In a previous post we learned how to draw Feynman diagrams by drawing lines and connecting them. We started with a set of rules for how one could draw diagrams:
We could draw lines with arrows or wiggly lines and we were only permitted to join them using intersections (vertices) of the above form. These are the rules of the game. We then said that the arrowed lines are electrons (if the arrow goes from left to right) and positrons (if the arrow points in the opposite direction) while the wiggly lines are photons. The choice of rules is what we call a “model of particle interactions,” and in particular we developed what is called quantum electrodynamics, which is physics-talk for “the theory of electrons and photons.”
Where did it all come from?
One question you could ask now is: “Where did these rules come from? Why do they prohibit me from drawing diagrams with three wiggly lines intersecting?”
The short answer is that those are just the rules that we chose. Technically they came from a more mathematical formulation of the theory. It is not obvious at all, but the reason why we only allow that one particular vertex is that it is the only interaction that both respects the (1) spacetime (“Lorentz”) symmetry and (2) internal ‘gauge’ symmetry of the theory. This is an unsatisfying answer, but we’ll gradually build up more complicated theories that should help shed some light on this. Just for fun, here’s the mathematical expression that encodes the same information as the Feynman rules above: [caution: I know this is an equation, but do not be scared!]
Without going into details, the 
represents the electron (the bar turns it into a positron) while the A is the photon. The number e is the ‘electric coupling’ and determines the charge of the electron. Because equations can be intimidating, we won’t worry about them here. In fact our goal will be to go in the opposite direction: we will see that we can learn quite a lot by only looking at Feynman diagrams and never doing any complicated math. The important point is that our cute rules for how to connect lines really captures most of the physics encoded in these ugly equations.
Now a quick parenthetical note because I’m sure some of you are curious: In the equation above, the 
is a kind of derivative. Derivatives tell us about how things change, and in fact this term tells us about how the electron propagates through space. The e?A term tells us how the photon couples to the electron. The m term is the electron’s mass. We’ll have more to say about this down the road when we discuss the Higgs boson. Finally, the Fs are the “field strength” of the photon: it is the analog of the derivative term for the electron and tells us how the photon propagates through space. In fact, these F’s encode the electric and magnetic fields.
[Extra credit for advanced readers: notice that the electron mass term looks like the Feynman rule for a two-electron interaction with coupling strength m. You can see this by looking at the electron-electron-photon term and removing the photon.]
What we can learn from just looking at the rules
We learned that we could use our lines and intersections to draw diagrams that represent particle interactions. If you haven’t already, I encourage you to grab a piece of scratch paper and play with these Feynman rules. A good game to play is asking yourself whether a certain initial state can ever give you a certain final state. Here are a few exercises:
- You start with one electron. Can you ever end up with a final state positron? [Answer: yes! Draw one such diagram.]
- If you start with one electron, can you ever end up with more final state positrons than final state electrons? [Answer: no! Draw diagrams until you’re convinced it’s impossible.]
- Draw a diagram where an electron and a photon interact to produce 3 electrons, 2 positrons, and 2 photons. Draw a few more to get a feel for how many different ways one can do this.
- If you start with a photon, can you end up with a final state of only multiple photons? [This is actually a trick question; the answer is no but this is a rather subtle quantum mechanical effect that’s beyond our scope. You should be able to draw a diagram think that the answer is ‘yes.’]
So here’s what you should get out of this: Feynman rules are a nice way to learn what kinds of particle interactions can and cannot occur. (e.g. questions 1 and 2) In fact, the lesson you should have gleaned is that there is a conservation of electric charge in each diagram coming to the conservation of electric charge in each intersection. You can also see how complicated interactions can be reduced to simple interactions with “virtual particles” (intermediate particles that don’t appear in the initial state). We are able to do this simply by stating the Feynman rules of our theory and playing with drawings. No math or fancy technical background required.
Summing diagrams: an analogy to summing paths
There’s a lot more one could do with Feynman diagrams, such as calculating probabilities for interactions to occur. Actually doing this requires more formal math and physics background, but there’s still a lot that we can learn conceptually.
For example, there were two simple diagram that we could draw that represented the scattering of an electron and a positron off of one another:
We recall that we can describe these interactions in words by “reading” them from left to right:
- The first diagram shows an electron and a positron annihilating into a photon, which then “pair produces” into another electron and positron.
- The second diagram shows an electron and a positron interacting by sending a photon between them. This is definitely a different process since the electron and positron never actually touch, unlike the first diagram.
Remember that these diagrams are actually shorthand for complex numbers. The numbers represent the probability for each these processes to occur. In order to calculate the full probability that an electron and a positron will bounce off of one another, we have to add together these contributions as complex numbers.
What does this mean? This is just quantum mechanics at work! Recall another old post about the double slit experiment. We learned that quantum mechanics tells us that objects take all paths between an initial observed state to a final observed state. Thus if you see a particle at point A, the probability for it to show up at point B is given by the sum of the probability amplitudes for each intermediate path.
The sum of diagrams above is a generalization of the exact same idea. Our initial observed state is an electron and a positron. Each of these have some fixed [and observed] momentum. If you want to calculate the probability that these would interact and produce an electron and positron of some other momentum (e.g. they bounce off each other and head off in opposite directions), then one not only has to sum over the different intermediate paths, but also the different intermediate interactions.
Again, a pause for the big picture: we’re not actually going to calculate anything since for most people, this isn’t as fun as drawing diagrams. But even just describing what one would calculate, we can see how things reduce to our simple picture of quantum mechanics: the double slit experiment. (more…)
