Renormalisation in the curved background Suppose we have some field theory on a curved background, and the metric tensor $g_{\mu \nu} (x)$ is a smooth function of the position. For simplicity, let's consider a scalar theory with Lagrangian:
$$
\mathcal{L} = -\frac{1}{2} g^{\mu \nu} \partial_\mu \phi \ \partial_\nu \phi + V(\phi)
$$
In general, the Green function for this operator may look inattractive, and the expressions for loop integrals are unlikely to be treated analytically.
However, renormalisation is a $UV$-effect, and looking at the physical processes at distances, much smaller that the characteristic scale, on which $g_{\mu \nu} (x)$ changes, it will look approximately constant.
Does it make sense to apply a renormalisation procedure locally, namely:

*

*At each point $x$ - set $g_{\mu \nu}$ to be a constant

*When integrating by parts to get a propagator neglect all terms with derivatives acting on $g_{\mu \nu}$

*Diagonalize the resulting matrix (Green function) in momentum space, which would now have the form $A^{\mu \nu} (x) k_\mu k_\nu$ (no summation over $\mu, \nu$ is assumed)

*Apply the Feynman rules in that basic locally

As a result, I expect to have coupling constants to depend on the position $x$ in a certain way.
Or one has to work with the exact Green function to obtain something meaningful?
 A: Regarding your 4 point procedure: The utility of the momentum-space Feynman rules comes from translation invariance of the action, which is lost in an action with a static metric $g_{\mu\nu}(x)$ (not to mention the overall factor $\sqrt{-g}$). For instance, we don't have any momentum conserving delta functions. And neglecting all terms with derivatives acting on $g_{\mu\nu}$ while computing perturbative corrections to the Green's function seems like an uncontrolled approximation.
However, renormalization is a UV effect and something from the flat-space procedure should survive, as you mentioned. I can't give a complete answer, but I see two possible ways to proceed:

*

*Standard QFT on a curved background (c.f. Carroll for instance). Pick a timelike direction, solve the classical Klein-Gordon equation (for the Gaussian truncated Lagrangian) and obtain a complete set of modes $f_i(x^\mu)$ orthonormal under the K-G inner product. The index $i$ can continuous or discrete. Expand the field $\phi = \sum_i (a_i f_i + a_i^* f_i^*)$ and quantize it as usual. The Green's function is $G(x,y) = \sum_i f_i(x) f_i^*(y)$. You can now proceed to do position-space Feynman rules to account for $\sqrt{-g} V(\phi)$ corrections.

*If $g_{\mu\nu}\approx\eta_{\mu\nu}$ then you could approximate your Lagrangian as $-\frac12 \eta^{\mu\nu} \partial_\mu \phi\partial_\nu \phi + \lambda(x) \tilde V(\phi,\partial \phi)$ where $\tilde V$ now contains pieces of the kinetic term and $V(\phi)$. It seems such theories haven't been studies much (one study). But in principle there is nothing stopping you from proceeding in position-space Feynman rules. If $|\lambda(x)|$ is bounded you could even argue that perturbation theory is valid (to whatever extent it typically is). The study I cited works out the 1-loop corrections to the $\lambda x^\kappa \phi^4$ quartic coupling perturbation, where the integrals aren't too difficult and finds an RG fixed point.

