Momentum Operator in curved spacetime (QFT)
In the paper [1], p.111, it is said that the operator of the square of momentum on a curved $n$-dimensional pseudo-Riemannian manifold is: $$ P^2 = -\hbar^2 \left( \square - \frac{n-2}{4(n-1)}R \right) $$ Here, $R$ is the scalar curvature. This $P^2$ is the analog of the classical one $g^{\alpha\beta}p_\alpha p_\beta = m^2 c^2$.
The inclusion of the scalar curvature $R$ is required by conformal invariance of zero mass scalar field. In the case where $n=4$, we have the analog of the Klein-Gordon equation on a curved space-time: $$ \left(\square - \frac16 R\right) \varphi + \left(\frac{mc}{\hbar}\right)^2\varphi = 0 $$ They say that this equation was considered by Penrose [3] in 1964. The "non-relativistic" limit of this should be Schrödinger equation where $R$ appears : $$ \hat{H}\psi = -\frac{\hbar^2}{2m} \left(\Delta - \frac16 R\right)\psi $$ This equation can also be found via geometric quantization of the hamiltonian responsible of the geodesic flow, see [2], eq. (7.114). Now, what is $\hat{p}_i$ on a curved space ? According to geometric quantization (see again [2], eq. (7.42) (7.82)) it is given, at least in the non-relativistic limit, by: $$ \hat{p}_i = -i\hbar \left(\partial_i + \frac12 \mathrm{Div}(\partial_i)\right) $$ Here, $\partial_i := \partial/\partial q^i$ and $\mathrm{Div}(\partial_i)=|\det[g]|^{-1/2}\partial_i(|\det[g]|^{1/2})=\Gamma^k_{ik}$ is the covariant divergence of the vector field $\partial_i$.
Now, what is $\hat{p}_\mu$ ? I guess it would be $-i\hbar \left(\partial_\mu + \frac12 \mathrm{Div}(\partial_\mu)\right)$, but I'm not sure.
So, to answer your question : Does $\hat{p}_\mu = -i\hbar\partial_\mu$ still hold in a curved spacetime? I would say no. There should be at least some divergence term $\mathrm{Div}(\partial_\mu)$ somewhere.
Remark : According to the Wikipedia sign convention : $$ \Gamma^k_{ij} := \frac12 g^{km}(\partial_i g_{jm} + \partial_j g_{im} - \partial_m g_{ij}) $$ $$ R^l_{kij} := \partial_i\Gamma^l_{jk} - \partial_j\Gamma^l_{ik} + \Gamma^l_{im}\Gamma^m_{jk} - \Gamma^l_{jm}\Gamma^m_{ik} $$ $$ R_{ij} := R^k_{ikj} = \partial_k \Gamma^k_{ij} - \partial_j \Gamma^k_{ik} + \Gamma^l_{ij}\Gamma^k_{kl} - \Gamma^l_{ik}\Gamma^k_{jl} $$ $$ R := g^{ij} R_{ij} $$
the "good" sign is $\square - R/6$ as in [2], p.180, and not $\square + R/6$ as in [1] and [3]. Also, even if [2], p.133, has a different definition of the Riemann curvature and scalar curvature, at the end of the day the scalar curvature according to [2] is the same as the one written here.
- [1] : Quantum theory of scalar field in de Sitter space-time, N. A. Chernikov and E. A. Tagirov, 1968
- [2] : Geometric quantization and quantum mechanics, J. Sniatycki, 1980.
- [3] : Conformal treatment of infinity, R. Penrose, 1964.
A short-ish answer to this : $\hat{p}_a = -i\hbar \partial_a$ is not appropriate for the momentum operator due to the fact that it is not self-adjoint. As a reminder, $\hat{p}$ is hermitian if
$$\langle \psi, \hat{p}_a \phi \rangle = \langle \hat{p}_a\psi, \phi \rangle$$
In a curved space, that will be
$$\langle \psi, \hat{p}_a \phi \rangle = \int \psi^* (-i\hbar \partial_a \phi) \sqrt{g}\ d^3 x$$
Integrating it by parts,
\begin{eqnarray} \langle \psi, \hat{p}_a \phi \rangle &=& -i\hbar [\psi^* \phi ] + i\hbar \int \partial_a (\psi^* \sqrt{g}) \phi d^3x\\ &=& -i\hbar [\psi^* \phi ] + i\hbar \int \left[ \sqrt{g} \partial_a (\psi^*) + \psi^* \partial_a (\sqrt{g}) \right] \phi d^3x\\ &=& -i\hbar [\psi^* \phi ] + i\hbar \int (-i\hbar (\partial_a + \frac{\partial_a \sqrt{g}}{\sqrt{g}}) \psi)^* \phi d^3x\\ \end{eqnarray}
It's not actually true that wavefunctions tend to 0 at infinity, but the boundary term can be neglected. We still have an extra term, though, which can be gotten rid of by using
$$\hat{p}_a = -i\hbar(\partial_a + \frac{1}{2} \frac{\partial_a \sqrt{g}}{\sqrt{g}})$$
From differential geometry, we also have that
$$\frac{\partial_a \sqrt{g}}{\sqrt{g}} = {\Gamma^b}_{ab} = \Gamma_a$$
so
$$\hat{p}_a = -i\hbar(\partial_a + \frac{1}{2} \Gamma_a)$$
This is both self-adjoint and obeys the canonical commutation relations.