zlacker

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+(OP)
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.

>>bhk+(OP)
What follows is very hand-wavy, and the renormalization sibling post may touch on it.

An answer is that the d->0 approaches infinity presumes a nice, continuous analytic function. If d->epsilon, you can't get to that singularity.

There was an equivalent problem in the E/M space with "The Ultraviolet Catastrophe" [1], which turned out to go away if you assumed quantization.

I'm not going to claim this is a perfect analog to the gravity problem, only that a lot of physics doesn't quite work right when you assume continuity. (The Dirac delta is a humorous exception that proves the rule here, in that doing the mathematically weird thing actually is closer to how physics works, and it required "distribution theory" as a discipline to prove it sound.)

[1] https://en.wikipedia.org/wiki/Ultraviolet_catastrophe

>>aberna+P1
This is definitely related, though not the whole story. Quantization did get rid of some infinities, but as GP kind of states, it also introduced others. My comment focuses on what we do for those.

>>knzhou+H2
> In the limit of d -> 0, the energy released -> infinity.

I believe the poster's general premise to be false. While renormalization may be useful in resolving infinities in general, I don't think it's necessary for this one.

You can't commute the dp*dx of a Hamiltonian to be zero in a quantized world, so if gravity has quantum properties, you don't need to worry about what happens when d -> 0. There is no "0" distance.

>>aberna+E3
I guess, but it depends on how one parses the poster's setup. You're correct that for particles interacting under a 1/r^2 force, the energy turns out finite in quantum mechanics. My comment was referring to the fact that once you quantize the field that gives rise to that force, the infinities return, but for a different reason.