Matter and energy have a very curious property. They interact with each other in predictable ways and the more energy an object has, the smaller length scales it can interact with. This leads to some very interesting and beautiful results, which are best illustrated with some simple quantum electrodynamics (QED).

QED is the framework for describing the interactions of charged leptons with photons, and for now let’s limit things to electrons, positrons and photons. An electron is a negatively charged fundamental particle, and a positron is the same particle, but with a positive charge. A photon is a neutral fundamental particle of light and it interacts with anything that has a charge.

That means that we can draw a diagram of an interaction like the one below:

In this diagram, time flows from left to right, and the paths of the particles in space are represented in the up-down direction (and two additional directions if you have a good enough imagination to think in four dimensions!) The straight line with the arrow to the right is an electron, and the wavy line is a photon. In this diagram an electron emits a photon, which is a very simple process.

Let’s make something more complicated:

In this diagram the line with the arrow to the left is a positron, and the electron and positron exchange a photon.

Things become more interesting when we join up the electron and positron lines like this:

Here an electron and positron annihilate to form a photon.

Now it turns out in quantum mechanics that we can’t just consider a single process, we have to consider all possible processes and sum up their contributions. So far only the second diagram we’ve considered actually reflects a real process, because the other two violate conservation of energy. So let’s look at electron-positron scattering. We have an electron and a positron in the initial state (the left hand side of the diagram) and in the final state (the right hand side of the diagram):

There are two easy ways to join up the lines in this diagram to get the following contributions:

There’s a multiplicative weight (on the order of a percent) associated with each photon interaction, so we can count up the photons and determine the contribution each process has. In this case, there are two photon interactions in each diagram, so each one contributes roughly equally. (You may ask why we bother calculating the contributions for a given pair of initial and final states. In fact what we find interesting is the ratio of contributions for two different pairs of initial and final states so that we can make predictions about rates of interactions.)

Let’s add a photon to the diagram, just for fun. We can connect any two parts of electron and positron lines to create a photon, like so:

A fun game to play in you’re bored in a lecture is to see how many unique ways you can add a photon to a diagram.

So how do we turn this into a fractal? Well we start off with an electron moving through space (now omitting the particle labels for a cleaner diagram):

Then we add a photon or two to the diagram:

Similarly let’s start with a photon:

And add an electron-positron pair:

This is all we need to get started. Every time we see an electron or positron line, we can replace it with a line that emits and absorbs a photon. Every time we see a photon we can add an electron-positron pair. We can keep repeating this process as much as we like until we end up with arbitrarily complex diagrams, each new step adding more refinement to the overall contributions:

At each step the distance we consider is smaller than the one before it, and the energy needed to probe this distance is larger. When we talk about an electron we usually think of a simple line, but real electrons are actually made of a mess of virtual particles that swarm around the central electron. The more energy we put into probing the electron’s structure (or lack of structure) the more particles we liberate in the process. There are many diagrams we can draw and we can’t pick out a single one of these diagrams as the “real” electron, as they all contribute. We have to take everything to get a real feel of what something as simple as an electron is.

As usual, things are even more complicated in reality than this simple picture. To get a complete understanding we should add the other particles to the diagrams. After all, that’s how we can get a Higgs boson out of proton- in some sense the Higgs boson was “already there” inside the proton and we just liberated it by adding a huge amount of energy. If things are tricky for the electron, they are even more complicated for the proton. Hadrons are bound states of quarks and gluons, and while we can see an individual electron, it’s impossible to see an individual quark. Quarks are always found in groups, so have the take the huge fractal into account when we look inside a proton and try to simulate what happens. This is an intractable problem, so need a lot of help from the experimental data to get it right, such as the dedicated deep inelastic scattering experiments at the DESY laboratory.

The view inside a proton might look a little like this (where the arrows represent quarks):

Except those extra bits would go on forever to the left and right, as indicated by the dotted lines, and instead of happening in one spatial dimension it happens in three. To make matters worse, the valence quarks are not just straight lines as I’ve drawn them here, they meander to and fro, changing their characteristic properties as they exchange other particles with each other.

Each time we reach a new energy range in our experiments, we get to prober deeper into this fractal structure of matter, and as we go to higher energies we also liberate higher mass particles. The fractals for quarks interact strongly, so they are dense and have high discovery potential. The fractals for neutrinos are very sparse and their interactions can spread over huge distances. Since all particles can interact with each other directly or through intermediaries, all these fractals interact with each other too. Each proton inside your body contains three valence quarks, surrounded by a fractal mess of quarks and gluons, exactly the same as those in the protons that fly around the LHC. All we’ve done at the LHC is probe further into those fractals to look for something new. At the same time, since the protons are indistinguishable they are very weakly connected to each other via quantum mechanics. In effect the fractals that surround every valence particle join up to make one cosmological fractal, and the valence particles just excitations of that fractal that managed to break free from their (anti-)matter counterparts.

The astute reader will remember that the title of the post was the seemingly fractal nature of matter. Everything that has been described so far fulfils the requirements of any fractal- self similarity, increased complexity with depth and so on. What it is that makes matter unlike a fractal? We don’t exactly know the answer to that question, but we do know that eventually the levels of complexity have to stop. We can’t keep splitting space up into smaller and smaller chunks and finding more and more complex arrangements of the same particles over and over again. This is because eventually we would reach the Planck scale, which is where the quantum effects of gravity become important and it becomes very difficult to keep track of spatial distances.

Nobody knows what lies at the Planck scale, although there are several interesting hypotheses. Perhaps the world is made of superstrings, and the particles we see are merely excitations of those strings. Some models propose a unification of all known forces into a single force. We know that the Planck scale is about fifteen orders of magnitude higher in energy than the LHC, so we’ll never reach the energy and length scales needed to answer these questions completely. However we’ve scratched the surface with the formulation of the Standard Model, and so far it’s been a frustratingly good model to work with. The interactions we know of are simple, elegant, and very subtle. The most precise tests of the Standard Model come from adding up just a handful of these fractal-like diagrams (at the cost of a huge amount of labour, calculations and experimental time.)

I find it mind boggling how such simple ideas can result in so much beauty, and yet it’s still somehow flawed. Whatever the reality is, it must be even more beautiful than what I described here, and we’ll probably never know its true nature.

_{(As a footnote, to please the pedants: To get a positron from an electron you also need to invert the coordinate axes to flip the spin. There are three distinct diagrams that contribute to the electron positron scattering, but the crossed diagram is a small detail might confuse someone new to these ideas.)}

Tags: Feynman Diagrams, physics