Sean Carroll

I would count that as bad.

And that's why I was not that excited about these ideas.

But what we realized, Sakshi and Nadia and myself, is that there's a loophole to this argument.

And it's kind of like a fussy technical loophole, to be perfectly honest.

But just the existence of a loophole, I think, is very interesting.

There's not, in other words, a knockdown argument that the universe can't be cyclic and described quantum mechanically in a finite dimensional Hilbert space.

And so the loophole is the following.

The way that you get the ordinary Poincaré recurrence theorem from Henri Poincaré at the end of the last century is to say, you know, there's some space of possibilities, some space of states through which the system evolves over time.

And if you don't really, really finely tune it, it will fill.

that space.

I mean, you have to define what you mean by the space that you're allowed to be in, so you can't change the energy of the system or other conserved quantities, but otherwise the system will wander around through the space of possibilities.

So if you look at the solar system and the relative arrangements of the planets, it's not just that they will come back to the arrangement they're in right now.

If you wait long enough, they will go into every possible arrangement given what the constraints are on the positions of the planets.

And so there's a footnote there, you know, unless you really fine tune things, right?

If, let's say, you only had two planets and the period of one planet was exactly twice the period of the other planet, then you wouldn't wait this very, very long time for everything to approximately line up.

They would actually, since their frequencies are what we call commensurate,

the frequency of one is twice the frequency of the other, since the frequencies are related by rational numbers, or by integers, in fact, in this case, then they repeat exactly, and they repeat exactly much, much more frequently than you would have figured out from this Poincare recurrence inevitability argument.

It turns out that exactly the same game can be played in quantum mechanics.

If the energies, if the energy eigenvalues, to be technical about it, if the allowed exact energies of your quantum system are related by integers, so if they're commensurable with each other, then all of the wave function will return exactly to where it started, and the period, the time for it to do that, is much, much shorter.

than the typical Poincaré recurrence time.