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Lev, yes, in Bohm’s theory the nonlocality is clear: if you have a pair of entangled particles, the velocity of one of them (according to the theory) can depend on a free choice (say, about some external magnetic field that you might apply) at the location of the other one. That is, something over here (the trajectory of this particle) depends on some distant intervention, according to the theory. So it’s not locally causal.
The thing I don’t understand about MWI is: what *is* there “over here”, according to the theory, that might or might not depend on some distant intervention? What really exists, physically, “over here” (in 3D space) according to the theory? If all you have is a quantum state (or wave function) it’s just not clear what ontology there is, if any, in 3D space.
I think maybe you meant to suggest that the local (i.e., existing-in-3D-physical-space) ontology is the local density matrices. Is that right? So then the idea is that what exists physically, in 3D space, “over here”, is some local density matrix/operator, and then it’s easy to show that this isn’t affected by distant interventions. Have I got that right? Assuming so, I might have follow up questions (such as how I should understand a mathematical operator as a description of something physically real, and whether the local density operators *exhaust* the ontology even though they don’t capture all of the structure that’s in the wave function (which I thought was what the theory said was real), etc.). But maybe I’ll pause there and just make sure I’m on the right track so far.