[Physics] Bell’s theorem and why nonlocality is problematic

bells-inequalitylocalitymeasurement-problemquantum mechanics

I generally hear it assumed that Bell's inequality implies violation of counterfactual definiteness, because locality is considered sacrosanct. I understand of course that measurable violations of locality are logically inconsistent. But what is so bad about "hidden" violations of locality? What are the reasons nonlocal hidden variable theories are frowned upon? Is it just because the ontologies currently on the table (such as de Broglie–Bohm theory) are considered kind of ugly?

Best Answer

Luboš as always gives a good account. There are many alternate accounts, however, some of which make some sense. Your question is asked in a way that suggests to me a specific type of answer.

Einstein locality of the dynamics is very well supported by experiment. If by locality you mean Einstein locality, then there are no "measurable violations of locality". On the other hand, there is no locality of initial conditions, by definition; consider, for example, a classical field that is zero everywhere in Minkowski space, or, equally nonlocally, the vacuum state of quantum field theory, which is by definition the same wherever and whenever you measure it. This kind of nondynamical nonlocality is the basis of one of the many ways of evading the derivation of Bell inequalities for random fields, which is usually pejoratively dismissed as the "conspiracy" loophole, but which, nonetheless, is there. [BTW, the conspiracy requires only a dynamically local deterministic evolution of probability distributions, not a deterministic evolution of trajectories.] Now, if you like, this is "hidden" nonlocality because it's "nondynamical", but I doubt almost anyone thinks there's anything wrong with it, insofar as we work with initial conditions all the time.

The distinction I make above between locality as a property of a dynamics and locality as a property of an initial condition is only one of many fine distinctions that have been made in the literature. Be careful how you use the word "locality".

Counterfactual definiteness has definitely been much made of in the literature on Bell inequalities for the particle case. The same idea (or perhaps it's just similar) can be put, less Philosophically, in terms of noncontextuality, the idea that one shouldn't have to say what experimental apparatus was used to measure a property. In these terms, the contextuality that is required to model Bell violating experiments classically is nonlocal in the sense that the whole measurement apparatus interacts both with the purported system that is measured and with the whole preparation apparatus, even if the system that is measured is purported to be two particles at the opposite ends of a light-years long fiber optic.

As a postscript to the above, which might or might not be a useful answer, according to your taste, the only way I have found to make this unproblematic is to take the "purported system" to be a (random) field in a coarse-grained equilibrium state (coarse-grained in the sense that the statistics of measurement events are invariant under time-like translations --in the sense, say, that they must be repeatable to get into a journal--, even though the events themselves are clearly not manifestations of a fine-grained equilibrium). Since the field is everywhere in the apparatus, and it's a commonplace that an equilibrium state is a nonlocal accommodation to whatever boundary conditions have been put in place by the experimenter, a nondynamical nonlocality is to be expected. Note too that the larger the experimental apparatus is, the longer we have to wait before we will record measurement events at the measurement apparatuses and the longer it will take to verify that the measurement events at the two ends in fact violate Bell inequalities (and the harder it will be to ensure that they do, despite the environment). Although it goes far into details that I won't expound here, particles in this view are modulations of the Vacuum Expectation Values, a generalization to the random field context of modulations of a classical field.

A large part of this approach is to apply ideas from quantum field theory as if they are signal processing mathematics. The violation of Bell inequalities can be derived for random fields only with assumptions that are not natural for a random field, whereas the assumptions required to derive the violation of Bell inequalities for classical particle models are generally deemed rather natural by most Physicists. It is, however, somewhat strange to ask classical physics not to use the resources of a random field.

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