Why are General Relativity and Quantum Mechanics incompatible?

Say we have a 1D random walk on a lattice. We want to take a continuum limit of the theory, so we consider finer and finer lattices, jumping at faster and faster times, and want to see what happens in the limit. This is basically a collection of independent systems at different length and time scales, and to take our limit we want to establish which 'finer systems' actually deserve to be considered as refinements of our coarser one. The requirement here is that they give the same macroscopic observables: that between two distant points, they take the same average time to travel between them. This requirement fixes how we have to scale things, and quickly leads to the diffusion equation, and a lot of stochastic calculus (because you find that dx2 is proportional to dt, and this gives you the Ito term and other things).

You could repeat this with a biased random walk, where right and left steps have different odds. In this case the previous scaling doesn't work, and in fact nothing will work. You have to change the bias as you scale as well to reproduce the same macroscopic results.

A common way to look at QFT is with some energy cutoff, and take the limit as the energy cutoff goes to infinity. If you calculate the observed mass of the electron, you find that it isn't the "bare mass" m that sits in your underlying theory, but that it gets corrections from the cutoff. To correctly scale QED down we need to adjust the 'fundamental mass' so that the observed mass is constant as we vary the cutoff.

Renormalization is just about correctly scaling your theory. The details in QFT are trickier, but the underlying point doesn't change. It's only about "cancelling infinities" in the sense that not adopting the correct scaling procedure spits out nonsense, but that shouldn't be particularly surprising. If you scaled the random walk down 'incorrectly' it would end up being ridiculous too.

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