Preface There are lots of times in physics when we use approximations of a more accurate or fundamental physics theory because it is easier to work with. For example, in quantum chromodynamics (QCD) lots of calculations are done using the Schwinger-Dyson equations rather than the actual equations of QCD directly, even though the Schwinger-Dyson equations are merely an approximation with a limited range of applicability. Similarly, lots of N-body simulations in astronomy use Newtonian gravity to model complex many body systems because the math is much easier and the benefits of using full general relativity (GR) are modest. Again, the physicists doing this know that it is a toy model and not a correct description of the universe, but sacrifice accuracy in exchange for a model that is easier to work with and is good enough for some purposes. Application to Quantum Gravity It is much easier to formulate something very similar to Newtonian gravity as a quantum gravity theory, than it is to do so with GR. It is simply a massless, non-self-interacting scalar boson (basically a spin-0 photon). This isn't exactly Newtonian gravity, because it doesn't propagate instantaneously and instead does so at the speed of light. But, the mathematical dilemmas of trying to model a massless, self-interacting, spin-2 boson are avoided. One could fairly easily make some generalizations of this almost Newtonian toy model of a quantum gravity theory without rendering the result impossible to deal with mathematically. For example, you could allow these spin-0 toy gravitons to couple to the total mass-energy of all particles other than gravitons, rather than merely to mass as in Newtonian gravity. If you were really bold, perhaps you could even generalize the theory further to allow the spin-0 toy graviton to couple to itself although this would make the mathematics much harder (but still much easier than GR with a spin-2 graviton - basically, this would be a static universe version of GR). Note that I'm not trying to do original theory here and honestly the details I what I've sketched out above may be inaccurate in one or more respects. I'm simply trying to communicate with an example the kind of approach that I am wondering about, because I would like to know if this general approach has been pursued, and if so, when and by whom. I am not intending to make any affirmative claims about these toy model theories. My Question My question is this: Is anyone aware of any efforts to come up with an imperfect version of quantum gravity by starting with a toy model that is a quantum version of Newtonian gravity and then generalizing it? I'm not really sure how to query the literature on the subject to find what I'm looking for. On the other hand, if no one has done this, is there a reason for that? For example, is there a no-go theorem of some kind that makes it clear that this is not useful for any purpose?