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Hi Lev, Thanks for the elaboration/clarification. I think we agree, about two things. First, yes, this setup probably should not be described as “counterfactual communication” according to Bohm’s theory since *something* (the empty wave) goes along the path through 1/2 and then causally influences the motion of the Bohmian particle below. And second, yes, I agree, different fans of Bohm’s theory (and perhaps sometimes the same one person at different times!) sometimes think/talk about the ontological status of the wave function differently. For me personally, one of the things that makes Bohm’s theory really attractive is the clear, intuitive, causal explanations it provides in exactly this kind of situation (similarly the 2-slit experiment, etc.). So I definitely tend to think of wave functions as physically real things — so much so that I am troubled by the ontological status of multi-particle (and ultimately the universal) wave function(s), and want to try crazy things like reformulating the theory so that only 1-particle wave functions appear as beables. That is to say, the things that attract me most to Bohm’s theory make it rather difficult for me to understand wave functions, not as physical waves, but instead as laws.
On the other hand, I can definitely understand the point of view of the people who (looking more at the fundamental formulation of the theory, in which there is only really one wave function, that of the whole universe) think of the wave function as more law-like (than beable- or field-like) in character. But then I am troubled at losing the intuitive/causal understanding of things like the 2-slit experiment and your example here. Of course, it’s not completely clear that one has to really choose one or the other. For example, one might say that the wave function of the Universe is best understood as a law, but then the one-particle (so-called “conditional”) wave functions that would figure in a Bohmian analysis of the 2-slit experiment, or your setup here, can perhaps be thought of as beable-/field-like. But to me that’s a bit weird. The “conditional wave function” in Bohm’s theory is just the universal wave function, evaluated at the actual (Bohmian) position of all the other particles. It’s like a single “slice” out of the Universal wave function. And… how can a single “slice” of a law, be a beable/field? That’s weird to me.
Anyway, yeah, if you are motivated here by the feeling that you sometimes get mixed signals or contradictory answers from Bohmians about whether one should think of wave functions as physical fields, or instead something like a law, that makes sense and I agree that this is worth discussing further. On the other hand, it should be noted here that these sorts of questions about how to understand the wave function aren’t as pressing for Bohmian mechanics as they are for some other theories — e.g., MWI/Everett, according to which the wave function is the only thing in the ontology. For Bohm’s theory, the matter we see around us (tables, chairs, trees, planets, etc.) is made of *particles* which are just unproblematically in 3D and can hence unproblematically build tables, chairs, etc.; the wave function has a somewhat secondary, background role; so it’s possible to understand quite a lot about what the theory says and how it works and how it explains the distribution of matter we observe, etc., without ever really resolving these questions about whether the wave function should be thought of as a law, or a field, or what. Whereas MWI/Everett, I think, has a long way to go to explain how we end up with tables and chairs and planets in 3D, starting *just* from a universal wave function.