Okay, I’m imagining a theoretical universe with only two particles of equal mass, at rest and touching each other. At some point in that universe’s history the two particles had collided head on with exactly equal and opposite momenta.

It seems to me the magnitude of their velocity is lost information…

Actually I don’t think you can tell from the present state if the particles had ever moved at all.

The simple answer is that particles (in the sense of indivisible subatomic particles) don’t work like that. If the particles were moving in the past, they had some kinetic energy. If they are stopped now, they no longer have that kinetic energy, so it must have been transitioned to some internal state of the particles, or carried away by some other particle, both of which “store” the information about the previous movement. If neither of those are possible, the particles don’t come to rest against each other, they have to bounce off elastically, which also retains the information about the previous movement.

We’re used to, at a human-scale level, thinking of interactions where energy is lost (e.g. non-elastic collisions like the one you mention). But really energy is not lost – we just decide to stop keeping track of it in detail since it gets spread over a whole bunch of matter that we’re not particularly interested in. But in particle physics we don’t have that luxury – if energy is lost, something carried it away.

Putting the question in terms of “transformations”, the assertion that information is never lost is equivalent to claiming that there are no singular transformations, and thus all transformations must be invertible. Is that correct?

Is it well established (or just assumed?) that there are no singular transformations?

The transformations of quantum mechanics (without collapse!) are unitary, hence invertible, hence non-singular. The requirement for unitarity comes from the preservation of the norm of the state, which in turn ensures that the probabilities for all possible outcomes add up to 1. Thus, unitarity is generally thought to be a pretty fundamental property of a viable theory, but this has been under discussion.

One reason for that is the apparent information loss in black hole evaporation: in Hawking’s original calculation, the radiation emitted by an evaporating black hole was exactly thermal (‘random’), and thus, carried no information about what went into the black hole, violating unitarity. But due to various model studies (most significantly using the AdS/CFT correspondence, where you can model the gravitational dynamics of a spacetime including a black hole by means of an ordinary quantum field theory at the boundary of that spacetime, which has unitarity baked in), people generally don’t believe that the evaporation process is information-lossy anymore (Hawking himself eventually accepted this conclusion, conceding a bet with John Preskill).

May be a nitpick, but “at rest” is meaningless here. There are only relative motions, and in your case only one relative motion (because there is only one body pair between which a motion could exist) whose velocity is zero. There’s no other relative motion and no absolute motion, so there’s also no information about those things.

Depends on whether Mach’s principle is true. It is likely that if those two particles were the entire mass distribution of the universe, then they would be non-rotating by definition.

Oh, yes, I didn’t think of that. I believe you’re right. If the two particles pull together with less than the gravitational and Coulombic attraction forces, you’d have to conclude there’s rotation – and you don’t need to reference any third mass to detect that.

There are some strange situations in physics that are hard to reconcile with conservation of information. For instance, it’s possible to create a dome shape such that if you shoot a marble up the side at the right velocity, it will come to rest at the peak in finite time. It’s then at the exact center of the dome. Haven’t you lost the information about the incoming direction? Worse, the marble is unstable in a strange way–after some unknown amount of time, it should slide back down in an arbitrary direction.

The best you can say is that for some reason, these situations are not physically realizable, but it’s not clear why.

This sounds like “spontaneous symmetry breaking”. To what extent is it not realizable, though? There are some examples if you click on “Goldstone boson”, but I would be lying if I said I was intimately familiar with them.

Is there actually a dome shape that will give you a finite time for the marble to come to rest? It seems to me that, classically at least, it should always take an infinite time to come completely to rest.

The link in my post further links to one such dome shaped h = \dfrac{2b^2}{3g}r^{3/2},\ 0\le r< \dfrac{g^2}{b^4}, though it is not smooth at the apex. r is the distance down the slope and h is the height.

Uncertainty principle? A marble with a tightly confined position at the top of the dome has a significant amount of spread in its momentum. Perfectly balanced at the top of the dome just means that it falls in all directions at once.

Classically, it will not slide down in some arbitrary direction after any amount of time, unless you include things like thermal motion of the atoms in the marble/dome/air, and if you include that, those motions are where the “erased” information hides out.

I thought that “classically” it is more of a thought experiment about (if you accept the physical setup) non-uniqueness of solutions, not a real problem where you need to take into account random thermal motion or quantum effects. Nobody thinks Newtonian mechanics accurately describes real motion in cases like singularities arising in the 5-body problem or via elastic collisions.

I agree with Chronos, I don’t think that you can make the marble reach the top of the dome and remain at rest under classic physics. Remember that everything is time reversible in classic physics. That means that a marble at rest on a perfectly level surface could spontaneously roll down. But resting at the top is an equilibrium (unstable but still an equilibrium) so you’d need some outside influence to make it move.

The example I gave came from the paper Causation as Folk Science by John D. Norton. The potential of the dome (sometimes called “Norton’s Dome”) is the same as what DPRK described. The paper has some other examples, like “space invaders”, but this one is the clearest I think.

Quantum mechanics does change the picture, though it raises more questions. In any case, DPRK is correct in that this is a thought experiment about classical physics. We should be careful about introducing new effects into our thought experiment because Newtonian mechanics is already thought to be time reversible–and yet here’s a situation where it looks like it isn’t.

Not true. It requires no perturbation to slide down. Although resting at the apex for infinite time is one solution, the class of solutions where it slides down spontaneously after an arbitrary time and in arbitrary direction are perfectly valid. There’s no obvious reason to favor one over the other. You can’t even set up a decent probability distribution, because the arbitrary time component goes off to infinity (you could set up a distribution where there is an exponential decay or something–but where does that come from?).

That’s just the thing. Even if you only consider the case where it stays at the apex, you have apparently broken time reversibility. The marble went up the dome–why doesn’t it come back down?

I’d like a cite on that. “An object at rest remains at rest unless acted on by some outside force.” In this case gravity is the only outside force, but the dome is “perfect” so that the force of gravity acts symmetrically so there is no net force in any direction. This must be true or the marble would never have stopped at the top.