I see. Nevertheless, I find it desirable to have a pure mathematical proof with clearly stated assumptions and without the need for physical input. After all, it's a purely mathematical question, whether there can exist a classical relativistic field theory that can reproduce quantum mechanics or not. There must be a purely mathematical answer to it. I think we shouldn't be satisfied below this level of rigor.My intention was to avoid the need to prove theorem 1'. The idea was that Herbert's proof avoids probabilities so there is no need to for theorem 1'. And the third assumption applies to particles and fields alike so there is no need for separate theorems if other two assumptions hold as well.
Yes, one must of course restrict to local observables. I find it non-trivial that no observables can possibly be constructed from relativistic fields that show non-local Bell violating correlations.I do not follow you. If do not we restrict any source of changes to past light cone then the changes propagate FTL (or retrocausally). This violates SR.
And I suppose there are observables that can be constructed from fields in past light cone. So the theorem 1' in general sense could not be proved. So we should consider only entangled state observables (conditional observations).