Solving the Impossible in Quantum Field Theory

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Matt O'Dowd (1:03)

This episode is supported by Curiosity Stream. Quantum field theory is stunningly successful at describing the smaller scales of reality, but its equations are also stunningly complex.

A lot of the genius in QFT's development was in finding 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 behavior of subatomic particles by expressing them as vibrations in quantum fields.

But even the most elegant and complete formulations of quantum field theory, 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 with 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 give 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 actually equations in disguise. Each part of the 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 fueled 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 involving only 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.