Is the Many-Worlds Interpretation of Quantum Mechanics Unitary? Proponents of the Many Worlds interpretation of quantum mechanics claim that according to this interpretation, the quantum state of the universe evolves unitarily, according to the Schrödinger equation. But how is this possible? After all, the state in MWI constantly branches into a huge number of new branches, which means that the number of possible states of the system grows (the Hilbert space increases), and this is not a unitary evolution. Maybe I'm misunderstanding something, please clarify.
 A: The number of worlds is the number of points in the domain of the wavefunction. For a single particle, this domain is, always was and always will be $\mathbb{R}^3$. The wavefunction may spread out and assign positive weights to the points (worlds) that used to have zero weight but a second copy of the wavefunction never gets made.
Popular science sources refer to this unitary evolution as "branching" because they want to make it sound like there's something special about the Many Worlds Interpretation. But really, the distinguishing features is that there isn't anything special about it. This is in contrast with the Copenhagen Interpretation where a non-deterministic collapse suddenly makes all but one point in $\mathbb{R}^3$ "no longer count".
A: OK, so, branching is a subtle issue as you can imagine but I think the core of your question can be addressed without digging deep into the issue of what constitutes a branching.
The dimensionality of the Hilbert space does not increase when a branching event happens for the same mathematical reason that $\vert \uparrow_x\rangle$ and $\vert\uparrow_x\rangle+\vert\downarrow_x\rangle$ belong to the same Hilbert space. Consider the following pre-branching state of the world:
$$\vert \uparrow_z\rangle\otimes\vert \mathrm{ready}\rangle_{S_x -\mathrm{measurement\ device}}\otimes \vert e_0 \rangle_{\mathrm{environment}}$$
Now, some interaction Hamiltonian evolves this state into the following state
$$\frac{1}{\sqrt{2}}\vert \uparrow_x\rangle\otimes\vert \mathrm{\uparrow}\rangle_{S_x -\mathrm{measurement\ device}}\otimes \vert e_\uparrow \rangle_{\mathrm{environment}}\\+\frac{1}{\sqrt{2}}\vert \downarrow_x\rangle\otimes\vert \mathrm{\downarrow}\rangle_{S_x -\mathrm{measurement\ device}}\otimes \vert e_\downarrow \rangle_{\mathrm{environment}}$$
This is a branching event corresponding to the measurement of $S_x$ over a state that was prepared in $\vert\uparrow_z\rangle$. As you can appreciate, the state simply evolved to an entangled superposition state in the same Hilbert space. We did not add a new Hilbert space or enlargen the Hilbert space to incorporate the branches.
