My understanding is that 1) SR doesn't consider acceleration (it's an extension of Galilean/uniform motion), and that 2) when Einstein considered acceleration as well as gravity via the equivalence principle which lead to GR [1]. The key insight of the equivalence principle was that the force from gravity (e.g. standing on the Earth) is no different to the observer in their frame of reference to them being in a room in a rocket accelerating at the same rate as gravity [1], [2].
Thus, if you extend QED/QCD/SM in a similar way (thinking of QED/QCD/SM extensions in terms of acceleration and curved space with the equivalence principle in mind) that may lead to a quantized theory of gravity. -- Sir Roger Penrose has a similar idea/thinking [3].
One of the key challenges with quantizing gravity is in how the terms in the expressions resulting from analyzing the Feynman diagram interactions behave [4] which prevent them being renormalized. For electromagnetism you can formulate the terms using the fine structure constant (via the coulomb potential, ħ, and c) which results in successive terms decreasing in value and thus stabilizing to a single value.
For gravity using Newton's relationship between two masses in a similar way to deriving the fine structure constant you get Gm^2/ħc. Applying E=mc^2 gives GE^2/ħc^5. Using the Planck energy constant gives (E/E_p)^2 for the energy coupling strength. This means that unlike electromagnetism, the successive terms in the Feynman diagram analysis grows exponentially instead of decreasing to 0. Thus, this approach to quantization doesn't work for gravity.
Note: you can still use this to analyze quantum gravitational effects at small energies by evaluating to a given number of terms.
> One of the key challenges with quantizing gravity is in how the terms in the expressions resulting from analyzing the Feynman diagram interactions behave [4] which prevent them being renormalized.
Not being renormalizable actually isn't a problem in itself if you view the theory as an effective theory, valid only up to some energy scale, not beyond that. You can still use the theory to make some predictions, as long as you're careful. But that does mean that the QFT of a massless spin-2 field can't be a fundamental theory of gravity; it can only be an effective theory, that approximates something deeper.
As I said, this is not correct. Early on, in the first years after Einstein published his papers, there were physicists who believed this (and IIRC Einstein was initially one of them), but that was well over a century ago. We've made a lot of progress since then, and part of that progress is understanding that SR can handle acceleration just fine, as long as spacetime is flat.
> when Einstein considered acceleration as well as gravity via the equivalence principle which lead to GR
The equivalence principle as Einstein first came up with it was actually about free fall. What he called "the happiest thought of my life" was "if a person falls freely, they will not feel their own weight". In modern terminology, we would say that, if you are dealing with a small enough piece of spacetime, you can treat it as flat, even if the spacetime globally is curved. And that means you can use all of the physics of SR in that small piece of spacetime. And that turns out to be a key piece of getting to GR, how to handle spacetimes that are globally curved.
> The key insight of the equivalence principle was that the force from gravity (e.g. standing on the Earth) is no different to the observer in their frame of reference to them being in a room in a rocket accelerating at the same rate as gravity
This was a further development of the equivalence principle from the free-fall version I described above. But note what it implies: it implies that, in modern terminology, treating a small enough piece of spacetime as flat works even if we adopt an accelerating reference frame in that small piece. Physics ultimately looks the same whether we adopt the frame of the object freely falling in the elevator/towards the Earth's surface, or the accelerated frame of the elevator/observer standing on the surface of the Earth. SR handles both just fine.
Where SR breaks down is when we need to extend our analysis beyond a small piece of spacetime--when the effects of spacetime curvature start to show up. That's when we need GR.
Again, all these implications were not necessarily clear to physicists a century ago. But they are now, and have been for decades.
Thus, if you extend QED/QCD/SM in a similar way (thinking of QED/QCD/SM extensions in terms of acceleration and curved space with the equivalence principle in mind) that may lead to a quantized theory of gravity. -- Sir Roger Penrose has a similar idea/thinking [3].
One of the key challenges with quantizing gravity is in how the terms in the expressions resulting from analyzing the Feynman diagram interactions behave [4] which prevent them being renormalized. For electromagnetism you can formulate the terms using the fine structure constant (via the coulomb potential, ħ, and c) which results in successive terms decreasing in value and thus stabilizing to a single value.
For gravity using Newton's relationship between two masses in a similar way to deriving the fine structure constant you get Gm^2/ħc. Applying E=mc^2 gives GE^2/ħc^5. Using the Planck energy constant gives (E/E_p)^2 for the energy coupling strength. This means that unlike electromagnetism, the successive terms in the Feynman diagram analysis grows exponentially instead of decreasing to 0. Thus, this approach to quantization doesn't work for gravity.
Note: you can still use this to analyze quantum gravitational effects at small energies by evaluating to a given number of terms.
[1] https://www.britannica.com/story/how-albert-einstein-develop...
[2] https://www.ebsco.com/research-starters/physics/equivalence-...
[3] https://www.youtube.com/watch?v=VQM0OtxvZ-Y "We need to 'gravitise' quantum mechanics, not quantise gravity | Roger Penrose | Full interview"
[4] https://www.youtube.com/watch?v=yTEPm5d6mrI "Why Quantum Gravity Doesn't Work"