I hope it's all right to ask a question about some very basic intuition in particle physics. I was looking into these (very nice) notes of Matthew Schwartz, I guess from his course on QFT:
Around the middle of the second page, one finds the lines
'The unitary representations of the Poincare group were classified by Eugene Wigner in 1939. As you might imagine, before that people didn’t really know what the rules were for constructing physical theories, and by trial and error they were coming across all kinds of problems. Wigner showed that irreducible unitary representations are uniquely classified by mass $m$ and spin $J$. $m$ can take any value and spin is a half integer $J = 0, 1/2, 1, 3/2,...$ Moreover, Wigner showed that if $m > 0$ then there are $2J + 1$ basis states in the representation, and if $m = 0$ there are exactly 2, for any $J > 0$. These basis states correspond to linearly independent polarizations of particles with spin $J$.'
Even though I've given a lengthy quotation for clarity of context, my question is about the last line. To avoid confusion, let's first note that the sentence about the number of basis vectors in the representation, taken literally, is wrong. The representation is infinite-dimensional. So I understand the 'representation' in that part of the sentence to mean the representation of the so-called 'little group' out of which one builds up the given representation of the Poincare group. Or, in other words, he is referring to the vectors in the representation corresponding to a fixed character of the translation subgroup. (We will ignore the fact that such a subspace doesn't exist rigorously.)
Now, what I find most interesting is his description of the basis vectors as 'linearly independent polarizations of particles of spin $J$'. I would like to know if this is a completely standard view, whereby the (complex) directions in the representations of the little group are thought of as 'polarizations'. If so, can one give an intuitive account of how they are natural extensions of, say, the polarization of light? Here, I would be grateful for some physical intuition, since the parallel at the level of mathematical formalism is reasonably clear. If there is good reference where this is discussed, I would be equally grateful.