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- TL;DR Summary
- Some beginners questions for the path to understand challenges of quantum gravity.

I'd like to understand how gravity does not combine with quantum mechanics. At least there is no accepted theory of quantum gravity, so I assume it is not solved? I'm only starting to learn QFT and eventually GR. Maybe, someone can already outline where those theories fail to combine and comment on the following thoughts (ideally in terms of concrete equations).

My understanding is that QFT has a Lagrangian describing all of the physics. And GR has the Einstein field equation. Shouldn't both describe some algebra to calculate "what happens" and the task is to find an algebra which encompasses both? Is that correct?

Or is there more than the QFT Lagrangian, because in canonical quantization there are these weird steps of making a coordinate dependent Hamiltonian and then replacing some terms by operators? Why are these operators not already in the Lagrangian anyway?

I've read answers that renormalization is a problem. So you can naturally combine QFT and GR in a Lagrangian? How concretely does it look then? When you calculate, then there will be infinities that you cannot remove? Is all this process written down somewhere? Is renormalization tied to a special way trying to solve the model (an additional assumption), or is the need for fixes already present in the original Lagrangian?

I also saw an answer that background dependency is an issue. How can I see this mathematically in the equations?

I've heard statements that string theory combines quantum mechanics and gravity successfully. So is it an algebra which does contain QFT and GR and can predict what happens? Or why is this not the accepted theory of quantum gravity?

My understanding is that QFT has a Lagrangian describing all of the physics. And GR has the Einstein field equation. Shouldn't both describe some algebra to calculate "what happens" and the task is to find an algebra which encompasses both? Is that correct?

Or is there more than the QFT Lagrangian, because in canonical quantization there are these weird steps of making a coordinate dependent Hamiltonian and then replacing some terms by operators? Why are these operators not already in the Lagrangian anyway?

I've read answers that renormalization is a problem. So you can naturally combine QFT and GR in a Lagrangian? How concretely does it look then? When you calculate, then there will be infinities that you cannot remove? Is all this process written down somewhere? Is renormalization tied to a special way trying to solve the model (an additional assumption), or is the need for fixes already present in the original Lagrangian?

I also saw an answer that background dependency is an issue. How can I see this mathematically in the equations?

I've heard statements that string theory combines quantum mechanics and gravity successfully. So is it an algebra which does contain QFT and GR and can predict what happens? Or why is this not the accepted theory of quantum gravity?