Doubt in hidden variable concepts

In summary, the conversation is about the concept of hidden variables in quantum mechanics and how it relates to the Bell inequality. The equation (1) and (4) are discussed and it is concluded that the key to understanding hidden variables is the inequality | P(a, b) - P(a, c) | ≤ 1 + P(b, c). The conversation also mentions the inconsistency between local hidden variables and quantum mechanics at certain angle settings. The conversation ends with a welcome to PhysicsForums.
  • #1
fpaolini
4
0
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) = [itex]\mp[/itex] 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.
 
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  • #2
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.
 
  • #3
fpaolini said:
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!
 

FAQ: Doubt in hidden variable concepts

1) What are hidden variables in science?

Hidden variables refer to hypothetical, unobservable quantities that are used to explain the behavior of complex systems in science, particularly in quantum mechanics. These variables are not directly measurable and are used to fill in gaps in our understanding of how these systems work.

2) Why is there doubt surrounding hidden variable concepts?

There is doubt surrounding hidden variable concepts because they are not directly observable and cannot be tested or proven. Some scientists argue that using hidden variables goes against the principles of scientific inquiry, which relies on empirical evidence and observable phenomena.

3) How do hidden variables relate to determinism?

Hidden variables are often used in theories of determinism, which suggest that the outcomes of events are predetermined and can be explained by a set of underlying causes. In this context, hidden variables are seen as the underlying causes that determine the outcomes of events.

4) Are hidden variables accepted by the scientific community?

There is no consensus within the scientific community about the validity of hidden variables. Some scientists argue that they are necessary to explain certain phenomena, while others reject their use altogether. The debate surrounding hidden variables is ongoing and continues to be a topic of discussion among scientists.

5) Can hidden variables be proven or disproven?

Since hidden variables are not directly observable, they cannot be proven or disproven in a traditional scientific sense. However, some scientists argue that their effects can be indirectly observed and used to support or refute theories that involve hidden variables. Ultimately, the existence of hidden variables remains a matter of interpretation and debate within the scientific community.

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