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I understand that for real world physics the current Theory of Quantum Mechanics, and current Theory of General Relativity are incompatible. I'm thinking about how one thing different about worldbuilding physics from real world physics is that in worldbuilding physics one only needs to make sure the physics is self consistent.
Is it possible to make up a self consistent quantum gravity for a fictional universe?
Well, yes, kind of at least.
Today's physicists are actively searching for a theory for quantum gravity. There are some theories (stringtheory and loop quantum gravity beeing the most prominent) which are mostly self-consistent. Why 'mostly'? All of these theories include some extremly rare special cases for which their consistency is not proven. But (and thats more important for a world creator) nobody was able to prove them wrong until now and they describe everything we can prove corectly. So I think for creating a futuristic world we can assume one of them to be correct by saying that the proof of consistency came later on by person X, without getting consistency problems.
The easy, or want of a better term method would be to simply choose one of the currently existing theories of quantum gravity and then expand on/play with that to suit the needs of your plot line. By default these theories already attempt to unify GR and QM thereby doing that part of your job for you.
Your plot is going to define your choice of theories however i.e. what specifically do you envisage your characters doing. Their ability to act will be obviously be bounded by the limitations your universe imposes. This means you are only going to have major issues if you insist on introducing things like FTL travel or communication via things like quantum entanglement etc or time travel into your setting.
If you actually want or need those things you might need to stretch you choice of theories out to some of the more fringe ideas which if don't at least outright ban whatever it is you want to introduce. Chances are lay readers won't know or care and even physicists will probably only get upset if is their 'pet' theory of Q G that your messing with.
The bad news is that if you really, really want to to make your setting revolve around 'hard' physics your going to have to do some reading on the topic so you can choose the right option.
All you have to do is decide how some corner cases work.
In most applications we only need to use one of (QM + Particle physics) or GR. For really small fast and energetic stuff like particles banging into each other the effect of gravity is small enough that it can be neglected. For really heavy stuff (Planets, Galaxies) the length scales are so vast that you can ignore any property about the matter except the mass.
If you want to make up a consistent system, you pretty much have to decide how a few corner cases like black holes work. After all those guys are very small and energetic and also very heavy. And if you think about your system hard enough, you might just be able to set it up so there are no obvious game-breaking mechanics.
If you really, really want to play games with gravity and base it purely on really basic quantum principles, while ignoring space/time warping, General Relativity and Einstein completely, then it would be fair game to take a probabilistic viewpoint of gravity, in the vein of Hitchhiker's Guide to the Galaxy.
That is, assume anything can be anywhere, at any particular time, without there being a defined path from point A at time A' to point B at time B', depending on some probabilistic function, without recourse to any specific time interval. That is, the world only exists in the immediate, from moment to moment. There is no 'before' and 'after', no 'cause and effect', only the 'immediate'. Then, make the probability function of a particle being in a particular place dependent on the mass of existing nearby objects. That is, the probability of a particle being in a particular place, becomes higher the closer that place is to a large mass. When the particle appears at this point, it increases the local mass, and increases the probability another particle will be there, and that this particle will be even closer to the center of the mass on the next iteration. Thus, over time, the probability of a particle being nearer a large center of mass increases, so it would appear that the particle is approaching the larger mass. In reality, what is happening is that it just becomes more probable that it is in that position as time goes on.
It explains black holes, but it does not explain the expansion of the universe. But then again, I am not sure quantum probabilistic principles alone will ever explain this. What would the probabilistic function look like for this? An anti-mass probabilistic function? That it is more probable for a particle to appear away from a large mass? Seems contradictory. Yin and yang. The ultimate cosmic tension. One probability in conflict with another.
Of course, explaining exactly why the probability changes with mass is left up to the reader.
