Is it mathematically as well physically alright (consistent) to not quantize metric when doing QFT in curved spacetime?
I am not well-versed with the theory of renormalization of QFT as of now (I just know that when you redefine your observable you get rid of UV divergence and then get finite predictions from your theory) so this question might not make any sense.
When we do QFT in curved spacetime as described in sec $2.1$ of <PERSON> and <PERSON> monograph: we replace
* $\partial$ by $\nabla$
* $\eta_{\mu \nu}$ by $g_{\mu\nu}$
* $d^nx$ by $|g|^{1/2}d^nx$
I am feeling a bit uneasy when I look at this minimal coupling prescription (infact it seems ad-hoc as well) because when I am doing QFT any field which is dynamic on spacetime is quantized either by defining a commutation relation or using path integral. The background here as in the language of QFT in curved spacetime is simply $\eta = diag(1-1-1-1)$ it doesn't vary with $x$ coordinate as $g_{\mu\nu}$ does.
So why are we justified to not quantize the metric which may in fact be dynamic as in the case of collapsing part of a collapsing star? Here dynamic might not be the right word because to define dynamic nature of something we need ruler and watch to see the change or mark the event which is given by the metric. 651
It's kind of circular reasoning here. And what's more mysterious to me is getting prediction out of this theory.
Maybe this mysticism and uneasiness is stemming out of my illiteracy of renormalization so can someone kindly explain to me why not quantizing the metric is a right choice till we probe curvature of order $(\frac{1}{l_p})^2$ where $l_{p}$ is <PERSON>'s length.
There is a similar question as well it doesn't justify why leaving metric un-quantized (classical) is a good approximation though my question is intimately connected to backreaction which is discussed there. I want to know a bit more explicitly why working with the classical metric is okay on physical as well as mathematical grounds. 651