Approaches towards quantum gravity

Question

Consider the following paragraph taken from page 30 of Thomas Hartman's lecture notes on Quantum Gravity:

Hawking radiation is a feature of QFT in curved spacetime. It does not require that we quantize gravity - it just requires that we quantize the perturbative fields on the black hole background. In fact we can see very similar physics in at spacetime.

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1. Does a QFT in curved spacetime simply replace a Lagrangian in flat Minkowski spacetime with the same Lagrangian in a given curved spacetime?

2. Does a QFT in curved spacetime not include the Einstein action?

3. Does a QFT in curved spacetime have limited predictive power than the non-renormalizable quantized Einstein-Hibert action coupled to the same QFT, because we have to choose a specific spacetime metric in the former case in order to draw predictions from the theory?

4. Does a QFT in curved spacetime quantize the matter fields which manifest themselves well below the Planck scale, but use a given classical spacetime metric?

Answer 1

1. To a certain extent. You have to correctly couple the terms to gravity. E.g. in the Yang-Mills Lagrangian we will have $$ -\frac{1}{2g^2} \text{tr} \; F_{\mu \nu} F^{\mu \nu} \rightarrow - \frac{\sqrt{-\det g}}{2g^2} \text{tr} \; F_{\mu \nu} F_{\sigma \tau} g^{\mu \sigma} g^{\nu \tau}; $$ and in the Dirac Lagrangian we will have $$ \bar{\psi} (i \gamma^{\mu}(\partial_{\mu}-ig A_{\mu})-m)\psi \rightarrow \sqrt{-\det g} \, \bar{\psi}\left( i e^{\mu}_I \gamma^I (\partial_{\mu}-igA_{\mu}+\omega_{\mu}) - m \right) \psi $$ with $e$ and $\omega$ the cotetrad and spin connection.

2. No, gravity is not quantized. You generally solve classical EFE equations and then quantize the matter fields on the given curved background.

3. Nonrenormalizable quantum gravity is actually a particular model of QFT in curved spacetime which also describes the quantized fluctuations of that very spacetime. So to some extent it is supposed to have more predictive power. But it is nonrenormalizable, meaning that it breaks down close to the Planck scale.

4. What happens below the Planck scale remains a mystery to us. QFTs in curved spacetime are approximate models. They describe physics far above the Planck scale.