Sean Carroll

coarse-graining the variables to get that emergent description.

So the claim here is that the emergence of space itself is exactly the same.

I can imagine a gajillion different ways to divide up Hilbert space into subsystems, but one of them has the property that, for example, a

Interactions are local, okay?

This was a result in a paper by Jordan Kotler, Jeff Pennington, and Daniel Renard several years ago, one of my favorite papers, called Locality from the Spectrum, where the spectrum is a way of specifying the Hamiltonian that gives you the dynamics.

And they said, if you didn't know about locality in quantum mechanics, could you find it?

And they said, look, in the set of all possible ways to take a big Hilbert space—

and subdivide it into little hilbert spaces representing what you would hope are regions of space almost all of them would have the property that every little subsystem interacts directly with every other subsystem so this little region of space right in front of my face if i poked on it if i defined space badly by doing the wrong coarse graining

by doing the wrong carving of Hilbert space at its non-joints than by poking the field, as I would attempt to think of it, right in front of me would affect the value of the field everywhere through the universe instantly.

So locality is very, very special.

Locality in the sense that when I have a little quantum field at one point in space and I change its value, I interfere, I measure it or do whatever, the implications of that perturbation only affect its immediate neighbors and then they affect their immediate neighbors and so forth.

That's a very, very delicately chosen way of dividing Hilbert space.

So the implications of this paper are that most Hamiltonians, most ways of writing down the dynamics of a vector in Hilbert space, have no local way of talking about them.

That is to say, you can't

Divide Hilbert space for a generic set of dynamics into a set of subsystems that resemble space, that have the property that if I poke it somewhere, then only its nearest neighbors feel the poking or the influence of it, okay?

Generically, if I poke anything, everything changes right away.

And then they go on to say that when there is a local way of subdividing Hilbert space, it's essentially unique.

Now, there there's been some pushback in the literature.

Some people disagreed and they say, well, you need a little extra ingredient.

Good.