Sean Carroll

The standard model of particle physics has an infinite dimensional Hilbert space, okay?

But it's one of the questions we don't know the answer to, is that really true for the universe as a whole?

So if the Hilbert space is infinite dimensional, that's kind of equivalent classically to an unbounded phase space.

So think about, you know, planets orbiting the sun.

If the planets orbit in closed orbits like circles or ellipses, there is what is called the Poincare recurrence theorem.

They will always eventually come back to where they started or at least arbitrarily close to where they started.

There's a time scale over which

For all intents and purposes, the system just repeats itself over and over again.

And that makes sense because in a bounded phase space, the system only has a finite number of things it can do and an infinite amount of time in which to do them, so it's going to end up doing the same thing over and over again.

The alternative is an unbounded phase space, and then you don't need to have recurrences or repetitions.

The system can just change in a different direction forever.

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.