Prelude to Feynman Diagrams

A lot of physics involves collisions: both the basic stuff you may have learned in high school or introductory physics in college and the really advanced physics used in high-energy particle colliders. Collisions in their simplest form go like this: two objects come into contact with each other, and two objects leave. Beyond that, there are a lot of variations: maybe the two objects are cars that collide on a highway, or maybe they’re pool balls on a billiards table. Perhaps one is an electron and the other a photon, and maybe the two objects that initially collide aren’t the same two that leave afterwards. In other words, collisions come in many varieties, but we can use a lot of the same language to describe all of them.

Let’s start off by laying out a few reasonable rules. We will assume Einstein’s theory of relativity is valid, so nothing with mass will move faster than the speed of light. (Pedants will point out that Einstein’s theory doesn’t technically forbid faster-than-light travel as long as the barrier of light-speed isn’t crossed. While this is true, there are problems with energy and causality – an effect happening before its cause – so we’ll leave that possibility out.) We’ll assume that energy, momentum, and electrical charge are conserved: no energy will come out of nowhere, and a particle starting at rest won’t spontaneously begin to move without a force acting on it. That’s enough rules!

A spacetime diagram containing several possible trajectories for particles with mass, and one impossible trajectory. A photon (which is massless) will follow a trajectory along the line marked "speed of light". The location of the various lines isn't particularly important: the slopes of the lines are.

We can put our rules into pictorial form in something called a spacetime diagram: time runs vertically, one dimension of space runs horizontally. (We can include the other spatial dimensions, but it gets messy and strangely enough we often don’t need to draw them as long as we remember that they’re present, and keep them in the math. Since we’re leaving the math out, we’re good.) Speed, which is the change in position in time, is the inverse of the slope. Something that doesn’t move will follow a vertical line on the spacetime diagram since it “moves” in time, but doesn’t change position in space. Lines with steep slopes represent something moving slowly, while shallower slopes indicate faster speeds…until you hit the barrier of the speed of light. Anything with a smaller slope than that is moving faster than light, and is forbidden by our set of rules above.

The figure shows a simple example: a rubber ball bounces off a wall and continues back the way it came with the same speed. Notice how similar our spacetime diagram is to the actual picture of what happens, but beware: the vertical direction is time, so it’s not exactly what you’d video with a camera. The little cartoon next to the spacetime diagram shows that the ball is only moving along a line: traveling left, hitting the wall, then heading back right.

A spacetime diagram for a ball bouncing off a wall, along with a comic strip showing the same thing.
Spacetime diagram for the collision between two distinguishable balls.

Now let’s make it more complicated: take two balls in a track or groove (so they can only move back and forth along a line) and send them toward each other. They bounce off each other, but they don’t have exactly the same speeds they did before the collision. However, they didn’t change character: the blue ball is still blue, the green ball is still green, and their masses are the same before and after the collision.

In our everyday macroscopic world, we can always tell objects apart. Even two cue balls for pool, which are milled to be as identical as possible, will turn out to have slightly different masses if you use a scale sensitive enough, and while being very spherical won’t be perfectly smooth, even straight from the factory. After a few games, their differences will only grow, with chalk marks and slight dings marring their surfaces. This is unavoidable, and is a metaphor for the futility of existence. (I guess you could also say it’s a sign that for all our similarities, we grow into unique individuals, but I’m in a glass-half-empty mood.)

The microscopic quantum world doesn’t follow those same rules: every electron is perfectly identical to every other electron, every hydrogen atom is identical to every other hydrogen atom. Though in an ideal world you could build up two cue balls to have exactly the same number of atoms in the same configurations, I defy you to do that in practice: the sheer number of atoms involved would defeat you. But there is no experiment that can distinguish between one electron and another: no colors, no unique shapes, no flavors or personalities. So, instead of taking two balls in a track, let’s take two electrons: they go in, they come out. But which is which after the collision? It doesn’t matter!

Actually, we need to ask what collisions even mean in this context: we know from elementary school physics that electrons are negatively-charged, so two electrons coming close to each other will repel (as two like charges always do). They won’t actually touch each other, though depending on how fast they’re moving, they might come very very close before veering away. If the energy is large enough, they might even change into some other kind of particle, but let’s leave that possibility out of the picture for now.

Electron-electron scattering: despite the simplicity of the basic diagram, a lot can happen inside that yellow circle! (I'm going to follow the usual convention from now on and drop the "space" and "time" axes, along with the "speed of light" line. Just imagine they're there, because they are.)

Because we are dealing with moving (dynamic) quantum particles that act on each other via the electric force (through their charges), we are working with quantum electrodynamics (QED). Inside the circle labeled “something happens” is the reason Richard Feynman won the Nobel Prize in physics, along with Julian Schwinger and Shin-Itiro Tomonaga, who worked out QED independently using different methods. (Freeman Dyson showed the different versions were the same, so all three physicists deserve equal credit. It just happens that Feynman’s version is a lot easier to explain in a blog post!)

Electron-electron collisions are also known as Møller scattering, and despite the simple picture I’ve drawn, a lot can happen inside that circle. Feynman had an idea, conceived during his graduate work at Princeton, that the best way to think about collisions and the like was through direct interaction. Particles act directly on each other, with a delay to keep things in line with relativity. If the electrons repel each other, the simplest way to get them to interact is if one of them emits a particle and the other absorbs it: the emission will make the first electron recoil and the absorption will make the second electron also recoil. Perfect, right?

One possible Feynman diagram for electron-electron scattering: electrons (labeled "e") repel each other because one spits out a photon ("γ") that hits the other. Our rules take care of everything else. The photon trajectory is horizontal because I'm still learning the graphics package to draw these diagrams.

That’s simple enough to start with, because we can figure out part of the character of the particles that are emitted and absorbed. They have to be massless and chargeless (to conserve energy and charge, according to our rules above), and there’s one particle that fits that bill: a photon. The image on the right is a hint, but only that, of what’s to come—a type of spacetime diagram known as a Feynman diagram.

We’re nowhere close to being done, even for Møller scattering, though. There’s no reason to prefer this particular version over others: you might have the electrons trading places, or maybe more than one photon is involved. What I found when I began to write this post is that I’m very rusty on quantum electrodynamics and Feynman diagrams, and so what began as a tutorial for you is turning into a refresher course for me.

Consider this a prelude, then, with the fugue to come, if you all are interested in listening.


9 responses to “Prelude to Feynman Diagrams”

  1. Ok, I have a question. It has always bothered me on some level (but when I was in grad school I kept my mouth shut… what a shame, I now realize) that electrons “absorb” photons. If an electron is really basically a point particle (well, it has no internal bits to rearrange at the very least, as far as we know), how does this absorption work? Or does it have something to do with the way virtual particles are different from particles (something I never actually clued into in grad school… I was an condensed matter experimentalist, as you’ve probably guessed by now). I guess I’m also a now little confused about Compton scattering, too (thought I totally understood that!). Is that an absorption/re-emission event? Or am I being way too macroscopic-analogy driven here?

    1. The photons exchanged in the simple collision here are very low energy, and yes, they’re virtual photons. (I was leaving that idea until my next post!) Even in classical physics, electrons can emit light, but you have to accelerate them first, and the energy comes from the electric field. Feynman cut out the middleman, so to speak, and got the extra energy from the wiggle room provided by the uncertainty principle. Virtual particles aren’t essentially different than real ones.

      Compton scattering is treated as absorption and reemission, yes. I’m covering that case in my next post, by the way: it actually relates to a process I used in my Ph.D. thesis. (I’m not a particle guy either, as you probably have gathered! I’m cosmologist and a gravitational physicist.)

      1. It wasn’t so much the emission of light by electrons that was bugging me (I can visualize that… wiggle the electron, the disturbance in the electromagnetic field propagates outward). It was more the absorption part, but I guess if I can envision an electron emitting a photon, I should be ok with it absorbing one? Seems like I took way too much at face value back in school.

        My current problem is that I have very pesky high school students (I taught them to be pesky, so what goes around comes around) asking lots of great questions about quantum processes, now that we are on Matter and Interactions Chapter 8. I’m trying not to bog down the class with my answers. Sometimes I do have to say “If we could answer everything here, college would be boring.” But still I would not like to lie to them TOO much.

      2. Ah, I see. Absorption is just emission in reverse. Photons can travel backwards in time, so there’s no difference in the mechanism from the point of view of Feynman’s rules. That should blow your students’ minds, I think.

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