# Spin and polarization, QM vs QFT

On page 34 of A. Zee's book QFT in a Nutshell, he states:

I expect you to remember the concept of polarization from your course on electromagnetism. A massive spin 1 particle has three degrees of polarization for the obvious reason that in its rest frame its spin vector can point in three different directions.

1. I understand that the spin of a spin-1 particle is a 3-vector, because it is the fundamental representation of $SO(3)$.
2. I also know from classical ED that the polarization of light can be quantitatively described using a 3-vector.

But why is this the same? Why is the second implication obvious:

$$\text{Spin}\,1 \implies \text{Spin} \in \mathbb{R}^3 \implies \text{Polarization} \in \mathbb{R}^3 \, ?$$

• Polarization is a 3-vector, but can't point in three directions. Oct 21, 2015 at 17:03
• The polarization states are the eigenstates of the S_z operator (for a massive particle, in its rest frame). Oct 21, 2015 at 18:28
• @BastianTreichler It applies to both QM and to (the single particle states of) QFT. From the QFT side the hardcore reference is Weinberg QFT vol 1 chapter 2, but that might be a bit much. I don't know a good reference off the top of my head. You can sort of see how it comes about from the Stern-Gerlach experiment. The different spots you see on the screen are the different polarizations, and they correspond to the different eigenstates of $S_z$. Oct 22, 2015 at 18:08
• @BastianTreichler Essentially yes that's what I'm saying. For a photon it's slightly trickier because the photon is massless and has no rest frame. The analogue of the z-component of spin in that case is known as helicity, it is the projection of the spin vector along the direction of motion of the particle. Oct 22, 2015 at 19:50
• @Bass I'm not 100% sure what you're asking but let's see if this helps. The number of polarization states depends on the spin and the mass. For a massive spin 1 particle, there are 3 polarizations. This is because in the rest frame of the particle you can apply the ordinary undergrad quantum arguments (S=1, S_z=-1,0,1). For a massless spin 1 particle there are 2 polarizations. The undergrad quantum arguments don't apply because you can't go into the rest frame of a photon. We characterize the polarization states by helicity (roughly S.P where P is momentum), and there are 2 helicity states. Jan 24, 2016 at 14:33