Sean Carroll

of quantum mechanics and classical general relativity.

So you also know, in fact, if you listened to last week's podcast, you know that I'm interested in taking quantum mechanics seriously for its own sake.

And what that means is not starting with space-time fields, all that stuff, but just starting with the quantum state, thought of as a vector in Hilbert space, obeying something like the Schrodinger equation.

And you can ask similar questions about the arrow of time in that context.

So the framework in which we're working is the following.

We imagine that the theory of everything is a vector evolving in Hilbert space, you know, some quantum state evolving in Hilbert space, maybe a mixed state for those of you who are experts out there, not a pure state, but it doesn't really matter.

And it obeys some version of the Schrodinger equation, okay?

And that's it.

That's the fundamental theory for some choice of the Hamiltonian of the universe.

And then there will be, hopefully, some map or some procedure where you can say, okay, given that the quantum state looks like this, that corresponds to a certain view of what space-time looks like.

And you might gesture toward what that looks like without knowing exactly all the details.

That's sort of the stage we're at right now.

The interesting thing is that just saying that simple picture, just saying the universe is actually a vector in Hilbert space, a quantum state evolving according to the Schrodinger equation, and not worrying about the details about spacetime and bounces and dark energy and all that stuff, still lets you say quite a bit.

First off, there's a very important distinction.

The very first question you ask is, is Hilbert space, the space of all possible quantum states, is it finite dimensional or is it infinite dimensional?

And both cases are very plausible and worth thinking about.

Traditionally, most people think of Hilbert space as infinite dimensional.

There's some reasons to think that maybe it's not from quantum gravity and things like that.

But you always start by thinking about infinite dimensional Hilbert spaces, the simple harmonic oscillator

has an infinite dimensional Hilbert space.