Thanksgiving

This year we give thanks for the spin-statistics theorem. (Previously we gave thanks for the Lagrangian of the Standard Model of particle physics, and for Hubble’s Law.)

You will sometimes hear physicists explain that elementary particles come in two types: bosons, which have a spin of 0, 1, 2, or some other integer, and fermions, which have a spin of 1/2, 3/2, 5/2, or some other half-integer. That’s true, but it’s hiding what’s important and emphasizing what’s auxiliary.

When it comes to classifying elementary particles, it’s not really the spin that’s important, it’s the statistics. And really, the word “statistics” in this context makes something deep and wonderful sound dry and technical. A boson is a particle that obeys Bose statistics: when you take two identical bosons and switch them with each other, the state you end up with is indistinguishable from the state you started with. Which only makes sense, really; if you exchange two identical particles, what else could you get? The answer is, Fermi statistics: when you take two identical fermions and switch them with each other, you get minus the state you started with. Remember that the real world is based on quantum mechanics, in which the state of a system is described by a wave function that tells you what the probability of obtaining various results for certain observations would be; when we say “minus the state you started with,” we mean that the wave function is multiplied by -1.

This difference in “statistics” seems a bit esoteric and removed from one’s everyday life, but in fact it is arguably the most important thing in the universe. This simple difference in what happens to the state of two particles when you interchange them underlies the most blatant features of how particles behave in the macroscopic world. Think of two identical particles that are in the same quantum state: sitting in the same place, doing the same thing, right on top of each other. If those two particles are bosons, that’s cool; we can switch them and get the same state, which just makes sense. But if they’re fermions, we have a problem; the two particles are purportedly in the same state, but if we switch them (which doesn’t really do anything, as they are in the same place) the state becomes minus what it used to be — seemingly a contradiction.

This seeming puzzle has a simple solution: in the real world, two identical fermions can never occupy the same quantum state! That’s the Pauli exclusion principle, and it has a simple translation into everyday English: fermions take up space. Electrons, which are fermions, can’t just be piled on top of each other as densely as we like; some of them would have to be in the same state, and that can’t happen. That’s why atoms take up a certain amount of space, which in turn is why ordinary material objects don’t simply collapse into themselves. Fermions — electrons, quarks, neutrinos, etc. — are matter particles, constituting the “stuff” of which the objects of our world are comprised.

Bosons, on the other hand, have no problem being in the same quantum state. So they will happily pile on top of each other. This is also important to our everyday lives. Bosons — photons, gravitons, gluons, etc. — are force particles, which pile on top of each other to form the classical force fields that hold fermions together. When you see light — a classical electromagnetic wave made of photons — or are held to the ground by gravity — a classical field made of gravitons — it’s only possible because of Bose statistics.

So the important distinction between bosons and fermions is not the “integer spin”/”half-integer spin” distinction, it’s the “pile on top of each other”/”take up space” distinction. The fact that these sets of features come hand-in-hand is the content of the spin-statistics theorem: particles that pile on have integer spins, particles that take up space have half-integer spins. Which is a deep and beautiful result that relies on the fact that nature is fundamentally quantum rather than classical, and on the topology of the group of rotations in three (or more) spatial dimensions, and on the features of relativistic field theory. None of which I’m going to explain right here, but John Baez has a fun “proof” of the theorem using ribbons which is worth checking out.

Rather, I will just reiterate that if the fermions comprising a turkey didn’t take up space, it would hardly constitute a filling meal; and if the gravitons from the Earth didn’t pile up to form a classical field, the traditional football game really wouldn’t work at all. So for the spin-statistics theorem, we should all be thankful.