The role of lambda in Bell (1964) and experiments

[Gordon Watson]

Moved from https://www.physicsforums.com/showthread.php?t=590249&page=3 to avoid confusion with the classical example in its OP. ThomasT and I are mainly discussing Bell (1964) here.

The individual outputs will be either that a detection has been registered, or that a detection hasn't been registered. You can denote that however you want, but the conventional notations are +1,-1 or 1,0, corresponding to detection, nondetection, respectively.

The example discussed relates to 2 spin-half particles in the original EPR-Bohm example, see Bell (1964). The outcomes are spin-up or spin-down. The typical notations then are +1 and -1. But in trying to sort out any confusion, imho, it helps to maintain the detector orientations and the orientations of the outcomes in your analysis. So a+ [= +1] is a spin-up output for Alice with her detector in the a direction; b- [= -1] is a spin-down output for Bob with his detector in the b direction; etc.

I don't know what you mean by the full physical significance of θ. θ just refers to the angular difference between the polarizer settings, afaik.

The angle between any Alice-Bob output combinations may also be expressed as a function of θ; see earlier example involving ∏. You seem to miss this important point?

I don't know what this means. The ab combinations are θ. I don't have any idea what the a+b- stuff means or where ∏ comes into it.

The ab outcome combinations are a+b+, a+b-, a-b+, a-b-. The angle between the outputs a+ and b- is θ + ∏; etc.

Well, I don't think I'm confused. P(A,B) is a function that refers to the independent variable θ. And, in the ideal, wrt optical Bell tests, P(A,B) = cos²θ.

How does this show that you are not confused?

Of course it's obvious. Because, in the ideal, this is the QM prediction. Rate of coincidental detection varies as cos² θ.

Well cos²θ in some experiments; other functions of θ in others.

The relation of λ to A is denoted as P(A) = cos² |a-λ| .

This is wrong; a big misunderstanding. This does not hold in entangled experiments. It would hold if λ denoted a polarisation but entangled particles are unpolarised (quoting Bell).

As I said, I don't think you understand what I'm saying. Namely, that the underlying parameter that determines rate of individual detection is not the underlying parameter that determines rate of coincidental detection.

The underlying parameters λ has given up the ghost, gone, been burnt off, in the production of each output. Having done its job, it exists no more. What remains are the outputs, which may be paired in 4 combinations: a+b+, a+b-, a-b+, a-b-. The angle between the output in each pair is a function of θ, and nothing else. It follows that, depending on the source, the overall output correlation will also be a function of θ alone; θ the difference between the detector orientations.

Plant a seed (input) λ; the seed λ is not in the subsequent fruit (output) a+ = +1; etc.

 

[Gordon Watson]

Afaik, wrt optical Bell tests, λ, the hidden variable denotes an underlying polarization that's varying randomly from pair to pair.

I guess I just don't understand your treatment here. As far as I can tell it's not going to get you to a better understanding of why BIs are violated formally and experimentally, and it doesn't disprove Bell's treatment which is based on the encoding of a locality condition which, it seems, isn't, in effect, a locality condition.

And now, since I am a bit confused by your presentation, I think I will just fade back into the peanut gallery. Maybe I'll learn something.

My apologies for any added confusion. I'm happy to do this via direct email for awhile to knock off some rough edges.

WRT your: "Afaik, wrt optical Bell tests, λ, the hidden variable denotes an underlying polarisation." IMHO, if you carried this analysis through (which I encourage you to do) you will get the classical example in https://www.physicsforums.com/showthread.php?t=590249. But note that such photons are not entangled.