How does Faddeev-Popov work for higher-spin fields? (or does it?) Take for example a spin $2$ field $h_{\mu\nu}$ and some gauge-invariant Lagrangian.


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*Does the Faddeev-Popov trick work here? by work I mean: does it lead to a consistent and unitary theory? is the theory tractable using the standard techniques (e.g., the ghost determinant exponentiates, etc.)?

*What is the gauge-fixing functional that would lead to generalised $R_\xi$ gauges? how many gauge parameters can/should we introduce? Disregarding convergence issues, are $S$-matrix elements $\xi$-independent?

*To what extent is the standard BRST theory applicable here? 
I am led to think Faddeev-Popov does not work, because the gauge algebra is open, so one must use Batalin-Vilkovisky. Is this correct?
Same questions about a Rarita-Schwinger field. 
 A: Let's consider perturbative quantum gravity as an example, with full metric $g_{\mu\nu}^f=g_{\mu\nu}+\kappa h_{\mu\nu}$. The Nakanishi-Lautrup auxiliary field and Faddeev-Popov ghost and antighost are vector fields. The BRST-quantised scalar Lagrangian density is$$R-2\Lambda+\frac{\xi}{2}B_\mu B^\mu-(\delta_\mu^\rho\delta_\nu^\sigma-kg_{\mu\nu}g^{\rho\sigma})(\nabla^\mu B^\nu \kappa h_{\rho\sigma}+i\nabla^\mu\bar{c}^\nu £_{c} g_{\rho\sigma}^f),$$where the covariant derivative is compatible with the unperturbed metric. You'll see the FP-ghost term contains a Lie derivative, which BRST-transforms the full metric. The most common gauge choice is $k=\frac{1}{2}$, for which the theory is anti-BRST invariant.
For more information, you may benefit from excerpts of my PhD thesis. In Secs 2.6.1-2.6.4, I explain the theory's BRST quantisation. (The formalism I have used above is not the more popular vielbein formalism, which is harder to compare by eye to BRST-quantised Yang-Mills theory; see 2.6.4.) In Appendix F (basically a rehash of Sec. 15.9 of Weinberg's The Quantum Theory of Fields, Volume 2: Modern Applications), I explain the motivation for the Batalin-Vilkovisky formalism, as well as why it wasn't ultimately needed for any of my thesis research. In short, you need BV when considering Hamiltonian constraints not originating in the Lie algebra (this is one way the case of perturbative gravity is unlike Yang-Mills), to repair BRST nilpotency due to an open algebra (which IIRC is not an issue here), or to address some quantum anomalies, especially in BRST or anti-BRST symmetries.
