I think that I have followed what Bell wrote, and how it is explained in Wikipedia, textbooks, and other references.
Apparently others have had the exact same dispute that I have had with the anonymous commenter. Travis Norsen, a coauthor of the Scholarpedia article I cited previously, writes in a 2008 paper:
J.S. Bell believed that his famous theorem entailed a deep and troubling conflict between the empirically verified predictions of quantum theory and the notion of local causality that is motivated by relativity theory. Yet many physicists continue to accept, usually on the reports of textbook writers and other commentators, that Bell's own view was wrong, and that, in fact, the theorem only brings out a conflict with determinism or the hidden-variables program or realism or some other such principle that (unlike local causality), allegedly, nobody should have believed anyway. ... Here we try to shed some light on the situation ...Yes, I am with those who say that Bell's theorem only presents a conflict with hidden variables or counterfactual definiteness. Others say that the theorem is stronger.
Norsen relies directly on Bell:
Here is how Bell responded to this first class of disagreement:No, his first definition in terms of light cones is much cleaner and sharper for mathematical analysis. That is the definition used in Maxwell's theory of electromagnetism, in quantum field theory, and in every other relativistic theory.
“My own first paper on this subject starts with a summary of the EPR argument from locality to deterministic hidden variables. But the commentators have almost universally reported that it begins with deterministic hidden variables.” (Bell, 1981, p.157) ...Bell’s fullest and evidently most-considered discussion of local causality occurs in his last published paper, La nouvelle cuisine (1990, 232-248). We will here essentially follow that discussion, supplementing it occasionally with things from his earlier papers.
Bell first introduces what he calls the “Principle of local causality” as follows: “The direct causes (and effects) of events are near by, and even the indirect causes (and effects) are no further away than permitted by the velocity of light.” Then, referencing what has been reproduced here as Figure 1, Bell elaborates: “Thus, for events in a space-time region 1 ... we would look for causes in the backward light cone, and for effects in the future light cone. In a region like 2, space-like separated from 1, we would seek neither causes nor effects of events in 1. Of course this does not mean that events in 1 and 2 might not be correlated...” (1990, p. 239)
After remarking that this formulation “is not yet sufficiently sharp and clean for mathematics,” Bell then proposes the following version, referencing what has been reproduced here as Figure 2:
“A theory will be said to be locally causal if the probabilities attached to values of local beables in a space-time region 1 are unaltered by specification of values of local beables in a space-like separated region 2, when what happens in the backward light cone of 1 is already sufficiently specified, for example by a full specification of local beables in a spacetime region 3...” (1990, 239-40)
What the heck are "beables", and how can anyone be sure about the probabilities?
Norsen drafted a Wikipedia article on beables in 2010:
The word beable was introduced by the physicist John Stewart Bell in his article entitled "The theory of local beables" (see Speakable and Unspeakable in Quantum Mechanics, pg. 52). A beable of a physical theory is an object that, according to that theory, is supposed to correspond to an element of physical reality. The word "beable" (be-able) contrasts with the word "observable". While the value of an observable can be produced by a complex interaction of a physical system with a given experimental apparatus (and not be associated to any "intrinsic property" of the physical system), a beable exists objectively, independently of observation. For instance, it can be proven that there exists no physical theory, consistent with the predictions of quantum theory, in which all observables of quantum theory (i.e., all self-adjoint operators on the Hilbert space of quantum states) are beables.So a beable is just Bell's notion of a hidden variable. It is not an observable, but somehow represents someone's opinion about what ought to be real.
While, in a given theory, an observable does not have to correspond to any beable, the result of the "measurement" of an observable that has actually been carried out in some experiment is physically real (it is represented, say, by the position of a pointer) and must be stored in some beable of the theory.
The mainstream interpretations of quantum mechanics say that the set of observables are what is important and real. Bell rejects this, and says that some other form of hidden variables must be what is real.
Bell also focuses on probability, as if that is something real. It is not, as I have explained here and elsewhere. It is not any more essential to quantum mechanics than to any other theory. It is a mathematical device for relating theories to the world, but it is not directly observable.
Thus when Bell defines causality in terms of beables, he is squarely and directly making an assumption about hidden variables. And that assumption contradicts the postulates, mathematical formulation, and spirit of QM.
Norsen himself is squarely in Bell's camp, as he proposed this rewrite of the Wikipedia article on Bell's theorem:
Bell's Theorem is a mathematical theorem first demonstrated by J.S. Bell in his 1964 paper "On the Einstein-Podolsky-Rosen paradox". The theorem establishes that any physical theory respecting a precisely-formulated locality (or "local causality") condition will make predictions, for a certain class of experiments, that are constrained by a so-called Bell Inequality. Since the particular theory called Quantum Mechanics makes predictions which violate this constraint, one can also think of Bell's Theorem as a proof that there is an inconsistency between (i) local causality and (ii) a certain set of QM's empirical predictions. ...This was apparently rejected because the Wikipedia editors agree with the textbook explanation of Bell's theorem, and do not accept Bell's own interpretation.
Bell's own interpretation of his theorem, however, is not widely accepted among physicists in general. Some physicists who have studied Bell's Theorem carefully point to alleged flaws or hidden assumptions in Bell's formulation of local causality and/or his derivation of (what Bell called) the Locality Inequality therefrom; such claims are controversial and will be addressed below. But most physicists fail to agree with Bell's statement above not because they think there is some flaw in the reasoning leading to it, but rather because what they have learned about Bell's Theorem (from textbooks and other sources) radically distorts the subject.
The core teachings of QM say that an electron is observed as a particle, but is not really the sort of classical particle that has a precise position and momentum at the same time. The Einstein-Bohm-Bell types refuse to accept this. They also refuse a positivist view that allows us to be silent about what cannot be measured. Instead they want to pretend to have values for that, and call them beables or hidden variable or reality or whatever sounds good. The consensus since 1930 has been that this approach does not work.
My critic says:
You seem to scrupulously avoid any cognitive engagement with the actual subject.I thought I did, but I guess he means that I avoid beables.
This is like discussing the twin paradox or some other subtle point in relativity theory, and someone objecting, "But what is the true time? You seem to avoid discussing what the real time is!"
Relativity teaches that different observers measure time differently. Defining some sort of universal real time is usually not helpful. Likewise, defining beables as the hypothetical result of unperformed measurements is usually not helpful either.
I remember taking a high school science class, and being told the history of the debate over whether light was a particle or a wave, with the arguments for each side. At the end of the course, I had a dissatisfied feeling because the teacher never told us which it was. I thought that maybe I skipped class that day, because surely one side was right and one was wrong.
No, nature does not always match our preconceptions. You have to let go of the idea that light has to match some intuitive classical model for a moving object. Relativity teaches us how clocks behave, but not what time really is. Quantum mechanics teaches us how to make and predict measurements of electrons and photons, but not what they really are.
If you want to understand electrons and photons in terms of classical joint probabilities of beables, then you will be disappointed, because nature does not work that way.
There is nothing in this Einstein-Bohm-Bell analysis but a failed attempt to prove QM wrong. The physicists who did the early Bell test experiments were convinced that they would win Nobel prizes for disproving QM. Instead they just confirmed what everyone thought in 1930.