The main problem facing physics today is how to unify quantum mechanics with general relativity using such theories as M-theory or loop quantum gravity. But even though I have a bachelor’s degree in physics, I never did understand why this was so difficult. Quantum electrodynamics and quantum chromodynamics have no problem unifying QM with special relativity, so why don’t they work with GR? I seem to recall something about QM assuming flat spacetime, but why?
More specifically, QFT (quantum field theory) assumes a “background” spacetime. When you want gravity to come into it, you need it to do two things: be a field and be a curvature.
In GR, gravity is a side-effect of the fact that spacetime is curved. The Earth moves in a straight line, which happens (given the curvature of spacetime) to wrap around the sun.
In QFT, interactions (forces are high-level epiphenomena of interactions) are mediated by particles, which are excitement states of a field. A photon is a solution of the field equations for the photon field. Imagine a vibrating guitar string. we know that it can only vibrate in so many ways because the ends have to be fixed points. quantum considerations show that it can only have so many amplitudes. The basic vibration types of the string (in one whole piece, in two pieces with a node in the center, etc…) are the states of “stringons”, while the amplitude measures how many stringons are in that state in a given vibration. That’s a 1-dimensional field theory.
In general, a field is just a function. The electromagnetic field assigns six numbers to every point in spacetime. The gravitational field assigns ten numbers to every point in spacetime. The problem is that at the same time, it changes the shape of spacetime. As soon as you’ve got gravity bouncing around, the “background” for the other fields is no longer flat. In fact, it’s sort of a hack to look at the one field as a background for the other fields. They should all enter on an equal footing. Worse yet, this argument applies to gravity itself: you can’t define a quantized field theory of gravity (as we traditionally understand QFT) without a background spacetime. Once you have the field around, you’ve changed the background.
There’s actually no problem with doing quantum mechanics in curved spacetime, in fact, here’s a whole book about it.
The problem arises when you try to write a quantum theory of gravity and spacetime. The short way to describe the problem is to say that gravity gravitates: in GR, any energy causes curvature of spacetime, and that includes the energy of gravity. This makes gravity a highly non-linear theory, and that turns out to be a problem when you try to quantize it.