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I have heard that gravity as described by General Relativity (GR) is non-renormalizable, so it is not possible to quantize GR. What about Newtonian gravity? Is it renormalizable? Is a quantum version of Newtonian gravity possible?

Qmechanic
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doctorsundar
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    Newtonian gravity isn't even a field theory, so it doesn't make sense to ask whether it's renormalizable. – knzhou Apr 26 '23 at 22:46
  • @knzhou You could argue that $\nabla U(\mathbf r)=4\pi G \rho(\mathbf r)$ is a field theory, where $U$ is the potential and $\rho$ the mass density. Although only in the classsical sense, not a QFT field theory. – AccidentalTaylorExpansion Apr 26 '23 at 23:10
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    Related : Can Newtonian Gravity be quantized ? – StephenG - Help Ukraine Apr 26 '23 at 23:13
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    @AccidentalTaylorExpansion That's an equation involving fields, but not a field theory. The field $U$ has no dynamics at all. It can trivially be integrated out to yield action at a distance between particles. – knzhou Apr 26 '23 at 23:23
  • It is possible, in a loose sense, to talk about 'renormalizability' of a theory that's not a typical field theory, because any theory is an effective field theory, and one can take a non-relativistic limit and see if any couplings have negative mass dimensions. This is also true for Newton's theory, where the force law has $G$, with dimension $-2$, just like GR. See Schwartz QFT, chap 22. – Avantgarde Apr 27 '23 at 00:20
  • Both gravity and classical electromagnetism can be represented by (almost) the same Poisson equation Delsquared U = constant times rho(r). Yet, apparently only electromagnetism can be quantized with the photon as the force carrier. Despite having the same governing equation, gravity, apparently, cannot be quantized with the graviton as the force carrier, because gravity is not renormalizable. I am trying to understand why. I thank those who have responded, so far. – doctorsundar Apr 27 '23 at 20:39

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