Bell's equalities

[Heinera]

For one thing, why are there no function B(...) on LHS? Have you completely forgotten about Bob?

Please take a look at Bell's paper again. Page 406.

What paper? Bell's original paper goes from pp 195-200.

[FrediFizzx]

For one thing, why are there no function B(...) on LHS? Have you completely forgotten about Bob?

Please take a look at Bell's paper again. Page 406.

What paper? Bell's original paper goes from pp 195-200.

I think he means page 198.

https://journals.aps.org/ppf/pdf/10.110 ... zika.1.195
.

[Heinera]

But this is of course not how the correlations are computed in experiments. Converting the integral into a sum and replacing with is meaningless, since and are two completely different things.

PS: But I retract my claim that the RHS in 8' can't be empirically measured. There is in fact a way to measure it, since we have .

[minkwe]

For one thing, why are there no function B(...) on LHS? Have you completely forgotten about Bob?

Please take a look at Bell's paper again. Page 406.

What paper? Bell's original paper goes from pp 195-200.

http://www.drchinese.com/David/Bell_Compact.pdf

[minkwe]

But this is of course not how the correlations are computed in experiments. Converting the integral into a sum and replacing with is meaningless, since and are two completely different things.

This is incorrect. The integral is converted into an average. The integral is simply a probability weighted average, which is exactly the same as the population average, which should be approached by the expression as N approaches infinity.
To see this, You can replace each distinct value with value with repetitions of that value, and since the integral of ,



as N approaches infinity. This is precisely the frequentist definition of probability/relative frequency which applies to experimental measurements.

[Heinera]

While gives perfect meaning as a mathematical integral, your RHS interpretation of it (by replacing with ) is meaningless empirically, since in experiment each particle is only measured once. The expression makes no sense. To get a meaningful empirical expression you have to first use the identity .

[minkwe]

While gives perfect meaning as a mathematical integral, your RHS interpretation of it (by replacing with ) is meaningless empirically

What you are saying makes no sense to me. I did not replace with , I replaced the probability weighted integral with the long running average. Surely you aren't arguing that there is a difference. If you insist, please write down the mathematical expression for how is calculated from experimental data.

The expression makes no sense. To get a meaningful empirical expression you have to first use the identity .

makes perfect sense. Have you looked at the equations in Bell's paper, I pointed you to? There are no B functions there, the identity is already applied long before we arrive at this expression. That is why the integral has a negative sign in front. See my equation (7) above.

is just the experimental outcome from the i-th iteration when the setting was , and is the same thing. If you like, you can express them as . Again, please write down the mathematical expression for how is calculated from experimental data.

I don't understand your concern about the average vs the integral.



Do you agree that this expression represents the expectation value of the paired product of outcomes at the setting pair (a, b)?

Is not also the case that experimentally the expectation value of the paired-product of outcomes at the setting pair (a, b) as is in fact



Where is the i-th outcome at setting (a) and is the i-th outcome at setting b?

Perhaps if you write down the mathematical expression for how is calculated from experimental data, I will be able to see what your objection is.

[Heinera]

Perhaps if you write down the mathematical expression for how is calculated from experimental data, I will be able to see what your objection is.

With settings a for Alice's station and b for Bob's station, the calculation from experimental data is

P(a, b) = [(Number of same results) - (Number of different results)]/(Total results)

The objection is of course that empirically you need to use results from both stations to estimate a correlation.

But this thread is derailing now. Back to topic: After a few attempts you derived the inequality

8'.

Okay, I give up, obviously do not have the time to focus on this. 8 should be absolute values not square brackets therefore 9 does not follow from 8 but this is irrelevant to the point I wanted to make. QM should easily violate 8'. Right?

So let me continue with this: What was the point you wanted to make with this inequality?

[minkwe]

With settings a for Alice's station and b for Bob's station, the calculation from experimental data is

P(a, b) = [(Number of same results) - (Number of different results)]/(Total results)

Please that is not a mathematical expression. I want to see symbols representing experimental outcomes at each station. Besides, that bears no resemblance to Bell's equation (14)



Please explain step by step, how you go from this expression to the one you suggested. There is no difference in that integral, explain how you arrive at that expression.

The objection is of course that empirically you need to use results from both stations to estimate a correlation.

That objection makes no sense because results from both stations are in-fact used in all the expressions I gave. Are you now saying you have the same objections about Bell's equation (14)?

[Heinera]

With settings a for Alice's station and b for Bob's station, the calculation from experimental data is

P(a, b) = [(Number of same results) - (Number of different results)]/(Total results)

Please that is not a mathematical expression.

Of course it's a mathematical expression.

Besides, that bears no resemblance to Bell's equation (14)

Who said it should bear any superficial resemblance to Bell's equation (14)? You asked how the correlations were calculated in experiments.

The objection is of course that empirically you need to use results from both stations to estimate a correlation.

That objection makes no sense because results from both stations are in-fact used in all the expressions I gave. Are you now saying you have the same objections about Bell's equation (14)?

Can you then explain to us how we are supposed to figure out that with the expression "results from both stations are in-fact used"? What kind of crazy notation is that?

[minkwe]

I give up, you are not making any sense.

Edit: Just in case there is any lingering confusion:

Bell's equation (14)

is an expectation value of the product of outcomes at two stations. That is the meaning of the above expression. An expectation value can be expressed with respect to any probability measure over the same event space. The expression above uses just one such measure .
Experimentally, the above integral can be estimated as the probability weighted average of the product of outcomes at two stations as the number of iterations approach infinity. The probability used does not have to be . It can be any valid probability measure over the space of events.

Obviously, the uniform probability measure is a valid measure over the space of N events. therefore, the expectation value of the product of outcomes at both stations can also be estimated as



Where is just the i-th outcome at one station when the setting is , and is the i-th outcome at the opposite station when the setting is . This is uncontroversial.

Another valid measure is the measure defined on the set of outcome pairs
Thus you could estimate





But and , ... etc where is the number of {+-} outcome pairs.
and and therefore,




Note that and which gives us

[Joy Christian]

***
My preferred notation would have been , where is a hidden variable for the run of the experiment.

***

[Heinera]

***
My preferred notation would have been , where is a hidden variable for the run of the experiment.

***

Bravo!

[FrediFizzx]

***
My preferred notation would have been , where is a hidden variable for the run of the experiment.

***

And of course the indexing follows for a and b. So Bell ends up with the inequality,

,

without any indexing. So the current argument by the Bell fans is that a, b and c don't have to happen all at the same time. But that is obviously wrong.
.

[FrediFizzx]

***
My preferred notation would have been , where is a hidden variable for the run of the experiment.

***

And of course the indexing follows for a and b. So Bell ends up with the inequality,

,

without any indexing. So the current argument by the Bell fans is that a, b and c don't have to happen all at the same time. But that is obviously wrong.

Back more on topic here. So if the pairs are independent like the Bell fans want to do, then the equality can be,

,

which of course then we have a higher bound of 3 for the inequality.
.

[Joy Christian]

***
My preferred notation would have been , where is a hidden variable for the run of the experiment.

***

And of course the indexing follows for a and b. So Bell ends up with the inequality,

,

without any indexing. So the current argument by the Bell fans is that a, b and c don't have to happen all at the same time. But that is obviously wrong.

Back more on topic here. So if the pairs are independent like the Bell fans want to do, then the equality can be,

,

which of course then we have a higher bound of 3 for the inequality.

Precisely. Without the assumption of "happen all at the same time", the stringent bound of 1 (or of 2 in the CHSH case) simply cannot be derived. That should be completely obvious even to a school child. And I suspect that Bell believers know this. So what can they do to fool the world? What they do is obfuscate this simple fact by invoking probability and statistics. In doing so they overplay (perhaps unwittingly) the EPR criterion of reality. See footnote 3 in my argument for a homely example: https://arxiv.org/pdf/1704.02876.pdf.

***

[FrediFizzx]

And of course the indexing follows for a and b. So Bell ends up with the inequality,

,

without any indexing. So the current argument by the Bell fans is that a, b and c don't have to happen all at the same time. But that is obviously wrong.

Back more on topic here. So if the pairs are independent like the Bell fans want to do, then the equality can be,

,

which of course then we have a higher bound of 3 for the inequality.

Precisely. Without the assumption of "happen all at the same time", the stringent bound of 1 (or of 2 in the CHSH case) simply cannot be derived. That should be completely obvious even to a school child. And I suspect that Bell believers know this. So what can they do to fool the world? What they do is obfuscate this simple fact by invoking probability and statistics. In doing so they overplay (perhaps unwittingly) the EPR criterion of reality. See footnote 3 in my argument for a homely example: https://arxiv.org/pdf/1704.02876.pdf.

***

Nice example. It blows my mind that they claim QM or the experiments "violate" Bell with independent pairs. How the heck has this nonsense gone on for over 50 years???????????????
.

[Joy Christian]

And of course the indexing follows for a and b. So Bell ends up with the inequality,

,

without any indexing. So the current argument by the Bell fans is that a, b and c don't have to happen all at the same time. But that is obviously wrong.

Back more on topic here. So if the pairs are independent like the Bell fans want to do, then the equality can be,

,

which of course then we have a higher bound of 3 for the inequality.

Precisely. Without the assumption of "happen all at the same time", the stringent bound of 1 (or of 2 in the CHSH case) simply cannot be derived. That should be completely obvious even to a school child. And I suspect that Bell believers know this. So what can they do to fool the world? What they do is obfuscate this simple fact by invoking probability and statistics. In doing so they overplay (perhaps unwittingly) the EPR criterion of reality. See footnote 3 in my argument for a homely example: https://arxiv.org/pdf/1704.02876.pdf.

Nice example. It blows my mind that they claim QM or the experiments "violate" Bell with independent pairs. How the heck has this nonsense gone on for over 50 years???????????????

"Extraordinary groupthink leads to extraordinary ignorance": https://blogs.scientificamerican.com/ob ... ore-likes/

***

[gill1109]

My preferred notation would have been , where is a hidden variable for the run of the experiment.

And of course the indexing follows for a and b. So Bell ends up with the inequality,

,

without any indexing. So the current argument by the Bell fans is that a, b and c don't have to happen all at the same time. But that is obviously wrong.

Back more on topic here. So if the pairs are independent like the Bell fans want to do, then the equality can be,

,

which of course then we have a higher bound of 3 for the inequality.

Precisely. Without the assumption of "happen all at the same time", the stringent bound of 1 (or of 2 in the CHSH case) simply cannot be derived. That should be completely obvious even to a school child. And I suspect that Bell believers know this. So what can they do to fool the world? What they do is obfuscate this simple fact by invoking probability and statistics. In doing so they overplay (perhaps unwittingly) the EPR criterion of reality. See footnote 3 in my argument for a homely example: https://arxiv.org/pdf/1704.02876.pdf.

There is no assumption that different measurements are made at the same time. The assumption of Local Realism is that a random realisation of is formed. Then, *if* Alice chooses setting a, she gets to observe the outcome A(a,). Similarly for Bob. But Alice and Bob are free to choose any settings they like, or even, to do nothing at all - they can switch off the *detectors* and go home early and watch TV ...

Then there is the no-conspiracy assumption that the average value of A(a,)B(b,) over many, many hypothetical repetitions is the same - up to statistical variation - as the average value of A(a,)B(b,) over only those repetitions in which Alice did choose a, and Bob did choose b

[Joy Christian]

There is no assumption that different measurements are made at the same time.

It is impossible to derive the bound of 2 on the CHSH correlator without that assumption. I am well aware that Bell's followers are blind to that assumption. But some in this forum are not.

***