# The role of lambda in Bell (1964) and experiments

• Gordon Watson
In summary, the conversation mainly discusses Bell's 1964 paper and the concept of hidden variables in entangled particle experiments. The outputs of these experiments are either detection or nondetection, denoted by +1/-1 or 1/0 respectively. The angle between the outputs of different detectors is denoted by θ. The parameter λ, which represents the underlying polarization, is not present in the outputs. The correlation between outputs is a function of θ, and this is how Bell's inequality is violated. However, this does not disprove Bell's treatment, which is based on the encoding of a locality condition.
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.

ThomasT said:
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.

ThomasT said:
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?

ThomasT said:
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.

ThomasT said:
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?

ThomasT said:
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.

ThomasT said:
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).

ThomasT said:
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.

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ThomasT said:
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.

## 1. What is the significance of lambda in Bell's 1964 paper?

Lambda, also known as the wavelength, plays a crucial role in Bell's 1964 paper on the concept of entanglement. It represents the distance between two entangled particles, and the experiments conducted by Bell were aimed at determining whether or not this distance had an effect on the correlated measurements of these particles.

## 2. How did Bell's experiments provide evidence for the role of lambda?

Bell's experiments involved measuring the correlation between two entangled particles at different distances from each other. These measurements were compared to the predictions of local hidden variable theories, which did not take into account the effects of lambda. The results of the experiments showed a violation of these theories, providing evidence for the role of lambda in entanglement.

## 3. What other factors, besides lambda, may affect entangled particles?

While lambda is a crucial factor in entanglement, there are other factors that may also affect the correlated measurements of entangled particles. These include the properties of the particles themselves, the measurement apparatus, and external influences such as magnetic fields.

## 4. How does the role of lambda tie into the concept of non-locality?

The concept of non-locality refers to the idea that entangled particles can influence each other's behavior instantaneously, regardless of the distance between them. Lambda plays a key role in this concept, as it represents the distance between the entangled particles and is a crucial factor in determining the strength of their correlation.

## 5. What implications does the role of lambda have on our understanding of quantum mechanics?

The role of lambda in Bell's experiments and the concept of entanglement has significant implications on our understanding of quantum mechanics. It challenges the classical notion of locality and suggests that there may be hidden variables at play in the quantum world. It also highlights the importance of considering the effects of distance in entanglement and the potential limitations of our current understanding of quantum mechanics.

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