I have learned (the basics) of how one can incorporate the principles of special relativity to quantum mechanics to obtain quantum field theory. Can the same be done with GR and QM to obtain a new theory? If not, what happends if one tries, what exactly goes wrong? Is there a mathematical inconsistency in any theory that combines them, such that either current GR or QM must be modified, or is it just that we dont know how to yet?
General Relativity + Quantum Mechanics = String theory.
The things that go wrong in a non string approach;
- the path-integral over metrics has a bad direction, so that the Euclidean continuation is suspect. You need a Euclidean continuation to define a quantum field theory in a mathematically sound way.
- You can't sum over high dimensional topologies without a cutoff on the allowed curvature, and to sum each topology once and only once cannot be done, because the classification of 4d topologies solves the halting problem. This means that any sum over 4d topologies is really an infinite process which defines a hyper-computation. If you needed to do this to define quantum gravity, this would be the end of physics, if "physics" means making computer programs to model reality.
- Nearly all gravity theories (all but the maximal supergravity in 4d) are perturbatively non-renormalizable. This wouldn't be so terrible, except that quantum gravity has no sound gauge invariant cutoff, you can't kill high frequency modes in any simple way, because the frequency is defined by the metric, which you are summing over.
- Quantum gravity path integrals naively require more degrees of freedom than the principles of black hole evaporation allow. In a path integral in a black hole background, there are infinitely many very very short-wavelength modes very close to the horizon whose energy is low because of time-dilation. These modes give black holes a divergent surface entropy, as noted by 'tHooft. You need to kill this divergence to reproduce the Bekenstein Hawking entropy, and this requires a special sort of nonlocality.
All these problems might be surmounted, but this requires a new idea, like loops. In 2d and 3d gravity, the same problems occur, but they are resolved completely by the topological nature of gravity. Loops give an interesting idea for 4d, but it is incomplete.
The way to solve all four problem at once is to move to an S-matrix theory for gravity. This idea, originally proposed by Heisenberg, but best attributed to Geoffrey Chew and Stanley Mandelstam, means that you define the theory asymptotically in terms of scattering states, not in terms of microscopic fields. This works to produce string theory.
Well, some people might say that string theory is the answer. If one takes the usual framework of quantum field theory (say, the standard model), we run into trouble if we want to add gravity. I think one can prove that it doesn't work (it's not renormalizable), and it's not just that we don't know how to do it. But finding the right theory that includes quantum mechancis and gravity is one of the big open problems in physics.
Someone more knowledgeable on the subject can surely point out exactly what the difficulties are.