A John Bell 1964 toy model for spin

[jf117]

TL;DR Summary

Description of the toy model for spin contained in the 1964 article by John Bell

I have never been able to fully grasp John Bell toy model for spin in his 1964 article. He starts with a particle with pure spin state p. There exists a hidden variable called λ with a probability distribution given by a uniform distribution over the hemisphere λ⋅p > 0. I guess this uniform distribution is ρ(λ)=1/(4π) if λ⋅p > 0, and 0 otherwise. Then it is written that the result of a measurement of spin along a direction a is sign λ⋅a' where I have absolutely no idea from where the a' come. I am missing something evident, but cannot understand what. Can anyone help me or point me to some in-depth and detailed reading on this toy model? Thanks.

 

[.Scott]

I believe the article you are referring to is this one.
That tick notation is described from equations 4 and 7. It is only used when a fixed (ie, "pure") particle polarization is presumed.

The sign λ⋅a' is the result of measurement using a hidden variable λ. I would call it a "straw horse" in contrast to a "toy model". It is a hidden variable that works perfectly well so long as you don't measure the entangled particle. But as Bell shows later in the article, it fails to account for the combined measurement results of both particles.

To be specific: sign λ⋅a' is the sign (ie, plus or minus) of the dot product of the hidden variable (λ) and the a specif angle between a and p.

 

[jf117]

Thank you for your comment, @.Scott. It says a' depends on a and p in a way to be specified. Fine, as it is to be specified, I can live with that until a' is used. But then <σ⋅a> is calculated and the result depends on θ', which is the angle between a' and p. Perhaps knowing how that result (equation (5) in Bell article) is obtained can shed some light on a'.

 

[.Scott]

The measurement angles at detectors A and B are a and b respectively. Given a known particle polarization angle, once a measurement angle is selected, the corresponding a' and b' can be computed. But this is only valid when the particle has a "pure spin state" - that is, it is not entangled.

As a note: during an actual experiment, the measurement angles are commonly selected from three values: roughly -15°, 0°, and +15°.

The final experiment being described involves many measurement with different a/b combinations. And each a and b is best chosen randomly an instant before the measurement is made.

Here's that sequence:
1) A pair of entangled particles are targeted at two measurement stations.
2) After it is too late to signal the other measurement station, each station randomly picks a measurement angle (a or b).
3) The measurements are made.
4) The experiment is repeated at least a few hundred times.
5) Those statistics are consistent with QM predictions, but they confound any attempt to use the "sign λ⋅a'" model (or any other model using a function of only a λ and the a's) to explain the result.

It says a' depends on a and p in a way to be specified. Fine, as it is to be specified, I can live with that until a' is used. But then <σ⋅a> is calculated and the result depends on θ', which is the angle between a' and p. Perhaps knowing how that result (equation (5) in Bell article) is obtained can shed some light on a'.

I have corrected my original post. Vector a' is an angle between a and the polarization angle of a "pure" (ie, unentangled) particle. It's an angle computed using equation 5 and the paragraph that precedes equation 5 in Bell's paper.