FrediFizzx wrote:Yeah, I don't know why Gordon keeps harping about some error in the derivation of the inequality. There is no error in the inequality itself as it is mathematically proven. The error arises when you say a, b and c can happen all at the same time. They can't. It is that simple.
.
Indeed, the inequality is trivial.
But nobody says that
a,
b and
c "happen" at the same time.
a,
b and
c are the names of three unit vectors. Alice and Bob are experimenters who may choose which of these three vectors to use as a setting in each of their apparatus. In any one trial of the series of experiments, Alice chooses one of the three, and Bob chooses one of the three.
Repeat many, many times. There are nine possible combinations. Suppose each combination occurs many, many times. Suppose that each time, anew, nature picks a value lambda of the hidden variable, according to the same probability distribution with probability density rho. Then the average of the product of the outcomes on each side, +/-1, when Alice's setting is
a and Bob's is
b, converges to the integral of A(a, lambda)B(b, lambda) rho(lambda) d lambda.
Not only is Bell's inequality an elementary inequality which can be proven in a hundred different ways and is known under many different names, but the theorem that quantum mechanics is incompatible with local realism is an elementary theorem which also has many different proofs.
I repeat: *nobody* says that those three settings happen at the same time. Any pair can happen at the same time (in either order, if they are different), many times, if we do independent repetitions whereby each time nature picks a value lambda according to the same fixed probability distribution rho (which, please note, does not depend on the settings taken by Alice and Bob).
In conventional probability theory, a real-valued random variable X is represented by a deterministic function X defined on a set Omega and taking values in the set of real numbers. The idea is that Fortuna, Goddess of Chance, picks a value omega in the set Omega. The random variable X then takes on the value x = X(omega). Fortuna's choices are ruled by a probability measure on Omega. The chance that omega is in some subset A of Omega is called P(A) and the function P (taking values in the closed interval [0, 1]) satisfies a little list of rules, or axioms, which I won't repeat here.
You are allowed to think of probability in any way you like. Some people like a subjective interpretation ("degree of belief"), some people like the so-called frequentist interpretation - limiting frequency in many independent repetitions. There is a huge literature about these interpretations and also about other interpretations, and for that matter, about alternative systems of axioms to the Kolmogorov axioms.
In Bell's paper he was thinking about lambda as standing for some kind of complete microscopic description of source, particles, and detectors which makes everything that happens deterministic. The initial conditions. Every time the experiment is repeated the initial conditions can be different, and that is why (with the same settings) the detectors give different outcomes. Think of statistical mechanics.