The Secrets of Feynman Diagrams

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Matt O'Dowd (1:00)

The Feynman diagrams revolutionized particle physics by providing a simple system to sort out the infinite possibilities when elementary particles interact. This incredible simplicity provides some stunning insights into the nature of reality.

Feynman's path integral shows us that to properly calculate the probability of a particle traveling between two points, we need to add up the contributions from all conceivable paths between those points, including the impossible paths. In fact, we can go even further. According to Feynman's approach to quantum mechanics, every conceivable happening that leads from a measured initial state to a measured final state does in a sense happen, at least in the math. To calculate the probability of any quantum system evolving between two states, we need to sum over every conceivable intermediate state.

This is impossible, because there are infinite possible intermediate states.

But, as we discussed in our episode on solving impossible equations, the Feynman diagrams allow physicists to quickly figure out which of the infinite possibilities need to be considered to get an answer that's good enough. Each diagram represents a family of interactions, and tells us the equation needed to calculate the contribution of that family to the total probability.

The miracle of Feynman diagrams is that an absurdly simple set of rules allows you to easily find all of the important interactions. Today, we're going to learn these rules. Then, you are going to apply them to do some quantum field theory yourself. There are space-time t-shirts at stake.

We're going to stick to quantum electrodynamics, the first and most predictably powerful quantum field theory. QED talks about the interaction of the electron field with the electromagnetic field. That means interactions between electrons, their antimatter counterparts, the positron, and photons. In Feynman diagrams, we depict the electron as an arrow pointing forwards in time, while the positron is an arrow pointing backwards in time. We'll soon see the power of representing antimatter as time-reversed matter. The photon is shown as a wavy line. Time direction is irrelevant for the photon. Throw these on a plot of space versus time, and we have a Feynman diagram.

A useless one. None of these particles are doing anything worth calculating. For this to be interesting, the electric and electromagnetic fields need to interact. This is where we start to see the power and simplicity of this approach. Particle slash field interactions are represented as a vertex. A point where the lines representing the different particles come together. It turns out that there's only one vertex that's possible in QED.

One with an arrow pointing in, an arrow pointing out, and a single photon connection. It looks like this. This vertex alone represents six very different seeming interactions, and it can be used to construct infinite Feynman diagrams. Let's look at the possibilities. Oriented like this, with time increasing upwards, this vertex represents an initial electron that emits a photon, after which both particles move off in opposite directions. But if we rotate this vertex so that photon is coming in from below, we have a picture in which an electron absorbs that incoming photon. The photon vanishes as its momentum is completely transferred to the electron. Rotate again, and the picture is of a photon coming in and giving up its energy to produce an electron-positron pair. A process we call pair production. Rotate again, and now we have a positron absorbing a photon, and a positron emitting a photon, and finally an electron and a positron annihilating each other to produce a photon. And that's it. That's all the ways that the electromagnetic and electron fields can interact. Every single QED interaction is built from these.

But why only this interaction? Well, because of conservation laws. Energy and momentum conservation requires that particles not just vanish or appear from nothing, which guarantees that if something goes in, then something else must come out. Charge must also be conserved. If one electron or positron goes in, then one electron or positron respectively must leave. If an electron and positron both go in, then their charges cancel, so a zero charged photon must leave. Similarly, if a photon creates a negatively charged electron, it must also create a positively charged positron. There are other more complex ways in which in-going and out-going particles can balance charge, but as we'll see, all of these can be built up from this one vertex.