The Secrets of Feynman Diagrams, QFT Pt. 5 (Script) (Patreon)

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 about 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’re 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 predictively powerful quantum field theory. QED talks about the interaction of the electron field  with the electromagnetic field. That means interactions between electrons, their anti-matter counterpart 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 anti-matter 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 the 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 emitting a photon, and a positron absorbing 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 the electromagnetic and electron fields can interact. Every single QED interaction is built from these. But why only this interaction? Because of conservation laws. Energy and momentum conservation require that particles not just vanish or appear from nothing, which guarantees that if something goes in then something else must come out. Charge must be also 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-charge 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 ingoing and outgoing particles can balance charge, but, as we’ll see, all of these can be built up from this one vertex.  

Before we look at those more complex interactions, Here’s another important rule. The overall interaction described by a set of Feynman diagrams is defined by the particles going in and the particles going out. These are the particles that we actually measure. We know their properties – for example their energy and momentum – and they obey Einstein’s mass-energy equation. We say that these particles are “on the mass shell” or just “on-shell – they sit on the shell structure you get when you plot Einstein’s equation for energy, momentum and mass.

On the other hand, everything that happens in between the ingoing and outgoing tracks has questionable reality; each possible diagram that results in the same ingoing and outgoing particles is a valid part of possibility space for that interaction. The particles that have their entire existence between vertices within the diagram, but don’t enter or leave, are called virtual particles. Their correspondence to anything resembling real particles is debatable.  They are also, by definition, unmeasurable – otherwise they’d be one of our ingoing or outgoing particles. These particles do NOT have to obey the mass-energy equation, so are “off-shell”. These particles aren’t even limited by the speed of light or the direction of time, which leads to all sorts of fun.

Let’s go back to the simple interaction we looked at in our recent episode. Electron scattering can be depicted as two electrons going into an interaction, and then two electrons going out. We know the momentum of the ingoing and outgoing electrons. Any combination of the fundamental 3-path vertex that can lead to this final result has to be considered. Simple examples are the exchange of a single photon to transfer momentum between electrons, or the exchange of two or more photons. But we can add as many of these vertices as we like; including the electrons exchanging photons with themselves at different stages in the process, or photons momentarily splitting into virtual electron-positron pairs. As long as the final result is the same, any of these are possible.

Part of the beauty of Feynman diagrams is that each of these diagrams themselves represents an infinite number of specific interactions. To start with, each of the particle paths are actually infinite paths, as well as infinite possibilities for particle momenta. We have to consider even impossible, faster-than light paths. And this is really important – for any particle BESIDES the ingoing and outgoing on-shell particles, ANY energy, speed, and even direction in time is possible. This last point is really bizarre and really powerful. For example, for two electrons exchanging a single photon, it doesn’t matter if we draw the photon going from the first to the second or the second to the first, even though this seems like a very different situation. We can think of the difference as being just the photon traveling forwards in time in one case, and backwards in the other. The math describing that transfer covers both cases.

Let’s look at an even weirder example of this. This is Compton scattering – an incoming electron and an incoming photon bounce off each other. One way that can happen is for the electron to emit a new photon and later absorb the incoming photon. In that intermediate stage between vertices the electron is a virtual particle, which means we include all possible paths it might take as long as they lead to it producing the same final electron and photon. That includes paths backwards in time. Mathematically, a time-reversed electron looks exactly like a positron. Like this. The same particles go in and out, but now the interactions look very different. Instead of an electron emitting and then absorbing a photon, we have on one side the incoming photon creating an electron-positron pair. That new electron becomes our outgoing electron, but the positron annihilates with the incoming electron to produce the outgoing photon.

These may seem like wildly different processes, but in the math of Feynman diagrams they are exactly the same! The interpretation of the interactions is irrelevant; all we care about is the topology of the diagram; in other words, how are the vertices connected to each other? This fact makes Feynman diagrams an incredibly powerful tool in simplifying quantum field theory calculations, vastly reducing the number of contributing interactions that need to be separately solved. The interpretation of anti-matter as time-reversed matter is one that some, including Richard Feynman, took quite seriously. We’re going to delve into that idea more deeply in an episode very soon. For now, I want to give you a chance to play with Feynman diagrams yourselves. So I have a challenge question for you.

When an electron and positron interact electromagnetically we call it Bhabha scattering. It’s an interesting case. The most important Feynman diagrams for Bhabha scattering are the two cases involving a single virtual photon, and include two-vertices each. Those diagrams seem to describe very, very different events, but lead to exactly the same result. Use the rules I described in this episode to draw both of the two-vertex diagrams for Bhabha scattering and describe what’s happening at each of the vertices. Then I want you to try to draw all of the possible four-vertex diagrams. For the latter, don’t bother with what we call the self-energy diagrams in which electrons or positrons emit and then reabsorb a photon.

Send your neatly drawn Feynman diagrams to pbsspacetime@gmail.com within one week of release of this episode. Include in the subject line the words Feynman Diagram Challenge, because we filter by subject line. We’ll randomly choose 5 correct answers to win a Space Time t-shirt, and that includes a choice from brand new t-shirt designs! One will be an exclusive for challenge winners and Patreon supporters - introducing the mighty Astrochicken Von Neumann, conqueror of the galaxy. And available to everyone, including challenge winners, The Heat Death of the Universe is Coming. A fun reminder in t-shirt form of the eventual cold, dark end of Space Time.