Multi-Bohm

I am exploring what seem to me to be under-explored parts of the logical space of interpretations of Quantum Mechanics. I may be wasting my time: there may be good reasons why those parts of logical space are not explored much. But I am also hoping that such exploration will broaden my mind.

So, here’s a curious interpretation: multi-Bohm. Assume no collapse as in Everett. At any given time t, there is the set S_t of all particle position assignments compatible with the value of the wavefunction ψ(t) (we can extend to spin and other things in the same way that Bohm gets extended to spin and other things). Typically, this set will include every possible position assignment, and will have continuum cardinality.

Now on Bohm’s interpretation, one member of S_t is privileged: it is the actual positions of the particles. But drop that privileging. Suppose instead that all the assignments of S_t are on par. Then S_t gives us a synchronic decomposition of the Everettian multiverse into "branches". Now stitch the synchronic decomposition into trajectories using the guiding equation: a position assignment s_t ∈ S_t is part of the same trajectory as a position assignment s_t′ ∈ S_t′ if and only if the guiding equation evolves s_t into s_t′ over the time span from t to t′ given the actual wavefunction ψ.

We can think of the above as a story with infinitely many (continuum many) parallel Bohmian universes. But that bloats the ontology by including infinitely many ensembles of particles. Since the wavefunction fully determines the sets S_t of position assignments (or so I assume—there are some worries about null-measure stuff that I am not perfectly sure of), we can stop thinking about real particles and just as a way of speaking superimposed on top of the many-worlds interpretation.

This means that we can interpret the many-worlds interpretation not as a branching-worlds story, but as a deterministic parallel worlds reading. For given the two-way (I assume) determinism in the guiding equation, the trajectories never meet: the branches always stay separate and parallel. Moreover, the probability problem of the many-worlds interpretation is unsolved, and so we cannot say that the story fits better with one set of experimental results rather than another.

This isn’t very attractive…