As they approach, they're pushing, not popping.
Yes. Then as they separate, they pop those forces back out again. When they reach separation X, which can be infinity if you like (or we can just define potential energy relative to that point) they have zero potential energy and a kinetic energy which cannot in general be equal to what they started with. The simplest way of seeing this is to have the coupling be constant A on the way in, then change to B at the point of closest approach. Then their total energy on again reaching the starting point is A integrated to the point of closest approach (which is equal to their starting kinetic energy) plus B integrated back out again; and the potential energy is zero since it has been fully popped.
What you mean is that the laws are no longer invariant under boosts.
Yes, you are correct. Still, the point stands that you are abandoning some symmetry or other, and therefore some conservation law or other.
Your example completely breaks the restrictions I gave it. The whole idea of pushing and popping is that going out is exactly the same as the last time they went in. Do you know what 'stack' means? GOing back out, you perfectly reproduce what you had going in.
Still, the point stands that you are abandoning some symmetry or other, and therefore some conservation law or other.
As I already said, if you constrain it that tightly, then you end up with our physics, period. Conservation of charge? That's a symmetry. Etc. if you hold those to be the same, you completely reconstruct our physics and of course there's no room for randomness.
This is a thread where people can ask questions that they would ordinarily feel embarrassed for not knowing the answer to. The previous thread is at close to 500 comments.