Pretty much what title says. We assume that the standard model is renormalizable (What do we mean when we say 't Hooft proved that Standard Model is renormalizable?), but this is for flat spacetime. What about curved spacetime version? Ignoring issues that come out of this not being quantum graviational theory and only approximative, can we still think about what a renormalizability theory means in curved spacetime, and whether if in flat spacetime a theory is renormalizable, it would also be renormalizable in curved spacetime as well?

By curved spacetime version, suppose that we are adding canonical quantum gravity into picture. Then we would transform action of QFT accordingly, along with adding Einstein-Hilbert action, which gives us action of QFT-plus-gravity. Drop Einstein-Hilbert action term, and we arrive at the curved spacetime version of QFT.

  • $\begingroup$ What, exactly, do you mean by its "curved spacetime version"? $\endgroup$
    – ACuriousMind
    Apr 15 '18 at 16:57
  • $\begingroup$ Since renormalization is about the short-range behaviour of the theory and that curved spacetime is locally flat, I would suppose that it does. $\endgroup$
    – Slereah
    Apr 16 '18 at 8:32
  • $\begingroup$ @ACuriousMind canonical quantization to curved spacetime? Like, action we use when we study canonical/naive quantization of gravity minus Ricci scalar term of action (contribution of pure gravity to action), which would leave us with the curved spacetime version of a theory... $\endgroup$
    – Vrizia
    Apr 16 '18 at 14:23
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    $\begingroup$ It might be better to ask "Suppose I have a QFT defined on (globally hyperbolic, etc) curved spacetimes. Is it renormalizable if its restriction to flat spacetime is?" Leave quantum gravity out of the mix. $\endgroup$
    – user1504
    Apr 16 '18 at 14:51

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