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It is said that renormalization in curved spacetime is difficult. But technically, renormalization procedures can be translated into a problem adding counterterms into Lagrangian. Can't this Lagrangian, translated to accommodate general spacetime metric, be used to define renormalization procedure?

I could understand that even if the above procedure is valid, since we do not in general know how to decompose solution to the equation of motion, we really do not know what we are really getting out of this procedure.

But my question is restricted to whether such Lagrangian procedure would technically be the valid approach.

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The issue with gravity is that it is not renormalisable - because the coupling constant ($G_\text{Newton}$) has mass dimension $-2$. You have to add an infinite number of counter terms to absorb all the UV divergences. See https://arxiv.org/abs/1209.3511 for more details

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    $\begingroup$ I think the OP is asking about renormalization of QFT in curved spacetimes (i.e. with classical GR background) and not about quantum gravity itself. $\endgroup$ – user1620696 Dec 5 '18 at 17:24
  • $\begingroup$ Oh fair cop. If that's the question I think there's no issue with absorbing divergences with counter terms; the only problem then is basically that the calculations are gross - the propagators for instance will be some complicated mess since the action depends on the metric. $\endgroup$ – user215135 Dec 5 '18 at 17:26

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