# 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.

1. 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.)?

2. 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?

3. 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.

• Consistency and unitarity depend too much on the interaction content of the perturbative QFT. Free linearized gravity is consistent and unitary, whereas if you add interactions – well, you know what happens. Add an $R^2$ interaction – you will recover renormalizability but lose unitarity. Faddeev-Popov formal manipulations don't require unitarity or renormalizability, they can be applied to any gauge-invariant path integral. – Prof. Legolasov May 24 '17 at 23:49
• @SolenodonParadoxus fair point. Im actually not that ambitious here: I don't really care about renormalisability yet. I could settle for a gauge invariant $S$ matrix. – AccidentalFourierTransform May 28 '17 at 14:14
• But isn't these two ultimately related? I've always seen proofs of unitarity and gauge-invariance of the renormalized $n$-loop S-matrix. Without renormalizability there isn't actually an S-matrix to begin with... But I see your point. In this case, provided that you have a gauge-invariant regularizer (like the one with higher-order covariant derivatives, for example) the Faddeev-Popov method works perfectly for the spin-2 field. You are aware that in this case you are fixing the remnant diffeomorphism invariance, right? – Prof. Legolasov May 28 '17 at 23:38

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.