Chapter 2 of the paper Symmetry of massive Rarita-Schwinger fields by T. Pilling 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?
edit:
To clarify, consider the Projector
$$P_x = \begin{pmatrix}1 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0\end{pmatrix}$$
The intution here is, that is projects the x-component of a three vector. Using this analogy, what is $P^{1/2}_{11}$ projecting?
Best Answer
It is apparent that the clear and explicit formal paper you are reading is not helping you systematize what you may have learned in your QFT theoretical course (this is distinctly not experimental HEP!), namely, the basic facts summed up in the paragraph following equation (12): at the end of the day, the independent conditions $\gamma \cdot \psi$ and $p\cdot \psi$ have no Lorentz indices anymore, and so they Lorentz- transform like plain spinors, which you know represent spin 1/2 s:
That is, of the original 16 d.o.f. of the reducible field $\psi^\mu$, you prune out 4 d.o.f. by the first condition, and another 4 by the second, being left with 8 for the spin-3/2 block: a parity doublet of the 4 spin states of the spin quartet. (Remember: we have not gauged out the intermediate helicity $\pm 1/2$ states, since susy-gauge-invariance has not been assumed, this not being supergravity; it might as well be a Δ baryon.)
Now the author spends quite some time giving you a formal (Ogievetskian) implementation of these maneuvers, through the projector $(P^{3/2})^2=P^{3/2} $, the only operator you really need to appreciate, $$ P^{3/2}_{\mu\nu}= \frac{1}{6p^2} \bigl ( 4(p^2 g_{\mu\nu} - p^\mu p^\nu) \\ -p^2[\gamma_\mu,\gamma_\nu ]+p^\mu p^\kappa [\gamma_\kappa,\gamma_\nu] - p^\nu p^\kappa [\gamma_\kappa,\gamma_\mu] \bigr ) \tag{7.1} ~~~~\leadsto \\ \gamma^\mu P^{3/2}_{\mu\nu}=0=P^{3/2}_{\mu\nu} \gamma^\nu = 0=p^\mu P^{3/2}_{\mu\nu}=0=P^{3/2}_{\mu\nu} p^\nu. $$
As a result, after the two piecemeal projections of the two spin 1/2 s, the pure spin 3/2 field you only need consider is $\tilde \psi^\mu = P^{3/2}_{\mu\nu} \psi^\nu$, which, now, automatically satisfies the conditions, by above!
You are projecting the spin 3/2 piece $P^{3/2}_{\mu\nu} \psi^\nu$ of the reducible field $\psi^\mu$ so you don't get distracted by the two irrelevant spin 1/2 spinors unfortunately also packaged in the vector-spinor; a demonstrably real risk, ipso facto. This is the intuition.
If, despite this, you really wished to look in the dross pile for the spin 1/2 pieces you discarded in two stages, 1.13 of van Nieuwenhuizen's review will furnish an overcomplete analysis of the polarization components of the two spin 1/2 s. In (1), $P^{1/2}_{22}$ of course projects onto the spin 1/2 of $(0,1/2)\oplus(1/2,0)$, while $P^{1/2}_{11}$ onto the remnant spin 1/2 spinor in the incompletely pruned $(1,1/2)\oplus (1/2,1)$, which still has 12, not 8, d.o.f.
For the simpler, massless, case, look at section II of van Nieuwenhuizen, P., Sterman, G., & Townsend, P. K. (1978): Unitary Ward identities and BRS invariance Phys Rev, 17 1501.