Preamble for clarity:
The many worlds interpretation is usually used to explain the measurement of a 2 level system ($|0\rangle$ or $|1\rangle$) as:
$$\frac{1}{\sqrt{2}}(|0\rangle+|1\rangle)|\text{device ready}\rangle|\text{env}_a\rangle\to\frac{1}{\sqrt{2}}(|0\rangle|\text{device says 0}\rangle|\text{env}_b\rangle+|1\rangle|\text{device says 1}\rangle|\text{env}_c\rangle)$$
where $|\text{env}_a\rangle$, $|\text{env}_b\rangle$ and $|\text{env}_c\rangle$ are orthogonal (or nearly orthogonal states of the greater environment).
The universe is then said to have essentially split into 2 "worlds", one in which the spin is in state $0$ and the device says it is in state $0$ and the other where the spin is in state $1$ and the device says it is in state $1$.
My question: This picture works for an interaction with a 2 level system but it seems to me that in general one is making an arbitrary discretisation (or coarse-graining of the wavefunction). How does one describe the same process for the measurement of a continuous variable, say of the location of a particle?
Secondary question:
Also, there seems to be an additional difficulty (or maybe its actually the same one in disguise) in that, in reality, we should really say
$$\frac{1}{\sqrt{2}}(|0\rangle+|1\rangle)|\text{device ready}\rangle|\text{env}_a\rangle\to\\\int\int(c^{(0)}_{\theta,\theta'}|0\rangle|\text{device 0}_\theta\rangle|\text{env}_{(b,\theta')}\rangle+c^{(1)}_{\theta,\theta'}|1\rangle|\text{device 1}_\theta\rangle|\text{env}_{(c,\theta')}\rangle)\mathrm{d}\theta\mathrm{d}\theta'$$
where $\theta$ is a variable which we use to enumerate the compatible device and environment states. This illustrates that in reality the device and environment will also become entangled.
Now the issue seems to be that even some of these states will actually be completely decohered from each other, and may have other observables be incompatible on a macroscopic scale. Hence it seems that we have not 2 "worlds" but 2 sets of "worlds", which it seems may even be continuously connected! (i.e. connected in the sense that for large separation in $\theta$ they correspond to macroscopically distinct worlds but for small separation, they are still coherently connected).
More explicitly I mean that if we project into the measured $0$ "world" in the simplified (standard) example and consider the reduced density matrix for the device we will get
$$\hat{\rho}^{(0)}_{\text{device}} = |\text{device says 0}\rangle\langle \text{device says 0}|$$
i.e. the device is in a pure state of having measured $0$ given that the spin is in state $0$, and so it feels reasonable to call this a unique "world". However, in the more realistic second example, we would find that
$$\hat{\rho}^{(0)}_{\text{device}} = \int\int c^{(0)}_{\theta,\theta'}|\text{device 0}_\theta\rangle\mathrm{d}\theta\mathrm{d}\theta'\int\int c^{*(0)}_{\phi,\theta'}\langle\text{device 0}_\phi|\mathrm{d}\phi $$
$$\hat{\rho}^{(0)}_{\text{device}} = \int\int \rho^{(0)}_{\theta,\phi}|\text{device 0}_\theta\rangle\langle\text{device 0}_\phi|\mathrm{d}\theta\mathrm{d}\phi $$
with
$$\rho^{(0)}_{\theta,\phi} = \int\mathrm{d}\alpha c^{(0)}_{\theta,\alpha}c^{*(0)}_{\phi,\alpha}.$$
This is clearly not, in general, a pure state and so the question arises does it correspond to multiple worlds (i.e. are there parts that are totally decohered and behave separately) or is there some way to explain this away and say it is just one? Hence the question arises more generally how does one define a "world"?
Any explanation to either question would be appreciated.
