Did Bell's Theory Contain a Mathematical Flaw?

[Gordon Watson]

I would hope this thread could be limited to Bell-mathematical questions and answers concerning just one given Bell paper -- though it may involve many mathematical questions as we follow it to resolution.

This thread concerns mathematical questions in one Bell paper and so it should NOT need a lot of words and diversion to resolve it. It is about quantum physics so I think it belongs here --- with help from mathematicians.

It might help us more if those helping us could say what level of mathematics and quantum physics they have reached. But I respect privacy.

I believe that logic is at its highest development in mathematics and probability theory -- so I should be not too slow in those areas.

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Let there be no question here about Bell's assumptions BUT
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Question 1. Is there a mathematical error in Bell's theory, as will follow?

After starting and following the threads "Understanding Bell's logic" and "Understanding Bell's mathematics", I would like to discuss and resolve a clash of the two in Bell's Bertlmann's socks paper -- which is available to all from CERN on-line -- [PLAIN]http://cdsweb.cern.ch/record/142461/files/198009299.pdfpapers [/URL] -- the discussion in those threads not appearing to resolve it for me.



One of Bell's latest papers on EPR, without excuse it should be one of his clearest?

Look at Bell's equations (11) and (12), and combine them to form

(Bell 12) = $$P(A, B|a, b) = \int d\lambda \rho (\lambda) P_ 1(A|a, \lambda) P_ 2(B|b, \lambda).$$

It is Bell's supposition (above his (11)) that the variables $$\lambda$$ allow this decoupling.

Question 2. Are $$P_ 1$$ and $$P_ 2$$ Probability Functions?


Question 3. Is it not the case that Probability Functions map a subset of the sample space to the real interval [0, 1]?


Question 4. If $$P_ 1$$ and $$P_ 2$$ are Probability Functions, how do we apply Bell's $$\rho (\lambda)$$ to such functions?

Question 5. If they are NOT Probability Functions, what are they, please?

Question 6. Could you provide an example of the Function that you believe them to be, please?

Question 7. Bell has $$P_ 1$$ and $$P_ 2$$. Why did he not have also $$\lambda_ 1$$ and $$\lambda_ 2$$ ?

Question 8. If he did the lambda-separation in Question 7 -- which is allowable under his theory -- how would he have written Bell (12) above?

Thank you very much.

 

[JesseM]

Question 1. Is there a mathematical error in Bell's theory, as will follow?

No.

One of Bell's latest papers on EPR, without excuse it should be one of his clearest?

Look at Bell's equations (11) and (12), and combine them to form

(Bell 12) = $$P(A, B|a, b) = \int d\lambda \rho (\lambda) P_ 1(A|a, \lambda) P_ 2(B|b, \lambda).$$

It is Bell's supposition (above his (11)) that the variables $$\lambda$$ allow this decoupling.

Question 2. Are $$P_ 1$$ and $$P_ 2$$ Probability Functions?

Yes.

Question 3. Is it not the case that Probability Functions map a subset of the sample space to the real interval [0, 1]?

Yes.

Question 4. If $$P_ 1$$ and $$P_ 2$$ are Probability Functions, how do we apply Bell's $$\rho (\lambda)$$ to such functions?

What do you mean by "apply"? $$\rho (\lambda)$$ is itself just a probability density function, the continuous analogue of a probability function on a discrete sample space. If λ only took a discrete set of values, labeled λ_i with i ranging from 1-N, then the integral you wrote above could be replaced by the sum $$P(A, B|a, b) = \sum_{i=1}^N P_3(\lambda_i) P_ 1(A|a, \lambda_i) P_ 2(B|b, \lambda_i).$$ The integral you wrote is just the analogue of this for a λ that can take a continuous range of values.

Question 7. Bell has $$P_ 1$$ and $$P_ 2$$. Why did he not have also $$\lambda_ 1$$ and $$\lambda_ 2$$ ?

What do you mean by λ₁ and λ₂? Are these just specific values of the random variable λ? If so, $$\rho (\lambda)$$ already assigns probability densities to each possible specific value of λ.

Question 8. If he did the lambda-separation in Question 7 -- which is allowable under his theory -- how would he have written Bell (12) above?

Unclear what you mean by "lambda-separation". $$P_1$$ and $$P_2$$ aren't based on "separating" a single random variable, rather they are based on separating a joint probability P(AB|a,b,λ) into individual probabilities for A and B. I don't see how you could do anything analogous for $$\rho (\lambda)$$.

edit: I suppose that depending on how λ is defined, and what type of local hidden variables theory we're imagining, it might be reasonable to split it into two parts λ₁ and λ₂, the first describing local hidden variables that influence the outcome of measurement A and the second describing local hidden variables that influence the outcome of measurement B. Is this the sort of thing you were thinking of? For example, if we define λ in terms of all local variables in cross-sections of the two past light cones of the measurements, with the cross-sections taken at some time after the last moment the two past light cones overlap, then λ₁ could just refer to all local variables in the cross-section of the past light cone of measurement A, and λ₂ could be all local variables in the cross-section of the past light cone of measurement B. In this case your integral could be replaced by the following double integral:

$$P(A, B|a, b) = \int \int d\lambda_1 d\lambda_2 \rho (\lambda_1, \lambda_2) P_ 1(A|a, \lambda_1) P_ 2(B|b, \lambda_2).$$

 

[DrChinese]

One of Bell's latest papers on EPR, without excuse it should be one of his clearest?

I have told you before, and I guess you will discover for yourself: Bell wrote a lot of things and he wrote in a lot of different styles. So no, you won't find much that is clear - or clearer - than the original paper. And for the original paper, like his later ones, you need to consider the audience in order to understand the argument. If you read it like an ordinary text or paper, it will not make sense.

Now, this is not an "excuse" for Bell. His paper has been cited thousands of times so I really don't think it needs much in the way of excuses. My point is that you will find it easier to follow if you consider the audience. Then the formalities won't matter so much. Also, don't forget that audience would know EPR too.

 

[Gordon Watson]

I have told you before, and I guess you will discover for yourself: Bell wrote a lot of things and he wrote in a lot of different styles. So no, you won't find much that is clear - or clearer - than the original paper. And for the original paper, like his later ones, you need to consider the audience in order to understand the argument. If you read it like an ordinary text or paper, it will not make sense.

Now, this is not an "excuse" for Bell. His paper has been cited thousands of times so I really don't think it needs much in the way of excuses. My point is that you will find it easier to follow if you consider the audience. Then the formalities won't matter so much. Also, don't forget that audience would know EPR too.

Thank you very much DrC. I do understand your view that Bell was preaching to the converted in so far as he spoke in a language clear to them -- like Danish. But my girlish intuition senses something not quite right in the state of Denmark. The mathematical point I am coming to is in all his EPR papers -- with the beauty of being the same in all languages, as in my next post here responding to JesseM's nicely detailed mathematical reply.

 

[DrChinese]

The mathematical point I am coming to is in all his EPR papers -- with the beauty of being the same in all languages, as in my next post here responding to JesseM's nicely detailed mathematical reply.

And nice it is!

The reason I mention this, when you clearly want to speak in the mathematical lingo (and excellent is that), is that you should definitely switch to definitions and representations that match how you think. Use the strongest arguments, not the weakest (as many do on this board). That way you don't go off topic because you are wandering around in the details and imagining Bell is wrong. (If you think Bell is wrong, it is simply one of your definitions that need to change.)

If you asked 10 different top scientists to derive Bell, you would probably see 10 different representations. And yet all 10, if shown the others, would agree all 10 are actually identical. They won't see the differences you might. If you go to my web site, I give 2 completely different approaches and neither match Bell directly. One follows Mermin (JesseM's scratch card example is identical), the other demonstrates negative probabilities:

Bell's Theorem and Negative Probabilities

I guess I should add another that closely follows Bell too, think?

 

[Gordon Watson]

No.

Yes.

Yes.

What do you mean by "apply"? $$\rho (\lambda)$$ is itself just a probability density function, the continuous analogue of a probability function on a discrete sample space. If λ only took a discrete set of values, labeled λ_i with i ranging from 1-N, then the integral you wrote above could be replaced by the sum $$P(A, B|a, b) = \sum_{i=1}^N P_3(\lambda_i) P_ 1(A|a, \lambda_i) P_ 2(B|b, \lambda_i).$$ The integral you wrote is just the analogue of this for a λ that can take a continuous range of values.

What do you mean by λ₁ and λ₂? Are these just specific values of the random variable λ? If so, $$\rho (\lambda)$$ already assigns probability densities to each possible specific value of λ.

Unclear what you mean by "lambda-separation". $$P_1$$ and $$P_2$$ aren't based on "separating" a single random variable, rather they are based on separating a joint probability P(AB|a,b,λ) into individual probabilities for A and B. I don't see how you could do anything analogous for $$\rho (\lambda)$$.

edit: I suppose that depending on how λ is defined, and what type of local hidden variables theory we're imagining, it might be reasonable to split it into two parts λ₁ and λ₂, the first describing local hidden variables that influence the outcome of measurement A and the second describing local hidden variables that influence the outcome of measurement B. Is this the sort of thing you were thinking of? For example, if we define λ in terms of all local variables in cross-sections of the two past light cones of the measurements, with the cross-sections taken at some time after the last moment the two past light cones overlap, then λ₁ could just refer to all local variables in the cross-section of the past light cone of measurement A, and λ₂ could be all local variables in the cross-section of the past light cone of measurement B. In this case your integral could be replaced by the following double integral:

$$P(A, B|a, b) = \int \int d\lambda_1 d\lambda_2 \rho (\lambda_1, \lambda_2) P_ 1(A|a, \lambda_1) P_ 2(B|b, \lambda_2).$$

Thank you JesseM very much for your time and accuracy. I welcome and appreciate your rigor.

It makes more sense to me to separate Bell's single lambda into two, so your "edit" is the way I see it.

Question 9. To build a common understanding between physicists and mathematicians, would it be acceptable to call the lambdas "spin-vectors" --- vectors related to angular momentum but unconstrained as to length or orientation?

Question 10. The lambdas would then constitute an infinite set of random variables in correlated pairs -- with a uniform distribution over 3-space -- which would not take Bell's mathematics in Bell (12) very far -- the result being one-quarter?

Question 11. Has Bell overlooked the fact that "measurement" is a 2-stage process? For photons, the randomized (but pairwise correlated) lambdas first impact the polarizers. The polarizers reduce an infinite input set to a countable output set which then becomes the local input to each associated analyzer -- Alice's analyzer (orientation a) being the device which delivers outcome A; Bob's analyzer (orientation b) delivers B?

Question 12. So, mathematically and physically, the density function for each analyzer input is the one Bell must require? NOT the density function for each polarizer?

In my view the exact QM expectations for any EPRB-experiment then follow readily -- consistent with local-realism?

Perhaps showing that neither Bell's assumptions nor his logic was at fault?

I look forward to your critical analysis,

With my thanks again,

JenniT

 

[JesseM]

Thank you JesseM very much for your time and accuracy. I welcome and appreciate your rigor.

It makes more sense to me to separate Bell's single lambda into two, so your "edit" is the way I see it.

Question 9. To build a common understanding between physicists and mathematicians, would it be acceptable to call the lambdas "spin-vectors" --- vectors related to angular momentum but unconstrained as to length or orientation?

Not sure why you say spin vectors--is the direction of the vector supposed to determine how the particle will respond to each detector setting? If so, how?

Like I said, I think the most rigorous way to approach Bell's proof is to imagine that each value of λ specifies the value of all local physical facts ('beables') in the cross-section of the past light cone of the measurement at some time t after the last moment that the past light cones of the two measurements overlapped, but before the experimenters have made a choice of what detector settings to use (this is essentially what Bell does on p. 242 of Speakable and Unspeakable in Quantum mechanics, although he actually has λ + c stand for all local variable in a series of cross-sections, the last of which are from a time after the last moment the two light cones overlap). If λ1 gives the complete state of some such cross-section of the past light cone of the measurement which might yield result A, it's then not too hard to see why locality implies that P(A|a,λ1) = P(A|B,a,b,λ1,λ2) [which together with the assumption that P(B|b,λ2)=P(B|a,b,λ1,λ2) implies that P(AB|a,b,λ1,λ2)=P(A|a,λ1)*P(B|b,λ2), so we can arrive at the equation you included in your original post], since outcome B has a spacelike separation from A (as does the choice of measurement setting b, and as does every event whose details are given by λ2), and any past events which could influence both A and B (creating a statistical dependence between them) would be in the overlap of the past light cones of A and B, and thus these events would be at a time earlier than λ1 (and in a local realist universe, the only way these earlier events can have an influence on A is by influencing λ1 which in turn influences A).

Question 10. The lambdas would then constitute an infinite set of random variables in correlated pairs -- with a uniform distribution over 3-space -- which would not take Bell's mathematics in Bell (12) very far -- the result being one-quarter?

I don't follow, how do you get any definite result from equation (12) if you don't know the value of P(A|a,λ1) and P(B|b,λ2)? Or are you assuming something specific about how the direction of your "spin vectors" interacts with the angle of the polarizer to determine outcomes A or B?

Also, why do you say the lambdas are in "correlated pairs"? Remember, at this stage in the proof Bell is no longer assuming that if both experimenters choose the same detector setting they are guaranteed to get identical (or opposite) results. On p. 12 he says:

It was only in the context of perfect correlation (or anticorrelation) that determinism could be inferred for the relation of observation results to pre-existing particle properties (for any indeterminism would have spoiled the correlation). Despite my insistence that the determinism was inferred rather than assumed, you might still suspect somehow that it is a preoccupation with determinism that creates the problem. Note well then that the following argument makes no mention whatever of determinism.

Question 11. Has Bell overlooked the fact that "measurement" is a 2-stage process? For photons, the randomized (but pairwise correlated) lambdas first impact the polarizers. The polarizers reduce an infinite input set to a countable output set which then becomes the local input to each associated analyzer -- Alice's analyzer (orientation a) being the device which delivers outcome A; Bob's analyzer (orientation b) delivers B?

Under the type of past light cone definition of λ I am using, and which Bell used in that section of "Speakable and Unspeakable in Quantum Mechanics", lambda does not refer to hidden variables at the time of measurement itself, but rather to all local variables in some cross-section of the past light cone of the measurement which might have a causal influence on the values of variables at the actual time of measurement. So, it makes no difference if the measurement is an extended process in which values of hidden variables associated with the particle can change based on interactions with the measurement-apparatus (or for any other reason compatible with locality).

Question 12. So, mathematically and physically, the density function for each analyzer input is the one Bell must require? NOT the density function for each polarizer?

No, as I've said the different values of lambda which appear in the density function need not specify anything about what is happening at the actual time of measurement.

 

[Gordon Watson]

Not sure why you say spin vectors--is the direction of the vector supposed to determine how the particle will respond to each detector setting? If so, how?

Thank you JesseM

1. Spin relating to angular momentum. So they need be vectors. What else could they be? Especially when nothing more is required to reproduce the QM dynamics?

2. Yes. "The direction of the vector determines how the particle will respond to each detector setting." For a realistic and relevant analogy, consider the dynamics involved with macroscopic wire-grid polarizers. We require nothing more.

3. You can see that my lambda specification is very broad, and very realistic. So I am not seeking to sneak something under the carpet. The lambdas are unconstrained as to length and orientation; and correlated by whatever angular momentum conservation setting was in place as the particles were co-produced, pairwise correlated.

4. Question 13. Many physicists resist agreeing with my lambda specifications and provide not one clue as to their reasons. I say that my very broad lambda specifications are NOT inconsistent with QM? Yet even you baulk at the, what is to me, obvious? Why?

5. Do you require my specifications strengthened or weakened? Why when they appear to be spot-on, realistic, adequate for the problem at hand?

Like I said, I think the most rigorous way to approach Bell's proof is to imagine that each value of λ specifies the value of all local physical facts ('beables') in the cross-section of the past light cone of the measurement at some time t after the last moment that the past light cones of the two measurements overlapped, but before the experimenters have made a choice of what detector settings to use (this is essentially what Bell does on p. 242 of Speakable and Unspeakable in Quantum mechanics, although he actually has λ + c stand for all local variable in a series of cross-sections, the last of which are from a time after the last moment the two light cones overlap). If λ1 gives the complete state of some such cross-section of the past light cone of the measurement which might yield result A, it's then not too hard to see why locality implies that P(A|a,λ1) = P(A|B,a,b,λ1,λ2) [which together with the assumption that P(B|b,λ2)=P(B|a,b,λ1,λ2) implies that P(AB|a,b,λ1,λ2)=P(A|a,λ1)*P(B|b,λ2), so we can arrive at the equation you included in your original post], since outcome B has a spacelike separation from A (as does the choice of measurement setting b, and as does every event whose details are given by λ2), and any past events which could influence both A and B (creating a statistical dependence between them) would be in the overlap of the past light cones of A and B, and thus these events would be at a time earlier than λ1 (and in a local realist universe, the only way these earlier events can have an influence on A is by influencing λ1 which in turn influences A).

There are a lot of words and inter-weavings here. Does what follows cut though:-

How can anyone object to my formulation when I insist that the lambdas are local realistic particle parameters -- they reside with each particle, one lambda-one-N accompanying the N-th particle that Alice will test, one correlated lambda-two-N accompanying the N-th particle that Bob will test?

I don't follow, how do you get any definite result from equation (12) if you don't know the value of P(A|a,λ1) and P(B|b,λ2)? Or are you assuming something specific about how the direction of your "spin vectors" interacts with the angle of the polarizer to determine outcomes A or B?

Of course. Interested in generality, I write a general formula for a general analyzer -- with a particle's intrinsic spin s in the argument. For spin-half "polarizers" [Stern-Gerlach magnets] s = one-half; for photon-polarizers s = 1.

Also, why do you say the lambdas are in "correlated pairs"? Remember, at this stage in the proof Bell is no longer assuming that if both experimenters choose the same detector setting they are guaranteed to get identical (or opposite) results. On p. 12 he says:


Under the type of past light cone definition of λ I am using, and which Bell used in that section of "Speakable and Unspeakable in Quantum Mechanics", lambda does not refer to hidden variables at the time of measurement itself, but rather to all local variables in some cross-section of the past light cone of the measurement which might have a causal influence on the values of variables at the actual time of measurement. So, it makes no difference if the measurement is an extended process in which values of hidden variables associated with the particle can change based on interactions with the measurement-apparatus (or for any other reason compatible with locality).

The particles ARE correlated. So I recognize it in the formalism.

Do you accept that this reflects a valid approach?

Would Occam?

For me, Bell wanders too far afield -- lost limping in a huge forest when a mere sapling will provide a crutch and escape.

No, as I've said the different values of lambda which appear in the density function need not specify anything about what is happening at the actual time of measurement.

But is it not the case that CRUCIAL measurement interactions and dynamics are involved "at the time of measurement"?

Crucial determined quantum dynamics as against quantum mechanics?

HOW does Bell allow for these dynamics?

Is he bound to thinking that each "measured" output identifies one-to-one with each input -- he being much opposed to "measurement" for this exact reason?

Especially considering ---

Huge numbers of incoming projectiles -- in correlated pairs -- their properties drawn from an infinite set -- reduced, by "measurement", to a countable set -- cardinality 4?

Do you see that my determined local realistic simplicity cuts through many words, assumptions and much irrelevant discussion?

Can you point me to something that I do that conflicts with QM or Bell's assumptions?


Many thanks again, and best regards,

JenniT

 

[Point Conception]

JenniT, I was wondering what level of mathematics and quantum physics you have reached.
thanks

 

[alxm]

1. Spin relating to angular momentum. So they need be vectors. What else could they be? Especially when nothing more is required to reproduce the QM dynamics?

But lambda here includes all vectors and all and any information you want, except the local detector settings. That's all. In my https://www.physicsforums.com/showpost.php?p=2743072&postcount=4" to you a month ago I said this, I don't know why you're back at this. Lambda can contain vectors and scalars and complex numbers and whatever you like except the detector settings, which are 'local'. The values of lambda can be as correlated as you like. P1 and P2 can depend on different subsets of lambda if you like. The point of Bell's (12) is that you can separate into P1 and P2 with respect to the local detector settings (or the local temperature in Lyons/Lille).

How can anyone object to my formulation when I insist that the lambdas are local realistic particle parameters -- they reside with each particle, one lambda-one-N accompanying the N-th particle that Alice will test, one correlated lambda-two-N accompanying the N-th particle that Bob will test?

I don't see that anyone did object to that. It's not at odds with what Bell asserts in (12). The lambda-dependence of P1 and P2 isn't given or even assumed to be known, so naturally they can depend on subsets of lambda. So introducing $$\lambda_1$$ and $$\lambda_2$$ or any other number is fine (but rho will of course then be a function of all lambdas, and you will have to integrate over all of them). What you're describing is a subset of what's allowed by (12).

But is it not the case that CRUCIAL measurement interactions and dynamics are involved "at the time of measurement"?
Crucial determined quantum dynamics as against quantum mechanics?
HOW does Bell allow for these dynamics?

I have no idea what you're talking about here. If you're referring to something specific, be specific.

Is he bound to thinking that each "measured" output identifies one-to-one with each input -- he being much opposed to "measurement" for this exact reason?

What do you mean, 'input'? The hidden variables? There's no assumption of a 1-to-1 correspondence at all.

Huge numbers of incoming projectiles -- in correlated pairs -- their properties drawn from an infinite set -- reduced, by "measurement", to a countable set -- cardinality 4?

So? And there are 6 billion people in the world and only one expectation value for their age. What would set theory have to do with it, anyway?

I do understand your view that Bell was preaching to the converted in so far as he spoke in a language clear to them -- like Danish.

Don't be silly, not even the Danes http://www.youtube.com/watch?v=s-mOy8VUEBk"

 

[Gordon Watson]

JenniT, I was wondering what level of mathematics and quantum physics you have reached.
thanks

Dear morrobay,

I am as a beginner in these things.

If there are things that I write that you do not understand, please let me know and I will do my best to correct, improve or explain.

And do not be misled -- I belong to a scattered minority group of convinced local realists -- more in Einstein's camp than Bell's.



JenniT

 

[Gordon Watson]

But lambda here includes all vectors and all and any information you want, except the local detector settings. That's all. In my https://www.physicsforums.com/showpost.php?p=2743072&postcount=4" to you a month ago I said this, I don't know why you're back at this. Lambda can contain vectors and scalars and complex numbers and whatever you like except the detector settings, which are 'local'. The values of lambda can be as correlated as you like. P1 and P2 can depend on different subsets of lambda if you like. The point of Bell's (12) is that you can separate into P1 and P2 with respect to the local detector settings (or the local temperature in Lyons/Lille).

Dear alxm,

Perhaps I am over sensitive but your reply here looks like you are building a straw woman and putting it down?

I will attempt to answer with respect and courtesy.

You start with a BUT. BUT What -- for me -- if I need less than Bell allows?

It seems that much [NOT all] of what you say here is irrelevant to anything that I have said or believe. Or reinforces agreed points as if I have missed them?

I suspect you might owe me an apology?

Your supposed reply to me https://www.physicsforums.com/showpost.php?p=2743072&postcount=4 is in agreement with what I said? So how is it a reply? And how am I "back at it"?

OK, so [see above]: "lambda here [my emphasis] includes all vectors and all and any information you want, except the local detector settings. ... includes all vectors and all and any information you want, except the local detector settings."

But the lambdas (plural) here are my "reduced" lambdas because I need nothing more that their representation as pairwise correlated spin vectors -- unconstrained in length and orientation.

Is this an acceptable general specification as far as a QM specialist is concerned?

As Science Advisor, please, tell me what else I will need --

1. To answer my question.

2. To tackle and understand Bell.

3. To be consistent with QM.

If Occam is more to my liking than Bell --- does it matter --- if my proof goes through?

And please consider this: The detectors are separated and local BUT the detector settings are simply orientations in 3-space -- a very important difference to me.

And I agree with the P1 and P2 separation -- we agree --- yet you make a point of it.

I don't see that anyone did object to that. It's not at odds with what Bell asserts in (12). The lambda-dependence of P1 and P2 isn't given or even assumed to be known, so naturally they can depend on subsets of lambda. So introducing $$\lambda_1$$ and $$\lambda_2$$ or any other number is fine (but rho will of course then be a function of all lambdas, and you will have to integrate over all of them). What you're describing is a subset of what's allowed by (12).

Thank you, Yes.

I have no idea what you're talking about here. If you're referring to something specific, be specific.

I am referring to the reduction of an infinite set to a countable set via the "measurement" interaction (dynamics).

What do you mean, 'input'? The hidden variables? There's no assumption of a 1-to-1 correspondence at all.

We agree. Thank you.

So? And there are 6 billion people in the world and only one expectation value for their age. What would set theory have to do with it, anyway?

Well 6 billion may be a big, but age is hardly an infinite set of lengths and orientations; and what "measurement" would you use to produce an equivalent countable set of cardinality 4?

Please answer this because I may be missing something -- because I do not see that we are yet at the stage of expectation values? For me, they come later.

Set theory? Perhaps relating to subsets of the sample space mapped by Probability Functions to the real interval [0, 1]?

Don't be silly, not even the Danes http://www.youtube.com/watch?v=s-mOy8VUEBk"

Taler du dansk? :!)

I tink so ... and maybe I have just come to like strong robust answers and discussion :!)

Thanks for them,

JenniT

 

[JesseM]

Thank you JesseM

1. Spin relating to angular momentum. So they need be vectors. What else could they be? Especially when nothing more is required to reproduce the QM dynamics?

You can't reproduce QM dynamics by assuming each particle has a local hidden variable corresponding to a spin vector with a well-defined direction, and that its deflection in a Stern-Gerlach device (up or down) is simply a function of its hidden spin vector and the orientation of the SG device. That would be an example of the type of local hidden variable theory that's ruled out by Bell's theorem. If you think it would be possible to come up with such a local model and reproduce all the QM dynamics, just provide the details of how the spin vector and the SG orientation are supposed to determine the result (or the probabilities of different results), so if I give you a particular angle for the spin vector and a particular angle for the SG orientation, you have a formula for calculating whether the result will be spin-up or spin-down (or what the probabilities of each will be).

Anyway, the point of Bell's theorem is to be as general as possible, and not make any specific assumptions about how the measurement outcomes are determined. It could be, for example, that each particle would have a large collection of hidden variables associated with it, and would "decide" whether to be deflected upward or downward by a given SG device by making use of some very complicated algorithm. This may not seem very physically plausible, but as long as it's a local hidden variables theory, Bell wants to rule it out. Also, keep in mind that if lambda represents the full state of all local variables in a cross-section from time t of the past light cone of the measurement (with t being after the last moment the past light cones of the two measurements overlap), then that will necessarily include any local hidden variables associated with the particle itself, so if the particle has something like a "spin vector" associated with it, the orientation of that vector at time t will already be included as part of lambda.

I gave more detail on the rationale for using cross-sections of the past light cones in post #61 here, starting with the paragraph that begins "Let me try a different tack." (the discussion then continued in post #62) I called the past light cone cross-sections PLCCS' there, and offered an analogy of two computers which can exchange date for a while, but after some time t their communication is cut off and afterwards they each simulate a measurement:

Suppose we have two computers A and B which will simulate the results of each measurement, and a middle computer M which can send signals to A and B for a while but then is disconnected, leaving A and B isolated and unable to communicate at some time t, after which they simulate both an experimenter making a choice and the results of the measurement with the chosen detector setting. Here the state of the information in each computer at time t represents the complete set of physical variables in the PLCCS of the measurement, while the fact that M was able to send each computer signals prior to t represents the fact that the state of each PLCCS may be influenced by events in the overlap of the past light cone of the measurement events.

After a discussion of this analogy in terms of how it can be used to understand the "no-conspiracy assumption" in Bell's proof, I continued in post #62:

If anyone proposes that a local hidden variables theory can explain the results of these experiments, there's no reason that such a theory could not be simulated in the setup I described, where a middle computer M can send signals to two different computers A and B until some time t when the computers are disconnected, and some time after t the experimenters (real or simulated) make choices about which orientation to use for the simulated detector (if the experimenters are real people interacting with the simulation they could make this choice by deciding whether to type 1, 2, or 3 on the keyboard, for example), and each computer A and B must return a measurement result. On p. 15 of the Jaynes paper you linked to, Jaynes seemed to acknowledge that if there was a local realist theory which could replicate the violations of Bell inequalities, then it should be possible to simulate on independent computers:

The Aspect experiment may show that such theories are untenable, but without further analysis it leaves open the status of other local causal theories more to Einstein's liking.

That future analysis is, in fact, already underway. An important part of it has been provided by Steve Gull's "You can't program two independently running computers to emulate the EPR experiment" theorem, which we learned about at this meeting. It seems, at first glance, to be just what we have needed because it could lead to more cogent tests of these issues than did the Bell argument. The suggestion is that some of the QM predictions can be duplicated by local causal theories only by invoking teleological elements as in the Wheeler-Feynman electrodynamics. If so, then a crucial experiment would be to verify the QM predictions in such cases. It is not obvious whether the Aspect experiment serves this purpose.

The implication seems to be that, if the QM predictions continue to be confirmed, we exorcise Bell's superluminal spook only to face Gull's teleological spook. However, we shall not rush to premature judgments. Recalling that it required some 30 years to locate von Neumann's hidden assumptions, and then over 20 years to locate Bell's, it seems reasonable to ask for a little time to search for Gull's, before drawing conclusions and possibly suggesting new experiments.

So, do you agree with the idea that this is a good way to test claims that someone has thought up a way to reproduce the EPR results with a local realist theory?

This last question was directed at the person I was talking to on that older thread, but I'd like to ask it of you too; if you think the analogy breaks down somewhere, can you point to where? Just as the two computers can no longer communicate after some time t, so there will be some time t that is after the last moment the past light cones of the two measurements overlap, but before the measurements are made (or the experimenter's random choice of which detector setting to use), and nothing in the past light cone of one measurement result at time t or after can have a causal influence on anything in the past light cone of the other measurement result after time t.

2. Yes. "The direction of the vector determines how the particle will respond to each detector setting." For a realistic and relevant analogy, consider the dynamics involved with macroscopic wire-grid polarizers. We require nothing more.

Well, again, can you please provide the formulas to show what the particle will do for any arbitrary combination of vector direction and detector direction? I'm not sure what "macroscopic wire-grid polarizers" are, but note that if you want a macroscopic analogue of particle spin, you can't just imagine something like a macroscopic charged spinning ball. As explained in the first section of this article, a macroscopic charged spinning ball traveling through an external magnetic field could be deflected at a range of angles depending on the angle between its spin axis and the direction of the external field; in contrast, quantum particles with spin like electrons traveling through the same magnetic field will all be deflected in one of two possible directions, corresponding to "spin-up" and "spin-down", regardless of the direction of the field.

3. You can see that my lambda specification is very broad, and very realistic.

But not as broad as mine, which will already specify all local variables associated with the particle at the time t which is before the measurements are done but after the last moment the past light cones of the measurement overlap (so there can be no causal influence of anything in the past light cone of measurement A at time t or after on anything in the past light cone of measurement B at time t or after, including a causal influence on local variables associated with the particle such as yours). The advantage of my definition is that it allows us to show rigorously why P(AB|a,b,λ1,λ2)=P(A|a,λ1)*P(B|b,λ2), whereas with your definition the rationale for this step isn't so clear.

4. Question 13. Many physicists resist agreeing with my lambda specifications and provide not one clue as to their reasons. I say that my very broad lambda specifications are NOT inconsistent with QM? Yet even you baulk at the, what is to me, obvious? Why?

Again, because the possibility of a spin vector associated with each particle is already included in my definition, and because my definition allows for a rigorous derivation of the fact that P(AB|a,b,λ1,λ2)=P(A|a,λ1)*P(B|b,λ2).

5. Do you require my specifications strengthened or weakened? Why when they appear to be spot-on, realistic, adequate for the problem at hand?

If by "strengthened" you mean "made more specific" and "weakened" you mean "made more general", then weakened. Reasons for this above.

How can anyone object to my formulation when I insist that the lambdas are local realistic particle parameters -- they reside with each particle, one lambda-one-N accompanying the N-th particle that Alice will test, one correlated lambda-two-N accompanying the N-th particle that Bob will test?

I don't object to this as a hidden-variables theory, I just say that for the purposes of making the derivation rigorous we should define λ in terms of the complete set of local facts in cross-sections of the past light cones of the measurements at some time t, which will naturally include an exact specification of your "local realistic particle parameters" at time t.

Of course. Interested in generality, I write a general formula for a general analyzer -- with a particle's intrinsic spin s in the argument. For spin-half "polarizers" [Stern-Gerlach magnets] s = one-half; for photon-polarizers s = 1.

So what is that "general formula"?

The particles ARE correlated. So I recognize it in the formalism.

Do you accept that this reflects a valid approach?

Sure, it's a valid approach to assume the particles will always give perfectly correlated results when measured with the same detector angle, since this is predicted by QM. Bell just wanted to generalize his proof a little to cover the possibility that QM's predictions about this might turn out to be imperfect--he showed that even if there wasn't a perfect correlation with identical detector settings, you'd still be able to use the assumption of local realism to derive some inequalities which are significantly violated by QM. But if you want to stick with the assumption of perfect correlation that's fine.

No, as I've said the different values of lambda which appear in the density function need not specify anything about what is happening at the actual time of measurement.

But is it not the case that CRUCIAL measurement interactions and dynamics are involved "at the time of measurement"?

All that's important to the proof is that if you already know the complete physical state λ1 of all local variables (including hidden ones) in a cross-section of the past light cone at time t, then the only events which could alter your estimate of different measurement outcomes A (given a known detector setting a) would be other events in the past light cone at times after t. That's enough to guarantee that P(A|B,a,b,λ1,λ2)=P(A|a,λ1), since B,b,λ2 all describe facts about events which lie outside this region of the past light cone of the outcome A. So certainly physical aspects of the measurement device during the measurement might have an effect, but these would be inside the past light cone of the final result after time t.

Again, just think of the computer analogy. If you already know the complete internal state λ1 of computer A at a time t after it can no longer communicate with the other computers, then whatever happens in the other computers should have no effect on your estimate of the probability of different final outputs for computer A. The computer may be performing all sorts of complicated calculations and simulations (like a simulated particle interacting with a simulated detector) between the time t and the time it gives its final output, and if its processes include a random element (like if it has an internal source of true random noise such as a geiger counter measuring some radioactive decay events) then learning information about its internal state between t and the final output might change your estimate of different possible final outputs, but what happens in the other computers besides A should be irrelevant as far as you're concerned.

HOW does Bell allow for these dynamics?

Is he bound to thinking that each "measured" output identifies one-to-one with each input -- he being much opposed to "measurement" for this exact reason?

If it is true that identical measurement settings always give identical (or always give opposite) results, then after we have defined λ1 and λ2 in terms of cross-sections of the past light cones of measurement outputs, and shown that local realism implies P(AB|a,b,λ1,λ2)=P(A|a,λ1)*P(B|b,λ2), then from this (and the 'no-conspiracy condition') you can derive the fact that the only way to explain the perfect correlation is to conclude that λ1 predetermined the result A for each of the three possible detector settings, and likewise λ2 predetermined the result B for each of the three possible detector settings. But this isn't an independent assumption, it's derived from the other definitions and local realism + no-conspiracy. And anyway, as I said, Bell also derived some inequalities that would hold even if we dropped the assumption of perfect correlations, in which case the value of λ1 need not give a predetermined measurement result for each of the three detector settings.

Do you see that my determined local realistic simplicity cuts through many words, assumptions and much irrelevant discussion?

Can you point me to something that I do that conflicts with QM or Bell's assumptions?

You are just defining lambda too narrowly, so that there is no rigorous way to see that the step P(AB|a,b,λ1,λ2)=P(A|a,λ1)*P(B|b,λ2) is actually justified. Again, Bell's assumptions already cover any type of local realist theory you can imagine, including one like yours where the local hidden variables are vectors associated with each particle. In the type of theory you describe, do you deny it would be possible the define a variable that gives the complete set of local variables associated with any specific point or set of points in spacetime, including the set of all points at time t in the past light cone of the final measurement result? You might say that this variable would include plenty of superfluous information that you think isn't relevant to determining what the final result will be, and that may well be true, but the definition is a coherent one and it allows us to see why the step above would necessarily be true under your (or any) local realist theory.

 

[DrChinese]

Dear morrobay,

I am as a beginner in these things.

If there are things that I write that you do not understand, please let me know and I will do my best to correct, improve or explain.

And do not be misled -- I belong to a scattered minority group of convinced local realists -- more in Einstein's camp than Bell's.



JenniT

I ask you: what are you really asserting if you say you are a local realist?

a) Do unentangled quantum particles have well defined attributes independent of observation?

b) Are entangled particles unaffected by observations on their distant partner?

If your answer to either of the above is YES, then how do YOU explain experimental results? You see, Bell is a tool to help us understand these questions. And to assist in framing new experiments around them.

Personally, I believe the answer to both of the above questions is NO. Experiments, while not altogether conclusive on each individually, strongly support that position. There is essentially no experimental evidence for a YES to either. (However, it is possible that one is a YES and the other is a NO.)

 

[Gordon Watson]

I ask you: what are you really asserting if you say you are a local realist?

Thank you DrC,

I assert that the real world is local and realistic.

There are many definitions of these terms.

In this arena, Einstein has said it all already for me.

That is: I am not aware of any statement by him, in this arena, that I would deny.

If there are such statements [by Einstein] which you, with your beliefs, would deny, please let me know.

a) Do unentangled quantum particles have well defined attributes independent of observation?

YES, of course.

b) Are entangled particles unaffected by observations on their distant partner?

YES, of course.

If your answer to either of the above is YES, then how do YOU explain experimental results? You see, Bell is a tool to help us understand these questions. And to assist in framing new experiments around them.

Very simply -- truly -- when constructing a physical theory, I let my mathematics follow the physics.

I see the interpretative (Feynman, understanding) issues which surround QM as arising from its brilliant origins in mathematics.

We have much more experimental data now -- so my theory builds only on the physics.

I have no quibble with any experiment in this area.

I question the conclusions drawn from them.

For me, NONLOCALITY is nonsense.

Personally, I believe the answer to both of the above questions is NO. Experiments, while not altogether conclusive on each individually, strongly support that position. There is essentially no experimental evidence for a YES to either. (However, it is possible that one is a YES and the other is a NO.)

DrC, your belief is beyond me.

Maybe you follow that Einstein error when he said -- "It is theory that decides what can be observed."

Does the inadvertent observation of the cosmic microwave background radiation (or the pigeons) rebut his statement here?

Not to mention the ether?

So maybe you are too influenced by Bell's theorem and his massive support base? As to what we observe?

PS: I very much respect your contributions here.

Would a paper supporting my claims likely be acceptable in Independent Research?

Would you help draft its terms in this thread so that we could all be sure of the papers to be cited and the terminology that would be used at IR?

If you would care to help with that draft we might find that I have missed some subtlety -- or that I have not?

If I cited just EPR and Bell and Aspect, would that be enough?

That is all I need.

In broad terms, would this be what you would like to see?

------------------------

Draft Abstract

Bell's theorem is widely regarded as a serious obstacle to a local-realistic theory of quantum mechanics. Critically examining Bell's formalism and its relevance to quantum dynamics, we refute Bell's theorem and show that local beables co-encode realism (and the binding transformation symmetries of some common properties) with locality (and the causal independence of space-like separated events). Accepting Bell's assumptions and logic (but not his mathematics), we show that stochastically independent micro-probabilities lead to stochastically dependent (entangled) macro-probabilities in full accord with quantum theory and practice. Our findings scotch all Bell-based claims that local-realists must abandon local-action or physical-realism or both.

Proposed papers to be cited

EPR, Bell, Aspect.

----------------------------------

DrC, I would sincerely welcome your involvement and assistance.

I believe the benefit for you might be "Yes, Yes" answers to your above leading questions.

For me, I might see something with new eyes and have to admit to having over-reached.

With best regards,

JenniT

 

[Gordon Watson]

You can't reproduce QM dynamics by assuming each particle has a local hidden variable corresponding to a spin vector with a well-defined direction, and that its deflection in a Stern-Gerlach device (up or down) is simply a function of its hidden spin vector and the orientation of the SG device. That would be an example of the type of local hidden variable theory that's ruled out by Bell's theorem. If you think it would be possible to come up with such a local model and reproduce all the QM dynamics, just provide the details of how the spin vector and the SG orientation are supposed to determine the result (or the probabilities of different results), so if I give you a particular angle for the spin vector and a particular angle for the SG orientation, you have a formula for calculating whether the result will be spin-up or spin-down (or what the probabilities of each will be).

Anyway, the point of Bell's theorem is to be as general as possible, and not make any specific assumptions about how the measurement outcomes are determined. It could be, for example, that each particle would have a large collection of hidden variables associated with it, and would "decide" whether to be deflected upward or downward by a given SG device by making use of some very complicated algorithm. This may not seem very physically plausible, but as long as it's a local hidden variables theory, Bell wants to rule it out. Also, keep in mind that if lambda represents the full state of all local variables in a cross-section from time t of the past light cone of the measurement (with t being after the last moment the past light cones of the two measurements overlap), then that will necessarily include any local hidden variables associated with the particle itself, so if the particle has something like a "spin vector" associated with it, the orientation of that vector at time t will already be included as part of lambda.

I gave more detail on the rationale for using cross-sections of the past light cones in post #61 here, starting with the paragraph that begins "Let me try a different tack." (the discussion then continued in post #62) I called the past light cone cross-sections PLCCS' there, and offered an analogy of two computers which can exchange date for a while, but after some time t their communication is cut off and afterwards they each simulate a measurement:

After a discussion of this analogy in terms of how it can be used to understand the "no-conspiracy assumption" in Bell's proof, I continued in post #62:

This last question was directed at the person I was talking to on that older thread, but I'd like to ask it of you too; if you think the analogy breaks down somewhere, can you point to where? Just as the two computers can no longer communicate after some time t, so there will be some time t that is after the last moment the past light cones of the two measurements overlap, but before the measurements are made (or the experimenter's random choice of which detector setting to use), and nothing in the past light cone of one measurement result at time t or after can have a causal influence on anything in the past light cone of the other measurement result after time t.

Well, again, can you please provide the formulas to show what the particle will do for any arbitrary combination of vector direction and detector direction? I'm not sure what "macroscopic wire-grid polarizers" are, but note that if you want a macroscopic analogue of particle spin, you can't just imagine something like a macroscopic charged spinning ball. As explained in the first section of this article, a macroscopic charged spinning ball traveling through an external magnetic field could be deflected at a range of angles depending on the angle between its spin axis and the direction of the external field; in contrast, quantum particles with spin like electrons traveling through the same magnetic field will all be deflected in one of two possible directions, corresponding to "spin-up" and "spin-down", regardless of the direction of the field.

But not as broad as mine, which will already specify all local variables associated with the particle at the time t which is before the measurements are done but after the last moment the past light cones of the measurement overlap (so there can be no causal influence of anything in the past light cone of measurement A at time t or after on anything in the past light cone of measurement B at time t or after, including a causal influence on local variables associated with the particle such as yours). The advantage of my definition is that it allows us to show rigorously why P(AB|a,b,λ1,λ2)=P(A|a,λ1)*P(B|b,λ2), whereas with your definition the rationale for this step isn't so clear.

Again, because the possibility of a spin vector associated with each particle is already included in my definition, and because my definition allows for a rigorous derivation of the fact that P(AB|a,b,λ1,λ2)=P(A|a,λ1)*P(B|b,λ2).

If by "strengthened" you mean "made more specific" and "weakened" you mean "made more general", then weakened. Reasons for this above.

I don't object to this as a hidden-variables theory, I just say that for the purposes of making the derivation rigorous we should define λ in terms of the complete set of local facts in cross-sections of the past light cones of the measurements at some time t, which will naturally include an exact specification of your "local realistic particle parameters" at time t.

So what is that "general formula"?

Sure, it's a valid approach to assume the particles will always give perfectly correlated results when measured with the same detector angle, since this is predicted by QM. Bell just wanted to generalize his proof a little to cover the possibility that QM's predictions about this might turn out to be imperfect--he showed that even if there wasn't a perfect correlation with identical detector settings, you'd still be able to use the assumption of local realism to derive some inequalities which are significantly violated by QM. But if you want to stick with the assumption of perfect correlation that's fine.All that's important to the proof is that if you already know the complete physical state λ1 of all local variables (including hidden ones) in a cross-section of the past light cone at time t, then the only events which could alter your estimate of different measurement outcomes A (given a known detector setting a) would be other events in the past light cone at times after t. That's enough to guarantee that P(A|B,a,b,λ1,λ2)=P(A|a,λ1), since B,b,λ2 all describe facts about events which lie outside this region of the past light cone of the outcome A. So certainly physical aspects of the measurement device during the measurement might have an effect, but these would be inside the past light cone of the final result after time t.

Again, just think of the computer analogy. If you already know the complete internal state λ1 of computer A at a time t after it can no longer communicate with the other computers, then whatever happens in the other computers should have no effect on your estimate of the probability of different final outputs for computer A. The computer may be performing all sorts of complicated calculations and simulations (like a simulated particle interacting with a simulated detector) between the time t and the time it gives its final output, and if its processes include a random element (like if it has an internal source of true random noise such as a geiger counter measuring some radioactive decay events) then learning information about its internal state between t and the final output might change your estimate of different possible final outputs, but what happens in the other computers besides A should be irrelevant as far as you're concerned.

If it is true that identical measurement settings always give identical (or always give opposite) results, then after we have defined λ1 and λ2 in terms of cross-sections of the past light cones of measurement outputs, and shown that local realism implies P(AB|a,b,λ1,λ2)=P(A|a,λ1)*P(B|b,λ2), then from this (and the 'no-conspiracy condition') you can derive the fact that the only way to explain the perfect correlation is to conclude that λ1 predetermined the result A for each of the three possible detector settings, and likewise λ2 predetermined the result B for each of the three possible detector settings. But this isn't an independent assumption, it's derived from the other definitions and local realism + no-conspiracy. And anyway, as I said, Bell also derived some inequalities that would hold even if we dropped the assumption of perfect correlations, in which case the value of λ1 need not give a predetermined measurement result for each of the three detector settings.

You are just defining lambda too narrowly, so that there is no rigorous way to see that the step P(AB|a,b,λ1,λ2)=P(A|a,λ1)*P(B|b,λ2) is actually justified. Again, Bell's assumptions already cover any type of local realist theory you can imagine, including one like yours where the local hidden variables are vectors associated with each particle. In the type of theory you describe, do you deny it would be possible the define a variable that gives the complete set of local variables associated with any specific point or set of points in spacetime, including the set of all points at time t in the past light cone of the final measurement result? You might say that this variable would include plenty of superfluous information that you think isn't relevant to determining what the final result will be, and that may well be true, but the definition is a coherent one and it allows us to see why the step above would necessarily be true under your (or any) local realist theory.

Dear JesseM,

Thank you very much again for this detail.

I will attempt to answer all your questions in follow-up posts.

I am not so good with so many words.

I will be trying to reduce your questions to mathematical ones.

Might it be possible to reduce your questions if I make some mathematical assertions which cut-through and satisfy you that I am a serious and valid local-realist in the full sense of all Bell's assumptions -- BUT NOT his mathematics?

I wonder if this next statement both puts your mind at ease and is an acceptable mathematical definition of local-realism to all in this thread:

(A) Local-realism defined: P(AB|a,b,λ1,λ2)=P(A|a,λ1)*P(B|b,λ2).

Or, in terms of Mermin's black-boxes, which I prefer -- where G = Green, R = Red. A = {G, R}. B = {G', R'}:

(A) Local-realism defined: P(GG'|a,b',λ1,λ2)=P(G|a,λ1)*P(G'|b',λ2).

In studying Bell's work, I start with this definition and then let my mathematics follow the physics of Aspect's experiments.

To help others and me who are more comfortable in mathematics with not so many words:

Is there a next definition or formalism that you would like to see asserted by me?

Mathematics and probability theory being our best developed languages?

Thank you,

JenniT

 

[DrChinese]

Thank you DrC,

1. I assert that the real world is local and realistic.

There are many definitions of these terms.

In this arena, Einstein has said it all already for me.

That is: I am not aware of any statement by him, in this arena, that I would deny.

If there are such statements [by Einstein] which you, with your beliefs, would deny, please let me know.


2. Very simply -- truly -- when constructing a physical theory, I let my mathematics follow the physics.

I see the interpretative (Feynman, understanding) issues which surround QM as arising from its brilliant origins in mathematics.

We have much more experimental data now -- so my theory builds only on the physics.

I have no quibble with any experiment in this area.

I question the conclusions drawn from them.

For me, NONLOCALITY is nonsense.


3. Would a paper supporting my claims likely be acceptable in Independent Research?

Would you help draft its terms in this thread so that we could all be sure of the papers to be cited and the terminology that would be used at IR?

If you would care to help with that draft we might find that I have missed some subtlety -- or that I have not?

If I cited just EPR and Bell and Aspect, would that be enough?

That is all I need.

In broad terms, would this be what you would like to see?

------------------------

Draft Abstract

Bell's theorem is widely regarded as a serious obstacle to a local-realistic theory of quantum mechanics. Critically examining Bell's formalism and its relevance to quantum dynamics, we refute Bell's theorem and show that local beables co-encode realism (and the binding transformation symmetries of some common properties) with locality (and the causal independence of space-like separated events). Accepting Bell's assumptions and logic (but not his mathematics), we show that stochastically independent micro-probabilities lead to stochastically dependent (entangled) macro-probabilities in full accord with quantum theory and practice. Our findings scotch all Bell-based claims that local-realists must abandon local-action or physical-realism or both.

Proposed papers to be cited

EPR, Bell, Aspect.


With best regards,

JenniT

1. Sure, Einstein believed the moon is there when we're not looking. And he did not believe in spooky action at a distance. I disagree with both of these statements.

I believe in what I refer to as quantum nonlocality. I believe the Heisenberg Uncertainty Principle (HUP) is fundamental QM and is an expression of something important about the nature of reality. An ordinary photon is not well localized, and could be said to exist in some respects at different points in spacetime; certainly that is true of its probability wave. Ergo, it is quantum nonlocal.


2. Math, experiments, these are good!


3. I don't know how much I can help, since my views are opposite. I would certainly encourage you, but honestly IR might be a tough sell since your view goes against established science. You might ask what new perspective you bring in drafting something. Here are some links to Local Realist papers which you may find of interest:

Nieuwenhuizen
Where Bell went wrong
http://arxiv.org/abs/0812.3058

Adenier
Quantum entanglement, fair sampling, and reality: Is the moon there when nobody looks?
http://arxiv.org/abs/0705.1477

Santos
Realistic interpretation of quantum mechancis
http://arxiv.org/abs/0912.4098


I have more too, but this would probably be a good start for background. I hope this helps.

 

[Gordon Watson]

1. Sure, Einstein believed the moon is there when we're not looking.

DrC -- seriously -- when ThomasT and many others meet with you for a celebratory drink -- and pleasantly drink you round-for-round into unconsciousness -- and we all look away -- are you yet with us?

Seriously, what happens to the moon when no one observes it?

More later,

JenniT

 

[DrChinese]

DrC -- seriously -- when ThomasT and many others meet with you for a celebratory drink -- and pleasantly drink you round-for-round into unconsciousness -- and we all look away -- are you yet with us?

Seriously, what happens to the moon when no one observes it?

More later,

JenniT

When I am uncounscious on the floor, I won't care.

As often as you look, you collapse the moon into an eigenstate. When no one is looking, nothing changes. Not really too tricky.

 

[Qubix]

I am very interested in this subject, and I found an article on arXiv that made me think a little :

http://uk.arxiv.org/abs/0904.4259

What do you make of it ?

 

[Gordon Watson]

When I am uncounscious on the floor, I won't care.

As often as you look, you collapse the moon into an eigenstate. When no one is looking, nothing changes. Not really too tricky.

DrC,

Trying to be truly serious for the sake of all beginners here --

A photon, reflected from the moon, hitting you in the eye, does something to the moon?

BUT a similar photon, hitting you between the eyes, does not wake you up?

And does nothing to the moon?

You really do believe in non-locality --- beyond anything Bell could have imagined?

Or was Bell crazy too? --- --- said in a very friendly way --- :!)

Are there fairies at the bottom of your garden?

Dial 911!

Love,

JenniT

 

[JesseM]

Dear JesseM,

Thank you very much again for this detail.

I will attempt to answer all your questions in follow-up posts.

I am not so good with so many words.

I will be trying to reduce your questions to mathematical ones.

Might it be possible to reduce your questions if I make some mathematical assertions which cut-through and satisfy you that I am a serious and valid local-realist in the full sense of all Bell's assumptions -- BUT NOT his mathematics?

I wonder if this next statement both puts your mind at ease and is an acceptable mathematical definition of local-realism to all in this thread:

(A) Local-realism defined: P(AB|a,b,λ1,λ2)=P(A|a,λ1)*P(B|b,λ2).

Or, in terms of Mermin's black-boxes, which I prefer -- where G = Green, R = Red. A = {G, R}. B = {G', R'}:

(A) Local-realism defined: P(GG'|a,b',λ1,λ2)=P(G|a,λ1)*P(G'|b',λ2).

In studying Bell's work, I start with this definition and then let my mathematics follow the physics of Aspect's experiments.

OK, but then do you think that these equations would still apply when λ1 and λ2 represent the "spin vectors" (a physical, not a purely mathematical, concept) that you assume characterize each particle? If you don't think they would apply, then that's exactly why I think it's necessary to define λ1 and λ2 in terms of past light cone cross-sections, because this allows us to see why the equations should apply under local realism. But if you are fine with accepting that these equations are a necessary consequence of local realism if λ1 and λ2 just represent some specific hidden variables associated with the particle, then you should be fine with accepting the equation I wrote earlier, $$P(A, B|a, b) = \int \int d\lambda_1 d\lambda_2 \rho (\lambda_1, \lambda_2) P_ 1(A|a, \lambda_1) P_ 2(B|b, \lambda_2)$$, and then we can continue to look at the rest of Bell's proof. In that case, what would be the first step Bell makes that you object to? (presumably you think there is an objectionable step, given the title of this thread?)

 

[Gordon Watson]

OK, but then do you think that these equations would still apply when λ1 and λ2 represent the "spin vectors" (a physical, not a purely mathematical, concept) that you assume characterize each particle?

Yes, of course -- that is what I have been saying.

And they must be physical in that I seek to have my mathematics follow the physics.

But please excuse my caution here -- are they "scare-quotes" that you are using in relation to my spin vectors?

Is there a more acceptable description for them within QM?

Do you deny or not understand their physical significance?

Do they not exist in QM?

This point must be clarified before we proceed too far?

If you don't think they would apply, then that's exactly why I think it's necessary to define λ1 and λ2 in terms of past light cone cross-sections, because this allows us to see why the equations should apply under local realism.

Well, as I have said -- they DO apply.

Do you not yet see that a very good definition of local realism follows -- without resort to the past -- and without resort to words?

I certainly want to be sure that we get off the start-line correctly.

But if you are fine with accepting that these equations are a necessary consequence of local realism if λ1 and λ2 just represent some specific hidden variables associated with the particle, then you should be fine with accepting the equation I wrote earlier,

$$P(A, B|a, b) = \int \int d\lambda_1 d\lambda_2 \rho (\lambda_1, \lambda_2) P_ 1(A|a, \lambda_1) P_ 2(B|b, \lambda_2)$$,

and then we can continue to look at the rest of Bell's proof.

I am happy to accept this equation HOWEVER since $$P_ 1, P_ 2$$ are probability Functions, their outputs will be discrete random variables. So the $$\rho$$ here will include delta functions!

Now some physicists ++ regard deltas as illegitimate, so (to be very safe) please rewrite your equation as a summation that is acceptable to you.

In that case, what would be the first step Bell makes that you object to? (presumably you think there is an objectionable step, given the title of this thread?)

I know what you mean -- but errors to me are not "objectionable" -- that is how we often learn best.

Depending on your reply, we should be ready for the next step -- asking you to understand my caution as I seek to avoid the need for backward steps.

Please take time, as necessary, to comment on my spin-vectors -- with their elegant simplicity, they are central to my case -- you may be able to save me from many laughs at my expense!

Please recall how I said that it is difficult to get meaningful comment on their validity.

Thank you,

JenniT

 

[JesseM]

Yes, of course -- that is what I have been saying.

And they must be physical in that I seek to have my mathematics follow the physics.

But please excuse my caution here -- are they "scare-quotes" that you are using in relation to my spin vectors?

Is there a more acceptable description for them within QM?

Do you deny or not understand their physical significance?

Do they not exist in QM?

This point must be clarified before we proceed too far?

In QM there are vectors describing the state of a given quantum system, and this vector determines the probabilities of each possible result for a given measurement. Here was a quick summary I wrote on another thread:

In QM the quantum state of a system is represented by a vector |Psi>, and for each type of measurement (position, momentum, energy, etc.) there is an operator A which can be understood as a matrix that multiplies the state vector, giving a new state vector. There's a certain select set of state vectors which are known as "eigenvectors" of that measurement operator, and they satisfy an equation of the form A|Psi> = x*|Psi>, where x is just a scalar (a single number rather than a vector with multiple components), which is known as the "eigenvalue" associated with that eigenvector (in other words, if we apply the operator to an eigenvector, it just gives us back the same state vector multiplied by a scalar). The eigenvalue is a possible result for a measurement with that operator, like a particular value of momentum if A is the momentum operator. If the current state |Psi> is an eigenvector of an operator representing a particular measurement, then if you actually make that measurement you're guaranteed to get the corresponding eigenvalue. If the current state |Psi> is not an eigenvector of what you're measuring, it can always be represented as a weighted sum of all the eigenvectors for that measurement operator (this is vaguely similar to Fourier analysis), and it's assumed in the standard interpretation of QM that when you make the measurement the quantum state "collapses" from the previous quantum state onto a random one of those eigenvectors, with the probability of collapsing onto each eigenvector being based on the weight assigned to each eigenvector in the sum for the state the system was in prior to the collapse.

Spin along a given axis is one of the measurement operators, so a system's quantum state can always be broken down into a weighed sum of spin eigenvectors, with the weights determining the probabilities of finding a given spin when the system is measured. The problem is that for an entangled quantum system, there is a single state vector |psi> for the whole entangled system rather than individual state vectors for the parts of the system, so this is not a local variable associated with a unique point in space at any given time (like the position of a single particle at that time). A measurement of one particle would cause an immediate change in the state vector representing both particles, even if there is a spacelike separation between the particles. The QM formalism gives no complete way to predict the behavior of both particle using only local variables associated with each particle (such as individual vectors associated with each one) that are not influenced by anything outside their past light cone.

The spin vectors you talk about also couldn't really be classical spin vectors describing the axes of rotation of two charged spinning objects, since as mentioned in this link which I posted earlier, classical charged spinning objects could be deflected at a continuous range of angles when traveling through a magnetic field, rather than always being deflected in one of two directions, "up" or "down". So yeah, I do think there's a need for "scare quotes" around your notion of a "spin vector" since it doesn't really seem to correspond exactly with any existing physics concept, it would presumably have to be defined in the context of a novel local hidden variables theory, and you'd need some definition of how the combination of the "spin vector" and the detector angle (along with any other properties of the detector you want to consider) determines the measurement result.

Well, as I have said -- they DO apply.

Do you not yet see that a very good definition of local realism follows -- without resort to the past -- and without resort to words?

I certainly want to be sure that we get off the start-line correctly.

Well, you still do need some words to point out that the equation only applies when A and B have a spacelike separation, and λ1 and λ2 contain enough detail that they screen out any correlation which exists between A and B due to events in their common past (like the fact that two particles may have been assigned correlated spin vectors by the source which created them). But if you want to minimize words, this equation is indeed a good start in helping define mathematically what "local realism" implies.

I am happy to accept this equation HOWEVER since $$P_ 1, P_ 2$$ are probability Functions, their outputs will be discrete random variables. So the $$\rho$$ here will include delta functions!

How do you figure? Consider the analogous but simpler equation $$P(X) = \int P(X|\lambda)\rho(\lambda) \, d\lambda$$. If X is a discrete random variable while λ is a continuous one, do you think this implies the probability density $$\rho(\lambda)$$ must include delta functions? Suppose λ could take any real value from 0 to 2, and $$\rho(\lambda)$$ was just a uniform probability density with density 0.5 everywhere (so if you integrate it from 0 to 2 you get 1, as expected for a probability density). Then if X could only take two values X1 and X2, and P(X1|λ) = 1 - (λ/2) while P(X2|λ) = λ/2, then for any given value of λ it's true that P(X1|λ) + P(X2|λ) = 1 (note that since λ can take a continuous range of values, this also means P(X1|λ) and P(X2|λ) are conditional probability density functions, not discrete probability functions). Then according to the integral, P(X1) would be given by $$\int_0^2 [1 - (\lambda/2)]*0.5 \, d\lambda$$ = $$0.5 * [\lambda - (\lambda^2 / 4)]^2_0$$ = 0.5*[(2 - 4/4) - (0 - 0/4)] = 0.5. Likewise P(X2) would be given by $$\int_0^2 (\lambda/2)*0.5 \, d\lambda$$ = $$0.5 * [\lambda^2 / 4]^2_0$$ = 0.5*[4/4 - 0/4] = 0.5. So, P(X1) + P(X2) also adds up to 1, all the probabilities make sense in spite of the fact that X is a discrete random variable and λ is a continuous one, with no need for $$\rho(\lambda)$$ to include any delta functions.

Now some physicists ++ regard deltas as illegitimate, so (to be very safe) please rewrite your equation as a summation that is acceptable to you.

What physicists regard delta functions as illegitimate? In any case there is no necessary reason to think $$\rho(\lambda)$$ should contain any delta functions as I explained. When you say "please rewrite your equation as a summation", are you asking what the equation would look like if we assumed λ could only take a discrete range of values (as Bell assumed in some versions of his proof) rather than a continuous range?

 

[DrChinese]

I am very interested in this subject, and I found an article on arXiv that made me think a little :

http://uk.arxiv.org/abs/0904.4259

What do you make of it ?

Christian's work has been rejected. But that is not likely to stop him. He fails test #1 with me: his model is not realistic.

 

[DrChinese]

DrC,

Trying to be truly serious for the sake of all beginners here --

A photon, reflected from the moon, hitting you in the eye, does something to the moon?

BUT a similar photon, hitting you between the eyes, does not wake you up?

And does nothing to the moon?

You really do believe in non-locality --- beyond anything Bell could have imagined?

Or was Bell crazy too? --- --- said in a very friendly way --- :!)

Are there fairies at the bottom of your garden?

Dial 911!

Love,

JenniT

Ha, I do have fairies in my garden. My mother gave them to me.

I believe in quantum non-locality, which is not at all the same as Bohmian non-locality. I mean, it might be the same, but the point is that I believe that the HUP implies a kind of non-locality. It makes sure that entangled particles act in a certain manner when they are spatially separated. It allows individual particles to have imprecise locations, which is automatically non-local. To me, this is just standard QM and nothing more. If you don't want to call it non-locality, that is fine by me. A lot of scientists do not consider that non-locality.

Since I think we live in an observer (observation actually) dependent universe: just, you could say I change the moon by observing it. Again, this is simply standard QM.

 

[alxm]

You start with a BUT. BUT What -- for me -- if I need less than Bell allows?

Well if you assume the system is described by something which is a subset of the more general conditions Bell set up, it doesn't contradict Bell.

I suspect you might owe me an apology?

Well I'm sorry if you feel I've misinterpreted you. But as I said earlier, it's not always clear to me what you mean.

But the lambdas (plural) here are my "reduced" lambdas because I need nothing more that their representation as pairwise correlated spin vectors -- unconstrained in length and orientation.

Is this an acceptable general specification as far as a QM specialist is concerned?

I'm hardly a QM specialist. I'm just one of those run-of-the-mill physicists who uses QM. All I need to do is go a bit outside my field and my knowledge will be pretty limited. I do think I can follow Bell's paper though. Anyhow, yes it's fine to have spin vectors as your choice of hidden variables. It's just that I don't see how this would contradict Bell in any way. You realize that it's a strength, not a weakness, that he allows for many more scenarios than any particular set of hidden variables?

If Occam is more to my liking than Bell --- does it matter --- if my proof goes through?

I haven't seen a proof though. You've made some arguments, but I don't feel you've shown there's anything wrong with Bell. On the contrary, since what you seem to be suggesting is a subset of what's covered by Bell's proof, it'd be subject to it - as long as Bell's logic is correct.

Bell's logic is correct, I'm certain. His original paper has almost 4000 citations by now, and has been pored over by all kinds of people, including many who probably were "local realists". 't Hooft's a bit of a hidden-variable fan, and he's probably forgotten more math than I've ever known, yet he doesn't dispute Bell's math. That's good enough for me. Rather he's criticized on the basis of other assumptions involved, e.g. how entanglement occurred in the first place. That's a legitimate line of attack in my opinion. Assuming Bell's math is wrong isn't. Especially since it's been verified experimentally.

(Similarly, many crackpots who dislike Special Relativity keep trying to "prove" Einstein's math wrong; often using high school-level math. And similarily I hold that Einstein's math is not wrong and if SR is ever disproved, it will be on the basis that some of the basic assumptions of the theory did not hold, not because of faulty logic)

And please consider this: The detectors are separated and local BUT the detector settings are simply orientations in 3-space -- a very important difference to me.

By the detector settings being "local" we mean that the settings at A do not affect the outcome at B and vice-versa, since they are assumed to be spatially separated. This is Bell's definition of a "local" theory.

I am referring to the reduction of an infinite set to a countable set via the "measurement" interaction (dynamics).

Uh-huh. It's still very vague, I assume you mean the decoherence process or some such.
In any case, when you ask "How does Bell allow for these dynamics" you have to explain why you think he needs to do so. Why/how does this have to be taken into account, and why/how do you think it would change the outcome?

Well 6 billion may be a big, but age is hardly an infinite set of lengths and orientations and what "measurement" would you use to produce an equivalent countable set of cardinality 4?

Yes so when you have a discrete number like 6 billion you can perform a summation. If you had a continuous number you'd perform an integration. You can integrate over an infinite number of vectors and get an average vector if you like as well. That's vector calculus. There's no 'measurement' involved in that.

Please answer this because I may be missing something -- because I do not see that we are yet at the stage of expectation values? For me, they come later.

I believe(d) we were talking about equation (12) in Bell, which is an averaged probability. (and an expectation value is an average of sorts). Hence the integral. Integration is just summation over a continuous number of values, right? And you need to perform a summation to calculate an average. And I don't need to know about all the possible variables to calculate an average of them. If I know there are 10 people in an elevator and they weigh 1234 pounds, I can tell you their average weight without knowing anything about what any individual weighs. There's an infinite number of possibilities but only one average weight.

Taler du dansk?

My "Danish" is about 30% Danish. The rest is Norwegian with Danish pronunciation (substitute 'k' for 'g', etc)

 

[Gordon Watson]

Well if you assume the system is described by something which is a subset of the more general conditions Bell set up, it doesn't contradict Bell.
Well I'm sorry if you feel I've misinterpreted you. But as I said earlier, it's not always clear to me what you mean.
I'm hardly a QM specialist. I'm just one of those run-of-the-mill physicists who uses QM. All I need to do is go a bit outside my field and my knowledge will be pretty limited. I do think I can follow Bell's paper though. Anyhow, yes it's fine to have spin vectors as your choice of hidden variables. It's just that I don't see how this would contradict Bell in any way. You realize that it's a strength, not a weakness, that he allows for many more scenarios than any particular set of hidden variables?
I haven't seen a proof though. You've made some arguments, but I don't feel you've shown there's anything wrong with Bell. On the contrary, since what you seem to be suggesting is a subset of what's covered by Bell's proof, it'd be subject to it - as long as Bell's logic is correct.

Bell's logic is correct, I'm certain. His original paper has almost 4000 citations by now, and has been pored over by all kinds of people, including many who probably were "local realists". 't Hooft's a bit of a hidden-variable fan, and he's probably forgotten more math than I've ever known, yet he doesn't dispute Bell's math. That's good enough for me. Rather he's criticized on the basis of other assumptions involved, e.g. how entanglement occurred in the first place. That's a legitimate line of attack in my opinion. Assuming Bell's math is wrong isn't. Especially since it's been verified experimentally.

(Similarly, many crackpots who dislike Special Relativity keep trying to "prove" Einstein's math wrong; often using high school-level math. And similarily I hold that Einstein's math is not wrong and if SR is ever disproved, it will be on the basis that some of the basic assumptions of the theory did not hold, not because of faulty logic)
By the detector settings being "local" we mean that the settings at A do not affect the outcome at B and vice-versa, since they are assumed to be spatially separated. This is Bell's definition of a "local" theory.
Uh-huh. It's still very vague, I assume you mean the decoherence process or some such.
In any case, when you ask "How does Bell allow for these dynamics" you have to explain why you think he needs to do so. Why/how does this have to be taken into account, and why/how do you think it would change the outcome?
Yes so when you have a discrete number like 6 billion you can perform a summation. If you had a continuous number you'd perform an integration. You can integrate over an infinite number of vectors and get an average vector if you like as well. That's vector calculus. There's no 'measurement' involved in that.
I believe(d) we were talking about equation (12) in Bell, which is an averaged probability. (and an expectation value is an average of sorts). Hence the integral. Integration is just summation over a continuous number of values, right? And you need to perform a summation to calculate an average. And I don't need to know about all the possible variables to calculate an average of them. If I know there are 10 people in an elevator and they weigh 1234 pounds, I can tell you their average weight without knowing anything about what any individual weighs. There's an infinite number of possibilities but only one average weight.
My "Danish" is about 30% Danish. The rest is Norwegian with Danish pronunciation (substitute 'k' for 'g', etc)

Dear alxm,

Thank you very much for all of this.

I would say we are in much agreement, especially as to my shortcomings.

But there is one crucial area of dissent:

Rather he's criticized on the basis of other assumptions involved, e.g. how entanglement occurred in the first place. That's a legitimate line of attack in my opinion. Assuming Bell's math is wrong isn't. Especially since it's been verified experimentally.

In my world, wherein I accept all the experimental results, the underlined statement is false!

Stay tuned.
JenniT

 

[Gordon Watson]

Ha, I do have fairies in my garden. My mother gave them to me.

DIAL 911!

I believe in quantum non-locality, which is not at all the same as Bohmian non-locality. I mean, it might be the same, but the point is that I believe that the HUP implies a kind of non-locality. It makes sure that entangled particles act in a certain manner when they are spatially separated. It allows individual particles to have imprecise locations, which is automatically non-local. To me, this is just standard QM and nothing more. If you don't want to call it non-locality, that is fine by me. A lot of scientists do not consider that non-locality.

Since I think we live in an observer (observation actually) dependent universe: just, you could say I change the moon by observing it. Again, this is simply standard QM.

Suggestion, with great respect: The photon in your eye changes the eigenstate of your brain -- well, assuming ---

Cheers,

JenniT

 

[Gordon Watson]

In QM there are vectors describing the state of a given quantum system, and this vector determines the probabilities of each possible result for a given measurement. Here was a quick summary I wrote on another thread:

Spin along a given axis is one of the measurement operators, so a system's quantum state can always be broken down into a weighed sum of spin eigenvectors, with the weights determining the probabilities of finding a given spin when the system is measured. The problem is that for an entangled quantum system, there is a single state vector |psi> for the whole entangled system rather than individual state vectors for the parts of the system, so this is not a local variable associated with a unique point in space at any given time (like the position of a single particle at that time). A measurement of one particle would cause an immediate change in the state vector representing both particles, even if there is a spacelike separation between the particles. The QM formalism gives no complete way to predict the behavior of both particle using only local variables associated with each particle (such as individual vectors associated with each one) that are not influenced by anything outside their past light cone.

The spin vectors you talk about also couldn't really be classical spin vectors describing the axes of rotation of two charged spinning objects, since as mentioned in this link which I posted earlier, classical charged spinning objects could be deflected at a continuous range of angles when traveling through a magnetic field, rather than always being deflected in one of two directions, "up" or "down". So yeah, I do think there's a need for "scare quotes" around your notion of a "spin vector" since it doesn't really seem to correspond exactly with any existing physics concept, it would presumably have to be defined in the context of a novel local hidden variables theory, and you'd need some definition of how the combination of the "spin vector" and the detector angle (along with any other properties of the detector you want to consider) determines the measurement result.

Well, you still do need some words to point out that the equation only applies when A and B have a spacelike separation, and λ1 and λ2 contain enough detail that they screen out any correlation which exists between A and B due to events in their common past (like the fact that two particles may have been assigned correlated spin vectors by the source which created them). But if you want to minimize words, this equation is indeed a good start in helping define mathematically what "local realism" implies.

How do you figure? Consider the analogous but simpler equation $$P(X) = \int P(X|\lambda)\rho(\lambda) \, d\lambda$$. If X is a discrete random variable while λ is a continuous one, do you think this implies the probability density $$\rho(\lambda)$$ must include delta functions? Suppose λ could take any real value from 0 to 2, and $$\rho(\lambda)$$ was just a uniform probability density with density 0.5 everywhere (so if you integrate it from 0 to 2 you get 1, as expected for a probability density). Then if X could only take two values X1 and X2, and P(X1|λ) = 1 - (λ/2) while P(X2|λ) = λ/2, then for any given value of λ it's true that P(X1|λ) + P(X2|λ) = 1 (note that this also means P(X1|λ) and P(X2|λ) are conditional probability density functions, not discrete probability functions). Then according to the integral, P(X1) would be given by $$\int_0^2 [1 - (\lambda/2)]*0.5 \, d\lambda$$ = $$0.5 * [\lambda - (\lambda^2 / 4)]^2_0$$ = 0.5*[(2 - 4/4) - (0 - 0/4)] = 0.5. Likewise P(X2) would be given by $$\int_0^2 (\lambda/2)*0.5 \, d\lambda$$ = $$0.5 * [\lambda^2 / 4]^2_0$$ = 0.5*[4/4 - 0/4] = 0.5. So, P(X1) + P(X2) also adds up to 1, all the probabilities make sense in spite of the fact that X is a discrete random variable and λ is a continuous one, with no need for $$\rho(\lambda)$$ to include any delta functions.

What physicists regard delta functions as illegitimate? In any case there is no necessary reason to think $$\rho(\lambda)$$ should contain any delta functions as I explained. When you say "please rewrite your equation as a summation", are you asking what the equation would look like if we assumed λ could only take a discrete range of values (as Bell assumed in some versions of his proof) rather than a continuous range?

Dear Jesse,

Much good material here. For now, to move us along on the mathematics, the BOLD section is the next move mathematically, please.

Regards,

JenniT