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Feynman Diagrams 101

Even the best physicists can get misled by Feynman diagrams occasionally.

A Feynman diagram is a physicist’s best friend and worst enemy.  They’re representations of a particle collision, so you draw lines coming in, you draw them meeting, and annihilating and then you draw a little wiggly line in the middle which could be a photon and then it goes off to something else, and they’re great.  They’re a cartoon representation of the actual equation you would use to calculate the probability of that happening.  For the incoming line there’s a term and equation, for the vertex where they meet there’s a term and equation, for the bit in the middle – the propagata we call it – they’re beautiful, they’re mathematically exact.  They’re a real help, they’re very intuitive and therein lies a little bit of a danger because they’re a little too intuitive because they are actually quantum mechanics and they’re a representation of quantum field theory and in fact they’re not like snooker balls, they’re not a drawing of a snooker shot.  It’s a drawing of a quantum mechanical amplitude.  

An amplitude is kind of half a probability, you have to, in quantum mechanics you add up amplitudes, square them and then, ah, you can never say ‘This Feynman diagram is what happened’ because all it’s doing is telling you a probability of one way you might have got from here to here.  In fact it’s telling you the mathematical sum of many probabilities of doing that, but there can be other ones which’ll be a different Feynman diagram, a different set of probabilities, if you actually want to get the answer, and predict the answer, you have to add those two amplitudes and square them to get the overall probability.

Now that sounds pretty technical, and it is, that’s what quantum mechanics is like.  What I’m trying to say is they’re a beautiful intuitive tool for what’s going on but if you’re not careful, I find even PhD physicists who say ‘I’m going to measure this Feynman diagram.  How can we work out whether this happened?’ but in quantum mechanics you never can, all you can work out is, you get from here to here and this is 90% of the chance that it went this way and 10% went another way or something, so it does mislead people sometimes.  But they’re the best tool of doing it.

And it’s even better than that because they’re also a representation of a series.  There’s a series of diagrams.  So if you say, I have an electron and a positron in, electron and a positron out, all the different ways you can go from here to here, you have to add them all up to get the right answer in quantum mechanics.  Feynman diagrams, the simplest one is the wiggle and a single photon in the middle, but there’s another one where that photon in the middle will suddenly split off into two and go back to a photon, or where two photons have swapped, or three photons, or x photons.  All those are repressible, in quantum mechanics they will all happen at some level but the simplest Feynman diagram is the biggest contribution.  Every time you have an extra particle, an extra vertex in there, the probability gets smaller but if you want the real full answer you’ve got to have them all in there so it’s all, that’s what we call perturbation theory, it’s a very common technique in all kinds of science where you collate the easiest, biggest bit of the thing first, but if you want to get more and more accurate you’ve got to add more and more corrections.  And, again, you have to always remember the physical thing, the measurable thing, is what went in and what came out, you can never measure the Feynman diagram.  The Feynman diagram is actually a mathematical tool for calculating the probability of going from here to here.  So it’s a beautiful tool, it’s very intuitive and it’s really helpful, but if you follow it too far it can actually mislead.  Even the best physicists can get can misled by Feynman diagrams occasionally.