It's well known that general relativity and quantum theory break down at the Planck distance. What I don't understand is why you can't use the Planck distance as a sort of cutoff.
As I understand it the infinities you get from just plugging in the various field equations is sort of like the infinities you get from using the quantum field equations with the electron that was discovered back in the 1930s. If you calculate all the various virtual particles of the electrons they are infinite but what happened is with renormalization you could completely cancel the infinities. So that's the electron. But from what I understand is if you try to renormalize gravity and quantum field theory this sort of renormalization doesn't work for gravity even if you use a cutoff at the Planck level. Why is this?