Thursday, 2 August 2012

Feynman Diagrams

In an earlier post, I talked about perturbation series, one of the most important tools in physics calculations.  In brief, there are very few problems in science that we can solve exactly.  Instead, we approximate the problem we have as a simple one that we can solve, plus terms that are "small".  The real solutions are the solutions of the simpler problem, plus an infinite set of corrections.

In that earlier post, I was pretty general.  In a sense this made things harder for me than necessary.  In particle physics, or relativistic quantum mechanics, there is a nice intuitive split between things we can solve exactly and things we can't.  The only problems we can solve exactly are non-interacting.1  We could only describe an electron exactly, if we turned of electromagnetism.2  The small numbers that we make our expansions in are the coupling constants.  For example, the natural definition of the charge of the electron is the through the fine structure constant,
$\alpha \equiv \frac{e^2}{4\pi\hbar c} \approx \frac{1}{137}.$
Here, e is the electron charge; h is Planck's constant; and c is the speed of light.

While the idea of a perturbation series is relatively simple, actually constructing one can be a bit tricky. The main problem is to ensure that you get all the relevant terms with the right factors.  This is where Feynman diagrams come in.  They are a concise way to construct perturbation series from a simple set of rules.  They can be easily coded into software programs, and a number of options exist.  And they have a really nice intuitive interpretation.
To set the ground work for a proper explanation, let's first consider the type of experiments done in particle physics.  I'm not actually going to talk about the LHC here, as there are a few subtleties that get in the way.  Instead, I'm going to go back to LEP, the previous CERN experiment that used the same tunnels currently containing the LHC.  LEP collided electrons with their antiparticles, positrons.  As far as we know, electrons and positrons are fundamental, meaning they are have no smaller components.

So we start with these particles far away from one another, and fire them at each other.  We don't actually see directly what happens when they collide, but we measure things produced in that interaction.  For example, one thing that could be produced are a pair of the particles known as muons.  Muons are basically copies of electrons, but two hundred times heavier.  The following is a basic sketch of this process:

The horizontal direction in this diagram is time, while the vertical direction is space.  Initially, we have an electron and a positron spatially separated.  As time progress, those two particles get nearer, because we have fired them at each other at nearly the speed of light!  Then, they get close and some complicated interaction occurs that we do not directly observe; that's the shaded circle.  Coming out of this circle are a muon and its antiparticle, which separate from each other and are eventually observed in the detector.

So what we actually measure are the energies and directions of the two muons, relative to the original energies and directions of the electron and positron.  We also measure what fraction of electron-positron interactions produced muons, as opposed to anything else.  For a given model of the interactions perturbation theory lets us calculate what those observables should be.

Feynman's realisation, however, was that we can represent the terms in the perturbation theory as a set of diagrams similar to that sketch I gave above.  The rules for constructing these diagrams are very simple:
  • For every particle/antiparticle pair in the theory, you have a different type of line.  For example, you have a line for electrons/positrons, a different line for muons and anti-muons, and a third line for photons.
  • For every interaction term in the theory, you have a vertex, a joining of several lines.  Which lines is given by which particles are involved in the interaction term.  For example, the electric charge of electrons means they interact with photons; so we have a vertex involving two electron lines and a photon line.
  • Then using only these lines and vertices draw all possible diagrams, such that for every initial and final particle we have one end of the associated line not joined to anything and no more.
There are a couple of further technical details relating to particles/antiparticles and diagrams that can be split into smaller pieces, that I'm sweeping under the rug for now.

Note that for any given set of initial and final particles, there will be infinitely many possible diagrams that can be drawn.  A natural way to order these diagrams is through the number of vertices; these correspond to the 'small number' that we expand in.  So the diagrams with the fewest vertices correspond to the largest terms, and as you add more vertices the contributions should get smaller.

To illustrate this, let's return to our original example: electron-positron annihilation to muons.  I've already noted the electron-photon coupling.  As muons are also electrically charged, there will also be a vertex involving muons and a photon.  The biggest contribution to predictions for the scattering process is then:

The use of straight lines and wavy lines is conventional; all fermions get straight lines, all gauge bosons wavy or curly ones.  The single diagram above is actually joined by another, when we consider the full electroweak theory:

The second diagram looks similar, but the internal line corresponds to a different particle, the Z boson.

We can construct more complex graphs to find the next largest terms, but this I defer to a later post.  Instead, compare our original diagram with a shaded circle with our two diagrams above.  It looks a lot like the Feynman diagram is showing us what 'really' happens during the interaction.  The process considered here is dominated by the electron and positron annihilating into either a photon or a Z, which in turn 'decays' back to a pair of muons.

This is actually a dangerous thought; the Copenhagen interpretation of quantum mechanics would say that the question of what goes on in that circle is unanswerable, and the question meaningless.  Other perspectives on quantum mechanics might say differently but they have their own costs.  Despite this warning, however, this way of thinking is useful.  Indeed, if you hang around particle physicists you'll often here people talk about these internal particles as real.  The intuition that comes from this is helpful; thinking of the Z boson above as really existing between the initial electron and final muon hints that something interesting might happen near the Z boson mass, and indeed it does.  It is this, as much as anything, that makes Feynman diagrams so helpful and ubiquitous in the field.  And now that I've talked about them, I'm going to feel free to add them to my posts liberally.

1. Strictly, this statement only holds in four dimensions; and even then there may be some exceptions I'm unaware of.  For practical purposes this is true, though.
2. And also the weak force, and we have to completely forget about gravity.

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