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From my understanding, QFT combines QM and special relativity. So doesn't QFT in curved spacetime combine QFT and general relativity? I realize that we need to quantize gravity to have the more accurate theory, but I want to know if GR and QFT are combined by QFT in curved spacetime. If it does, why do people say it's difficult to combine them?

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    $\begingroup$ QFT in curved spacetime can be done, but in a fixed curved spacetime. I.E. it can't be done when you allow the metric to change due to the curvature coupling with the stress-energy tensor (Einstein's equations). In that case you run into the standard issues with renormalizability. $\endgroup$ – Bobak Hashemi Jun 20 '20 at 8:54
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The problem arises due to the fact that if one wants to consider the quantum effects of gravity itself, then one has to compute graviton loop corrections. These are famously non-renormalizable, which means that one needs to include an infinite number of counter terms in the Lagrangian in order to be able to cure the divergences in the theory.

So yes, you can consider a quantum theory in the background of some curved space-time, you could derive quantum effects which are special to curved space time (e.g. Hawking radiation, Unruh effect etc) but you cannot truly call such a theory "quantized gravity", since by doing this, you are effectively only considering gravity at tree level. And that is, by definition, the classical approximation. So in a way studying fields in a curved background is a type of semi-classical approximation.

Gravity is classical but the rest is quantum.

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  • $\begingroup$ Is that the same as saying that the current quantum theories combined with GR are not fully-dynamical? $\endgroup$ – Akerai Jun 20 '20 at 19:30
  • $\begingroup$ @Akerai I'm not sure what it is that you mean by "fully-dynamical". Tree level approximations are not necessarily "not dynamical". It's jut that they are not quantum. $\endgroup$ – Stratiev Jun 20 '20 at 21:47
  • $\begingroup$ I See, I was just curious if the current approximations require the background to be fixed, or if it's dynamical in any way $\endgroup$ – Akerai Jun 21 '20 at 12:26
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To echo what others have said: in QFT on curved space you allow the metric to be non-flat, but the metric itself (and thus spacetime) is not a quantum variable.

It is perhaps also worth emphasizing that although graviton interactions are non-renormalizable, you can still compute quantum corrections from gravitons at energies much lower than the Planck energy (see for instance ch. 22.4 of this textbook). It's once you reach energy scales on the order of $10^{19}$ GeV that the non-renormalizable nature of graviton interactions becomes a serious obstruction.

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QFT on curved spacetime describes the evolution of quantum fields defined on a curved spacetime, but this means abandoning some of the fundamental mathematical requirements of quantum theory, notably unitarity and with it the probability interpretation. In my view this is not just difficult, it is not even possible. I think a different approach altogether is necessary, one which recognises quantum mechanics itself does not require a fundamental spacetime. As Dirac put it,

“In the general case we cannot speak of an observable having a value for a particular state, but we can … speak of the probability of its having a specified value for the state, meaning the probability of this specified value being obtained when one makes a measurement of the observable.”

In particular, since position does not exist in the general case in quantum mechanics, nor does spacetime. This was also encapsulated in von Neumann's treatment of quantum mechanics as a theory of measurement results. We should therefore consider spacetime as an emergent property from the mathematical structure of quantum mechanics, not as a fundamental on which quantum mechanics is based.

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  • $\begingroup$ Why do we abandon unitarity? Couldn't we require unitarity for the wave-function restricted to every Cauchy surface? $\endgroup$ – Cam White Jun 21 '20 at 1:14
  • $\begingroup$ @CamWhite, We can derive the general form of the Schrodinger equation from unitarity. Minkowski metric appears in the solution. I have given a demonstration in academia.edu/40217917/Mathematical_Implications_of_Relationism $\endgroup$ – Charles Francis Jun 21 '20 at 5:46

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