What exactly are the high energy problems between general relativity and quantum field theory?

Question

I know there are many questions on physics stack exchange devoted to the question of quantum gravity and reconciling GR and QFT, but don't mark this as a duplicate because my question is a bit more specific.

In the PSE page A list of inconveniences between quantum mechanics and (general) relativity? it was explained that the crucial problem in uniting general relativity and quantum mechanics comes when probing high-energy conditions at the Planck scale. However, this answer was not quite detailed enough for me, for I still do not fully understand why the theoretical frameworks clash at these conditions.

Can someone explain thoroughly (preferably conceptually and with minimal math) why general relativity and quantum mechanics implicitly do not work together to describe these high-energy interactions and a few examples of them.

I have a good conceptual understanding of both theories, but have little formal mathematical training.

Answer 1

GR is a theory that predicts e.g. the Big Bang as a singularity; more precisely, a singularity where for the length scale $\Delta x$ holds $\Delta x \mapsto 0$. For classical GR, such a singularity is ok, but in Quantum mechanics you would have a momentum uncertainty

$\Delta p \geq \frac{\hbar}{2 \Delta x} \mapsto \infty$

which means you can't measure any momenta properly and that would be physically impossible. Therefore you must find a theory where the length scale also at the Big Bang has a finite value. Merging GR and QFT is therefore not easy, because in a continuum (which is also used for Special relativistic QFT without gravity) you have $\Delta x \mapsto 0$ for Big Bang and other cosmological singularities.

There are several ways, where length scales are treated finite, e.g. in string theory, all particles are not pointlike, These are strings that have a nonzero length.

Moreover, many other physical observables like scattering probabilities will turn out to have infinite values when GR and QFT are naively combined, which don't makes sense. These infinities arise from the Quantum correction of GR (Loop orders of the corresponding feynman diagrams); one has to modify the theories, such that finite values are obtained.