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Einstein insisted that nature is locally real, which is also know as counterfactual defeniteness and means that results of experiments are predetermined. But, if everything is predetermined in every region of space(-time), if everything is predetermined by some hidden variables, then we do not have any room for stochastism (randomness/free will) anywhere. This means that we have superdeterminism -- our space-time is static and everything is know in advance. Is it right?

I mean that if result of your measurement is predetermined by the region of space you are located in, then the property you are chosen to measure is also predetermined by your region properties. You cannot have free will for yourself and determinism for the object that you measure.

I am asking because the famous Bell theorem is known to be inappropriate in case of superdeterminism. Yet, it is used to rule out the local realism. But how can you do that if local realism implies the superdeterminism?

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You are right that Bell's inequalities do not rule out "superdeterminism" (Bell's term), as he himself acknowledged:"...if our measurements are not independently variable as we supposed...even if chosen by apparently free-willed physicists... then Einstein local causality can survive. But apparently separate parts of the world become deeply entangled, and our apparent free will is entangled with them". Bell's inequalities are instead conditional: if experimenters are free to choose which experiments they perform then the outcomes of those experiments as predicted by quantum mechanics (and confirmed in labs) are inconsistent with local realism. Testing determinism requires assuming indeterministic freedom. So at first glance the Bell's inference appears circular, as Brown puts it in Von Neumann's Postulate and Bell’s Freedom:

"He proved, assuming the predictions of quantum mechanics are valid (which the experimental evidence strongly supports), that not all events can be strictly consequences of their causal pasts, and in order to carry out this proof he found it necessary to introduce the assumption that not all events are strictly consequences of their causal pasts!"

But all is not as circular as it appears. The catch is that even the strictest of mechanical determinists typically applied their determinism to the experiments they conduct, not to their ability to conduct them, even though their determinism dictates that they may not be able to conduct them at will. If they applied their determinism consistently to themselves as well (which is what Bell calls "superdeterminism"), then it becomes not only empirically unfalsifiable, but also vacuous: no counterfactual can ever be tested because all setups are pre-arranged. So what Bell's inequalities bring into sharp relief is that one can not be a determinist about nature without also being a determinist about their ability to "interrogate" it, in Galileo's word. And that determinism is highly implausible to a point of fantastic for the reason mentioned by Bell, it requires "deep entanglement", fine tuned correlations between vastly distant regions of the universe, correlations that eventually force spatially remote experimenters to make one measurement rather than another. In practice (that is applied to anything other than dubiously to the universe as a whole), environmental decoherence will quickly wipe out any such correlations.

The irony is that it was quantum "spooky action at a distance" that originally motivated Einstein's demand of local realism, yet in view of Bell's inequalities he'd have to accept something far spookier to keep it. But the good news is that while Bell's inequalities do not rule out superdeterminism, they do not mandate it either, they are perfectly consistent with measurement choices being undetermined, along with their outcomes. And that is by far a more plausible interpretation.

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There are two real problems with the notion that deterministic physics means determinable outcomes:

1) If the functions that determine the behavior of everything involve feedback, and are nontrivial in complexity, they will exhibit chaotic dynamics.

2) We cannot distill all of the contributing information that leads to a current state. Nor can we determine the exact current state itself without destroying that state. But the details below our ability to measure remain.

Randomness does not just decrease over time as things settle into consistent patterns, as traditional mechanistic thinking once imagined. Nor does it "play itself out": entropy is not a form of energy, it is a different thing. Our inability to predict outcomes, especially at small scales, spreads and recombines with itself. And it tends to increase.

Chaotic dynamics shows how very small changes can have disproportionate effects. So there is a pervasive basic randomness on a small order that cannot be removed, just pushed around. And we cannot assume all of its effects will be small.

So there is a large gap between the idea that "things are determined", and the idea that "there is no randomness". The outcomes of experiments may be as determined as their inputs. But at some scale, those inputs are not determined either.

The remaining uncertainty leaves plenty of room for free will up and down the scales of behavior, even if the rules are perfectly obeyed at every point in space. Determinism may be theoretically true, but irrelevant.

  • How is it irrelevant when we speak about the real world, not your illusion about it? When Einstein says that the outcome is predetermined he means that the world is absolutely predictable, despite you cannot measure below Plank scale and demonstrate happiness about such "free will". – Valentin Tihomirov Apr 7 '16 at 5:59
  • If there is randomness from the start, then there is randomness forever, not merely what we do not know, but real randomness -- irreversible in nature due to entropy. If Einstein says that things can be determined absolutely by any actually existing or future being or process, he is denying thermodynamics. If not, then declaring it 'predictable' when neither we nor any other creature can actually ever do the prediction is just a word game without content. – jobermark Apr 7 '16 at 17:18
  • What this has to do with thermodynamics? Einstein says a simple thing: thermodynamics is a statistical theory. It is incomplete. The latest video from Susskind support this view. The elementary physics, the QM is reversible, he says. It does not matter whether there is a randomness from the start as long as we care about determinism. "Word game without content"? Brilliant. Reduce your view on EPR debate to that single sentence. We need your opinion. – Valentin Tihomirov Apr 7 '16 at 17:36
  • It is not my turn of phrase -- it is Wittgenstein's. There are language games that have content and word-games that only seem to have content. This is one of the latter, I think. The whole topic seems to be 'underhanded God-talk'. What cannot be determined by anyone, is not determined. And the only player here besides everyone bound by physics is God. – jobermark Apr 8 '16 at 16:14
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The term local realism is used in practise to mean the following: any measurable quantity has a single measurable value at any time, and the single value of that quantity is determined by local equations of motion. Bell's theorem explains that any theory that satisfies this constraint is non-local.

You could imagine that the entire history of the universe was laid out in advance and includes outcomes that violate Bell's theorem, but these take place in a way that satisfies local realism. However, this violates another principle of science: the evolutionary principle that knowledge only arises by processes involving variation and selection among replicators. The point of Bell-type experiments is to come up with a situation that tests local realism versus quantum mechanics. If the laws of nature are conspiring to thwart such tests, then the evolutionary principle is false. Not only was no variation and selection required to achieve this conspiracy, it was not even necessary for the relevant physical processes to use any information storage to bring about this outcome, so there were no replicators either. There is a shorter way to summarise superdeterminism: shit happens. It doesn't explain the outcomes, or even propose a hint at any explanation, it just says shit happens.

There is a local explanation of the Bell inequalities that is realistic in the sense that it accepts the existence of an objective reality: it's called quantum mechanics. In quantum mechanics, meaaurable properties of systems are not characterised by single numbers, but, rather, by Hermitian operators. There is an entirely local explanation of how the correlations arise in terms of properties of systems represented by such operators. For an explanation of how the correlations arise, see

http://arxiv.org/abs/quant-ph/9906007

and

http://arxiv.org/abs/1109.6223.

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