Quantum Gravity is one of those cool project names that doesn't necessarily have a set definition.
This is because the Quantum Gravity guys know roughly where they're trying to get to, but are still exploring different potential routes that might be able to get us there. It's not necessarily about "quantising gravity" in the graviton sense (although some people are following that path) – more generally, it's any research that tries to work out how the heck we can reconcile quantum mechanics with general relativity.
Currently, the situation with gravitational horizons is that quantum theory lets us prove, absolutely definitely, that horizons radiate (Hawking radiation), while Einstein's general theory lets us prove, absolutely definitely, that they don't. We can superimpose QM "sprinkles" onto a GR background to retrofit the desired QM effect onto a nominal GR metric, but that's really a "handcrafted" response to the problem. It'd be better to have a single overarching theory with a single consistent set of rules that let us incorporate the best parts of both models in a single internallyconsistent scheme, and that's the end result that quantum gravity research tries to take us towards.
A common approach to QG research is seeing what happens when we add additional dimensions, since this increases the range of geometrical possibilities beyond those already defined for the individual models that we're trying to unify ... the hope being that we might find one special extended geometry that has a special claim to owning the subtheories that it needs to include. The string theory guys tend to do this sort of work (although some folk feel that they may have gotten a little carried away with the extent of their attempt to catalogue all the possible solutions!).
Another is to look at ways of extending the QM approach of carrier particles as mediators for force to include gravitation, and see what happens. So we have things like Higgs Field research. But since classical gravitation is usually considered to be a curvature effect, not everyone agrees that we need a separate "curvature particle", since the idea seems to be a little selfreferential.
We now also have people studying acoustic metrics as a way of generating curvedsurface descriptions of Hawking radiation.
Different approaches to QM can suggest different approaches to QG. The "stochastic" approach to QM suggests that perhaps we can average the random fluctuations in a model to produce an effective classical geometry, and perhaps we might even be able to leave the dimensionality of these fluctuations unspecified, and see whether our own 3+1 dimensional universe emerges naturally over larger scales once everything's smudged together in the correct statistical proportions.
So, lots of approaches. Folk are working on a lot of potential theories, in the hope that one of them might end up being (or leading to) the winner ... or perhaps a few different methods might succeed if they turn out to be dual to each other. It's probably a bit like being back in the early days of quantum mechanics, where there were multiple approaches and nobody really knew which were good, which were bad, which were dead ends, and which were equivalent.
Meanwhile ... while we're waiting for "the theory of quantum gravity" to arrive ... we have some nice logos to look at.
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