- I argue that there are good reasons to expect that QG will be indeed a standard quantum theory, the general principles of QT remaining unchanged, instead of a modified, generally covariant modification of quantum theory.
Sunil said:The name given for the not yet existing solution is "quantum gravity", suggesting a quantum theory, but no relation to GR. If there are problems of GR which prevent the use of standard quantum theory with the Schrödinger equation, and they can be solved in a modified interpretation of relativity (say, using the neo-Lorentzian interpretation of SR or the field theory version of GR or a neo-Lorentzian interpretation of it) to discuss it in the Special and General Relativity sounds appropriate.
I think there is nonetheless a difference, and it is in favor of quantum theory.Not at all. Most physicists who use the term "quantum gravity" mean finding a quantum theory that has classical GR as an approximation in some appropriate limit. Discussion of such proposed theories belongs in the Beyond the Standard Model forum.
Having the GR equations as an approximation in some appropriate limit is, of course, required by the correspondence principle, as well as simply of empirical viability. In this sense, the same is obligatory as for GR, as for QT.
But this is something very different from being a general-relativistic theory, that means, a theory with a four-dimensional spacetime manifold. As in LQG, as in string theory there is no such manifold. I know only about a single approach to quantum gravity following this line - the proposal of Penrose (see references in https://en.wikipedia.org/wiki/Penrose_interpretation ) which has a classical GR solution and quantum theory is replaced by an objective collapse theory where the collapse happens when the parts become distinguishable by gravity.
Instead, all other approaches I know about modify GR but leave the principles of quantum theory unchanged. LQG is standard quantum theory of some lattice, string theory is a standard quantum field theory, Schmelzer's proposal would be a standard quantum condensed matter theory.
In principle, one could not exclude that QG modifies as GR, as QT. But I'm not aware of such an approach.
Moreover, I think that there is also a justification for this asymmetry. Classical GR has singularities, so it is known from the start to be wrong, it has to be replaced by a different theory. Quantum theory in the minimal interpretation has no such singularity problems. Then, GR has problems with local energy conservation laws for the gravitational field. But a Hamilton operator defining energy seems obligatory in a quantum theory.
The only counterargument I know about is that in realistic interpretations of quantum theory QT has infinities too, namely the Bohmian velocity, with is the average velocity in other such interpretations like Nelsonian stochastics, becomes singular near the zeros of the wave function. Thus, one has to modify QT as well to get rid of those singularities. But in interpretations of QT which do not give that velocity any physical meaning (given that it is unobservable and its equation violates Einstein locality) this argument would fail. Moreover, it seems not very plausible that such a modification, even if necessary, would give anything for the quantization of GR and the unification of SM and GR into some TOE.