Sean Carroll

Well, here is the recipe.

for taking two systems electron one electron two and creating the combined system of both electron one and electron two and likewise you can do this for other particles for spins whatever there's a recipe that you're given if you really want to know it involves tensor products but what matters is you can start with subsystems and there's a very definite way in quantum mechanics to aggregate them together into bigger systems

We're trying to go the other way.

We're trying to reverse engineer.

We're trying to say we have the whole system.

but no one is divided up into subsystems for us.

Are there better and worse ways of dividing the whole of Hilbert space into subsystems, carve Hilbert space at its joints?

There are an infinite number of possible ways to divide up Hilbert space, okay?

But there are some that will be more useful to us.

And what we have in the back of our minds is basically space.

That's the most important thing to get out of this game that we're trying to play.

Space, again, doesn't show up as part of the fundamental definition of quantum mechanics, but we want to locate it somewhere.

So what does that mean?

And there's actually kind of two different ways in which space shows up.

There's a long story here in which these two different ways are closely connected to each other, but we don't need to go into that.

It'll be completely acceptable, I think.

This is not one of the hard things to accept.

One way is if you just have kind of things like an electron or a macroscopic thing like Schrodinger's cat or, you know, some big buckyballs or, I don't know, some big solid superconductor, whatever quantum system you want to talk about, you want to cast it as something moving in a three-dimensional world that we call space.

So really what you want to do is divide up Hilbert space and to say, here's the cat and

Here's the box it's in.