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We can, in fact, get a low energy effective field theory for quantized general relativity (that’s a mouthful). But I think this is a case of physicists talking past each other. When we say that GR and QM are incompatible, we tend to have their physical content in mind.
The low energy EFT is just classical gravity as a background field
Yeah, that’s one of the pieces of ‘physical content’ I had in mind. Background independence is very rich in substance, and simply applying quantization methods successfully doesn’t make the problem go away.
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