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).

A standard model for what stuff is made from

Reading an MSc dissertation is not a task for the faint-hearted. Even for an expert in the field, wading through all the detail that’s necessarily part of a two-year project is a substantial undertaking. However! It seems a pity that two years of (often) publicly-funded work should be buried in the obscurity of that part of the library. Publishing papers on the work helps some, but does nothing to clear the traditional (and admittedly useful-in-context) fog of jargon.

As a result of these regrettable complications, it seems like there could be value in a step-by-step, fifteen-minutes-a-week, minimally-technical walkthrough of such a dissertation. It occurred to me (no doubt purely by happenstance) that I am particularly well situated to produce such a walkthrough, having just submitted a dissertation that contains such phrases as

The simplest, although not necessarily the most elegant, solution is to simply factorise the average (the so-called large-Nc approximation) so that the dependence is instead over a set of 2-point functions.

The attempts at introducing humour or otherwise amusing elements into the narrative will no doubt render the entire project repugnant to beings of higher taste than the author, but I hold out hope that a few people may nonetheless find it instructive, or at least entertaining. Thusly, I present step the first: A standard model of what stuff is made from (AKA “The Standard Model of Particle Physics”).

A “model” is simply a “tool for describing something,” so our task is to elucidate this standard tool for describing what stuff is made of. saltLet’s use the example of table salt, since it’s a relatively simple material (if you’re reading this, you’re definitely a lot more complicated than table salt), but it shares the features we want with all those more complicated systems.

We can zoom in to look at the underlying structure of the salt crystals. Since we’ll rapidly get to the stage where photographing the things I”m talking about is downright impossible, I’ll make use of a cartoon microscope right from the start. We can immediately see that the salt crystal is made of two kinds of stuff: in the picture, the small purple balls and the big green balls. The big green balls are chlorine (like the stuff that goes into swimming pools) and the purple balls are sodium (which is why salty food is sometimes described as having “high sodium content”). It turns out that you can’t get chlorine or sodium in smaller chunks than these balls, so the balls are called atoms. ‘Atom’ is really just a name for the smallest chunk you can have of any particular chemical. For a long time this was considered a good model for what stuff is made of: stuff is made of atoms.

Because scientists can never leave well alone (that’s why they’re scientists, after all), people began trying to shoot things through atoms to see what happened. In general atoms seemed to be pretty fuzzy — it wasn’t too difficult to get things through them. However, sometimes this didn’t happen at all and the things that were shot at the atom bounced right back. I like the way Ernest Rutherford described this:

It was quite the most incredible event that has ever happened to me in my life. It was almost as incredible as if you fired a 15-inch shell at a piece of tissue paper and it came back and hit you.

One might begin to suspect that the tissue paper was not entirely tissue paper after all. A more detailed model of what things are made of was in order. It was pretty well established that there was some part of the atom that makes electricity work. This electricity part of the atom is made up of electrons (as far as I can tell, ‘electron’ just means ‘thing that makes electricity work’). The electrons turn out to be (more or less) the fuzzy part of the atom. The less fuzzy part of the atom was called the nucleus (which means ‘the thing in the middle’). orbitalsThe rule of thumb is that each electron contributes a particular fuzzy area. Where exactly is the electron itself? That’s rather contentious, actually — it turns out that the fuzzy cloud is all we can really see of the electron anyway. (We’re seeing it through a cartoon microscope, so perhaps I mean all we can know about it — although recent experiments have captured more ‘real’ images of these clouds.) The various electron clouds combined produce the green and purple balls that we saw earlier. Now that we’ve zoomed in, we see that they aren’t perfectly round, but it wasn’t a bad approximation.

So much for the fuzzy electrons. What about the thing at the centre, that seemed to be responsible for bouncing things back at the bemused Ernest Rutherford? The nucleus turns out to be pretty fuzzy too, although it’s in some sense ‘more concentrated’ than the electron cloud. (One of the fundamental insights of quantum physics is that everything is a bit fuzzy if you look closely enough — this comes from Werner Heisenberg’s famous uncertainty principle.) One of the first things we can discover about the nucleus is that it has clumps. Some clumps change the way the electron cloud behaves (and determine the number of electrons in the typical atom) — these are called the protons (which means something like ‘first thing’, although they aren’t). Other clumps seem to ignore the electron cloud: the neutrons (‘neutral things’, since they don’t interact with the electrons). This is a good start to an explanation, but it doesn’t answer all the questions one might ask. For one thing, if the neutrons are so neutral, why do they insist on hanging around with the protons? This leads to the idea that being electrically neutral (and ignoring electrons) doesn’t necessarily mean being completely neutral — there must more than electricity involved. Another question that physicists doing experiments began to ask is, “Why are there so many threes?” This turns out to be an important question.

Whenever a proton or a neutron — one of the chunks inside the nucleus (middle bit) of the atom — can do something, it seems to be able to do it three times. The more one does different experiments, the more threes one finds. Eventually Murray Gell-Mann produced an explanation based on the idea that protons and neutron were made up of three smaller things. He thought it would be amusing to call the smaller things ‘quarks’ and so he did. (You can make different people think you’re wrong depending on quarkswhether you make ‘quark’ rhyme with ‘squawk’ or with ‘mark’, but that’s a story for another day.)

This explanation of having three quarks inside each proton and neutron explains the threes so perfectly that it’s generally accepted to be correct. However, it’s impossible to break a proton or a neutron up into quarks, which makes them very strange objects indeed. In fact, putting enough energy into a proton (or neutron) to separate out a quark tends to produce antimatter instead. However, since antimatter isn’t a defining feature of most everyday stuff (like our original example of table salt), I’ll gloss over that for now.

To describe the way that quarks are always found in groups of three, they were named after the three primary colours: red, green and blue. A package of quarks always contains one of each ‘colour’ (we can make this slightly more complicated by including antimatter). This deals with most of the threes in the experiments, but not quite all of them. To deal with the remaining threes, the quarks are also named after various flavours. The flavour pairs (yes, flavour now comes in pairs) are up and down, strange and charm, and top and bottom. Very strange flavours indeed, but then quarks are very strange objects. The quarks in the picture above  are up and down quarks: two ups and a down form a proton. Two downs and an up would form a neutron. The other flavours form more exotic things, which I’ll skip over on account of their not being required for table salt.

In fact, at this point, we can begin to think of making table salt from scratch. Salt is made of chlorine and sodium atoms. Those atoms are made of fuzzy electron clouds and a things at the centre. The thing at the centre has neutron chunks and proton chunks. Those chunks (like bureaucrats, perhaps) do everything in triplicate, which leads us to believe that there are even smaller things, called quarks, inside. Since quantum physics tells us everything is fuzzy, the quarks must be fuzzy too, but we expect them to be fuzzy in groups of three. And that’s it. Stick it all together and you have table salt.

How do we stick it together? Well, that’s a good question. As we noticed earlier, there seem to be several ways for things to interact (or not interact). Come back next week for photons (‘light things’), gluons (‘glue things’), W’s and Z’s (‘we ran out of names things’). If you have questions, feel free to ask in the comments and I’ll do my best to answer!