in Scientific American in 2008 with Dr. R. Loll. In a second, mainly survey paper entitled “On a class of general theories for high energy particle physics”, CS&F, 14 (2002), pp. 649-669, El Naschie consider the dilaton again. For instance in Table 5 on page 657 he reports on the values of S® and T® estimated in a paper by F. Quevedo. These were found to be S (R) 25 and T(R) 1. These are quite near the exact transfinite values found by El Naschie, namely S(R) = 26 + k. It is tempting at this point to speculate that S(R) + T(R) = 26 + k. This is supported by the numeric (S(R) 25) + (T(R) 1) = 26 but the more important fact is that the frequent appearance of 26 in E-infinity theory must have some hidden reason. We recall the 26 + k = 26.18033989 was found to be the value for all of the following. It is the value for the bosonic dimension of Heterotic string theory, the inverse super symmetric unification of all fundamental forces, the curvature of the core of E-infinity spacetime, the Euler characteristic of the fuzzy K3-Kähler manifold of E-infinity theory as well as the instanton density of the same. Now that even the real part of the dilaton turned out to be 26, we think the reason is the topology and dynamics of the holographic boundary of E-infinity. This boundary is a compactified Klein modular curve with 336 + 16k degrees of freedom equal nearly to 336 + 3 = 339. Two of the most important orbits in this modular curve are the 42 and the 26 orbit.
In conclusion we think incorporating the dilaton in quantum field theory ‘t Hooft is essentially realizing very deeply that his deterministic quantum mechanics is a homomorphism of E-infinity indetermanistic, chaotic and fractal mechanics.
E-infinity Group