Images in space and time: Feynman diagrams

So far in this series, we’ve introduced the standard model of particle physics: we’ve made a list of the fundamental particles that make up everyday matter (plus a few extras), called fermions, and the particles that bind them together as forces (plus the Higgs particle), called bosons. All of that can be summarised in a diagram suspiciously similar to the periodic table:


This week, I’ll talk about describing collisions and interactions between these particles. This is almost universally done by using a system of diagrams invented by and named after Richard Feynman. These Feynman diagrams absorb all the calculus and group theory and complicated mathematical notation into pictures, making their interpretation quite accessible (although calculating the rules for Feynman diagrams is another story). For example, this is a photon, labelled as γ:


Once we introduce axes and directions to the diagrams, we can talk about what the photon is doing, but for now, that’s a photon. This is a gluon:


And this is a Higgs boson:


Fermions are all represented by straight, solid lines, so the labels become especially important. These particles also come with arrows — if the arrow goes backwards, we’re dealing with antimatter, not ordinary matter! (We’ll go into that a little more next week.) For the moment we haven’t specified forward and backward, but I’ll make the arrows consistent with the notation we’ll adopt by the end of the article. Here are a quark and an electron:



We could go on and draw all the particles of the standard model, but for this project we’ll stop here, since we only need four: the quark, the electron, the photon and the gluon. There’s one more thing we can do with them before we start thinking about axes and directions. Two weeks ago we said protons and neutrons must be made of quarks because of all the threes. So we can put three quarks together to represent a proton:


Of course, three quarks could also be a neutron, so labels matter. (Unless things are crystal clear from context, in which case the labels are sometimes left out.)

To go much further, we need to introduce the axes of Feynman diagrams. Bearing in mind that we’re generally trying to describe collisions between particles, we use one axis to describe the separation between the relevant particles. In fact this is the only information about position that the diagrams explicitly include. So far, two colliding particles would look like this:


Clearly the ability to represent the separation at different times is crucial. We use the diagram’s other axis to represent passing time. Then we can represent two electrons interacting by exchanging a photon like this:


Sometimes it’s easier to represent the maths behind the Feynman diagrams by starting with early times on the right and flowing towards later times on the left:


Although the left-to-right notation is more common, for all the reasons you’d expect, I’m going to need specialised Feynman diagrams that assume the right-to-left convention later, so I’ll draw time progressing from right to left from the start. Here’s a photon undergoing a process called pair production in which it turns into an electron and an anti-electron (called a positron):


And here’s a real live working diagram of a collision between an electron and a proton, mediated by a photon:


This one’s pulled directly from my thesis, so it may need a little more explanation. Since I’ve told you that the quarks make up a proton, you should be able to figure out that they’re quarks, not antiquarks and put the arrows on for yourself. The photon is labelled as γ* because it’s a “virtual particle” — we can’t measure it, since it disappears before the collision is over. This allows it to have some unusual properties that wouldn’t make sense otherwise. The blob between the photon and the quark represents the fact that while they could interact directly, we also want to consider more complicated processes. For instance, the photon could produce an electron and a positron (like in the earlier diagram) and that electron could produce another photon, which interacts with the quark. Since we haven’t specified exactly what the interaction is, we don’t know exactly what comes out the other end, so there are just a bunch of lines collectively labelled “X”. This process is called Deep Inelastic Scattering (DIS) and it’s the context for most of the work done in my MSc project (although my work is based on a slightly different diagram that also falls under the DIS heading).

That’s Feynman diagrams! Next week we’ll talk about the Feynman rules, which tell us which diagrams make sense and which don’t. Along the way we’ll also talk more about matter, antimatter and virtual particles.

It’s a force — it’s a particle — it’s a quantum field theory!

In this series of posts, I’m talking about the science that went into my MSc thesis, hopefully in a readable, comprehensible way. (Occasionally I will mildly abuse technical terminology to achieve this.) Last week I wrote about the particles that everything is made of, using table salt as an example. (For the completionists out there, I should mention that I left a few particles out: the muon and the tau are similar to the electron, but much, much rarer and not used in building up atoms. Neutrinos are particles associated with electrons, muons or taus that almost never interact with anything, making them virtually invisible.) This week I’ll talk about the forces that hold those particles together (or push them apart). But before we get into that, it’s helpful to think about what we mean by a particle.

The common sense definition of a particle goes something like this: there’s empty space for a while, then there’s a thing and immediately afterwards there’s empty space again. The part where there’s a thing is the particle. Since we’re doing Science!, let’s draw a  graph of that:


(In the full mathematical description, the spike becomes a Dirac delta function, named after Paul Dirac who did a lot of work in this area of physics.) However, last week we said that quantum physics means particles are more like fuzzy clouds than solid objects. We can update the graph to account for that:


That gives us a better picture of what we’re talking about when we use the word ‘particle’: it’s not necessarily what we’d normally call a particle, but it’s a necessary consequence of quantum physics.

Once we start to think of particles as clouds, we have to allow them to do other strange things. (Doing the maths and the experiments confirms that talking about particles this way does help us to predict what will happen in a given situation.) For example, a single particle may have a cloud that is split into two parts:


Frequently the cloud will be much more fragmented than this. Perhaps even worse, multiple particles might have overlapping clouds that can’t be distinguished. (At this point somebody might bring up Wolfgang Pauli and his exclusion principle: the rule that two particles can’t be in exactly the same state. That almost helps, but  while it prevents particles from being exactly the same, it doesn’t mean they can’t have some properties in common.) Our original definition of a  particle doesn’t seem much good any more.

We had said that

There’s empty space for a while, then there’s a thing and immediately afterwards there’s empty space again. The part where there’s a thing is the particle.

Now we need something more along the lines of

There’s a thing somewhere — or maybe everywhere — that you’re most likely to bump into in certain places.

Admittedly this doesn’t sound very much like a particle, which is where ideas like wave-particle duality come in. (You can think of how ripples in a pond might fill the whole pond, but all have properties determined by a single disturbance.) The important thing for our purposes is that while a particle may have some very definite properties, such as a particular energy or charge, it’s not at all constrained to a particular position or required to act like a hard little ball. This makes our idea of a particle more difficult to work with, but perhaps we can make up for it a little by getting more use out of the idea. Let’s talk about forces.

Our new definition of a particle says that

There’s a thing somewhere — or maybe everywhere — that you’re most likely to bump into in certain places.

There’s nothing to stop the thing in question from being a force. If we treat forces this way, we’ll have to treat them as coming in chunks of some sort — one chunk per particle — but quantum physics requires us to do that anyway (‘quantum’ is just a fancy word for ‘chunk’, after all). In this new way of talking about forces, we can’t say that one particle exerts a force on another particle. Instead, the first particle produces a force particle, which interacts with the second particle. It’s helpful to remember that when we say ‘particle’, we’re not talking about a hard little ball. We simply mean a chunk of something that may not have a very specific position: in this case, a force.

Treating force as a particle requires some tweaking of the mathematics involved. These new particles with slightly tweaked maths are all called bosons, after Satyendra Nath Bose; the particles from the last post are called fermions, after Enrico Fermi. One of the biggest differences between bosons and fermions is that while two fermions cannot have exactly the same properties (according to Wolfgang Pauli’s exclusion principle), there is no such restriction for bosons. This makes possible things like Bose-Einstein condensation, which are fascinating, but tangential to our purposes. For today we’ll stick to cataloguing how the fermions we identified last time interact.

Electrically charged particles interact — push and pull — by exchanging force-carrying particles called photons (that’s ‘light things’). Photons are also responsible for interacting with particles in our eyes, allowing us to see things and particles in our radio devices, allowing us to send messages. The theory of electromagnetic waves is essentially an approximation to a full theory of photons — a useful and often very accurate approximation.

Particles that have colour charge interact by exchanging gluons (yup, we went there). The fermions that have colour charge are just the quarks (the particles that make up protons and neutrons). However quarks are not the only particles with colour charge: gluons themselves have colour. This makes the theory of colour interactions (quantum chromodynamics, to give it its technical name) relatively complicated — as do features like the quarks’ insistence on appearing in threes. These complications mean that quantum chromodynamics is still very much an area of active research (including my own MSc work).

All fermions also interact via an additional force called the weak force. (The charge involved here is called flavour.) This is mostly used to describe how the nucleus of an atom breaks up, as in a nuclear fission reaction. The particles that mediate weak force interactions are named W-bosons and Z-bosons. There are two bosons for the weak force: the W and the Z do basically the same job, but they have different masses and electric charges, so they must be different particles.

That’s almost all, but there are two necessary corrections. One is the (in)famous Higgs boson. The Higgs boson doesn’t really describe a particular force. Rather, Peter Higgs, together with a number of other physicists, found a way of writing the equations of particle physics that made the particle masses make a lot more sense. Doing so required introducing new elements to the equations: elements which would exactly correspond to a new kind of boson. Subsequent experiments at CERN’s LHC have detected a particle which seems to have just the properties of the one invented to fix the mass problem — they ‘found’ the Higgs boson.

The other problem is that I’ve omitted gravity from the list of forces. It’s easy enough to make up a name for the particle that mediates gravitational interactions — it’s usually called the graviton — but writing down a consistent mathematical description is another story. Gravity has very little effect at the scale of particle physics experiments and thus far, the most effective tactic has been to ignore it. It’s not very satisfying, but it’s all we have — so far, at least.

And that really is all. Here’s the fundamental particle summary diagram, as seen in particle physics talks everywhere (click through for source):

Standard_Model_of_Elementary_Particles.svgNext week we’ll start drawing these particles and their interactions using Feynman diagrams.

† Technically Pauli’s exclusion principle also only applies to particles in the fermion (‘thing named after Enrico Fermi’) category (which includes all the particles we’ve discussed up to this footnote).