The quantum sets of D. Finkelstein, E-infinity sets and the paper of D. Benedetti

10th December, 2010.
E-infinity communication No. 51

The quantum sets of D. Finkelstein, E-infinity sets and the paper of D. Benedetti

Prof. David Ritz Finkelstein is one of the earliest pioneers of quantum set and the construction of spacetime in a way similar to deriving thermodynamics from the motion of atoms. This is the original idea of El Naschie’s work which he acknowledges to have taken over from Finkelstein and Wizecker. Finkelstein is a deep thinker who was described in one of the acclaimed books of Lee Smolin as one of the seers in science today. The monad of Finkelstein space is the null set. The null here is not the dimension and we can consider his null set to be our null set as well as our empty set. We recall that E-infinity follows Suslin set theory and constructs everything from a Suslin tree starting with the empty set, the null set and the unity set. In E-infinity these are ( ̶ 1, ϕ2), (0, ϕ) and (1, 1). We note that in E-infinity we have Mandelbrot-El Naschie notion of the degrees of emptiness of an empty set leading to the totally empty set ( ̶ ∞, 0). The monad of E-infinity is random Cantor sets in general. Monad in the terminology of ‘t Hooft are called the building blocks of spacetime. ‘t Hooft does not work with sets. However a scientist somewhat close to him and very close to Prof. R. Loll, namely Prof. Fay Dowker works with partially ordered sets as explained in earlier communications. The work by D. Finkelstein was developed considerably by Heinrich Saller (see Quantum space-time-gravity by J. Baugh, D. Finkelstein, M. Shiri-Garakani and H. Saller). A particularly good summary of D. Finkelstein’s pioneering work is “Quantum sets and Clifford algebras”, Int. J. of Theoretical Physics, Vol. 21, No. 6/7 (1982). In the abstract of this paper Finkelstein says “… Quantum set theory may be applied to a quantum time space and quantum automaton.” In the introduction of his lecture in the presence of Feynman and Wheeler he said “several of us here including Feynman, Fredkin, Kantor, Moussouris, Perti, Wheeler and Zuse suggest that the universe may be discrete rather than continuous and more like a digital than analog computer. C.C. von Weizaecker (the student of Heisenberg and the elder brother of the past President of the Federal Republic of Germany) worked this path since the early 1950’s and we have recently benefitted from the relevant work of J. Ford (Ford was an expert in deterministic chaos and together with two of El Naschie’s friends, J. Casati and B. Cherekov pioneered the science of quantum chaos which is sometimes confused with the theory of Cantorian-fractal spacetime).
In the rest of this communication we address the technical aspect of E-infinity and show how to derive all the results of the work of D. Dario Benedetti from first principles in a far simpler and transparent manner using E-infinity. It is difficult to do this without writing equations but we will try our best.
First we consider our Cantorian E-infinity spacetime to be the union of infinitely many elementary monads Cantor sets. A single random Cantor set has, by the theorem of Mauldin-Williams, a Hausdorff dimension equal to the golden mean. The higher order Cantor sets will have a Hausdorff dimension equal ϕ to the power of n where n is 0, 1, 2… Adding all these dimensions together one finds the finite dimension of the large Cantorian space. Thus from summing from zero to infinity of ϕ to the power n, one finds 1 + ϕ + ϕ2 + ϕ3 + … and so on to infinity. The sum is exactly equal 2 + ϕ. This corresponds in the Connes-El Naschie dimensional function or bijection to a Menger-Urysohn dimension of exactly 3. However this has not been gauged in terms of the original monad ϕ. Therefore we have to divide 2 + ϕ by ϕ and this gives us the famous dimension 4.23606799 which corresponds exactly to the Menger-Urysohn topological dimension equal to 4. To show that 2 + ϕ corresponds to the topological dimension of 3 we just exclude the non-fractal dimension 1 from our summing. That means we start from n = 1 to n = ∞. That way the total dimensions become 2 + ϕ ̶ 1 = 1 + ϕ. To obtain the gauged dimension we divide by ϕ and find 2 + ϕ again which means the topological dimension is 3. Thus we have obtained from first principles the 4 and 3 dimensions of spacetime and space only with a single assumption which is that the union of all the elementary monads represents our spacetime. The monads themselves are formed by intersections of two and more monads. Unlike other theories our monads are explicit and well defined. They are random Cantor sets. The zero set for instance is given by two dimensions. First the topologically invariant Menger-Urysohn dimension is zero and second the Hausdorff-fractal dimension which is in this case ϕ. Thus the zero set is fixed by (0, ϕ). Using a simple elementary cobordism argument for (0, ϕ), the neighborhood or border is a wave given by the empty set ( ̶ 1, ϕ2). We could go on that way indefinitely. That means we have ( ̶ 2, ϕ3), ( ̶ 3, ϕ4) and finally (0, ̶ ∞) as mentioned earlier on. In fact we can determine the world sheet of fractal string theory. We did that when we summed from n = 1 to n = ∞ and found the total Hausdorff dimension to be 2 + ϕ ̶ 1 = 1 + ϕ. This corresponds in the Connes-El Naschie dimensional function bijection to a two dimensional object. The fractal world sheet is topologically two dimensions but Hausdorffly more than one dimensional and less than two dimensions. It is a fractal area-like with a Hausdorff fractal dimension equal 1 + ϕ = 1.618033989.
In fundamental scientific research there are two directions. We either keep improving and even patching and fixing our older theories or we start afresh from different or slightly different basic assumptions and principles. E-infinity is the second possibility. Dario Benedetti is the usual first possibility. Einstein’s relativity and quantum mechanics as well as string theory and loop quantum gravity started as the second possibility like E-infinity. In fact ‘t Hooft’s quantum field theory started exactly as E-infinity theory. Now string theory and quantum field theory are the establishment and the improvements and patching are characteristic for these important developments. In real life we need both methods and both philosophies. When we reach the same conclusions, this is then the promised land.
E-infinity group.