Sean Carroll

And that's basically what you get with an infinite dimensional Hilbert space.

The universe can simply change, or the quantum state can simply change forever.

There's no need to recur or anything like that.

So that kind of description

would work in principle with that kind of space-time interpretation of a universe that expands in both directions in the past and the future, okay?

A universe that sort of has a U-shaped curve for entropy and a double-headed arrow of time.

where entropy increases both toward the far, far past and the far, far future.

So that's the picture that you would, the quantum version of the picture where you explain the arrow of time dynamically.

Whereas if you have the finite dimensional Hilbert space, then that's like the planets moving in the sun in circular or elliptical orbits.

There's going to be a certain time after which the universe just returns to its pre-existing quantum state.

So this

Fact is unsurprisingly known as the quantum recurrence theorem, and people have written papers about it.

It's a well-known thing.

In finite dimensional Hilbert spaces, you start the quantum state wherever you want, you will eventually come back to where you began.

Now, the time it takes to come back to where you began, the recurrence time, is generally hilariously long.

It's very, very, very, very long.

The details depend on what you count as really recurring.

But to give you a very vague idea, the area of our cosmological horizon

in desider space.

So just to make sense of those words, we live in a universe with dark energy, which is accelerating.