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Dear Bob, I Think we should may be continue this discussion by email since the page of the forum is sometimes blocked for unknown reasons.
First, concerning the ontology: BM uses also the Hilbert space for defining the guiding waves otherwise the theory will not work. The ontology is double : particles and waves. Both are necessary and are on an equal footing.
Now, in your CH interpretation the key ingredient is the counterfactual interpretation of the many-time correlation formula also known as ‘Wigner’ formula (for example trace [P2P1rhoP1P2]] where P2 and P1 are projectors associated with measurements at different times ). You want to use this formula to speak about things which were not observed but could have happened. Since you didn’t observe them I am pretty sure that a Bohrian like Zeilinger would class your theory in the ‘hidden variable’ drawer together with BM. They are some differences since the ontology are different but I am affraid that they are both metaphysical in the sense of Heisenberg. I am convinced that, in the present state of our knowledge, in order to solve the problem of which ontology is ‘better’ one would need ‘god eyes’. In other words, if we dont find new data in the future the game is over for making any progress. I can not believe in that of course but this can not be excluded. May be we have already reached the end of ontological science. If quantum mechanics is rigorously true for ever there is no hope to solve the dilemma.
Second, I have your fantastic book in front of my eyes and I went through the chapters you mentioned specially the chapter 14 and 20. What I do not understand is that if you think about an history you can update for sure your information and select was is necessary like for an historian analyzing the Bastille attack. However, the historian can not change the past : facts are facts and he can only put filters to select the good information from the full mess (which was already written in a block world picture to which I subscribe). Now, the choice of the projector families used to define your ontology is forced by your consistency condition, which is by the way not exactly the same as the one used by Gellmann and Hartle, and not all the sets of correlators are consistent if you want to find the usual classical sum rules for probabilities. However, by doing that you introduce some very strong constraints which can lead to contradiction like in the Wheeler delayed-choice experiment involving retrocausality (a point which is absent in BM at least in the preferred frame: each theory has is own surrealism). Your stochastic theory analyzes the interferometer with the two BSs using consistent histories which are not the same as the ones used without the second BS. This is a contradiction for me if you don’t involve retrocausality. Did I missed something? In the same vain I have also the feeling that even worst contradictions could come from Hardy’s experiment watched in different Lorentz frames since what is allowed for an observer in a reference frame is not necessarily possible for an observer in a different frame when the causal order of events is changed.
with best regards Aurélien Drezet
PS: I like very much this discussion and may be we will find a way to disagree or agree in a short text for this forum or the IJQF journal at the end.