Sean Carroll

The number of particles in the observable universe is only like 10 to the 88th by comparison.

So 10 to the 122 is a big number.

If you have a quantum system in thermodynamic equilibrium with a certain entropy, then there's a well-known fact that the dimensionality of Hilbert space is roughly speaking e to the entropy, okay?

So that's e to the 10 to the 122, which is a preposterously big number.

And I'm building up to the fact that the recurrence time is roughly again, very roughly speaking, because who cares?

These numbers are too big to be precise about it.

the recurrence time is something like e to the dimensionality of Hilbert space.

So e to the e to the 10 to the 122.

And by the way, when the numbers are big enough, it doesn't matter whether you use e or 10 or 2 as your base.

e to the e to the 10 to the 122 is approximately equal to 10 to the 10 to the 10 to the 122.

That's just how these exponentials really work out.

So all of which is to say,

If the universe as a whole were described by a finite dimensional Hilbert space, there would be a very, very, very long time between recurrences, but they would eventually happen.

And the reason why I've never been excited by that possibility is the Boltzmann brain problem.

If you imagine that, okay, I'm going to imagine a universe with a finite dimensional Hilbert space.

Its quantum state will just cycle through Hilbert space until it eventually comes back to where it left.

And I'm going to invent an interpretation

for that quantum state in terms of space time.

And I'm gonna say, okay, at some moment, there was kind of a big bounce, sort of a big bang slash crunch phase.

And from there, my quantum state does things which I interpret as the universe expands and cools and structure forms and all those things.