# Difference between QFT In curved spacetime, semiclassical, and quantum gravity?

Could someone describe the difference, qualitatively, between QFT in curved spacetime, semiclassical gravity, and quantum gravity? I know that each is an approximation to the next and the end goal is a unification of QFT with GR. I've often heard that the first two are well-formulated but quantum gravity is not? How well formulated are theories of semiclassical gravity vs. say, string theory? What is the critical step to go from semiclassical to quantum gravity? Are there experimental confirmations of any of these?

• Nobody knows what the theory of quantum gravity is. String theory is a candidate. If it turns out to be the theory, it is well-formulated, and, unlike QFT in curved spacetime complete in the UV and unlike semiclassical gravity also able to describe the other interactions.
– user178876
Jul 16, 2019 at 21:59

• Semiclassical gravity: This name often refers to an approximation that attempts to account for how the spacetime metric would react to the quantum fields, but still using a classical description for the metric (so that the metric commutes with all operators on the Hilbert space). The stress-energy tensor $$T_{ab}$$ is a non-commuting operator, so to describe the behavior of the metric field using something like Einstein's field equation, this approximation uses the expectation value of $$T_{ab}$$, like this: $$R_{ab}-\frac{1}{2}g_{ab}R\propto\langle T_{ab}\rangle$$. This approximation is not strictly self-consistent , but it's usable as long as we avoid situations involving quantum superpositions of very different values of $$T_{ab}$$. It has been used, for example, to try to account for the fact that a black hole should lose mass as it radiates. The inconsistency is explained in this blog post: Luboš Motl (2012), "Why 'semiclassical gravity' isn't self-consistent," https://motls.blogspot.com/2012/01/why-semiclassical-gravity-isnt-self.html
• Quantum gravity: This refers to a fully quantum theory with no fixed background metric (except for its asymptotic structure)$$^*$$, but which nevertheless exhibits gravity in a way that is well-approximated by general relativity under appropriate conditions. The correct theory of quantum gravity remains to be determined. Currently, the best-understood example of this kind of theory comes from the AdS/CFT correspondence. This can be regarded as defining a theory of quantum gravity (string theory) in asymptotically-AdS spacetime in terms of a conformal quantum field theory (CFT) associated with the asymptotic structure. CFT is relatively well-understood mathematically, but exactly how spacetime and gravity manage to emerge is still an active research area, as is the question of how to extend the idea to spacetimes with more realistic asymptotic conditions (namely de Sitter).
$$^*$$ I don't know of any well-developed proposals for how to get by without a fixed asymptotic spacetime structure, and I have seen general arguments that such a structure is necessary. Of course, any general argument is only as good as the assumptions that go into it, and since we are on unfamiliar territory when dealing with quantum gravity (due to the lack of any prescribed geometry or even any prescribed causal structure in the bulk of spacetime), it is difficult to know how watertight those general arguments are.