Brian Bi

Why is there so much mention of the 'spin' of a particle? Why is it important?

Hmmm. Spin is important because it's like the "shape" of a particle. To describe a spin-0 particle you only need one complex number at each point; its squared magnitude gives the probability density of finding the particle near the point. (And of course it has a phase.) But a spin-1 particle can be thought of as being shaped like a tiny arrow [1], so you need four numbers at every point (space-time is four-dimensional). Right off the bat, the mathematical descriptions of spin-0 (or "scalar") and spin-1 (or "vector") fields differ.

The spin is relevant to how particles interact. Spin-0 particles are like billiard balls [2] so there's no question of relative orientation when they collide. Spin-1 particles are more like... fencing swords, maybe. They can collide at different angles to each other, with different results. So the interactions of spin-1 particles are more complicated than the interactions of spin-0 particles. A spin-2 particle is like... a pair of nunchucks (okay, this is really a stretch now); the two ends can have different orientations, so there are two directions per object. Collisions are, again, correspondingly more complicated.

There's also spin-1/2, which is perhaps the most intriguing at all. I can't give you an example of an object with an analogous shape, because spin-1/2 particles have the curious property that you have to rotate them by 720 degrees, rather than 360, before they return to their original state. To describe their orientation we need not a vector but a different kind of object called a spinor.

Here's a simple example of how spin is relevant to dynamics. The total spin angular momentum of a spin-1/2 particle is \(\hbar/2\). Orbital angular momentum is quantized in units of \(\hbar\). This implies that particles of half-integer spin have to be created and destroyed in pairs, because if, say, you could create just one particle of half-integer spin, then you would change the total angular momentum by \(\hbar/2\) and no change in orbital angular momentum would be able to compensate for that, so you couldn't conserve total angular momentum. Electrons are spin-1/2 and photons are spin 1; the QED vertex has two electron lines and one photon line. You could never have a quantum field theory with a vertex with two spin-1 lines and one spin-1/2 line.

[1] not really, but for the sake of analogy
[2] again, not really