Sean Carroll

The real numbers are a bigger infinity than countable.

Those are uncountable in number.

So in Hilbert space, the number of dimensions can either be finite, depending on what system you're looking at, or infinite but countable, so similar to the number of integers.

And when Hilbert space is infinite dimensional, unsurprisingly, there's a whole bunch of new mathematical subtleties that come into play.

And this is where the idea of specifying what observables you have becomes super important.

And people think that it's necessary.

I suspect it's not.

I'm writing a paper with a student right now, Hongzhu Liu, here at Johns Hopkins, where we're investigating this question.

I don't think that the observables are actually necessary.

I think they emerge from everything else.

But there's some subtleties there that we'll get into in the paper.

Right now, that's just a footnote.

For sake of the rest of this podcast, let's make our lives easy and imagine that Hilbert space is finite dimensional.

It might be.

Like, we actually don't know.

This is one of the embarrassing things about approaches to the theory of everything.

People don't even know how big Hilbert space is.

Isn't that sad?

But that's the state of the art right now.

We have to work with what we have.