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    $\begingroup$ I edit my definition of Poincare group to include the double covering $\endgroup$
    – LolloBoldo
    Commented Sep 2, 2023 at 15:18
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    $\begingroup$ The reason i wanted to use the Poincarè instead of the Lorentz group (or the double covers) is that in the SM you need the Poincare one to fond the representations of massless fields, see the Weinberg definition of massless spin 1 states $\endgroup$
    – LolloBoldo
    Commented Sep 2, 2023 at 15:20
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    $\begingroup$ I edit the $\mathcal{P}$ definition. Yes i saw the answer, i'm reasoning over it a bit before accepting :) $\endgroup$
    – LolloBoldo
    Commented Sep 2, 2023 at 16:59
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    $\begingroup$ Very nice answer. I would only add that one can really use quantized GR as an EFT with a tower of UV cutoffs and it will never run into an inability to predict anything at current and near-future experimental capability/accuracy. Quantum gravity is not a problem for essentially any observational or experimental area of physics. It is just that people are trying to repeat the same story with this EFT as they did with Fermi's EFT of weak interactions (see chapter 1 of Hořejší's excellent electroweak textbook for the history, arxiv.org/pdf/2210.04526). $\endgroup$
    – Void
    Commented Jun 10 at 12:11
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    $\begingroup$ @Moguntius Yes, start with the assumption that quantized gravity is an EFT that was obtained as a low-energy limit of some renormalizable theory and try to find that theory, possibly unifying gravity with other interactions along the way. This is the route that took one from Fermi's four-fermion theory to the Weinberg-Salam theory and what people tried to repeat with string theory etc. $\endgroup$
    – Void
    Commented Jun 14 at 10:38