akvadrako said:
I mean an instance of a string theory landscape. As in you can describe a landscape as existing on some Hilbert space. If this isn't true and a landscape requires multiple Hilbert spaces to be defined, then this is my misunderstanding
Generally separate vacua live in different Hilbert spaces in QFT and string theory. Placing the landscape in an eternally inflating background permits tunneling into other vacua across a bubble. Of course what vacuum a given bubble has depends on the state of its neighbour at the point of nucleation. In MWI thus when a bubble ##A## nucleates into a bubble ##B## then ##B## will have all vacua since ##A## will have explored all of its states under MWI. Under Copenhagen ##B## will nucleate only into one vacua. However even under Copenhagen as space expands the whole landscape is explored as each vacua will eventually be present in some bubble.
It can be hard to know "properly" the quantum structure of things in String Theory because often the calculations are semi-classical and we don't know the underlying structure of the theory.
To give an example of two Hilbert spaces in a QFT, the ##\phi^4## Hamiltonian:
$$H = \int{\left[\frac{\pi^2}{2} + \frac{\left(\nabla\phi\right)^2}{2} + \frac{m^2 \phi}{2}+ \left(\phi^2 - \nu^2\right)^{2}\right]}$$
can be defined on two separate Hilbert spaces ##\mathcal{H}_{\nu}## and ##\mathcal{H}_{-\nu}## corresponding to the vacuum expectation value of the field, i.e. ##\langle\phi\rangle = \pm\nu##. The Hilbert space of one vacuum is utterly disconnected from the other and states cannot unitarily evolve from a state in one Hilbert space to a state in the other. So in a MWI account of this theory the multiverse would not contain both vev values.
You can form a Hilbert space as their direct sum ##\mathcal{H} = \mathcal{H}_{\nu} \oplus\mathcal{H}_{-\nu}##. However a state like:
$$\Psi = \Psi_\nu + \Psi_{-\nu}$$
with ##\Psi_\nu \in \mathcal{H}_{\nu}## and ##\Psi_{-\nu} \in \mathcal{H}_{-\nu}##, will not represent quantum superposition but simply classical ignorance as to the vacuum. Unitary evolution decomposes on this space (is not ergodic), this what I meant earlier by unitary evolution not giving a transition between vacua. It's a sign that the Hilbert spaces are disconnected and vector sums across them don't correspond to quantum superposition.
So in a QFT the Hamiltonian does not completely fix the quantum theory, there is a choice of Hilbert space. Non-Relativistic QM has only one Hilbert space by the Stone-VonNeumann theorem.
This is also what is going on in QED. Technically states with different charges are actually separate Hilbert spaces.