What are the first steps in converting a metric into a quantum field theory? I know roughly what to do once I have a pair of non-commuting operators, but how do I get to that point?
Specifically, I'd like to start with the weak-field ($r \gg r_s$) low-speed ($v = \frac{dr}{dt} \ll c$) limit of the Schwarzschild metric:
$${ds}^2 = -c^2{dt}^2 + {dr}^2 + r^2{d\Omega}^2 + 2\frac{GM}{r}c^2{dt}^2$$
which is flat Minkowski spacetime plus time dilation, and gives Newtonian gravity. (It is amusing to note that Misner-Thorne-Wheeler Gravitation states (in Box 12.2 on p.296) that a "spacetime metric cannot be defined" for Newtonian spacetime. Apparently this metric violates one or more of their assumptions (see exercise 12.10), such as playing nicely with the covariant derivative. This metric is also almost completely ignored in the physics literature; one exception is that Sean Carroll discusses it briefly in Chapter 4 of his online lecture notes on GR, specifically around equations 4.10 to 4.22 on pages 105-106.)
Since all the curvature is in the time dimension, is this likely to run into the "there is no time operator in quantum mechanics" problem?