Doubt in hidden variable concepts

[fpaolini]

Hi, I am trying to read the chapter 2 of the book "Speakable and Unspeakable in Quantum Mechanics" from J. Bell. I have got some doubts and I would be glad if someone could help me.

I see that λ is a vector that makes the role of the hidden variable. All the tricks about the angles between a, a' and λ I think that I could understand.

Now some doubts:

The equation (1)

A(a, λ) = \mp 1

means that A may take just 1 or - 1? So is A not an average, but a measure itself ( like spin 1/2 or - 1/2), right?

In the equation (4)

sign λ.a'

Is it means sign( λ.a' ) ? And in this case the author is just saying that A(a, λ ) = sign( λ.a' ) ? And so is it in that sense that λ would define the result of the measure? If λ.a' > 0 the measure will show the spin up, otherwise spin down.

That is all for a while.
I will go on reading the rest of the chapter.
Thanks in advance.

 

[fpaolini]

That is me again.

After some time studying the subject I made some improvement over my ideas.

At the start of my study I have spent some time trying to understand an illustration about the concept of hidden variables ( The section 3 of chapter 2 of the "Speakable and Unspeakable in QM" from J. Bell).

Now I see that the inequality

| P(a, b) - P(a, c) | ≤ 1 + P(b, c)

is really the key of the concept.

For example, in a system with two particles with 1/2 spins, if the wave function representing the singlet state could represent a state where the two particles were already in a defined state ( after the two particles are enough apart ) then it should be possible to associate a set of hidden variables to each of these states.
However, as some quantum mechanics correlations does not respect the above inequality then we are forced to conclude that it is just after a measure has been done that the spin state of each of the particles can be defined.

After all, now I thing I finally found the general ideas behind Hidden variables.

 

[DrChinese]

That is me again.

After some time studying the subject I made some improvement over my ideas.

At the start of my study I have spent some time trying to understand an illustration about the concept of hidden variables ( The section 3 of chapter 2 of the "Speakable and Unspeakable in QM" from J. Bell).

Now I see that the inequality

| P(a, b) - P(a, c) | ≤ 1 + P(b, c)

is really the key of the concept.

For example, in a system with two particles with 1/2 spins, if the wave function representing the singlet state could represent a state where the two particles were already in a defined state ( after the two particles are enough apart ) then it should be possible to associate a set of hidden variables to each of these states.
However, as some quantum mechanics correlations does not respect the above inequality then we are forced to conclude that it is just after a measure has been done that the spin state of each of the particles can be defined.

After all, now I thing I finally found the general ideas behind Hidden variables.

You have done a good job going through the material! Yes, the above is really the key equation. This relates the outcomes of measurements at 3 angle settings, a b and c. I think it is best to imagine those settings related to ONE single photon. Then you can see that there must be predetermined outcomes for an infinite number of angles IF you believe there are local hidden variables. But 3 is enough to get the Bell result. Clearly, Bell shows us that there are inconsistencies at some angle settings. 0, 120 and 240 degrees for photons is the easiest one to follow. You cannot have outcomes for these angles that will also be consistent with local observer independence AND the predictions of QM (in this case a 25% match rate).

By the way, welcome to PhysicsForums!