Could fractals be used in fundamental physics, if not, why not?

Question

Perhaps naively I feel there could be many ways in which fractals could explain the strangeness of Quantum mechanics (or quantum gravity). There are some papers on the subject but they are few and far between, and I'm just not sure why this is. As someone just entering into an Msc in theoretical physics, I'm guessing there's an obvious reason I am not aware of as to why this is not an effective strategy. Any ideas, perspectives, or references on the subject would be great.

Answer 1

Define Fractal?

Here's the real answer, it depends. Do you want non trivial fractals? Or, mabye you want self-similarity? Is it about the fractal dimension?

Basically, and object $S$, is a fractal if it is the non-trivial attractor of some dynamical system, with generator $T$. So then,

$$S=T(S)$$

What do we mean by nontrivial? Well, we should have a large number of sets $X$ that evolve towards our attractor. We should also demand that the attractor takes on a non-trivial form. I.E it can't be a point or something similar.

Compare this to the fundamental objects of physics. For instance, consider the unitary $U(1)$ group. Roughly speaking, it consists of the set of numbers such that,

$$S=R(S)$$

Where $R$ is a rotation operator that acts on the set $S$. Here's the big difference...the attractor $S$ of this system is trivial. It's simply the whole plane. One has to add extra conditions to get moderately less trivial attractors. Even in this case we simply get the unit circle for the set $S$ that makes up the unitary group.

We can keep going, another fundamentally important symmetry is the $SO(4)$ group. Once again the "attractor" is mostly trivial from a complexity stand point. Sure, it's elegant and simple, but not in that fractal way that makes things so interesting.

What's happening? Simply put, Lie Groups are natural in a way Strange Attractors aren't. For instance, a large number of strange attractors are stiched together copies of themselves. Lie Groups aren't stiched together copies of themselves. This allows for them to be simpler and less fragile as an object and thus more robust as a mathematical object. However, it's this very property that makes Strange Attractors interesting in the first place!

This isn't to say that there don't exist truly bizarre Generalized Lie Groups that are truly Strange Attractors. However, it seems as though nature prefers robust groups like U(1), SO(4), etc. rather than the fragile groups that generally make up the set of strange attractors.

Answer 2

There is lots of work on fractals in mathematical physics. For example, you might start with this paper