Chapter 2 of the paper [Symmetry of massive Rarita-Schwinger fields](https://arxiv.org/abs/hep-th/0404131) by [T. Pilling](https://inspirehep.net/authors/1021422) mentions "the usual" spin projection operators. However, to me, they are not usual and I struggle with intuition and notation. I understand that we find the correct Lorentz representation of the RS vector-spinor by taking the tensor product of a bispinor and vector representation (eq 1 in the paper): $$\left[(1/2,0)\oplus(0,1/2)\right] \otimes (1/2, 1/2) \quad\quad\quad$$ $$\quad\quad\quad = (1, 1/2) \oplus (1/2, 1) \oplus (1/2, 0) \oplus (0, 1/2) \tag{1}$$ My main question is about the later mentioned projection operators. Clearly a RS-spinor $\psi_\mu$ has a mixture of 3/2 and 1/2 degrees of freedom. We are not interested in the 1/2 background, so we need a projection operator $P^{3/2}$ to get rid of them. Fine. Likewise, I can define an operator $P^{1/2}$ to get the spin-1/2 background. That's just a mathematical excercise. Now, what are the extra indices at $P^{1/2}_{11}$, $P^{1/2}_{12}$, $P^{1/2}_{21}$ and $P^{1/2}_{22}$? In the paper they are described as > the individual projection operators for the two different spin-1/2 components of the Rarita-Schwinger field This is where I'm lost. What am I projecting? For what reason? Can someone explain this to me and give me some intution?