From Osher Doctorow Ph.D.
I'm going to introduce a question and a theory - although the theory is still being developed in the early stages. If readers aren't interested in this first post, I'll take the question and theory else- where. QUESTION. Are fractals Deformations, and if so, are they in particular Quantum Deformations? The Theory involved is Deformation Theory. If you look under the key- word Deformation on the internet, you'll find hundreds of articles in many different fields and disciplines and quite a few definitions of Deformation that seem to have nothing in common. My interest is to unify these different theories but in a Simple way and an Intuitive way that doesn't require years of study but applies almost directly to fields like Fractals. Deformation intuitively is "change". Unlike the concept of the "infinitesimal" in calculus, which is a "very small change," a Deforma- tion can be large or small or intermediate, but it is specified in some way both globally and locally. Deformations have revolutionized Quantum Theory and Mathematics, and for example Quantization in Quantum Theory is presently of two main types: Deformation Quantization and Geometric Quantization (look them up as keywords on the internet). The simplest example of Deformation is probably also one of the most fundamental in mathematics and physics: Non-Euclidean Geometry (NEG for short here). NEG is the Geometry of curved ("bent") space while Euclidean Geometry (EG for short here) is the Geometry of flat space. So we get from Eucidean Geometry to Non-Euclidean Geometry by the very Simple and Intuitive idea of Bending or Curving something. What's so great about that? Well, Non-Euclidean Geometry was discovered in the 1800s by Gauss, Riemann, Lobachewski, and Bolyai, and both Mathematics and Physics were fundamentally changed afterward. Physics used NEG as the basic tool (in what is called tensor formulation) for General Relativity of Einstein, and General Relativity is in turn involved either in classical or modified forms in Quantum Gravity and Superstring and other Quantum-Related research. Before getting to Fractals, I'll give one more example of "what's so great about that". "Very small" Deformations were discovered as "Infinitesimals" by Sir Isaac Newton and Leibniz in the 1600s, although they actually built upon pioneering work of Pierre De Fermat of 1600s France and Archimedes of later Ancient Greece. In other words, they are the essence of Calculus. Mathematics and Physics and other Sciences were not the same after Calculus. They changed fundamentally. Yet, like Non-Euclidean Geometry, Calculus was a type of Deformation, both very Simple and very Intuitive when formulated as Deformations (not so simple when presented in complicated textbooks designed for Math or Physics majors who have mastered many prerequisite courses). Most of us, I think, have an intuition that Fractals and Chaos are as Revolutionary as anything around, so a "natural" choice is to ask whether they can be regarded as Deformations of anything. Fractals change the shapes of objects by Self-Similar Contractions. In simple language, the original shape of a "large" object is shrunken into many smaller copies of that object, and that is the Change involved. The new, smaller objects are similar to the origin ones. Everywhere you look, so to speak, there are smaller copies of the original object - smaller and smaller on smaller and smaller scales. In Computer and Mathematics language, the way in which the original object changes is kept track of and specified by Iteration and repetition of the previous operation (the opera- tion often called Composition in Mathematics). What could "Deformation" give FractInt that current methods don't give? In my opinion, it will probably eventually yield Quantum Fractals and Quantum Knowledge-Information from a different route than current Quantum Computer and Quantum Information theories. That route, as I indicated above, is through Deformation Quantization among other methods. But as important as that, it will probably enable us to understand the relation- ship between "shrinking" large objects on the human scale and the nature of the apparently very diferent microscopic Quantum world where waves or fields and particles have a curiously different nature. If no reader is interested, I'll take this thread elsewhere, and I'll have no hard feelings since I won't give up my interests. Hopefully, readers won't have hard feelings about being disturbed from their other interests in this posting. Incidentally, nobody has to make a "major" contribution to be interested - just indicate that it interests you. Osher Doctorow