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Thank you for the different answers. My point concerning the precision of the path concerns the ontology used in the CH model. If your aim is to define counterfactuals associated with paths not observed but supposed to exist then it seems that your CH model is less precise than BM since you don’t have determinism, i.e no precise position x(t) and no trajectory. If you accept that of course this is OK otherwise the theory is incomplete.
Any way, by reading some of your previous papers and re-reading the book by Roland Omnes and one by Bernard Despagnat I realized that the usual Mach Zehnder interferometer experiment is analyzed in a very ‘orthodox’ way in your CH model. This leads to curious properties no so intuitive for a realist.
Indeed, consider a state ‘psi_0’ sent on a beam splitter 1 (BS1) and splits into two paths ‘left’ and ‘right’. These path are subsequently sent on a second beam splitter BS2 and all the state finishes its journey in the exit door 3 with probability P(3)=1; the exit door 4 being always empty. I convinced my self this morning in front of a good coffee (before the heat of the day start :)) that the set of histories ‘Psi0–>Left–> 3’, ‘Psi0–>Left–> 4’, ‘ Psi0–>right–> 3’, ‘Psi0–>right –> 4’, is not a consistent set in your interpretation (as you indeed wrote many times) because the different wave functions involved are not orthogonal (sorry if I repeat some obvious things for you but this theory is quite new for me). Now, the probability to observe 3, i.e. P(3)=1, is the sum of
P(3,Left)+P(3,right)+2Real[ <Psi0–>Left–> 3|Psi0–>right–> 3>]=1/4+1/4+2*1/4=1 and the probability to find ,the particle in the gate 4 P(4)=0 is similarly P(4,Left)+P(4,right)+2Real[ <Psi0–>Left–> 4|Psi0–>right–> 4>]=1/4+1/4-2*1/4=0. The non always positive diagonal terms induce interference and allow to obtain the good probability P(3)=1 and P(4)=0 from histories which have all a weight 1/4>0. I understand that you consistency condition prohibits such an history set by definition and that the question ‘which path was followed by a particle in an interferometer’ is a wrong question in the CH interpretation. But then what is good history set and what does it implies concerning realism?. A good set of histories in the CH interpretation is obviously associated with ‘Psi0–>sigma+–> 3′,’Psi0–>sigma+–> 4’, ‘Psi0–>sigma- –> 3′ ,’Psi0–>sigma- –> 4’ where the state sigma+= (right +i*Left)/sqrt2 and sigma-=(right -i*Left)/sqrt2 which are associated with cat states have been introduced.
Now, we have P(3)=P(3,sigma+)=1 and P(4)=P(4,sigma+)=0 since all other terms, including the non diagonal elements, vanish. What I find most striking here is that by allowing a second beam splitter BS2 you find, in full agreement with orthodox ‘copenhagenists’, that between the preparation and the final actual measurement at gate 3 or 4 the good histories can not include paths ‘left’ and ‘right’ while it can do it of course if only the first BS1 is present. How can that be in an ontological theory? Your reality is thus changing as a function of the context? This is what we indeed learn in text books but in the context of Bohr approach which refuse to speak about the nature of the system between measurements (counterfactuals have no meaning for Bohr). I was convinced that you tried to save realism while your theory seems better to destroy it unless you admit some non contextual features that your theory can not explain. This implies the paradoxical fact that the presence of the second BS2 which can be introduced after the wave packet crossed BS1 can change the nature of the object even retrodictively by acting into the past (see the Wheeler delayed choice experiment). So your theory is not so intuitive as I expected. BM do not need backward causation and the nonlocality is induced thanks to a preferred frame. I dont say that this is better but both approach are ‘surrealistic’ contrarily to your claim. Clearly it means for me that CH is not better than BM for explaining QM in an ‘intuitive’ way (this is not so surprising finally).
I hope that my explanation was not too detailed with best regards Aurelien.
PS: I agree with you that the disagreement with a ‘plausible answer’ in BM is in cm not in nanometer. I could even add in meters or kilometers since beam splitters and optical fibers are used on large scale nowadays. Bujt the same is now also true for the CH interpretation is wou can act retrodictively over kilometers it seems that the notion of past becomes anbiguous in your approach.