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:

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

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    $\begingroup$ Polarization is a 3-vector, but can't point in three directions. $\endgroup$
    – Ryan Unger
    Oct 21, 2015 at 17:03
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    $\begingroup$ The polarization states are the eigenstates of the S_z operator (for a massive particle, in its rest frame). $\endgroup$
    – Andrew
    Oct 21, 2015 at 18:28
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    $\begingroup$ @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 $S_z$. $\endgroup$
    – Andrew
    Oct 22, 2015 at 18:08
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    $\begingroup$ @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. $\endgroup$
    – Andrew
    Oct 22, 2015 at 19:50
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    $\begingroup$ @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. $\endgroup$
    – Andrew
    Jan 24, 2016 at 14:33


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