# Approaches towards quantum gravity

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.

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?

• Yes/No questions are a bit difficult to handle because "Yes to all" is to short to even submit as an answer. What do you really want from an answer here? Can you try to ask the question in a way that it necessarily elicits longer answers? Would perhaps reading an introduction to QFT in curved spacetime clear up your questions better than asking a question here? We can't substitute textbooks and review articles, here. – ACuriousMind Apr 17 '17 at 11:20

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.