Sean Carroll

You have things like position and momentum and maybe they

have some relationship.

They don't commute with each other.

That's the uncertainty principle.

So maybe you need to specify what all those observables are and exactly how they relate to each other in some way.

And indeed, let me say that here is a technicality I'm not going to get into.

When you do have a Hilbert space, there's a nice feature and a sad feature.

The nice feature is Hilbert spaces are all the same.

You don't need to say like which Hilbert space you're talking about.

If you have the space of two-dimensional surfaces,

That doesn't tell me what you have.

You might have a sphere or a torus or a higher genus Riemann surface or something.

There's many, many two-dimensional surfaces.

Once you tell me the dimension of Hilbert space, you're done.

All Hilbert spaces of the same dimension are the same space, okay?

So that's nice.

That's a good feature.

The bad feature is that that dimensionality might be infinity, right?

It might be what we call countable, which means it's the size of infinity related to the number of real numbers, or sorry,

not the real numbers, the integers or the counting numbers.