Gordon Watson wrote:Esail wrote:Gordon Watson wrote:How then do you move, in a mathematically correct way, from Bell's 1964:(14a) to his (14b)?
Gordon
With A(b,lambda) *A(b,lambda)=1 see Bell's eq.(1) you will easily arrive from Bell's 14a at 14b.
With A(a,lambda)*A(b,lambda) <1 you will arrive from 14b at 14c provided A(a,lambda) exists.
But A(a,lambda) exists unambiguously only with noncontextual models. Thus, with a noncontextual model Bell's equation is correct but does not apply to QM.
Eugen
Thanks Eugen, I should have been more specific. We accept the truth of Bell (14a) but differ over the truth of (14b).
So please show me the (14aa), (14ab), (14ac) — say, if you need 3 steps — that YOU use to arrive at (14b).
.
Gordon:
Bell defines A(a, lambda) to be the result “A” of measuring the observable “sigma_1 cdot a” when, just before measurement, the state of particles and measurement devices is lambda. That result is either +1 or -1. It’s a number.
“sigma_1 cdot a” is a self-adjoint operator on a Hilbert space, represented mathematically by a 2x2 complex matrix.
The derivation from (14) to (15) is irrefutable, elementary. After some preparatory explanation, Bell assumes given: a set Lambda of points lambda endowed with a probability measure rho, a function A taking the values +/-1, and he defines P(a, b) = - int d lambda rho(lambda) A(a, lambda) A(b, lambda).
In the language of probability theory, he has assumed existence of a collection of random variables A_a on a single probability space, taking values in {-1, +1}, and studies the expectation values P(a, b) = E( A_a A_b). He considers only three values of “a” so we are talking about three binary variables. There are 2x2x2 = 8 elementary events in play, by which I mean subsets of Lambda on which A_a, A_b and A_c take on each if the 8 possible combinations of values.
His inequality is an exercise to the reader in Boole (1853). Boole shows that the 6 one-sided Bell inequalities (I.e., single inequalities without an absolute value sign) are necessary and sufficient for existence of 8 non-negative probabilities adding up to 1 which reproduce the three expectations of products P(a, b), P(a, c), P (b, c).
P(a, b) - P(a, c) - P(b, c) is not greater than +1
P(a, b) - P(a, c) - P(b, c) is not less than -1
P(a, c) - P(b, c) - P(a, b) is not greater than +1
P(a, c) - P(b, c) - P(a, b) is not less than -1
P(b, c) - P(a, b) - P(a, c) is not greater than +1
P(b, c) - P(a, b) - P(a, c) is not less than -1
Eugen:
Bell defines A(a, lambda) to be the result “A” of measuring the observable “sigma_1 cdot a” when, just before measurement, the state of particles and measurement devices is lambda. That result is either +1 or -1. It’s a number. Bell’s model is as contextual as you could possibly desire. It is “local”. Bell assumes A(a, lambda), not A(a, lambda; b). The latter would be contextual, taking the whole system of both measurement devices to be the context for one joint measurement depending on a joint setting (a, b).