## Monday, October 06, 2014

### Fermions and Bosons

It occurred to me that I should maybe define some terms, especially when they represent such curious features of the world.

It all goes back to the photoelectric effect. It turns out that light comes in little “chunks” (photons). These have different energies, which correspond to different frequencies. The higher the energy, the higher the frequency; they’re directly proportional. But there’s another little detail about them that emerges: they have angular momentum.

That’s kind of weird at first glance, but electromagnetic fields can have momentum and angular momentum, so we can live with that. It turns out that not just photons have angular momentum but also other elementary particles, including the electron. Since as far as we can measure it doesn’t seem to have any size, it is hard to imagine how an electron could have angular momentum, which is the product of the radius and the component of momentum perpendicular to the radius. Radius=0 means angular momentum is 0, right? Except that it isn’t, here. There’s some intrinsic angular momentum to the electron. And the proton. And the photon.

And that the angular momentum comes in units, it isn’t continuous. Each photon has the same total angular momentum. It can point in different directions, but it is always the same total. So if a particle emits a photon, that particle’s intrinsic angular momentum has to change by one unit.

That unit is, of course h, Planck’s Constant, the pivot of the Heisenberg Uncertainty Principle. You might think that this just applied to photons, since that was where it was discovered, but it turns out to be more universal than that: gluons too, and the W/Z of the weak interaction--all have 1 unit of angular momentum.

So if something has an intrinsic angular momentum, what sizes can it be?

You can guess right off the bat: 0, 1, 2, 3, 4 ... units of h. If these are elementary particles we call them bosons. If they are composite (like a Helium atom), we generally still call them bosons but with the caveat that they are complicated.

But, there’s another possibility. 1/2 h. If something with that angular momentum emits a photon, you get (1/2-1)=-1/2: the same size, just pointing in the opposite direction. Particles with spins 1/2, 3/2, 5/2 etc. we call fermions.

That different spin might not seem to matter much, but it turns out to be very important. Each electron is exactly like every other electron. If you have two, their joint wave function has to be anti-symmetric. In other words, if you swap their positions, directions and their spins, you get the opposite sign. So what happens if they’re both at the same place with the same spin? When a function is equal to its negative, it has to be 0. So you can’t have two fermions in exactly the same state. Which is fortunate, or else atoms would collapse as all the electrons emitted photons and fell into the nucleus.

So if angular momentum comes in discrete chunks, does that mean that the angular momentum of a spinning wheel is a finite (but large) number of those chunks instead of a continuum? Or that your angular momentum relative to the door knob as you walk down the hall is limited to particular values? Probably. And if you try to wrap your mind head around how that works, you begin to see why some researchers are working with models of spacetime/momentum space that use a grid rather than the old faithful number lines. In everyday life you can't tell the difference between a quadrillion h and a quadrillion and one, so it looks continuous to you. But things look different in the small world when the lumps become obvious.