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It's not all hidden-variable models that are proven to not work to be honest. E.g., the Bohmian interpretation of non-relativistic QM is a deterministic non-local interpretation. There are no hidden variables thought. The well-known ones are sufficient ;-)).georgir said:At least intuitively any model involving randomness can be replaced with a deterministic model with extra variables controlling that randomness. Not knowing those variables can be called "incomplete understanding", though I think I'd be content and even consider our understanding complete for such models where certain variables are not even theoretically knowable, as long as their effect is well defined.
Problem with QM is that even such hidden variable models are proven to not work. For me my "incomplete understanding" stems not from the randomness itself, but from the way the random outcomes under different parameters are related. But I guess this is a whole other topic.
Only any local deterministic hidden-variable theory, as defined by Bell, is ruled out by the may Bell tests done up to now. All demonstrate the violation of Bell's inequality with astonishing significance and confirm the predictions of QT. The problem with non-local deterministic HV theories is to formulate them in accordance with (special) relativity. That's the reason, why Bohmian QT is not (yet?) satisfactorily formulated for relativistic QFT. It's of course not clear, whether there's some non-local deterministic HV theory consistent with relativity. At least there seems to be no proof for such a no-go theorem. On the other hand up to now nobody has found any such non-local theory yet.