Sean Carroll

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.

And so therefore, if you can imagine a finite dimensional quantum mechanical system that has a spacetime interpretation as universe expands, recontracts, crunches, bounces, and then expands, and the cycle returns again and again, you can number one, avoid the Boltzmann brain problem,

because the period is much, much less than what you had in the typical case, the generic case.

And because the period is much, much less, there's just not enough time to make that many Boltzmann brains.

And number two, you're not explaining the arrow of time, but you're accommodating the arrow of time because your cycles, unlike the traditional semi-classical cosmological cyclic universes,

in this model the cycles are exactly repeating each other okay so in this model which is the model we proposed in our paper um what is happening to you right now just like friedrich nietzsche warned you about has already happened to someone like you an infinite number of times in the past and will happen to someone like you an infinite number of times in the future in our model the universe expands from a big bang it branches because it's ever ready in quantum mechanics so there's different branches where different things are going on

It then sort of approaches thermal equilibrium, which looks like empty space, just a sitter space with a positive cosmological constant.

It lingers there for what to you and I would count as a very long time, but still a very short time compared to the naive recurrence time.

And then it starts to…

Matter starts to appear, and it looks like the universe is re-collapsing, but of course it's just the expansion played backward in time, statistically speaking.