zlacker

>>qqqqqu+(OP)
When two particles get closer, their mutual gravitational attraction increases. As the distance approaches zero, the force approaches infinity. In the limit of d -> 0, the energy released -> infinity. Obviously at some scale the notion of a point mass breaks down, but even quantum theory would be problematic if we think of a wave function as describing a probability distribution, wouldn't it? What's the "official" story on this?

>>bhk+An
The official way this is handled is called renormalization.

Basically, we just declare that we have no idea what is going on at such short distances, and put in some regulator by hand to get rid of the infinity. One very crude regulator (which nobody uses, but which is suitable for demonstration) would just be to say that particles are simply not allowed to get any closer than some fixed tiny distance.

But what about the effects that occur when particles actually do get that close? Well, in most theories, whatever is happening can be parametrized in terms of a few numbers (e.g. it could shift the observed mass of the particles, or their charge, etc.). Our ignorance of what is actually happening prevents us from computing these numbers from first principles. But we can still make scientific progress, because we can treat them as free parameters and measure them -- and after that measurement, we can use the values to crank out perfectly well-defined predictions.

Repeating this process through several layers was crucial to building the Standard Model, which currently has about 20 free parameters.