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Today, nobody knows how GR is truly supposed to be married with QFT. As a result, the standard model as it is typically presented does not include gravity. Could it be modified to include Newtonian gravity instead? Although we know that GR is true, the addition of a quantized Newtonian gravity to the standard model would expand the number of scenarios that it could give correct answers for.

Qmechanic
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  • Newtonian gravity is non-relativistic. However, QFT of the standard model is relativistic. I don't see any reason why QFT could be modified to contain a non-relativistic interaction. – Valac Oct 11 '19 at 14:24
  • @LibertarianMonarchistBot That's a good point. You would have to make Newtonian gravity relativistic to fit with QFT, but I would consider that a fair modification. – Display Name Oct 11 '19 at 14:27
  • but Newtonian gravity by definition is non-relativistic. I guess you want to include the linearized Einstein gravity as S. McGrew pointed out. – Valac Oct 11 '19 at 14:29
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    I guess you can also consider non-relativistic QFT instead of QFT of the standard model. For example, the 2nd quantized Schrodinger field theory is such a QFT. – Valac Oct 11 '19 at 14:32
  • Apparently, linearized gravity can be quantized: Gupta 1952 – S. McGrew Oct 11 '19 at 14:19
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    For the record: Gupta is considering linearized Einstein gravity, not Newtonian gravity. – Qmechanic Oct 12 '19 at 12:10

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That's an open problem. Let us here just consider the traditional approach to perturbative (but non-renormalizable) quantum gravity (QG). In relativity this approach can be based on the action for Einstein-Cartan (EC) theory by gauging the Poincare group. To carry out OP's program, we should presumably instead consider an action (if it exists!) for Newton-Cartan (NC) theory by gauging the Bargmann group (which is a central extension of the Galilean group). Needless to say that all the usual problems with relativistic QG are only bigger in non-relativistic QG. Good luck!

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