Solving the Impossible in Quantum Field Theory (Script) (Patreon)

Quantum field theory is stunningly successful at describing the smallest scales of reality. But its equations are also stunningly complex. A lot of the genius in QFTs development went into the development of brilliant hacks to make these equations workable. The most famous of these are the incredible Feynman diagrams.

The equations of quantum field theory allow us to calculate the behaviour of subatomic particles by expressing them as vibrations in quantum fields. But even the most elegant and complete formulations of quantum physics – like the Dirac equation or Feynman’s path integral - become impossibly complicated when we try to use them on anything but the most simple systems. But physicists tend to interpret “that’s impossible” as “I dare you to try!” And try they did. First they expressed these impossible equations in approximate but solvable forms. Then they tackled the pesky infinities that kept appearing in these new, approximate equations. Finally, the entire mess was ordered into a system that mere humans could deal using the famous Feynman diagrams.

To give you an idea of how messy quantum field theory can be, let’s look at what should be a simple phenomenon: electron scattering, when two electrons repel each other. In old-fashioned classical electrodynamics, we think of each electron as producing an electromagnetic field. That field then exerts a repulsive force on the other electron. At least in the simplest cases, the Coulomb equation governing this subatomic billiard shot is really easy to solve.

But in quantum field theory – specifically quantum electrodynamics, or QED – the story is very different. We think of the electromagnetic field as existing everywhere in space whether or not there’s an electron present. Vibrations in the EM field are called photons – what we experience as light. The electron itself is just an excitation – a vibration in a different field - the electron field. And the electron and EM fields are connected. Vibrations in one can cause vibrations in the other. This is how QED describes electron scattering: one electron excites a photon, and that photon delivers a bit of the first electron’s momentum to the second electron. It’s arguable exactly how real that exchange photon is. In fact we call it a virtual photon and it only exists long enough to communicate this force. There are other types of virtual particle whose existence is similarly ambiguous. We’ll get back to those in another episode.

This is a good time to introduce our first Feynman diagram. The brilliant Richard Feynman developed these pictorial tools to organize the painful mathematics of quantum field theory. But they also serve to get a general idea of what these interactions look like. In a Feynman diagram, one direction is for time – in this case up. The other axis represents space, although the actual distances aren’t relevant. Here we see two electrons entering in the beginning and moving towards each other. They exchange a virtual photon – this squiggly line here – and the two electrons move apart at the end.

But Feynman diagrams aren’t really just drawings of the interaction; they’re equations in disguise. Each part of a Feynman diagram represents a chunk of the math. Incoming lines are associated with the initial electron states, and outgoing lines represent the final electron states. The squiggle represents the quantized field excitation of the photon, and the connecting points – the vertices – represent the absorption and emission of the photon. The equation you string together from this one diagram represents all of the ways that two electrons can deflect via the exchange of a single virtual photon. And from that equation it’s possible to perfectly calculate the effect of that simple exchange. Unfortunately real electron scattering at a quantum level is a good deal more complicated than this. For that reason this simple calculation gives the wrong repulsive effect between two electrons.

If we observe two electrons bouncing off each other, all we really see is two electrons going in and two electrons going out. The quantum event around the scattering is a mystery. There are literally infinite ways that scattering could have occurred. In fact, according to some interpretations, all infinite intermediate events that lead to the same final result actually DO happen. Sort of. We talked about this weirdness when we discussed the Feynman path integral recently. Just as with the path integral, to perfectly calculate the scattering of two electrons we need to add up all the ways the electrons can be scattered.

This is where Feynman diagrams start to come in handy, because they keep track of the different families of possibilities. For example, the electrons might exchange just a single virtual photon, but they also might exchange 2 or 3 or more. The electrons might also emit and reabsorb a virtual photon, or any of those photons might do something crazy like momentarily split into a virtual particle-antiparticle pair. Those last two events are actually hugely complicating, as we’ll see.

With infinite possible interactions behind this one simple process, a perfectly complete quantum field theoretic solution is impossible. But if you can’t do something perfectly, then maybe near enough is good enough. This is the philosophy behind perturbation theory – an absolutely essential tool to solving quantum field theory problems. The idea is that if the correct equation is unsolvable, just find a similar equation that you CAN solve. Then make small modifications to it – perturb it – so it’s a bit closer to the equation that you want. It’ll never be exact, but it might get you pretty close.

In the case of electron scattering, the most likely interaction is the exchange of a single photon. Every other way to scatter the electrons contributes less to the probability of the event. In fact the more complicated the interaction the less it contributes. Here, Feynman diagrams are indispensable. It turns out that the probability amplitude of a particular interaction depends on the number of connections – vertices in the diagram. Every additional vertex in an interaction reduces its contribution to the probability by a factor of around a hundred. So the most probable interaction for electron scattering is this simple case of one photon exchange, with its two vertices. 

A three-vertex interaction would contribute about 1% of the probability of the main two-vertex interaction. However it turns out for electron scattering there are no three-vertex interactions. However there are several interactions that include four vertices, and each contributes about 1% of 1% of the two-vertex interaction. This is true even though those complex interactions are very different to each other. They include exchanging two virtual photons, one electron emitting and absorbing a virtual photon, or the exchange photon momentarily exciting a virtual electron-positron pair. And more complicated interactions add even less to the probability. So with Feynman diagrams you very quickly get an idea which are the important additions to your equation and which you can ignore. Perturbation theory with the help of Feynman diagrams make the calculation possible.

But that doesn’t mean we’re done. Including all of these weird intermediate states really opened up a can of worms. This is especially true for so-called loop interactions, like when a photon momentarily becomes a virtual particle-antiparticle pair and then reverts to a photon again, or when a single electron emits and absorbs the same photon. This later case can be thought of as the electron causing a constant disturbance in the EM field. Electrons are constantly interacting with virtual photons. This impedes the electron’s motion and actually increases its effective mass. The effect is called “self-energy”. But if you try to calculate the self-energy correction to an electron’s mass using quantum electrodynamics, you get that the electron has … infinite extra mass.

This sounds like a problem. To calculate the mass correction due to one of these self-energy loops you need to add up all possible photon energies, but those energies can be arbitrarily large, sending the self-energy and hence the mass to infinity. In reality something must limit the maximum energy of these photons, but we don’t know what the limit is. The answer probably lies within a theory of quantum gravity, which we don’t yet have. But just as with perturbation theory, physicists found a cunning trick to get around this mathematical inconvenience. It’s called renormalization.

Obviously electrons do not have infinite mass. We know because we’ve measured that mass. Although any measurement we make actually includes some of this self energy, so our measurements are never of the fundamental or “bare” mass of the electron.  And that’s where the trick lies. Instead of trying to start with the unmeasurable fundamental mass of the electron and solve the equations from there, you fold in a term for the self-energy-corrected mass based on your measurement. In a sense you’re capturing the theoretical infinite terms within an experimental finite number.

This renormalization trick can be used to eliminate many of the infinities that arise in quantum field theory; for example the infinite shielding of electric charge due to virtual particle-antiparticle pairs popping into and out of existence. However you pay a price for renormalization. For every infinity you want to get rid of, you have to measure some property in the lab. That means the theory can’t predict that particular property from scratch; it can only make predictions of other properties relative to your lab measurement. Nonetheless, renormalization saved quantum field theory from this plague of infinities.

Feynman diagrams successfully describe everything from particle scattering, self-energy interactions, matter-antimatter creation and annihilation, to all sorts of decay processes. We go further into the nuts and bolts of Feynman diagrams in an upcoming challenge episode. A set of relatively straightforward rules governs what diagrams are possible, and these rules make Feynman’s doodles an incredibly powerful tool for using quantum field theory to predict the behavior of the subatomic world. The results led to the standard model of particle physics. In future episodes we’ll talk more what is now the most complete description we have of the smallest scales of space time.