Physics

From Tetration.net

1 Tetration in Physics
2 Continuous Iteration in Physics
3 Continuous Iteration in Renormalization
- 3.1 The Problem of Efficiently Computing Iterated Functions
  - 3.1.1 The General Case of Discrete Iteration
  - 3.1.2 Hyperbolic Continuous Iteration
4 Resources

Tetration in Physics

Does tetration play a role in physics? Consider the Julia set of exponential map for $e z$ . The Julia set is periodic due to the periodic nature of the exponential function.

Julia set of exponential map	Julia set of exponential map magnifiged
Bands - Plank's constant

If this Julia set were some type of universe, what type of universe would it be? Well, the Julia set is periodic creating bands, so a reasonable question would be ask, how could the one band move to another band? The exponential function maps each band into the entire complex plane. Because of the chaotic nature of tetration and the exponential map, the paths a point can take from one band to another is inconceivably complex. But there is one system in physics that does have the same type of inconceivably complex dynamics, quantum field theory.

Let the "movement" of one band to another in the Julia set be defined in terms of the amount of area on the destination band covered by points that had once been on the original band. Then the movement could be equated with the idea of a transition amplitude of a particle in quantum mechanics from one point to another being computed from the sum of all paths.

If tetration exists in physics, then it is almost certainly as complex as quantum field theory. That leads to a second question, could tetration be the ultimate cause of quantum weirdness and a possible mathematical foundation for quantum field theory? Tetration grew out of research on the power tower, but the Feynman Path Integral that is used to compute the dynamics of quantum fields appears to be an odd variant on the power tower with integrals and multiple dimensions added. Since tetration is a sub-branch of arithmetic, unwillingness of academia to tackle problems like extending tetration to the real and complex numbers could hamper placing quantum field theory on a proper mathematical foundation.

Continuous Iteration in Physics

One could argue that continuous iteration is physics since abstract dynamical systems in physics can be defined as measure preserving continuously iterated functions.

Continuous Iteration in Renormalization

Faà di Bruno's formula and the associated Bell polynomials are the foundation of my work with continuous iteration. I have longer wondered whether there is any connection between the principles of continuous iteration and renormalization. Now a new paper, Combinatorial Hopf Algebras in Quantum Field Theory I. points out that Faà di Bruno's formula is a very important Hopf algebra. The paper explains how Hopf algebras are important in several different areas of quantum field theory including renormalization.

What follows is a description of a problem I'm working on in continuous iteration. A superficial look at the problem shows that it raises issues about systems without infinities being modeled by equations with infinities. This is also the same issue that renormalization deals with, analyzing models based on equations with infinities in them and then systematically being able to remove the infinities because they never actually existed, they were merely artifacts of a powerful model and not what was being modeled. The question will be is the following problem like renormalization or is it a new view of the problem of renormalization itself?

The Problem of Efficiently Computing Iterated Functions

I have two generations of Mathematica software for finding the Taylor series of iterated functions. The older software deals with iterated functions that have a hyperbolic fixed point, which is by far the most common case in dynamics. This software is used to compute hyperbolic continuous iteration and was capable of computing thirty terms of the Taylor series for iterated functions that have a hyperbolic fixed point on PCs in the mid-nineties. The newer software is completely general, but as a result it is limited to discrete iteration. But placing appropriate constraints on the first derivate of the function being iterated results in the general discrete expression simplifying to an expression where iteration is expressed continuously.

The General Case of Discrete Iteration

$f^{\,t}(z)=f_0 +f_1^t (z-f_0) + \frac{1}{2!}(f_2 \sum_{k_1=0}^{t-1}f_1^{2t-k_1-2}) (z-f_0)^2$ $+ \frac{1}{3!}( f_3\sum_{k_1=0}^{n-1}f_1^{3n-2k_1-3} +3f_2^2 \sum_{k_1=0}^{n-1} \sum_{k_2=0}^{n-k_1-2} f_1^{3n-2k_1-k_2-5}) (z-f_0)^3 + \ldots$

Hyperbolic Continuous Iteration

$f^{\,t}(z)=f_0 +f_1^t (z-f_0) + \frac{1}{2!} f_2 \frac{{{f_1}}^{-1 + n}\, (-1 + {{f_1}}^n) } {-1 + {f_1}} (z-f_0)^2$ $+ \frac{1}{3!} ( f_3 \frac{{{f_1}}^{-2 + n}\, \left( -1 + {{f_1}}^n \right) \, \left( -{f_1} + {{f_1}}^n \right) } {{\left( -1 + {f_1} \right) }^2\, \left( 1 + {f_1} \right) } + 3f_2^2 \frac{{{f_1}}^{-1 + n}\, \left( -1 + {{f_1}}^n \right) \, \left( 1 + {{f_1}}^n \right) }{ \left( -1 + {f_1} \right) \, \left( 1 + {f_1} \right) } ) (z-f_0)^3 + \ldots$

The new software gives results for hyperbolic continuous iteration like the older software did, but it also gives results for parabolic continuous iteration which is associated with Abel’s equation. The problem is that cranking out just the first eight terms of the Taylor series for iterated functions is time consuming on today’s PCs. This raises the question of if it is possible to move from the continuous hyperbolic form back to the more general discrete form. If this was possible then the older software could be adapted to computer the more general discrete form.

Here is a problem broadly related to iterated function as a possible foundation for mathematical physics as per A New King of Science. It employs Faà di Bruno's formula, which is a Hopf algebra and leads to a problem of systematically removing infinities from useful models because the infinities are known to not actually physically exit. Note that the infinities that appear in hyperbolic continuous iteration only appear when the first derivative is a root of unity, these exceptional cases are from the existence of other types of fixed points than hyperbolic, parabolic and rationally neutral. These different types of fixed points force symmetries that the model can’t express, so it blows up. In renormalization the existence of symmetries allow the infinities to be removed. So is my problem to efficiently compute iterated functions a type of renormalization problem?

Resources

Figueroa, H. and Gracia-Bondía, J. M. Combinatorial Hopf Algebras in Quantum Field Theory I. 19 Mar 2005.