Dear Prof. McHarris,
you present an intriguing discussion of the quantum 'cut' between observer and observed as, if I understand you correctly, originating in the boundaries of predictability imposed by nonlinear dynamics of the observed system. I think this is not too far away, in spirit at least, from the approach of Tim Palmer in this essay contest, who considers the ultimate state space of quantum systems to be a fractal attractor of some nonlinear dynamic system, with the resulting picture excluding 'counterfactual' states we would otherwise expect to be 'close' to the actual states---since, as you note, chaotic dynamics may introduce arbitrary separation between arbitrarily close initial values.
However, you, again if I understand you correctly, propose that the quantum formalism is a statistical description of the underlying, nonlinear dynamics. I think this sort of approach has certain problems. For one, it's not quite clear to me how interference effects are supposed to be explained---in a sense, the fact that there is no detection in the 'dark' detector of an interferometer certifies that the photon cannot be taking either one route or the other, as in that case, half the time it ought to be detected there.
But perhaps more importantly, the Pusey-Barrett-Rudolph theorem seems to significantly dampen the chances for any such statistical interpretation of quantum mechanics---given some mild assumptions, mostly the possibility to repeat the same experiment, they show that there can be no set of quantities ('hidden variables') such that they yield more information than the quantum state, and hence, as the original titel of their paper had it, 'the quantum state cannot be interpreted statistically'. (https://arxiv.org/abs/1111.3328)
How does your approach connect with this (if it does)?
Anyway, thanks for an intriguing and eminently readable essay. I wish you best of luck in the contest!
Cheers
Jochen
