Jason Cawley
Wolfram Science Group
Phoenix, AZ USA
Registered: Aug 2003
Posts: 712 
You posit (guess) any underlying variable's existence and some formal way it behaves (e.g. an unobservable state vector and a wave equation), with some relation, potentially quite complicated or "lossy", to any overlying observable or set of them (e.g. some probabilistic map from state vectors to phenomenal states of detectors or event counters, perhaps sensitive to phase). Then you formally deduce within your model what the underlying variable does (solve underlying equations in some situation). It predicts sets (e.g. sequences perhaps) of observables (event counts e.g., and how they vary with distances or energies).
Then you look at real sets or sequences of observables and see if they "track". If they don't, you deduce something was wrong with your guess and guess again. If they do, you can't be sure but stick with your guess until something better comes along. We say, you have related one set of observables (system parameters you solved your equation for, meaning physical sources, positions of detectors, etc) and another set of observables (event counts) via a hidden variable model (state vectors and phases nobody has ever touched or seen, just posited)  updating in this formally defined way, aka according to this law.
When we speak of a hidden variable model in the specific context of QM, we usually mean in addition, some theory about underlying real constituents or relations that cause the "manyvaluedness" seen in results of QM experiments. That is, we set up what we think is the same experiment, and get a dispersion of results rather than the same thing every time. You might think that is some objective chance function in the underlying law. Or you might think there is an additional hidden variable differentiating between the cases that come out A and the cases that come out B. The latter is then a hidden variable explanation of the remaining manyvaluedness of the (previous) theory.
There are theorems that show no classically local (no action at a distance) hidden variable theory is consistent with QM observations. Which means hidden variable explanations of the remaining variance are possible, if and only if they give up classical locality as a model property.
In other words, some QM effects can't have a hidden *local* cause (meaning, all effects of that cause are restricted in space and time to the distance a light signal could travel from one observed result to another), but they can have a hidden cause. Indeed, since these instances typically arise from entanglements where both observables depend on a prior event, it is natural to think of both observations as "symptoms" of some underlying common fact. The set of things casually dependent on that fact just can't be localized in space. It is intrinsically extended.
The alternative is simply to view QM transitions (or world to world transitions for manyworlds QMers, which just shifts the problem without explaining it) as inherently multivalued or objectively random. They still aren't independent  nonlocal correlations (if not causes) still appear  that is observable, so no theory can get rid of that degree of nonlocality.
Probably more than you meant to ask about. FWIW.
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